Arithmetic, practically applied, for advanced pupils, and for private reference, designed as a sequel to any of the ordinary text-books on the subject. By Horace Mann ... and Pliny E. Chase ...
Mann, Horace, 1796-1859., Chase, Pliny Earle, joint author. 1820-1886,

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Page  I PRACTICALLY APPLIED9 FOR ADVA.NCED PUPILS, AND FOR PRIVATE REFERENCE, DESIGNED AS A EQUEL TO ANY OF THE ORDINARY TEXT-BOOKS ON THE SUBJECTt BY HORACE MANN, LL.D., THE FIRST SECRETARY OF THE MASSACHUSETTS BOARD OF EDUCATION, AND PLINY E. CHASE, A.M., AUTHOR OF'THE COMMON-SCHOOL ARITHMETI-;2.) PHILADELPHIA: PUBLISHED BY E. H. BUTLER & CO. 1857.

Page  II Entered, according to the Act of Congress, in the year 1850, by HORACE MANN AND PLINY E. CHASE. in the Clerk's Office of the District Court of the United States, in and for the Eastern District of Pennsylvania. JUST PUBLISHED, THE ELEMENTS OF ARITHMETIC, PART FIRST, for Primary Schools. The SECOND PART, for Grammar Schools, which is in course of preparation, will complete the following Arithmetical Series:ELEMENTS pF ARITHMETIC, PART FIRST. By HORACE MANN and PLINYVE. VHASE. ELEMENTS Mt,'FARITHMETIC, PART SECOND. By HoRAcE MANN and PLIN'Y,'. 5HASE. COMMON-S-CHOOL ARITHMETIC. By PLINY E. CHASE. ARITHMETIC PRACTICALLY APPLIED. By HORACE MANN and PLINY E. CHASE. E. B. MEARS, STEREOTYPER. C. SHERMAN, PRINTER.

Page  III MR. MANN'S PREFACE. THE, appearance of my name on the title page of this Arithmetic, requires me to state the extent of my connection with its authorship, and of my responsibility for its execution. Believing the idea of the work to be original, I will attempt its elucidation. In seeking for the elements or materials of its questions, it proposes to take a survey of all the vocations of life, of all the facts of knowledge, and of all the truths of science, and to make a selection from each department of whatever may be most interesting and valuable. It does not confine itself to the playthings of the nursery, or to the commodities of the market place, and to the mnoney they will cost, or make, or lose. On the contrary, the present work proposes to carry the student over the wide expanse of domestic and social employments; to introduce him to the various departments of human knowledge so far as that knowledge has been condensed into tables, or exhibited in arithmetical summaries, and to make him acquainted with many of the most wonderful results which mathematical science has revealed. Insteadc of groping along the mole-path of an irksome routine, with little other change than from dollars and cents, to pounds and pence, or some other familiar currency, and with little other variety than from cloth to corn, or some other common-place commodity, it derives its examples from biography, geography, chronology, and history; from educational, financial, commercial, and civil statistics; from the laws of light and electricity, of sound and motion, of chemistry and astronomy, and others of the exact sciences. Trades, handicrafts, and whatever pertains to the useful arts, so far as they are the subject of numerical statement, and their facts possess arithmetical relations, together with all the ascertained and determinate results of economical or political knowledge, and of scientific discoveries, are laid under contribution, and are made to supply appropriate elements for the questions on which the youthful learner may exercise his arithmetical faculties. (iii)

Page  IV iv MR. MANN' S PREFACE. In this way, and without departing from the most rigid rules on which an arithmetical text-book should be constructed, I have supposed that a work may be prepared which shall exemplify in the best manner the science of numbers, and be full of useful knowledge also; and which, while it exercises the student's powers of calculation, shall enlarge his acquaintance with the varied business of the world, and with many of the most interesting results of applied science. In a universe like this, where every star has been weighed in a mathematical balance, and all inter-stellar spaces have been measured by a mathematical line; where the orbits of all the planets have been traced as by a compass, and their velocities graduated to their distances by an unchanging law; where not only wind and tide, but every particle of dust in a hurricane, and every drop of water in a cataract, know their exact places by an infallible rule; where the gravitation of matter, the radiation of heat, and the diffusion of light, at all times and instantaneously, adjust their force to their distance with unerring precision; where every chemical combination is formed on some fixed principle of proportion, and the atoms of every crystal arrange themselves around their nucleus in geometric lines; and where, whatever other contemplations or volitions occupy the Infinite Mind, it is still true, as was said by the old Greek philosopher, that "'God geometrizes;"-I say, in such a universe, built, weighed, measured, compounded, and arranged on mathematical principles, why should not the arithmetical exercises of those minds which have entered it, to dwell in it forever, embrace something more than the market price of commodities, the gain or loss in trade, and the interest and discount of banks? Why, for instance, cannot a child be taught to count the bones in his hands, as well as the nuts in his pocket; to add together the number of bones in all the different parts of his body, or to subtract the number of those contained in his head or in his hands, from the number of those contained in his feet, as well as to add or subtract the number of apples or of cakes in the possession of James, John, and Joe; and why can he not, by such exercises, be led to enrich his mind with anatomical or physiological facts, instead of stimulating his imagination with the provocatives of appetite? Why cannot a child add together the population of the different States of this Union, or of the different nations of Europe or of the world, and thus learn the sum of the population of the whole earth and of its parts, as well as to

Page  V MIR. MANN'S PREFACE. V add naked columns of abstract figures? In addition, subtraction, multiplication, and division, why cannot the pupil use examples whose elements or data are drawn from the distance between different historical epochs, from the ages of distinguished men, from the date of one discovery or invention to that of another discovery or invention, or from the rise to the fall of dynasties, or from a comparison of the heights of different mountains, or the lengths of different rivers, or of degrees of longitude on different parallels of latitude, or the distance from city to city by land, or from port to port by sea; and thus live in the perpetual presence and company of most important truths pertaining to history, chronology, biography, and geography, and so familiarize himself with these classes of facts, without devoting special time or effort to their acquisition, just as he becomes familiar with the faces and the names of his school-fellows and his townsmen, merely because he has always lived amongst them? If the arithmetical exercises of the pupil direct his attention almost exclusively to the shop of the retailer or the countingroom of the merchant, then he does not enjoy even a pedlar's opportunity to become acquainted with men, events, times, places, and things,-with the great results of business and of civilization, as they now exist in the world. Instead of wearying the learner with endless reiterations about bales, boxes, barrels, and bushels, or dollars, dimes, cents, and mills, why not open before him some of the vast storehouses of truth, and display some specimens of their endless variety and beauty? Let the teacher, taking the learner by the hand, follow the farmer, the craftsman, the architect, the manufacturer, the road-maker, the mill-wright, the ship-wright, the watch-maker, and the long catalogue of others who are employed in the mechanic arts, or other branches of useful industry; or, rising into the sphere of the educated or professional laborer, let him observe the optician, the electrician, the mathematical instrument maker, the astronomical observer, the telegraph operator, and borrow from them all some of the curious facts pertaining to their respective arts and professions, and convert them into the pleasing and instructive elements of arithmetical problems. I can conceive of a work so replete with the facts of technology and science, that it shall be examined with interest and profit by any one who is only seeking after valuable information, and which, at the same time, shall be perfectly adapted to the mere student who only seeks after the best means of arithmetical practice.

Page  VI vi iMR. MANN'S PREFACE. Among the advantages of such a work as is here proposed, the two following seem to me unquestionable: 1. The pupil, while studying Arithmetic for its own sake, will acquire some knowledge of many other things. Cicero observes th'at the face of a man will be tinged by the sun, for whatever purpose he may walk abroad. So, by daily familiarity, the mind of the student will be replenished with useful facts, and imbued with a scientific spirit, although the acquisition of the facts and the spirit be not the direct object of his study. So far as the examples of his text-book have been drawn from the actual business of life, as it goes on around him, the learner cannot but see that his studies have a practical bearing and are connected with obvious realities. A great number of facts, such as dates, sums, quantities, distances, will be impressed on his mind; and thus a species of most valuable knowledge, which, as we all know, it is most difficult to acquire in after-life, will be gratuitously bestowed upon him. Doubtless great differences will exist among pupils, in regard to the amount of information they will obtain from being domiciled, as it were, among the determinate truths of business, art or science; but even the most dull and stupid will be constrained to learn something, not only of the existence of various sciences, and of various kinds of business, of which they would otherwise be forever ignorant, but also of the nature and distinctive characteristics of those sciences or employments. All will be saved from great misconceptions; and doubtless the curiosity of many will be awakened for further information. These advantages embrace not only an increase of positive knowledge, but an enlargement of the mind's scope in regard to the subjects of knowledge. It is unnecessary here to remind the observing man, how the understanding of any one thing, by the self-activity of the faculties, generates the power of understanding many other things, and each of these, in their turn, of many more, and so on in geometrical progression. The knowledge of any one truth acts as an introducer and interpreter between us and all its 7cindred truths. 2. There are advantages of another kind, which appear to me of not inferior magnitude, though I am fully aware that the views I am about to present, will affect different minds very differently, according to their Theory of AMind. My belief is that Arithmetic, in its strict and technical sense, addresses but one faculty of the mind, or, at most, but a very limited group of the faculties. No

Page  VII MR. MANN'S PREFACE. Vii other study pursued in our schools is so restricted, either in regard to the mental powers, which it calls into exercise, or the objects which it brings under their cognizance. Hence, during the hours devoted to Arithmetic in our schools, most of the mental faculties lie dormant, or play the truant by employing themselves upon forbidden objects. Doubtless this intense exercise of a single faculty, or of a limited group of faculties, and the non-exercise of all the rest, is one of the main reasons why Arithmetic, when not taught with great ability, is so often an irksome study. Reading and Geography, for instance, cover the widest field of interesting subjects, and it is impossible for any teaching to be so dull, or any circumstances so repulsive, as wholly to despoil them of their charms. If we would invest Arithmetic with similar attractions, we must draw its examples from as wide and as rich a field. Then will it interest new faculties,-faculties which, otherwise, it never addresses. All metaphysicians know, from the principles of their science, and all laborious students know, from their own experience, that nothing refreshes or re-creates a wearied faculty so certainly and so speedily as the genial exercise of some other faculty. Such exercise is more restorative than absolute quiescence. It is with the faculties of the mind as with the muscles of the body, they should have alternate exercise and rest; and the most healthful and agreeable relaxation for any organ that is tired of exercise, is the exercise of another organ that is tired of repose. The footman who travels over a long and level road, where the same muscles are subjected to a perpetual recurrence of the same strain, rejoices at the sight of a hill in the distance; for he prefers to put a new set of muscles to the hard service of carrying his body up a hill, rather than to compel the fatigued ones to continue their lighter task. Equally cheering and recuperative must the alternation be, when, after addressing one set of faculties with one combination of agreeable truths, we appeal to another set of faculties to determine certain arithmetical relations which exist between those truths. Should any teacher dissent from this doctrine, that alternate exercise and rest, like the ever-alternating systole and diastole of the heart, is the law of all our powers, both bodily and spiritual; or should any pupil be found, whose senses are such perfect non-conductors of truth, as to exclude all information though the atmosphere by which he is surrounded is saturated with it, still, no loss or harm would be incurred; for the single process

Page  VIII Viii MR. MANN' S PREFACE. of arithmetical training can be carried on by the aid of the examples contained in this book as well as by those in any other. Such is an outline of the present work. Since its conception first flashed upon my mind, I have pondered upon it much, and have conversed respecting it with many gentlemen not only of great mathematical attainments, but of varied scholarship, and both reflection and conversation have deepened my conviction of its value. Justice to my associate, Mr. CHASE, requires me to say a word in regard to the shares of responsibility and of merit which attach to us respectively as the joint authors of this work. In communicating my plan to him, I unfolded its whole scope and purpose as it lay in my own mind; I indicated the various sources whence materials for its construction might be drawn, and I have rendered him some aid in the collection of those materials. The residue of the work,-the definitions, the rules, the statement of the questions, and the answers, so far as answers are given, together with the arrangement of the topics, or subjects,-is substantially his. So far as there is skill in their selection, or science in their statement, or accuracy in their results, I shall gladly join with others in awarding the merit to him. HORACE MANN. MARCH, 1850.

Page  IX P. E. CHASE'S PREFACE. THE following work does not profess to be a mere Arithmetic of the ordinary stamp. It takes ground that has hitherto been unocccupied, and its plan in most respects is entirely new. In the " Common-School" Arithmetic the rules and principles that are usually taught, have been very fully explained and illustrated, but in that work, as in all of the other similar treatises that are now in general use, the primary obj ect is merely the inculcation of processes,-the practical application of those processes being introduced only incidentally. But every teacher is aware that practical exercises, more numerous than it would be possible to insert in an elementary work, without rendering it of an inconvenient size, are necessary in order to make a thorough arithmetician. This necessity can often be but partially supplied, and consequently most pupils find whenever they enter into active life, that the calculations of their business, whatever that business may be, are all to be learned anew, and that all the Arithmetic they have studied at school is of little value, except for the expertness it may have given them in the simple operations of the fundamental rules, I have long believed that a Sequel to Arithmetic,-a work not designed to take the place of any of the ordinary text-books, but absolutely requiring a familiarity with some one or more of them before it can be studied at all,-might be so prepared as to supply the want to which I have alluded. By giving numerous examples similar to those which are constantly occurring in the various walks of life, the student may be enabled to prepare himself better at school, for his future employment, and by the incidental introduction of subjects of general interest, the study may be made pleasant, and the thoroughness which is of the first importance in every undertaking may be more readily secured. There are many difficulties connected with the selection and arrangement of materials for such a treatise, which may be urged in excuse for any deficiencies that exist in the present volume. I trust, however, that there will be few- objections brought (ix)

Page  X X PRYEFACE. against the execution of our undertaking, for which the book itself does not afford a remedy. Any fault of arrangement may be easily corrected, by taking up the chapters in a different order from the one I have adopted; any redundancy of examples may be avoided, by omitting such portions as appear of the least importance; any deficiency may be supplied, by framing additional questions from the abundant materials, in the shape of tables and remarks, which are interspersed throughout the book. This latter peculiarity of our plan, we regard as one of its greatest recommendations. Teachers can make new questions for their classes, to any extent they deem expedient, and they may find it a most valuable exercise to allow their pupils to form questions for each other, from the data that are given in the text-book. In many places, the explanations have been simplified by the use of letters. If their use should seem, at first, too algebraical, I think all objection will be removed upon examination of the manner in which they are employed. Every scholar who has any tolerable degree of familiarity with Arithmetic, should be able to reason as readily with as and bs, as with apples and beans; and I flatter myself that the manner in which I have applied the simplest rules of analysis to operations with letters and symbols, will be of great use to all who study the following pages. The character of the work having rendered it necessary that it should be, like other school-books, principally a compilation, I have endeavored to search for the best authorities, and to give due credit to the various writers that I have consulted. I amn also indebted to officers of the government, and to teachers of some of our leading institutions of learning, for many valuable suggestions. I have added many rules and illustrations that are entirely original, and I hope that the result of our joint labors will be found a valuable companion in every schoolroom into which it may be introduced. I consider myself most fortunate in having obtaz,ed the valuable assistance of my associate, HORACE MANN. His Proposal, that the work should be rendered highly instructive as well as practical, will doubtless commend itself to general favor; and for any merit that may be found in the execution of the plan indicated in his preface, I am greatly indebted to the suggestions and criticisms with which he has favored me, during his revision of the manuscript and proof-sheets. PLINY E. CHASE.

Page  XI TABLE OF CONTENTS. ARTICLE I.-SYMBOLS. ART. VII.-THE HOUSEHOLD. Section Page Section Page 1. Significations of Symbols...... 15 24. General Information............ 129 2. Examples for the Pupil........ 16 25. Household Mensuration.......129 26. Examples for the Pupil........131 ART. II.-TEsT QUESTIONS. 3. Questions on the Theory of ART. VIII.-ARTIFICERS' WORK. Arithmetic................. 17 27. The Carpenter and Joiner......136 4. Questions on the Practice of 28. The ason.......... 140 Arithmetic................. 25 29. The Bricklayer..............142 30. The Plastere.................143 ART. III.-THE FUNDATMENTAL 31. The Painter and Glazier.......144 RULES. 32. The Paver, Slater, and Tiler.. 141 5. Examples in Addition and Sub- 33. The Plumber.................1... 15 traction..................... 33 34. Specification and Estimates.... 146 6. Examples in Addition, Subtraction, Multiplication, and Di- ART. I.-STRENGTH OF MATERIALS. vision.................. 3..... 335. Flexibility and Strength of Tim7. Table for Additional Exercises 46 ber..........................153 8. Fractions and Compound Num- 36. Problems on the Strength of bers......................... 48 Timber.....................153 9. Table of Latitudes and Longi- 37. Examples for the Pupil........156 tudes........................ 55 38. Strength of Prisms one inch 10. Miscellaneous Examples...... 59 square..........157 11. Test Examples............. 64 39. Lateral Strength of Bars......157 40. Cohesive Force of Iron........158 ART. IV.-MEASURES, )VEIGHTS, 41. Resistance to Crushing.........158 AND CURRENCIES...............42. Table of Cables............. 159 12. Standards of the United States 66 43. WVeight of Stone..............159 13. Standards of Great Britain.... 81 44. Problems on the Strength of 14. Standards of France.......... 84 Iron................... 160 15. Miscellaneous Table......... 87 45. Examples for the Pupil........163 16. Ancient Measures, W;Veights, ART. X.-SPECIFIC GRAVITY. and Coins............... 104TPEIFI GRAIT. 17. Examples for the Pupil........106 46. Table of Specific Gravities....165 47. Problems in Specific Gravity.. 169 ART. V. —THE FARtMd. IS. Examples for the Pupil.........170 18. Rules for determining the Weight of Cattle..110 ART. XI.-THE ROAD. W~eight of Cattle........... 110 19. Rules for Measuring Grain....111 49. General Remarks................173 20. WVeight of' Grain and Hay.... 112 50. Examples for the Pupil........174 21. Measurement of Land....... 1.......13 22. Examples for the Pupil........ 13 51. The Steam Engine. 178 AnT. VI. —TiE GARDEN. 52. The Water Wheel............. 180 23. Examples for the Pupil......123 53. Pumps.......18.3 (111)

Page  XII Xii TABLE OF CONTENTS. ART. XIII.-T E LABORATORY. ART. XVIII. —INvOLUTION AND EVOSection Page LUTION. 54. Chemical Combinations........ 4 Section Page 55. Table of Chemical Equivalents 186 84. Involution................... 274 56. Examples for tile Pupil........17.. 85. Evolution...................276 86. Roots of All Povers........... 278 ART. NXIV.-GENERAL ANALYSIS. 87. Application of tle Square Root 282 57. Remarks on the Solution of 88. Application of the Cube Root 281 Questions................... 189 58. Examples illustrating the First ART. XI.-PROGRSION1 ON Method..................190 59. Examples illustrating the Se- 89. Arithmetical and Geometrical cond Method................195 Progression............... 286 60. Miscellaneous Exanples......196 90. Harmonical Progression......289 91. Compound Interest..... 291 ART. XV. —THE COUONTI1NGO-HOUSE:. 92. Annuities................ 292 61. Percentage.................... 208 62. Problems in Percentage........209 63. Examples in Percentage....... 211 93. Single Position................300 64. Percentage on Sterling 1MIo- 94. Double Position.............301 ney.................. 213 65. Banking........2.15... ART. XXI.-APPROXIMATIONS. 66. Partial Payments.............218 95. Multiplication.....305 67. Legal Interest................. 219 96. Division.. 306 68. Equation of Payments........223 97. Continued Fractions.......307 69. Accounts Current.............. 226 98. Evolution......311 70. Practice.................... 229 99. Examples in Approximation..313 71. Cause anld Effect............... 236 72. Exchange.....................2 A. II. —PROPERTIES OF NM73. Arl)itration of Exchalge......255 BERS. 74. Alligatiol..................... 258 100. Properties of Squares and 75. General Average.............. 2G63 Cubes......................313 101. Prime and Composite Numbers 316 102. Figurate Numbers...........320 76. Product of Mines............. 265 103. The Fundamental Rules......321 77. Agricultural Products.......... 266 104. Curious Problems............ 325 78. Agricultural Products, continuet.......................2G7 ART. XXIII.-IIMIscELLANEOIjs PROB79. Occupa;tions.................. 268 LEMS. 80. State of Education............269 105. Chronology............ 327 81. Products of Industry.......... 270 1013. The Moon's Age and Southing 335 107. Mensuration.................338 ART.. XVII.-PERSIUTATION AND 108. Tonnage of Vessels..........342 COmBINaTIOn. 109. Gau ging.....................344 82. Permutation.............. 271 110. Miscellaneous Examples.....347 83. Combination...........2.......273 111. Table of Prime and Composite Numbers...376

Page  XIII LIST OF AUTHORITIES. Allen,W., American B iographical Chambers's Educational Course. and Historical Dictionary. Information for the People. American Almanac. Colman, I.,Agricultural Reports. American Temperance Society, - European Agriculture. Reports. Crossley & Martin, Intellectual Babbage, C., Economy of Ma- Calculator. chinery and Manufactures. Daniell, J. F., Chemical Phi-- Bridgewater Treatise. losophy. Bache, A. D., Reports onWeights, Deane, S., New England Farmer. Measures and Balances. Desaguliers, J. T., Course of Barlow, P., New Mathematical Experimental Philosophy. Tables. Draper, J. W., Text Book on Barnard, H.,School Architecture. Chemistry. Beckman, J., History of Dis- Emerson, G. B., Report on the coveries and Inventions. Forests of Massachusetts. Belknap,J.,American Biography. Encyclopntdia Americana. Benjamin, A., Practice of Archi- Britannica. tecture. Edinburgh. Bigelow's Technology. Rees'. B1iographia Americana. Evans, O., Young Mill-Wright Biographie Portative Universelle. and Miller's Guide. Bonnycastle's Arithmetic. Ewing, A., Practical MatheBoston Custom-House, Table of matics. Currencies. Ferguson, J., Lectures on Select Bowditch, N., Practical Navi- Subjects. gator. Gillespie, W. M., Manual of British Husbandry. Road Making. British Sanitary Reports. Gregory, G., Dictionary of Arts Budge, J., Practical Miner's and Sciences. Guide. Griffiths, T., Chemistry of the Buist, R., American Flower Gar- Four Seasons. den Directory. Hare, R., Chemistry. Carpenter, W. B., Mechanical Hatfield, R. G., American House Philosophy, Horology, and Carpenter. Astronomy. IHerschel, Sir J., Outlines of Carpenters' and Builders' As- Astronomy. sistant. Hunt, F., Merchants' Magazine. Cavallo, T., Elements of Natural Hutton,C., Recreations in MathePhilosophy. matics and Natural Philosophy. (13)

Page  XIV Xiv LIST OF AUTHORITIES. Ingram, A., Concise System of Newton, I.,Universal Arithmetic. Mathematics. Nicholson, J., Operative MIcJackson, C. T., Report on the chanic. Geology and Mineralogy of Norie, J. W., Epitome of PracNew Hampshire. tical NaVigation. Jamieson, A., Mechanics of Owen, R. D., Hints on Public Fluids. Architecture. Johnson, C. W., Farmer's Ency- Parnell, E.A.,Appliecl Chemistry. clopoedia. Partington, C. F., Account of Johnson, G. W., Dictionary of the Steam Engine. Modern Gardening. Patent Office Reports. Johnson,L.D., Memoria Technica. Pierce, B., Elementary Treatise Johnson, W. R., Report on on Sound. American Coals. Pilkington, J., Artists' Guide and Keith, T., Complete Practical Mechanics' Own Book. Arithmetician. Pratt, J. H., Mechanical PhiLavoisne, C. V., Genealogical, losophy. Historical, Chronological, and Priestley, J., Description of a Geographical Atlas. System of Biography. Leslie, J., Philosophy of Arith- Ranlett, W. H., The Architect. metic. Rogers, J., Vegetable Cultivator. Library of Useful Knowledge. Scientific American. Liebig, J., Animal Chemistry. Shaw, E., Practical Masonry. Lloyd, H., Lectures on the Wave Smeaton, J., Experimental InTheory of Light. quiry concerning the natural London, Edinburgh, and Dublin powers of water. Philosophical Magazine. Somerville, M., Connexion of the Loudon, J. C., Encyclopaedia of Physical Sciences. Agriculture. Tredgold, T., Principles of WarmM'Culloch, J. R., Commercial ing and Ventilating Public Dictionary. Buildings. M'Gauley, J. W., Lectures on Tucker, G.,Progress of the United Natural Philosophy. States in fifty years. Mahan,D. H., Elementary Course United States Almanac. of Civil Engineering. Ure, A., Dictionary of Arts, Mass. Board of Education, An- Manufactures and Mines. nual Reports. The Cotton Manufacture Mint, U. S., Manual of Coins. of Great Britain. Moseley, H., Illustrations of Me- Wade, J., British History, Chrochanics. nologically Arranged. Murray, H., Encyclopamdia of Walsh,MI.,Mercantile Arithmetic. Geography. Wistar, C., System of Anatomy. Nesbit, A., Treatise on Practical Year Book of Facts. Arithmetic.

Page  15 ARITHMETIC. I. SYMBOLS. 1. THE following characters, or symbols, are frequently employed in Arithmetic to represent the operations that are to be performed upon quantities: The sign + (plus or and) shows that the numbers between which it is placed, are to be added together. Ex.: 5+6; 3+10+27. The sign - (minus or less) shows that the quantity which follows it, is to be subtracted from the one which precedes it. Ex.: 11-8; 29-16. The sign = (equal) shows that the quantity which follows it, is equivalent to the quantity which precedes it. Ex.: 4 + 19 =27 —18 + 14 (read 4 plus 19 equal 27 imntus 18 p2us 14); 65-27+3=41. The sign X (times) shows that the numbers between which it is placed, are to be multiplied together. Ex.: 4x 15=60; 3x10=6x5. Multiplication is also sometimes expressed by a point, thus: 4 ~ 15=60; 3 ~ 10 —6 5. The sign - or: (divided by) shows that the former of two quantities is to be divided by the latter. Ex.: 24015=16; 108: 9=36: 3. Division is also represented by placing the dividend above, and the divisor below a horizontal line. EX.: 276-95+15 _ 14. A vinculum, or a parenthesis ( ); shows that several (1.)

Page  16 16 SYMBOLS. [ART. II. quantities are to be collected into one. Ex.: (8 -4 + 16 -- 2) x 3=36; (2x7-1)x3= —14-11x 3=3x3=9. A small figure placed above any quantity, at the right hand, denotes a power of that quantity. Ex.: 92=81; 1+3)3=64; ((2x 5-6 x3)-o10)= (10-6x sl - i0)5 =(4x 3 -o10)5- (12- 10) = 2 =32. The radical sign,/ prefixed to a quantity, is used to denote some root of that quantity. If no figure is written above the sign, the square root is indicated. To express the 3d, 4th, or nth root, 3, 4, or n is placed at the left hand. Ex: V/6-2= V4=2; /18+9 = /27-=3; (8 x3)-(7 - - V24-8_= V 16-2.,. EXAMPLES TO BE READ AND SOLVED -BY THE PUPIL. 1. 8+9+7-3+6-5-11=? Ans. 11. 2. 4-2+6 —1+13 Xa5=? 3. 84-3 + 5 +16-27=? 4. 27x 3 x 5E.45=? Ans. 9. 5: (6x 8)+(42: 14x 10) —(65 5)=-? 6. (~27 X 3-2 X 3: 2)+ (15 x 12-10)=? Ans. 206, 7. (4 x 11 x 3) —(6'3 x 144: 36 x,LSr)=? 9. /2D7x V64- /-24+8=? 10. 42+22-(37+3 x 5)-87 (2 x 3 x 4+10)=? II. TEST QUESTIONS. EvERY pupil should be required to give written answers to as many of the questions in this article as he is able to a The pupil should be taught to say, 13 times 5, 8 times 4, rather than 13 multiplied by 5, 8 multiplied by 4.

Page  17 ~ 3.] THEORY CF ARITHMETIC. 17 solve. The teacher -will this understand the proficiency cf each member of the class, and can give such explanations as may be most desirable. The questions should be reviewed from time to time, until they can all be answered without any mistakes." $. QUESTIONS ON THE THEORY OF ARITHMETIC. 1. What is ARITHIrErTIC? What is a number? In what different ways may numbers be expressed? What are the digits, and why are they so called? Whatt different names has the character 0? Can you give any reason for calling naught the figure of place? What are the fundamental operations of Arithmetic? What is given, and what is required in each? 2. What is NUMEruATION? What method of numeration is generally adopted? What is the peculiarity of that method? What is a unit? What is meant by the simple value of a figure?-by the local value? What is the difference between a place and a period? What names are given to the places in each period? How do you read units?-tens?-tens and units?-hundreds?hundreds, tens, and units? Repeat the names of the first eight periods.b How do we read whole numbers? Can the Arabic method represent numbers less than a unit? What are such nunmbers called? How are they distinguished from whole numbers? How do we read decimals? a, A teacher ought not only to instruct his pupils, but also to interrogate themn fiequently, and test their proficiency." QinLtilian. b'lhe Numeration table may be continued to any extent we please. The periods above Sextillions, to Vigintillions, are, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions. The following illustrations will show how difficult it is for us to form any distinct idea of the value of very large numbers. If every man, woman, and child, on the face of the globe, were to count at the rate of four every second, without ceasing day or night, the whole amounzt of all that they could coun0t inl 5000 years, would be less thano one sextillion. If the sun, all the planets, and all the fixed stars that are visible through the most powerful telescope, were reduced to a powder so fine that 100000 grains would be less in bullk than a drop of water, and if one of these grains were destroyed in every million years, the vwhole visible universe would probably be annihilated in( less thaos one vigintillion years. 2

Page  18 18 TEST QUESTIONS ON THE [ART. II. 3. What is the effect of removing a figure two places to the left? —two places to the right? —one period to the left? —to the right?-two periods to the left?-to the right? What is meant by Notation? How do you write whole numbers? In writing five sextillion, thirty billion, and seven thousand, what would you place in each period? If you were to omit the naughts in any one period, what effect would it have on all the figures above it? How do you write decimals? How would you make 5702 represent hundredths?-millionths?-thousands?-thousandths? What is the effect of placing naughts at the right hand of decimals?-at the left hand?-at the right hand of integers?a-at the left hand? What is the value of tens multiplied by thousands?hundreds by hundreds?-ten-thousands by tens?-hundred-thousands by hundreds?-tens by hundreds?-thousands by thousands? -tenths by tells?-thousandths by hundreds?-thousandths by hundredths?-thousands by hundredths?-millions by millionths? -How would you represent 9000 by adopting 9 for the base of the numerical system? —by adopting 5 for the base? 4. What is FEDERAL MONEY? How may it be written? What is the probable origin of the sign that is usually prefixed to dollars? How may dollars be reduced to cents?-to mills? How may cents be reduced to dollars?-to mills? How may mills be reduced to cents?-to dollars? If there are decimals below mills, how may they be read? 5. Since removing the decimal point in any number changes the place of each figure, what is the effect of removing the decimal point two places to the left?-to the right?-three places to the left?-to the right'?-seven places to the left? —to the right? Then how can you most readily find 10, 100, or 1000 times any number?-.1,.01,.001, or.0001 of any number? 6. What is ADDITION? What sign is used to denote addition? What is the sign of equality? How do you arrange numbers that are to be added together? Where do you commence the addition? What do you do with the sum of each column? Why do you not always place the whole sum of a column underneath the column? How do you find whether your answer is correct? Can any one of a series of numbers be greater than the sum of the whole series? How can you prove addition by casting out 9s? 7. What is SUBTRACTION? What is the remainder?-the subtrahend?-the minuend? What is the signification of the termia An integer is a whole number: as seven, forty-nine.

Page  19 ~ 3.] THEORY OF ARITHMETIC. 19 nation nd in many arithmetical terms? If you add the difference of two numbers to the less number, what will you obtain? If you take the difference from the greater number, what will you obtain? How do you find the minuend, if the remainder and subtrahend are given? —the subtrahend, if the remainder and minuend are given? What is the sign of subtraction? Of what operation is subtraction the opposite? How do you write numbers in subtraction? Where do you begin to subtract? If any figure is greater than the one above it, what may be done? Explain the reason of this. What may be done when the subtrahend has more decimal places than the minuend? How do you prove subtraction? In finding the difference between two numbers, which must be the minuend? How can you tell which is the larger number? Can you think of any method of proving addition by subtraction? How can you prove subtraction by casting out 9s? 8. What is MULTIPLICATION? Of what is it an abbreviation? What is meant by the multiplicand?-the product?-the multiplier?-the factors?-a composite number? Name some composite numbers. What is the sign of multiplication? Can the multiplier and multiplicand exchange places? Give an illustration in proof of your answer. How do you multiply by a single figure? How may you multiply by a composite number? How do you proceed when the multiplier consists of a number of figures? How many decimals must there be in the product? What do you do if there are not decimals enough? What may be done, if there are naughts at the right hand of either factor? How can you most readily multiply by 10, 100, 1000, &c.? Explain the reason for the several modes of multiplication. How do you prove multiplication? What must you know before you can find the value of any number of things? How can you prove multiplication by casting out 9s? 9. What is DIVISION? Of what is it an abbreviation? What is meant by the dividend?-the quotient?-the divisor?-the remainder? Of what are the divisor and quotient factors? What are the different modes of expressing division? Give, in your own words, a rule for division, and explain the reason for each step of the process. Of what operation is division the opposite? How many decimals must the quotient contain? Why? How do you find the dividend, when the divisor and quotient are given? -the divisor, when the dividend and quotient are given? What is the difference between short and long division 9 What may be done, if there are naughts at the right hand of the divisor? How

Page  20 20 TEST QUESTIONS ON THE [ART. II. can you most readily divide by 10, 100, 1000, &c.? How do you prove division? How can you avoid frequent trials, in finding the true quotient figure? When the value of any number of things is given, how do you find the value of one? When the value of a number of things is given, how can you find the number of things? Can you think of any method for proving multiplication by division? How can you prove division by casting out9s? 10. When the SuIn oF Two NUMBERs and one of the numbers are given, how do you find the other? When the greater of two numbers and their difference are given, how do you find the less? When the less of two numbers and their difference are given, how do you find the greater? When the factors are given, how do you find the product? When the product and one of the factors are given, how do you find the other factor? When the value of any number of things is given, how do you find the value of any other number? By which of the fundamental rules do you increase a number? By which do you diminish a number? In what cases would you employ addition? —subtraction? —multiplication?division? What is meant by the 3d,-7th,-15th power of a number? 11. What is a PRIMiE NITMBER?-a composite number? Name all the prime numbers less than 100. How can you tell whether any number can be divided by 2, 5, 3, 9? How do you find the prime factors of any number? How do you determine whether any number is a prime? What is meant by a multiple?-a common multiple?-the least common multiple? How do we find the least common multiple of two or more numbers? Show that this process will give the least common multiple. What is a common divisor? —the greatest common divisor? How do we find the greatest common divisor? Why do we proceed in this manner? What is cancelling? How do you cancel? When you have cancelled all the numbers in either the dividend or divisor, what must be put in their place? When the divisor is a composite number, how may we obtain the quotient? 12. What are FRACTIONS? In how many different ways may they be read? Show why each of these ways may be adopted. What is meant by the numerator?-the denominator?-the terms of a fraction? What is shown by the numerator?-by the denominator? How is the value of a fraction affected by increasing the denominator?-the numerator?-by diminishing the numerator? -the denominator? What is a mixed number? How would you find what part 324 is of 187? Why?

Page  21 ~ 3.] THEORY OF ARITHMETIC. 21 13. What is REDUCTION? HOw do we reduce fractions to whole numbers?-to mixed numbers?-to decimals? Explain these reductions. What are proper fractions?-improper fractions? When is a fraction greater than 1?-less than 1?-equal to one? How would you write a decimal in the form of a fraction? Explain the rule for reducing whole or mixed numbers, or decimals, to fractions. Explain the method of reducing compound to simple fractions; fractions to their lowest terms; fractions to a common denominator; complex to simple fractions; fractions to others having any given numerator or denominator. Explain the rules for addition, subtraction, multiplication, and division of fractions. What is meant by an integer? —by the reciprocal of a number? Take examples of your own, and form rules for finding the sum and the difference of two fractions, each of which has 1 for its numerator;-for finding the sum and the difference of two fractions which have the same numerator? 14. What are INFINITE DECIMALS? What is the repetend? How is it distinguished? To what is every repetend equivalent? Show that it is so. How then would you reduce any infinite decimal to a fraction? How may addition and subtraction of infinite decimals be performed without reducing them to fractions? 15. What are COMPouND NUMBEnRs? Repeat the table of Federal Money; English Money; Troy Weight; Apothecaries' Weight; Avoirdupois Weight; Long Measure; Cloth Measure; Square Measure; Cubic Measure; Dry Measure; Liquid Measure; Circular Measure; Time Mleasure. How many shillings in a guinea? How many dollars in a pound sterling? Ihow do we find the areaa of a rectangular surface? How many cubic feet make a foot of wood, or a cord foot? How many cord feet in a cord? How many cubic feet in a ton of round timber?-a ton of square timber? —a ton of shipping or storage? How do we find the solid contents of any rectangular solid? 16. For what purpose are each of the weights and measures used? How many cubic inches in a half peck?-in a common gallon?-in a gallon of milk or of malt liquor? For what purpose is the hogshead (measure) used? How many degrees in a quadrant?-in a sign of the zodiac? How many days in a leap year? What years are leap years? Repeat the number of days in each month. In mercantile business how many days are considered as making a month? a The number of square feet, rods, or acres, &c., in any surface, is called its area.

Page  22 22 TEST QUESTIONS ON THE lART. 11. 17. Give, in your own language, a rule for reducing higher denominations to lower; lower denominations to higher; for compound addition; compound subtraction; compound multiplication; compound division. Explain each rule. Can fractions and decimals be reduced by the same rules as whole numbers? 18. What is meant by PER CENT.? How do we compute any required percentage? Why? What is meant by commission? - insurance? - premium? - policy? - underwriters? - taxes?stocks?-the par value?-the dividend on a stock? When is a stock said to be above par?-below par?-at a discount? —at an advance? What part of the original cost is stock worth, that sells at 162 per cent. below par?-at 12} per cent. advance?-at a discount of 177 per cent.?-at 25 per cent. above par? What is meant by duties?-specific duties?-ad valorem duties?-draft? -tare?-gross weight?-net weight? In custom-house business, what allowance is made for leakage and breakage?-for draft? How do we find what per cent. is gained or lost by any transaction? 19. What is INTEREST?-simple interest?-compound interest? What is the principal?-the rate?-the amount? Give, in your own language, the General Rule for Interest;-the Bank and Business Rule. Show how each rule is obtained. What is a promissory note?-an endorsement? What is the usual mode of computing interest on notes that are settled within a year from their date? What is the Leyal Rule? How do we find the amoinzt of any sum at compound interest? -How do we find the compozund interest? 20. How do you find THE RATE, when the principal, interest, and time are given?-the time, when the principal, interest, and rate are given?-the principal, when the time, rate, and interest are given?-the principal, when the time, rate, and amount are given? Explain the reasons for each of these methods. 21. What is DISCOUNT?-equitable discount?-bank discount? How do you find equitable discount?-bank discount? What are days of grace? How many days of grace are usually allowed? 22. What is ANALYSIS? What is meant by known and unknown quantities? What should we endeavor to do, in solving difficult questions?- How may this usually be done? Give a General Rule for analysis. 23. What is RATIO? How is it expressed? -Iow may it be indicated? Which is the usual mode, and how is it read? What is the antecedent?-the consequent? How do you find the ratio between different denominations?

Page  23 ~3.] THEOtY OF ARITIIMETIC. 23 24. What do you understand by REDUCTIONT OF CURRENCIES? DO the people of the United States ever have occasion to make such a reduction? Why? Name the currencies in common use. How may each be reduced to Federal Money? How may Federal Money be reduced to each currency? How do you reduce English Money to Federal Money, including the premium of exchange? Federal Money to English Money? Give your reasons. What is the cause of the premium? 25. What do you understand by PRACTICE? Repeat the table of parts of a dollar. Give some examples illustrating the use of the table. What is meant by' given terms?-by terms of demand? Give the rule for complex analysis, and illustrate it by an example of your own. 26. What are DlUODEcIMALS? In what are they used? What are the lenominations of duodecimals? How are they marked? What are those marks called? How do you add or subtract duodecimals? How do you find the product of any two denominations? Why? Multiply 4 7' 10" by 3" 5"'. How do you find the quotient of any two denominations? Why? Divide 8' 2" 6"' 10"" 4""' by 2' 8" 4"'. 27. What is a PROPORTION? How is it usually written, and how read? In what other ways may it be written? Why? What are the extremes of a proportion?-the means? Prove that the product of the extremes is equal to the product of the means. If any three terms of a proportion are given, how may the other be found? Which term usually represents the unknown term? How do yoifind the fourth term? What class of questions may be stated in the form of a proportion? How would you arrange the first and second terms, if you wished the fourth term to represent the multiplication of the third term by a ratio?-the division of the third term by a ratio? What do you understand by direct, and by inverse ratio? 28. What is placed as the THIRD TERM in a proportion-? State the rule for simple proportion. What is this rule sometimes called? Is the value of the third term ever affected by more than one ratio? Illustrate your answer by an example, and state that example in the form of a compound proportion. Why do you invert the several ratios by which the third term is to be multiplied? State the rule for compound proportion. 29. What is the origin of ARBITRATIoN OF EXCHANGE? Give an example. In what different ways may arbitration be effected? Which is the usual method? Repeat the Chain Rule.

Page  24 24 TEST QUESTIONS ON TIIE [ART. II. 30. What is FEIaxowsSIIP? How may it be solved by Proportion? What is Compound Fellowship? Give the rule. Explain examples of your own, both by analysis and by proportion, in Simple and Compound Fellowship. 31. What is meant by AVERAGE? Give an example. How do you find the average of a series of quantities? What is Alligation? -alligation alternate? What is the rule for alligation medial? What is Equation of Payments? How does it differ from alligation medial? Give, the rule. On what supposition is this rule founded? What is done with fractions of a day? Give the rule for alligation alternate. Explain the rule by an example of your own. 32. What is INVOLUTION? What is the first power of a number? -the second power?-the fifth power?-the eleventh power? What is the exponent? What is the second power often called, and why?-the third power? 33. What is EVOLUTION? What is the root of a number? What is the radical sign, and how is it used? What is the square root? How may we determine the number of figures that the square root will contain? Explain the method by which the square root is found, by multiplying 39 by 39 and extracting the square root of the product. What must be done when any trial divisor is not contained in the dividend? — when any figure obtained for the root proves too large? How may approximate roots be obtained? 34. What is the CUBE ROOT of a number? How may we determine the number of figures that the cube root will contain? Explain the method by which the cube root is found, by raising 39 to the third power, and extracting the cube root of the result. What must be done when the trial divisor is not contained in the hundreds of the dividend?-when any figure obtained for the root is too large? How may approximate roots be obtained? How may the trial divisors, after the first, be conveniently found? 35. What is a SERIES?-a natural series?-an arithmetical series? —a geometrical series?-a harmonical series? What is meant by Arithmetical Progression? What' are the extremes?the means?-the common difference? Explain the rule for finding one of the extremes and the sum of all the terms; for finding the common difference and sum of the terms; the number of terms and the sum of the terms; the number of terms and common difference; one extreme and the common difference. 36. What is meant by GEOMETRICAL PROGRESSION?-by the ratio?

Page  25 ~ 4.] PRACTICE OF ARITHIMETIC. 25 -the extremes?-the means? Explain the rule for finding the last term, when the first term, ratio, and number of terms are given; for finding the sum of all the terms. How may compound interest be found by series?-by the table? How may we find by the table, the present worth of any sum at compound interest?the time in which any principal will amount to a given sum?the rate at which the principal will amount to a given sum in a given time? 37. What is an ANNUITY?-an annuity certain?-a contingent annuity?-a perpetual annuity?-an annuity in possession?-an annuity in reversion? —the present worth of an annuity? How do you find the amount due on an annuity?-the present worth of an annuity certain?-the present worth of a perpetual annuity? -the present worth of an annuity in reversion?-an annuity, the present worth being given? What does penrmutation show? Explain the rule. 4. PRACTICAL QUESTIONS. All the information necessary to enable the pupil to answer these questions, will be found in the body of the work. 1. How do you find the price at which goods must be sold, in order to gain any required amount? 2. How do you find the original cost, the selling price and the loss being known? 3. How do you find the whole amount of an invoice, from the origin:ll cost and the charges upon the merchandise? 4. How do you find the amount gained by any sale? 5. The loss upon a sale? 6. The net proceeds of a sale? 7. The balance of an account? 8. The average price of several ingredients? 9. The quantity of several ingredients, that will make a mixture of a given average value? 10. How would you determine the number of feet of boardsa in the floor of a room? 11. The number of bricks in the walls of a house? 12. The number of shingles or slatesb on a roof'? 13. The number of clapboardsc on a house? 14. The amount of painting and plasteringd in a house? 15. The quantity of timber in the frame of a house? 16. The cost of glazinge a house? a Lumber is measured by Board Measure, 1 foot= -l~ cub. foot. b 1000 shingles, or 500 slates, are allowed to 100 square feet. c Give the data requisite to determine the number. d Painting and plastering are estimated by the square yard. e Glazing is estimated by the square foot. Builders usually furnish windows by the piece.

Page  26 26 TEST QUESTIONS ON THE [ART. II. 17. How would you find the amount of earth to be removed in excavating a cellar,a when the surface of the ground is level? 18. When the surface is uneven?b 19. The quantity of gravelc required for making a road? 20. For an embankment? 21. For an embankment on uneven ground, the top of the bank being level? 22. The quantity of stone in a wall?d 23. In estimating walls, how would you allow for corners? 24. For windows, doors, gates, or other openings? 25. How would you estimate the number of yards of carpeting required for covering a floor? 26. The number of rolls of papere necessary for papering a room? 27. The quantity of canvass in the sails of a ship? 28. The weight of a stone wall or pillar? f 29. The weight of iron pipes or pillars? f 30. The weight of lead pipe?f 31. The weight of a brick wall?f 32. The weight of a wooden bridge?f 33. The weight of wire cables? f 34. The weight of a ship in the water? 35. How many days from June 3d to August 19th? 36. From May 24th to September 9th? 37. From Jan. 7th, 1848, to March 6th? 38. From Feb. 27th, 1859, to Nov. 11th? 39. Give the shortest method that you know for finding the number of days between any given dates. 40. If you give a note on MIonday, payable in 90 days, on what day of the week will it become due?s 41. On what day would a note at 60 days become due, if given on Saturday'? 42. On Thursday? 43. On Tuesday? 44. On Friday? 45. On Wednesday? 46. A note at 30 days, given on Wednesday? 47. On Tuesday?h 48. On what day of the month would a note fall due, if dated June 27th, at 30 days? 49. March 9th, at 60 days?i 50. Aug. 1st, at 90 days? 51. May 18th, at 3 months? a Excavations are estimated either by the cubic yard, or by the "square",of 216 cubic feet. b In excavating uneven ground, the depth may be measured at several different points, and the average of all the measurements taken. c Gravel is estimated by the " square" (see note a.) d Stone is measured by the " perch" of 241 cubic feet. e A roll of paper-hangings contains 4- square yards. f See Table of Specific Gravities-. g 93 days= 13 weeks and 2 days. The note will therefore become due on Wednesday.'1 If the last day of grace falls on Sunday, or on a holiday, the note must be paid on the day previous.' 9 and 63 are 72. Deducting 31 days for March, 41 remains. 13cunct 30 days for April, 11 remains. The note will be due May 11.

Page  27 ~ 4.] PRACTICE OF ARITIIMETIC. 27 52. October 31st, at 4 months?a 53. Sept. 25th, at 6 months? 54. If the 1st of June falls on Monday, how lmany days' interest shall I lose by giving my note at 60 days, dated May 3d? 55. How would you determine the distance of a cannon, or a thunder cloud, by observing the flash and report?b 56. Suppose the nearest fixed star to be suddenly destroyed, how long would its light continue to reach us?c 57. How long would it take a locomotive, at the rate of 30 miles an hour, to travel the same distance that light goes in a single second? 58. To travel as far as the distance from the earth to the sun?d 59. How far does the earth move in its orbit, while a ray of light is coming froom the sun?e 60. Can you tell why a difference of 150 in longitude, makes an hour's difference in time? f 61. In latitudes where a degree of longitude is equivalent to 45 miles, how far must you travel to find 10 seconds difference of time? 62. If you travel east, will you find the time earlier or later than in the place from which you start? 63. Why? 64. When it is noon at Philadelphia, what time is it at New York? 65. Boston? 66. St. Louis? 67. London? 68. Paris? 69. Washington? 70. Portland? 71. St. Petersburg? 72. Canton? 73. Astoria? 74. Suppose the chronometer of a vessel to be regulated by the Boston time, in what longitude is the vessel, if the sun passes the meridian at half-past nine by the chronometer? 75. If an unbroken telegraph wire should be extended from Boston to Oregon City, at what time could the people at the latter place hear of a transaction that occurred in Boston at 20 minutes past 3 P. M.?, 76. What place would the same news reach by telegraph, at noon of the same day? 77. If your watch keeps accurate time, and is correct when you leave a A note for months, falling due on the 31st, is considered as due on the last day of the month. 1) Sound moves at the rate of a mile in about 5 seconds.'The distance of the nearest fixed star is upwards of 21000000000000 miles. Light moves at the rate of' 192000 miles in a second. d The mean distance of the earth from the SUIn, is about 95 million miles. The circumference of the earth's orbit is about 600 million miles. f Iow many degrees does any point on the earth's surface pass over ill a day? g No perceptible time will elapse during the passage of electricity over the wire.

Page  28 28 TEST QUESTIONS ON THE [ART. II. Boston, how would you set a watch right that is five minutes slower than your own when you reach Buffalo? 78. How would you determine the area of any piece of land, if you had no measure but a yard-stick? 79. Knowing the width of a rectangular field, how would you find the length of a strip that would contain an acre? 80. How would you estimate the area of a field by pacing?a 81. If a road 3 rods wide, runs 380 feet through my land, what amount of damages can I claim, the land being worth $150 an acre?b 82. The length of a certain field is four times the breadth, and the area is ten acres;c how many rods of wall will enclose it? 83. What must be given, in order to determine the altitude of a triangle? 84. The base? 85. The area? 86. How would you estimate the contents of a load or a pile of woo?d 87. Of a pile of boards?e 88. Of a pile of bricks? f 89. Of a stack of hay?g 90. Of a heap of earth? 91. Of a wall? 92. Name some object that you think would measure about 1 foot;1 yard;-3 yards;-1 rod;-some distance that you think about a furlong. 93. Draw a line about 1 inch long; 3 inches; 6 inches; 9 inches; 1 foot; 4 inches; 8 inches. 94. How much do you suppose a hogshead' of water would weigh?i 95. If you should wish to weigh 15 pounds, but had mislaid your weights, how could you form an estimate with water? 96. Give an estimate of all the dimensions of the room you are in? 97. Suppose a note to be given for $1000, interest payable semiannually, to what would it amount in 4 years, at compound interest? 98. At simple interest? 99. Allowing simple interest on the a A pace is estimated at 3 feet. An ordinary step is about 24 feet. b $64.77. c The field can be divided into four squares, and the side of one of those squares will be the shorter side of the field. d Give the answer for piles of regular and irregular shapes. e A cubic foot= 12 feet Board Measure. f 27 bricks measure 1 cubic foot. The number of bricks required to build a house, may be estimated by dividing the number of cubic feet in the walls by.04. g The area of a circle may be conveniently found by squaring.8 of the diameter. Hay is sometimes sold in the stack, by the cubic foot. 400 feet of trodden hay, weigh about one ton. h A hogshead of measure, is intended. A cubic foot of water weighs abouit 1000 ounces.

Page  29 ~ 4.] PRACTICE OF ARITHMETIC. 29 principal, and also on the interest after it becomes due?a 100. Allowing compound interest when notice is given, supposing the debtor to be notified at the end of each year?a 101. How would you determine the face of a note, that would yield $1000 in 3 years, at simple interest?b 102. At compound interest?b 103. At annual interest, allowing simple interest on the interest?b 104. How would you determine the face of a note to be discounted at bank, in order to obtain any required sum?b 105. How many shillings are equivalent to.1 of a pound? 106. To what decimal of a pound, is 1 shilling equivalent? 107. To what decimal of a pound, are 24 farthings equivalent? 108. Then how many must you add to any given number of farthings, to represent their value in thousandths of a pound? 109. Can you give any rule, deduced from the answers to the preceding questions, for reducing shillings, pence and farthings, to the decimal of a pound, by inspection? 110. Can you reverse the process, and give a rule for reducing any decimal of a pound to shillings, pence and farthings, by inspection? 111. In computing interest at 6 per cent., in how many months will an investment gain.01 of itself? 112. In how many days will it gain.001? 113. Canll you give a convenient rule for determining by inspection, the interest on $1 at 5 per cent., 6 per cent., and 7 per cent., for any given time? 114. Give similar rules for finding the amount, and the present worth of $1 for any given time. 115. Can you think of any other abbreviations in computing interest?c a The laws of diflerent states vary with regard to compound interest. In many places it is collected on all notes; in some cases the note is renewed each year and the interest is included in the new note; by the laws of some states, compound interest may be collected, provided notice is given when the interest becomes due; and in some states, simple interest is allowed on the principal, and also on all the interest from the time each payment becomes due till the final settlement. b After finding how much $1 would yield in the given time, how would you find the number of dollars required to yield the desired amount? C The following rule will be found very convenient in computing interest on notes and accounts, when the rate is 6 per cent. lultiply I per cent. of the principal by one-half the even number of mnonths, and if there is an odd month, add 30 to the number of days. Divide the days by 6, and multiply.001 of the principal by the quotient. If there are any re,

Page  30 30 TEST QUESTIONS ON TIE [ART. I. 116. If you intrust a certain sum to a factor, to cover the whole amount of his purchases and commission, how would you find the amount he can lay out? 117. In selling a certain invoice of merchandise at wholesale, a discount of 15 per cent. was made from the retail price. The clerk, in making out the account, calculated 15 per cent. on the whole cost, and deducted it from the bill. Could you have told him any readier mode of obtaining the result? 118. How would you determine the present value of a widow's dower?a 119. The value of a pension, payable during the life of one or more individuals?a 120. The amount that should be annually contributed to secure the payment of any desired sum, at a person's decease?a 121. The amount of an annual payment, for securing a weekly contribution during illness?b 122. The amount of a legacy, sufficient to erect a bridge and provide funds for all the repairs that it will ever probably need? 123. The amount of weekly savings necessary, to make a young man worth %5000 in 20 years? 124. The amount of weekly savings necessary for cancelling a debt, with all the interest, in any given time? 125. How would you reduce Sterling to Federal Money, at 9 per cent. premium? 126. Federal to Sterling Money, at 7-1 per cent. premium? 127. How would you find the value of stock at a discount of 27 pei cent.? 128. At an advance of 18a per cent.? 129. H-Iow are interest and discount computed in Banks? 130. How do you compute percentage on English Money? 131. If you have the diameter of a circle given, how would you find the diameter of a circle that is 16 times as large? 132. i as large? 133. 2.56 times as large? 134. 49 times as large? 135. If you know the area of a field, what would be the area of a similar field, each side of which is ~ as long? 136. 3 times as long? 137. 7 as long? 138. If a ball 2 inches in diameter, weighs 11- pounds, what would be the weight of a similar ball, 6 inches in diameter? 139. What would be the diameter of a similar ball that would weigh 96 pounds? 140. If a tree 1 foot in diameter, yields 2 cords of wood, how much wood is there in a similar tree that is 3 mainizlg days, take as many 6Oths of 1 per cent. Add the p'roducts thU obtained, and their sum will be the interest at 6 per cent. a All these questions are solved by estimating the probable duration of life. After that is determined, what remains to be done? b T'he average amount of sickness is supposed to be known.

Page  31 ~ 4.] PRACTICE OF ARITHMIETIC. 31 feet 6 inches in diameter? 141. If a hollow sphere 3 feet in diameter and 2~ inches thick, weighs 12 tons, what are the dimensions of a similar sphere that weighs 324 tons? 142. Knowing the original cost of any article, how would you determine the price at which it must be sold, in order to gain any given per cent.? 143. How do you find the percentage gained or lost in any transaction? 144. The dividend that a bankrupt can pay upon each dollar of his debts? 145. The percentage of increase in the population of a place? 146. The original cost, by knowing how much per cent. is gained or lost by selling at a given rate? 147. The entire value, by knowing the value of any given percentage? 148. How would you determine the time at which a debt could be cancelled by a note for any particular amount? 149. The time at which several debts can be cancelled by a single payment? 150, The amount of interest due on an unsettled account, there being debits and credits embraced in the account? 151. The average time for settling an account, in which there are charges with different times of credit? 152. Can you give more than one method for averaging, or equating an account? 153. Why do we begin at the left hand in division? 154. Do you know of more than one mode of proof for each of the simple rules. 155. What is meant by the Arithmetical Complement of a number? 156. Can you perform a number of additions and subtractions at a single operation, by using the Arithmetical Complement? 157. Can you multiply by three or more figures at a single operation? 158. How do you find the cost of articles sold by the hundred or thousand? 159. Knowing the difference of longitude between two places, how do you find their difference of time? 160. Knowing the difference of time, how do you find the difference of longitude? 161. If d represents the diameter of a circle, c the circu1mference, and a the area, how would you find d, c and a being known? 162. Knowing c and d, how would you find a? 163. Knowing a and d, how would you find c? 164. How would you find the rate of insurance, knowing the premium and the amount insured? 165. The amount necessary to insure, in order to cover the premium and expenses of collecting, in addition to the loss? 166. The cost and rate per cent. of profit or loss being given, how would you find the amount of profit or loss? 167. The cost given, how would you find the selling price to gain or lose a specified

Page  32 32 TEST QUESTIONS. [ART. II. rate per cent.? 168. Cost and selling price given, how would you find the rate per cent. of profit or loss? 169. Selling price and rate of profit or loss given, how would you find the cost? 170. From the prime cost how would you find the selling price so as to gain any proposed percentage, and allow a discount for ready money? 171. I-ow would you estimate the quantity of grain in a rectangular bin? 172. In a circular bin? 173. In a pile against the side of a building? 174. In a pile in the corner of a building? 175. How would you find the true weight, by a pair of false scales? 176. How would you find the value of any estate, knowing its annual rent? 177. Knowing the cost, rent, and annual outlay for taxes and repairs, how would you find the rate of interest that any estate yields? 178. How would you find what quantity of stock may be purchased for any given sum, allowing for brokerage? 179. How would you find the rate of interest gained by money invested in stocks at any given price? 180. How would you find what sum must be laid out in any kind of stock to produce a given annual income? 181. What must be given, and in what manner would you proceed, to determine the area of a rectangular field? 182. To determine either side? 183. To find either side of a right angled triangle? 184. To find the area of the surface of a sphere? 185. The solidity of a sphere? 186. The solidity of a cone? 187. Of a cylinder? 188. How would you find the area of an irregular field? 189. The solidity of an irregular body? 190. The specific gravity of a body? 191. The tonnage of a ship? 192. The contents of a pan with slanting sides? 193. Of a cylindrical pail? 194. Give the dimensions of a box that would hold a bushel? 195. A peck? 196. A wine gallon? 197. A wine quart? 198. A beer gallon? 199. A gill? 200. For what purpose is the hogshead measure used? 201. How would you find the area of a roof that would fill a cistern of given dimensions, with a fall of 4 inch of rain? 202. What is meant by gross weight? 203. Net weight? 204. Tare? 205. What is meant by a common year? 206. A sidereal or periodic year? 207. A leap year? 208. A civil year? 209. How would you find the number of gallons equivalent to one foot in depth of a cistern? 210. The number of plants in an acre of ground, their distance apart being given? 211. What do you understand by an engine of forty horse power?

Page  33 ~ 5.] MISCELLANEOUS EXAMPLES. 33 III. THE FUNDAMENTAL RULES. N. B. The pupil should solve all the questions that he can, mentally. Let him never use the slate in obtaining an answer, except when it is absolutely necessary. B. EXAMPLES IN ADDITION AND SUBTRACTION. 1-4. Find from the following table, the population and extent of the globe, according to each of the authorities mentioned.~ Gr Divisions ccording to Balbi. | Weimar Alm. 1840. English English of the Globe. Population English Population... Sq. Miles. Sq. Miles. Europe - - - -227700000 3700000 238240043 3807195 Asia - - - 890000000 16045000 608516019 17805146 Africa - - - - 60000000 11254000 101498411 11647428 America - - 39000000114730000 48007150 13542400 Oceanica - - - 20300000 4105000( 1834194 3347840 5. The population of China in 1743, according to the French missionaries, was 150029855; in 1825, according to Dr. Morrison, 352866002. What is the difference between these two estimates? 6-9. The skull has 8 bonesb the face 14,c the ear 4,d the tongue 1,e and there are 32 teeth.f How many bones American Almanac, 1842. b Osffrontis, two ossa parietalia, two ossa temporum, os occipitis, os spihenoides, and os etlnoides. c Two ossa maxillaria superiora, two ossa nasi, two ossa unguis, two ossa malarsum, two ossa palati, two ossa spongiosa iniferiora, the vomer, and the os maxillare inferius. d ll1alleus, incus, os orbiculare, and stapes. e Os hyoides. f Sixteen in each jaw, viz: four incisores, two cuspidati, four bicus. pides, and six molares. 3

Page  34 34 FUNDAMENTAL RULES. [ART. III. in the whole head? Which are the most numerous? By how many do they exceed all the others? If you subtract each class of the bones, named above, from the entire number, how many will be left after each subtraction? 10-13. In the trunk there are 24 spinal bones,a 24 ribs, the sternum, or breast-bone, the os sacrum, the os coccygis, and two ossa innominata. Required the number in the entire trunk? In the trunk and head together? How many more in the trunk than in the skull? How many more in the head than in the trunk? 14-17. Each of the'upper extremities contains the following bones, viz: 2 in the shoulders,b 3 in the arm,c 8 in the carpus, or wrist, 5 in the metacarpus, or palm, 2 in the thumb, and 3 in each of the fingers.d How many in all? How many in both of the upper extremities? In the head, trunk, and upper extremities? How many more in one of the upper extremities than in the face? 18-21. Each of the inferior extremities contains the following bones, viz: 4 in the leg,e 7 in the tarsus, or ankle, 5 in the metatarsus, or foot, and 3 in each of the toes, except the great toe, which has but two. Required the number in each of the lower extremities? In both? In all the extremities? In the whole body? Ans. to the last, 240.f 22-25. How many bones in the whole body, besides those of the head? Besides those of the trunk? Of the upper extremities? Of the lower extremities? 26-29. The skin of the cranium has 3 principal musa The 24 true vertebrae, 7 cervical, 12 dorsal, and 5 lumbar. b The clavicle and scapula. C Humerus, radius, and ulna. d Called phalanges. Os femoris, tibia, fibula, and patella. t Besides the bones enumerated, are the sesamoid bones, which vary in number. Marjolin counts five for each of the upper, and three for each of the lower extremities, making two hundred and fifty-six in the whole body.- Wistar's Anatomy.

Page  35 ~ 5.] MISCELLANEOUS EXAMPLES. 35 cles,a each ear has 4, the lids of each eye 2, each eye 6, the nose 2, the mouth 19, the tongue 8, and the lower jaw has 4 pairs. How many in all? - How many less than the whole does the mouth contain? How many less than half? WThat is the difference between the number of bones and the number of muscles in the head? 30-33. The neck and throat contain 50 principal muscles, the trunk 116, each of the superior extremities 46, and each of the inferior extremities 51. How many in all? How many in the entire body.? How many less in the head than in the rest of the body? How many more muscles than bones in the whole system? 34-37. Find the difference between the following numbers, commencing the subtraction at the left hand:b 84.9108757944362 and 190027.08; 67490083574.00882 and 3375109884726.37095; $16438 and $9728.87758403; 7 and.0019547998027184920625. 38-44. Determine from the following table, the entire population of the United States and Territories, at each of the given dates. In obtaining the results, add two columns of figures at a time.c a One pair and one single muscle. Most of the muscles are found in pairs, one on each side of the body. There is sometimes a slight difference inrthe number, in different individuals. b In beginning to subtract at the left hand, if at any point the remaining figures of the subtrahend are greater than those of the minuend, we must add one to the figure we are subtracting. For example, in subtracting 92824 from 164809, say 9 from 16 164809 leaves 7; then as 824 is greater than 809, say 3 from 4 leaves 92824 1; as 24 is greater than 09, say 9 from 18 leaves 9; 2 from 10 71985 leaves 8, 4 fromn 9 leaves 5. Let the pupil explain the ra. tionale of this process. c The teacher will find this a valuable exercise, to be performed aloud, at the recitation of the class. In adding two or more columns at once, it will be found most convenient to commence at the left hand, and observe at each step whether there will be one to bring from the right hand figures. Thus in adding 972 and 645, instead of saying 5 and 2 are 7, 4 and 7 are 11, 9 and 6 are 15 and 1 are 16,-say 9 and 6 and 1 are 16, 7 and 4 are 11, 5 and 2 are 7. After a little prac.

Page  36 36' FUNDAMENTAL RULES. [ART. IIT. TABLE OF POPULATION-FROM THE AMERICAN ALMANAC. STATES. 1790. 1800. 1810. 1820. 1830. 1840. 1850 Maine 96,540 151,719 228,705 298,335 399,955 501,793 New Hampshire 141,899 183,7621214,360 244,161 269,328 284,574 Vermont 85,416 154,465 217,713 235,764 280,652 291,948 Massachusetts 378,717 423,245 472,040 523,287 610,408 737,699 Rhode Island 69,110 69,122 77,031 83,059 97,199 108,830 Connecticut 238,141 251,002 262,042 275,202 297,665 309,978 New York 340,120 586,756 959,949 1,372,81211,918,608 2,428,921 New Jersey 184,139 211,949 249,555 277,575 320,823 373,306 Pennsylvania 434,373 602,365 810,091 1,049,458 1,348,233 1,724,033 Delaware 59,098 64,273 72,674 72,749 76,748 78,085 Maryland 319,728 341,548 380,546 407,350 447,040 469,232 Dist.ofColumbia... 14,093 24,023 33,039 39,834 43,712 Virginia 748,308 880,200 974,622 1,065,379 1,211,405 1,239,797 North Carolina 393,751 478,103 555,500 638,829 737,987 753,419 South Carolina 249,073 345,591 415,115 502,741 581,185 594,398 Georgia 82,548 162,101 252,433 340,987 516,823 691,392 Alabama........... 20,845 127,901 309,527 590,756 iMississippi... 8,850 40,352 75,448 136,621 375,651 Louisiana...... 76,556 153,407 215,739 352,411 Florida.............. 34,730 54,477 Texas.............................. Kentucky 73,077 220,955 406,511 564,317 687,917 779,828 Tennessee 35,791 105,602 261,727 422,813 681,904 829,210 Ohio 45,365 230,760 581,434 937,903 1,519,467 Indiana 4,875 24,520 147,178 343,031 685,866 Illinois............. 12,282 55,211 157,455 476,183 Michigan............. 4,762 8,896 31,639 212,267 Iowa......................... 43,112 Wisconsin.................................. 30,945 Missouri............ 20,845 66,586 140,4451 383,702 Arkansas......... 14273 30,388 97,574 Minesota Terr. Missouri " Oregon................................. Indian ~ New Mexico California 45-57. Find the increase in the population from 1790 to 1800; to 1810; 1820; 1830; 1840; 1850; from 1800 to 1810; 1810 to 1820; 1820 to 1830; 1830 to 1840; 1840 to 1850; 1790 to 1820; 1820 to 1850. 58-181. Find the amount of each of the following cobl umns, and also the total amount of receipts and, expenditures in each year. Add two columns of figures at a time. tice the pupil will say at once, 972 and 645 are 1616, as naturally as he would say 8 and 6 are 14, without stopping to count his fingers. The numbers to be added should not be mentioned in performing the addition. Thus in adding 8, 5, 9, 3, 7, 9, 6, 4, say 8, 13, 22, 25, 32, 41, 47, 51, instead of saying 8 and 5 are 13, and 9 are 22, and 3 are 25, &c. In adding 65, 48, 27, 92, say 65, 113, 140, 232, and proceed in like manner in all cases.

Page  37 ~ 5.] MISCELLANEOUS EXAMPLES. 37 STATEMENT OF THE RECEIPTS AND EXPENDITURES OF THE UNITED STATES FOR SIXTY YEARS. FROM THE AMERICAN ALMANAC, AND PUBLIC DOCUMENTS. RECEIPTS. I EXPENDITURES. Internal and Lands and Civil and Years. Customs. direct taxes. Miscellan. Miscellan. Army. Navy. 1789-91 $4,399,473 $1,083,401 $835,618 $570 179 3,4413,071 $208,943 654,257 1,223,594 53 1793 41255,306 337,706 472,450 1,237,620 1791 4,801,065 274,090 705,598 2,733,540 61,409 1795 5,588,461 337,755 1,367,()37 2,573,059 410,562 1796 6,567,988 4.75,290 $4,836 772,485 1,474,661 274,784 1797 7,549,650 575,491 83,541 1,246,904 1,194,055 382,632 1798 7,106,062 644,358 11,963 1,111,038 2,130,837 1,381,348 1799 6,610,449 779,136 1,039,392 2,582,693 2,858,082 1800 9,080),933 -1,543,620 444 1,337 613 2,625,041 3,448 716 1801 10,750,779 1,1582,377 167,726 1,114,768 1,755,477 2:111,424 1802 12,438,236 828,464 188,628 1 462,929 1,358,589 915,562 1803 10,479,418 287,059 165,676 1,842,636 944,958 1,215,231 1804 11,098,565 101,139 487,527 2,191,009 1,072,017 1,189,833 1805 12,936,487 43,631 540,194 3,768,588 991,136 1,597,500 1806 14,667,698 75,865 765,246 2,891,037 1,540,431 1,649,641 1807 15,845,522 47,784 466,163 1,697,897 1,564,611 1,722,064 1808 16,363,550 27,370 647,939 1,423,286 3,196,985 1,884,068 1809 7,296,021 11,562 442,252 1,215,804 3,771,109 2,427,759 1810 8,583,309 19,879 696,549 1,101,145 21555,693 1,654,244 1811 13,313,223 9,962 1,040,238 1,367,291 2,259,747 1,565,566 1812 8,958,778 5,762 710,428 1,683,088 12,187,046 3,959,365 i813 13,224,623 8,561 835,655 1,729,435 19,906,362 6,446,600 1814 5,99S,772 3,882,482, 1,135,971 2,208,029 20,608,366 7,311,291 1815 7',282,942 6,840,733 1.,287,959 2,898,871 15'394,700 8,660,000 1816 36,306,875 9,378,344 1,717,985 2,989,742 16,475,412 3,908,278 1817 26,283,318 4,512,288 1,991,226 3,518,937 8,621,075 3,314,598 1818 17,176,335 1,219,613 2,606,565 3,835,839 7,019,140 2,953,695 1819 20,283,609 313,244 3,274,423 3,067,212 9,385,421 3,847,640 1820 15,005,612 137,847 1,635,872 2,592,022 6,154,518 4,387,990 1821 13,004,447 98,377 1,21.2966 2,223,122 511.81114 3,319,243 1822 17,589,762 88,617 1,803,582 1,967,996 5,635,187 2,224,459 1823 19,088,433 44,580 916,523 2)022,094 5,258,295 2,503,766 1824 17,878,326 40,865 984,418 7,155,308 5,270,255 2,904,582 1825 20,098,714 28,102 1,216,090 2,748,544 5,692,831 31049,081 1826 23,341,332 28,228 1,393,785 2,600,178 6,243,236 4,218,902 1827 19 712,283 22,513 1,495,945 2 314,777 5,675,742 4 263,878 1828 23,205,524 19,671 1,018,309 2,886,052 5,701,203 3'918,786 1829 22,681,966 25,838 1,517,175 3,092,214 6,250,530 3,308,745 1830 21,922,391 29,141 2,329,356 3,228,416 6,752,689 3,239,429 1831 24,224,442 17,440 3,210,815 3,064,346 6,943,239 3,856,183 1832 28 465,237 18,422 2,623,381 4,574,841 7,982,877 3,956,370 1833 29,032,509 3,153 3,967,682 5,051,789 13,096,1.52 3,901,357 1834 16,214,957 4,216 4,857,601 4,399,779 10,064,428 3,956,260 1835 19,391,311 14,723 4,757,601 3,720)167 9,420,313 3)864,939 1836 23,409,940 1,099 4,877,180 5,388,371 18,466,110 5,800,763 1837 11,169,290 6,863,556 5,524,253 19,417)274 6,852,060 1838 16,158.800 3 214 184 5,666,703 19 936,312 5 975,771 1839 23,137,925 7,261.,118 4 994,562 14,268,981 6,225,003 1840 13,499,502 3,494,356 5,581,878 11 621,438 6,124,456 184i 14,487,21.7 1,470,295 6,490,881 13,7041 882 6,001,077 1842 18,187,969 1,456 058 6,775 625 9,188,469 8,397,243 1843 a 7,046,8144 1,018,482 2,867,289 4,158,384 3,672,718 1844 b 26,183,571 2,320,948 5,231,747 8,231,317 6,496,991 1845 b 27,528,113 2,241,021 5)61.8,207 9,533)203 6,228,639 1846 b 26,712,668 2,786,579 6,783,000 13,579,428 6 450,862 1847 b 23,747,864 2,598,926 6,715,854 41,281,606 7,931,633 1848 b 31,757,071 3)679,680 5,585,070 27,820,163 9,406,737 1849 b I a 6 months of 1843. b For the year ending June 30.

Page  38 38 FUNDAMENTAL RULES. L[AT. IE. 6. EXAMPLES IN ADDITION, SUBTRACTION), MULTIPLICATION, AND DIVISION. 1. Invoice of 6 bales of Dry Goods:Boston, November 27, 1849. Messrs. Thompson & Allen, Bought of William MIansfield,a T. & A. A bale containing 15 pieces Lint Strelitz Osnaburgs, No. 35. each 130 yds., ( 10c. Wrapper, 20 yds., ( 6c. Bale, cording and packing, 1 25 36to40 Five bales, containing No. 36, 464 yds. No. 37, 481 No. 38, 437 " No. 39, 475 " No. 40, 470 " 2327 " ( 11c. Bales, cording and packing, 5 00 $458 42 2. Bill and Receipt:Philadelphia, March 25, 1848. Benjamin Stabler, To John Farmer, Dr. To 4 tons of hay, at $25 50 115 bushels of oats, "A 30 95 A" corn, " 94 Received payment, JOHN FARMEt. a The items should be extended in the left hand column, and the amounts of each package (or all the packages that are included together, as in Nos. 36 to 40), carried out in the right hand column. This is the usual form of an American invoice. The pupil should give te amount for each of the items that is left blank.

Page  39 ~ 6.] MISCELLANEOUS EXAMPLES. 39 3. Bill, unreceipted:New York, July 7, 1849. George Lenox, Bought of Carlisle & Williams, 1 doz. Long Shawls, @ $6.75 3 pieces Sheeting, 30, 31, 33 yds. @.09 1 piece Mousseline de Laine, 28" @.17 2 pieces Broadcloth, 35 and 40" ( 4.75 6 doz. Linen Hdkfs. @.25 12 doz. Cotton Half Hose, ( 1.50 4 months. $486 47 4. Statement of Account:Albany, Jan. 5, 1849. Mason, Hamilton & Co. 1848. To Johnson & Brooks, Drs. Aug. 27. To Stoves, &c., per bill rendered, 6 mo. $843 75 Sept. 5. v " " 375 69 " 13. - cc " "C 118 50 "" 30. " v " " 97 11 Oct. 8. " C " 103 88 " 23. " " " " 491 35 Nov. 19. " " i sC 85 67 Contra, Cr. Oct. i. By Cash, 450.00 Nov. 7. " acceptance of draft, 200.00 Dec. 12. " note at 4 months, 562.50 Balance, $ Due by average, June 21, E. & O. E.a 5. The population of the Chinese Empire has been estimatedb as follows: China proper, 148897000; Corea, 8463000; Thibet and Boutan, 6800000; Mandshuria, Mongolia, &c., 9000000; Colonies, 1000000. Its territory con. a Errors and omissions excepted. b Murray's Encyclopedia of Geography.

Page  40 40 FUNDAMENTAL RULES. [ART. III; tains about 5350000 square miles. Required the entire population, and the number of inhabitants to a square mile. 6. Bill of Lading:tiftpelb, in good order and condition, by Norton & Phillips, in and upon the Brig called the 1adrgaret, whereof -' 1 to 8 lJeriam is master for this present voyage, and now B. Van Pelt, lying in the port of Charleston, and bound for New Care of York, L. Haines, Three Boxes and Five Bales of Goods, 116 Broadway, Weight 1835 lbs. New York. Being marked and numbered as in the margin, to be delivered in the like good order and condition, at the aforesaid port of New Yorik (the danger of the seas only excepted), unto Benjamin Van Pelt, or to his assigns, he or they paying freight for the said goods, at the rate of 28 cts. per 100 lbs., with Primage and Average accustomed. IN TESTIMONY WHEREOF, the Master or Purser of the said Brig hath affirmed to three Bills of Lading, all of this tenor and date, one of which being accomplished, the others to stand void. Dated at Charleston, the 30th day of March, 1849. J. M. MERIAr. Required the amount of freight on the above merchandise. 7. Receipt in full: Worcester, June 5th, 1849. James A. Chase, 1848. To Moses Allen, Dr. Oct. 27.- To Mdse. per bill rendered, 27.35 Nov. 3. " Hire of Horse and Chaise to Leicester, 1.50 1849. March 11, "1 ton of Coal, 9.75 Received payment in full, $ MOSES ALLEN. a A receipt in full is understood to cover all demands.

Page  41 ~ 6.] MISCELLANEOUS EXAMPLES. 41 8. Publisher's Estimate for one thousand copies of a book: Printing and corrections, $513.00 Paper and certificate of copyright, 326.00 Binding, 175.00 Advertising, 200.00 $ 1 copy to be deposited in the Clerk's Office, 10 copies to the author, 989 copies for sale at $1.75, Deduct cost Balance to cover commissions, interest on capital, and profit to author and publisher, $516.75 when all are sold. j 9. Order for Goods Providence, August 18, 1849. Messrs. Anthony.& Smith, Please ship to us by first packet: 25 cwt. Sugar (Brown Havana) about $5.75 per cwt. 13 hhd. W. I. Molasses, $23. 27 boxes Louisiana Oranges, $3.80. Old Java Coffee, say 1300 lbs. at.09. Honey, about 250 gallons at.67. 18 chests Black Tea, $5.50 per chest. 49 bbls. Genesee Flour, $4.75. And oblige, Yours, &c., L. KENTON & RAY. What would be the amount of the above order, if all the articles were sent at the prices affixed to them? 10. Determine by the Roman Notation, how many years elapsed from the discovery of America by Columbus, Anno Domini MCCCCXCII, to the adoption of the United States Constitution, A. D. MDCCLXXXIX.a a The pupil will probably find no difficulty in adding or subtracting Roman numerals, except when such numbers as IV, IX, XL, XC,

Page  42 42 FUNrDAMENT.AL RULES. [ART. III. 11. The Pilgrims landed at Plymouth in MIDCXX, and Benjamin Franklin was born LXXXVI years afterwards. Find by the Roman Notation, in what year he was born. 12. Multiply by the Roman method, and determine the number of wecks that elapsed between the landing of the Pilgrims and the birth of Franklin, allowing LII weeks to each year, and adding XII to the product. 13. Settlement of Account, exhibiting different forms of receipts:Account rendered Jan. 1, 1848. Lowell, Jan. 1, 1848. Thomas Lawrence 1846. To Henry Appleton, Dr. July 11. To Mdse. per bill, $984.32 t" 29. 1" 5 bbls. Flour, ( $85.25 Aug. 31. " 49 yds. Broadcloth, @ 4.50 Oct. 17. " 56 yds. Carpeting, @( 1.15 Ad Balance of interest from Jan. 1, 1847, 16.70 1847. Cr. Jan. 13. By Cash, 100.00 MIar. 4. " Mdse. 27.42 127.42 Balance, $1184.75 occur. The difficulty may be removed by changing them to the forms IIII, VIIII, XXXX, LXXXX. Thus, if it were required to add DCCXLIV, CCXCIX, and DCCXXXXIIII MDLXVIII, writing them as in the margin, we CCLXXXXVIIII commence at the right hand,'and say, eleven Is MDLXVIII are equivalent to two Vs and I; two Vs and two to be added from the Is make four Vs or two MMDCXI Xs; nine Xs and two to be added make eleven Xs, or two Ls and X; two Ls and two to be added make four Ls, or two Cs; four Cs and two Cs make six Cs, or one D and C; two Ds and one D are three Ds, or M and D; M and M are MM. In multiplication, the multiplier maybe divided into any convenient number of parts, and after multiplying by each part, the several partial products should be added together.

Page  43 ~ 6.] MISCELLANEO US EXAMPLES. 43 On settling the account, August 5, 1849, Thomas Law. rence presented the following receipts, viz:I. A Receivt on account. "Received of Thomas Lawrence Seventy-five Dollars, on account. Lowell, Jan. 25, 1848. $75. HENRY APPLETON." II. A Receipt through a third cperson. "Received of Thomas Lawrence, per hands of James Brown, One Hundred and Ten Dollars, on account. $110. HENRY APPLETON, Lowell, March 4, 1848. per WILLIAM ALLEN." rII. Receiptfor Note on account. "Received of Thomas Lawrence his note dated July 5, 1848, @ 8 months date my favor, for Seven Hundred and Fifty Dollars, on account. Lowell, July 7, 1848. Note $750. HENRY APPLETON." The receipts being all correct, there is found to be due to Henry Appleton $78.46 interest, in addition to the unsettled balance of account. Required the amount that has been paid, the balance outstanding, and the amount paid at settlement, for which H. A. gives the following Receipt in fcll. "Received of Thomas Lawrence, I-6 Dollars, in full of all demands to this date. $ HENRY APPLETON." Lowell, August 5, 1849. 14. Allowing 365 days and 6 hours to each year, what length of time would be required to count the number of tons of carbonic acid contained in the atmosphere, counting 3 every second, for 12 hours a day?a Ans. 111697y. 124d. 5h. 26m. 40sec. 15. A horse-power is generally estimated as sufficient to raise 330001bs. I foot high in I minute; and Desaguliers a See Example 17.

Page  44 44 FUNDAMENTAL RULES. [ART. III. estimates the power of 1 horse as equivalent to that of 5 men. According to the latter estimate, how much would 1000 men be able to raise to the height of 20 feet in 5 minutes? Ans. 16500001bs. 16. Account Sales: AccoUNTr SALES. 8 Hhds. Molasses, per M3ayflower, from Trinidad, for account of Williams & Tasistro. gals. qt. CIARGES. W T # 29, 113 Insurance on $150 at 29 30, 112 3 1 per cent. $1.50 to 31, 115 3 Policy,.50 32, 125 - 2.00 83, 119 1 Freight and storage, 19.50 34 117 Duties, 65.30 35, 120 3 35, 120 3 Brokerage, 3.88 36,'121 2 Commission & Guaianty, 16.33 Net proceeds, due Gro. 945 gals. Oct. 5, 1849. $219.54 Net 933 gals. 13ostofi, Aug. 28, 1849, E. E. 35 cts. $ WELLINGTON, CARTEIt & Co. 17. It has been estimated' that the atmosphere contains 3994592925000000 tons of nitrogen, 1233010020000000 tons of oxygen, 5287350000000 tons of carbonic acid, and 54459705000000 tons of aqueous vapor. Required the weight of the whole, in pounds. An's. 11843664 trillion pounds, according to the French Notation.b a Griffiths. b The old English Notation is still employed in some works, but the French method, which is simpler and more easily learned, is generally adopted, even by the more modern English writers. In the English system, each period consisted of six places. A billion was therefore equivalent to a million million; a trillion, was a million billion, and so on.

Page  45 ~ 6.] MISCELLANEOUS EXAMPLES. 45 18. Invoice of 2 cases merchandise, shipped at Havre on board the ship Duchesse d' Orleansj Richardson, Master, bound for New York, purchased for account and risk of Messrs. Hamilton & Co., of Philadelphia, and to them consigned. 226 One Case. No. 54, 107 Q, 214 doz. Braces 6.b 55, 1" 2 " " 9.50 Discount 2 (jo c 26.05d 108 Boxes,.60 Packing, 25.90 - F.1367.65 226 One Case. No. 72, 29V 58 doz. Braces, 9.50 80, 20" 40" " 10. 87, 15" 30 " " 14. Discount 8010 109.70 64 Boxes,.60 Packing, 22.80 Commission 3 ~010 80.70 F. 2770.85 E. E. Paris, 25 November, 1849, LEioux & CIE. 19. The bulk of carbonic acid produced by a healthy adult in 24 hours, is about 15000 cubic inches, and weighs a Cartons. b In French currency the Franc is considered as the unit, and Centimes are written as hundredths. Calculations are therefore made as in Federai Money. Thus 214 doz. at 6 Fr. = 1284 Frs.; 11 doz. at 1.25, would be 13.75, &c. c This character ( ~01o ) is an abbreviation for per cent. d No fraction of a franc is counted, less than 5 centimes.

Page  46 46 FUNDAMENTAL RULES. [ART. III about 6 ounces.a If this were the average rate, how much would be produced daily by the population of a town containing 5000 inhabitants. Ans. 43402c. ft. 1344c. in.-16cwt. 2qr. 271b. 7. TABLE FOR ADDITIONAL EXERCISE. In every kind of business, correctness and readiness in the use of figures are of the first importance. These can only be obtained by practice, and to give an opportunity for such practice, a page of figures is here inserted, which may be so divided by the teacher as to make many thousands of examples. Thus, in addition, the pupil may add a single column, or any two, three or more contiguous columns, or parts of columns, or the entire page; and he may be exercised in adding two, three, or more columns at once.b In subtraction (in which it is recommended that he should always be required to commence at the left hand),c he may take parts or the whole of any two adjoining rows, or he may subtract, (without using his slate,) part of the figures in any row, from the remaining figures. In multiplication and division the examples may be varied in like manner to any desired extent. By this arrangement a whole class may be engaged in the same operation, without having the question written on the board, and the teacher can save the time that might otherwise be required for setting down examples on the slate. Thus, should any scholar require additional exercise in either of the fundamental rules, he may be told to add columns 19, 20, and 21, or to subtract row 15 from row 14, prefixing a 7 to the upper row; or to multiply row 7 by row 11; or to divide row 3 by the first five figures in row 18; or he may be required to form examples for himself., Griffiths. b See note c, page 35. c See note b, page 35.

Page  47 ~7.] TABLE FOR ADDITIONAL EXERCISE. 47 1 2 3 4 5 6 7 8 9 0 12 3 4 5 6 7 8 9 0 123 4 5 6 7 8 2 8740517493952157-80693426792 3463791036734597185046803593 4 7 1 2 59 5 1 0 7 6 7 8 9 7 7 0 0 6 5 6 9 5 5 4 1 3 4 553189605740256970896463291,5 6 790456157880045678943156366 7429170836020162968730769357 8 630690579380083968036970168 9975021890679214607359207429 105968107529574 1 02156481963 1 0 116620715 6 8 6 1562795 6210 84621 12 1465309519362680 I 6951606742 13620986814610840168368516103 14 6 0 5 6 3 6 8 5 6 9 6 0 6 8 3 2 8 6 0 9 6 2 4 1 6 3 4 15 2 9 6 2 9 6 0 7 9 6 3 8 0 0 3 6 7 2 9 5 617 3 3 0 5 16308410466806417794626047906 17 96 708 69316790 6 7 2 4 1 609 706247 1816434902816079217426770264 8 19 0753591165026917692 4 6902619 2023167046160715 8520795 2 0260 21 2 9 0 5 917 0 5 7 9 3 7 0 1 3 60 7 9 2 5 1 9 5 61 22560735790245600982603791572 235975907086979086 348 4 39 7335 3 24586806964804687904790869064 2565097608297680 6470987808 7 8 0 8 0 85 26'9 7 8 4 9 6 9 990 6 7 9 6 8 06 0 9 7 6 8 9 0 7 8 9 6 2778792697060 87078730 8 7 3 0 1 6 3 91 757 283501.69517401583925365495818 2946370 17 30 43681360320726039 30036563263596420547390253640 31 561941357359640 473759 1630 3 7 5 9 1 6 3 0 1 3203570485514026720 6 4 8 6 2 5 1 6 52 33 0542560734653615036 72564063 34 4 8 4 0 513 3 0 5 2 7 4 0 3 8 4 2 60 31 5 3 0 5 4: 35 630484738540168102816214105 36601698356109213084572946136

Page  48 48 FUNDAMENTAL RULES. [ART. ITI. 8. FRACTIONS AND COMPOUND NUMBERS. 1-10. What part of the entire number of bones in the human body is contained in the skull? [See ~ 5, Ex. 6-21, and express all the answers in the lowest terms.] What part in the face?-the ear?-the tongue?-the whole head? -the spine?-the ribs?-the whole trunk?-the head and trunk?-the extremities? 11. Set down the answers to the questions in the preceding paragraph, and add all the fractions together. 12-13. If you were to count three every second, for ten hours a day, how long would it take you to count a million? How long would it take to count a billion, at the same rate? 14. The population of Washington in 1840 was 23364. If all the inhabitants were to commence counting at the rate mentioned in the last example, how long would it take them to count a trillion dollars? 15. The salary of the President of the United States is $25000 per annum. How many years' salary would be equivalent to a million dollars? 16. Pure silver is worth at the mint $15.50 a pound. How many tons would it take to be worth a million dollars? Express the answer in tons and fractions of a ton, and also in tons, cwt., qrs., &c. 17. Standard gold being worth $18.60 per ounce, what weight of gold would be required to furnish a half-eagle to every inhabitant of the United States, estimating the population at 21000000? Express the answer as in the last example. 18. What is the amount of the President's salary per day?-per hour?-per minute? 19. The average pulsation of the heart is in childhood about 105 beats in a minute, in youth 90 beats, in middle

Page  49 ~ 8.] FRACTIONS AND COMIPOUND NUMBERS. 49 age 75, and in old age 60. Estimating the duration of childhood at 14 years, youth at 11 years, middle age at 25 years, and old age at 25 years,a how many times would the heart beat in each period? Iow many times in the whole life? 20. TABLE OF DISCOVERIES AND IMPROVEMENTS.b B. C. Astronomical observations first made in Babylon. 2234 Lyre invented......1004 Sculpture... 1900 Agriculture by Triptolemus. 1600 Chariots of War....... 1500 Alphabetical letters introduced into Europe... 1500 The first ship seen in Greece, arrived at Rhodes from Egypt 1485 Iron discovered in Greece, by the burning of Mount Ida. 1406 Seaman's Compass invented in China.. 1120 Gold and Silver money first coined by Phidon, king of Argos 894 Parchment invented by Attalus, king of Pergamus.. 887 Weights and measures instituted.... 869 First astronomical observation of an eclipse.. 721 Ionic order used in building...... 650 Maps and globes invented by Anaximander 600 Sun-dials invented..... 558 Signs of the Zodiac, invented by Anaximander. 547 Corintlian order of architecture... 540 First public library established at Athens. 526 Silk brought from Persia to Greece... 325 The art of painting brought from Etruria to Rome, by Quintus Pictor..... 291 Solar quadrants introduced. 290 Mirrors in silver invented by Praxiteles. 288 Silver money first coined at Rome. 269 Water-clocks...... 245 Hour-glass invented in Alexandria. 240 Burning mirrors invented by Archimedes.. 212 First fabricating of glass.... 200 a Encyclopedia Americana, Rees' Encyclopmedia. b IMemoria Technica, Rees' Encyclopaedia, Beckmann, Scientifio American. 4

Page  50 50 FUNDAMENTAL RULES. [ART. III. B. C. Brass invented * 146 Paper invented in China.. 105 Rhetoric first taught at Rome..87 Blister-plasters invented.... 60 Julian year regulated by Caesar... 45 Apple trees brought from Syria and Africa into Italy 9 A. C. Vulgate edition of the Bible discovered. 218 Porcelain invented in China..... 274 Water-mills invented by Belisarius.. 555 Sugar first mentioned, by Paul Eginetta, a physician. 625 Quills used for writing. 636 Stone buildings first erected in England, by Bennet, a monk 670 The system of couriers, or posts, invented by Charlemagne 808 Figures used by the Arabs, borrowed from the Indians 813 Lanterns invented by king Alfred..890 High towers first erected on churches.. 1000 Musical notes invented by Guido, of Arezzo. 1021 Heraldry originated..... 1100 Distillation first practised.. 1150 Glass windows first used in England. -. 1180 Chimneys built in England..... 1236 Leaden pipes for conveying water, invented. 1252 Magic lanterns invented by Roger Bacon. 1290 Tallow candles first used... 1290 Fulminating powder invented by Roger Bacon. 1290 Spectacles invented by Spina...... 1299 Wind-mills invented...... 1299 Alum discovered in Syria.... 1300 Paper made of linen..... 1302 Coals first used in England......1307 Saw-mills at Augsburg. 1322 Woollen cloths first made in England.. 1331 Gold first coined in England. 1344 Painting in oil colors....... 1410 Muskets used in England. 1421 Pumps invented...... 1425 Printing invented.... 1440 Glass first made in England..... 1457 Wood-cuts invented..... 1460 Almanacs first published in Buda.. 1460

Page  51 ~ 8.] FRACTIONS AND COMPOUND NUMBERS. 51 Printing introduced into England by Caxton. 1470 Watches invented at Nuremberg 1477 Stages and post horses established. 1483 Tobacco discovered in St. Domingo o 1496 Shillings first coined in England..1505 Stops in literature introduced 1520 Spinning-wheel invented at Brunswick. 1530 Variation of the Compass first observed 1540 Pins first used in England...... 1543 Needles first made in England by an East Indian. 1545 Sextant invented by Tycho Brahe.. 1550 Lace knit in Germany..1. 1561 Coaches first used in England.. 1580 Telescopes invented by Jansen. 1590 Stocking weaving invented... 1590 Decimal arithmetic invented at Bruges. 1602 Microscopes used at Naples. 1618 Circulation of the blood discovered by Harvey 1619 Coins made with dies in England. 1620 Thermometers invented by Drehel.. 1620 Barometer invented by Torricelli, an Italian. 1626 Newspapers first published. 1630 Regular posts established in London. 1635 Coffee brought to England.. 1641 Steam engines invented by the Marquis of Worcester 1649 Pendulums for clocks invented..1649 Air-pumps invented,. 1650 Air-guns invented by Guter... 1656 Spring pocket watches invented by Dr. Hook. 1658 Engines invented to extinguish fires.. 1663 Bayonets invented at:Bayonne. 1670 Micrometer invented..... e 1677 Telegraphs invented...... 1687 New style adopted in England.... 1752 Spinning frame by Arkwright.. 1761 Cotton first planted in the United States. 1769 Steam engine improved by Watts.. 1769 Georgium Sidus discovered by Herschel.. 1781 Power looms invented by Cartwright.. 1783 Steam cotton-mills first erected.. 1783 Steam grist-mills first erected.... 1785 Stereotype printing invented by Mr. Ged, Scotland. e 1785

Page  52 52 FUNDAMENTAL RULES. [ART. IIT. Cotton first spun in America. 1787 Mesmerism, or animal magnetism, discovered by Mesmer. 1788 Sunday schools established in Yorkshire, by Rotert IRaikes 1789 Galvanism, 1767,-its extraordinary effects on animals discovered by Mrs. Galvani.... 1789 Steam woollen factory first erected, at Leeds. 1792 Flax spun by steam...... 1793 Vaccination introduced by Jenner.. 1798 Ceres. discovered by Piazzi...1801 Pallas discovered by Olbers. 1801 Life-boats invented...... 1802 Juno discovered by Harding. 1804 Vesta discovered by Olbers......1807 Steam first used to propel boats by Fulton, in America. 1807 Engraving on steel first invented by Perkins, an American 1818 Gas first used for lighting streets in the U. S., at Baltimore 1821 Egyptian hieroglyphics first decyphered by Champollion 1822 Macacldamizing streets commenced in London by McAdam 1824 First locomotive at Liverpool... 829 Electro-magnetic Telegraph invented by Morse, America. 1832 Daguerreotype impressions first taken by Daguerre in France 1839 The existence of the planet Neptune predicted by Adams and Leverrier..1846 Magnetic pendulum used for measuring longitude.. 1848 21-30. How many years elapsed between the invention of the lyre and the invention of musical notes? The earliest known astronomical observations, and the invention of the signs of the Zodiac? The first eclipse on record, and the invention of the telescope? The use of the mariners' compass in China, and the introduction of solar quadrants? The arrival of the first ship in Greece, and the building of the first steamboat? The establishment of the first public library at Athens, and the invention of printing? The in. vention of the hour-glass, and the invention of watches? The use of paper in China, and the manufacture of paper out of linen? The first manufacture of glass, and the invention of spectacles? The discovery of the Vulgate edition of the Bible, and the establishment of Sunday schools? 31-40. Allow 365{- days for each year, and reduce to weeks the number of years that elapsed between the manu

Page  53 ~ 8.] FRACTIONS AND COMPOUND NUMBERS. 53 facture of porcelain in China, and the manufacture of glass in England? The invention of pendulum clocks, and the use of the magnetic pendulum for measuring longitude? The introduction of figures by the Arabs, and the invention of decimal arithmetic? The invention of the steam engine, and the introduction of locomotives? The invention of wind-mills, and the application of steam to grist-mills? The first astronomical observations known to have been made at Babylon, and the prediction of the planet Neptune? The invention of the sun-dial, and the invention of watches? The use of the compass in China, and the discovery of the variation of the needle? The introduction of the art of painting into Rome, and the invention of the daguerreotype? The regulation of the Julian year, and the introduction of the new style into England? 41-43. Give the answer to each of the remaining ques. tions on this page, in years, months, and days,-in years and fractions of a year,-and in years and decimals of a year. What length of time elapsed between the birth of John Calvin, July 10, 1509, and the birth of Oliver Crom. well, April 25, 1599? 44-52. What length of time elapsed between the birth of Fenelon, August 6, 1651, and the birth of William Penn, October 14, 1644? Between the birth of Galileo Galilei, February 19, 1564, and the birth of Sir William Herschel, November 15, 1738? Between the birth of Sir Francis Bacon, January 22, 1561, and the birth of Benja> mrin Franklin, January 17, 1706? 53-67. What length of time elapsed between the birth of Tycho Brahe, December 19, 1546, and the birth of Sir Isaac Newton, December 25, 1642? The birth of George Washington, February 22, 1732, and Patrick Henry, May 29, 1736? The birth of William Shakspeare, April 23, 1564, and John MIilton, December 9, 1608? The birth of John Adams, October 19, 1735, and Thomas Jefferson,

Page  54 54 FUNDAMENTAL RULES. [ART. III. April 2, 1743? The birth of Roger Sherman, April 19, 1721, and Benjamin West, October 10, 1738?a 68: How many days have elapsed since the birth of Sir Humphrey Davy, December 17, 1779? 69. The equatorial diameter of the earth is 7970 miles, and the circumference is 34- times the diameter. If a man 6 feet high were to travel round the earth, how many yards farther would his head travel than his feet?b 70.. A pound Avoirdupois = 14oz. 1ldwt. 16gr. Troy. What part of a Troy ounce is an ounce Avoirdupois?.lns. i 7 5 71-75. Find the difference of latitude and longitudec between Boston and Philadelphia. New York and St. Louis. Charleston and New Orleans. Cape Horn and the Cape of Good Hope. Paris and St. Petersburg. 76. Assuming 1 ton as the average amount of carbonic acid produced by 6100 persons in 24 hours, and estimating the total population of the globe at 759999000, how many tons would be produced daily by human respiration? Ans. 124590 tons.'77 The average length of the tropical year is 365d. 5h. 48m. 49 - sec.d How many days in 4 centuries, and what fraction of another day? 78. A balance made by Ramsden, for the Royal Society, is capable of weighing 10 pounds avoirdupois, and turns with.01 of a grain.e What part of the weight is required to turn -the scale? a Encyelapiedia Americana, Rees' Encyclopadia, Belknap, Biog. American, Allen. b Keith. c See Table, Q 9. d Somerville. The tropical or civil year is the time that elapses between'two consecutive returns of the sun to the same equinox or solstice. e tUre'6s Dictionary.

Page  55 ~ 9.] LATITUDES AND LONGITUDES. 55 9. TABLE OF LATITUDES AND LONGITUDES, AND DISTANCES FROM WASHINGTON, OF THE PRINCIPAL CITIES, OBSERVATORIES, NAVAL. STATIONS, ETC. FROM THE UNITED STATES AND AMERICAN ALMANACS. Longitude'Dlst, Place. State. Object, from Latitude. from Greenwich. Wash 0 I/1 0 II Mites Aberdeen........ Scotland..... Obs. W. 2 5 42 57 8 57.8 Abo............... F inland...... Obs. E. 22 17 12 60 26 56.8 Albany...........N. Y.......Capitol.... W. 73 42 49 42 39 3 370 Alexandria.......C.............. 77 6 45 38 49 7 -Altona,............ Denmark..... Obs....... E. 9 56 39 53 32 45.3 Amherst..........Mass.........College Ch... W. 72 31 28 42 22 15.6 383 Annapolis.......... Md..........................6 33 38 58 35 40 Antnapolis....., Md..76 33 38 58 35 40 Apalachicola Bay.. Fa..........Light.. 85 5 15 29 37 25 883 Armagh....... Ireland... Obs. " 6 38 53 54 21 12.7 Auburn........... N. Y.............. 76 28 42 55 333 Augusta...........Ga.............81 54 33 28 575 Augusta.......Me...........State House. cc 69 50 44 18 43 595 Baker's Island.....Mass.........Light........" 70 47 37 42 3211 452 Baltimore.....d....Md...........St. Mary's C. " 76 37 39 17 55 40 Bangor............ Ie...........State House.., 68 47 44 47 50 663 Barnstable......... Mass......... New C. H_ ic 70 18 34 41 42 6 466 Batavia........... N. Y........... 7 13 42 59 374 Beaufort.-....... S. C.........Arsenal...... 80 41 23 32 25 57 635 Bedford........ England...Obs........., 8 0 52 8 27.6 Berlin.......... Prussia......Obs. Old..... E. 13 23 53 52 31 13.5 Blackheath.... England.....Obs......... 41 5128 2 Bologna...........Italy.................. 1120.5344295 Bolognla.Italy,....| 11 20 53 44 29 54 Bonn.............Germany.... Obs.......... 7 6 45 50 44 9.1 Boston........... Mass......-. State House.. W. 71 4 20 142 21 23 440 Bremnen........... Germany.....Obs.......... E. 8 48 59 [53 4 36 Breslau............ Germany.....Obs......... " 17 2 30 51 6 30 Bridgeport......... Conn.........Baptist Ch.... ~V. 73 11 46 41 10 30 284 Bristol.............R. I... Episcopal Ch. 4, 71 17 19 41 40 3 409 Brooklyn...........N. Y.. Navy Yard.. 74 0 3 40 42 0 226 Brunswick........ Me...........College...... 6955 4353 570 Brussels.......... Belgium....Obs.......... E. 4 22 8 50 51 10.8 Buda............. Hungary. Ob s......... 19 3 11 47 29 12.2 Buffalo.......... N. Y,,,,.................. Vr. 78 55 42 53 381 Burlington......... N.J... Obs....... " 74 52 6 40 4 52 156 Burlington........ Vt................. 73 10 44 27 513 Bushy Heath.......England.....Obs..........,s 20 14 51 37 44.3 Calais..............M...................... 67 12 15 45 11 18 786 Cambridge.........Mass...Obs. Old....' 71 7 21 42 22 15 437 Cambridge.........England.....Obs. E. 5 51 52 12 51.8 Camden...........S. CV. 80 33 34 17 473 Canandaigua......N. Y......,............ 44 77 1.7 42 54 341 Cape Ann.......... Mass......... N. Light..... 4 70 34 44 42 38 18 470 Cape Elizabeth..... Me..... Light...... " 70 11 6 43 33 6 Cape Cod...........Ae..........Light........ " " 70 4 9 42 2 22 507 Cape Horn.......S. America............. 4 67 16 8 55 58 40 s. Cape of Good Hope.Africa.......Obs.......... E. 18 28 45 33 56 3 s. Castine............Me......... Fort......... W. 68 59 33 44 22 45 671 Charleston.........S. C..........St. M. Ch.... "; 79 57 27 32 46 33 540 Charlestown........ Mass......... Navy Yard..'~ 71 3 33 42 22 441 Charlottesville...Va......... Univ. Obs.. 78 31 30 38 2 3 121 Chicago...........Ill................ 87 3030 42 0 717 Christiania.........Sweden..... Obs.. E. 10 44 57 59 54 42 4 Cincinnati...W...Obs......... 84 24 39 5 54 492 Columbia......... S.C.....................; 81 7 3357 506j Columbus.......... Ohio........ " 83 3 39 57 393 Concord........... N. H..S......tale House.. " 71 29 43 12 29 481 IConstantinople..... Turkey..... A S. St:Sophia E. 28 59 /41 016

Page  56 56 FUNDAM.ENTAL RULES. [ART. IIIr. Longitude Dist1 riace. State. Object. from Latitude. from Greenwich. Wash j o, i o;; Miles Copenhagen.....Denmark..... Obs E 12 34 57 55 40 53 Coteau des Prairies. Iowa........Red Quarry.. W. 94 19 15 44 0 52 Coteau du Missouri. Iowa...................... 15 15 38 3728 Cracow...........Poland......Obs.......... 19 57 47 50 350 Dayton............ Ohio.............................11.... 44 461 Dayton~~~~~~~~~X.OhttW84 11] 39 44 /461 Dedham............Mass.........1st Cong. Ch. 7110 59 421457 422 Detroit..............Mich.........................c 82 58 42 24 524 Detroit. Mich.~~~~|~82 58] 42`24 I524 Dorchester.........Mass........ Ohs....... 71 4 30 42 19 10 438 Dorpat.............Russia....... Obs.. ]E. 26 43 45 58 22 47.1 Dover.............. N 1-1...................W. 70 5.. 43 13 490 Dover~~~~~~~~~' ~Iv.Ni. W70 54 43 13 t490 Dover. Del.753 91 2 Dover............. Del........... ~... 1" 75 30 { 39 10 120 Dover - - - -... - -Ohio....................... 29 40 30 52 372 Dover. Ohio.~~~~~/~81 29[ 40 39 52 [37'2 Dublin.............Ireland......Obs.6 29 30 53 23 1.3 Durham.............Scotland.....Obs......... DErham. scotland. Ohs... 1 34 30 54 46 14.9 Easton.t.M.......... Court I-IHodse.[~, 76 8 38 46 10 80 Eastport...........Me......................... 9 6 56 44 54 769 Edenton.........3. N. C.................. it 7640 36 5 274 Edinburgi.........Scotland Obs.... 3 10 54 7 23.2............. 55 57 23.2/ Exeter...N......N H... 70 55 42 58 474 Florence........... Italy.......................E. 11 15 54 43 46 40.8 Florence. Italy. ~~~~~E. 11 13 54143 46 49.8/ Frankfort........Ky....y.................... 40 38 14 542 Frankfurt. Ky.W~~~~~]~. 8- 410 138 14 t542 Fredericksburg.... Va..... 77 38 38 34 56 Frederickton.......N. B....................... 66 45 46 3 Frederick..........Md........................." 7718 39 24 43 Galveston..........Texas..................... 494 47 15 129 15 Germantown.....Pa.......... Obs......... 7510 9 40 240 144 Geneva.............Switzerland.. Ohs..........IE 6 9 22 46 11 59.4 Georgetown........D. C.. Obs..... 77 5 5 38 5 2 Georgetown........S........................ 79 17 133 21 48, Gloucester.........rMass.........Univ. Church]. 70 40 19 42 36 44 462 Gotha..............Germany..,..................5 5. Gotha.L Germany............... E. 10 44 6 50 56 5.2 Gottingen..........Hanover.....O s..........O 9 56 38 51 31 47.9 Greenfield.:...Mass.........2d tong. Ch.. IW.72 36 32 42 35 1.6 396 Greenwich...England..Obs...0 51 28 39 Hagerstown M........Mid................. 77 35 39 37 68 Halifax..N. S................. 63 36 40 144 39 20 936 IHallowell..........0Me........ ~ 69 52 30 144 17 593 Hamburg...........Germany.....Ohs..........IE. 9 58 32 53 33 5 IIarrisbtrg.W. Pa..............V. 76 50 40 1.6 110 Hartford.......Corn.......... State House..c 72 40 45 41 43 59 335 Havana............. W'AT.. I.. Moro Castle.. c 82 22 21 23 9 26 Haverford.......... Pa.........School75 18 36 40 1 12 130 HIelsingfors.........Finnland... Obs.......... E 24 57 53 60 9 42.3 Holmes's Hole......Mass.. Wn.....Vi dmill.... 70 36 38 41 27 15 457 Hudson............N. Y...................| 73 46 42 14 3 Hudson.............Ohio.........O s. 812454 41 14 42.6 335 Huntsville........... Ala........................1c86 57 34 33 708 Indianapolis........d............ 86 5 39 55 571 Ipswich.... iass.........Eastern Li.lt 70 46 17 42 41 8 462 Jackson......M...... Miss................... 0 8 32 23 010 Jefferson.............. 8 3836 936 Kasan..............Russia.....Ohs.......IE. 49 6 38 55 47 30 Kensington....... England Obs..........W 11 43 51 30 12.7 Z) ~~~~~~~~11 43151 30 1'2.7 Key West.......... Light......../ht 81 48 30 24 32 32 Kingston..........U. C................... 76 40 44 8 456 Knoxville...... Tenn.............. 83 54 35 59 498 Konigsberg.........East Prussia.Obs E 2.......E 0 30 8 }54 42 50.4 Kremsmunster....;. Germany.....Ohs........ c 14 8 9 48 3 24 Lancaster........Pa.......' WV. 76 20 33 40 2 36 111 Lexington..........Ky........................../ 84 18 138 6 522 Leyden............Holland...... s.........E. 4 38 53 52 9 28.2 Little Rock.........Ark......................W. 92 12 314 40 1065!ockport...........N.Y..............:........ 78,16 443 11 41)3 L.~~~~~~~~~~~~~~~~~~~~~~~~~~,4)

Page  57 ~ 9.] LATITUDES AND LONGITUDES. 57 Longitude Dist. Plaee. State. Object. from n Latitude. iom LGreenwicta. Washo F M~~~~~~~~~~~~~~~~~~~~~~3ites Louisville...........Ky.......................... 1 31i _1e Louisville...........Ky.. W.85 30 38 3 590 London....,England S.t..St. Pauls.. 5 45 5 30 49 Lowell............Mass........ St. Ann's Ch. 4 71 18 57 142 38 48 444 Lyncubutg Va. c 79 22 37 36 198 Lynn iss........ Church. 70 57 25 42 27 51 441 Machias................................. 67 22 44 3 Madas d............. aa nd Obs. E. 80 15 57 13 4 9.2 Man-iesta...t...... Getrmany..... Obs- cc8 27 51 49 29 13.7 Marblehead.........s 1ass t....... Light l. 70 50 39 42 30 14 458 Marseilles 1tnsO.......... France 01.ba.. 5 22 15 43 17 49 yMatinacus Rock...i elut.... L)l....... 68 57 43 4 42 Mexico.............Mexico...................... 192545 Mrlddletown..... Con V......... AV. Ueuv...' i6d7 39 41 33 8 432 M1illedgeville.....2. Ga.......5.. 83 20 33 4 30 618 Milaa............. Italy........ Obs........... 11 45 28 0.7 MtNobite -..... Ala.,,,.,,,.. IP]V. 88 11 30 40 1.013 noble Point.....................ilt....... 88 0 36 30 1.3 38 Modena........Obs. E. 10 55 48 44 38 52.8 Monlegan Jblaud e.........................V. 69 18 5 i43 45 52 MR~~ononclv ~~.. nManss......... Pt. Lilght.... 70 0 61 4 33 3" 500 Monstpeliera.......Vt. 72 36 44 17 516 M[ontlreal............L. C 49 7...... t.3 35 45 31 601 Mutinich..... v. ria. b3ne 3.63......8. 4 E. 11 36 38 41 845 Nantucket.......... Mass......... South TowerN IV. 70 12 41 16 56 490 Naples............. Italy.........Obs.......... 4 5 5 40 5 4.6 Nashville........Te.Tnn.........Univ Obs.... 86 49 3 36 9 33 684 Naotcliez... M.C...... I VItSstle 44 91 21 42 31 33 48 1110 NTewark............N. J...................4.... ~ 34 10 40 45 215 New Bedford..05.....5t...4.... 70 55 49 41 38 7 434 Newbern. N........ N C.... 77 5 35 20 348 NeMiburig 4................ 74 1 41 31 286. Nerwbuy r yp, ort. -Mauss....... Liadit 4.. ( 70 49 30 42 48 23 478 Newcastle..........Del......... 32 340 1.05 New Haven........Conn.......... Obs. 72 57 30 41 18i 30 300 Now London.......Conn.........,2 9 1 22 353 New Orleansa..... La.....,..N. E. Light.. c4 80 1 50 29 8 32 1172 Newiport........... R. I... Court House ii 71 21 14 41 28 20 408 Newport........ o....M.. a......91 715 38 3358 New Yorki..........N. Y.....C.... ity Hall..74 1 6 4042 35 225 Nicolae f...........Russia...... Obs.......... 31. 58 47 46 5 20.6 Norfolki.........Va. Fai.. - X arm. Bank.c IWV. 6 18 47 36 50 50 230 Northanpton.. -Mass......... Ist Cone. Cli. i4 2 38 21 42 19 8 3s0 Norwich............ Conan......b...........1172 7 41, 33 357 Ornsukirk..........England Obs.......... To e2 541 53 34 18 Oxford............. Enland...... Ohs......... 1 1523 51 45 40 Padua............... Italy........ Obs..... 1 52 18 45 24 2.5 Palermo............Italy.........Obs. Old 9 13 21 24 38 6 44 Paramatta....N. S. Wales. Obs..... 151 1 35 33 48 50 s. Paris.................France. Obs........ 20 22 48 50 13 Pensacola.......................Navy ard... 15 21 30 20 30 105 PeNersburg.........Va......................... 77 20 37 13 54 1.40 Pliladelphia........ Pa.. _State House. 75 9 45 30 56 58 138 Pittsburgyp.........7. P....... 7958 40.26.15 22 PittesfielM.. ass..-. 1st Cong. Cl. 11 73 15 36 42 26 55 380 Plattsbt e..........N........................ 7326 4442 539 Plymouth.....Ms....Court.House. 4 70 40 28 41 57 28 447 Portland..................... Tn House.. 70 20 30 43 326 545 Portsmouth.......N H......U...Unit. Church.. 70 45 50 43 4 35 493 Poughreepsie...N. Y.........R. 73 55.41 41. 301 Prague............. Boemia.....Obs..........E. 1.4 25 29 50 5 1.5 Princeton..........N......... Nassau Hall.. I. 74 30 33 40 20 41 177 Provience........... Rai.....Univ..all.... 7124 48 41 49 32 400 Quebec.L... C.....Citadel....... 7.7b1 16 46 49 12 781

Page  58 58 FUNDAMVENTAL RULES. [ART. Irl. Longitude'Dist, Place. State. Object. from Latitude. fromn Greenwich. Wash Q / Mites Raleigh........... N. C... 78 48 3547 288 Regent's Park.... London......Obs.... 9 17 51 31 30 Richmond..........Va......... Capitol....... 77 27 28 37 32 17 117 Rochester.......N. Y...Roch. House., 7751 43 817 369 Rome...............Italy.........Obs.........IE. 12 2841 41 5354 Roscoe.............Ohio.................... W 8152 4016 40 338 Sackett's HarbourN Y...N. 75 57 43 55 415 Saco........-Me............. 7026 43 31 528 St. Augustine........Fa. 8135 29 48 30 821 St. Croix........... I......... Obs. 64 41 15 1744 32 St. Fernando.... Spain........Obs. it...6 12 17 36 27 43 St. Helena........Africa.....Obs.... 5 42 30 15 55 26 s. St. Joseph's Bay....Fa..........Light........ 85 23 15 29 52 808 St. Louis........... Mo. 90 Cathedral. 90 15 10 38 37 28 8541 St. Mark's...........CFater........ l 84 11 0. 30.2 62 St. Petersurg....Russia... Pulkova Obs. -.,30 19 46 59 46 18.7 Salem...............Mass..E. I. M. Hall W. 7053 57 4231 19 448 Sandwich........ Mass........ st Cong. Ch. ] 70 30 13 41 45 31 456 Savannah...Ga........ Exchaiae.... ] 81 7 9 32 456 662 Schenectady........ NY.73 55 42 48 391 Slough........ N. England.... Obs.......... 36 0 513020 South Bend........L. Michian..... 87 1.9 34137 6 Southwick.....-. nan...........Mss.. 724857 42 0 47 35o Speyer.e..r....... M ichGermany.. Obs......... 8 26 38 49 18 55.2 Springfield..... -..Mass.........Court House. W. 72 3547 42 6 1 363 Springfield.........11......................... 8933 12 3948 780 Stockholm..........Sweden......Obs....... E. 18 3 30 59 20 31 Stratford............Conn............ W..73 8 45 4111 7 287 Tallahassee F..................... 84 36 30 28 896 Tampa Bay.........Fa...........Egmont Key. I" 82 45 15 27 36 4 Ti Tanka Tam. Lake Iowa...................... 3 20 54 44 16 41 Taunton.............Mass....71 6. 41 54 8 420 Toronto........U. C............. 79 20 43 33 500 Tortugas..................... Light....... 82 52 22 21 37 20 Treaton............ N. J.............. 4 39 4014 166 Troy............... N.7340 42 44 376 Turin.............Italy................... E. 7 42 6 45 4 6 Turtle Island.......Lake Erie.................W.83 23 35 4145 9 Tuscaloosa.........Ala..... O.. s. 87 42 33 12 858 University of Va.....a.V....... 78 31 29 38 2 3 124 Upsala.........;... Sweden......Obs..........E. 17 38 42 59 51 50 Utica..............N. Y.... Dutch Church/W. 75 13 43 649 383 Utrecht...........Holland. Obs.. 5 7 23 52 5 11 Vandalia. Ill................ W..89 2 38.50 781 Vera Cruz.......... Mexico..................... 96 830 191152 Vevay.............I84 59 38 46 544 Vienna......Austria......Obs. E. 16 22 59 48 12 35.5 Vincennes,............I 87 25 3843 688 WASHINGTON......D. C........Capitol. 77 1 30 38 53 23 0 Washington..... Miss......91 20 31 36 1104 I'Warsaw............Poland......Obs..........E. 205823 5213 1 Weasel Mountain....N. J.........C. S. St......V. 74 1215 405235 West Hills....N..Y.C.S.St. 732517 404849 260 West Quoddy Head Me........... Light.....16657 19 44.49 4 Wheeling..a......................... 80 42 40 7 264 NTilli:tmstown.....ass..C Church i... t 73 13 20 42 42 51 396 Wilmington........Del......................'~ 7528 39 41 110 Wilminsgton........ N C....................... 78 10 34 11 405 Wilna...........Poland.......Obs..........E. 25 17 59 54 41 0 Worcester..... Mass.. Antiq. Hall.. W. 71 48 10 42 16 13 398 York...............7040 4310 502 Yoric...,.,,Pa..................76 40 39 58 90 Y'orktown.........Va....................76 34 37 13

Page  59 ~ 10.] MISCELLANEOUS EXAMPLES. 59 1. MIISCELLANEOUS EXAMPLES IN INTEGERS, DECIMALSn FRACTIONS) AND COMPOUND NUMBERS. 1-4. Fifteen degrees of difference in longitude are equivalent to an hour's difference of time. What difference of longitude would make a difference of four minutes in the time? A difference of one minute? Of four seconds? Of one second? 5-6. Give a rule in your own words for determining the difference of time when the difference of longitude is known. For the difference of longitude when the difference of time is known. 7-10. What is the difference of longitude between two places, if the difference of time is 2h. 15m. 27sec.?. If the difference is 4h. Om. 16sec.? 39m. 45sec.? 101h. 48m. 49sec.? 11-15. What is the difference of time between Boston and Philadelphia? New York and St. Louis? London and St. Petersburg? Paris and New Orleans? Washington and Stockholm? 16-20. When it is noon at Cincinnati, what time is it at Charleston? At Little Rock? At Florence? At Jefferson? At Raleigh? 21-25. When it is lh. 20m. 25sec. P. M. at Providence, what time is it'at Paramatta? At Vera Cruz? At Albany? At Harrisburg? At Hartford?a 26. Estimating the pound sterling at $4.84, in how many years would the agricultural products of the United States pay the British National Debt, the annual amount of the former being about $252,240,779,b and the latter amounting in 1847 to ~764,608,284?c a The table in ~ 9, furnishes materials for an indefinite number of questions, similar to these. b Patent Office Report, 1847. e Annual Register.

Page  60 60 FUNDAIENTAL RULES. [ART. II1. 27. French Invoice:Lyon, 9 Avril, 1848. Messrs. Merrit & Lamb Doivt. a Perigord & Lubin. No. 40, Une caisse par roulage et paquebot Bavaria P L du 24 Avril. 568a Cravates longues 6-1 42 " 290 50 569 " 4 ecossais 118 36 " 570 "4 e 20 42 " 571 "c 2 0 " " 572 6s 574 44 4 "8 573 " e 19 5 54 576 4 5 48 v 577 " 2-" 36 6 5788 e l 30 4530 Emballage 45 Comm. 301o 137 25 Comm ~F. 4712 25 28. How many centuries in a trillion seconds, allowing 95 leap years in each 4 centuries, and 24 leap years in each of the remaining centuries, and how many years, days, hours, minutes, and seconds of another century? 29. If a cannon-ball could be fired from the earth to the sun, and move uniformly, in a straight line, at the rate of 1600 feet per second, in what time would it reach the sun, the mean distance being estimated at 95000000 miles? a No. of design.

Page  61 ~ 10.] MISCELLANEOUS EXAMPLES. 61 30. Invoice of Goods shipped by M. Naylor & Sons, of London, in the lVestmninster, Warren, Master, for Boston; by order and for account and risk of Messrs. W. Appleton & Co. 86 Two Chests Asafcetida, s. d.le. s. d. ~. s. d. _ Gr. 5 0 7 tr. 2 24 87 " 5 0 24 " 2 24 Gr. 10 1 3 tr.1 1 20 1 1 22 dft. 2 8 3 9 50 Disct. 2~ per cent. 11 Cording, 4 88 Two boxes Kino, - Gr. 2 21 tr. 21 89 " 2 25 " 23 1 1 18tr. 44. 1 18 dft. 2 10 0 45 Disct. 2~ per cent. 1 2 Cording, 2 One Cask Litharge, M. N. 90 Gr. 618 Tr. 1 1 6 0 7 22 6 Cording, 1 6 Charges, Customs entries, 6 Cartage, wharfage, and shipping, 15 B. of Lading 2s. 6d., Postages 2s. 4 6 32 3 9 Brokerage and corn., 3 per cent. 19 4 Insurance ~40. 35 14 Stamp 2s. Comm. Is. 3 17 E. E. London, 27th Jan., 1850. M. NAYLOR & SONS.

Page  62 62 FUNDAMENTAL RULES. [ART, IIi. 31. TIME-BOOK. JUNE, 1849. NAMES. 125 26 27 28 29 30 Total. per Amount. EMARKS. George Brooks, 1 1 1 1 1 1 6 10.50 Ex. workm'n. Frank. Jones, a 1 1 1 4 - 6.00 Rather slow. John Smith, 1 1 1 1 1 1 5 I 5.00 Intemperate. Wm. Brown, 1 1 a 1 1 a 4 9.00 Good hand. Thos. Martin, Il1 1 1 1 1 5 4.00 Careless. Find the amount of each man's wages, and the total amount of wages for the week. 32. If 150 leaves of paper make a pile an inch high, and each leaf is twice the thickness of a hair, what would be the extent of a quadrillion hair-breadths? 33. In what length of time would light, which moves at the rate of 192000 miles per second, reach us from a star that is one quintillion miles distant? 34-41. Express 1853 by a scale of notation that shall have 9 for its base, instead of 10.a Express the same number by a scale of 7; of 5; of 3; of 2; of 8; of 6; of 4. 42-44. In the first three weeks of June, 1849, the number of gallons of water consumed in the city of Philadelphia was as follows: June 1st. 6439650; 2d. 6163665; 3d. 3955785; 4th. 6041005; 5th. 6102335; 6th. 5887015; 7th. 4569085; 8th. 5887680; 9th. 5489035; 10th. a If 9 had been adopted as the base of our system, 9 units of any order would have made one of the next higher order. Therefore if we wish to reduce 14646 to the scale of 9, we divide by 9 9)14646 and find that there are 1627 units of the 2d order, and 3 - of the 1st. The 1627 units of the 2d order are equiva- 9)1627-3 lent to 180 units of the 3d order, and 7 of the 2d. The 9)180-7 180 units of the 3d order make 20 units of the 4th order and 0 of the 3d. The.20 units of the 4th order make 2 units of the 5th order and 2 of the 4th. The whole 9)2-2 number would therefore be expressed thus: 22073.

Page  63 ~ 10.] MISCELLANEOUS EXAMPLES. 63 3557140; 11th. 4753075; 12th. 4395096; 13th. 5703690; 14th, 5550365; 15th. 6346655; 16th. 6623640; 17th. 5550365; 18th. 6224995; 19th. 5918345; 20th. 6565310; 21st. 7239945. What was the total consumption of the three weeks? The average daily consumption? If the whole quantity were put into 100-gallon casks, and the casks were arranged side by side, how far would they extend, each cask occupying a space of 3 feet? 45-50. Determine from the following table, the area of the United States and Territories, the amount of exports and imports, in the years 1840 and 1847, and the number of inhabitants to a square mile in each state and territory, by the census of 1840. SNo. Square Exports Exports Iports Imports STATES. Miles. 1840. 1847. 1840. 1847. Maine a35,000 b$1,018,269 b$1,634,203 b$628,762 b$574,056 New Hampshire 8,030 20,979 1,690 114)647 16,935 Vermont 8)000 305,150 514 298 404)617 239,641 Massachusetts 7,250 10,186,261 11,248,462 16,513,858 34,477,008 Rhode Island 1,200 206,989 192,369 274,534 305 489 Connecticut 4 750 518)210 599)192 277,072 275,823 New York 46,000 34,264,080 49,844,308 60,440,750 84,167,352 New Jersey 6,851 16,076 19,128 190,209 4)837 Pennsylvania 47,000 6,820,145 8,544,391 8,464,882 9,587,516 Delaware 2)120 37,001 235,459 802 12,722 Maryland 11,000 5,768 768 9,762,244 4,910,746 4,432,314 Dist.of Columbia 50 753,923 124 269 119,852 25,049 Virginia 61,352 4,778,220 5,658,374 545685 386 127 North Carolina 45,500 387,484 284,919 252,532 142,384 South Carolina 28,000 10,036,769 10,431,517 2,058,870 1,580,658 Georgia 58,000 6,862,959 5 712,149 491,428 207,180 Alabama 50 722 12,854,694 9,054,580 574,651 390)161 Mississippi 47,147.................... 336 Louisiana 46,431 34,236,936 42,051,633 10,673,190 9,222,969 Florida 59,268 1,858,850 1,810,538 190,728 143,298 Texas 325,520............................... 29,826 Kentucky 37,680................ 2241 26,956 Tennessee 44,000................ 28,938 1,256 Ohio 39,964 991,954 778,944 4,915 90,681 Indiana 33,809......................... Illinois 55 405....... 52,100......... 266 Michigan 56,243 162,229 93,795 138,610 37,603 Iowa 50,914........................... WvVisconsin 53,924............................................ Missouri 67%380........... 10,600 167,195 Arkansas 52 198........................................... MIinesota Terr. 166,000............................... Itissouri " 579,000.......................................... Oregon " 311,463................................... Indian " 248,851................................ New Mexico 77,387......................................... California 448)691............................ a U. S. Land Office Documents, 1849; see also, 12th Ann. Rep. Mass. Board of Educ., pp. 34, 35. b Am. Almanac.

Page  64 64 FUNDAMENTAL RULES. [ART. mIl. 51. Water is composed of two volumes of hydrogen and one volume of oxygen, a volume of oxygen being of the same weight as 16 volumes of hydrogen. Required the weight of each gas in a cubic foot of water, which weighs, at the temperature of 620 Fahrenheit, 62.5 lb. 52-55. Give abbreviated rules for multiplying, and for dividing" by 5; by 25; by 125; by 625. 1 1. TEST EXAMPLES. 1. From the sum of 287.53 + 195.7 + 6008 + 7.975 + 3092.06, subtract by a single operation the sum of 948 ~ 27.008 + 1090.3 + 4726.87 + 95.953. 2-10. Reduce to whole numbers -4-o09; 4.ss; 2s_ 1 1001. 100. 1001 9870. 9875 131043 11' 3U 7 Y T25. 1 25. 3613 11-15. Reduce 137 to 8ths; to 4ths; to liths; to 125ths; to 1016ths. 16-25. Reduce to simple fractions i of 7; 1 of 4 of 31; I of 3' of2 -2 i11 of 34 o f 90; -4; 16 3 2 - I Ii 1 61, Y9 -.' 9' 424' 17 of 1{- of 11 of 4 of 161. 85 a Since -2-.04 and 25=. S, we may obtain 21 of a number, (which is equivalent to dividing the number by 25,) by multiplying by.04, and we may obtain 25 times a number, by dividing by.04. By finding the decimal value of 1, &c., similar rules may be formed for each of the other cases. b The arithmetical complement of a number is the difference between the number and some power of ten. Such examples as the 1st in this section can be most readily solved by addition, taking the arithmetical complement mentally, of each of the 4692.8 numbers to be subtracted. This may be done by taking -271.04 each of the figures from 9, except the right hand figure, -2637. which should be taken from 10. After adding all the 1384.2 arithmetical complements, we must deduct 1 fiom the next place at the left. For example, if the sum of 4692.8 3168.96 - -271.04-2637+1384.2, were required, writing the numbers as in the margin, we commence by saying 4 from 10 leave 6; 2+9+8 =19; 1+4+3q-8+2-=18; 1+8+6+2+9 =26; 2+3+:3+7+6=21; 2+1+7-1+4 = 13; 1-1 =0.

Page  65 ~ 11.] TEST EXAMPLES. 65 26-30. Find the value of 15-+37+3 11275+~5+ 41-~;7ofof 3 -2 of1 a4 of 72; 9 f fI f8 5 X a of ll; 5 *9 7Y; 2 13 13 5. 31-35. Reduce to mixed numbers 87s 187; 330; 2002; 4,0 1 7 5 186 36-40. Reduce to a fractional form 491; 1082-;.0932; 2314f; 8.087. 41-45. Reduce to the lowest terms 144. 219. 264. 532. 492 46-55. Find the numerators and denominators tnat are indicated by a blank in the following fractions; ~=-11; 47-.3. 19-21 3-8 534; 2; f 3of -=3; 2': 6=56-60. Reduce to fractions or mixed numbers.18; 2.05;.82; 27.854;.0563. 61-65. Reduce to decimals II; 5; -4 9, *6 252 66-70. Reduce to the least common denominator 27 and 1 9; - 43and 4; - 1 and-; 9 -T3 a-nd -; 9-6 2 4 and 9. 71. A clerk, in balancing his books, found an error of $407.25. What was the probable cause of the error?72-75. Divide.0072 by 576000; 27.9 by.00124; 46500 by.0002976; 87.002 x 1.008 by 963 x 72000. 76-80. Reduce 8s. 7d. 2qr. to the decimal of a ~; 2yd. 2ft. 3in. to the fraction of a furlong; 8s miles to fur., r., yd., ft., and in.; 7mo. 21d. to the fraction of a year; 4-d. to seconds. a The difference between any number, and the same number transposed in any way, is divisible by 9. Thus, 723-327, 723-372, 723273, 723-237, are each exactly divisible by 9. Therefobre, if we find in comparing the books of a counting-room or banking-house, that they do not agree, and the amount of their disagreement is divisible by 9, we know that it may have arisen from a transposition. We shall thus frequently be enabled to discover an error readily, which would otherwise have required a long and tedious ex'amination. 5

Page  66 66 MEASURES, WEIGHTS, ETC. [ART. IV. 81-85. Multiply in a single linea 4793 by 27; by 103; by 251; by 19; by 874. IV. MEASURES, WEIGHTS, AND CURRENCIES. THE standard weights of nearly every country, are derived from the linear measures. Coins are made of platina, gold, silver, or copper. As gold and silver are too soft to be used by themselves, some other metal is mixed with them before coining. The metal which is added is called alloy. i f. STANDARDS OF THE UNITED STATES.b Congress has never fully exercised the power granted to it by the constitution, of establishing a uniform standard of weights and measures. The standards used at all the Custom-Houses were prepared by Mr. F. R. Hassler, in 1835-6, a Multiplication in a single line will be found a very valuable exercise, and in many cases it is much 4 8 7 2 more expeditious than the ordinary 6X4+X 8+6X 76X2 method. The ac- 5X 4+5X 8S+- X 7+5X 2 companying exam- 9X 4+9 X 8+9 X 7+9 X 2 pie, of the multi- 3X4+3 X 8+3 X 7+3X2 plication of 4872 plication of 4872 12 + 60 -+ 113 +- 133 +' 101 +- 52 -+- 12 by 3956, will show m. h. th. t. th. th. h. tens un. the several products that are to be added to obtain each figure of the entire product, and will perhaps render the process as intelligible as it could be made by any formal rule. To obtain the product in a single line, we say6X2=12. Set down 2 and carry 1l. 1+6t X7+5- X 4872 2=53. Set down 3 and carry 5. 5+6XS+5X7+9x2 3956 =106. Set down 6 and carry 10. 10+-6X4-i-5XS+9 19273632 X7+3X2 143. Set down 3 and carry 14. 14+55X4 +9X8+3X7= 127. Set down 7 and carry 12. 12+9X4+3X8 = 72. Set down 2 and carry 7. 7 and 3 X4 =19, which we set down, making the entire product 19273632. b A. D. Bache. Report on Weights, Measures, and Balances.

Page  67 ~ 12.] STANDARDS OF THE UNITED STATES. 67 and are similar to those used in England, anterior to the passage of the "Act of Uniformity," in May, 1834. Many of the states have attempted to establish uniformity within their own limits, and have passed laws for that purpose. There is, therefore, a slight diversity in the usages of different sections of the Union, but, as nearly all the laws have assumed the English system for their basis, it does not seem desirable to attempt making an abstract of them. The teacher, however, should make his pupils familiar with the laws that have been passed on the subject by the legislatures of their own state. There is a great discrepancy in the statements of different writers on arithmetic, relative to the government standard. Many of those who have alluded to the subject, seem to have regarded the local customs of their own neighborhood as identical with the practice of the United States' officers, and probably no one has given correct information as to the standards in actual use at the Mint, the Custom-Houses, and in all the departments of the General Government. In the joint resolution of June 14, 1836, the Secretary of the Treasury is " directed to cause a complete set of all the weights and measures adopted as standards, and now either made, or in the progress of manufacture, for the use of the several Custom-Houses, and for other.purposes, to be delivered to the governor of each state in the Union, or such person as he may appoint, for the use of the states respectively, to the end that a uniform standard of weights and measures may be established throughout the United States." In order further to secure this uniformity, Congress directed, in 1838, the preparation and distribution to the states, of balances for adjusting weights and capacity measures. In 1848, twenty-one of the states had received these standards, and asufficient number had been prepared to meet the demand from the remaining states.

Page  68 68 MEASURES, WEIGHTS, ETC. [ART. IV. But in these very standards, there is a great want of system. The foot is subdivided decimally, instead of being divided into inches; the decimal multiples of the Troy pound, and the decimal sub-multiples of the Avoirdupois pound are given, although they are never used. The whole matter is, therefore, in great confusion, not only in this country, but in every other country, except France. Some steps have been taken towards the adoption of a uniform international standard, and it is not improbable that some modification of the French system will eventually come into general use throughout the civilized world.a 1. LONG MEASURE. The denominations are Leagues, Miles, Furlongs, Rods, Yards, Feet, and Inches. Le. m. f. r. yd. ft. in. 1 3 24 = 960 = 5280 = 15840 = 190080 1- 8-320 — 1760 - 5280 = 63360 1= 40 — 220 - 660 - 7920 1 = 5s= 16 = 198 1 — 3 - 36 1 - 12 The English standard unit of Long Measure is the yard, which is equivalent to -33 5 ~ - of the length of a " pendulum vibrating seconds of mean time in the latitude of London, in a vacuum at the level of the sea."b The United States standard, the original, of which the state standards are copies, is a brass scale of 82 inches in length, prepared for the survey of the coast of the United States, by Troughton, of London, and deposited in the Office of Weights and Measures. The IRod is sometimes called Perch, or Pole. The Yard, for CLOTH MEASURE, is subdivided into Quar-,ers and Nails. a Prof. McCulloh, U. S. Mint. b McCulloch.

Page  69 ~ 12.] STANDARDS OF THE UNITED STATES. 69 yd. qr. na. in. 1- 4 - 16 = 36 1 — 4 9 1= 2 Surveyors use CHAIN MEASURE, in which the unit is a Chain of 4 Rods. It is subdivided into Poles and Links. Mile. fur. ch. poles. 1. in. 1 -- 8 -- 80 - 320 - 8000 - 63360 1 - 10- 40 — 1000=- 7920 1= — 4= 100- 792 1 - 25=- 198 1= 7.92 A lot of land measuring 10 chains in length and 1 in breadth, contains an acre. Apalm - 3 inches; a hand -- 4 inches; a span - 9 inches; a pace - 3 feet; a fathon, - 6 feet; a knot, or geographical mile, is I of a degree, or 21 -, of the earth's circumference, and is equivalent to 1.15257 statute miles, or 6085.56 feet;" a degree at the equator, is 69- miles. The inch is generally subdivided on scales into l0ths or decimal parts, but sometimes into halves, quarters, eighths, and sixteenths. In the work of carpenters and other mechanics, the duodecimal division is sometimes employed, The inch = 12 lines, or primes; the prnime -12 seconds; the second = 12 thirds, &c., &c. a This is the value of the geographical or nautical mile, employed in the Topographical Bureau at Washington, in 1849.* The rate of a ship's sailing is determined by the half-minute glass and log-line. The log is a piece of board, loaded on one end so that it will stand vertically in the water. The intervals between the knots on the line, are intended to bear the same proportion to a sea mile, as a half minute to an hour. Thus if 8 knots on the line run off of the reel, while the sand is running out of the half-minute glass, the vessel is moving 8 knots an hour. The length of the knots on the log-line would accordingly be ~of a nautical mile or 50.713 ft., if perfect accuracy were required. But in order to keep the ship " behind her reckoning," and avoid the danger of running ashore, they are made three or four feet shorter. The length of a knot for a 28 seconds' glass is usually 6.t fathoms. * Determined, for the Bureau, by John Downes, from Bessel's Elements of the Terrestrial Spheroid.

Page  70 70 MEASURES, WEIGHTS, ETC. [ART. IV. 2. SQUARE MEASURE. The Square Mile is subdivided into Acres, Roods, Rods, Yards, Feet, and Inches. M.< A. R. sq. r. sq. yd. sq.ft. sq. in. 1 = 640 = 2560 = 102400 = 3097600 = 27878400 = 4014489600 1 = 4 = 160= 4840 = 43560 = 6272640 1 = 40= 1210 = 10890 = 1568160 1 (5~)2 301 = (16)22 2721-(198)2 39204 1= (3)2 9:(36)2 1296 1 =(1272 144 A square piece of land, measuring 209 feet (or nearly 70 paces), on each side, is about equivalent to an acre. The value of the several denominations in Square and Cubic Measure, is determined by the standards employed in Long Measure. 3. CUBIC MEASURE. The denominations are Cubic Yards, Cubic Feet, and Cubic Inches. c. yd. c.ft. c. in. 1 - (3)3 or 27 (36)3 or 46656 1 (12)3 or 1728 A foot of wood is 16 cubic feet. 8 feet of wood, or 128 cubic feet, make a cord. A ton of timber, storage or shipping, is 40 cubic feet. A perch of stone is 24{ cubic feet 1 perch square and lh feet thick. A square of earth is a cube measuring 6 feet on each side, and is equivalent to 216 cubic feet. In measuring round timber, a deduction is sometimes made from the diameter of each stick, to allow for waste in sawing. Therefore, a ton of round timber, although nominally but 40 feet, often contains about 50 feet.a a In England 40 c. ft. of round timber, or 50 c. ft. of hewn timber, make 1 load or ton. In the American lumber yards, the custom is nearly or quite universal of estimating the ton at 40 ft. for both hewn and round timber. But as there are different modes of measurement

Page  71 1~ 2.] STANDARDS OF THE UNITED STATES. 71 Boards are measured by the superficial foot; but if they are more than 1 inch thick, allowance is made for the additional thickness. A board 1I ft. wide, 20 ft. long, and of any thickness not exceeding I inch, contains 20 x 1=30 ft. But if it is 1- inches thick, it will contain 20 X 1~ x 1l+ =37a ft. 4. LIQUID MEASURE. The denominations are Hogsheads, Gallons, Quarts, Pints, and Gills. hhd. gall. qt. pt. gi. 1I 63 - 252- 504 -2016 1- 4- 8- 32 1- 2- 8 1 — 4 The United States standard for measuring liquids, is the gallon, which is a vessel containing 58372.2 grains (8.3389 pounds avoirdupois) of the standard pound of distilled water, at the temperature of 39~.83 Fahrenheit, the vessel being weighed in air in which the barometer is 30 inches at 620 Fahrenheit. This corresponds very nearly with the English wine gallon, which contains 231 cubic inches. Milk and malt liquors are sold by Beer Measure in many places, the beer gallon containing 282 cubic inches. The hogshead (measure) is used only in estimating the contents of cisterns, wells, or large bodies of water. The Gallon (Cong.a) is subdivided by apothecaries, into Pints (Ob), Fluidounces (f,), Fluidrachms (f3), and Minirs (nl). Cong. 0. f.5 f;3 Di 1 = 8 = 128 = 1024 = 61440 1= 16= 128= 7680 1= 8= 480 1 - 60 in use, a ton of round timber will often contain 50 c. ft. Most timber is now sold by Board Measure, the " ton" being nearly obsolete. a From the Latin, Congiarium, a gallon. b From Octars, an eighth part.

Page  72 72 MEASURES, WEIGHTS, ETC. [ART. IV. It is sometimes desirable to make an estimate of the weight of fluids. A pint of water weighs a pound;a 45 drops make about a fluidrachm; a common teacup holds about 4 fluidounces; a common tablespoon about half a fluidounce; a teaspoon about 1 fluidrachm.b 5. DRY MEASURE. The denominations are Bushels, Pecks, Quarts, and Pints. bu. pk. qt. pt. 1 = 4 = 32 = 64 1= 8=16 1= 2 The United States standard is the bushel measure, containing 543391.89 standard grains, (77.6274 pounds Avoirdupois,) of distilled water, at'the temperature of 39~.83 Fahrenheit, and barometer 30 inches at 620 Fahrenheit. This corresponds very nearly with the Winchester bushel, which is a cylinder 18~ inches in diameter, and 8 inches deep, containing 2150.42 cubic inches. 6. TROY WEIGHT.c The denominations are Pounds, Ounces, Pennyweights, and Grains. lb. oz. dwt. gr. 1 = 12 = 240 = 5760 1 = 20 = 480 1= 24 The standard Troy pound' is equivalent to the weight of a A pint of distilled water at 620 Fahrenheit weighs 1 lb. The difference between distilled water and well-water at any ordinary temperature is so slight, that in making estimates, a pint of water may always be considered as weighing a pound. b United States Dispensatory. c " Troy" Weight is said to signify London weight, the name being derived from Troy Novant, the ancient name of London.-MlcCulloch. d The Troy pound was declared to be " the standard and Troy pound of the Mint of the United States, conformably to which the coinage thereof shall be regulated," by an act of Congress of 19th May, 1828.

Page  73 ~ 12.] STANDARDS OF THE UNITED STATES. 73 22.79442 cubic inches of distilled water at its maximum density, the barometer standing at 30 inches; or to 22.8157 c. in. of water at 62~ Fahrenheit, barometer at 30 inches.a It was copied by Captain Kater, in 1827, from the English Imperial Troy pound. In APOTHIECARIES' WEIGHT, which is used in compounding medicines, the Troy pound is subdivided into Ounces (3), Drachms (3), Scruples (3), and Grains. lb. 3 gr 1 =12= 96 = 288 = 5760 1= 8= 24= 480 1= 3= 60 1 20 In DIAMOND WEIGHT, 4 quarters = 1 grain; 4 grains = 1 carat; 7~ carats = 1 Troy dwt. A diamond weighing 1 carat is worth about $9 if rough, and $36 if cut. The value increases as the square of the weight, unless the weight exceeds 20 carats, in which case the increase is not so rapid. Thus a cut diamond weighing 3 carats, would be worth about 32 X 36 = 324 dollars.b The gold carat grain = 2- - dwt. At the UNITED STATES MINT, the Troy ounce is adopted as the standard, and all weights are expressed in decimal multiples and submultiples of the ounce. Thus 951b. 8oz. 15dwt. 15gr. of bullion, would be credited on the Mint books as 1148.65625oz.c 7. AVOIRDUPOIS WEIGHT. The denominations are Tons, Hundred-weights, Quarters, Pounds, Ounces, and Drams. This is the only direct legislation in regard to the adoption of standards, but the joint resolution of June 14th, 1836, indirectly recognises the weights and measures used at the Custom-Houses, as having been "adopted as standards." a McCulloch. b Encyclopedia Americana. c Professor McCulloh, of the U. S. Mint.

Page  74 74 MEASURES, WEIGHTS, ETC. [ART. IV. 7'. cwt.a qr. lb. oz. dr. gr. Troy. 1 = 20 = 80 = 2240 = 35840 - 573440 = 15680000 1= 4=- 112- 1792-= 28672 = 784000 1 = 28= 448 = 7168 = 196000 1 = 16 = 256 = 7000 1 = 16 = 4837 1 = 27 The standard Avoirdupois pound is equivalent to 7000 Troy grains, or to the weight of 27.7274 c. in. of distilled water, at 620 Fahrenheit, barometer at 30 inches.b In many of the states, statutes have been enacted, fixing the ton at 2000 lb., the hundred at 100 lb., and the quarter at 25 lb. But in the standard of the general government, the ton of 2240 lb. with, its subdivisions, is still retained.c Even where the legal ton is 2000 lb., 2240 lb. are often allowed in weighing bulky or cheap materials, such as iron, coal, plaster, &c. 8. FEDERAL MONEY.d The denominations are Eagles, Dollars, Dimes, Cents, and Mills. E. $ di. ct. mi. 1 = 10= 100 = 1000= 10000 1= 10= 100= 1000 1= 10= 100 1= 10 The only coins in circulation are the double eagle, the a From c. for centurm (which signifies one hundred), and wt. weight. b The old English pound, which is said to have been the legal standard of weight from the timb of William the Conqueror, to that of Henry VII., was derived from the weight of grains of wheat; 32 grains gathered from the middle of the ear, and well dried, made a pennyweight, 20 pennyweights an ounce, and 12 ounces a pound. Henry VII. altered this weight and introduced the present Troy pound, which is w of an ounce heavier than the Saxon pound. The Avoirdupois pound was introduced by a statute of 24 Henry VIII.-Brande. c See the different Tariff Acts. d Manual of Coins.

Page  75 ~ 12.] STANDARDS OF THE UNITED STATES. 75 eagle, hlalf-eagle, quarter-eagle, and dollar, of gold; the dollar, half-dollar, quarter-dollar, dime, and half-dime, of silver; and the cent and half-cent, of copper. The gold and silver coins contain -o pure metal, and 1 alloy. The alloy of gold is composed of about I- silver and 4 copper (not to exceed e of silver); the alloy of silver is pure copper. The metal thus alloyed is called standacrd. The eagle contains 258 grains of standard gold, the dollar 412~ grains of standard silver, and the cent 168 grains of copper, and the multiples and subdivisions of all the coins the same proportion. Previous to 1834, the eagle contained 270 grains, of which 22a grains were alloy. By the act of Congress, June 28th, 1834, it was provided that all gold coins minted anterior to the 31st of July, of that year, should be receivable in all payments at the rate of 94.8 cents per pennyweight. The old eagle is therefore worth $10.665. By the same act, the following coins were rendered current in the United States:The gold coins of Great Britain, Portugal, and Brazil, of not less than 22 carats fine, at 94js cents per pennyweight. The gold coins of France, 19 fine, at 931I cents per pennyweight. The gold coins of Spain, Mexico, and Colombia, of the fineness of 20 carats 3 7 grains, at 88- 9 cents per pelmyweight. The silver dollars of Mexico, Peru, Chili, and Central America, of not less weight than 415 grains each, and those re-stamped in Brazil, of the like weight, and of not less fineness than 10 oz. 15 dwt. in the Troy pound, of standard silver, at $1.00 each. The Five-franc piece of France, when of not less fineness than 10 oz. 16 dwt. in the Troy pound, of standard silver, and weighing not less than 384 grains each, at the rate of 93 cents. The following table exhibits nearly the value of the principal gold coins of different countries. But, as most of the coins that circulate in the community are more or less worn, the current value is generally a few cents less.

Page  76 76 MEASURES, WEIGHTS, ETC. [ART. IV. Contents New NAMES OF COINS. Weight. in pure Assay. value. Gold. dw. gr. grains. c..r. gr. d. c, mn. United States. —Eagle coined before July 31, 1834. 11 6 247.5 22 10 66 5 Do. coined after July 31, 1834, shares in proportion... 10 18 232 21 214 10 Austrian Dom i-iozs.-Souverain 3 14 78.6 21 3- 3 38 7 Double Ducat...... 4 12 106.4 23 24 4 59 3 Hungarian dl)... 2 5T 53.3 23 34 2 29 7 Bavaria.-Caroil.. 6 51 115 18 2 4 95 7 Max d'or, or Maximilian 4 4 77 18 14 3 31 Ducat......... 2 54 52.8 23 24 2 27 5 Berne.-Ducat, double in proportion 1 23 45.9 23 1 T 1 97 7 Pistole....... 4 21 105.5 21 24 4 54 2 *Brazil.-Johannes,.4 in proportion 18 21 34 17 6 4 Dobraon...34 12 759 22 32 70 6 Dobra.. 18 6 401.5 22 17 30 1 Moidore, 4 in proportion 6 22 152.2 22 6 55 7 Crusado. 16 14.8 21 34 63 8 Brunswick.-Pistole, double in pro'n. 4 21~ 105.7 21 24 4 55 2 Ducat......... 2 54 51.8 23 O0 2 23 1 Cologne.-Ducat...... 2 54 52.6 23 2 2 26 7 *Colombia. —Doubloon... 17 9 360.5 20 3 15 53 5 Denmark.-Ducat, current.. 2 42.2 21 04 1 81 5 Ducat, specie...... 2 5-4 52.6 23 2 2 26 7 Christian d'or.4 7 93.3 21 3 4 2 1 East India.-Rupee, iBomebay, 1818 7 11 164.7 22 04 7 9 6 Rupee of Madras, 1818.. 7 12 165 22 7 11 Pagoda, Star..... 2 4T 41.8 19 1 79 8 *England.-Guinea, 4 in proportion 5 84 118.7 22 5 7 5 Sovereign, do. 5 24 113.1 22 4 83 8 Seven Shilling Piece... 119 39.6 22 1 69 8 *France.-Double Louis, coin. b. 1786 10 11 224.9 21 2 9 68 8 Louis, do. 5 54 112.4 21 2 4 84 3 Double Louis, coined since 1786 9 20 212.6 21 21 9 16 2 Louis, do. 4 22 106.3 21 21 4 58 1 Double Napoleon, or 40 francs 8 7 179 21 21 7 70 3 Napoleon, or 20 francs.. 4 34 89.7 21 2' 3 86 6 Frankfort on the lliain.-Ducat. 2 5T 52.9 23 22 2 27 9 Geneva.-Pistole, old..... 4 7' 92.5 21 2 3 98 5 Pistole, new..... 3 154 80 21 34 3 44 6 Genoa.-Sequin....... 2 53 53.4 23 34 2 30 2 IEamburg.-Ducat, double in pro'n. 2 5T 52.9 23 2-1 2 27 9 Hlanover.-George d'or.. 4 64 92.6 21 2T 3 99 Ducat. 2 543 53.3 23 34 2 29 7 Gold Florin, double in proportion 2 2 39 18 3 1 69 4 iIolland.-Double Ryder... 12 21 283.2 22 12 20 5 Ryder........ 6 9 140.2 22 6 4 3 Ducat.... 2 54 52.8 23 2' 2 27 5 Ten Guilder Piece, 5 do. in pr'n. 4 74 93.2 21 24 4 1 6 l3Ialta.-Double Louis.... 10 16 215.3 20 04 9 27 8 Louis. 5 8 108 201 4 65 3 Demi Lolis. 2 16 54.5 20 14 2 34 8 5*11exico.-Doubloons, shares in pr'n. 17 9 360.5 20 3 15 53 5 ililan.-Sequin...... 2 54 53.2 23 3 2 29 3

Page  77 ~ 12.] STANDARDS OF THE UNITED STATES. 77 Contents NeW NAMES OF COINS. Weight. in pure Assay. value. do,. gr. glains. car. gr. d. c. m. Doppia or Pistole.4 1- 88.4 21 3 3 80 7 Forty Lire Piece 1808.. 8 8 179.7 21 2[ 7 74 2 Naples.-Six Ducat Piece, 1783 5 16 121.9 21 1 5 24 9 Two do. or Sequin, 1762.. 1 20 37.4 20 14 1 61 3 Three do. or Oncetta, 1818. 210~ 58.1 23 31 2 49 6 Nethlerlands.-Gold Lion or 14 Florin Piece.... 5 74 117.1 22 5 4 6 Ten Florin Piece, 1820.. 4 74 93.2 212- 4 1 6 Parma. —Quadruple Pistole, double in proportion... 18 9 386 21 16 62 7 Pistole or Doppia, 1787.. 4 14 97.4 21 1 4 19 6 Do. do. 1796.. 414 95.9 20 34 413 5 Maria Theresa, 1818... 4 34 89.7 21 24 3 85 1 Piedmont.-Pistole, coined since 1785 half in proportion. 5 20 125.6 21 24 5 41 2 Sequin, half in proportion 2 54 52.9 23 24 2 27 9 Carlino, coined since 1785, half in proportion..... 29 6 634.4 21 2 27 33 4 piece of 20 Francs, ealled Marengo........ 4 34 82.7 20 3 56 4 Poland.-Ducat...... 2 5 52.9 23 24 2 27 9 *~Posrtugal.-Dobraon..... 34 12 759 22 32 70 6 Dobra... 18 6 401.5 22 17 30 1 Johannes.... 18 17 6 4 Moidore, half in proportion 6 22 152.2 21 34 6 55 7 Piece of 16 Testoons, or 1600 rees 2 6 49.3 22 2 12 1 Old Crusado of 400 rees.. 15 13.6 21 3 58 8 New Crusado of 480 rees.. 16 14.8 21 34 63 7 Milree, coined in 1755... 19 18.1 21 38 78 New Dobra...... 17 6 22 16 25 3 Joannese, double in proportion 9 64 21 3; 8 76 3 Half in proportion.... 4 15 21 34 i 4 37 1 Piece of 12 Testoons,or 1200 rees 1164 21 3 1 57 4 Piece of 8 Testoons, or 800 rees 1 4 21 3 1 1 12 Prussia.-Ducat, 1748.... 2 54 52.9 23 22 2 27 9 Ducat, 1787.... 2 5 52.6 23 2 2 26 7 Frederick, double, 1769.. 8 14 185 21 24 7 97 5 Do. do. 1800.. 8 14 184.5 21 2 7 95 1 Do. single, 1778.. 4 7 92.8 21 24 3 99 9 Do. do. 1800 4 7 92.2 212 3 97 5 Rome.-Sequin, coined since 1760 2 4' 52.2 23 34 2 25 Scudo of the Republic... 17 04 367 21 24 15 80 4 Russia.-Ducat, 1796.. 2 6 53.2 23 24 2 29 Ducat, 1763...... 2 5 52.6 23 2 2 26 71 Gold Ruble, 1756... 1 04 22.5 22 96 7 Gold Ruble, 1799.... 184 17.1 21 34 73 7 Gold Poltin, 1777.... 9 8.2 22 35 5 Imperial, 1801...... 17 181.9 23 24 7 83 6 Half do. 1801...... 3 204 90.9 23 24 3 91 3 Do. do. 1818. 4 34 91.3 22 04 3 94 2 Sardinia.-Carlino, half in proportion 10 74 219.8 21 14 9 47 Saxony.-Ducat, 1784.... 2 5 52.6 23 2 2 26 7 Ducat, 1797...... 2 54 52.9 23 24 2 27 9

Page  78 78 MIEASURES, WEIGHTS ETC. [ART. IV. Contents New NAMES OF COINS. Weight. in pure Assay. value. Gold. dw. gr. grains car. gr. d. n Augustus, 1754 4 64 91.2 21 1 3 92 7 Do. 1784.... 4 64 92.2 21 24 3 97 4 Sicily.-Ounce, 1751.... 2 20 58.2 20 1 2 50 5 bouble do. 1758..... 517 117 20 2 5 4 2 *Spain. —Quadruple Pistole, or Doubloons, 1772, double and single, and shares in proportion. 17 81 372 21 2 16 3 8 Doubloon, 1801 17 9 360.5 20 3 15 53 5 Pistole, 1801 4 8- 90.1 20 3 388 4 Coronilla, Gold Dollar, or Vintemn, 1801...... 1 3 22.8 20 1 98 3 Sweden.-Ducat...... 2 5 51.9 23 2 2 23 6 Switzerland.-Pistole of Helvetic Republic, 1800. 4 214 105.9 21 2 4 56 Treves.-Ducat. 2 5 52.6 23 2 2 26 7 Turkey.-Sequin Fonducli, of Constantinople, 1773. 2 54 43.3 19 1 186 8 Do. 1789........ 2 5 42.9 19 0 1 84 8 Half Misseir, 1818.... 18 12.2 16 0 52 1 Sequin Fonducli..... 2 5 42.5 19 1 1 83 1 Yermeebeshlek.'.. 3 1 70.3 22 3 3 2 8 Tuiscainy.-Zechino, or Sequin. 2 5 53.6 23 34 2 30 9 Ruspone of the k'm. of Etruria 6 174 161 23 3 6 93 9 FVenice.-Zechino or Sequin, shares in proportion.... 2 6 53.6 23 31 2 31 f'rMrtembuirg.-Carolin.... 6 3 113.7 18 2 4 89 8 Ducat......... 2 5 51.9 232 2237 Zurich. —Ducat, double, and half in proportion... 2 5 52.6 23 2 2 26 7 I _-_ The foregoing Table is copied from the American Almanac for 1835. It was originally compiled from the "Manual of Coins," published at the United States Mint, Kelly's Cambist," and " Moore's Philadelphia Price Current." The gold coins of the countries to which the star is prefixed, if possessed of the fineness prescribed, are made, by the act already referred to, to "pass current as money, and to be receivable in all payments, by weight, for all debts and demands, from and after the 31st day of July, 1834." The other coins in the Table are not made a legal tender; but they are sold at a certain rate per dwt., according to the purity of the gold.

Page  79 ~ 12.] STANDARDS OF THE UNITED STATES, 79 TABLE OF THE PRINCIPAL SILVER COINS OF DIFFERENT COUNTRIES. Contents NAMES OF COINS. in pure Value. Silver. grains. $ cts. Austria. —Rix Dollar, or Florin, Convention.. 179.6a 48 Copftsuck, or 20 Creutzer Piece... 59.4 16 Halbe Copftsuck, or 10 Creutzer Piece. 28.8 08 Baden.-Rix Dollar....... 358.1 96 Bavaria.-Rix Dollar of 1800...... 345.6 93 Copftsuck.59.4 16 Brunswicck.-Rix Dollar, Convention.... 359.2 96 Denmnarlk.-Ryksdaler...... 388.4 1 05 Mark, specie, or HIalf Ryksdaler... 64.4 17 East Indies.-Sicca Rupee, Calcutta.... 175.9 47 Company's Rupee (1835)...... 165. 44 Bombay, new, or Surat (1818).. 164.7 44 Fanam, Cananore...... 32.9 09 " Bombay, old........ 35. 09 " Pondicherry.,.. 22.8 06 Gulden, Dutch East India Company ~ ~ 148.4 40 England.-Crown (old)....... 429.7 1 15 Shilling........ 85.9. 23 Crown (ze)........ 403.6 1.09 Shilling"........ 80.7 22 Prance.-Franc........... 69.4 19 Genoa.-Scudo of 8 Lire...... 457.4 1 23 Hambug. —IRix Dollar, specie..... 397.5 1 07 Double Mark, 32 Schilling..... 210.3 57 8 Schilling Piece......... 50.1 13 Hanover. —Rix Dollar, Constitution.... 400.3 1 08 Florin, or piece of 2, fine...... 200.3 54 Hollanzd.-Florin, or Guilder...... 146.8 40 12 Stiver Piece.......... 92.4 25 Florin of Batavia....... 141.6 38 Lubeck.-Rix Dollar, specie........ 391.9 1 05 Mark............ 105.1 28 Lucca.-Scudo........ 372.3 1 00 Malta.-Ounce of Eamman. Pinto..... 337.4 91 2 Tari Piece.......... 17.7 05 XAilanz.-Scudo of 6 Lire.... 319.6 86 Lira........ 52.8 14 Jifodena.-Scudo of 1796........ 287.4 77 Naaples.-Ducat, new...... /295.4 80 Piece of 10 Carlini........ 295.1 80 NYetherlands.-Florin of 1816...... 148.4 40 Polanzd.-Florin, or Gulden....... 84. 23 Portugal.-New Crusado (1809) 198.2 53 Seis vintems, or Piece of 120 Rees... 46.6 13 a Brande.

Page  80 80 MEASURES; WEIGHTS; ETC. [ART. IV. Contents NAMES OF COINS. in pure Value. Silver. grains. ~ cts. Testoon. 42.5 11 Portuyuese Colonies.-Piece of 8 Macutes, of Portuguese Africa... 159.8 43 Prussica.-Rix Dollar, Convention.... 359. 96 Florin, or Piece of...... 198.4 51 Rorne.-Scudo, or Crown........ 371.5 1 00 Paolo.......... 37.2 10 Rulssia.-Rouble.......... 312.1 84 Rouble of Alexander (1805).. 278.1 75 20 Copeck Piece......... 62.6 17 Sardinia.-Sctudo, or Crown....... 324.7 87 Saxolzy.-Rix Dollar, Convention.. 358.2 96 Sicily.-Scudo........... 348.2 93 Spain.-Dollar........... 370.9 1 00 S&veden.-Rix Dollar...... 38. 5 1 05 Switzerland.-Ecu of 4 Franken.... 407.6 1 10 TU1rklyJ.-Piastre of 1818........ 67.7 18 TaLscany.-Lira....... 53.4 14 Wurtembuag.-Rix Dollar, specie.... 359.1 96 Copftsuck.... 59.8 1 6 9. ASTRONOMICAL MEASURES. The denominations of CIRCULAR MEASURE, and of TIMIE, are the same in all civilized countries. Every circle is divided into degrees (~), minutes (/), and seconds ("). Seconds are usually subdivided decimally. A Sign in Astronomy, is 30 degrees. A Quacdrant is 90 degrees. Cire. 0 1 = 360 = 21600 = 1296000'1= 60 = 3600 1= 60 The denominations of Time are Years, Months, Weeks, Days, Hours, Minutes, and Seconds. Yr. Dy. h. mnin. sec. 1 = 365 = 8760 - 525600 = 31536000 1 = 24 = 1440 = 86400 1 = 60 = 3600 1= 60

Page  81 ~ 13.] STANDARDS OF GREAT BRITAIN. 81 A common year is 365 days. A bissextile, or leap year, is 366 days. A Julian year is 3651- days. A tropical, solar, or civil year is 365d. 5h. 48m. 49.7sec.a 1 3. STANDARDS OF GREAT BRITAIN.b 1. MEASURES. The linearo, superficial, and cubic measures are the same as in the United States. The old wine, beer, and dry measures have been supplanted by the IMPERIAL LIQUID AND DRY MEASURE. The denominations are Quarters, Cooms, Bushels, Pecks, Gallons, Pottles, Quarts, Pints, and Gills. Qr. C. bt. pkc. gal. pot. qt. pt. gi. 1 = 2 - 8 = 32 = 64 = 128 = 256 = 512 = 2048 1= 4-= 16 = 32 = 64-= 128-= 256= 1024 1 = 4 = 8= 16 = 32 = 64 = 256 1 = 2 = 4= 8-= 16 = 64 1= 2= 4= 8= 32 1= 2= 4= 16 1i= 2= 8 1- 4 The imperial standard gallon contains 10 lb. Avoirdupois of distilled water, temperature 620, barometer 30 inches. Its capacity is therefore 277.274 c. in. The imperial bushel is a cylinder, of which the inner diameter is 18~ inches, and the depth 8- in. The following denominations of WINE MEASURE have been discarded under the new system, viz.: the Tun = 2 Pipes, the Pipe or Butt = 2 Hhd., the IHogshead = 63 Gal., the Puncheon = 2 Tierces, and the Tierce = 42 Gallons. In the old BEER MEASURE, 1 Tun = 2 Butts, 1 Butt = 2 Hhd., 1 Hogshead = 1 Barrels, 1 Puncheon - 2 Barrels, 1 a Somerville. b McCulloch, and Man. of Coins. c The Irish mile = 3038 yd.; the Scotch mile = 1984 yd. 6

Page  82 82 MEASURES, WEIGHTSI ETC. [ART. IV. Bbl. = 2 Kilderkins, 1 Kilderkin = 2 Firkins, 1 Firkin = 8 Gal. of Ale, or 9 Gal. of Beer. Coals were formerly sold by the Chaldron, which was subdivided into Vats, Sacks, and Bushels. The coal bushel held 1 qt. more than the Winchester bushel. Twenty-one chaldrons made a Score. Score. Chal. Vats. Sacks. Bu. 1 = 21 = 84 = 252 = 756 1 - 4 = 12 = 36 1= 3= 9 1= 3 In DRY MEASURE, a Last is 2 Weys, and a Wey or Load is 5 Quarters. 2. WEIGHTS. All weights are derived from the Troy and Avoirdupois pounds. The Imperial standards, and the denominations of Troy, Apothecaries', and Avoirdupois weight, are the same as in the United States. Not only have the English no natural standard of weight, but at the present time they have no standard, the Imperial Troy pound having been destroyeda by the fire which consumed the Houses of Parliament, Oct. 16, 1834.b But the bulk of water to which it is equivalent has been so accurately determined (27.7274 c. in.), that it could easily be restored. The length of the seconds' pendulum at Greenwich, may, therefore, be very properly regarded as the present basis of the entire system of weights, measures, and currencies, both of Great Britain and the United States. Avoirdupois weight may be readily converted into Troy, or Troy into Avoirdupois. 144 lb. Avoirdupois 175 lb. Troy. 192 oz. Avoirdupois = 175 oz. Troy. A stone is generally 14 lb. Avoirdupois. But a stone of a Brande. b Wade.

Page  83 ~ 13.] STANDARDS OF GREAT BRITAIN. 83 butcher's meat, or fish, is 8 lb. A stone of glass is 5 lb. A seam of glass is 24 stone. A truss of hay = 56 lb.:A truss of new hay, until the 1st of September = 60 lb. A truss of straw - 36 lb. 36 trusses make a load. In weighing wool, 1 last = 12 sacks, 1 sack = 2 weys, 1 wey = 61 tods, 1 tod = 2 stone, 1 stone = 2 cloves, I clove = 7 lb. A pack of wool contains 240 lb. 8 lb. = 1 clove of cheese or butter, and 56 lb. = 1 firkin of butter. A wey is 32 cloves in Essex, or 42 cloves in Suffolk. 3. ENGLISH OR STERLING MONEY. The denominations are Pounds, Shillings, Pence, and Quarters, or Farthings. ~ s. d. qr. 1 = 20 = 240 = 960 1 = 12 = 48 1= 4 A guinea is 21 shillings. A crown is 5 shillings. The coin which represents the pound is called a sovereign. The coins are the five-guinea piece,- the guinea, half-guinea, quarter-guinea, seven-shilling piece, double sovereign, sovereign, and half-sovereign of gold,-the crown, half-crown, shilling, sixpence, fourpence, threepence, twopence, one-and-ahalfpence, and penny of silver,-and the penny, halfpenny, farthing, and half-farthing of copper., A pound of silver of the English mint standard, contains lloz. 2dwt. of pure silver, and 18dwt. of alloy. This pound is coined into 66 shillings. A shilling therefore weighs 87.27 grains, and contains 80.727 grains pure silver. The standard for gold is 11 parts fine gold and 1 part alloy. The sovereign weighs 123.274 grains, and contains 113.001 a In Scotland, a bodle = 4 of an achison = ~ of a bawbee 1= of a plack = i of a penny.-Gregory.

Page  84 84 MEASURES, WEIGHTS, ETC. [ART. IV. grains of pure gold. A sovereign of full weight is there. fore worth $4.866.a The guinea and its subdivisions have not been coined since 1816. 14:.-STANDARDS OF FRANCE.b 1. MEASURES. The standard unit of linear measure is the Metre, from which all measures and weights are derived. It is intended to be equivalent to 1 of- the distance from the pole to the equator, and was determined by measuring the distance from Dunkirk to Rhodes.c The names of the multiples and submultiples of the unit, in all the tables, are formed by the following prefixes: -Deca prefixed, signifies 10 times. D)eci denotes -j0. Ilecto a 100 " Centi " 1 Kilo " " 1000 " Ijilli " 1ifJyria " " 10000 " Thus the Decametre = 10 M6tres; the Decistere = -11 Stdre; the Hectogramme = 100 Grammes. The mdtre is equivalent to 39.371 inches. The millim6tre is sometimes called trait (line), the centimdtre, doigt (finger), the decim6tre, palme, and the decamdtre, perche. The unit of superficial measure is the Are, which is a square decamdtre, and is equivalent to 119.6046 square yards. The hectare is often called a)pent (acre). a In 1838 a dispute arose in the settlement of an account, which was submitted to arbitration. The charges were all made in English money, and the point in dispute was the value of the pound sterling, in our own currency. It was decided by the referees, to be $4.858.iler. 3'oag. At the Custom-House, the sovereign is estimated at $4.84; the pound of the British Provinces, Nova Scotia, New Brunswick, Newfoundland and Canada, at $4.00; the pound of Jamaica, Honduras, and Turk's Island, at $3.00; the pound of Nassau, at $2.50. I McCulloch, Enc. Amer., Hunt's Mer. Mag. ~ About 570 miles.-Hunt's lIer. M11ag.

Page  85 ~ 14.] STANDARDS OF FRANCE. 85 The unit of solid measure is the Stere, which is a cubic mAtre, and is equivalent to 35.3174 cubic feet, or 1.30805 cubic yards. The unit of measures of capacity is the Litre, which is a cubic decimdtre, and is equivalent to 1.05676 quarts of our standard. The setier is equivalent to the hectolitre, and the mufid, or barrel, to a kilolitre. In the Systdme Usuel,a the names of the ancient weights and measures are retained, together with the subdivisions into halves, quarters, eighths, &c. The toise usuelle = 2 metres; the pied or foot = mtre; the aune = 1- m&6 tres; the boisseau = 12.5 litres. 2. WEIGHTS. The unit of weight is the Gramme, which is derived from the cubic centimdntre, and is equivalent to 15.434 Troy grains. In the Systdme Usuel, the half-kilogramme is called a livre, which is thus subdividedl: 4 gros make 1 once; 16 onces make I livre usuelle; 2 livres make I kilogramme. A mnillier = 1000 kilogramlmes = 2205.48 lb. It is used for marine tonnage. 3. MONEY. Accounts are kept in Francs and Centimes. The old denomination, Sols or Sous, is sometimes used in accounts, 20 sous being rated as a franc. Ten centimes make a decime. F. D. C. 1 = 10 = 100 1 = 10 The subdivision of the franc, therefore, resembles our subdivision of the dollar into dimes and cents. Prior te the revolution, the money was divided into louis-d'ors, a Called " usuel," because the common subdivisions are retained but their value is altered to correspond with the new standard.

Page  86 86 MEASURES, WEIGHTS, ETC. [ART. IV. ecus or crowns, livres tournois, sous, and deniers; the livre, when newly coined, being equivalent to the franc. L.d'or. e. lii. s. d. 1 = 4 = 24 = 480 = 5760 1 = 6 = 120= 1440 1 = 20= 240 1= 12 The gold coins now in circulation, are the double louisd'or, the louis-d'or, the double Napoleon (worth 40 francs), the Napoleon, the 40 franc piece, and the 20 franc piece; the silver coins are the crown, half-crown, 30 sous, 15 sous, 6 livres, 5 francs, 2 francs, franc, and quarter-franc; the copper coins are the 5 sous, 2 sous, sol, decime, 5 centimes, 2 centimes, and centime. The mint standard for both gold and silver, is -19 fine metal, and 1 alloy. A kilogramme of standard gold is coined into 155 twenty-franc pieces. A kilogramme of standard silver is coined into 200 francs. The Custom-House valuation of the franc is $0.186. 4. SUBDIVISIONS OF THE CIRCLE AND OF TIME. With the introduction of the decimal system of measures, weights, and money, an attempt was also made to change the divisions of the circle and of time. Each quadrant of the circle was divided into 100 degrees (making 4000 in the entire circle, instead of 360~), each degree into 100 minutes, and each minute into 100 seconds. A series of logarithmic trigonometrical tables, to correspond with this system, was computed by M. Borda, and a compendium of his work, with the logarithms extended to seven places, has been published. Each of the 12 months was composed of three decades of 10 days each, and at the end of the year, five intercalary days, (in leap years six days,) were added. The day was divided into 10 hours, the hour into 100 minutes, and the minute into 100 seconds. After a short trial, the attempt to introduce these changes entirely failed.

Page  87 ~ 15.] IMISCELLANEOUS TABLE. 87 1 5. MISCELLANEOUS TABLE a OF MEASURES, WEIGHTS, AND MONEYS OF ACCOUNT. 1. ACAPULCO.-See MExico. 2. ALEXANDRIA.-The yard orpik = 26.8 inches. The measures for corn are the rhebebe = 4.364 Eng. bushels, and the quillot or kisloz = 4.729 bushels. The cantaro or quintal = 100 rottoli; but the rottolo has four different values. The rottolo foforo =.9347 lb. Av.; 1 rottolo zaidino = 1.335 lb. av.; 1 rottolo zauro = 2.07 lb. Av.; 1 rottolo mina = — 1.67 lb. Av. MoNEY.-Accounts are kept in currentpiastres. 1 piastre = 40 paras or medini; 1 medino = 30 aspers, or 8 borbi, or 6 forli. A purse contains 25000 medini. A piastre is worth about 6 cents. Large payments are generally made in Spanish dollars. 3. ALICANT. —The yard or vara = 4 palmos = 29.96 inches. In liquid measure 1 cantar o = 8 medios = 16 quartillos = 3. 05 Eng. wine gallons. The tonnelada or ton = 2 pipes = 80 arrobas = 100 cantaros. In dry measure I cahiz = 12 barchillas = 96 medios = 192 quartillos = 7 Winch. bush. The cargo — 2' quintals - =- 10 arrobas. The arroba =- 27 lb. 6oz. Av., and contains 24 large pounds of 18 Castilian ounces, or 36 small pounds of 12oz. each. At the Custom-House, the arroba = 25 lb. of 16oz; each. MONEY.-Accounts are kept in libras of 20 sueldos, each sueldo containing 12 dineros. The libra or peso = 10 reals = 272 maravedis of plate or 512 maravedis vellon = 78 cents. 4. AmsTErDAMn.-In 1820, the French system of measures and weights was introduced into the Netherlands, the names only being changed. The unit of LONG MEASURE is the elle, which equals the French mstre. Its decimal divisions are the palm, duim, and streep (corresponding to the decimetre, centimietre, and millimetre), and its decimal multiples, the roede and mijle (corresponding to the decamhtre and kilometre). The unit of SQUARE MEAsuRE is the vierkante elle or square elle, which equals the French centiare or metre carr6. Its divisions and multiples are the vierkante palm, vierkante duimt, vierkante streep, vierckante roede,and vierkante bunder. The vierkante bunder = 1 Are. In MEASURES OF CAPACITY 1 kubicke elle = —-1 stWre. Its divisions are the kubic7ce palm, duim, and streep. A k. elle of firewood is called wisse. In a McCulloch, Hunt's Mer. Mag., Am. Al., Boston Custom-House Table, Enc. Amer.

Page  88 88 MIEASURES, WEIGHTS, ETC. [ART. IV. DRY MEASURE, 1 k2op =1 litre. The maa(je, schepel, and mudde, or Oak, correspond to the decilitre, decalitre, and hectolitre. In LIQUID MEASURE, the kan, maatfje, vingerhloede, and vat, are respectively equivalent to the litre, decilitre, centilitre, and hectolitre. The last, or measure for corn, =- 27 mudden. The aam, liquid measure, = 4 ankers 8 steckans _ 21 viertels = 64 stoopen = 128 mingles = 256 pintes - 180 litres. The WEIGHTS are the wigtje, iorrel, lood, ons, and pond, corresponding to the gramme, decigramme, decagramme, hectogramme, and kilogramme. The last for marine tonnage - 2000 ponds. The apothecary's new pound = 5787 grains Troy, or 375 grammes, and is subdivided as the English apothecary's pound, into ounces, "drams, scruples, and grains. By the old method of calculating, 100 lb. Amsterdam - 108.923 lb. Avoirdupois. MONEY.-100 cents — 1 florin -= 0.40. Accounts are sometimes kept in Flemish money. 1 pound = 6 florins = 20 schillings = 120 stivers = 240 groats = 1920 pennings. 5. ANTWERP.-The same as Amsterdam. Of the old weights, which are still occasionally referred to, the quinztal of 100 lb. 103k lb. Av. A schippound is 3 quintals. A stone is 8pounds. Of the old measures, 1 last — = 371 viertels - 150 macken = about 9- 7 imperial quarters. A barrel - 261- Eng. gallons. 6. ARABIA.-See BussoRAII, DJIDDA, MOCHA, MUSCAT. 7. AvsTRIA. —See TRIESTE. 8. BANGIOK. —1 fathom = 4 cubits -- 8 spans = 96 finger-breadths about 61 ft. Eng. 20 fathoms = 1 sen, and 100 sen = 1 yuta. In weighing, 1 piuczl = 50 catties = 133~ lb. av. MoNEY. —The currency consists only of silver and cowrie shells. 1 bat or tical = 4 salsungs = 8 fuags = 16 sing-p'hais = 32 p'hainusgs == 6400 bia or covwries == 55 cents nearly. 80 ticals make 1 catty, and 100 catties make 1 picul. Gold and silver are weighed by small weights which have the same denominations as the coins. The p'hai-nung is then divided into 32 sagas. 9. BARcELoNA. -The yard or cana - 8 palmos = 32 quartos = 21 inches nearly. The quartera, or measure for grain =12 cortanes 48picolizs =.235 Winch. quarters. The carga or liquid measure 12 cortanes or arrobas = 24 cortarinas =- 72 mitadellas = 32.7 wine gallons. 4 cargas = 1 pipe. MONEY.-1 libra - 20 steeldos = 240 dinzeos -- 480 smallas - 53

Page  89 ~ 15.] MISCELLANEOUS TABLE. 89 cents. The libra is likewise divided into 62?reales de plata Catalan, or 10 reales ardites. 10. BATAVIA.-The Chinese weights are used, (the picul and catty), but the picul is considered equal to 136 lb. Av. Accounts are kept in florins or guilders, and centimes. The rupee = $0.44. The rix dollar -- 48 stivers -- $0.75. See AiSTERDADI and CANTON. 11. BELGIUM.-See ANTWERP. 12. BENGAL. See CALCUTTA and MADRAS. 13. BILBAO.-See CADIZ. 14. BOIBA~. —1 gYuz = 1 c haths = 24 tussoos = 27 inches. In salt measure, 1 rash = 16 annas = 1600 parahs = 16800 adowlies 2572176 cub. inches or 40 tons. In grain measure, 1 candy = 8 parahs = 56 pailies = 224 seers = 448 tipprees = 1561b. 12oz. 12.8dr. av. In liquor measure, 1 mezaund = 50 seers - 3000 rupees -761b. lloz. 13dr. Av. In weighing all heavy goods except salt, 1 smaund = 40 seers = 2880 tanks = 281b. Av. In pearl weight, 1 tank = 24 ruttees = 330 tuclks = 72 Troy grains. In gold and silver weight, 1 tola = 40 swalls = 179 gr. Troy. MONEY.-Accounts are kept in rupees. 1?zupee = 4 qzuaters400 reas = 16 annas = 50 pice = $0.45. An eurdee is 2 reas; a doreea, 6 reas; a dooguney, 4 reas; a fuddea, 8 reas; a paunchea, 5 rupees; a gold mohur, 15 rupees. The annas and reas are imaginary moneys. 15. BRAZIL. -Same as LISBON. 16. BREiIEN. —1 ell = 2 feet = 22.76 Eng. inches. 1 last = 4 quarts = 40 schlefels = 160 viertels = 640 spints = 80.7 Winchester bushels. 1 oxhoft= 1~ tierces = 6 ankers = 30 viertels - 264 quarts = 58 Eng. wine gal. An ahm = 4 ankers. The commercial pound = 2 marks = 16 ounces = 32 loths. 100 lb. Bremen - 109.8 lb. Av. A shippound = 2~ centners = 290 lb. A waage of iron = 120 lb. MONEY. —1 thaler or rix dollar = 72 grootes = 360 swares = $0.78-. 17. BUENOS AYREs.-The same as CADIZ. 18. BURmAIH.-See RANGOON. 19. BusmIIRE.-The league or parasang = 3m. 3fur. 25r. The royal guz, or cubit = 37- inches. The common guz = 25 inches.

Page  90 90 MEASURES, WEIGHTS, ETC. [ART. IV. The artaba or principal corn measure 16 bushels nearly. Pearls are weighed by the abbas = — 2-gr. Troy; gold and silver by the miscal = 3dwt. very nearly. The maund shaw = 2 maunds tabree = 134 lb. Av. at the Custom-House, or 124 lb. at the bazaar. It is used by dealers in sugar, coffee, copper, and all sorts of drugs. The maund coprac is 7A lb. at the Custom-House, and from 7- to 74 lb. at the bazaar. It is used by dealers in rice and other provisions. MONEY. —1 toman = 50 abasses - 100 mamoodis = about $2.50. The toman of Gombroon= $5. 20. BussoAr. —The Arabian mile = 2148yd. The Aleppo yard, for silks and woollens = 2ft. 2.4in.; the Hadded yard, for cottons and linens, = 2ft. 10. 2in.; the Bagdad yard, for all purposes, = 2ft. 7.6in. Gold and silver are weighed by the cheki = 100 miscals = 7200gr. Troy. 1 oke of _Bagdad -- 24 vakis =- 47'oz. Av.; 1 maund atteree = 281 lb. Av.; 1 maund sofy or sesse = 90 l1b. Av.; 1 cutra of indigo = 138 lb. 15oz. Av. These are the weights used by the European merchants settled at Bussorah; they differ a little from those used by the Arabians. MONEY.-Accounts are kept in mamoodis of 10 daniems, or 100 floose. 1 toman = 100 mamoodis = $8 nearly. 21. CADIZ.-100 yards or varas - 92+ English yards. The common legua = 800 varas; the legal legua = 500 varas. In corn measure 1 cahiz = 12 fanegas = 144 celeminas = 576 quartillas = 1.576 bushels. In liquid measure 1 cantaro or arroba- =8 azumbres. 32 quartillos. There are two arrobas, the greater and the less, the former =- 4, the latter = 3 3 wine gallons. A moyo of wine - 16 arrobas. A botta= 30 arrobas of wine, or 38- of oil. A piepe 27 arrobas of wine, or 34 of oil. 100 lb. Castile = 101- lb. Av. The ordinary guintal is divided into 4 arrobas, or 100lbs. of 2 marcs each. MONEY.-Accounts are kept by the real of old plate, of which there are 104- in the peso duro or hard dollar. The real = 16 quintos or 34 maravedis. The ducado de plata, or ducat of plate, is worth 11 reals. At the U. S. Custom-House the real of plate is estimated at $0.10, and the real vellon at $0.05. 22. CAGLIARI.-The palm= 104 inches. The starello, or corn measure = lbu. 1Jpk. Eng. 1 cantaro= 4 rubbi= 104 Ibs. = 1248oz. - 93 lb. Ooz. 8dr. Av.

Page  91 ~ 15.] MISCELLANEOUS TABLE. 91 MoNEY.- 1 lira= 4 reali= 20 sofi= $0.186. 19 reali make 1 scudo. 23. CALCUTTA. — coss = 1000 fathoms = 4000 cubits = 8000 spans = 24000 hands = 96000 fingers = 288000 barleycorns or jozws = Im. Ifur. 3-171yd. In CLOTH MEASURE 1 gUz =2 hauts or cubits = 16 gheriahs = 48 angullas -- 144 jorbes = lyd. Eng. In SQUARE MEASvRE 1 biggah= 20 cottahs =320 chittacks =1440 sq. ft. A chittack is 5 cubits or hauts in length, and 4 in breadth. In GRAIN MEASURE 1 khahoon = 16 soallies = 3200 pallies = 12800 raiks 51200 khaonks =30 bazaar mnaunds. In LIQUID MEASURE 1 bazaar maund= 8 pussarees or measures = 40 seers = 160 pouahs or pice = 640 chittacks = 3200 sicca weight. WEIGHTS. —1 maund= 40 seers = 640 chittacks = 3200 siccas. The factory maund = 74 lb. 10oz. 102-dr. Av.; the bazaar maund=82 lb. 2oz. 1-25dr. In GOLD AND SILVER WEIGHT 1 anna = 6 rutties= 25 dhans orgrains= 100punknhos; 1 sicca =10 massas=80 rutties; 1 tolah=100 rutties = 224.588gr. Troy. 1 rohur = 1661 rutties. MoNEY.-Accounts are kept in sicca, or in current rupees, with their subdivisions, annas and pice. The sicca rupees bear a batta (premium) of 16 per cent. over the current. 1 gold mohur=16 sicca rupees = 64 cahauns = 256 annas = 3072 pice or 1024 punns = 20480 gundlas. 4 cow'ries (a species of shell) make 1 gunda, and 2560 cowries = 1 current rupee. A current rupee is worth about 44 cents, and a sicca, rupee, $0.50. A lac of rupees —100000 rupees. A crore = 100 lacs. 24. CANADA.-See QUEBEC. 25. CANTON. —1 li= 180 fathoms= 1800 Chinese feet = —1897k Eng. ft. 1 covid or cobre = 10 punts= 145 inches. There are no liquid or dry measures; all articles that are usually sold by those measures, being sold in Canton by weight. 1 picul-= 100 catties or gins =1600 taels or lyangs = 16000 mnace or tchens = 160000 candarines or fivans = 1600000 cash or lis -= 133 lb. Av. The mace, candarine, and cash are money weights. MOrNEY.-Accounts are kept in taels, mace, candarines, and cash. The cash is the only coin made in China. It is composed of 6 parts of copper and 4 of lead, and is cast with a square hole in the middle, so as to be strung on a wire or string. The circulating medium consists principally of cut Spanish dollars. In calculations of prices, and of accounts between foreigners and native merchants, 720 taels = $1000. In weighing money for payments, 715 taels = $1000, except to the Company's treasury, when 718

Page  92 92 MEASURES, WEIGITS, ETC. [ART. IV taels -- $1000, or to native merchants, not of the co-hong, or to ship and house compradors, when 715 taels = $1000. A tael of fine silver should be worth 1000 cash, but on account of their convenience their price is often so much raised that only 750 are given for a tael. In the Custom-House estimate, the tael = $1.48. 26. CAPE TOWN.-12 Rhynland inches =1 lRhynland foot; 27 Rhynland inches = 1 Dutch ell. In square measure, 1 morgen = 600 roods = 86400 square feet = 12441600 square inches. In corn measure, 1 load= 10 muids = 40 schepels = 30bu. 2pk. 5qt. very nearly. In liquid measure 1 leaguer- =4 aams = 16 ankers = 256 flasks = 152 wine gallons. The weights are derived from the standard pound of Amsterdam, the loot, =-31- of a Dutch pound, being regarded as the unit. MoNEY.-Accounts are kept either in pounds, shillings, pence, and farthings, (as in Great Britain,) or in rix dollars, schillings, and stivers. 1 rix dollar = 8 schillinys — 48 stivers = is. 6d. 27. CEYLON.-See COLUMIBo. 28. CHILI.-See VALPARAISO. 29. CHINA.-See CANTON. 30. CHRISTIANIA.-Measures and weights, same as at Copenhagen. MONEY.-1 species dollar = 120 skillings =-1.06. There are no gold coins made in Norway. 31. CIITA VEccIIIA. —The Roman foot = 11.72 inches English. The canna c 78.34in. The builders' canna - 87.96in. The barrel= 12.841 imp. gallons of wine, or 12.64 imp. gals. of oil. The soma of oil= 36.13 imp. gals. The rubbio of corn=8.143 imp. gals. The libra or pound = 12 onci = 6912 grani = 5234gr. Troy. There are three cantaros or quintals,-of 100, 160, and 250 lb. The migliajo = 1000 libre. MONEY.-l scudo = 10 paoli= 100 bajocchi= $1.00. 32. CoLUnBO.-The principal dry measures are the seer, which is a perfect cylinder, 4.35in. deep, and 4.35in. diameter,-and the parrah, which is a perfect cube, its internal dimensions being 11.57in. on every side. The liquid measure, the weights, and the money, are the same as in Great Britain. A leaguer or legger =150 gallons. A candy or bahar =5001b. Av. A rix dollar= Is. 6d. 83. CONSTANTINOPLE.-The pik or pike is generally estimated at

Page  93 ~ 15.] MISCELLANEOUS TABLE. 93 - of a yard Eng. The berri= 1826yd.; the Turkish mile = 1409yd. In corn measure 1 fortin= 4 kisloz= 3.764 bushels. Oil and other liquids are sold by the alma or meter = gal. 3pt. WEIGIHTs. -1 rottolo =176 drams; 1 oke = 2.272 rottoli; 1 quintal or canztaro- 7- batmans= 44 okes =1241b. 72oz. Av. The quintal of cotton is 45 okes -= 127.2 lb. Av. MONEY..-1 piastre = 40 paras = 120 aspers = about 4 cents. The piastre is exceedingly variable in its value, those coined in 1764 being worth $0.60,'and those coined in 1832 being worth only 0.03. A bag of silver=500 piastres. A bag of gold= 30000 piastres. 34. COrENHAGEN. —The Danish ell =2 Nhineland feet = about 25 inches. The Danish mile = 8244yd. In dry measure 1 last =12 toendes or tons = 96 -scheffels = 384 viertels = 47- bushels. In liquid measure 1 quarter = 2 pipes = 4 hogsheads = 6 ahms or ohms = 24 ankers = 240 gallons, very nearly. A fitder of wine = 930 pots = 237- gallons. 1 shippound = 20 lispounds = 320 pounds = 352.8 lb. Av. MONEY.-I six dollar = 6 mares = 96 skillings. The rigsbank dollar = $0.52. But the money generally used in commercial transactions is bank money, which is at a heavy discount. The old rix dollar, or species daler, is worth $1.05. 35. CUBA.-See HAVANA. 36. DANTZIC.-1 ell = 2 Dantzicfeet = 22.6 Eng. inches. The Rhineland or Prussian foot = 12.356 Eng. in. The Prussian or Berlin ell = 25/- Prussian inches. The Prussian mile = 4.8 English miles. The last of grain = 3- znalters = 60 scheffels 2= 40 viertels = 960 metzen = 91 bushels. The last of beer = 2 fudelr 4 both = 8 hogsheads = 12 ahms = 48 ankers = 240 quarts = 620gals. 1 pipe 2 ahms. The ahm of wine = 39z gallons. 1 lispound= 16 pounds = 264 ounces = 8448 loths = — 17 lb. Av. 100 lb. Dantzic = 103.3 lb. Av. 1 shippound =3 centnhrs= 330 pounds. MONEY. — thaler or dollar =- 30 silver groschen -- 360 pfennings -$0.69. Accounts are still sometimes kept in guldens, guilders, or flormizs. 1 s'ix dollar = 3 florins = 90 g7roschen -- 270 schillings = 1620 pfennings = $0.49. 37. DENMARsK.-See COPENHAGEN, ELSINEUR. 38. DJIDDA.-See ALEXANDRIA. 39. EAST INDIES.-See BOMBAY, CALCUTTA, MADRAS, TATTA. 40. EGYPT. -See ALEXANDRIA.

Page  94 94 MEASURES, WEIGHTS, ETC. [ART. IV. 41. ELSINEUR.-The same as COPENHAGEN, except that the rix dollar is divided into 4 orts instead of 6 mares. 42. GALACz.-See CONSTANTINOPLE. 43. GENOA.-The palmo=9.725 inches. The canna piccola, used by tradesmen and manufacturers,-=9 palmi; the canna grossa, used by merchants, =12 palmi; the Custom-House canna - 10 palmni. The braccio = 21 palmi. In dry measure 1 minae8 quarte- 96 gombette -3- bushels nearly. Salt is sold by the nzondino of 8 mine. In liquid measure 1 mezzarola — 2 barilla200 pinte- 391 gallons. The barilla of oil.-17 gallons. The pound is of two sorts; thepeso sottile=48911gr. Troy, for weighing gold and silver, and commodities of small bulk,-and the peso grosso, for weighing bulky articles. The cantaro of 100 lb. peso grosso 7= 76 lb. 14oz. Avoir. MONEY.-1 lira Italiana = 100 centesimi= — 1 French franc. The lira was formerly divided into 20 soldi, and the soldo into 12 denari. Sales of merchandise continue to be made, for the most part, in the old currency. 6 old lire di banco =5 new lire very nearly. 44. GERMANY.-For weights and measures, see BREMEN, DANTZIC, and TRIESTE. The German short mile= 6859 yards; the German long mile=10126yd.; the Hanover mile 11559yd.; the Hessian mile = 10547yd.; the mile of Saxony= 9905yd. MONEY.-The gold ducat==$2.24. The forin or guzilder=60 kreutzers =240 pfennings. By the Custom-House valuation, the florin of Nuremburg, Frankfort, and the Southern States of Gernmany — $0. 40; the florin of Augsburg, Austria, and Bohemia, = $0.485; the florin of St. Gall — = $0.4036. The thaler or rix dollar = 30 groschen = 360 pfennings. The thaler of Saxony, Prussia, and the Northern States of Germany = $0.69. The guilder of Surinam, Curagoa, Essequibo, and Demarara, is divided into 20 stivers of 12 pfennings each. Its value is fluctuating, but does not differ materially from that of the German florin. 45. GIBRALTAR.-Weights and measures same as in England, except the arroba-=251b. Av., and the fanega for grain, =1abu. Wine is sold by the gallon, 100 of which=109.4 U. S. gallons. MONEY.-Accounts are kept in current dollars, (pesos,) divided into 8 reals of 16 quaros each. 12 reals currency make a cob, or

Page  95 ~ 15.] MISCELLANEOUS TABLE. 95 hard dollar, by which goods are bought and sold; and 3 of these reals = 5 Spanish reals vellon. 46. GREECE.-See PATRAS. 47. HAMBURInG. —The Hamburgh foot=11.289 inches. The Rhindland foot, used by engineers and land surveyors, =-12.36in. The Brabant ell, used in the measurement of piece goods, =27.585 in. A ton of shipping — 40c. ft. DRY MEASURES.-1 stock-=1l last = 3 wisps = 30 scheffels = 90 fass = 180 himtems -- 720 spints = = 134.4bu. LIQUID MEAsuRs S. — lfuder = 6 ahms = 24 ankers or 30 eimers = 120 viertels = 240 stubgens = 480 kanens = 960 quartiers = 1920 oessels —229gal. U. S. A fass of wine= 4 oxhofts -=6 tierces. An oxhoft or hogshead of French wine=-62 to 64 stubgens; an oxhoft of brandy = 60 stubgens. A pipe of Spanish wine = 96 to 100- stubgens. A tun of beer = 48 stubgens. A pipe of oil = 820 lb. Whale oil is sold by the barrel of 6 steckan=32 gallons U. S. WEIsHTS. —1 shippound=21- centners =20 lispounds =280 pounds =4480 ounces = 8960 loths; 100 Hamburgh pounds = 106.8 lb. Av. In estimating the carriage of goods, the shippound is reckoned at 3801bs. In things sold by number, a gross thousand=1200; a ring=2 gross hundred=240; a smnall thousand- 1000; a shock -3 steigs = 60; a gr'oss =12 dozen. MONEY. —1 marc = 16 sols or schillings lubsa = 192 pfennings lubs. Accounts are also kept, particularly in exchanges, in pounds, schillings, and pence, or grotes Flemish. The pound consists of 2~ crowns, 33 thalers, 71 snarcs, 20 schillings Flem. or 240 yrotes Flem. The moneys in circulation are divided into banco and current money. The former consists of sums credited by the bank to those who have deposited bullion or specie, and is worth an agio or premium over current money. This agio is usually about 23 per cent., but is constantly varying. Of the coins in circulation, the rix dollar bcnco = about $1, and the rix dollar current = -$0.80, are the most common. The Hamburgh gold ducat= $2.07. The mare banco = $0.35, according to the Custom-House valuation. 48. HAVANA.-108 varas = 100 yards. 1 fanega = 3 bushels nearly, or 100 lb. Spanish. 1 quintal =4 arrobas = 101 lb. Av. An arroba of wine or spirits = 4.1 U. S. gal. nearly. MONEY. —1 dollar = 8 reals plate = 20 reals vellon = $1.00. A doubloon =- 17. 49. HAYTI.-See PORT AU PRINCE. a Lubs is a contraction for money of Lubeck.

Page  96 96 MEASURES, WEIGHITS) ETC. [ART. IV. 50. HOLLAND. -See AmSTERDtAM. 51. JAPAN.-See NANGASACKI. 52. JAVA. -See BATAVIA. 53. KONIGSBERG.-See DANTZIC. 54. LAGUAYRA. —Weights and measures the same as in SPAIN, with the exception of the British Imperial gallon. MONEY. —The currency consists of silver money, called nacuquena. l dollar = 8 reals =$0.75. The money is very unequal in weight and purity. 55. LEGHORN.-1 braccio=20 soldi =60 quattrini = 240 denari -=22.98 inches. The canna- 4 bracci. The Tuscan mile=- 1808 yards. 1 barile = 20 fiaschi = 40 boccali = 80 7mezzette =12 U. S. gal. The barile of oil =16fiaschi=8.83 U. S. gal.; it weighs about 16 lb. Av. A large jar of oil contains 30 gallons; a small one 15; and a box with 30 bottles contains 4gal. Corn is sold by the sack or sacco =2.0739bu. U. S. The pound is divided into 12 ounces, 96 drachms, 288 denari,and 6912 grani, and is equal to 5240gr. Troy. The quintsl or centinajo = 100p2ountds= 74.8841b. Av.; but in mercantile transactions, on account of tares and other allowances, it is usual to estimate 100 lb. of Leghorn = 77 lb. Av. The cantaro is generally 150 lb., but a cantaro of sugar = 151 lb.; of oil =88 lb.; of brandy = 120 lb.; of stock fish, and some other articles = 160 lb. The rottolo = 3 lb. MONEY,.-Accounts are principally kept in pezze di otto reali, (or dollars of 8 reals,) the pezza being divided into 20 soldi or 240 denari. The lirca is another money of account, chiefly used in inferior transactions, and subdivided like the pezza. 1 pezza = 5lire. The moneys of Leghorn have two values, moneta bzuona, or the effective money of the place, and suoneta lunga, which is worth 12 more than nmoleta buona. The pezza of account = $0.87; the pezza lunga-= $0.9076. The Tuscan scutdo or crown $1.05. It was subdivided like the pezza. 56. LIMA.-See CADIZ. 57. LISBON.-1 branga=2 va~ras= 31 covados or cubits=10 poales = 86.4 inches. The palnme — 2 pes or feet. The legoa= — 6760yd. In liquid measure, 1 tonnelada 2 pipes = 52 alnudes; 1 baril = 18 alnudes; 1 alinude=2 potes 12 canadas =48 quartellos = 4.37gal. U. S. The value of the almude in different parts of Portugal varies from 41 to 61 gallons. The principal dry meas

Page  97 ~ 15.] MISCELLANEOUS TABLE. 97 ure is the moyo = 15 fanJzeas 60 alquieres =240 qartos = 480 selemnis= 23.03bu. U. S. The alquidre varies in different parts of Portugal from 3.07 to 3 87 dry gallons. 1 quintal = 4 arrobas = 88 arratels or pounds 1 76 mares - 1408 ounces = 89.047 lb. Av. MONEy.- 1 rnilree = 1000 rees = $1.12 in silver. The milree of the Azores = $0.83~; the milree of Madeira = $1.00; the milree of Brazil is fluctuating in value. In the notation of accounts, the milrees are separated from the rees by a crossed cypher (D), and the milrees from the millions by a colon (:), thus, Rs. 2:700~500 -2700 milrees and 500 rees. The crusado of exchange or old crusado = 400 rees; the new crusado = 480 rees; the testoon = 100 rees; the vintem = 20 rees. 58. MADRAs.-The garce, corn measure, = 80 parahs = 400 marcals = 137bu. U. S. nearly. The mnarcal = 8 puddies = 64 olluckcs. When grain is sold by weight, the garce=9256~lbs. Goods are weighed by the candy of 20 maunds = 500 lb. Av. 1 maund= 8 vis = 40 seers = 320 pollams = 3200 pagodas. These are the weights adopted by the English, but those used in the Jaghire, (the territory round Madras belonging to the Company,) and in most other parts of the Coromandel coast, are called the Malabar weights. They are the gursay or garce =20 baruays or candies = 400 manLunzghs or mnands = 9645- lb. Av. The maund= 8 visay or vis = 320 pollams = 3200 varahuns. MIONEY.-The East India Company and European merchants keep their accounts at 12 fanams the rupee. 1 pagoda= 3- ru2pees =42 fanams = 3360 cash. The star (or current) pagoda = $1.84. 59. MALABnAR.-See MADRAS. 60. MALACCA.- See SINGAPORE. 61. MAiLAGA.-The arroba or cantara = 4.19gal. U. S. The regular pipe of Malaga wine contains 35 arrobas, but is reckoned only at 34; a bota of Pedro Ximenes wines=53~ arrobas; a bota of oil is 43, and a pipe 35 arrobas; a carga of raisins is 2 baskets, or 7 arrobas; a cask contains as much, though only called 4 arrobas. For other measures, weights, and coins, see CADIZ. Accounts are kept in reals of 34 maravedis vellon. 62. MALTA.-1 canna= 8palmi=22-yd.a The Maltese foot = 11 inches. The caffiso or measure for oil= 5gal. U. S. The salma of corn, stricken measure,=8.22bu. U. S.; heaped measure is a This is the allowance usually made by merchants, in converting Malta into English measure. In reality 1 canna = 8.19 inches; 1 cantaro re= 1741 lb. Av. 7

Page  98 98 MEASURES, WEIGHITS, ETC. [ART. IV. reckoned 16 per cent. more. The cantaro=100 r ottoli or pounds = 3000 oncie= 175 lb. Av.a MONEY.-In 1825 British silver money was introduced into Malta; the Spanish dollar being made legal tender at 4s. 4d; the Sicilian dollar at 4s. 2d.-and the Maltese scudo at is. 8d. The scudo = 12 tari = 240 grani = $0.40. 63. MANILLA.-The same as CADIz, except that weights are estimated by piastres. 16 piastres are estimated =1 Spanish pound, though they are not quite so much. 1 tale of silk=11 piastres or ounces; 1 catty=22 piastres; 1 macrc of silver = 8 piastres; 1 tale of gold = 10 piastres; 1 picul=100 catties= 1338 lb. Av. 64. MAURITIUs.S.-See PORT Louis. 65. MECKLENBURG.-See ROSTOCc. 66. MEXICO.-See CADIZ. 67. MOCHA. —The guz-=25 inches; the land covid=18in.; the long iron covidl = 27in. 1 cuda, liquid measure, = 8 znusseahs = 128 vakias = about 2gal. U. S. Grain is measured by the kellah, 40 of which = 1 tomnand= about 170 lb. Av. 1 bahar = 15 frazels = 150 maunds=400 rottoli=6000 vakias=4501b. Av. There is also a small maund of only 30 vakias; I1 Mocha bahar =164 Bombay maunds = 13 Surat maunds= 15.123 seers. MONEY.-The current coins of the country are carats and commnassees; 1 Spanish dollar = 8 Mocha dollars = 60 commassees = 420 carats. 68. MoGADORE.-The canna or cubit=21 inches. The corn measures are, for the most part, similar to those of Spain. The commercial pound is generally regulated by the weight of 20 Spanish dollars, therefore the quintal of 100 lb. = 119 lb. Av. The market pound for provisions is 50 per cent. heavier. MONEY.-1 znutkeel or ducat = 10 ounces= 40 blanlkeels =960 faice -$0.75. 69. MoLDAVIA.-See GALACz. 70. MONTEVIDEO.-For weights and measures, see CADIZe. The current coins are the Brazilian patacon and Spanish dollar. 1 hard dollar -=1 current dollars = 960 centesinmos or cents = $1.00. 1 real = 100 centesimos. 71. MoRocco.-See MOGADORE. a See Note on Page 97.

Page  99 ~ 15.] MIISCELLANEOUS TABLE. 99 72. MuscAT.-1 maund= 24 cuchas = 8, lb. AV. MONEY.-1 mamoody = 20 goz = about $0.05. The coins in circulation are generally sold by weight. 73. NANGASsACiI.-The inc is about 4 Chinese cubits, or 6,ft. 21 Japanese leagues are computed to be about 1 Dutch league. The revenues are estimated by two measures of rice, the man and ko~f; the former contains 10000 kolfs, each 3000 bales or bags of rice. The picul=100 catties=1600 taels=16000 mace=160000 candalrines = about 130 lb. Av. It is, however, generally estimated at 133k lb. MoNEY.-Accounts are kept in taels, mace, and candarines. The Dutch reckon the tael at 3- florins, or $1.40. The coins in circulation, are the old and new itjib, and cobangs or copangs, of gold,-the?andioyin, itaganne, and kodama, of silver,-and the seni, of copper, brass, and iron. \Most of them are without any determined value, and are therefore always weighed by the merchants. The schluit is a silver piece, lloz. fine, weighing 4oz. 18dwt. 16gr. Troy. 74. NPrLEs. —1 cannac = 8 palmi-= 96 onzie = 6ft. 1lin. 1 salman of oil= 16 staje =256 quarti-= 1536 mismette. At Naples, the salma = 42gal.. U.S.; at Gallipoli it is from 3 to 4 per cent. less; at Bari it is a little larger. The carro of wine = 2 hotti or pipes =24 barili = 1440 caraffe = 264gal. U. S. The carro of corn = 36 tomoli- 52.2bu. U. S. The cantaro grosso = 100 rottoli = 1961 lb. Av. The cantaro piccolo = 106 lb. Av. MONEY. —1 ducato di regno = 10 carlini = 100 graie $0.80. The scudo of 12 carlini = $0.95. 75. NoRWAY.-See CHRISTIANIA. 76. PALEnRIo. —The yard or canna 8 palmi=3-yd. U.S. The tonna of liquids — 2 caffi = 4 barili= 8 quartare = 160 gquartucci - 93gal. U. S. The salma grossa = 9.48bu.; the almea gelnale = 7.62bu. The cantaro grosso = 100 rottoli grossi of 33 onzie, or 110 rottoli sottili of 30 onzie. The cantaro sottile = 100 sottoli sottili, or 250 lb. of 12 onzie. The rottolo grosso == 1.93 lb. Av.; the rottolo sottile = 1.75 lb. Av.; 100 Sicilian pounds of 12oz. = 701b. Av. MONEY.-1i ducato = 10 piccioli = 100 bajocchi = $0.80. Accounts are generally kept in oncie, tari, and grani. I oncia= 30 tao _ 600 grani = 3 ducati. 77. PAPAL STATEs.-See CIVITA VECCHIA.

Page  100 P100 MEASURES, WEIGHTS, ETC. [ART. IV. 78. PATRAS. —The long pic, for measuring linens and woollens, =27in. The short pic, for measuring silks, =25in. The staro of corn =2-bu. U. S. The quintal is divided into 44 okes, or 132 l1b. 100 lb. of Patras = 881b. Av. Silk weight is - heavier. MONEY.-1 phcenix or drachm= 100 lepta. The phoenix is a silver coin, which should contain 9U of pure metal, and be worth about 16 cents. The lepton is a copper coin. The coinage has been greatly debased. 79. PERSIA.-See BUSHIRE. 80. PERU.-See LIMIA. 81. PETERSBURG.-1 sashen or fathom- 3 arsheens -=48 wershok -7ft. 100 Russian feet= — 1141 Eng. ft. The verst, or Russian mile = 500 sashen = 5fur. 12r. The Polish short mile = 6075yd.; the Polish long mile = 8101yd. The English inch and foot are used throughout Russia, chiefly, however, in measuring timber. In liquid measure, 1 sorokovy = 40 wedros = 320 krashkas = 3520 tshkcyss== 130gal. U. S. 1 jpipe=2 oxhofts=12 ankers= 36 wedros = 480 bottles. 1 chetwert of corn =- 2 osmins = 4 pajocks = 8 chetwivericks = 64 garnitz = 5.952bu. U. S. 1 berkovitz = 10 poods = 400 pounds = 12800 loths = 38400 zolotnicks = 360 lb. Av.a MONE Y.-Accounts are kept in bank roubles'of 100 copecks. The silver rouble = $0.75, and was declared, by a ukase issued in 1829, to be worth 360 copecks, but the value of the paper rouble fluctuates with the exchange. At the Custom-House, it is estimated at $0.214. 82. PHILIPPINE ISLANDs.-See MANILLA. 83. PoRT AU PRINCE.-The measures are the same as in FRANCE. They are divided as in Avoirdupois and Apothecaries' weights, but are about 8 per cent. heavier. The value of the dollar is about $0.33, but is constantly fluctuating. 84. PORT LoIs. —The measures and weights are those of FRANCE,. previous to the Revolution. 100 lb. Fr. =-108 lb. Eng.; 16 Fr. ft. - 15 Eng. ft. The commercial velte = 2 gallons. MONEY. —Government accounts are kept in sterling money, the franc being received for 10d., and the Spanish dollar for 4s. 4d. a The pood is reckoned by merchants at 36 lb. 100 lb. Russian = 90.26 lb. Av., according to Dr. Kelly, or 90.19 lb. according to Nelkenbrecher.

Page  101 ~ 15.] MISCELLANEOUS TABLE. 101 Merchants keep their accounts in dollars and cents, or in dollars livres, and sous. 85. PORTO RIco. —See HAVANA. 86. PORTUGAL.-See LIsBON. 87. PRUssIA. -See DANTZIC. 88. QUEBEC.-The Paris foot is used for all measures of lands granted previous to the conquest, and all measures of length, unless a contrary agreement is made. The English foot, fox measuring lands granted since the conquest, and whenever specially agreed upon. The English yard for cloth measure. The English ell of 5qr. when specially agreed upon. The Canada 7minot -11bu. for dry measure, except when it is specially agreed that the Winchester bushel shall be used. The Eng. imp. gallon is used for liquids. Weights and currency, as in England. The pound = $4.00. 89. RANGoo N.-1 ten or basket = 4 saits = 8 sarots = 16 pyis= 64 sale's= 128 lanze's = 256 lamyets. A ten of clean rice ought to weigh 16 vis or 58.4 lb. Av. 1 pailctha or vis = 100 cyats or licals - 400 moat'hs = 800 auns = 1600 bais = 6400 las-re mevds = 12800 snmall rwe's - 3.65 lb. Av. MONEY.-Lead is used for small payments; gold and silver, (principally the latter,) for larger ones. There are no coins, but the metal must be weighed, and, very generally, assayed at every payment. Every new assay of silver costs the owner 21 per cent. 90. RIGA. —1 clafter = 3 ells = 6 feet= 64.74in. 1 fuder, for liquids, = 6.hms = 24 anlkers = 120 quarts = 720 stoofs = 248gal. U. S. The loof for grain= 1.9375bu. A last= 48 loofs of wheat, barley, or linseed,-45 loofs of rye,-or 60 loofs of oats, malt, or beans. 1 shippound = 20 lispozunds =400 poulnds= 800 nzmacs 12800 loths c 368.68 lb. Av. MhoNFY.-See PETERSBURG. The current rix dollar of Riga-= 90 groschen = $0.69. 91. ROSTOCI.-1 ell-=2 feet = 22.76 Eng. in. The last=96 schetfels = 116 imp. bu. of oats, or 104 imp. bu. of other grain. The commercial weights are the same as those of Hamburg. There are other weights, 5 per cent. heavier than these, which are principally used in the trade with Russia. iMONEY.-The cix dollar, new,- 32 schillings, and contains 199.lgr. pure silver.

Page  102 102 MEASURES, WEIGHTS; ETC. [ART. IV. 92. RUSSIA. —See PETERSBURG and RIGA. 93. SALONICA.-Weights and measures same as at Constantinople, except the kisloz or killow, which = 3.78 kisloz of Smyrna. MoNEY.-Accounts are kept in piastres of 40 paras, or 120 aspers. The coins are those of Constantinople. 94. SARDINIA.-See CAGLIARI and GENOA. 95. SIAM. —See BANGKOK. 96. SICILY.-See PALERMO. 97. SINDE.-See TATTA. 98. SINGAPORE.-English weights and measures are frequently used in reference to European commodities. Piece goods and many other articles, are sold by the corge or score. A coyan of rice or salt = 40 piculs. Nearly everything is sold by weight, as in China. The picul = 100 catties = 1331 lb. Av. Gold dust is sold by a Malay weight, called the bungkal, = 832gr. Troy. Bengal rice, wheat, and pulses, are sold by the bag, containing 2 Bengal maunds, or 1644 lb. Av. MoNEY. —Merchants' accounts are kept in Spanish dollars, divided into 100 parts, represented either by Dutch doits or by English copper coins of the same value. 99. SMnYRNA.-See CONSTANTINOPLE. The kisloz =-1.456 bushels. 100. SOUTH AFRIcA.-See CAPE TOWN. 101. SPAIN.-See ALICANT, BARCELONA, CADIZ, GIBRALTAR, and MALAGA. Vellon is the old copper coin of Castile, and is of but half the value of the plate or silver currency. 102. STOCKlHOLM.-The rod = 22 fathoms = 8 ells or lnas = 16 feet =15.58 Eng. ft. 1 pipe = 2 oxhofts = 3 cahms = 6 einers = 12 ankers = 180 kannor = 360 stup = 124-1-gal. U. S. In corn measure 1 tun or barrel = 2 spann = 8 quarts = 32 kappor = 41bu. U. S. The victuali or commercial weights are 1 skippund = 20 lispunds = 400 punds = 375 lb. Av. The iron weights are four-fifths of the victuali. 1 iron skipp2und = 20 marlk punds = 400 marks = 300 lb. Av. MONEY. —For many years, there were no coins except copper in circulation, but both silver and gold are now coined. 1 rix dollar - 48 skillings = 576 rundstycks or'ore. The banco currency is worth 30 per cent. of the silver currency. A rix dollar banco is worth about $0.40. A silver rix dollar = 1.06 - 2 dollars 3 2 skillings banco.

Page  103 ~15.] MISCELLANEOUS TABLE. 103 103. SWEDEN.-See STOCKHOLMI. 104. TATTA. —1 gUZ = 16 garces = 32 inches. But 1 guz of cloth, at Tatta = 34 inches. 1 carval of wheat = 60 cossas = 240 twiiers = 960 pzuttoes = 22 Pucca maunds, or 21 Bombay parahs. The gross weights are 1 mzaund= 40 seers = 640 aenas = 2560 pice = 741b. 5oz. 7dr. Av. The small weights, 1 tolah = 12 massas = 72 r'uttees = 576 hubbahs = 1728 moons = about 6 dwt. Troy. MoNEY. —1 rupee = 50 carivals = 600 pice = 28800 cowries-= 50.55. 105. TRIESTE. —The Bohemian mile = 10137 Eng. yd.; the Hungarian mile = 9113 Eng. yd. The ell for woollens = 26.6in.; the ell for silk = 25.2in. The orna or eimer, liquid measure, = 40 boccali = about 15gal. U. S. The barile = 1738gal. U. S. The orna of oil = 5b ccaigsi = 17gal. U. S. The principal dry measure is the stajo or staro = 2.34bu. U. S. Sometimes the Vienna metzen is used, = 2 polonicks = 1.723bu. The commercial pound = 4 quarters — 16 ounces = 32 loths'- 8639gr. Troy. MoNFY. — Contracts are usually in silver; gold coins, not being legal tender, pass only as merchandise. Mercantile accounts are usually kept in convention money, so called from an agreement made by some of the German princes in 1763. The current coins are dollars, lhalf-dollars or florins, and zwanzigers, or pieces of 20 kreutzers. Ten dollars are coined out of the Cologne marc = 3608gr. Troy, of pure silver. The florin = 60 klreutzers = 240 pfenrinrgs = $0.484. 106. TuNIS.-The pie for woollens = 26.5in.; the silk pic = 24.8in.; the linen pic =18.6in. The principal oil measure is the metal or mettar = about 54-gal., but it is of different dimensions in different parts of the country. The wine measure is the millerolle of Marseilles=6a- mitres=.14.1 imp. gal. The principal corn measure is the cafiz = 16 whibas = 192 sahas = 14~ imp. bu. The cantaro =100 rottoli or pounds = 111.05 lb. Av. Gold, silver, and pearls are weighed by the ounce of 8 meticols; 16 of these ounces make the Tunis pound= 7773.5gr. Troy. MONEY. —1 piastre = 16 carobas = 52 aspers -- 624 burbine $0.24. The piastres coined since 1828 are worth but {0.125. 107. TURKEY.-See CONSTANTINOPLE and SALONICA. 108. TuscANY.-See LEGHORN. 109 URUGUAY.-See MONTEVIDEO.

Page  104 104 MIEASURES) W-EIGHITS, ETC. [ART. IV. 110. VALPARAISO.-The vara = 33. 384in. The fanega, or principal corn measure, contains 3439 c. in. The quintal = 4 arrobas = 100 pounds= 101.44 1b. Av. See CADIZ. 111. VENEZUELA.-See LAGUAYRA. 112. VENICE.-The woollen braccio=26.6in.; the silk braccio =24.8in.; the Venice foot = 13.68in. The botta=2 migliaje-5 bigonzi== 80 miri. The miiro of oil = 4.028ga1. U. S., or 25 lb. peso grosso. The anfora of wine =4 bigonzi = 8 mastelli -48 secchii 192 bozze=768 qulartuzzi = 137ga1. U. S. The moggio, for corn, =4 staje=16 quarte = 64 quartaroli = 9.08bu. U. S. 100 lb. peso grosso = 105.186 lb. Av.; 100 lb. peso sottile = 66.428 lb. Av. The liba Italiana is sometimes used, and is equivalent to the French kilogramme. IMoNEa.-1 lira Italiana =100 centesimi= 1000 szillesini. The lira is supposed to be of the same value as the franc, but the lire actually in circulation, are worth only about $0.09. At the Custom-HIouse, the lira is estimated at $0.16. 16. ANCIENT MEASURES, WE1,JIGHTS, AND COINS.a 1. SCRIPTURE LONGe MEASURE.-1 schLcnus =10 Arabian poles = 131 Ezelciel's seeds =20 fathoms = 80 cubits= 160 spans =480 palmls =1920 digits= 46.2 yards. A day's journey=8 parasangs = 94 eastern miles =48 Sabbath-day's journeys = 240 stadia = 96000 clrbits 33.264 miles. LIQUID MEASURE.-1 coroZ or chomer= 10 bath or epha=30 seah=60 lhn=180 cab=720 log=960 caph=103.35 gallons. DRY MEASURE.-1 coron0 or chomeer=2 latech= 10 epsha = 30 seah =100 gaomor = 180 cab = 3600 gachal =10.9616 bushels. WEIGHTS.-1 talent = 50 naneh8 3000 shekcel = 114 lb. Troy, nearly. MoNEY.-i1 talent = 60 maneh = 3000 shekels 6000 bekcah = 60000 gerahz=~342 3s. 9dC. A talent of gold==5475. The solidcs azureas or sextula=12s. Od. 2qr. The siclss aureus== 1 16s. 6d. 2. GRECIAN LONG MIEASURE.-1 milions or mile =8 stadios, dselos, or furlongs = 800 orgya or paces = 3200 pechys or larger cubits = 3840 pygon=4266* pygme or cubits =4800 pous or feet 6400 spitame = 6981-f9 orthodoron -7680 dichas -19200 doron or dochme = 76800 dactylos or digits = 4835 feet. LIQUID MEASURE.1 metretes = 12 chous = 72 xestes = 144 cotyle= 576 oa.ybaphlon = 864 a MIcCulloch, Gregory, Brande, Lavoisne, Enc. Amer.

Page  105 ~ 16.] MISCELLANEOUS TABLE. 105 cyathos - 1728 conche = 3456 mnystron = 4320 cheme = 8640 cochliarion = 10.335 gallons. DRY MEASURE. —1 1edimnos = 48 chcenix 72 xestes = 144 cotyle = 576 oxybaphon = 864 cyathos = 8640 cochliarion — 1.0906 bushels. WEIGHTS.-1 talent - 60 mince-_ 6000 drachma -- 36000 obolos = — 56 lb. Av. MONEY. —1 tetradrachnma or stater = 2 lidrachma - 4 drachnma - 6 tetrobolon - 12 diobolonz 24 obolos == 48 hemiobolos -- 96 dichalcos = 192 chalcos - 1344 lepton=2s. 7d. The pentadrachma-=5 draclaza. The drachmza and its multiples, were of silver,-the other coins, mostly of brass. The stater aureuss =25 Attic drachmas of silver. The stater Cyzicenus, stater Philippicus, and stater Alexandrinus = 28 silver drachmas. The stater Daricus and stater Crcesius = 50 silver drachmuas. 3. ROMrAN LONG MEASURE. —1 milliare = 8 stadia = 1000 passus 2000 graduls = 33333 cubits = 4000 pallipedes - 5000 pedes or feet - 20000 palhmi nziores = 60000 uncice = 80000 digiti transversi =4835 feet. SQUARE MEASURE.-l as = 1i- deunx- c1 dextans - 1 dodrans- 1=- bes = 1 septunx =2 semis or acts nmajor —=2' quincunx 3 triens -4 quadrans- =6 sextans or actuzs eininus z= 8 clima or sescucia- = 12 uncia -— 28800sq. feet. LIQUID MEASURE.1 culeus =20 anzphorx =- 40 urznce = 160 congii = 960 sextarii = 1920 hemince 3840 quartarii = 7680 acetabeula = 11520 cyathi= 46080 liguzlce = 143.4258 gallons. DRY MEASURE.-1 nzodius = 2 seminzodlii = 16 sextarii = 32 hemeince = 128 acetabula- = 192 cyathi = 768 liyglce =1.0141 pecks. WErIGHT. — libra = 12 uncico = 36 duellcea = 48 sicilici = 72 sextulce = 5246gr. Troy. MONEY.-I denarius =-2 quinarii or victoriati = 4 s)estertii = 10 libellce or asses = 20 sembellce =40 teruncii = 7d. 3qr. All above the as, (and sometimes the as also,) were of silver, the rest of brass. The gold coin was the auereus, which generally weighed double the denarius, and was about equivalent to 18 denarii in value. 4. VARIous ANCIENT MEASURES. —The Olymzpic stadium =202 yards; the Alexandrian stadium = llOyd.; the Egyptiasn stadiuml = 245yd.; the stadizum of Aristotle — 115yd.; the Persian parasang 6440yd.; the Egyptian schcene = 11120yd.; the ancient Spyanish mile = 1374yd.; the ancient British mile = 148yd.; the league of Gaul = 2446yd.; the rasta of Germany = 4705yd. [These measures are given principally on the authority of Lavoisne.]

Page  106 106 MEASUI.XES, WEIGHTS, ETC. [ART. IV. ~ 1 7. EXAMPLES FOR THE PUPIL 1. The great bell of Moscow was cast in 1653, during the reign of the Empress Anne. Its weight is estimated at 198tons 2cwt. lqr.a To what would this be equivalent in Troy weight? 2. In the first year after the discovery of gold in California, the amount collected is supposed to have been about $5000000. Estimating its average value at $16 per oz. Troy, what would be the weight of the whole in Avoirdupois? 3. Find the weight in Troy grains of the French twentyfranc piece of gold, and also of the silver five-franc piece. 4. The distance from Boston to Liverpoolb is about 2883 statute miles. At the average rate of 8 knots an hour, in what time would a vessel sail from one of these places to the other? 5-10. On the Cathedral of Paris a bell was placed in 1680, which weighed 340cwt.; a bell was cast in Vienna, in 1711, weighing 354cwt.; in Olmiitz is one of 358cwt.; the Susanne, a fine-toned bell at Erfurt, which has a large proportion of silver in its composition, weighs 275cwt., and has a clapper weighing llcwt.a Reduce each of these weights to kilogrammes, and find the amount of the whole. Ans. for the amount, 67966.308kil. 11-15. Great Tom, of Christ Church, Oxford, weighs 17000 lb.; Great Tom of Lincoln, 9894 lb.; the bell of St. Paul's, London, 8400 lb.; a bell at Nankin, China, is said to weigh 50000 lb.; and seven at Pekin, 120000 lb. each.a Required the weight of each in Amsterdam ponds. 16-25. Reduce the span and height of each of the following bridges,c to the measure required in the right-hand columln. a Enc. Amer. b Cunard Steamers. c Brande's Enc.

Page  107 ~ 17.] MISCELLANEOUS EXAMPLES. 107 Wi lest Arch. Name of Bridge. Material River.'lace. - Date. AIeasure. Spa1o. tlei att. Colosss..... xWood Scahuylkill lPhiiaclelphia 340 ft.1 20 ft. 1813 Tur'h piks. Piscataqua.. e' Piscataqua Portsmoutlh 250 "' 27 t3 / 1794 SSedish ft. Southwark... Iron iThames iLondon 240'' 24 181SI Ans.palms. Sunderland...' iWear Sunderlandc 240,' 30' 1796 Prussian ft. Bamberr..... Wood;Regnitz Germany 208 1 17k * 1809 Bremen ft;. Trenton...... "'DelAware INew Jersey [200 "32 " 1804 Venice ft. Vielle Brioude Stone!Allier Brioude'183Y," 70' " 1454 Fr. metres. Ulm........ " Danube Ulm 181'" e22y " 1 806 Dantzic ft. Waterloo ".... " Thames London 120 " 32 ", 1816 Script. cub. Blackfriars... Thames London 1100 1"41 " 1771 Russian ft. 26. Reduce 2 tons 13cwt. 3qr. 171b. to Canton weight. 27. Reduce 159 taels 3 mace 5 candarines 7 cash to Troy ounces. Ans. 193.663oz. 28. In some places milk is sold both by beer measure and by wine measure. To how many wine gallons would the difference amount in a year, in a family that takes 2qt. per day? 29-38. At $4.84 per pound sterling, what is the cost per c. ft. in English money, of each of the following public buildings? U S. Pulblic Buildings. Contents.b Cost.b Capitol............ 4147400 c. ft. $2250000 Treatsury Building...... 1944740 " 648743 Patent Office........ 1466660 " 417550 Generlt Post Office..... 1071252 " 452765 Girard College........ 2.545485 " 1427800 New York Customn-House.. 906000 " 960000 Boston Custom-House.... 730000 " 776000 Philadelphia Custom-House. 530613 " 257452 Trinity Church, New York. 821070 " 338000 Smithsonian Institution... 1545000 " 215000 39. If you had a pair of scales, but no weights, how would you weigh 31b. 7oz. by using water? 40. How many cents weigh a pound Avoirdupois? How many eagles? How many half-dollars? a This was the largest single span in the world. The bridge was destroyed by an incendiary, September 1, 1838, and a wire suspension bridge has sinace been erected in its place. b R. Dale Owen.

Page  108 108 MEASURES, WEIGHITS, ETC. [ART. IV. 41. At $0.186 per franc, what would be the value in francs, of a diamond weighing 15 carats, at $36 for the first carat? Ats. 43548fr. 40c.a 42. What is the value in Federal Money of the French twenty-franc piece, provided the gold is of the present mint standard, and of full weight? 43-44. The largest known diamond was found in Golconda, in 1550, and is in the possession of the Great Mogul. It is half the size of a hen's egg, and is said to weigh 900 carats.? In the usual mode of estimating diamonds, what would be its value, at $9 for the first carat? What is its weight in ounces Avoirdupois? 45-50. The Greeks reckoned by the era of the Olympiad, which began at the summer solstice, 776 B. C. The Roman era commenced with the building of the city, April 24, B. C. 753. The Julian era dates from the reformation of the calendar by Julius Cwssar, B. C. 45. The Mohammedan era dates from the Hegira, July 16, A. D. 622. The era of Sulwanah, used in a great part of India, corresponds with A. D. 78. The era of Yezdegird, used in Persia, began June 16th, A. D. 632.' Find the time that elapsed between each of these dates and the commencement of the Jewish era, 3760 B. C. 51-67. Find the value of 100 lb. Av. in the commercial weights of each of the following places: Alexandria, Amsterdam, Bbmbay, Bremen, Cadiz, Calcutta, Canton, Civita Vecchia, Constantinople, Hamburg, Mtadras, Naples, Patras, Petersburg, Stockholm, Trieste, Venice. 68. Newly burned charcoal will absorb 90 times its bull of ammonia, or 35 times its bulk of carbonic acid.c Hovw many cubic inches of each of these gases would be absorbed by 17 cubic feet of new charcoal? a No friaction of a franc is counted, less than 5 centimes. For the method of estimating the value of diamonds, see ~ 12, p. 73. bBrande. Carpenter.

Page  109 ~ 17.]:MISCELLAN'EOUS EXAMPLES. 109 69. A spider's thread is in some instances no more than 1 of an inch in diameter. Each thread is formed of from 4 to 6 filaments, and each filament of not less than 1000 fibrils.a How many fibrils would occupy a foot in breadth, allowing 1100 to a filament, 5 filaments to a thread, and 3000 threads to an inch? 70. A large sunflower has been observed to lose 1 lb. 4oz. during 24 hours by evaporation, and a cabbage lost 1 lb. 3oz. during the same time.b At this rate, in what length of time would the evaporation from each plant amount to 1cwt.? Ans. Sunflower, 89d. 14.~h. Cabbage, 94d. 7~9-h. 71. Find the value of each of the French weights and measures, in the weights and measures of the United States. 72-81. Only 4 of the 55 chemical elements a seem to be essential to the constitution of living matter, viz: Carbon, Hydrogen, Oxygen, and Nitrogen. What weight of each element is contained in lcwt. of each of the following substances, the weight of 1 part of Carbon being represented by 6, 1 part of Hydrogen by 1, 1 part of Oxygen by 8, and 1 part of Nitrogen by 14? Gum. Sugar. St arch. Lig7in. TVax. Carbon, 414 parts. 421 428 500 806 Hydrogen, 65 " 64 63 56 114 Oxygen, 521 " 515 509 444 80 1000 " 1000 1000 1000 1000 Gluten. Gelatin. Albumen. Fibrin. Fat. Carbon, 557 parts. 483 516 520 790 Hydrogen, 78 " 80 75 72 118 Oxygen, 220 " 276 259 250 92 Nitrogen, 145 " 161 150 158 00 1000 " 1000 1000 1000 1000 a Carpenter. b Griffiths.

Page  110 110 THE FARM. [ART. V. 82. Human hair is friom 1o to - of an inch in diameter. How many hair's breadths in 49 yards, if there are 3 70 in an inch? 83. Estimating the entire population of the world at 1000000000, what would be the size of a field that would hold the whole, if each person occupied I sq. yd.? Ans. 322 sq. m. 531A. 2760 sq. yd. 84. The average weight of the lignin, or woody fibre, of an oak, has been estimated at 60 tons, 30 tons of which are carbon.a According to this estimate, how many square miles of oak forest would yield an amount of carbon equivalent to that contained in the atmosphere, supposing the oaks to stand 1 rod apart,-carbonic acid consisting of -3 carbon, and -Ta oxygen?b 85. What is the value of the labor expended in manufacturing 1 lb. Avoirdupois of watch-springs, each spring being worth $2, and weighing -1 of a grain, and the value of the iron employed, being 1 cent? V. THE FARM. S. RULES FOR DETERMINING THE WEIGHT OF LIVE CATTLE. 1. Measure in inches, the girth round the breast, just behind the shoulder-blade, and the length of the back from the tail to the fore-part of the shoulder-blade. Multiply the girth by the length and divide by 144. If the girth is less than 3ft., multiply the quotient by 11; if between 3ft. and 5ft., multiply by 16; if between 5ft. and 7ft., multiply by 23; if between 7ft. and 9ft., multiply by 31.c If the animal is lean, deduct I5 from the result. Or, a Carpenter. b See Ex. 17, Sect. 6. c Chambers's Information for the People, —Brit. Husbandry.

Page  111 ~19.] MEASUREMENT OF GRAIN. 11 2. Take the girth and length in feet. Multiply the square of the girth by the length, and multiply the product by 3.36. The result will be the answer in pounds. Either of these rules will be found useful in making approximate estimates of the weight of cattle, but experienced judges depend more upon observation than upon arbitrary rules. The live weight, multiplied by.605, gives a near approximation to the net weight. 1 9. RULES FOR MEASURING GRAIN. 1. If the grain is heaped on a floor, in the form of a cone, measure the depth and slant height in inches. )Multiply the difference of the squares of these two measurements by.0005 of the depth, for the contents in bushels. 2. If the grain is heaped against the side of a building, take i of the result obtained by the first rule. If heaped against an inner corner of a building, take 1,-and if against an outer corner, take - of the same result. 3. If the grain is in barrels or common casks, the quantity may be found with tolerable accuracy by the following rule: " To.4 of the square of the bung diameter, add i of the square of the head diameter, and i of the product of the two diameters; multiply the amount of these three quantities by the length of the cask, and divide the product by 265 for wine gallons, by 324 for ale gallons, or by 309 for half-pecks."a 4. If the grain is in a bin or crib,.8 of the number of cubic feet, will give the number of bushels. Two bushels of corn on the cob will give about 12bu. when shelled. A "barrel" of shelled corn, (in the Southern States,) is 5 bushels.b a Simplified from Dr. tutton's rule. For more accurate rules, see the article (on gauging. b Scientific American.

Page  112 112 TIIE FARMI. [ART. V. 5. If the grain is not of uniform depth or breadth, take the dimensions in several different places, and take the average length, breadth, and height, with which proceed as in Rule 4. 6. To estimate the yield per acre, measure the quantity produced on 1 scq. rod, and multiply by 160, or meastue the produce of 11 yards square, and multiply by 40. 20. WEIGHT OF GRAIN AND HAY. 1. The standard weights of grains in Great Britain, and in many parts of the United States, are as follows: Wheat, 60 lb. per bushel; Indian corn and rye, 56 lb.; barley, 48 lb.; oats, 32 lb. In Indiana, a bushel of rye is 46 lb., and a bushel of oats, 33 lb.b Beans and clover seed weigh nearly the same as wheat; castor oil beans, and timothy seed, the same as corn; potatoes, 40 lb. per bushel; buckwheat, 52 lb.; salt, 50 lb.; bran, 20 lb.; dried peaches, 24 lb.; dried apples, 22 lb.; blue grass seed, 14 lb.c 2. Ten solid yards of hay which has settled in the stack during winter, weigh about 1 ton. In stacks more than a year old, 9yd., and sometimes 8yd. make a ton. Clover takes 11 or 12yd., and sometimes when it has been stacked very dry, 13yd. to a ton. In the barn, 400 c. ft. in the bay, or 500 c. ft. on the scaffold, make about 1 ton.d If a stack of hay has a circular base, its contents may be found very nearly by the following rule: Measure the circumference at the bottom of the stack, and the height of the stack. Multiply the square of the circumference by the height, and divide the product by 40. The contents may also be measured by finding the average length, breadth, and height, and forming the continued product of the three dimensions.e a Brit. Husbandry. b Mer. Mag. C Scientific Amer. d Colman, and private information. e A product that is formed by multiplying together three or more factors, is called the " continued product" of those factors. Thus, the continued product of 3, 17, and 5, is 3X17X 5=255.

Page  113 ~ 22.] MISCELLANEOUS EXAMPLES. 113 2 1. MEASUREMENT OF LAND. 1. If the field is square, or oblong, multiply the length by the breadth, and the contents will be the area. If the dimensions are taken in paces, the area will be in square yards. Multiply the number of square yards by 4, and divide the product by 121, and you will obtain the area in square rods; or divide the number of square yards by 484, and point off one figure from the right of the quotient, and the result will be the area in acres. 2. If the field is irregular, divide it into triangles. In each triangle, multiply the length of the longest side by the shortest distance from that side to the opposite angle. Add all the results, divide their sum by 2, and the quotient will be the area. If any portion of the boundary is curved, straight lines should be drawn in such manner as to enclose the same area as the curve. 2?2. EXAMPLES FOR THE PUPIL.' 1. An ox measured 6ft. 3in. in girth, and the length of his back was 5ft. 4in. Determine his weight by each of the rules. 2-5. There are four heaps of grain; one in an open field, one against the side of a granary, one against an inner corner, and one against an outer corner of a barn. The depth of each heap is 4ft. 8in., and the slant height 6ft. 7in. Required the contents in bushels. 6. A bin 8ft. 6in. long, 3ft. 4in. wide, and 4ft. 9in. deep, is filled with ears of corn. How many bushels are there, and how many will there be when shelled? 7. How much hay in the bay of a barn, the bay being 18ft. square, and the hay 10ft. deep? a Colman, Loudon, G. B. Emerson, Johnson, Brit. Husb., and private information. 8

Page  114 114 THIE FARM. [ART. V. 8. Hfow many furrows 9 inches in width, will make an acre in a field that is 6 rods wide? In a field 10 rods wide? 8 rods wide? 40 rods wide? 9. The amount of wheat annually sown in France, on 10863959 acres, is estimated at 32491978 bushels. If, under improved cultivation,,- of this amount could be saved, and the crops could be increased 5 bushels per acre, what would be the total amount gained? 10. If, by expending $50 per acre in the cultivation of potatoes, one farmer can secure a crop of 300 bushels, worth 25 cents a bushel, and another by expending $43.60 on an acre of corn, obtains a crop worth $67, how much per cent. does the former gain, by cultivating potatoes instead of corn'?a 11. HI-ow much net weight must a pig gain per day, to pay the expense of keeping, when corn sells at $1.00 per bushel, and pork at 10 cents per lb., supposing him to consume 3qt. of corn and 2cts. worth of other food per day? 12. If 3 lb. live weight are equivalent to 2 lb. net weight, what would be the weekly profit of a pig that gains 2 lb. live weight per day, the expense of keeping and the price of pork being the same as in the last example? 13. Produce of 3I acres in 1841: 21bu. wheat, at 8s.; 44bu. oats, at 2s. 9d.; 80bu. potatoes, at is.; fodder for the support of 2 calves, at ~2 15s.; 423~ lb. butter, at is.; milk sold or given to the pigs, ~10. Required the value of the produce per acre, at $4.84 per ~. 14. It is estimated that in Great Britain and Ireland, 7085370 acres are annually sown with wheat, at the rate of 2Ibu. per acre. How many quarters of 8 bushels are required for seed? If 3 i-pk. per acre would be sufficient, a Although the crop of potatoes may be more profitable for a single year than corn, yet a rlotation of crops is necessary to avoid exhausting the land.

Page  115 ~ 22.] MISCELLANEOUS EXAMPLES. 115 how mluch is annually wasted, and what is its value, at $1.12~ per bushel? 15. Find, from the following estimate of the expense of an acre of corn, the balance in favor of the crop. Expenses.-Ploughing, 82.50; harrowing, $2.50; holeing, 50cts; 6bnu. leeched ashes, POcts; lbu. plaster, 65cts.; 1Oqt. seed, ( 10cts.; putting on ashes and plaster, and planting, $1.20; harrowing, 30cts.; weeding, $1.50; cultivating, 45cts.; second and third hoeings, $2.30; gathering and husking, $5.00; -gathering stalks, $1.50. Poceecs.-Crop.-50bu. corn @ $1.00; corn fodder, equal to 1 ton of hay, $10. 16. How many hop-vines, and how many poles, will be required to an acre, allowing two poles to each hill, and two vines to each pole, the hills being eight feet apart;- and what would be the expense of the poles at $83 per hundred?a Pcartial Als. 2720 vines. 17. Required the net returns on an acre for a six years' rotation of crops, raising corn for two years, wheat the third year, and hay for three years; the balance in favor of corn each year being $17.75, the balance in favor of wheat, $21, hay, fourth and fifth years, each $17.50, sixth year, $11.50;expenses to be deducted,-grass-seed, $1.875, and six years' compound interest on land, valued at $100 per acre. 18. A yoke of oxen weighed December 15, 42201b.; January 15, 44101 b.; March 7, 47301b. Required the total gain from December 15, to March 7, and the average gain per day from December 15 to January 15, from January 15 to March 7, and from December 15 to March 7. 19. If each ox in the preceding example consumed daily 14 lb. of hay, zebu. of potatoes, and 8qt. of Indian meal, a To determine the number of hills or plants in a field, divide the area of the field by the area occupied by each hill or plant.

Page  116 116 THE FARM. [ART. V. what was the cost of the gain per lb., supposing hay to be worth $10 per ton, potatoes, 20cts. per bushel, and Indian meal 60cts. per bushel? 20. If an acre of land yields 1y' tons of hay, or 15 tons of carrots or Swedish turnips, or 60 bushels of Indian corn; and if a working horse would consume 3cwt. of hay, or 3cwt. of carrots, or lbu. of Indian corn, per week, how many acres of each crop would be required to support a horse for the year? 21. Bought December 1, a pair of oxen for $65, and sold the same February 26 at $5 per 100 lb., net weight, their net weight being 1846 lb. Required the amount lost on the sale, the following being the expenses of keeping:-73bu. turnips at l0cts.; 361bu. Indian meal at 60cts.; 65-bu. potatoes at 25cts., and 25 lb. hay per day at 25cts. per 100 lb.; interest on cost, at 6 per cent., 94!cts. Ans. $24.441. 22. What is the value of pasture land per acre, 50 acres of which will pasture 8 cows at 25cts. per week, 4 oxen at 50cts., and 75 sheep at 3cts. per week for twenty weeks in the year, the pasturage being estimated at 6 per cent. of the value of the land? 23. Required the yearly expense of keeping a sheep, allowing 155 days for the time of foddering in the barn, and 30 weeks pasturage at 3cts. a week, the food consumed while in the barn being 2 lb. of hay per clay at $8 per ton of 2000 lb., and 1-bu. of rutabaga at l0cts. a bushel. 24. Required the amount of profit in the following experiment in stall-feeding sheep. Bought 118 at $2.50, 2 at $3, and 60 at $3-ji; commissions for purchase and driving, 25cts. each; interest and risk estimated at $9.27; produce consumed, 519bu. turnips at 8cets., 151bu. corn at 75cts., 2 lb. of hay each per day, at $8 per ton of 2000 lb. The sheep were all put up on December Ist, and 125 were sold February 11th at $5 each, and the others, February 18th, at $5.25 each.

Page  117 ~22.] MISCELLANEOUS EXAMPLES. 117 25. Estimating the expense of cultivating Indian corn at $25 per acre, wheat at $10, oats at $5, rye at $8, hay at $2; and the harvest per acre of corn, 40bu. at 65cts., and fodder worth $10; wheat, l4bu. at $1.50, and straw worth $3; oats, 56bu. at 42cts., and straw worth $2.40; rye, 12bu. at $1, and straw worth $2.40; and hay, 2 tons per acre, at $10; what would be the profit on a farm containing 7~ acres of corn, 5- acres of wheat, 5 acres of oats, 5 acres of rye, and 21 acres of meadow? 26. When hay sells at $16 per ton of 2240 lb., and cornstalks at let. per bundle, how much can be saved per week with 6 cows, by substituting corn-stalks for hay, if the average consumption of each cow is 5 bundles of stalks, or 25 lb. of hay per day? 27. Two hogs were fed from April 30th to May 20th, exclusively upon Indian hasty-pudding. The pudding used during the interval, took 4- bu. of meal, at 78cts. a bushel, and the weight gained by the hogs was 105 lb. Allowing i of the gain to be net weight, what should be the price of the pork per lb. to gain 20 per cent. on the cost? 28. From the following account of sales, on a farm of 25 acres, for ten years, find the average amount received annually for each item. Date. Vegeta's. Fruit. Vinegar. Meat. Hay. Stock. Milk, Barley. n and Total. 1811 $132.06 $126.76 $183.94 $14.38............................ 1812 181.91 145.48 12.83 81.92 $130.18 $32.00 $17.13 $32.05 1813 112.93 68.07 93.47 69.67 202.68 30.00 44.97.75 1814 180.38 63.71 254.92 41.21 253.08 112.00 15.00 103.50 1815 162.38 151.56 206.60 61.37 506.69....... 16.99....... 1816 165.36 169.73 187.17 60.69 399.69 37.00 12.71 128.50 1817 132.49 240.37 295.31 48.24 329.93...... 24.75. 1818 84.34 116.71 246.30 77.99 162.21 30.02 98.47. 1819 103.85 280.68 131.95 25.84 185.00...... 94.04. 1820 111.62 248.88 191.83 87.21 207.43.. 128.27... To.... i. 29. What was the average net annual income of the above farm, estimating the amount of farm produce consumed by the family, and not included in the above account, at $454, and the expenses at $638.30 per annum?

Page  118 118 THE FARM. [ART. V. 30. What is the annual gain on a five-acre wood lot, which cost $50, the yearly growth being 1 cord of wood per acre, the market price of wood averaging $3.12~ per cord, and the expense of cutting and hauling 62'cts. per cord? 31. Required the profit on an acre of hops, the yield being 700 lb., worth 15~cts. a lb., and the expenses of cultivation as follows: Renewal and setting of poles, $12; planting, $1; tying up, $1; hoeing 3 times, at $1.50; 4 loads of manure, at $1.121-; picking, let. a lb.; a man to tend the pickers, 87; board of the laborers, $1.50; kiln-drying and packing, $1 per 100 lb.; bale, 45cts. 32. If a horse consumes the produce of 6 acres of land before he is fit for work, and can afterwards be kept constantly employed for 12 years, by annually consuming the produce of 4 acres,-and a pair of oxen consume 10 acres' produce before they are fit to work, and can be employed only 3 years by consuming the produce of 2 ~ acres per annum each, the produce of how many acres will represent the difference between the expense of keeping a horse and a yoke of oxen for 12 years? Ans. 46 acres. 33. Estimating the value of a horse after 12 years' farm service, at $15, and supposing that every ox can be fattened after 3 years' service by the produce of 1~ acres, so as to be worth $65, what should be the value of 1 acre's produce, to make the expense of keeping, the same for a horse as for a yoke of oxen? Ans. $8.71 nearly. N. B. Add to 46( acres, the number of acres' produce necessary to fatten the oxen, ancl you will obtain the number of acres' produce equivalent to the difference of value between the horse and the oxen. 34. Find the expense of keeping 8 oxen one year, estimating their total consumption at 1 ton of hay per week from January 1 to Mlay 15, and from October 30th to January 1, the hay being worth $10.75 per ton, pasturage

Page  119 ~ 22.] MISCELLANEOUS EXAMPLES. 119 from May 15 to October 30, at 62~cts. per week each, repairs of yokes and bows $3.75, wear of ploughs, chains, &c. $15. 35. What should be the price of charcoal per bushel, to make 25 per cent. on the following estimate of the cost of burning a kiln of 30 cords, the wages of labor being $1.25 per day? Cost of standing wood, $1.75 per cord; 21 cords make 100 bushels of coal; 1 man can chop 2 cords for the kiln in a day; collecting, drawing together, covering, and burning, require 20 days' work of 1 man; expense of hauling the whole to market, and selling, $8.75. _Ans. 10-1-5 cents. 36. The whole amount of hay purchased and used at a stage stable, from April 1 to October 1, 1816, was 32T. 4cwt. 10 lb. at an average cost of $25 per ton. At the same stable, from October 1, 1816, to April 1, 1817, there was consumed by the same number of horses, 16T. 13cwt. 3qr. 10 lb. of straw, at $9.75 per ton, and 9T. 14cwt. lqr. of hay at $25 per ton. A straw-cutter was employed during four months of the latter period. Required the amount of money saved by its use. 37. In an experiment on transplanting the layers of a single stalk of wheat, made by 3Mr. Miller, of Cambridge, Eng., one grain of wheat produced 3- pecks, weighing 47 lb. 7oz. Supposing a cubic inch to contain 200 grains, how many grains were there in the whole? How many in a pound? 38. If 6 men, with cradles, can cut as much grain in a day as 12 men with common sickles, and if the 6 cradlers require 3 men to bind, and 3 boys to assist, while the 12 reapers require only 2 men to bind, how much can be saved per day by cradling, allowing $1.25 for each man, and 50 cents for each boy? 39. If 105 gallons of milk yield 36 lb. of butter, worth

Page  120 120 THE FARM. [ART. V. 183cts. a pound, and 60 lb, skim-milk cheese, worth 6cts. a pound, and if 140 gallons of milk of the same quality, with the same amount of labor, yield 141 lb. of cheese, worth 9ets. a lb., and whey worth $1~, which is the more profitable, the manufacture of butter or cheese? 40. It is supposed that a pair of tame pigeons consume a pint of grain per day, for 280 days in the year. As it has been estimated that there are 1250000 pairs in Great Britain, how much would they consume per annum at this rate, and what would be its value, at an average of 50 cents a bushel? 41. The cost of improving a 12-acre field was as follows: Blasting large stones, $44.25; trenching, $20; drains, $3.87~; lime, 64bu. per acre, at 40cts. per bu.; ditch for intercepting hill water, $37.62 -. Required the total cost, and the average per acre. 42. What should be the market price of milk per quart, when butter is worth 25cts. a pound, and skim-milk cheese is worth 5cts. a pound, -supposing that 100 gallons of milk will make 34 lb. of butter, and 74 lb. of skim-milk cheese? 43. The late Duke of Athol planted 6500 Scotch acres a of mountain ground with the larch, which, in 72 years from the time of planting, will be a forest of timber fit for building ships of the largest class. Supposing the plantation by that time to be thinned out to 400 trees per acre, and each tree to contain one load of 50 c. ft., what will be the value of the wood per acre, and the value of the whole plantation at 25cts. per c. ft., which is about - its present value? 44. A hectare in the Isle of Bourbon, produces 76000 kilogrammes of cane, which give 9200 kilogrammes of sugar, at an expense of 2500 francs for labor. A hectare of beet-root produces 40000 kilogrammes of roots, which a A Scotch acre = about 1- English acres.

Page  121 ~ 22.] MISCELLANEOUS EXAMPLES. 121 will yield 2400 kilogrammes of sugar, at an expense of 354 francs. What per centage of sugar is yielded by the cane, and by the beet-root? What is the cost of each kind of sugar per lb., estimating merely the labor bestowed on each crop? 45. Two men mow a square meadow, but one being a faster mower than the other, agrees to take the outside swath, and cut off all the corners. What part of the whole will each mow, there being twelve swaths in each side of the field? Ans. -31 41 The following rule will furnish the answer to all questions of this kind. Let the pupil endeavor to prove its accuracy. Square the number of swaths in the side of the field for a denominator. Then if the number of swaths is odd, multiply it by 2, and diminish the product by 1; or, if it is even, multiply it by 2, and diminish the product by 4, for a numerator. The fraction thus obtained, will show what part of the field the outer man will mow more than the inner one. 46. The amount of hay necessary to sustain oxen, is about.02 of their weight, daily. When fattening, they require about.04 of their weight per day. At $10.50 per ton of 2000 lb., what would be the cost per week of the extra hay consumed in fattening two oxen, one of which weighs 834 lb., and the other 917 lb.? 47. In England and Wales, it has been estimated that there are 3252000 acres of wheat, 1250000A. of barley and rye, 3200000A. of oats, beans and peas, 1200000A. of clover and artificial grasses, 1200000A. of field-roots, 2100000A. fallow, 48000A. of hops, 17300000A. of meadow and pastnre, 1200000A. of hedgerows, copses, woods, and wastes, and that the annual agricultural income is ~216817624 At $4.84 per pound, what is the average income per acre? 48. What would be the cost of seed in the last example, supposing that there are 500000 acres of barley, and that the following is the average quantity used, per acre? Wheat,

Page  122 122 TIHE ARM [ART. V. 5pk. @ $1.10 per bu.; barley, 3bu. @( $0.50; rye, lbu. @, $0.80; oats, beans, and peas, 3bu. ~ $0.95; grasses, 3pk. ( 82.00 per bu.; field-roots, 13bu. ~, $0.45. 49. A farmer wishing to determine the area of his field, finds that it may be divided into four triangles, the dimensions of which are as follows: 1st. Base 240 paces, altitude 87 paces. 2d. Base 213 paces, altitude 95 paces. 3d. Base 107 paces, altitude 28 paces. 4th. Base 92 paces, altitude 79 paces. Required the entire area. Ants. 5A. 1R. 9 -Ir. nearly. 50. 1 have a barrel 20 inches in diameter at the middle point, 16 inches at the head, and 27 inches long. Required its contents in ale, wine, and dry measure. Ans. 31gal. nearly, wine inceas. 25gal. 1-qt. beer meas. 3bu. ipk. 2qt. dry meas. 51.a A heap of grain piled against the outer corner of a barn, contains 63 bushels, and its depth is 5ft. Required the slant height. Ans. Oft. Sin. 52. A granary holds 1310.4 bushels. The dimensions of the floor are 15ft. 9in. by 12ft. Required the average depth of the grain. Ains. Sft. Sin. 53. A stack of hay with a circular base, has settled so much that 250 c. ft. are estimated as equivalent to 1 ton, and according to this estimate, it is found to contain 1.44 tons. What is the height of the stack, the circumference of the base being 30ft.? Ans. 16ft. 54. The area of an irregular field is 2A. 3R., and the average length is 121 paces. What is the average breadth? Ans. 110 paces. a The remaining examples in this section are introduced to test the skill of the pupil in reversing the rules.

Page  123 ~ 23.] MISCELLANEOUS EXAMPLES. 123 VI. THE GARDEN.,O3. EXAMPLES FOR THE PUPIL.a 1-5. How many plants would be required for an acre, if they were placed 1I ft. apart? If placed 2ft. apart? 3ft. apart? 6ft. apart? 16iGft. apart? Ans. to the first, 19360. 6-10. How long a strip of land will be required to contain an acre, if the width is 1OOft.? If the width is 3- rods? 97.6ft.? 181-3ft.? 2r. 5~ft.? 11. Allowing Ipt. of early peas to a row 20yd. long, ipt. of marrowfat peas to a row 32yd. long, lpt. of string beans to a row 27yd. long, ipt. of runners to a row 36yd. long, and ipt. of dwarf kidney beans to a row 26yd. long, how mnchel seed would be required to plant 4 rows of early peas, 6 rows of marrowfats, 3 rows of string beans, 5 rows of runners, and 8 rows of dwarf kidney beans, each row measuring 28 yards? 12. A quarter acre of land cost $25. Expended for labor in improving it, $157; for seed potatoes, $15; rye and grass seed, $1.17; 6 cords of manure, at $5 per cord; 2 casks of lime at $1; cost of other seeds and gathering in the crop, $17.50. The products of the first year were 327 bushels of potatoes, at 60cts.; 5bu. rye, at $1.25; 8bTu. corn, at $1; 100bu. rutabaga, at 30cts.; hay, $12; 600 cabbages, at 50cts. a dozen; 2000 lb. squashes, at let. j fuel taken off, $25.b Estimating the value of the land as doubled, what was the first year's balance in favor of the improvement? Ans. $125.28. 13-15. How many times must a spade be thrust into the ground in digging a square rod, supposing the surface removed a Buist, Colman, Loudon, Johnson. b Colman.

Page  124 124 THE GARDEN. [ART. vI. at each time to measure 7 by 8 inches? The spade weighing 8 lb., and a spadeful of earth 17 lb., how many pounds must the gardener lift in a day, 10 sq. rods being a day's work? How many tons must he lift in spading an acre? 16. What amount isannually gained by expending $16 per acre to drain a garden of 14 acres, money being worth 6 per cent. per annum, and the land renting before it was drained for $3 per acre, but after it was drained, for $8 per acre? 17. What distance would a man walk in ploughing a garden containing - of an acre, if each furrow was 9 inches wide, adding I rod to every 18 for the ground travelled over in turning? 18. What must be the length of a strawberry bed, to contain 3 — of an acre, the width being 33 feet? 19. Required the cost of digging a trench 200ft. long, 3ft. broad, and 3ft. deep, a laborer being able to remove 1 cubic yard per hour,-allowing for wages $1.25 per day of 10 hours. 20. Money being worth 6 per cent. simple interest, what is the present value of an orchard containing 150 trees, which will be all in full bearing in 5 years, and will then be worth $20 apiece, the land which they occupy being now worth $1450? Ans. $3757.69. 21. In a garden of 1~ acres the following articles were raised in the year 1836:-3500bu. onions, at 5cts.; 45bbl. beets, at $1.50; 14bu. parsnips, at 75cts.; 2bu. beans, at $2; 20bu. potatoes, at V$; $100 worth of cabbages; and vegetables for the family, estimated at $100. Allowing one-half of the proceeds for the expense of cultivation, and estimating the value of the land at $450 an acre, what profit was made on the whole?

Page  125 ~ 23.] MIISCELLANEOUS EXAMPLES. 125 22. When ordinary pears are sold at 50 cents a bushel, and choice varieties bring 3 cents apiece, how much more profitable is a tree grafted with one of the best kinds, than a common pear-tree, the yield of each being eight bushels, and the choice pears averaging 45 to a peck? 23. In a garden 225ft. long, and 144ft. broad, is a gravel walk 6ft. wide, extending around the whole, at a uniform distance of 8ft. from the boundary line, intersected by three walks each 4ft. wide, parallel to the shortest sides. What amount of land is taken up by the walks? Ans. 650 x 6 + 348 X 4 = 5292 sq. ft. 24. Throughout the whole extent of the above walks is a trench 10in. wide and 8in. deep, filled with broken stone, and covered with a layer of broken bricks and other rubbish, 3in. thick over the whole width of the walks, the top-dressing of gravel being of the average depth of 4 inches. How many loads of 27c. ft. each, were required of the broken stone? How many of rubbish? How many of gravel? What was the cost of the whole at 50cts. for each load? Ans. 2041 o loads of stone, 49 of rubbish, 651 of gravel; cost, $ 25. What would be the expense of making a hedge of roses 468ft. long, setting the bushes with a distance of 61 inches between the centres, the cost of the bushes being 28cts. apiece, and the expense of setting $2.25 per hundred? 26. I wish to make a hot-bed, composed of 3 parts manure and 1 part oak leaves. The frame is 12ft. long and'6ft. wide. The bed is to be 3 I ft. deep, and is to project 8 inches beyond the frame, on every-side. How many cubic yards of manure will be required, and what will it cost at $1.75 per yard? 27. Two banks, each 104ft. long, and 5Rft. high, and a grass plat 162ft. long and 157ft. wide, are to be covered with turf. Estimating the average thickness of the turf at 3in.,

Page  126 126 THE GARDEN. [ART. VI. what would be the cost of the whole at $1.25 per c. yd? At lcts. per sq. ft.? 28. An English market-gardener received for the produce of a single acre in one year, for radishes, ~10; cauliflowers, 60; cabbages, ~30; celery, ~90; endive, ~30. Allowing t3 lOs. for rent, and ~90 for expenses of cultivation and marketing, what was the profit per sq. rod, in Federal Money, estimating the value of the sovereign at $4.84? 29. Frederick Tudor, Esq., of Nahan't, Mlass., in the year 1841, raised 42284 lb. of sugar-beets on 93 rods of land. Allowing 561b. per bushel, how many bushels could be raised per acre at this rate, and what would be the value of the whole at 50cts. per bushel? 30. If 2 men digging;can keep 1 man employed in wheeling to the distance of 20 yards, and if each digger removes 15 c. yd. per day, what will, it cost to remove 3340 c. yd. to the distance of 160 yards, the wages of each laborer being 75cts. per day? Ans. $835. N. B. First find the number of men required to wheel the dirt away, and add the two diggers, and you will obtain the whole number employed. 31. WVhat number of bulbs will be required for a crocus bed, the bed being 23in. wide, and the rows 6 inches apart, allowing 150 bulbs to each row; and what will they cost,. of the whole number being bought at 75ets. per 100, 4 at $1.374, and the rest at 1.G624? 32. WVhen the double hyacinths were first brought into notice, some of the roots were sold at 2000 guilders apiece. Equally fine varieties can now be bought for $4 a dozen. At 40cts. per guilder, how large a bed could now be stocked with the choicest varieties, at the original cost of a single root, the roots being placed 8 inches apart, and allowing Gin. for border? 33. What will be the cost, at $12.75 per 100, of stocking a tulip bed 20ft. long and 4ft. wide, the bulbs being planted

Page  127 ~ 23.] MISCELLANEOUS EXAMPLES. 127 in rows, allowing Gin. between the plants, and 7in. between the rows? Ans. $35.70. 34. In order to introduce enough fresh air to fill a hothouse twice in 24 hours, how many feet of air heated to the proper temperature, must be introduced hourly into a house 4Oft. long, and 16ft. wide, the height in front being 6ft., and at the back 1Sft.? Ails. 320 c. ft. 35. What would be the daily expense of raising the temperature of the hot-house in the last example 15~, if 2no- lb. of coal will raise the heat of a c. f. of air 10, coal being worth $6.50 per ton of 2000 lb.? 36. A mechanic has a vacant space in his garden, 18ft. long and 4ft. wide, which he wishes to occupy as an asparagus bed. He trenches the whole to the depth of 2ft., and fills the trench half full of manure before returning the earth. He afterwards plants the whole with roots, setting the roots 9 inches apart. The land is worth 12~cts. a square foot, and he will be obliged to wait 3 years before the bed will be in a condition to be cut. Estimating the labor bestowed on the bed at 4 hours, at lOcts. per hour, the cost of manure at $1.25 per solid yard, the cost of plants at $1.0( per 100, and interest on the whole at 6 per cent. per annumn, what will be the entire cost of the bed when it comes into bearing? 37. Allowing loz. of onion, carrot, or parsnip seed for sowing 15 sq. yd.,'oz. of cabbage or cauliflower seed for 4 sq. yd.,'oz. of turnip seed for 11 sq. yd., and 160 asparagus plants for a bed 5ft. by 30, what quantity of each will be required to plant a garden of 40 sq. rods with onions, carrots, parsnips, cabbages, cauliflowers, turnips, and asparagus, allowing the same quantity of ground to each? 38. In 1637 a collection of 120 tulips was sold in Holland for 9000 guilders; in England, at the present day, ~50 is frequently given for a single bulb no finer than solme

Page  128 128 THE HOUSEHOLD. [ART. VII. of the varieties which can be purchased for less than a dollar. Estimating the florin at 40 cents, and the sovereign at $4.84, how many of the Dutch bulbs would amount to the same as 15 of the English bulbs, at the above rates? 39. Apples should stand 35 feet apart, pears 20ft. apart, and plums 18ft. apart. How large an orchard would be required for 180 apple-trees, 315 pear-trees, and 350 plumtrees, there being 5 rows of each sort, allowing 28ft. between the apples and pears, and 20ft. between the pears and plums? Ans. 11A. 3R. 31r. 12-yd. 40-42. How many hills of corn in a rectangular piece of land containing a quarter of an acre, if the hills are 3ft. apart one way, and 2ft. 9in. the other? if the hills are 2ft. 6in. x 3ft? —2ft. x 2ft. 3in.? VII. THE HOUSEHOLD. 24:. GENERAL INFORMATION. 1. THE grains yield nearly the following quantities of meal and bread per bushel.a Wheat weighing 60 lb. flour, 48 lb.; bread, 64 lb. Rye weighing 54 lb.-meal, 42 lb.; bread, 56 lb. Barley weighing 48 lb.-meal, 37 7 lb.; bread, 50 lb. Oats weighing 40 lb. —meal, 22 lb.; bread, 30 lb. 2. The following weights and measures nearly correspond.b Wheat flour, 14oz.=lpt. Indian meal, 18oz.= lpt. Butter, 15oz.=lpt. Loaf sugar, broken, Ilb.=lpt. White sugar, crushed, 17oz.=lpt. Brown sugar, 18oz.=lpt. Eggs, 10-1 lb. A chaldron of soft coal-58- c. ft. Stone coal, lbu. =70 lb., or 42 c. ft.=I ton. A common tumbler holds apt. or ~lb. of water. A common wine-glass holds a Brit. Husbandry.'Sci. American, Brit. Husb., Enc. Brit., and private information.

Page  129 ~ 25.] HOUSEHOLD MENSURATION. 129 A gill or 2oz. of water. Currants, 14oz.=lpt. Cherries, 12oz.= pt. Honey, 23oz. = lpt. Lard, tallow, or spermaceti, 15oz.= lpt. Milk, 1 lb. =pt. Oil, 15oz. = ipt. Salseratus, dry, 23oz. =lpt. 25. HOUSEHOLD MENSURATION. 1. To find the contents of any cylindrical vessel, in pints. -Measure the diameter and height in inches. Then, for wine measure, multiply the square of the diameter by the height, and divide the product by 37; for beer measure, multiply the square of twice the diameter by twice the height, and divide the product by 359; for dry measure, multiply the square of twice the diameter by the height, and divide the product by 171. Let D represent the'diameter, H the height, and C the contents in pints. Then, if the contents are taken in wine measure, -I = 37C — D2; D= /37C — H. For beer measure substitute 44-, and for dry measure 42a, in the place of 37, in each formula. 2. To find the contents of any vessel with a cireular base, and tapering sides.a —Take one half of the sum of the greater and less diameters, for a mean diameter, and proceed with this mean diameter and the height, as in the preceding problem. If great accuracy is required, proceed as in Problem 1, substituting for "the square of the diameter," " the product of the greater and less diameters, added to one third of the square of their difference." In beer and dry measure, use four times this product, in the place of " the square of twice the diameter." The height and mean diameter, may be found as in Prob. 1. Let G be the greater diameter, L the less diameter, and NM the mean diameter. Then G=/ 3gM2__ of L2 -IL; L = V/3MTa - 4 of G0 -G. a Such a vessel forms afrustum of a cone. 9

Page  130 -]130 THE HIOUSEHOLD. [ART. VII. 3. To find the contents of a bowl, in pints.-Measure the diameter of the top and the depth, in inches. To three times the square of half the top diameter, add the square of the depth; multiply this sum by the depth, and divide the product by 55 for wine measure, by 67 for beer measure, or by 64 for dry measure. 4. To find the contents of a well or cylindrical cistern in khogsheads.-Measure the dimensions in feet, multiply the square! of the diameter by the depth, and divide the product by 11. If great accuracy is required, measure the dimensions in inches; multiply the square of the diameter by the depth; multiply, the product by 54, and cut off six figures from the right hand. Let D be the diameter in inches, d the depth, and C the contents in hogsheads. Then D = -1000000 C —54d; d=1000000 C- 54 D2. 5. To determine the height to which any cylindrical vessel wvill be filled by a gallon.-For wine measure, divide 294; for beer measure, divide 359; for dry measure, divide 342, by the square of the diameter, measured in inches. One quart would evidently fill the vessel I as high; one pint would fill it j as high; and one gill would fill it 3 as high. Any cylindrical vessel can therefore be marked by this rule, so as to serve the purpose of a set of measures. Let N be the number of the required measure which will be equivalent to a gallon, D the diameter, and H the height. Then, for wine measure D= /294. (N x H); for beer measure D = / 359 — (N x H); for dry measure D = 342 — (N x H.) 6. To test the accuracy of cylindrical dry measures.3Measure the dciameter in inches, and divide 2738 by the square of. the diameter. The quotient will be the depth for 1. bushel. The depth for a half bushel, will be ~ the

Page  131 ~ 26.] MISCELLANEOUS EXAMPLES. 131 quotient; for a peck, l of the quotient; for a half peck, i of the quotient, &c. 26. EXAMPLES FOR THE PUPIL. 1-3. I have a cylindrical tin dish, 6 in. in diameter, and 6in. deep. Required its contents in wine, in beer, and in dry measure. Ans. 3qt. 3.4gi. wine meas.; 2qt. 1.65pt. beer meas.; 3qt. nearly, dry meas. 4-6. At what height should marks be placed on the inside of the above dish, to indicate a quart of each measure? Ans. 1-& in. nearly for wine meas.; 2iin. nearly for beer meas.; 2in. for dry ineas. 7-9. Find the contents in each measure, of a pan, the height being 5 inches, the top diameter 17in., and the bottom diameter 9in. Ans. by the accurate rule, 23.56pt. wine meias.; 19.42pt. beer meas.; 20.39pt. dry meas. 10-12. Find the contents in each measure, of a bowl, the diameter of which is 6in., and the depth 3in. Ans. 1.96pt. wine meas.; 1.61pt. beer meas.; 1.69pt. dry meas. 13. A cylindrical cistern is 5 feet in diameter, and 6ft. deep. How many hogsheads does it hold? 14-15. A cylindrical cistern is 7ft. 6in: in diameter, and Gft. 9in. deep. Find its contents by each rule. Ans. by Rule 1, 34-9-~Lhhd.; by Rule 2, 35.43hhd. 16-17. The house to which the above cistern belongs, is 40ft. long, and 34ft. wide, and is supplied with eave-troughs which convey all the water that falls on the roof to the cistern. To what depth would the cistern be filled by a single shower, in which there is a fall of -in. -of rain, allowing 6in. on each side for the projection of the eaves? How many inches of rain would fill the cistern? 18. A man has the following cylindrical measures: One

Page  132 132 THE HOUSEHOLD. [AIXT. VII. designed for a bushel, the diameter of which is 18~ inches; one designed for a half-bushel, diameter 14'in.; one designed for a peck, diameter 12in.; one designed for a halfpeck, diameter 94in.; and one designed for a quarter-peck, diameter 7'in. What should be the depth of each? Ans. for the half-bushel, 64in. 19. I send a barrel of flour to a baker, and he agrees to furnish me an equal weight of bread in return. The flour weighs 1cwt. 3qr., and cost me $7.50; the barrel, (which the baker keeps,) is worth 25cts. How much do I pay the baker for his trouble? Ans. $2.12~. N. B. If 48 lb. of flour make 64 lb. of bread, how many lb. of flour will make 1cwt. 3qr. of bread? And what will be the value of the flour that is left, at $7.50 per bbl.? 20. A cow gave 350 gallons of milk in a year, and consumed in the same time two tons of hay, at $9.75; 12bu. of Indian meal, at 60cts.; 50bu. of beets, at 40cts.; and her pasturage cost $8.75. Estimating the value of her milk at 5cts. a quart, and allowing 10 per cent. on $25 for the interest of her worth, and risk of keeping, and $2.50 for labor and attendance, what amount of profit did she yield.? 21. Find the cost of 100 lb. of bread, made from each of the following grains, the grain being of the full weight given in ~24, and the cost of manufacture in each instance, being 56cts.; wheat at $1.10 per bushel; rye, at 90cts. per bu.; barley, at 75cts. per bu.; oats, at 40cts. per bu. 22. How much crushed sugar, by measure, should be used in preserving 6qt. of currants, in order that there may be equal weights of sugar and fruit? a 23. How would you measure the following ingredients?2 lb. flour, 2 lb. sugar, I lb. butter, and 1 lb. eggs.a 24. A common watch vibrates 5 times in a second; if from any cause each vibration is S-.1 less than its proper time, how much will the watch gain per day? a See Section 24.

Page  133 ~ 26.] MISCELLANEOUS EXAMPLES. 183 25. A cloak containing 6wyd. of broadcloth that is 1 yd. wide, is to be lined with silk that is iyd. wide. How much silk will be required? 26. A parlor 35ft. 6in. long, and 19ft. 6in. wide, is to be carpeted. How much carpeting, that is a yard wide, will be required, provided there is no waste in matching the figures? How much will be required, if ~yd. is lost in matching each breadth? 27. A man wishes to build in his cellar a potato bin that will hold 40 bushels. It is to be 6ft. long, and 3ft. 6 in. high. What must be the width of the bin? 28. If a family of 3 persons consume a barrel of flour in 11 weeks, what is the average amount of bread eaten daily by each person, 3 pounds of flour being sufficient to make 4 pounds of bread? Ans. 1 3lb. 29. Required the entire quantity, and the value at 22cts. a gallon, of the milk taken by D. N. Breed, of Lynn, Mass., from one cow, in 11 months of 1839-40. The daily average was as follows:-from April 15 to April 30, 1839, 6qt.; in May, 14qt.; in June, 16qt.; in July, 13qt.; in August, 12qt.; in Sept., llqt.; in Oct., lOqt.; in Nov., 10qt.; in Dec., 9qt.; in Jan., 1840, 9qt.; in Feb., 7qt., and from March I to March 15, 2qt. 30. Alfred the Great had large candles made with marks upon them, so that he might judge of the time by the quantity that had burned.a If a candle were lighted at noon, of such size that only 9in. would be consumed in 24 hours, what would be the time when 2~in. had burned? 31. How much coal, at 2240 lb. to the ton,. would fill a bin that holds 160 bushels of potatoes? 32. How many wine gallons in a pail 12in. in diameter, and 143in. deep? a Carpenter. The Clepsydra had not been introduced into England.

Page  134 134 THE HOUSEHOLD. [ART VII. 33. A cylindrical -vessel is 4 inches in diameter. At what heights must marks be placed for Iqt. wine measure; for ipt. beer measure; and for 1 half-peck dry measure? 34. Required the contents in wine measure, of a waterpail, the height being 8jin., the top diameter h11in., and the bottom diameter 9in. 35. How many rolls of paper hangings, each 9yd. long, and ]yd. wide, would be required to paper one side of a room that is 19ft. long and 9ft. 6in. high? 36. A saving of $1 per annum, invested at 6 per cent, compound interest, will amount in 40 years to $154.761966. If a young man at 20 years of age, commences laying up Gicts. per day, how much will he be worth when he is 60 years old? 37. A man expends all his income, but in looking over his accounts at the end of the year, he finds that $125 has been laid out foolishly. If he resolves to retrench, and saves that amount annually, what will he be worth in 40 years? 88. Find the ratio of illumination between two lights, which throw shadows of equal intensity, if placed at the distances of 9ft. and 5~ft. respectively.a a " The following method of measuring the comparative illuminating power of different lights, is founded on the law that the amount of rays thrown on a given surface, is inversely as the square of the distance of the illuminating body. Place two lights, which are to be compared with each other, at the distance of a few feet, or yards, from a screen of white paper, or a white wall. On holding a small card near the wall, two shadows will be projected on it. Bring the fainter light nearer the card, or remove the brighter light farther from it, till both shadows acquire the same intensity. Measure now the distances of the two lights from the wall or screen, and the squares of these dis tances will give the ratio of illumination. In this experiment the spectator should be equidistant from each shadow." —Bigelow's Technology.

Page  135 ~ 26.] MISCELLANEOUS EXAMPLF.S. 135 39. If a cylindrical cistern is 7ft. 6in. in diameter, what must be its depth, to hold 45 hogsheads? To hold 30 hogsheads? 40. If wood is sawed 2ft. long, how many cords will there be in a pile 50ft. long, and 8ft. 3in. high? 41. How many potatoes are there in a cellar, there being 5 barrels full, and one half-full, each barrel holding 2bu. 3pk., and a bin that is 10ft. long and 5ft. wide, being filled to the height of 3ft. 6in. 42. An imperial gallon of sperm oil, burned in an Argand lamp, which yields a light equivalent to 5 candles, (6 to a lb.) will burn about 100 hours.- The solar lamp, with an imperial gallon of whale oil, yielding a light equivalent to 4a candles, (6 to a lb.) will burn about 90 hours.a When sperm oil is $1.25 per gallon, and whale oil 62~cts. per gallon, how much can be saved on every gallon by the use of whale oil? Ans. 44] cents. 43. A cylindrical cup contains 3qt. 3.4gi. wine measure. Required the height, the diameter being 6~ inches. Ans. 6 inches. 44. What is the greater diameter of a pan, the less diameter being 9in., the height 5in., and the contents 19.42pt. beer measure? Ans. 17 inches. 45. A cylindrical cistern, 6ft. 9in. deep, contains 35.43 hogsheads. What is its diameter? 46. A cylindrical cistern, 71ft. in diameter, holds 47.24 hogsheads. What is its depth? Ans. 9ft. 47. What should be the diameter of a cylinder that is 6~in. deep, to hold a half-bushel? Ans. 144 inches. 48. What should be the diameter of a cylinder that is 2'in. deep, to hold a half-pint, beer measure? Ans. 3in. nearly. a Parnell.

Page  136 136 ARTIFICERS' WORK. [ART. vmI VIII. ARTIFICERS' WORK.a 2 7. THE CARPENTER AND JOINER. 1. The contents of a board are found by multiplying the length by the mean breadth, provided the thickness does not exceed 1 inch. If the board tapers regularly, the mean breadth is half the sum of the two end breadths. If it is of irregular shape, the breadth should be taken at a number of different points at equal intervals, and their average should be regarded as the mean breadth. If the thickness exceeds 1 inch, multiply the number of feet in the area by the number of inches in the thickness. If the contents are given, and either of the dimensions are required, divide the contents by the product of the given dimensions. Examples.-A board 12ft. 6in. long, ift. 3in. broad, and tin. thick, contains 12 x 11 —15sft. board measure. A board 13ft. 4in. long, ift. 8in. broad, and 1DI in. thick, contains 13~ X 1 x 1~=33.Ift. board measure. 2. The contents of square or hewn timzber are found by multiplying the mean breadth by the mean thickness, and their product by the length. The mean breadth and thickness are found in the same manner as in measuring boards. Sometimes the contents are found by squaring i of the girt, and multiplying by the length. This method is erroneous, and always gives the,contents too great. Timber is often sold by board measure. Cubic feet can be reduced to board feet by multiplying by 12, (provided the thickness exceeds L inch,) or the contents can be found at once in board measure, by taking one of the dimensions in inches, and the other two in feet. If the contents are given, and any two of the dimensions, the other dimension may be found, a Ingram, Gillespie, Crossley and Martin, Nicholson, Pratt, and private information.

Page  137 ~ 27.] THE CARPENTER AND JOINER. 137 by dividing the contents by the product of both the given dimensions. Examples.-A hewn log 19ft. 6in. long, 2ft. 2in. broad, and 2ft. lin. thick, contains 19~ x 2] x 212z =2 884 c. ft., or 1056~ft. board measure. A hewn log 30ft. 8in. long, ift. 10in. wide, and ift. 3in. thick, contains 30N x 22 x 11 = 1681 board feet. 3. The contents of roound timber are usually found by squaring I of the mean girt, and multiplying it by the length. If the tree is covered with bark, lin. should be deducted from the quarter-girt before squaring. If the bark is very thick, more than lin. is sometimes allowed. No rough timber is considered measurable, if the diameter is less than 6 inches.a This rule gives the contents too small, nearly in the proportion of 11 to 14. A ton of timber is considered as equivalent to 40 c. ft., but 40 c. ft. of round timber as generally measured, really contains 50 c. ft. The statement usually given without any explanation in Arithmetical tables, that "40ft. of round timber, or 50ft. of hewn timber make 1 ton," is therefore erroneous. The allowance was originally introduced as a partial compensation to the purchaser of round timber, for the waste occasioned in squaring it. If the true content is required, it can be found very nearly, by squaring -5 of the girt, and multiplying by twice the length.b Example.-A. log 47ft. 8in. long, and girting at the ends 18ft. and Gft., has for its mean quarter girt l of (18 + 6) -2-=3. Its contents are therefore 32x 47= —429 c. ft. 4. Flooring, partitioning, and roofing, and all large and plain work, in which a uniform quantity of materials a Let G be the mean girt in feet, L the length, and C the contents in c. ft. Then L=166 — G2; G=4X 4x/C L. b In employing this rule, as well as in the usual method, lin. should be deducted from the girt if the tree is covered with bark.

Page  138 138 ARTIFICERS' WORK. [ART. VI..' and labor is expended, are generally measured by the "square"=100 sq. ft. Some work is measured by the linear foot or yard, some by the square foot or yard, and some by the cubic foot. For some of the more difficult kinds of work, it is usual to allow " measure and half," or " double measure," but the custom varies so much in different places, that it is impossible to give any general rules of measurement. Shingles are generally 18 inches long, and of the average width of 4 inches. When nailed to the roof 1 is usually left out to the weather, and 6 shingles are therefore required to a square foot. But on account of waste and defects, 1000 should be allowed to a "square." The weight of a square of partitioning may be estimated at from 1500 to 2000 lb.; a square of single-joisted flooring, at fromn 1200 to 2000 lb.; a square of framed flooring, at fiom 2700 to 4500 lb.; a square of deafening, at about 1500 lb.a When a floor is covered with people, 120 lb. per sq. ft. should be added to the weight.b 5. The sliding r~ule is often used by carpenters and other artificers, in the measurement of timber and work. The foot is divided in the usual way, into inches and eighths of an inch, and it is also subdivided decimally and logarithmically, so as to facilitate the labor of computing. The use of the rule can only be learned by practice. The carpenter's square is used in determining whether the corners of boards or buildings are square. In framing buildings, the corners are sometimes "' squared," by measuring 8ft. on one timber, and 6ft. on the other, and placing the extremities of the measured lines O10ft. apart. This mode of operation is founded on the property of right-angled triangles, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. A roof is said to have a true _pitch, when the length of each rafter is I of a Hatfield. b Tredgold.

Page  139 ~ 27.1 THE CARPENTER AND JOINER. 139 the breadth of the building. The two sides of the roof then form nearly a right angle. 6. The area of posts.-All rules for determining the resistance of timber, should be based on the supposition that the timber is of " merchantable" quality, straightgrained, seasoned, and free from large knots, splits, decay, or other defects. When the height of a piece of timber exceeds about ten times its thickness, it will bend before crushing. To find the area of a post that will safely bear a given weight, when the height of the post is less than ten times its least thickness; —Divide the given weight in pounds by 1000 for pine, or by 1400 for oak, and the quotient will be the least area of the post in inches. EXAMPLES. 1. Required the contents of a board 13ft. 7in. long, iin. thick, and ift. 6in. wide, and its value at N9cts. per foot. Partial Ats. aTalue $1.94. 2. The length of aboard is 11ift. 8in., the thickness Ir'in., and the breadths measured at five different points are as follows; ift. 6in., 2ft. 3in., ift. 9in., 2ft. 6in., and 2ft. What are the contents? Aens. 35ft. 3. How many cubic feet, and how many feet board measure, in a log 21ft. 6in. long, the mean breadth being Ift. 4in., and the mean thickness ift. 3in.? 4. A log is 42ft. 9in. long, and the girts at four points, outside of the bark, are 56, 451, 58, and 64- inches. What are its contents by the ordinary rule, and by the correct rule? 5. How many squares of flooring in a four story house, 42ft. 6in. by 28ft 4in. within the walls, deducting from each floor the vacancy for the stairway, 13ft. by 7ft. 6in.; and what is the cost of the whole, at $3.87~ per square? 6. A board measures at five different points, lft. Gin.,

Page  140 140 ARTIFICERS' WORK. [ART. VIrI. lft. 9in., ift. 8in., ift. 4in.5, and lft. 3in., in breadth. What is its length, the area being 181ft.? Ans. 12ft. 6'. 7. A stick of timber 9ft. 6in. long, measures 1 ton 141 c. ft. What is its mean area? Ans. 5ft. 9'. 2S. THE MASON. 1. Rubblea walls are generally measured by the perch, which is 160ft. long, Ift. deep, and l1ft. thick, and is therefore equivalent to 241 c. ft. In some places, 25 c. ft. is allowed to the perch, in measuring stone before it is laid, and 22 c. ft. after it is laid in the wall. When the wall is not of uniform height, the height should be measured at several places, from the bottom of the foundation to the top of the wall, and the mean height employed in computing the solid contents. Nine pecks of good lime and 3 one horse loads of sand, will make mortar for 3 perches of wall. Rough stone and marble are often measured by the cubic foot. In measuring workmanship, linear feet and yards, and square feet and yards are employed. 2. The rood of 36 sq. yd. is sometimes employed in measuring walls that are more than 18in. thick. The wall should first be reduced to 2ft. thick. Thus a wall 90ft. long, 8ft. high, and 21in. thick, is equivalent to I of a wall 90ft. long, 8ft. high, and 2ft. thick; i x 90 X 8=630 sq. ft; 6309=70 sq. yd.; 70+ 36=1 rood 34yd. 3. Cisterns can be measured accurately by finding the solid contents in cubic inches, and dividing by the number of cubic inches in a hogshead, (63 x 231.) But, in measuring circular cisterns, the rules given in Sect. 25, are much more convenient than this method, and are sufficiently correct for ordinary purposes. If great accuracy is required, measure the diameter and depth in inches, multiply the square of the a Rough stone work is called rubble work.

Page  141 ~28.] THE MASON. 141 diameter by the depth, and multiply the product by either of the following numbers, to obtain the contents:In cubic feet In ale gallons In wine gallons In cubic inches.00045451.002785.0034.785398 In hogsheads In bushels In lbs. of water In Imperial gal..000054 e 13-.028326.0028326 4. Arches are measured by applying a line close to the surface in taking the dimensions. If the arch is not of uniform length, breadth, and thickness, the dimeLsions may be measured at several points, and the mean of all the measurements taken. A pointed arch will sustain almost any weight on its crown, provided the lowest stones do not give way. Therefore the Gothic arch is stronger for lofty buildings than the circular, but the circular arch is far better adapted than the Gothic, for bridges or other works, where every part of the arch may be exposed to equal, or nearly equal pressures. EXAMPLES. i. How many cubic feet in a block of marble, 4ft. 6in. long, 3ft. 8in. wide, and 2ft. 4in. thick? 2. How many perches of 243 c. ft. in a wall 97ft. long, and 2ft. thick, the heights at five different points being 4ft. Sin., 3ft. 8in., 3ft. 9in., 4ft., and 5ft. 2in. and how much lime will be required to make mortar for the whole? 3. Required the cost at 50cts. per perch of 25 c. ft., of making a cellar wall 6ft. 3in. high, and ift. 6in. thick, the outside measurement being 47ft. long, and 22ft. 6in. wide? 4. How many hogsheads in a cylindrical cistern 13ft. 6in. in diameter, and lift. 7in. deep? 5. How much hewn work in 18 lintels and sills, each 4ft. by Ift. 6in., and in chimney coping, 58ft. 6in. by 19in., and what is the cost of the whole at 10~cts. per foot? Partial Ans. Cost, $21.06.

Page  142 142 ARTIFICERS' WORK. [ART. VIIr. 6. Find the contents in cubic feet, in hogsheads, and in Imperial gallons, of a cylindrical cistern 10ft. 6in. in diameter, and 9ft. 2in. deep. 1st Ans. 793.738 c. ft.,29. THE BRICKLAYER. 1. Breickwork is measured either by the square yard, or by the square rod, and is usually estimated at 1- bricks thick,= 12 inches, the actual thickness being reduced to the standard of 1- bricks. 272 sq. ft. is generally counted as a rod, the fraction of 1 sq. ft. being rejected.a The rood of 18Sft. sq., or 324 sq. ft., and the rod of 164 sq. ft., are also used. But the common practice is now to reckon bricks by the 1000, the number required depending on the size of the bricks. In estimating the number, an allowance of a of the solid contents should be made for the space occupied by the mortar. 2. The usual dimensions of bricks, are; length 8in., breadth 4in., thickness 2in. Whatever the length may be, the breadth is generally 4, and the thickness X as great; thus a brick 9 inches long, would be 44 in. wide, and 21 in. thick. In some places, all walls are charged as solid, no allowance being made for doors, windows, or other openings; in others, the openings are deducted in charging for materials, but the workmanship is estimated as if the walls were solid; in others an allowance of one half is made for all openings; and in others the actual materials employed, and workmanship expended, are charged. 3. The number of rods in any wall may be found by multiplying the area of the surface by ~ of the number of half-bricksb in the thickness, and dividing the product by the number of square feet allowed to a rod. A load of sand= 30 struck bushels; a ton =- 24 e. ft. a The weight of a rod of brickwork may be estimated at 16 tons. b A half-brick - 4 inches.

Page  143 ~ 30.] THE PLASTERER. 143 EXAMPLES. 1. How many bricks of the usual size, will be required to make a wall 40ft. long, 16ft. high, and 2 bricks thick, making allowance for mortar? Ans. -9 of 40 x 16 X 1 27=20736. 2. How many roods of 324ft. in a wall 96ft. long, 10ft. high, and 3 bricks thick, and what is the cost of the bricks at $7.75 per NI? Ans. Cost, $361.58. 3. A garden contains 11 acres, and is 150ft. wide. Required the cost of enclosing it with a brick wall 9ft. 4in. high and 3 bricks thick, at $7.50 per M., deducting 2 doors, each 6ft. 3in. by 4ft., and a gateway 12ft. wide? Ans. $3404.19. 30. THE PLASTERER. Plain plastering is measured either by the square foot, the square yard, or the " square" of 100 sq. ft. The number of coats, and the quality of the finishings, should be stated in the bill. Cornices and mouldings, if 12 inches or more in girt, are sometimes estimated by the square foot; if less than 12 inches, they are usually measured by the linear foot. Plastering on walls is called rendering; ceiling, is plastering on laths. The custom varies as to the proper allowance for doors and windows. EXAMPLES. 1. What will be the cost of plastering a ceiling 21ft. 6in. long, and 19ft. 6in. wide, at olets. per sq. yd.? 2. How much plastering on a partition 22ft. 3in. long, and 7ft. 9in. high, deducting two doors, each 6ft. by 3ft. 2in., and what will it cost at 12 cts. per sq. yd.? Ans. Cost, $1.87. 3. How many squares of ceiling, and of rendering, and

Page  144 144 ARTIFICERS' WORK. [ART. VIII. how many feet of cornice, in a hall 60ft. long, 28ft. 6in. wide, and 10ft. high, deducting 17yd. 5ft. 6' for doors and windows? Ans. 16 sq. lyd. 2ft. 6' of rendering. 17 sq. lyd. Ift. of ceiling. 177ft. of cornice. 31. THE PAINTER AND GLAZIER. Painting is measured by the square yard. In taking the dimensions, the measuring line is laid into all the mouldings, so as to reach every point which the brush touches. Glazing is sometimes measured by the square foot, sometimes by the piece, or by the light. In estimating by the square foot, it is customary to include the whole sash. Circular or oval windows are measured as if they were square. EXAMPLE S. 1. A room is 24ft. long, l8ft. 6in. wide, and 9ft. 6in. high. How many yards of painting are in it, deducting a fire-place 4ft. 6in. by 4ft., and 3 windows, each 6ft. by 3ft. 3in.? Ans. 81yd. 2ft. 2. At 12~cts. per yard, what will it cost to paint a wall 15ft. 8' by 8ft. 3'? Ans. $1.78. 3. How many feet of glazing in an oval window 4ft. 6' by 2ft. 8'o? 4. At 18l-cts. per foot, what will it cost to glaze 3 stories of a house, with 8 windows in each story, the breadth of each window being 3ft., and the height 5ft. 4in.? Ans. $72. 32. THE PAVER, SLATER AND TILER. Paving is measured by the square foot, the square yard, or the rood of 36 sq. yds. If the pavement is grooved, the grooves are added to the surface measure. Slating and Tilifng are measured by the square yard, by

Page  145 ~ 33.] THE PLUMBER. 145 the rood, or by the "square" of 100 sq. ft. It is not usual to make any deductions for chimneys, skylights, or other apertures. In measuring the girt of slate roofs, allowance must be made for the double row at the eaves. EXAMPLES. 1. A yard is 80ft. long, and 28ft. 6in. wide. What will it cost to pave it at 70cts. per square yard? 2. The side of a square court measures 120 feet. What will it cost to pave it, leaving a flagged walk Gft. wide around the outside, at 75cts. per yd., and paving the rest with bricks at $8.00 per M., allowing 4 bricks to a square foot? Ans. $601.25. 3. How many roods of tiling in a roof 52ft. 8' long, and 45ft. 6' in girt? 4. At $1.30 per yard, what will be the expense of slating a roof 49ft. 6' long, and girting 46.ft.? Anls. $3332. 33. THE PLUMBER. Pllmbers' work is generally done by the pound or hundred-weight. A square foot of sheet lead, -l-in. thick, weighs 5.899 lb. From this value the weight of a square foot of any other thickness can readily be determined. Lead pipes of -in. bore, weigh about 10lb. per yd.; lin. bore, 121b.; 1-in. bore, 161b.; 1tin. bore, 181b.; lEin. bore, 211b.; 2in. bore, 24 lb. The contents of lead pipe of any given dimensions, may be found by the table in Sect. 28, for measuring cisterns. EXAMPLES. 1. What is the weight of 1 sq. ft. of lead, the thickness being i of an inch? a The weight of a square foot of slating may be estimated at 111 lb. The greatest force of the wind on a roof, is about 40 lb. per square:fot. —Tredgold. 10

Page  146 146 ARTIFICERS' WORK. [ART. VIII. 2. Find the weight of lead necessary to cover one side of a roof 36ft. 9in. long, and 18ft. 3in. wide, at 8~ lb. per square foot. 3. What is the cost of covering and guttering a roof, at 18s. per cwt., the length of the roof being 43ft., and the girt 32ft.; 57ft. of guttering 2ft. wide; weight of roofing, 9.831 lb. per sq. ft., and of guttering 7.373 lb. per sq. ft.? Ans. ~115 9s. lid. 4. Required the expense of a leaden pipe, lain. bore, and 185ft. long, at 1lets. per lb. 5. A roof 40ft. long, and 57ft. girt, is covered with lead'in. thick; the water pipe is lain. bore and 52ft. long, and the waste pipe is 2in. bore and 40ft. long; the water cistern is 4ft. 3in. long, 3ft. 6in. wide, and 3ft. deep, and lined with lead {in. thick. What is the amount of the plumber's bill, rating the sheet lead at $7.50 per cwt., and the pipe at lOcts. per pound? 34. SPECIFICATION AND ESTIMATES. 1. Specification of materials to be provided, and labor to be performed,in the construction and finishing of a Schoolhouse for the City of Worcester, to be erected on Summit Street in said City, according to plans drawn by E. Boyden, Architect, and herewith presented. Size of House.-58 feet long by 50 feet wide, not including the projection of the pilasters. Height of stories as figured on Section of Front Elevation. The location of the cellar, and depth of excavation, to be determined by the building committee. A well to be dug upon the lot in such location as directed by the building committee, and also to be stoned up so as to leave a diameter of three and a half feet in the clear. All earth dug out of the cellar and well, to be deposited upon the lot, as may be directed by the building committee. The foundation walls to be three feet thick at the bottom, 2.} feet thick at the top, and of such height as may be determined by

Page  147 ~ 34.] SPECIFICATION AND ESTIMATES. 147 the building committee. The walls to be made of large square block-stone, well faced and bonded.a Underpinning 2 feet wide, and not less than 8 inches thick, to be made of rough split South Ledge stone, with a rough-hammered bevelled washb between the pilasters. The face of the underpinning to project as far forward as the face of the pilasters. Stone steps, of fine-hammered South Ledge granite, located and of such size as represented on the plan. All the foundations for piers and partition walls, to be not less than 8 inches thick, and to be placed as represented on the plan of the cellar. Stone lintels over all the cellar windows, and four stone thresholds of fine-hammered South Ledge granite, to be made of the dimensions marked on the ground plan. Outside wall of brick to be one foot thick, with pilasters projecting 4 inches beyond the face of the wall, and Corbel-Coursec and Frieze,d as shown on elevation. Four brick piers in the cellar, each one foot square, and a brick partition 8 inches thick to be carried from the bottom of the cellar to the attic floor, as represented on the plans. The chimneys to be located and constructed as represented on the ground plan. Building to be lathed throughout, and plastered with two coats, except the play-room in the basement. Walls not to be plastered underneath the ceiling. All the windows to have four-course,e tooled sandstone caps, and two-course sills. The doors to have five-course caps of the same material. Twenty ventilating registers, each one foot in diameter, to be furnished and inserted, two in each ventilating flue, one near the floor and the other near the ceiling. All the rooms to be thus ventilated except the play-room. For the arrangement and sizes of timber in the first three floors, see plan of flooring, as represented on the basement plan. a Laid like bricks, so that the joints will not come over each other. b The wash of the stone is the inclined surface for water to run off. c Projections in a wall to sustain the timbers of a floor or roof, are called corbels. a The part of the wall above the pilasters. e Of the thickness of four courses of brick.

Page  148 148 ARTIFICERS' WORK. [ART. VIIL Frames to be made entirely of good spruce framing timber. Joists in 4th or attic floor to be 2 X 9, framed 15 inches between centres. All other timber in roof and observatory to be of the size figured on the plan of roof. Joists in first three floors to be Jointed in. crowninga to 15 feet in length. All floorings to be bridged with good X bridging where marked on plan of flooring. All large timber to be well and properly secured to the brick walls' by suitable anchor irons. The joists to be also secured in a similar manner, as often as once in every ten feet. Roof to be boarded with suitable 9in. boards, planed, jointed, matched, and suitably tinned. Roof to be bracketed and project as represented on elevation. Observatory to be framed and finished in every respect as represented and figured on plan of observatory, front elevation, and plan of roof. Lintels 6 X 7'in. to be furnished for all windows and doors. All floors to be lined with suitable Uin. lining boards laid edge to edge, and nailed with 8d. nails. The floor to the upper schoolroom to be deafened in the centre of the floor joists with a suitable coat of coarse mortar. All top floors to be of suitable southern hard pine 7in. thick, and not to exceed 6in. in width, well laid and'nailed with 12d. floor nails. Five iron columns to be placed as represented on plans of basement and second story, of size and quality like those in the Pleasant Street Schoolhouse. Partitions to be arranged as represented on plans. Those in basement and 2d story, to be constructed of 2X5in. partition plank, bridged once with 14 X 5in. herring-bone bridging. Those in 3d story to be of 2 X 6in. partition plank, bridged twice with herring-bone bridging as above. Allpartition plank to be jointed and set edgewise, so as not to exceed one foot between centres. The contractor is also to make all necessary arrangements in the partitions for pipes to convey heated air to all the different apartments, and to do all necessary wood-work preparatory to putting in registers to admit the hot air. The house to be furred throughout with 4 X2~ inch furrings, placed at the distance of one foot between their centres. Teachers' platforms to be elevated 6 inches above the floor, of the situation and sizes represented on plans. a Long timbers are usually made " crowning" in the centre, so as to allow for settling.

Page  149 ~34.] SPECIFICATION AND ESTIMATES. 149 All window-frames to be constructed as represented by the drawings, with hard pine pulley styles.a Four cellar window frames, to be made of 2in. chestnut plank, each large enough for a window with 4 lights of 9X 12 glass. All sash to be of first quality Eastern pine stock, lip sash, ogee b style, 1-2in. thick, to be double-hung with suitable weights, cords, and pulleys, and glazed with best-quality German glass of such sizes as figured on front elevation. All windows in 2d and 3d stories to have blinds to slide into the walls upon each side of the window, as shown on detail of window frame. Four outside doors with side lights, as represented on front elevation, each 8ft. high by 3'ft. wide, and 2in. thick, 4 panels with bevel joints, to be hung with 3 sets 4in. loose joint butts, and trimmed with suitable mineral knobs. The two front doors to have suitable mortice locks, and the two other doors suitable bolts inside. All other doors to be 31X 7ft. 4 panels, 1Hin. thick, with bevel joints, suitably hung with 3 sets of 4in. loose joint butts, and trimmed with suitable mineral knobs and mortice locks. A flight of cellar stairs, with hard pine treads ain. thick, placed as represented on cellar plan. All other stairs located as represented on plans, with hard pine risers lin. thick, and treads l in. thick, and to have cherry newelsc and hand-rails. All staircases to be ceiled up on the well-roomd side as high as the hand-rail, and all rooms to be ceiled as high as the window stools, with suitable Eastern pine stock tin. thick, not to exceed 6 inches in width, jointed, matched, and beaded. Cleats to be put up in entries and recitation rooms, sufficient to contain 30 doz. glazed clothes-hooks, placed 6 inches apart, and to be provided with said hooks. All doors and standingwood-workinside, to be grained in imitation of oak. Jet and brackets to be painted and sanded in imitation of sandstone. Observatory, window frames, and door frames, to be painted white and sanded. All painting to be done with three coats of pure white lead and linseed oil, colored as above specified. Tinning upon the roof to be painted with one-coat upon the under side, and two upon the upper side, of spruce yellow and boiled linseed oil. a The strips in which the window pulleys are placed. b An ogee is a moulding resembling, the letter S in its outline. c The posts into which the hand-rail is inserted are called newels. d The space occupied by thestairway is called the well-room.

Page  150 150 ARTIFICERS' WORK. [ART. vrI A piece of raised tin to be placed on each side of each front door to turn the water off outside of the door. A single floor to be laid in the attic, of common white pine boards Zin. thick, planed, jointed, matched, and nailed with 8d. nails.a 2. ESTIMATE.b 45 squares of roof. at- $12 33 " ceiling... 8 32 windows...... 16 12 " 10 5 iron columns. 12 4 outside doors with frames and trimmings 16 6 flights of stairs. 50 30 squares attic flooring.. 3 3 hard pine floors. 900.00 Iron work for building... 45 Observatory.... 250.00 40 brackets...... 2.25 14 inside doors trimmed.... 8 14 door casings..... 4.75 35 window casings. 1.12~ 2000ft. boards for furring... 16 Labor in furring.... 75.00 Painting....... 250.00 32000ft. timber in frame.... 22 Deafening $150. Nails $10 2200yds. plastering...22 200000 brick.... 8.50 Caps and sills.... 225.00 175 seats... 3.25 175 "........80 125 perch of stone.... 1.00 500yd. excavation....15 $7991.62~ a The foregoing specification furnishes materials for a great number of useful questions, which the teacher may frame so as to adapt them to the wants of his pupils. b In each item, the expense of labor is included.

Page  151 ~ 34.] ESTIMIATES. 151 3. ESTIMATE OF THE MATERIALS AND LABOR REQUIRED IN A COTTAGE. -From Ranlett's Architect. $ 296 cubic yds. excavation @.09 2538 cubic ft. stone work @I.10 7 stone sills.50 24 linear ft. steps (.14 Cistern work $3.50; 4 hearths @ 3.00 1 marble mantel $50; 2 veined mantels @ 25.00 2 brown stone chimney caps @ 14.00 1451 square yds. plastering (J.26 36500 brick @ 9.50 per M. 268 linear ft. cornice ~.24 16196ft. timber in frarme ( 2.00 per lund. 474 joist, set in frame and partitions @.18 4782 sq. ft. of sheathing and siding @.07 2283 " " and iron roof (.16 60 linear ft. 3in. leader, @ 12~c., 114ft. ~.11 3485 square ft. of interior floor (.04 1002 " " veranda" (1I.08 1184 "c " " roof @.09 172 lin. ft. main cornice (.85 83 " "wing " @.70 199" " veranda" @.55 15 veranda columns@($10.50; 3 antee @ 3.00 164ft. veranda cornice and filling @.14 15 steps and rises-back stairs @ 1.00 18 " "'" principal" @ 3.50 2 front doors, with side and head lights @ 30.00 11 doors in principal story ~ 11.00 10 " second " (I 8.00 10 " wing ( 7.00 7 double windows, first story ~ 14.00 8 " " second " ( 10.00

Page  152 152 STRENGTH OF MATERIALS. [ART. IX. $ 7 double windows, wing (@ 9.00 8 single " @ 6.50 6 cellar " 2.50 3 wood mantels @ $4.50; 7 bells @ 3.25 400 square yds. tight furring @.07 197 linear ft. of blinds (.80 1152 " " of base (~.04 12 closets, to shelve and put in hooks @ 4.50 1000 lb. white lead in oil' @ 7.00 per hund. 43 gallons linseed oil @ 80 4 " " boiled @.90 5 " spirits turpentine.50 " varnish @ 4.00 30 lb. putty, ~ 4ct.; 10 lb. litharge ~.06 3 lb. glue, @ 20ct.; 2 lb. lampblack in oil @.40 6 lb. chrome yellow in oil @.30 60 days painters' labor @ 1.75 Hardware;-locks, bolts window weights, &c. 99.39 $4550.00 IX. STRENGTH OF MATERIALS. ALL solid substances may be exposed to four kinds of strains. 1st, they may be torn asunder, as in the case of ropes, tie-beams, king-posts, &c. 2d, they may be crushed, as in the case of columns, posts, &c. 3d, they may be broken across, as in the case of joists, beams, &c. 4th, they may be twisted or wrenched, as in the case of wheel-axlesj the screw of a press, the rudder of a vessel, &c. Numerous experiments have been made to determine the strength to which the materials in common use may safely be subjected, and tables have been compiled from the results of those experiments.

Page  153 ~36.] STRENGTH OF TIMBER. 153 35. TABLE OF THE FLEXIBILITY AND STRENGTH OF TIMBER.a Name of woo(l. G vi Specific Value Value of Value Name of wood. Gravity of U.b alue of C. Teak 745 818 9657802 2462 15555 Poon 579 596 6759200 2221 14787 English Oak 969 598 34947830 1181 9836 Do. specimen 2 934 435 5806200 1672 10853 Canadian Oak 872 688 8595864 1766 1I428 Dantzic Oak 756 724 4765750 1457 7386 Adriatic Oak 993 610 3885700 1583 8808 Ash 1 760 395 6580750 2026 17337 Beech 696 615 5417266 1556 9912 Elm 553 509 2799347 1013 5767 Pitch Pine 660 588 4900466 1632 10415 Red Pine 657 605 7359700 1341 10000 New England Fir 553 757 5967400 1102 9947 Riga Fir 753 588 5314570 1108 10707] Do. specimen 2 738 3962800 1051 Mar Forest Fir 696 588 2581400 1144 9539 Do. specimen 2 693 403 3478328 1262 10691 Larch 531 411 2465433 653 Do. specimen 2 522 518 3591133 832 Do. specimen 3 556 518 4210830 1127 7655 Do. specimen 4 560 518 4210830 1149 7352 Norway Spar 577 648 5832000 1474 12180 36. PROBLEMS IN DETERMINING THE STRENGTH OF TIMBER.a 1. To find the strength of direct cohesion of a piece of timber of any given dimensions. Rule.-Multiply the number of square inches in the transverse section by the value of C in the table, (~35,) and the product will be the strength required in pounds. N. B. If the specific gravity differs from the mean specific gravity of the table, multiply the product by the actual specific gravity, and divide by the tabular specific gravity for the correct strength. a Ingram. b Ultimate deflection. c Elasticity. d Strength. e Cohesion.

Page  154 154 STRENGTH OF MATERIALS. [ART. IX. If 6 =the breadth in inches, and d-= the depth, h XdXC W. If either b or d is required, divide W by the product of the remaining factors. 2. To find the deflection of a beam fixed at one end, and loaded with any given weight at the other. Rule.-Divide 32 times the weight multiplied by the cube of the number of inches in the length of the beam, by the continued product of the tabular value of E, the number of inches in the breadth, and the cube of the number of inches in the depth of the beam. N. B. When the beam is loaded uniformly throughout, multiply the cube of the length by 12 times the weight, instead of 32 times the weight. 3. To find the deflection of beams supported at both ends, and loaded in the middle with any given weight. Rule. —Multiply the cube of the number of inches in the length, by the number of pounds in the given weight, and divide by the continued product of E, the number of inches in the breadth, and the cube of the number of inches in the depth of the beam. N. B. When the beam is not only supported, but is fixed at both ends, the deflection is i of that given by the rule. If the weight is distributed uniformly throughout the length of the beam, the deflection will be i of that given by the rule. 4. To find the ultimate deflection before their fracture of beams or rods supported at both ends. Rule. —Multiply U by the number of inches in the depth of the beam, and divide the square of the number of inches in the length by the result. 5. To find the ultimate transverse strength of any rectangular beam of timber, fixed at one end and loaded at the other. Rule.-Find the continued product of S, the number of inches in the breadth, and the square of the number of inches in the depth, and divide the continued product by the number of inches in the length.

Page  155 ~ 36.] STRENGTH OF TIMBER. 155 If W represents the number of pounds that would produce fracture, I the length in inches, b the breadth in inches, and d the depth in inches, l=SXbxd2 X W; b lXW (SX d2); dVlXw —(S X b); S = I X W-(b X d2). 6. To find the ultimate transverse strength of any rectangular beam when supported at both ends, and loaded in the centre. Rule.-Find the continued product of S, 4 times the number of inches in the breadth, and the square of the number of inches in the depth, and divide the product by the number of inches in the length. -=4 X S X b X d2W; b= X W- *(4 X S X d2); d=_ -/1 X W-(4 X S X h); S=4 X W — (4 X b X d-). N. B. When the beam is fixed at each end and loaded in the middle, the result obtained by the rule must be increased onehalf. When the beam is loaded uniformly throughout its length, the result obtained by the rule must be doubled. When the beam is fixed at both ends, and loaded uniformly throughout, the result obtained by the rule must be trebled. If the load is to be permanent, it should not exceed - of the amount obtained by the rules. 7. To find the weight under which a given column will begin to bend, when placed vertically on a horizontal plane. Rute. —Find the continued product of E, the cube of the number of inches in the least thickness, the number of inches in the greatest thickness, and.2056. Divide this continued product by the square of the number of inches in the length. If W represents the number of pounds that the column can sustain, d the number of inches in the greatest thickness, b the number of inches in the least thickness, and I the number of inches in the length; d W X 12. (E X 3 X.2056); b = /W X 12 (.2056 X E X d); I=-/Exd b3 X.2056; E c W X 12 — (d X b3 X.2056).

Page  156 156 STRENGTH OF MATERIALS. [ART. IX. 37. EXAMPLES FOR THE PUPIL. 1. What weights will be required to tear asunder two pieces of beech, each 4in. wide, and 3in. thick, the sp. gr. of the 1st being 696, and that of the 2d 678? 2d Ans. 115868 lb. 2. A red pine beam 8Sft. long, 4in. broad, and 6in. deep, is fixed at one end, and loaded with a weight of 5001b. Required the deflection, when the weight is suspended at its extremity, and also when it is distributed uniformly throughout the length of the beam. 1st Ans. 2.51in. 3. A beam of Canadian oak, 5in. broad, 8in. deep, and 25ft. long, is supported at both ends, and loaded with a weight of 3000 lb. Required the deflection when the weight is placed in the centre, and also when it is distributed uniformly throughout the length. 2d Ans. 2.23 inches. 4. Required the deflection at the instant of fracture, of an ash beam 30ft. long, 9in. wide, and 6in. deep. Ans. 54.68 inches. 5. What weight will be required to break a beam of larch, sp. gr. 531, fixed at one end and loaded at the other, the breadth being 3in., depth 6in., and length 6ft? Ans. 9791 lb. 6. What weight will be required to break a beam of pitch pine, supported at both ends and loaded in the middle, the length being 16ft., the breadth 6in., and the depth 9in.? What weight would be required if the beam were fixed at both ends, and loaded uniformly throughout? 1st Ans. 16524 lb. 7. What weight will bend a column of New England fir, 8ft. 4in. long, 8in. wide, and 6in. thick, placed vertically on a plane, the weight being applied to its upper extremity? Ans. 212007.8776 lb. 8. What weight would have been required to break the beam mentioned in Ex. 6, if it had been fixed at each end,

Page  157 ~'38] TABLES. 157 and loaded in the middle? If it were merely supported at each end, and loaded uniformly throughout its length? 38 S. TABLE SHOWING THE NUMBER OF POUNDS THAT WILL PULL ASUNDER A PRISM ONE INCH SQUARE; OF DIFFERENT MATERIALS, ACCORDING TO THE EXPERIMENTS OF M. MUSCHENBROEK.a Cast gold... 22000 Bismuth.2900 Cast silver.... 41000 Good brass... 51000 Anglesea copper... 34000 Ivory.. 16270 Swedish copper... 37000 Horn.. 8750 Cast iron..... 50500 Whalebone. 7500 Bar iron, ordinary. 68000 Ditto, best Swedish. 84000 COMPOSITIONS. Bar steel, soft... 120000 Gold 5, copper 1,.. 50000 Ditto, razor temper. 150000 Silver 5, copper 1,.. 48500 Cast tin, Eng. block. 5200 Swedish copper 6, tin 1, 64000 Ditto, grain... 6500 Block tin 3, lead 1,. 10200 Cast lead... 860 Tin 4, lead 1, zinc 1,. 13000 Regulus of antimony. 1000 Lead 8, zinc 1,. 4500 Zinc.2600 ACCORDING TO THE EXPERIMENTS OF Mr. RENNIE. No. of lbs. that would Length in ft. that tear asunder a prism wnnuld breako itb 1 in. square. its own weight. Cast steel. 134256 39455 Swedish iron. 72064 19740 English iron... 55872 16938 Cast iron..... 19096 6110 Cast copper. 19072 5092 Yellow brass. 17958 5180 Cast tin.....4736 1496 Cast lead..... 1824 384 Good hemp rope 6400 18790 Ditto, lin. in diameter 5026 18790 39. THE LATERAL STRENGTH OF BARS ONE FOOTLONG, AND ONE INCH SQUARE.0 Weight that will bleak theon. with sa.fet y Bre a kin g w eig ht. with safety. Blealiing meieht. weight. Cast iron 3270 lb. 1090 lb. Memel fir 390 lb. 130 lb. Oak 627 lb. 209 lb. Am. white pine 206 lb. 69 lb. a Ingram.

Page  158 158 STRENGTH OF MATERIALS. [ART. IX 40. THE COHESIVE FORCE OF A SQUARE INCH OF IRON OF DIFFERENT KINDS.a Iron wire.... 1130771b. English iron... 65772 lb. Dittob... 93964 " Welsh iron... 64960" Swedish iron.. 78850" Ditto.... 55776" Ditto... 72064" German iron... 69133 " Ditto. 54960" French iron... 61000" Ditto... 53244" Russian iron... 59472" English iron... 66000" Cast iron.... 18295" Ditto... 55000" Ditto.19488" Ditto.... 61600" Ditto, Welsh... 16255" 4L1. THE NUMBER OF POUNDS NECESSARY TO CRUSH C(UBES OF 1~ INCHES.a Aberdeen granite, blue 24536 Craigleith stone, White-veined Ital. marble 21788 with the strata 15560 Very hard freestone.. 21254 Ditto across ditto 12346 Purbeck limestone... 20610 Cornish granite..... 14302 Limerick limestone, black 19924 White statuary marble. 13632 Peterhead granite... 18636 Fine brick....... 3864 Compact limestone... 17354 Yellow baked brick... 2254 Yorkshire paving stone. 15856 Red brick........ 1817 Dundee sandstone.... 14919 Pale red brick..... 1265 Chalk......... 1127 ONE-INCH CUBES WERE CRUSHED BY THE FOLLOWING WEIGHTS: Elm...... 12841b. English oak... 3860 lb White deal.... 19281b. Craigleith stone.. 8688 lb. CUBES OF ONE-FOURTH OF AN INCH WERE CRUSHED BY THE FOLLOWING WEIGHTS: Iron, cast vertically. 111401b. Cast tin......... 9661b. Ditto horizontally. 10110" Cast lead........ 483 " Cast copper..... 7318" a Ingram. b The different numbers represent the different results obtained by the most careful experimenters.

Page  159 ~ 43.] TABLES. 159 422. TABLE OF IRON AND HEMPEN CABLES OF EQUAL STRENGTH.a Iron Cables. Hemp Cables. Resistance. Diameter of Iron Rod. Circumference of Rope. Inches. Inches. Tons. 79 12 1 10 18 41 11 26 l~~ ~ ~12 32 13 35 1 14 to 15 38 16 44 1-,g 17 52 1 8 18 1 a 1 8 60 1 7`20 70 2 22 to 24 80 The stress given in the above table is the greatest to which the cables should be exposed, and is about ~ the breaking strain.b A " cable's length," is 120 fathoms. 43. MEAN WEIGHT OF A CUBIC FOOT OF STONE, AND THE WEIGHT IT WILL SUSTAIN WITH SAFETY.c Weight. Pressure. Sandy Bay granite 168.48 1b. 1197000 lb. Quincy. 167.04 " 156000 " Concord. 159 " 149000 " Frankfort ".162 " 148000 " New York white marble 173 185000 " N. Haven variegated " 175 " 89000 " Penn. dove marble 170 " 86000 " Vt.,,.....168 " 86000 " Thomaston blue marble 179 " 90000 " Connecticut sandstone 164 " 118000" North River ".16. 108000 " Potomac.. 153 " 98000 " a Ure. b A common rope Ift. long, and 1 inch in circumference, weighs.044 to.046 lb. In a cable, it weighs.027 lb. To find the number of pounds which a rope will sustain, square the number of inches in its girt. and multiply by 200 fobr common ropes, and by 120 for cables.-Tredgold. c Shaw.

Page  160 ~160 STRENGTH OF MATERIALS. [ART. IX, 4z4, PROBLEMS ON THE STRENGTH OF IRON. 1. To find the breadth of a uniform cast-iron beam, to sustain a given weight in the middle. Rule.-Multiply the number of feet in the length, by the number of pounds to be supported, and divide the product by 850 times the square of the number of inches in the depth. The quotient will give the number of inches in the breadth. When neither the breadth nor depth is known, but merely the proportion that exists between them; find the continued product of the number of ft. in the length, the number of lb. in the weight, and the ratio of the depth to the breadth; divide the continued product by 850, and extract the cube root of the quotient. The result will be the number of inches in the depth. If W = the weight to be supported in pounds, 1 = the length in feet, b - the breadth in inches, and d = the depth in inches, W - 850 X b X d2 l d / (1 X W) (850 X b); I = 850 X b X d2 * W. 2. To find the proper breadth and dep'th of a beam of cast iron supported at both ends, when the load is not in the middle. Rule.-Measure the number of. feet from the point at which the weight is applied, to each support, and find the product of the two numbers; divide 4 times this product by the whole length between the supports, and proceed with the quotient in the same manner as with the length in Problem 1. N. B. When the load is uniformly distributed over the length of the beam, the depth need be only -4 as great as when it is all placed in the middle. 3. To find the proper breadth and depth of an iron beam fixed at one end, and the load applied to the other, or of a beam supported upon a centre of motion. Rule.-Take the length from the point at which the beam is fixed, or from the centre of motion, to the point where the

Page  161 ~44.] PROBLEMS ON THE STRENGTH OF IRON. 161 load is applied, and calculate the strength by the rules in Problem 1, using instead of 850, 212 for cast iron, 238 for wrought iron, or 425 when the weight is uniformly distributed over the length of the beam. 4. To find the proper depth of the teeth of wheels. Rule.-Divide the number of pounds which represents the stress of the wheel at the pitch-circle,a by 1500, and extract the square root of the quotient. The result will give the thickness of the teeth in inches. N. B. The length of the teeth ought not to exceed their thickness. The breadth should be in proportion to the stress upon them, and this stress should not exceed 400 lb. for each inch in breadth. 5. To find the proper thickness of the teeth of a wheel, when the power of the first mover is given in pounds, and the velocity per second in feet. Rule.-Form the continued product of the numbers which represent the power and velocity per second of the first mover, and.073; multiply the number of revolutions the wheel is to make per minute, by the radius the wheel should have if its pitch were two inches; divide the first of these products by the second, and the cube root of the quotient will give the thickness of the teeth in inches. 6. To find the proper diameter of a solid cylinder of cast iron to sustain a given weight, when supported at both ends, and the weight applied at the middle of the length. Rule. —Take the weight in pounds, and the length in feet; multiply the two numbers together, and divide the product by 500; the cube root of the quotient is the number of inches in the diameter. If W represents the weight in pounds, l the length in feet, and d the diameter in inches, W= 500 X ds 3 1; 1= 500 X d 3 * W. a The pitch of the teeth of a wheel, is the distance between the middle paints in the bases of two adjacent teeth. It should be at least 2.1 times the thickness of the teeth. 11

Page  162 !162 STRENGTH OF MATERIALS. [ART. IX. 7. To find the proper diameter of a solid cylinder of cast iron supported at both ends, to bear a given weight when the strain is not in the middle. Rule.-Take the weight in pounds, and the distances in feet, from the point where the weight is applied to each of the points of support; divide the continued product of these three numbers by the number of feet in the distance between the points of support, and cut off three figures from the right hand. The cube root of the result will give one half of the diameter of the cylinder in inches. 8. To find the proper diameter of a solid cylinder of cast iron when supported at both ends, to sustain a load uniformly distributed over its length. Rule.-Multiply the number of feet in the length by the number of pounds in the weight, and 1 of the cube root of the product will be the number of inches in the diameter. W=1000 d 1; 1000 X d3 -.W. 9. To find the proper length of a solid cylinder of cast iron, when fixed at one end and loaded at the other; also when the cylinder is supported on a centre of motion. Rule. —Take the weight in pounds, and the distance of the weight from the point of support in feet, multiply the two numbers together, and - of the cube root of the product will be the number of inches in the diameter. W=125 X d3 — 1; 1=125 X ds. W. 10. The strength of direct cohesion, of the materials in tables, ~38 and 40, may be found by Problem 1, ~36, using the numbers in those tables opposite to the given material, instead of the value of C. 11. The lateral strength of iron may be found by the rules for that of timber, using the number opposite to iron in ~ 39, instead of the value of S in ~ 35. 12. The strength of a column to resist being crushed, is proportioned to the area of its transverse section. Hence,

Page  163 ~ 45.] MISCELLANEOUS EXAMPLES. 163 to find the weight which will crush any column, multiply the number of inches in the area of its transverse section by the proper number in ~ 41, and divide the product by 2{; or multiply the number of feet in the area of the transverse section, by the pressure given in ~ 43. The area of a cylindrical column may be estimated, in making this calculation, at - of the square of the diameter. The true area of a circle, is found by multiplying the square of the diameter by.7854. The result is the number of pounds. 45. EXAMPLES FOR THE PUPIL. 1. What is the breadth of a cast-iron beam 30ft. long, and 8in. deep, that will support a weight of 8 tons, placed in the middle? If the length is 30ft. and the breadth 6in., what must be the depth to support 1.0 tons? ist Ans. 9.88in. 2d'" 11.47" 2. The front of a house is to be broken out to make shops, and the front wall, which is 40ft. long, is to be supported by 2 cast-iron beams, with a prop under the middle of the wall. If there are 4000 c. ft. of wall to be supported, weighing 140 lb. per c. ft., what must be the breadth and depth of each beam, tlie depth being 5 times the breadth? Ans. Depth 20.352in.; Breadth 4.0704in. 3. The second story of a building is to project 3ft. over the first. What must be the depth of the fixed iron beams, which are 4 inches broad, supposing the weight supported by each to be 33600 lb.? Ans. 7.7in. 4. If the greatest stress at the pitch-circle of a wheel is 6500 lb., what should be the thickness of the teeth? Ans. 2.08in. 5. If the effective force of the piston of a steam engine is 10000 lb., and its velocity 5 ft. per second, what should

Page  164 164 STRENGTH OF MATERIALS. [ART. IX. be the thickness for the teeth of a wheel, which is to make 20 revolutions in a minute, and to have 120 teeth? a Ans. 1.46in. 6. What weight will a cast-iron cylinder, supported at both ends, sustain in the middle of its length, the diameter being 8 inches, and the length 16~ ft.? Ans. 15515 lb. 7. What must be the diameter of a cast-iron cylinder, which is 20ft. long, to sustain a weight of 33600 lb., the weight being applied at the distance of 5ft. from one end? Ans. 10.026in. 8. A load of 25000 lb. is to be uniformly distributed over a solid cast-iron cylinder. Required the length of the cylinder, the diameter being 9 inches. Ans. 29.16ft. 9. A solid cylinder of cast iron, 8 inches in diameter, is supported in the middle. What may be the length of the arms, to support 10000 lb. at the extremity of each? Ans. 6.4ft. 10. What weight would pull asunder a hemp rope 21 inches in diameter? Ans. 31412 lb. 11. What weight distributed uniformly over a cast-iron beam, 20ft. long, 6in. broad, and 8in. deep, will break the beam, it being supported at both ends? Ans. 41856 lb. 12. What weight can be sustained with safety by each of the marble pillars of Girard College, estimating their strength as equivalent to that of Pennsylvania Dove marble, the least diameter of the pillars being 5ft? b Ans. 1688610 lb., or, estimating the area at 7- of the square of the diameter, 1672222 lb. 1.3. The length of each arm of the wrought-iron beam of a, balance is 3ft., and the depth is 8 times the breadth. a The circumference of a wheel with 120 teeth, and a pitch of 2in., is 120 X 2 = 240in. The radius of such a wheel would be 240 - (2 X 3.1416)= 38.197in. See the article on Mensuration. b Architect's Report;

Page  165 ~46.] TABLE OF SPECIFIC GRAVITIES. 165 Required the depth and breadth necessary to enable the balance to weigh ~ a ton. Ans. Depth, V3 8 x 3 x 1120-. 238 =4.834in. Breadth,.604in. 14. What should be the depth and breadth of a cast-iron beam, 30ft. long, to support a weight of 20 tons, placed 10ft. from one end, the beam being supported at both ends, and the depth being 4 times the breadth? Ans. Depth, 17.7812in.; Breadth, 4.4453in. X. SPECIFIC GRAVITY. THE Specific Gravity of a body, is the ratio of its weight to the weight of an equal volume of some other body assumed as a standard. The standard usually adopted for this purpose is pure distilled water at a given temperature. In England, the temperature of 62~ Fahrenheit is generally taken; the French take 320, or the temperature of melting ice.a 46. TABLE OF SPECIFIC GRAVITIES. Compiled from the Encyclopedia Britannica and other sources.b AcAcIA, inspissated juice 1513 Acid, muriatic 1284.7 Acid, acetic 1007 to 1009.5 nitric 1271.5 to 1583 acetous 1009.5 to 1025.1 phosphoric, liquid 1417 arsenic 3391 solid 2852 boracic, scales 1475 sulphuric 1840.9to2125 citric 1034.5 Agate 2348 to 2666.7 fluoric 1500 Air 1.2308 a Brande. b The specific gravity of water is fixed at 1000. As a cubic foot of water weighs 1000 ounces avoirdupois, the specific gravity of each article named in the table will represent the weight of 1 cubic foot.

Page  166 166 SPECIFIC GRAVITY. [ART. X. Alabaster 2611 to 2876.1 Chalke 2252 to 2657 Alcohol, absolute 791 Cherry 715 mixed with Chestnuta 610 water 829.3 to 991.9 Chromiumd 6900 Alder wood 800 Citron wood 726.3 Alum 1750 Claya 2000 Amber 1078 to 1085.5 Coal, bituminousf 1262 to 1364 Antimony, fused 6624 to 6860 anthracitef 1500 Apple tree 793 Cobalt, fused 7645 to 7811 Arsenic, fused 8310 Cocoa wood 1040.3 glass of, (arsenic Cokea 744 of the shops) 3594.2 Copal 1045.2 to 1139).8 Asbestos 2577.9 to 3080.8 Copper,g native 7600 to 8508.4 Do. mountain cork 680.6 to 993.3 fused 7788 to 8607 Ash 727a to 845 wiredrawn 8878 Asphaltum 1070 to 2060 Coralh.2680 BASALT 2421 to 3000 Cork 240 Beech 696a to 852 Cypress 644 Beryl, oriental 3549.1 DIAMOND 3444.4 to 3550 aquamarine 2650 to 2759 EARTH, commons 1520 to 2000 Bismuth, molten 9756 to 9822 mean densityi 5670 Blood, human 1054 Ebony 1209 to 1331 Bone of an ox 1656 Elm 671 Borax 1740 Emerald 2600 to 3155.5 Boxwood 912 to 1328 Emery" 4000 Brass, common 7824 to 8395b Ether, acetic 866.4 wiredrawn 8544 muriatic 729.6 Brazil wood 1031 nitric 908.8 Brick 1557a to 2000 sulphuric 716 to 745 Brickworkc 1872 FAT 923.2 to 936.8 Butter 942.3 Felspar 2438 to 2704 CADMIUM 8604 to 8694.4 Filbert tree 600 Camphor 988.7 Fir 498 to 553 Cannel coal 1270 Flint 2243.1 to 2664.4 Caoutchouc 933.5,X GARNET, common 3576 to 3688 Castor oild 970 precious 4085 to 4352 Cedar 457a to 561 Gas, ammonia.73459 a Cavallo. b 369 c. in. = lcwt. c Benjamin. d Ingram. e 13 c. ft. =1 ton. f W. R. Johnson. s A square foot of sheet copper, tin. thick, weighs 11 lb. 12oz. h Barlow. i Cavendish.

Page  167 ~ 46.] TABLE OF SPECIFIC GRAVITIES. 167 Gas, atmospheric air 1.2308 Gum ammoniac 1207.1 carbonic acid 1.87 Arabic 1452.3 carbonic oxide 1.1777 guaiacum 1228.9 carburetted hydro..73848 lac 1139 chlorine 3.0401 tragacanth 1316.1 cyanogen 2.2228 Gunpowder, solid 1745 fluosilicic acid 4.3984 shaken 932 hydriodic acid 5.4684 Gypsum 1872 to 3310.8 hydrogen.0898484 HAZEL. 606 muriatic acid 1.5353 Hone 2876.3 to 3127.1 nitrogen 1.928 Honey 1450 nitrous 1.2786 Hornblende 3333 to 3830 nitrous acid 3.908 ICEb 930 nitrous oxide 1.9865 Indigo 769 olefiant 1.20377 Iodine 4948 oxygen 1.3588 Iridium, fused 18680 phosph. hydrogen 1.0708 Iron,c bar 7600 to 7788 steam.76739 cast 6953 to 7295 sulph. hydrogen 1.4661 magnetic 4518 sulphurous acid 2.6097 meteoric 6480 Glass, bottle 2732.5 Ivory 1825 to 1917b crown 2487 to 2520 JARGON, of Ceylonl 4416 flint 3000 to 3437 Jasper 2358.7 to 2816 green 2642.3 Jet 1259 plate 2520 to 2760 Juniper tree 556 Gold, not hammered 19258.7 LARD 947.8 hammered 19342 Lead 11352 to 11445 Am. standard 17350a Lignum Vitro 1333 Eng. " 18888 Limestone 1386.4 to 3183 French" 17486 Linden 604 Granite 2613 to 2760.9 Linseed oil 940.3 a The specific gravity of the Mint standard gold varies from 17250 to 17500, according to the greater or less proportion of copper used in the alloy. The average specific gravity is about 17350. b Barlow. c A square foot of cast iron,'in. thick, weighs 9 lb. 10.6oz; a sq. ft. of malleable iron of the same thickness, 9 lb. 15.2 oz.; a bar lft. long and 1. in. square, of cast iron, 9 lb. 8oz.; a bar of malleable iron, of the same size, 9lb. 11oz.; a round iron rod, Ift. long and l'in. in diameter, 7 lb. 9.2 oz. d Ingram.

Page  168 168 SPECIFIC GRAVITY. [ART. X. Living mena 891 Pine 540 to 683 Loadstone 4200 to 4900 Pitcha 1150 Logwood 913 Platina 14626 to 22069 MADDER 765 Plumbago 1987 to 2267 Magnesia, sulphate of 1797.6 Poplar 360 to 529.4 Mahogany 1063 Porcelain, China 2384.7 Manganese 6850 do. European 2145 to 2545 Maple 755 Potash,, carbonate 1459.4 Marble, Carrara 2716 Potassium 972.23 Egyptian 2668 Proof spirit 916 various 2516 to 2858 Pumice stone 914.5 Mercury 13568 QUARTZ 2652 Mica 2883 Quince tree 705 Milk 1020.3 to 1040.9 RHODIUa1 11000 Molybdena, native 4738.5 Rock crystal 2581 to 2888 Mortar, dryb 1384 to 1893 Rosina 1100 Muriatic acid 1284.7 Ruby 3531 to 4283.3 NAPHTHA 847.5 SANDb 1454 to 1886 Nickel, metallic 7421 to 9333.3 Sandstone 2142 to 2483.5 forged 8600 Sapphire 3130 to 4283 Nitre 1900 to 2246 Scythe stone, fine 2609 Nitric acid, (aquafortis) 1500a Serpentine 2264 to 3000 Nitrous acid 1452a Silver 10474 to 10510 OAKb 748 to 993 Slate 26718 to 2752b heart of 1170 Soda, sulphate 1439.8 Obsidian 2348 carbonate 1000 to 1500 Oil, of turpentine 870 Sodium 865.07 olive 915.3 Spar, heavya 4430 whale 923.3 Spermaceti 943.3 various 857.7 to 1044 Steel 7767 to 7840.4 Opal 1958 to 2600 Stone, common 2000 to 2700 Opium 1336.5 rotten 1981 Orange tree 705.9 Sugar, white 1606 PALLADIUM 11800 Sulphur, native 2033.2 Pear tree 661 Sulphuric acid 1841 Pearls 2683 TALLOW 941.9 Peat 600 to 1329 Tara 1015 Pewtera 7471 Tellurium, native 5700 to 6100 Phosphorus 1770 Tin 7063 to 8487 a Ingram. b Cavallo.

Page  169 ~ 47.] PROBLEMS IN SPECIFIC GRAVITY. 169 Topaz 3531 to 4061.5 Water, sea 1026.3 Tourmaline 3086 to 3362 well 1001.7 Tungsten 4355 to 6066 Wax, bees' 964.8 Turpentine 991 shoemakers' 897 spirits of 870 Whaleb one 1300 ULTRAMARINE 2860 Willow 585 Uranium 7500 Wine, Burgundy 991.5 VAPOR of alcohol 2.58468 Canary 1033 do. absolute 1.985 Champagne 997.9 hydriodic ether 6.7884 Malaga 1022.1 muriatic ether 2.731 Malmsey 1038.2 sulphuric ether 3.182 Port 997 iodine 10.6089 Tokay 1053.8 oil turpentine 6.17 Wolfram 5705 to 7333 sulph. of carbon 3.255 Wootz, hammereda 7787 water.76739 YEw, Dutch 788 Vinegar 1013.5 to 1080 Spanish 807 WALNUT 671 ZINC, common 6862 Water, distilled 1000 pure & compressed 7190.8 Dead Sea 1240.3 47. PROBLEMS IN SPECIFIC GRAVITY.a 1. To find the magnitude of a body fromn its weight. Find the weight of the body in ounces, and divide by the specific gravity. The quotient will be the number of cubic feet in the contents. II. To find the weight of a body from its magnitude. Find the number of cubic feet in the body, and multiply by the specific gravity. The product will be the number of ounces in the weight. III. To find the specific gravity of a body. CASE I. When the body is heavier than water. Weigh the body both in air and in water. Annex three a Ingram.

Page  170 170 SPECIFIC GRAVITY. [ART. X; ciphers to the weight in air, and divide by the difference of the weights. The quotient will be the specific gravity. CASE II. When the body is lighter than water. Having weighed the light body in air, and a body heavier than water both in air and water, fasten them together with a slender tie, then weigh the compound in water, and subtract its weight from the weight of the heavy body in water; to the remainder add the weight of the light body in air, and by the sum divide one thousand times the weight of the light body in air. The qu'otient will be the specific gravity of the light body. IV. To find the quantity of each zng/reclient in a zmixtuzre of two substances.a 1. Multiply the specific gravity of the mass by the difference between the specific gravities of the two ingredients, for a first product. 2. Multiply the specific gravity of that ingredient whose quantity is desired, by the difference between the specific gravity of the mass, and that of the other ingredient, for a second product. 3. Multiply the whole weight of the mass by the second product, and divide by the first product. The quotient will be the weight of the ingredient sought. 48. EXAMPLES FOR THE PUPIL. 1. How many cubic inches in 1 lb. of white sugar? Ans. sg c. ft.=17.215 c. in. 2. A keg is found to contain 13790 cubic inches. What weight of butter will it hold? Ans. 470 lb. 3. A piece of Quincy granite weighs 25 lb. 12.-oz. in air, and 161 lb. 1 oz. in water. What is its specific gravity? Ans. 2661. " The student will observe that this is a case in Alligation.

Page  171 ~ 48.] MISCELLANEOUS EXAMPLES. 171 4. A piece of copper weighs 18 lb. in air and 16 lb. in water. A piece df elm, which weighs 15 lb. in air, is fastened to the copper, and the compound weighs 6 lb. in water. What is the specific gravity of the elm? Ans. 600. 5. What quantity of gold, sp. gr. 19258, and of silver, sp. gr. 10474, must be mixed to form a mass weighing lewt. 3qr. 4 lb., and having a specific gravity of 16000? Ans. 151.44 lb. gold, 48.56 lb. silver. 6. What are the cubical contents of a pillar of Pennsylvania marble, sp. gr. 2720, the weight being 63T. 8cwt. 211b.? Ans. 835 c. ft. 884 c. in. 7. Each of the pillars of Girard College is 55ft. high, and 6ft. in diameter at the base. What would be the weight of a square block of marble from which a column of the same size could be cut, the specific gravity being 2716? Ans. 150T. 3qr. 211b. 8. It is proposed to float 500 c. ft. of granite on a pine raft 50 ft. long, and 20ft. wide. What must be the depth of the raft in order that it may float at least 6 inches above the water, the specific gravity of the granite being 2620, and the sp. gr. of the pine 600?a 9. A raft of elm is 3ft. 6in. thick. To what depth will it sink? Ans. 2ft. 4.182in. 10. What must be the depth of a cedar raft, 16ft. long and 10ft. wide, to float 10000 lb. of bricks, the cedar being a All floating bodies sink till they have displaced a quantity of fluid equivalent to their own weight., In this example, the granite would cause the raft to displace 1310 c. ft. of water, to do which, it must sink 1310. — (50 X 20) = 1.31ft. to which the 6in. =-.5ft. should be added, making 1.81ft. to represent the buoyancy of the pine. As the sp. gr. of the pine is - that of water, it will sink - of its depth, leaving 2 for buoyant force. 1.81ft. must therefore be 52 of the dc:pth of the raft, and the depth must be 1.81.- -4.525ft.

Page  172 172 SPECIFIC GRAVITY. LART. X. of the sp. gr. 550, and the raft floating 3in. out of water? Ans. 2ft. 9jin. 11. What weight will a raft 30ft. long, 16ft. wide, and 3ft. deep sustain, and float 6 inches above water, the specific gravity of the raft being 580? Ans. 22800 lb. N. B. First find how much above water the raft would float if it were not loaded, and subtract 6 inches to find how much it is sunk by the load. The weight of the load will then be equivalent to the weight of the quantity of water displaced by it. 12. How many inches above water will a raft float, if loaded with 8500 lb., the raft being 12ft. wide, 20ft. long, and 2ft. deep, and of the specific gravity of 625? Ans. 2.2 inches. N. B. First find the entire weight of the raft and load, and see what depth of water must be displaced to yield the same weight. Subtract the depth of water displaced,from the depth of the raft, and the remainder will give the part out of water. 13. What is the weight of a sheet of malleable iron, 3ft. 6in. wide, 8ft. long, and Lin. thick? Ans. 69 lb. 10.4oz. 14. What is the weight of an iron rod, 3in. in diameter, and 162ft. long?a Of a rod lin. in diameter, and 10+-ft. long? Ans. 4cwt. lqr. 23 lb. 15.2oz.; lqr. Gilb. 8.1oz. 15. What is the weight of a sheet of copper, 3ft. wide, 6ft. 8in. long, and min. thick? Ans. 1cwt. 5 lb. 8oz. 16. The amount of water displaced by a loaded ship, is found to be 96000 cubic feet. Required the weight of the vessel and cargo, the water being of the specific gravity of 1020. Ans. 2732T. 2cwt. 3qr. 12lb. a The weight of iron rods of the same length, is proportioned to the squares of their diameters.

Page  173 ~ 49.] GENERAL REMARKS. 173 XI. THE ROADa 49. GENERAL REMARKS. WHEN it is proposed to construct a road, the engineer first makes himself acquainted with the face of the country through which the road is to pass, and selects what he considers as the best general route. An instrumental survey is then made of the country along the proposed route, taking levels'from point to point, throughout the whole distance, to determine the requisite inclinations of the slopes of the cuttings and embankments, and making borings in all places where excavations are required, to determine the strata through which the cuttings are to be made. A plan and section are then drawn, exhibiting the results of this investigation. In selecting the route, regard should be had to the supply of materials for constructing the road, and for keeping it in repair. Therefore the position of gravel pits and quarries in the neighborhood of the proposed line, should be well ascertained. The expense of construction should be proportioned to the traffic expected on the road. If the amount of travel will be great, all steep acclivities should be avoided, either by cutting down the hills and filling up the valleys, or by passing around the base of the hills. It is recommended by some writers to avoid a dead level, as a moderate inclination of the surface facilitates drainage, and tends to keep the road dry. But, if proper attention is paid to the form of the road, there will be no difficulty in keeping it properly drained, without resorting to any expedient that will be necessarily attended with a loss of power. a Brande, Gillespie, Mahan.

Page  174 174 THE ROAD. [ART. XI. The top should be slightly rounded, being made highest in the middle, and gradually sloping to a trench at each side, so that all the water may be carried off. The surface should be as hard and smooth as possible, and, whenever repairs are required, broken stone, pebbles, or hard gravel should be used, if it is possible to obtain either of them. o50. EXAMPLES FOR THE PUPIL. 1. How many acres per mile will be taken up by a road that is 2 rods wide?-by a road 40ft. wide?-by a road 60ft. wide? 2. In 1678, a contract was made to establish a coach between Edinburgh and Glasgow, a distance of 44 miles. The coach was to be drawn by 6 horses, and the journey between the places, to and from, was engaged to be completed in 6 days.a At what average rate did the coach move, if it travelled 9 hours per day? 3. In the year 1763, there was but one line of stage-coaches between Edinburgh and London, which started once a month from each place. It then took a fortnight to perform the journey, which is now completed in less than 48 hours.a The number of persons travelling between the two places did not probably exceed 50 per month, but the present intercourse is supposed to amount to at least 300 per day. What has been the rate of increase, both in the rate of travel and in the number of passengers? 4. What will be the cost of a plank road per mile, for a single track 8ft. wide and 3in. thick, with two sills, each 4in. square, at $4.50 per M.,b the laying and grading being 75cts. per rod, and superintendence $75 per mile? a Brande. Even so recently as the year 1750, the stage-coach fromr Edinburgh to Glasgow took a day and a half to make the journey. b " Wood is paid for by the cubic foot, unless some one of its diMnensions is as small as 4 inches, when board measure is used."-Gillespie.

Page  175 ~ 50.] MISCELLANEOUS EXAMPLES. 175 5. Wishing to know my distance from the foot of a steeple which is 125ft. high, I hold a foot rule at arm's length, (which I have found to be equivalent to 2ft. 3in. from the eye,) and find that 1'l inches on the rule intercepts the rays from the top and base of the steeple. What is the distance? Ans. 1928A ft. 6. The mean velocity of sound through the atmosphere is about 1090ft. per second. What is the distance of a hill on which a cannon is fired, if 8} seconds elapse between the flash and report? 7. If two supports of a rail that are 3ft. apart, vary i of an inch from an exact level, to what elevation per mile would the ascent be equivalent? 8. Estimating the cost of a railroad at $30000 per mile, and the annual repairs and expenses at $2000 per mile, how much might be profitably expended to shorten the road 1 mile, the rate of interest being 6 per cent.?-cto shorten it 2m. 6fur. 23r.? 2d Ans. $178718.75. 9. An embankment of 27000 c. ft. is to be made. It is estimated that 1 man can loosen 20 c. yd. per day, or load in barrows 25 c. yd., or transport 30 c. yd., or spread and level 80 c. yd. According to this estimate, what will be the cost of the whole, allowing $1.25 per day for wages, and 10 per cent. for shrinkage of the earth, and adding - of the amount of wages for tools and superintendence, and a The length marked on the rule, is to the distance of the rule from the eye, as the height of the object is to its distance. Distances may also be conveniently measured by pacing, or by noting the time that elapses between the flash and report of a gun. b Pierce. c In laying out a road of any kind, the preference never should be given to the longer of two proposed routes, merely because it can be constructed at a less expense. In order to shorten the distance, the difference in the cost of making, together with a sum, the interest of which would defray the annual repairs and expenses of the road, for the distance saved, may be profitably expended. In the remaining examples of this section, the rate of interest is understood to be 6 per cent.

Page  176 176 THE ROAD. [ART. XI. I for contractor's profit, the earth costing 10 cents per c. yd.? Ans. $324.79. 10. The average power of draft of a horse, moving 3 miles per hour for 10 hours a day, being 100 lb., what will be the annual cost of transportation over a road 30 miles long, on which the average friction is 2 of the weight, estimating the amount transported at 50000 tons, and the value of a days' labor of a horse at 75 cents? Ans. $42000. 11. If the road in the preceding example should be improved by macadamizing, or otherwise, so that the friction should be reduced to -5 -I of the weight, what would be the annual cost of the transportation, and how much might be profitably expended in making the improvement? 2d Ans. $420000. 12. The annual cost of transportation over a road 15 miles long, being estimated at $35000 per mile, what amount of saving can be effected by expending $20000 to shorten the road 2 miles, and $1000000 to reduce the friction to i its present amount, the annual cost of repairs being the same in both cases? Ans. $3938333-i. 13. If a hill by friction and gravity, causes 5000 days' work of a horse, at 75 cents per day, which can be avoided by a road along the base of the hill, that will require only 2300 days' work, and if the new road will require an extra annual outlay of $375 for repairs, how much can be saved by expending $10000 in making the improvement? Ans. $17500. 14. It is calculated that, in locomotives, the evaporation of 1 cubic foot of water per hour, produces a mechanical force of about two horse power, and that a horse on a railway can pull 10 tons with ease.a According to this estimate, what load can be drawn by a locomotive which evaporates 175 c. ft per hour? a Chambers.

Page  177 ~ 50.] MISCELLANEOUS EXAMPLES. 177 15. If an engine has sufficient force to draw 92T. 19cwt. lqr. over level ground, what additional power must be exerted on an ascending grade of 37ft. per mile? a Ans. 13cwt. 31wt. -- lb. 16. Determine the amount of excavation and embankment in the following example, by taking the average of the end areas of each section as the true area of the section,b and find the cost of the whole at 10cts. per cubic yard. Station. )Distance. End Areas. Excavation. Embankment. Cubic feet. Cubic feet. tair is.... 0... 2 561 ft. 1386 sq. ft. excav. 388773 O 3 858 " 1600" " ".... 0 4 825 " 0" ".. it 0 5 820 " 1672 " " emb. 0 685520 6 825 528 " " 0 7 330 " 0' "'t' 0. ____ _________I - _____2329767 1680140 Cost, $14851.51. On all inclined planes,,the power is to the weight, as the length of the plane is to the height. b This method, which is the one usually employed, gives a result greater than the true contents. Sometimes the calculation is performed by deducing the middle area of each section from the arithmetical mean of the heights at the two extremities, but the result thus obtained is too small. The true contents may be found by the PRISMOIDAL FORMULA. Find the area of each end of the mass, and also the middle area, corresponding to the arithmetical mean of the heights of the two ends. Add together the area of each end, and four times the middle area. Multiply the sum by the length, and divide the product by six. The quotient will be the true cubic contents required. The reduction of the contents to cubic yards would be greatly facilitated if the distances of the stations were always some multiple of 54 feet.-Gillespie. 12

Page  178 178 THE ENGINEER. [ART. XII XII. THE ENGINEER. 5 1. THE STEAM ENGINE. A POUND of steam at 2120 will raise the temperature of a pound of water 9700.a But as some of the heat is wasted, the increased temperature may be considered, in practice, as equivalent to 9000. From this fact the following rule is derived: " To find the quantity of steam required to raise a given quantity of water to any required temperature. —Multiply the number of gallons to be warmed by the number of degrees between the temperature of the cold water and that to which it is to be raised, for a dividend; and to the excess of the temperature of the steam above 2120 add 900 for a divisor. The quotient will be the number of gallons formed into steam, required."b A " horse power" was estimated by Boulton and Watt as sufficient to raise 32000 lb. avoirdupois 1 foot high in 1 minute, but in estimating the force of their engines, they used 44000 as a divisor instead of 32000. Desaguliers's estimate was 27500; Smeaton's, 22916; some of the modern English engines are computed at 66000,c but the number commonly used is 33000,d or 44000 when an allowance of i is made for friction. To calculate the power of an engine. Form the continued product of the number of square inches in the area of the cylinder, the number of pounds which represents the effective pressuree per square inch, and the number of feet through which the piston moves per minute, and divide by the number of pounds a horse can raise I foot per minute. a Daniell. b Pilkington. c Sci. American. i It is customary to consider the friction of the machinery as equivaient to - of the effect produced. The effective pressure is the force remaining after making allowance for the waste and fifiction of the steam.

Page  179 ~ 51.] THE STEAM ENGINE. 179 The following rule is simpler, and in most cases it will be found sufficiently accurate. When the usual estimate of a horse power is employed, and the effective force is 8.4 lb. per square inch, and the distance traversed by the piston 200ft. per minute, the same result will be obtained by either rule. Square the number of inches in the diameter of the cylinder, and multiply by.04. The product will give the number of horse power. EXAMPLES. 1. What quantity of water converted into steam at 2200 will raise 100 gallons of water at 500 to the boiling point? Ans. 171 —I7 gallons. 2. What is the power of a steam engine with a cylinder 37 inches in diameter, making the usual estimate of the effective force of the steam and the stroke of the piston? Ans. 54.76 horse power. 3. Find by each rule, the power of a steam engine that makes 10 strokes per minute, each stroke being 8ft., allowing for friction, ~ of the force, which is 15 lb. per sq. in., the diameter of the cylinder being 40 inches? a Ans. by Rule 1, 60.928 horse power.'" " 2, 64 " " 4. Two steam engines constructedfor the island of Ceylon, working 10 hours per day for 300 days in the year, will convert 576000 lb. of paddyb into rice worth ~116035, while by the common method, the same quantity of paddy converted into rice, would yield only ~64799.c Allowing ~20000 per annum, for interest, repairs, and the extra expense of working the machinery, what is the average daily amount saved by each engine? Ans. 1041. 2s. 4.8d. a As the piston must move forward and back at each stroke, the dis. tance passed through per minute is 160ft. b Rice in the husks. c Partington.

Page  180 180 THE ENGINEER. [ART. XIi. 5. An engine at the Wheal Hope mine, in Cornwall, works 3 pumps, the length of stroke of each being 8ft.; their pistons support and lift at each stroke, columns of water, whose joint weights are 277661b., and in the month of December, 1826, they made 261890 str9kes.a Required the velocity per minutes and the total dynamical effect.b Ans. Velocity 46.93ft.; effect 1303058.38 lb. 6. The working effect of 1 bushel of coals, or the number of pounds which could be raised ift. high by 1 bushel is called the Dut&y of the engine. Required the duty of the engine in the preceding example, the amount of coal consumed being 1242 bushels. Ans. 46838246. 7-10. A cubic foot of water makes 1689 c. ft. of steam, at the temperature of 212~.0 Estimating the pressure of the atmosphere at 2120 lb.a on a square foot, what will be the dynamical effect of 1 lb. of each of the following kinds of fuel? d Weight to evaporate Dynamical effect Fuel. Spec. gravity. I c. ft. of water. of 1 lb. of fuel. Antlracite coal 1500 6.534 lb. 548007 lb. ift. Va. bituminous coal 1364 7.82 " lb. " Pa. " 1262 9.37 " lb. " Dry pine wood 336 15.4 " 2325121b. " 5;2. THE WATER-WHEEL. The pressure of water on any surface is equal to the weight of a column of water with the same base as the surface pressed, and of a height equivalent to the depth of the centre of gravity. If the surface is of any regular shape, the centre of gravity corresponds with the centre of the surface. The actual velocity of water flowing from an orifice, or a Moseley. b The dynamical effect is found by multiplying the velocity by the weight. a Daniell. d W. R. Johnson.

Page  181 ~ 52.] THE WATER-WHEEL. 181 falling, will generally be.6 or.7 of the theoretic velocity.a To obtain the theoretic velocity per second, find the number of feet in the perpendicular fall of the water, (or in the depth below the surface to the middle of the orifice,) extract the square root, and multiply by 8.018. The effective force of a water-wheel, may be estimated at i of the power applied to it, if the wheel is overshot, and at J of the power if the wheel is undershot.b To determine the power of a stream of water, measure the breadth and depth of the stream, the height of fall, and the velocity per minute, all in feet; form the continued product of these four numbers, and divide by 792..c For undershot wheels take ~ this result. To calculate the power of machinery or wheelwork, multiply together the effective power, and the lengths of all the driving levers, (or the radii, circumferences, cogs or rounds of the driving wheels,) and divide by the continued product of the lengths of all the leading levers, (or the radii, &c., of the leading wheels.) For the velocity, multiply the velocity of the power by the lengths, radii, circumferences, cogs or rounds of the leading levers or wheels, and divide by the product of the like dimensions of all the driving levers or wheels. The maximum effect will be produced in machinery of any kind, when the load, or resistance, is 4 of the power, and when the velocity of the machinery at the point of action, is i of the greatest velocity of the power.d:EXAMPLES. 1. What is the amount of pressure on a dam 75ft. by 12ft., the depth of water being 8ft.?F Ans. 225000 lb. a Nicholson. b Pilkington. c 2 D.B.H.V. D.B.H.V. 3X 33000- X 62.5- 792 ( Brande. e The depth of the centre of gravity is 4ft.

Page  182 182 THE ENGINEER. [ART. XII. 2. What is the theoretical velocity of water flowing through an orifice, the centre of which is 6ft. 3in. below the surface? Ans. 20.045ft. per second. 3. Find the bottom pressure, and the entire pressure upon the bottom and sides of a cube, each side of which measures 8 feet. Ans. Bottom pressure, 32000 lb. Entire " 96000 lb. 4. If the area of an orifice is 2k sq. ft., and the velocity of the water flowing through it is 15ft. per second, what will be the weight of the water discharged in 1 minute? Ans. 131250 lb. 5. A stream 12in. deep, and 22in. broad, moves with a velocity of 88ft. in 15f". Required its effective force, with a fall of 50ft. Ans. 402-~- horse power. 6. How many cuts are made per minute by the beater of a paper mill, which has 60 teeth, each of which passes by 24 cutters at every revolution, when there are 150 revolutions per minute? Ans.' 216000 cuts.a 7. What force will be exerted at the distance of 2 feet from the centre of a millstone, by a water-wheel with a power of 1000 lb., the diameters of the driving wheels being 8ft., 2ft., and ift., and the diameters of the leading wheels being 4ft. and 3ft.? -As. 666k lb. 8. If the power in the preceding example moved with a velocity of 12ft. per second, what would be the velocity of the millstone? Ans. 18ft. 9. What should be the area of the section of a canal, to deliver 90000 c. ft. per hour, the water moving with a velocity of 4ft. per second? Ans. 6k sq. ft. 10. Using 10 c. ft. per second, what time would be necessary to exhaust a pond, the area of which is 10 acres, and a This rapid motion makes a coarse musical note, that can be heard at a great distance from the mnill.-U1re.

Page  183 ~ 53.q] PUMPr. 183 the average depth 3 feet, the pond being supplied by a stream that furnishes 150 c. ft. per minute? Ans. 48h. 24min. 53. PuMPs.a The power employed in working a pump is estimated by multiplying the number of pounds discharged per minute, by the number of feet that the water is raised above the reservoir. The weight of water in a yard of pipe may be found very nearly by squaring the number of inches in the diameter of the pipe, and increasing the square by 4 of itself.b The result will be the weight in pounds avoirdupois. The number of ale gallons in a yard of pipe may be found very nearly by squaring the number of inches in the diameter, and dividing by 10. In estimating the power necessary to overcome resistance in pumps, I should be added for the friction of the water. The diameter of the pipes should be at least as great as the diameter of the pump. If it is greater, the friction will be diminished. EXAMPLES. 1. At the height of 225 ft. above the level of a reservoir, 250 ale gallons are discharged per minute. Required the power of the engine, the weight of one ale gallon of water being 10 lb.e Ans. 13.1 horse power. 2. Find the weight of water, and the number of ale gallons, in a pipe 15 rods long, and 3 inches in diameter. Ans. 759- lb.; 741 gallons. 3. On the top of a hill 75 feet high, is a reservoir 40 feet a Nicholson, Pilkington, Ferguson. b.7854 X 3 X 62.5 45 _ _144_ -= U- very nearly. c In all the examples of this section, an allowance of -I is made for the friction of the machinery. See Sect. 51.

Page  184 184 THE LABORATORY. [ART. xIII. square, and 12ft. deep. What power is necessary to fill the cistern in 45 minutes? 40 x 40 x 12 x 62.5 x 75 6 A4ns. x 54-L horse power. 45 x 44000 1 1 4. What power is necessary to fill a cistern 30ft. long, 22ft. wide, and 10ft. deep, in 25 minutes, the water being raised 100ft.? Ans. 45 horse power. 5. What should be the diameter of the pump in each of the two preceding examples, if there are 40 strokes per minute, the length of the effectivea stroke being 2ft.? Ans. Ex. 3: 40 x 40 x 12 _ 426*- c. ft. per minute; 45 426* - (2 X 40) = 5* sq. ft. area of pump; / 5* -.7854 - 2.6ft. diameter of the pump. Ex. 4: Diameter 2.05ft. 6. A town of 25000 inhabitants, is to be supplied with water from a river 200 feet below the proposed reservoir. Estimating the average daily consumption at 9 ale gallons for each individual, what must be the power of an engine working 10 hours per day, and what will be the size of the pump, making 30 strokes per minute, the effective stroke being 3ft.? Ans. Engine 17.47 horse power. Area of pump 98 sq. in. nearly. Diam. of pump 11.17in. XIII. THE LABORATORY. 5J4. CHEMICAL COMBINATIONS.? IN chemical compounds, the following curious facts have been observed. a An allowance is made from the stroke of-the piston rod for the escape of water through the valves. Pilkington states this allowance at 3 inches. b Draper.

Page  185 ~ 54.] CHEMICAL COMBINATIONS. 185 1. The constitution of a compound body is always the same. Thus it has been found that 9 grains of water contain 8 grains of oxygen and 1 grain of hydrogen; and however often the analysis is repeated, this proportion is found to be invariable. 2. The proportions in which bodies are disposed to unite with each other, can always be represented by certain numbers. Thus water is composed of one atom of oxygen and one atom of hydrogen, and as the oxygen atom is 8 times as heavy as that of hydrogen, it follows that in 9 parts by weight, of water, there are 8 parts of oxygen and 1 of hydrogen. 3. If two substances unite with each other in more proportions than one, those proportions bear asimple arithmetical relation to each other; thus 14 grains of nitrogen will successively unite with 8, 16, 24, 32, 40 grains of oxygen, forming five different compounds, which contain respectively 1 atom, 2 atoms, 3 atoms, 4 atoms, 5 atoms of oxygen, and 1 atom of nitrogen. There are three ways in which the composition of a substance may be expressed: 1, by atom; 2, by weight; 3, by volume. Thus water is composed, by atom, of oxygen 1, and hydrogen 1; by weight, of hydrogen 1, and oxygen 8; and by volume, of hydrogen 2, and oxygen 1. Elementary bodies are represented in chemistry by letters or symbols. A list of the elements and symbols is given in the next section. A symbolic letter standing alone, represents one atom of the element. Thus C denotes one atom of carbon; 0, one atom of oxygen. To denote more than one atom, we may either repeat the symbol, or a figure may be placed either before or after the symbol: thus 000, 30, or 03,would each represent 3

Page  186 186 THE LABORATORY. [ART. XIII. atoms of oxygen. The latter method is usually adopted. Nitric acid, which is composed of 1 atom of nitrogen and 5 of oxygen, is denoted by NO5. To denote a compound formed of several compounds, we employ one or more commas, thus: SO,,HO, which is the formula of strong oil of vitriol. The terms, combining proportion and chemical equivalent, have the same meaning as atomic weight. 455. TABLE OF CHEMICAL EQUIVALENTS.a Names and Symbols of Hydro- Names and Symbols of HydroElements. gen= 1. Elements. gen =1. Aluminum.. Al 13.72 Manganese. Mn 27.72 Antimony.. Sbb 129.24 Mercury.. Hgh 101.43 Arsenic... As 75.34 Molybdenum. Mo 47.96 Barium... Ba 68.66 Nickel.. Nk 29.62 Bismuth... Bi 71.07 Nitrogen.. N 14.19 Boron... B 10.91 Osmium.. Os 99.72 Bromine... Br 78.39 Oxygen... 0 8.01 Cadmium.. Cd 55.83 Palladium.. Pd 53.36 Calcium... Ca 20.52 Phosphorus. P 31.44 Carbon... C 6.04 Platinum.. Pt 98.84 Cerium... Ce 46.05 Potassium.. Ki 39.26 Chlorine-... C1 35.47 Rhodium.. R 52.2 Chromium.. Cr 28.19 Selenium.. Se 39.63 Cobalt. ~. Co 29.57 Silicon... Si 22.22 Columbium. Tae 184.9 Silver... Agk 108.31 Copper... Cud 31.71 Sodium.... Na 23.31 Didymium.. D? Strontium.. Sr 43.85 Erbium... E? Sulphur.. S 16.12 Fluorine... F1 18.74 Tellurium. o Te 64.25 Glucinum. G 26.54 Terbium.. Tr? Gold.... Aue 199.2 Thorium.. Th 59.83 Hydrogen.. H 1. Tin... Snm 58.92 Iodine... I 126.57 Titanium.. Ti 24.33 Iridium... Ir 98.84 Tungsten.. W' 94.8 Iron...... ef 27.18 Vanadium.. V 68.66 Lantanum.. La? Uranium.. U 217.2 Lead.... Pb; 103.73 Yttrium.. Y 32.25 Lithium. Li 6.44 Zinc... ~ Zn 32.31 Magnesium.. Ma 12.89 Zirconium. Z 33.67 a Compiled from Parnell and Draper. b Stibium. c Tantalum. d Cuprum. e Aurum. f Ferrum. g Plumbum. " Hydrargyruln. i Kalium. k Argentum. 1 Natrium. m Stannum. n Tungsten or Wolfram.

Page  187 ~ 56.] MISCELLANEOUS EXAMPLES. 187 If compounds are united by a feeble affinity, the sign tis sometimes used. Thus the composition of sulphuric acid may be indicated by SO0, or by S02+ 0, the latter formula showing that one of the atoms of oxygen is held by a feebler affinity than the other two. 1-58. Find the value of each of the foregoing equivalents, assuming oxygen=100.,Ans. Al 171.28. Sb 1613.48, &c. &c. 5; 6. EXAMPLES FOR THE PUPIL. 1-33. Find the atomic weightsa of each of the following Acids:b Acetic, C4H303; Arsenic, AsO,; Arsenious, AsO,; Benzoic, CJH503; Boracic, B03; Bromic, BrO5; Carbonic, CO2; Chloric, C105; Chromic, CrO,; Citric, CH41204; Formic, C311HO3; Gallic, C17HO3; Hydriodic, HI; Hydrobromic, HBr; Hydrochloric, HC1; Hydrocyanic, H + C2N; Hydrofluoric, HF1; Hydrosulphuric, HS; Hypermanganic, MIn20,; 1-yposulphurous, S,,02; Hyposulphuric, S205; Iodic, 10,; Lactic, C6H505; Malic, CHO,; Manganic, MnO3; Nitric, NO5; Oxalic, C203; Phosphoric, PO,; Silicic, SiO3; Sulphuric, S03; SUlphurous, S02; Tannic, C1811509; Tartarie, C8H401o. 34-58. Find the atomic weights of each of the following Bases: Alumina, A1203; Ammonia, NH3; Oxide of Antimony, SbO,; Barytes, BaO; Oxide of Chromium, Cr203; Oxide of Cobalt, CoO; Protoxide of Copper, CuO; Suba The equivalent of a compound body is always equal to the sum of the equivalents of its constituents.-Parnell. b," Compound bodies may, for the most part, be divided into three groups; acids, bases, and salts. By an acid, we mean a body having a sour taste, reddening vegetable blue colors, and neutralizing alkalies; by a base, a body which restores to blue the color reddened by an acid, and possessing the quality of neutralizing the properties of an acid; by a salt, the body arising firom the union of an acid and a base. These definitions, however, are to be received with considerable limitation." -Drlper.

Page  188 188 THE LABORATORY. [ART. XIII. oxide of Copper, Cu2O; Peroxide of Iron, Fe20,; Protoxide of Iron, FeO; Protoxide of Lead, PbO; Lime, CaO; Magnesia, MaO; Protoxide of Manganese, MnO; Oxide of Mercury, HgO; Suboxide of Mercury, HgO0; Oxide of Nickel, NiO; Oxide of Platinum, PtO; Potash, KO; Oxide of Silver, AgO; Soda, NaO; Strontian, SrO; Protoxide of Tin, SnO; Peroxide of Tin, SnO2.; Oxide of Zinc, ZnO. 59. Hydrochloric acid consists of equal volumes of chlorine and hydrogen, united without condensation. Required its specific gravity, the specific gravity of hydrogen being.069, and that of chlorine 2.47.a Ans. 1.2695. 60. If one volume of carbonic acid gas contains 1 volume of oxygen and 1 volume of carbon vapor, what is the specific gravity of carbon vapor, the specific gravity of carbonic acid being 1.5238, and. that of oxygen being 1.1025? IAns..4213. 61. The vapor of alcohol is composed of 8 volumes of carbon, 12 volumes of hydrogen, and 2 volumes of oxygen, the whole being condensed into 4 volumes of vapor. Required its specific gravity; the specific gravity of carbon being.4213, that of hydrogen.069, and that of oxygen 1.1025. Ans. 1.60085. 62. According to the experiments of Despretz, loz. of carbon evolves, during its combustion, as much heat as would raise the temperature of loz. of water 14067~. I-ow many pounds of water could be raised from the freezing point to the mean temperature of the humnanbody, (98.3~,) by the 13.9oz. of carbon, which are daily converted into carbonic acid in the body of an adult?b Ans. 184.3 lb. a In determining the specific gravity of gases, air is generally assumed as the standard, = 1. b Liebig.

Page  189 ~ 57.] GENERAL REMARKS. 189 XIV. GENERAL ANALYSIS. 57. REMARKS ON TIE SOLUTION OF QUESTIONS. IN most treatises on arithmetic, after the fundamental rules have been taught, the various applications of those rules are arranged under different heads, such as Interest, Discount, Practice, Proportion, Profit and Loss, Fellowship, Bankruptcy, &c. This division of the subject is convenient for beginners, but the expert arithmetician must be entirely independent of formal rules; he must be able, by analyzing any question that is proposed to him, to determine what operations are necessary for its solution. Accountants, men of business, and nearly all who are required to make frequent calculations, perform most of their work by analytical processes, and not by the rules that they learned at school. Mistakes are rarely made in determining the proper mode to be pursued when numbers are to be merely added or subtracted; but when multiplication or division is required, care is often necessary to avoid multiplying when we ought to divide, or dividing when we ought to multiply. Practice and careful attention to the conditions of the question will generally remove all difficulty. Much unnecessary labor may often be avoided, by stating the operations as we proceed, and not performing any of the multiplications or divisions until the statement is complete. Thus, if it were required to multiply 27 by 19, to divide the product by 2 times 171, and to multiply the quotient by 2 of 8 of - of 594, the readiest method of arriving at the result, would be 27 x 19 2 x 7 x 4 x 594 to express the operations thus, 27 X 19 x x x 594 2x171 3x8x9xl 1 then by cancelling, wce readily determine the answer, which is 7 X 33 or 231. One of the two following methods, will furnish the answer to nearly all questions that admit of an analytical solution.

Page  190 190 GENERAL ANALYSIS. [ART. XIV. 1st. In the majority of cases, we should endeavor to find from the terms that are given, the value which would correspond to ONE of each of the terms concerning which an answer is required. Having found the answer for ONE, we can easily determine it for the REQUIRED NUMBER in each of the terms of demand. 2d. We are sometimes required to reason from a result to its origin. In such cases it is generally best to reverse all the operations by which the result was obtained. Examples illustrating both of these methods, will be found in the two following sections. Whenever fractions or decimals are involved in the conditions of a question, the -same operations must be performed on them, that would be required if their place were occupied by whole numbers. Therefore, if we are ever at a loss whether to multiply or divide by a fraction, all difficulty may be removed by substituting a small integral number, and considering what would then be required. S. EXAMPLES ILLUSTRATING THE FIRST METHOD. The pupil should be required to repeat the analysis of all the following examples, and encouraged to give a solution of his own, whenever a different one occurs to him. 1. What is the interest of $635.50 for 3y. 8mo. 24dy. at 6 per cent.? The interest of ONE dollar for ONE year, is $.06. For 3y. 8mo. 24dy. it would be 831 times as much, or $6 X 06. The interest 1~5 ~15 of $635.50 will be 635.50 times as much as the interest for ONE, dollar, or $635.50 X 56 X.0Ga_ $? 15 a The object of these examples is merely to show how the general principles of analysis can be applied to all classes of questions. All the short processes and contractions that the pupil may have learned, and can retain in his memory, he may be allowed to employ.

Page  191 ~ 58.] MISCELLANEOUS EXAMPLES. 191 2. How much per cent. is gained by selling at 90cts. a pound, tea that cost 75 cents? A gain of ONE per cent. on a pound, would be $.0076. The actual gain was 15 cents, which is (.15 -*.0075) times ONE per cent., or per cent.? 3. If $400 at simple interest amounts to $440.50 in 2y. 3moo., what is the rate per cent.? The interest is $40.50. The interest of ONE dollar for ONE year at ONE per cent., would be $.01. The interest of $400 for 2{y. at ONE per cent., = 00 X - The NUMBER of per cent.40.50 1 4 1:L X C ~X 9X oper cent.?.Give an analytical solution of the above question, by first finding the interest for one year. 4. In what time will $900 amount to $1044 at 6 per cent., simple interest? We wish to find in how many years $900 will yield $144 interest, at 6 per cent. In ONE year it would yield $54. The NUMBER of years is therefore 144 —. 54- yr.? 5. If a man's property yields 5~ per cent. simple interest, and his annual income is $1093.122, what is he worth? We wish to find how many dollars he is worth. If he was worth ONE dollar, his income would be $.05~. He is therefore worth as many times ONE dollar, as are equivalent to 1093.12,-.05, or $? 6. When money yields but 5 per cent. interest, what is the present worth of $5316.84 due in lyr. 7mo. 6dy.? We wish to find how many dollars will amount to $5316.84 in 13 yr. at 5 per cent. ONE dollar would amount to $1.08. The number of dollars required, is therefore as many times ONE dollar, as are equivalent to $5316.84. $1.08, or {? 7. What must be the face of a note at 90 days, to be discounted at Bank, at 6 per cent., to yield $500? We wish to find how many dollars would yield $500, and we

Page  192 192 GENERAL ANALYSIS. [ART. XIV. first find that ONE dollar would yield $.9845. The number of dollars required is therefore as many times ONE dollar, as are equivalent to 500 -.9845 = $? 8. If I gain 25 per cent. on the original cost, by selling merchandise for $1718.75, how much did it cost me? How many dollars did it cost? ITf it had cost ONE dollar, to gain 25 per cent. it must have been sold for $1.25. It must therefore have cost (1718.75 -1.25) times ONE dollar, or $? 9. When exchange on England is at a premium of 8i- per cent., what is the value in sterling money, of $112? We cannot so easily find the value of $1 in English Money, as the value of ~1 in Federal Money.a If ONE ~ = $1.081 X -4~_, how many ~ = $112? As many times ONE pound, as there are times 1.08- lX 4Oin 112. 112X X =~? 9 1 2.17 40 10. If 36 men, in 1275 days of 138 hours, dig a trench 33- yd. long, 10-ft. deep, and 15 3ft. wide, how many men in 7U days of 12s-r hours, will dig a similar trench 821 -8yd. long, 7{ft. deep, and 10ft. wide? First, to find how many men in ONE day, of ONE hour, would dig a trench ONE yd. long, ONE ft. deep, and ONE ft. wide, we must multiply 36 by 1271- and by 1'81, and divide by 33, 10, and 15 5, or' X X 227 X 40 X22- X' Second, in 7Y days it will not require so many men as in one day; working 12-8 — hours a day, will not require so many men as working one hour a day, &c.; we must therefore divide our first result by 7 2 and 12i-8, and multiply by 82s, 7 5, and 10, which will give 6 X 2 X 27 4 2 x xx x >( x x 10 mn~. X 135 X X 5 X 51 X 140 X 91 X 357 X 1 men 11. How many square yards in a room 28ft. 4in. wide, and 37ft. 6in. long? The answer is to be in yards, and therefore the dimensions must be reduced to yards. The width is 9gyd., and the length 12}yd. If the room was ONE yard long and ONE yard wide, it would conaSome attention will often be required, to determine which of the quantities should be taken to find the value of ONE.

Page  193 ~ 58.] IISCELLANEOUS EXAMPLES. 193 tain one square yard. But the width is 94yd., instead of 1yd.; and the length 12lyd., instead of lyd. We must therefore multiply 1 X 5 X = sq. yd.? 12. If a -man receives $30 for building 8 rods of wall, and he can purchase 3 barrels of flour for $14, and 5cwt. of sugar for 6 barrels of flour, and 126 lb. of tea for 10cwt. of sugar, how many pounds of tea can he purchase by building 24 rods of wall? We cannot find directly how many pounds of tea he can purchase by ONE rod of wall, but for ONE rod he will receive $3~. For ONE dollar he can buy 13 of a barrel of flour, and for $3a0 he can buy 38o X -3- of a barrel.'For ONE barrel of flour he can buy;cwt. of sugar, and for 30o X y3- bbl., he can buy 3_0 X 3 X bcwt. For ONE cwt. of sugar he can buy?on —0 lb. of tea, and for 30 X134 X;cwt. he can buy 380 X -3- X X l2 lb., which is therefore equivalent to 1 rod of wall, and 24 rods will purchase ~- Xl 0 X X X X 116 lb. of tea? 13. A.'s stock in a partnership is $450, B.'s $350, and C.'s $500. How must a loss of $169 be divided between them? If $1300 loses $169, how inuch does ONE dollar lose? If ONE dollar loses $1 1ao90, $450 will lose $ 4 X = $, 350 will lose 5-o X 1o9 $, and $500 will lose $a a 6 69 14. Divide 650 into four parts, which shall be to each other in the proportion of -, -1, 3-, and 1 If one part has ~ of a share, another l of a share, another a of a share, and the other -7- of a share, then the whole will be 2} shares. ONE share is therefore 650 -. 2 —. Then I share-, share =-> X -- X i3 =, - shares -3-X5~X =,and 7 shareX= -7 X' ~ = 15. A bankrupt owes $15600, and his property is worth only $10600. How much can he pay on a debt of $450? 13

Page  194 194 GENERAL ANALYSIS. [ART. XIV. On ONE dollar he can pay T6'1 3 and on $450 he can pay $49X.=$? 16. A. invests a certain sum for a certain time, B. twice as much for -3 of the time, and C. three times as much for 2 of the time. How should they share the gain, which was $559? Find the value for ONE sum for ONE time, which will be A.'s share. 2 sums for - of a time = f2 sum for ONE time = B.'s share. 3 sums for -2 of a time = sums for ONE time —= C.'s share. Then the whole is equivalent to 1 + 2 + 15 2 -35 sums for ONE time. ONE sum for ONE time = $59 2-1-5 =, A.'s share; 2 sums for -1- of a time, or 2 of a sum for ONE time =$2 X 5 1-9 X 4-3 3 3 3tim4 r $, B.'s share; 3 sums for 5 time, or 1- sums for ONE time = [~6`9 5 $-...... X 5 —=$, C.'s share. 17. A goldsmith mixed 3 lb. of gold 22 carats fine, 5 lb. 20 carats fine, 8 lb. 24 carats fine, and 4 lb. of alloy. What was the fineness of the mixture? We wish to find how many carats there are in ONIE pound. There are 20lb. in the whole, containing carats, and therefore in each pound there are - 20 = carats? 18. Find the equated time for the payment of $500 due in 3 months, and $700 due in 5 months. The question is, in how many months should $1200 be paid, and we first find how many dollars could be used for ONE month, to be equivalent to the two debts. If $5000 could be used ONE month, how many months could $1200 be used? 19. A rectangular piece of ground contains 5 acres, and the width is 20 rods. Required the length. The field contains 800 square rods, and we wish to find how many rods long it is. If it were only ONE rod long and 20 rods wide, it would contain 20 square rods. But as it contains 800 rods, it must be 8s?2O times one rod long, = rods? 20. A stick of hewn timber 11in. wide and 10in. thick, contains 40 cubic ft. What is its length?

Page  195 ~ 59.] MISCELLANEOUS EXAMPLES. 195 How many feet long? If it were oNE ft. long, I lft. wide, and lift. thick, it would contain I 11 X X -= 5 cubic ft. But as it 6'"' I ~a, iu rrv-2 6cui c f t. 72 contains 40 c. ft. it must be (40'- 52) times ONE ft. long, =- ft.? 21. If A. can do ~ of a piece of work in 5 days, B. can do i of it in 4 days, and C. can do i of it in 2 days, in how many days can they do the whole by working together? In ONE day they can do 1-1y + -T + 1g of the 1 piece of work. Then they can do the I piece of work in 1. (-1 +- -y'- + g)" days? 59. EXAMPLES ILLUSTRATING THE SECOND METHOD. 1. The greater of two numbers is 5~ times the less, and the sum of the numbers is 52. What are the numbers? 52 is produced by adding the less number to 5~ times the less, and is therefore 6~ times the less. 2. Divide the number 582 into four such parts that the second may be twice the first, the third 21 more than the second, and the fourth 54 more than the first. 582 is produced by adding 21 and 54 to 6 times the first number. Therefore if we subtract 75 from 582, the remainder will be 6 times the first. 3. A farmer bought some horses, cows, and calves for $1250, giving $50 apiece for the horses, $23 apiece for the cows, and $9 apiece for the calves, and there were three times as many calves as cows, and half as many horses as calves. How many were there of each? If there had been 2 cows, 6 calves, and 3 horses, they would have cost $250. But he paid $1250, then how many times could he repeat the purchase of 2 cows, 6 calves, and 3 horses? 4. If from 5~ times a certain number 83 be subtracted, 12* added to the remainder, and the sum divided by 61, the quotient will be 30. What is the number? (30 X 6I — 12~ + 81).5 =? 5. Five-eighths of a certain number exceeds * of it by 21. What is the number?

Page  196 196 GENERAL ANALYSIS. [ART. XIV. 5 —~ = 7. If 21 is > X the number, the number itself is 21 7 6. What number is that from which if we deduct 3 of itself, and 2 of the remainder, there will be 18 left? 18 is - of what is left after deducting 3- of the number. Then the whole number is - of 4 of 18 -? 7. If 30 per cent. is lost by selling shoes at 87~cts. per pair, at what price should they be sold to gain 10 per cent.? If 30 per cent. is lost, they must be sold at 70 per cent. of the cost. To gain 10 per cent., they should be sold at 110 per cent. of the cost. tiiO of.871=? 8. B. is 2 years older than A., C.'s age is 4 years more than the sum of A.'s and B.'s, and D.'s age, which is 48, is equal to the sum of the other three. What is the age of each? 48 is the sum of A.'s, B.'s, and C.'s. B.'s is 2 more than A.'s, and C.'s is 6 more than twice A.'s; the three ages are therefore 8 more than 4 times A.'s age. 69. MIISCELLANEOUS EXAMPLES IN ANALYSIS. 1. What is the interest of $1872.88 for 7yr. 10mo. 15dy. at 7 per cent.? Ans. $1032.43. 2. Bought 18.75yd. of broadcloth, at $4 per yard, and sold the whole for $83.50. What did I gain, and how much per cent.? Ans. $8.50 = 11 per cent. 3. At what rate per cent. will $421.50 amount to $674.40 in 6yr. 8mo.? 4. A note of $431 amounted, at its settlement, to $546.291. How long had it been on interest, the rate being 6 per cent.? Ans. 4y. 5mo. 15dy. 5. What is the face of a note which, at 7 per cent., will yield $111.65 interest in 3y. 8mo.? Ans. $435.

Page  197 ~ 60.] MISCELLANEOUS EXAMPLES. 197 6. What is the present worth of $2000, due in 3y. 6mo., interest at 7 per cent.? Ans. $1606.43. 7. What must be the face of a note at 60 days, to be discounted at Bank at 7 per cent., to yield $375? Anis. $379.65. 8. A commission merchant received 21 per cent. for the sale of an invoice of merchandise. What was the amount of the invoice, the total amount of the sale and commission being $1666.24,? Ans. $1625.60. 9. When exchange on England is at a premium of 91 per cent., what is the value in sterling money, of $137.75? Ans. 281. 7s. 44d. 10. How many days of 821 hours, will 42 men require, to build a wall 98-ft. long, 7ift. high, and 2Mft. thick, if 63 men can build a wall 45]ft. long, 6-7-ft. high, and 3lft. thick, in 68 days of 11 hours? Ans. 297 days. 11. Determine from the following table, the degree on each thermometer-scale, that corresponds to 0~ of Fahrenheit. Name of Thermoreter. Where used. Freezing point. Boiling point. Fahrenheit's... Great Britain & a ~, 012 United States. -~-320a + 2120 renneits2~France. Centigrade, or Celsius's France.. 0 + 10~ Reaumur's.... GermanyItaly, 00 + 800 Spain& France. 80 Russian, or Delisle's Russia. -1500 00 Ants. Centigrade, 17"~ below 0. Reaumur's, 140~'" 0. Russian, 176-~ " 0. 12. How many square yards in a room 19ft. 6in. wide, and 34ft. 8in. long? Ans. 759 sq. yd. a The degrees above zero are indicated by'le sign -b-; the degrees below zero by the sign -.

Page  198 198 GENERAL ANALYSIS. [ART. XIV. 13. If 33 copecks are equal to 5 English pence, 11 English pence are equal to 3 piasters, 13 piasters are equal to 1 florin, and 5 florins are equal to 29 francs, how many francs are equal to 11000 copecks? Ans. 2021 —4 francs. 14. A. contributed $380 to an adventure, B. $420, C. $500, and D. $700. What was each man's share of the gain, which was $900? Ans. A.'s share $171; B.'s $189; C.'s $225; D.'s $315. 15. Divide 7500 into 5 parts, in the proportions of ~, j, 2 6' Ants. 258618-; 17241-; 1293 9; 10344; 862%51. 16. An echo on the north side of Shipley Church, in Sussex, England, repeats 21 syllables.a If the speaker utters 3 syllables a second, at what distance from the echo does he stand, the velocity of sound being 1090ft. per second? Ans. 3815ft. 17. A man failing in trade owed $75000, to meet which he had property valued at $14500. HI-ow much can he pay A., who is a creditor for $10000, B., who is a creditor for $3750, and C., who is a creditor for $12362.50? Ans. A. $19331; B. $725; C. $2390.08. 18. The amount contributed by the United States in 1847, for the relief of Ireland and Scotland, has been estimated at $591313.29.b In what time would 10 men-of-war consume the same amount, supposing each vessel to have 3 officers, 20 midshipmen, and 1000 sailors; estimating the wages and rations of each officer at $30, of each midshipman at $20, and of each sailor at $16 per month? Ans. 3mo. 17.57-4-dy. 19. A farmer mixed 18~ bushels of wheat, at $1.00 per bushel; 1693bu. at $1.12i- per bushel; 13Tbu. of barley, at a Pierce. b Am. Almanac, 1848.

Page  199 ~ 60.] MISCELLANEOUS EXAMPLES. 199 62~cts. per bushel, and 10bu. of oats, at 37'}cts. per bushel. What was the mixture worth per peck? Anzs. $0.21 +. 20. At what temperature does the mercury indicate the same degree on Fahrenheit's and on the Centigrade scale? [See Ex. 11.] Ans. - 400~. 21. Manson & Hill, of Liverpool, have given to Thomas Morton & Co., of New York, a note of ~225 1Os., payable in 60 days; one of ~196, payable in 60 days; one of ~218 7s. 6d., payable in 90 days; and one of ~300, payable in 120 days. At what time may the notes all be equitably cancelled by a single payment of ~939 17s. 6d.? Ans. 86 days. 22. The length of a floor is 37ft. 6in., and the area is 93{ sq. yd. What is the width? Ans. 22ft. 6in. 23. A stick of hewn timber ift. 2in. wide and 9in. thick, contains 60-2- solid feet. What is its length? Ans. 69ft. 4in. 24. A grain of musk is said to be capable of perfuming for several years, a chamber 12ft. square, without sensible diminution of volume or weight.b If the chamber is 8ft. high, and constantly contains an average of 1 particle to every cubic tenth of an inch, how many particles must there be in the grain, supposing it to have lost IT-o of its weight after the air has been changed 5000 times? Ans. 9953280000000000. 25. A father's age is 63 times his son's age, and the sum of their ages is 34yr. 2mo. 12dy. What is the age of each? 26. Divide $5000 into four such parts, that the first may be twice the second, the third $50 less than 4- the second, and the fourth $100 more than the sum of the first and third. 27. At what temperature does the mercury indicate the a If 1000 gain 800, how many degrees will gain 320~? b Moseley.

Page  200 200 GENERAL ANALYSIS. [ART. XIV. same degree on Fahrenheit's and on Reaumur's scale? [See Ex. 11.] Ans. - 25.6~. 28. A farmer hired a certain number of boys, and twice as many men, agreeing to pay each man 75 cents a day, and each boy 25 cents. The daily wages of the whole amounted to $5.25. How many were there of each? Ans. 3 boys; 6 men. 29. -A man's age is such that if it be multiplied by 3, and if 2 of the product be tripled, X of the result will be 16. Required his age. Ans. 28. 14 30. What is that number 3 of of which exceeds 67 of 7 _9 ~16 - itself by 21? Ans. 147 31. At what temperature would the mercury indicate the same degree on the Centigrade and Russian scales? On the Russian and German scales? [See Ex. 11.] Ans. +300~; + 1713~. 32. What number is that, from which if we deduct 3 of 12 of itself, and A of - of the remainder, there will be II 6 2, willbe 10-: left? Ans. 34. 33. If 18 per cent. is lost by selling merchandise at $2050, at what price should it have been sold to gain 10 per cent.?-to gain 25 per cent.?-to lose 10 per cent.? 3d Ans. $2250. 34. B. is 5 years older than A.; C.'s age is 5 years more than the sum of A.'s and B.'s; and D.'s age, which is 55, is equal to the sum of the other three. What is the age of each? Ans. A. 10; B. 15; C. 30. 35. In examining a piece of charcoal through a microscope, Dr. Hook counted 150 pores in -1 of an inch.a At this rate how many pores would there be in one square inch of surface? Ans. 5760000. a Moseley.

Page  201 ~ 60.] MISCELLANEOUS EXAMIPLES. 201 36. A. and B. can do 1-1- of a piece of work in 1 day; B. and C. can do -9 of it; A. and C. can do 5 of it in the 10 ) 6 same time. In what time will they all do it working together? By adding', 1o90, and 5, and dividing the sum by 2, we find the part that they will all do in 1 day. Ans. -30- of a dlay. 37. Three men traded in partnership. A. contributed $1500, B. $2250, and C. the remainder. The whole gain was $2700, of which C. received $1200. How much did C. contribute, and what did A. and B. gain? Ans. C. contributed $3000; A. gained $600; B. gained $900. 38. An estate of $15000 is to be divided among three persons; A. is to receive $5+ as often as B. receives $4~, and B. is to receive $8~ as often as C. receives $4,. What is the share of each? Ans. A. $6620.69; B. $5586.21; C. $2793.10. 39. A cistern has three pipes; the first can fill it in - an hour, the second can fill it in i of an hour, and the third can empty it in an hour. In what time will the cistern be filled, if they all run together? Ans. 15 minutes. 40. At what temperature is the mercury as many degrees above zero of Fahrenheit, as it is below zero of the Centigrade? a [See Ex. 11.] Ans. 117~. 41. A bathing-tub that holds 147 gallons, is filled by a pipe that brings 14 gallons in 9 minutes, and emptied by a pipe that discharges 40 gallons in 30 minutes. Both pipes having been left open for 3 hours, it is required to find in what time the tub will be filled if the discharging pipe is closed? Ans. lh. 8 11 minutes. 42. What sum of money will amount to $1500, in 5 years, at 5 per cent. simple interest? Ans. $1200. a If the Centigrade moves 100~ when the sum of the motions is 2800, how much will it move when the sum of the motions is 32?

Page  202 202 GENERAL ANALYSIS. [ART. XIV 43. At what rate per cent., simple interest, will $700 amount to $1300 in 11 years? Ans. 76- per cent. 44. In what time will $1100 amount to $1750, at 6 per cent. simple interest? Ans. 928 years. 45. Supposing the weight of a molecule of light to be 1l0-o6- of a grain, what should be the velocity of a ball, weighing loz. avoirdupois, to have the same momentum.a Ans. 2.317 + ft. per second. 46. A laborer received $1.50 for every day he worked, and lost 50 cents every day he was idle. He worked twice as many days as he was idle, and at the end of the time he received $42.50. How many days did he work? Ans. 34 days. 47. A floor is laid with boards U1in. thick. How many feet are required, the room being 18ft. 6in. wide, and 20ft. 8in. long? Ans. 477-1ft. 48. How many clapboards would be required to cover an area of 1376 sq. ft., the clapboards being 4ft. long, and laid with 4 inches to the weather? Ans. 1032. 49. A cellar is to be made 40ft. long, and 25ft. wide. How many squares of earth must be removed, the depths at six different points being 8ft., 7ft. 6in., 7ft. 3in., 8ft. 4in., 8ft. 9in., and 5ft. 2in.?b Ans. 341 squares. 50. At what temperature is the mercury as many degrees above zero of Fahrenheit, as it is below zero of Reaumur? [See Ex. 11.].Ans. 91 — 1 51. What is the weight of a stone wall, the solid contents being 2760 c. ft., and the specific gravity 2500? Ans. 431250 lb. a The momentum of a body is determined by multiplying the weight by the velocity. The velocity of light is 192000 miles per second. Such considerations are supposed to prove that light is without weight. b Assume as the true depth of the cellar, the average of the six measured depths.

Page  203 ~ 60.] MISCELLANEOUS EXAMPLES. 203 52. 1Wishing to estimate the contents of an irregular pile of wood, I take the dimensions in several places, and find that the average length is 34ft. 6in., the average breadth 27ft. 4in., and the average height 2ft. 3in. If it were piled regularly, I judge that it would occupy only ] of the space that it now does. Required its estimated contents. Ans. 11 cords 61 c. ft. 53. How many cubic feet in a stack of hay, which is estimated to be equivalent to a cylinder 12ft. in diameter and 15ft. high? Ans. 1696 c. ft. 54. What must be the area of a roof that would fill a cistern holding 40 hogsheads, with a fall of i- inch of rain, the roof being of a true pitch? The roof being of a true pitch, the area of the roof will be 1times the area of the building, but the water that falls upon it will be the same as if the roof were fiat. Ans. 24255 sq. ft. 55. A chronometer usually vibrates 4 times in a second. How much must the length of each vibration be increased, in order that it may lose 1 second per day? Ants. 31 IU6" 56. The area of a cistern is 381 sq. ft. How many gallons would fill it to the depth of 1ft.? Ans. 288 gallons. 57. If a package of sugar weighs 6 lb. 2oz. in one scale of a balance, and 8 lb. in the other, what is its true weight? Ans. 7 lb. 58. A man performed a journey of 135 miles, going twice as far the second day as on the first, and three times as far the third day as on the second. How far did he travel each day? a The true weight of any body may be found by a false balance, by weighing the body in each scale, and taking the mean proportional between the two weights. It may also be obtained by first balancing the body with shot or some other article, and then removing the body and placing weights in the scale till the equilibrium is restored.

Page  204 204 GENERAL ANALYSIS. [ART. XIVo 59. A., B. and C. entered into partnership, contributing in the whole, $4833. B. paid twice as much as A., and C, paid twice as much as A. and B. How much did each contribute? 60. In a certain school of 70 scholars, three times as many study Arithmetic as study Latin, and twice as many learn to read, as study Arithmetic. How many are there in each study? 61. What degree of temperature would be indicated in the same manner on the scales of Fahrenheit and Delisle? [See Ex. 11.] Ans. — 1060~. 62. An estate of $7000 was so divided that the widow received $500 more than the daughter, and the son $1100 more than the widow. What was the share of each? 63. Divide the number 97 into four such parts that the second may be twice the first, the third 7 more than the second, and the fourth 18 more than the first. 64. A thief travels at the rate of 6 miles an hour, and after he has been absent 5~ hours, a constable starts in pursuit, at the rate of 9 miles an hour. In what time will the thief be overtaken? 65. A man when he was married, was three times as old as his wife, but after they had lived together 15 years, he was only twice as old. How old was each at the time of marriage? Ans. 45 years; 15 years. 66. At a certain election, the successful candidate had 163 votes more than his opponent, and the whole number of votes polled was 1125. How many did each receive? 67. What sum of money will yield $123.50 in 2 years, at 5 per cent. simple interest? 68. A gentleman distributed $1.95 among 3 beggars, giving the second 25 cents more than the first, and the

Page  205 ~ 60.] MISCELLANEOUS EXAMPLES. 205 third twice as much as the second. How much did each receive? 69. If from three times a certain number 17 be subtracted, the remainder will be 112. What is the number? 70. At what temperature is the mercury as many degrees below zero of Delisle, as it is above zero of Reaumur?of Celsius? —of Fahrenheit? [See Ex. 11.] Ans. 52-43o; 600; 96-4o 71. A merchant owes two of his creditors $1575, and he owes the second but * as much as the first. What is the amount of each debt? 72. In a certain school -1 tPle boys learn to read, -l learn to write, lo learn Algebra, -5- learn drawing, and the remaining 4 study Latin. How many are there in the school? 73. One-third of a certain pole is painted green, and 9of it is painted white, the remainder, which is 8 feet, being in the ground. What is the length of the pole? 74. A man going to market, was met by another, who said: " Good morrow, neighbor, with your hundred geese." H-le replied: " I have not a hundred; but if I had as many more, and half as many more, and two geese and a half besides, I should have a hundred." How many had he? 75. A man bought 38 pounds of coffee and 95 pounds of sugar; he gave 2 cents per lb. more for the coffee than for the sugar, and the sugar cost twice as much as the coffee. WVhat was the price of each per pound? Ans. Sugar Sets.; coffee 0cts. 76. If 4 men can saw 15 cords of oak in the same time that 5 men saw 14 cords of hickory, and if 3 men saw 18 cords of hickory in 3 days, by working 9 hours a day, how many hours a day must 7 men work, to saw 84 cords of oak in 6 days? Ans. 61 8 hours. 77. There are two such numbers, that if 21 be added to

Page  206 206 GENERAL ANALYSIS. [ART. XIV. the first, the sum will be 5 times the second, and if 21 be added to the second, the sum will be 3 times the first. What are the numbers? 25$ is the difference between 1 of the first, and 3 times the first. The first, therefore, is 9, and the second 6. 78. Two stages are travelling towards each other, one at the rate of 5} miles an hour, the other 6- miles an hour. In what time will they meet if they are now 38-4 miles apart? Ans. 3 — 4 9 hours. 79. Two men start from the same place and travel in opposite directions, one at the rate of 4- miles an hour, and the other 5a-3- miles an hour. In what time will they be 100 miles apart? Ans. 10{-P hours. 80. With what velocity must a battering ram, weighing 2000 lb., be moved, to have the same momentum as a cannon ball, weighing 20 lb. and moving 1200ft. per second? Ans. 12ft. per second. 81. There is a number to which if -,, and of itself be added, the sum will be 2 of 87 less than 16. What is the number? Ans. 3-.-2 41 82. If:A. can do i of a piece of work in 5 days, B. can do, of it in 4 days, and C. can do i of it in 2 days, in what time will they all do I of it by working together? Ans. 2' days. 83. There are two men of equal ages, but if one was 5 years older, and the other 9{1 years younger, the former would be twice as old as the latter. Required their ages. Ans. 24yr. 84. If a man can do 7- of a piece of work in 3 days, and a boy can do 7. of it in 5 days, how long will it take them both to do the whole? Ans. 4-594 days. 85. A hare starts 5 rods before a greyhound, and runs at the rate of 12 miles an hour. After running 48 seconds,

Page  207 ~ 60.] MISCELLANEOUS EXAMPLES. 207 the hound starts in pursuit, and runs 20 miles an hour. In what time will the hare be overtaken? Ans. lm. 19 see. 86. If 300 tiles that are 9in. long and 6in. wide, will pave a court-yard, how many tiles would be required that are 6in. long and 4in. wide? Ans. 675 tiles. 87. How many men will build a wall 240yd. long, 6ft. high, and 3ft. thick, in 8 days of 9 hours, if 7 men can build a wall 40yd. long, 4ft. high, and 2ft. thick, in 32 days of 7 hours? AAns. 294 men. 88. A wall which is to be built to the height of 27 feet, has been raised 9 feet in 6 days, by 12 men working 13 hours a day. How many men must be employed to finish it in 2 days, working only 12 hours a day? Ans. 78 men. 89. Amsterdam exchanges with London, at 34 schillings 4 pfennings per ~, and with Lisbon at 56 pfennings for 400 reas. What is the arbitrated exchange between London and Lisbon, by way of Amsterdam? Ans. ~1 = 2 ~ 942-. 90. Find the number of vibrations in one second, of each of the rays of light, the velocity of light being 192000 miles per second,a and the lengths of the waves being, for red light,.0000256 of an inch. For orange...0000240 For blue..0000196'C yellow...0000227 " indigo...0000185 e" green...0000211 " violet...0000174b Ans. Red, 475200000000000 vibrations. Violet, 699144827586207 &c. &c. &c. 91. The force available for mechanical purposes in an adult man, is reckoned, in mechanics, equal to j- of his own weight, a Herschel. b Draper.

Page  208 208 TI-IE COUNTING-IHOUSE. [ART. XV. which he can move during 8 hours, with a velocity of 2;ft. per second.a What is the momentumb for the day's work of a man who weighs 160 lb.? Ans. 2304000. XV. THE COUNTING-HOUSE. 61. PERCENTAGE. 1. THE term per cent. is an abbreviation of the Latin per centum, which signifies by the hundred. Any number of per cent. of a quantity is therefore equivalent to as many hundredths of that quantity. Thus 7 per cent. is.07; 41 per cent. is.04~ or.045; 184 per cent. is.18i or.1875. 2. Per cent. should not be confounded with any of the denominations of Federal Money. Thus, 6 per cent. is not 6 cents, or 6 dollars, but simply yOO. 6 per cent. of 125 dollars, is T6- of 125 =$7.50; but 6 per cent.,of 125 apples is 7.50, or 7~ apples, and 6 per cent. of 125 lb. is 7 lb. 8oz. 3. Any fraction may be reduced to per cent., either by reducing it to a decimal, and stopping the decimal at hun. dredths' place, or by multiplying the fraction by 100. Thus a7 reduced to a decimal, gives.463, or 46i per cent.; the same fraction reduced to hundredths by multiplying by 100, gives 71050 hundredths, or 7~00 per cent., or 46fi per cent. 4. To determine the value of any quantity when there is a specified gain or loss per cent., we may add or subtract the given percentage from f -Qa or 1. Thus, if stock is sold a Liebig. b The momentum is obtained by multiplying the weight by the distance.

Page  209 ~ 62.] PROBLE0MIS IN PERCENTAGE. 209 at 25 per cent. advance, it is sold for,2 0 or 1.25 times its par value; if goods are sold at a loss of 18 per cent., they are sold for -s of their cost. 62. PROBLEMS IN PERCENTAGE. I. To find the gain or loss per cent. Make the gain or loss the numerator, and the prime cost the denominator of a fraction, and multiply the resulting fraction by 100. [See Sect. 61, 3.] a. To find what percentage must be gained on the selling price, to yield any desired profit per cent. on the cost. Divide the desired percentage of gain, by 100 plus that percentage, and reduce the quotient to hundredths. b. To find the percentage of profit on the cost of merchandise, the percentage gained on the selling price being known. Divide the percentage gained on the selling price by 100 minus that percentage, and reduce the quotient to hundredths. II. To deternmine the value of a quzantity, when tlhe value of any percentage is known. Divide by the percentage expressed decimally. EXAnIPLE. —If 2500 is 16 per cent. of a certain number, what is that number? Since.16 of a number =.16 timnes that nu mber, 2500 must be.16 times the number sought. The answer, therefore, is 2500.16 =15625. a. To find the amount which should be added to an insurance, to recover the amount of premium paid. Divide the premium by the difference between the rate and 100 per cent. III. To find the result that any quantity will yield in percentage. Multiply the quantity by the result that ONE will yield. a. To find the interest, or the amount, of any principal for any given time. Find the interest, or the amount, of ONE dollar for the given time, and multiply by the NUMBER of dollars in the given principal. This rule will hold good, both for simple and compound interest. 14

Page  210 210 THE COUNTING-HOUSE. [ART. XV. To find the interest of $1 for any given time: To!- as many cents as there are months, add i as many mills as there are remaining days, and the amount will be the interest at 6 per cent. For any other rate, take such part of the interest at 6 per cent. as may be requisite. EXAMPLE. —What is the interest of $287.75 for 3y. 7mo. 23dy., at 5, per cent..? The interest of $1 at 6 per cent. = - of 43cts. _+ _ of 23 mills, or $.2185. At 51- per cent. it will be 5- sixths, or as much, or l- X 1-.L3. The interest of $287.75$287.75 X -1 X 1.= $57.72. b. To find the selling price, to make any proposed gain or loss per cent. Multiply the prime cost by 1 with the percentage added which is to be gained, or the percentage subtracted which is to be lost. c. To reduce Sterling to Federal Money. Multiply the par value of ~1, ($4 0,) by 1 + the premium. The product will be the exchange value of t1. Multiply the exchange value of ~1, by the number of pounds. IV. To find the quantity that will yield any given result in percentage. Divide by the result that ONE would yield. EXAMPLE.-What principal will amount to $500, in 2y. 6mo., at 5 per cent.? $1 would amount to $1.125, and 500 * 1.125= $444. a. To reduce Federal to Sterling Money. Divide by the exchange value of ~1. b. To find the face of a note to be discounted at bank, in order to obtain any required sum. Divide the sum required, by the amount that would be received by discounting $1. The quotient will be the number of dollars for which the note should be drawn. c. To find the amount that a factor can lay out of a sum intrusted to him, and reserve a specified percentage for his commission. Divide the sum intrusted to him, by 1 + the percentage which is allowed for his commission. d. To find the prime cost, when the selling price and the gain or loss per cent. are known. Divide the selling price by the value of 1 with the proposed gain or loss per cent.

Page  211 ~63.] EXAMPLES IN PERCENTAGE. 211 e. To find the selling price so as to allow a discount for cash, and gain any proposed rate per cent. Multiply the prime cost by 1 + the proposed gain per cent., and divide by 1- the proposed discount per cent. f. The gain or loss per cent. at any given price being known, to find the gain or loss per cent at any proposed price. Multiply the percentage of the prime cost which corresponds to the given price, by the proposed price, and divide by the giver price. The quotient, (reduced to hundredths,) will be the percentage of the prime cost which corresponds to the proposed price. The difference between this percentage and 100 per cent., will be the gain or loss per cent. 63. EXAMPLES IN PERCENTAGE. A. Examples illustrating ~ 61. 1. Find 25 per cent. of $13.50; 7A per cent. of 2cwt. 3qr. 12 lb.; 16* per cent. of 252 miles; 4* per cent. of ~60; 135 per cent. of 10mo. 2. How many per cent. are equivalent to a; to 2;'7 1 5. 8. 37. 19. 4. 278? 45 95 80' kG6) 7P06 13 3. What part of the original value is stock worth, when it is 152 per cent. below par? When it is at a premium of 8-: per cent.? When it is at a discount of 12~ per cent.? 4. fIow much was received for a farm, bought for $1350, and sold at an advance of 33* per cent.? Ans. $1800. B. Examples under Problem I. & 62. 5. How much per cent. was lost on flour, which was bought at $5, and sold at $4.62~ per barrel? Ans. 7~ per cent. 6. What percentage must a merchant gain on the total amount of his sales, to be equivalent to a gain of 10 per cent. on the cost? Ans. 9 -yr per cent. 7. A tradesman finds that his profits on a year's business amount to 16-* per cent. of the sales. What percentage of the cost has he gained? Ans. 20 per cent

Page  212 212 THIE COUNTING-HOUSE. [ART. XV. C. Examples under Problem IL. ~ 62. 8. Twenty-seven and a half is 18 per cent. of what number? Ans. 1521. 9. What sum should be insured to cover the amount paid for premium and policy, if I wish to insure $1800 on merchandise, at a premium of J per cent., the charge for the policy being $1? Ans. $1814.61. D. Examples under Problem III. ~ 62. 10. Find the interest of ~27 7s. 6d. for ly. 7mo. 18dy. at 5 per cent. Ans. ~2 4s. 8~d. 11. At what price should I sell broadcloth, which cost $38 per yd., in order to gain 11 per cent.? Ans. $3.75. 12. At 9~ per cent. premium, what is the value in U. S. currency, of ~27 13s.? Ans. $134.56. E. Examples under Problem IV. ~ 62. 13. What principal, at 7 per cent. compound interest, would yield an interest of $500 every third year? Ans. $2221.80. 14. At 9 per cent. premium, what will be the value in English money, of a Bill of Exchange for $1250.25? Ans. ~258 Is. 7d. 15. Required the face of a note at 90 days, to yield $500 when discounted at bark. Ans. $507.87. 16. A factor receives 5 per cent. commission on the amount that he purchases. If I send him $1000, how much can he lay out, after reserving enough to pay his own commission? Ans. $952.38. 17. If I gain 15 per cent. by selling land at $402.50 per acre, what did the land cost per acre? Ans. $350. 18. At what price must I sell molasses that cost 25cts. per gallon, in order to gain 10 per cent., after discounting 5 per cent. for cash? Ans. $.28-8 per gallon.

Page  213 ~ 64.] PERCENTAGE ON STERLING MONEY. 213 19. If 12] per cent. is gained by selling a house foi $3825, what percentage would be gained or lost by selling it for $3230? Ans. 5 per cent. lost. F. Miscellaneous Examples. 20. A bill of goods is purchased on 6 months credit, amounting to $175.75. How mouch should be paid in cash, at the time of purchase, if the buyer is allowed 5 per cent. for his money? Ans. $167.38. 21. At what rate, simple inteQrest will any principal be doubled in 12y. 6mo.? 22. In what time will $2700 amount. to $3132, at 6 per cent. simple interest? 23. What sum of money at 6 per cenb. compnund interest, will amount to $2750, in 3y. 6mo.? Ans, $2241.70. 24. If a merchant receives his usual profit, by selling a quantity of sugar for ~46 5s., how much must he raise the price, in order to allow a discount of 7- per cent.? Ans. ~3 15s. 041. PERCENTAGE ON STERLING MONEY. In computing interest or discount on English money, for any length of time less than a year, it is customary to omit the shillings and pence in the principal, when they are less than ten shillings; but if they amount to ten shillings or more, they are considered as another pound. If it is desired to compute the percentage exactly, it may be done either by reducing the shillings and pence to the fraction or decimal of a pound,a or by multiplying 1 per a Shillings, pence, and farthings may be reduced to the decimal of a pound by inspection, as follows:-Multiply the number of shillings by 5, and call the product hundredths. Reduce the pence and farthings to far'things, increasing their number by 1, when it exceeds 12, and by 2, when it exceeds 36, and call the result thousandths. The sum of these two values will be the decimal required. To reduce the decimal of a pound to shillings, pence and farthings,

Page  214 214 THE COUNTING-HOUSE. [ART. XV. cent. of each denomination, by the number of per cent. required. The latter method is generally the readiest. EXAMPLE ILLUSTRATING EACH METHOD. Find 3~ per cent. of ~480 10s. 3d. First Method. Second Method. lOs. 3d. =.5125~ s. d. 480.5125 1percent. = ~4.80.10.03.035 31 = 2- 7 24025625 2) 33.60.70.21 14415375 16.80.35.105 16.8179375 20 20 16.35 16.3587500 12 12 4.305 4.30500 4 4 1.220 1.220 Ans. ~16 16s. 4~d. Ans. ~16 16s. 41d. The legal rate of interest in England is 5 per cent. For computing interest at this rate, the following rules are convenient. In each case, the principal is supposed to be expressed in pounds, and parts of a pound. 1. Multiply the principal by the number of years, and the product will be the interest in shillings. 2. Multiply the principal by the number of months, and the product will be the interest in pence. 3. Multiply the principal by the number of days, and divide the product by 30; the quotient will be the interest in pence. For any other rate than 5 per cent., first compute the Multiply the number of tenths by 2, and the product will be shillings. From the remainder of the decimal subtract g-i of itself, a*nd thefigures which stand in the hundredths' and thousandths' places will befarthings.

Page  215 ~ 65.1 BANKING. 213 interest at 5 per cent., and multiply the result by.2 X the number of per cent. For 3 per cent., multiply by.6; for 4~ per cent., multiply by.9; &c. &c. EXAMPLES. 1. What is the interest of ~487 10s. 8d., from March 4th, to Dec. 17, at 6 per cent.? Mercantile Ans. ~23 Os. 4Dd. Correct Ans. ~22 19s. 11d. 2. What per cent. is gained by selling, at ~1 1Os., velvet that cost ~1 2s. 6d., per yard? Ans. 33k per cent. 3. How much was received for 25 shares of stock, sold at a premium of 84 per cent., the par value being ~50 per share? Ans. ~1359 7s. 6d. 4. What was the interest of six India Bonds, of ~100 each, at 3a per cent., calculating from Sept. 30, 1849; the Bonds having been sold Jan. 15, 1850? Ans. ~6 2s. Gd. 5. What amount of 4 per cent. stock, will yield an income of ~150 per annum? Ans. ~3750. 6.5. BANKING. In computing interest at Bank, the time is usually determined in days. When the rate is 6 per cent., the interest is found by multiplying the principal by as many thousandths as are equivalent to the number of days, and dividing the product by 6. Thus the Bank interest of $175.50, for 63 days, is $175.50 x.063 - 6 = $1.84. For any other rate than 6 per cent., we may first compute the interest at 6 per cent., and add or subtract such part as may be required. For 4 per cent., subtract -k of the interest at 6 per cent.; for 4k per cent. subtract -1; for 5 per cent. subtract k; for 7 per cent. add i, and so on.

Page  216 216 THE COUNTING-ITOUSJE. [ART. XV, If a note is given, or a bill drawn for any number of months, calendar months are always understood. A note. at 4 months, dated on the 29th, 30th, or 31st of October, would expire on the last day of February, and would be legally. due on the 3d of March. The 3d of 3March is, therefore, a heavy day at bank, as~ in leap years there are 3 days' payments, and in common years 4 days' payments, which fall due on that day. If either the 3d or 4th of March, in any year except leap year, falls upon Sunday, there will be 5 days' payments falling due on the Saturday previous. There are two modes of estimating the time that elapses between different dates. The first, is by compound subtraction, which is the method almost invariably adopted in computing interest on notes, payable on demand. The second, is by determining the number of entire calendar months, and then finding how many days are left. This mode is adopted in many counting-houses, and in all banks. Thus, from Oct. 27th, 1850, to March 15th, 1853, would be, according to the 1st method, 2y. 411no. 18dy. " " " 2d " 2y. 4mo. 16dy. From Oct. 31st, 1850, to March 15th, 1853, would be, according to the 1st method, 2y. 4mo. 14dy. c'" " "' 2d " 2y. 4mo. 15dy. Bank discount is the same as Bank interest. If a note is discounted at bank, the bank takes off the interest for the time the note has to run, and pays the balance only to the holder of the note. The number of days which elapse between two given dates, may be found as in the following example: Required the number of days between 3March 23d, and Sept. 5th. We find by adding the number of days in all the intervening time, that Sept. 5th would correspond to March 189th, if the days were all regarded as belonging to March. Between 3MIarch

Page  217 ~ 65.] BANKING. 217 23d, and March 189th, would be 166 days, or March has 31 days 23 weeks and 5 days.a April; 0 8 May " 31 " If we wish to find on what day of the week June " 30'6 any given date will fall, we may proceed in July " 31 a similar manner. Thus, if March 23d comes August 31 Add for Sept. 5 " on Friday, as there are 23 complete weeks, and 5 additional days between March 23cl and 189 Sept. 5th, Sept. 5th will fall 5 days after Friday, or on Wednesday. 166 dy. A. note at 30 days, will fall clue in 33 days =4wk. 5d.; a note at 60 days, in 63 days, =9wk.; a note at 90 days, in 93 days,13wk. 2d.; a note at 120 days, in 128 days, -17wk. 4d. Therefore a 30 days' note becomes due 5 week days later than the day on which it is given; a 60 clays' note, on the same day of the week as the day on which it is given; a 90 days' note, 2 week days later; a 120 days' note, 4 week days later. It is sometimes important to date a note so that it will not mature on Sunday, or on a holiday. This can easily be done by one of the methods above given. EXAMPLES. 1. How much would an English merchant receive on a note for ~600 at 4 months, discounted at 4 per cent.? An~s. ~591 16s. 2. When will a 3 months' note fall due, if dated Aug. 31? -a note at 8 months, dated June 30th? a The number of days may also be found by the following table, when the time is less than a year: TABLE FOR ASCERTAINING THE NUMBER OF D)AYS FROM ANY DAY IN THE YEAR, TO ANY OTHER DAY. 1st Mo., Jan.. 0 5th Mo., May. 120 9th Mo., Sept. 243 2d Mo., Feb. 31 6th Mo., June 151 10th Mo., Oct.. 273 3d Mo., Mar.. 59 7th Mo., July.181 11th Mo., Nov. 304 4th Mo., Apl.. 90 8th Mo., Aug. 212 12th Mo., Dec.. 334 RULE. To the given day of each month, add the tabular number for the month, and subtract the less sum from the greater. if the two dates are in different years, subtract the result thus found fromn 365. Isn leap years, add 1 to the number after the 28th of February.

Page  218 218 THE COUNTING-IIOUSE. [AlT. XV. 3. Determine by each method, the time that elapsed ]between July 4th, 1776, and June 2d, 1850; between Aug. 18, 1820, and Feb. 29, 1848. 4. How many days were there between Jan. 27, and June 16, 1844?-between Feb. 20, and Oct 8, 1849? 5. New Year's day, A. D. 1850, fell on Tuesday. On what day of the week was Christmas of the same year? 6. Required the availsa of a note at 4 months, for $275.50, dated Feb. 12th, 1850, and discounted at a bank in New York, where the legal rate of interest is 7 per cent. Ans. $268.91. 66. PARTIAL PAYMENTS. When partial payments are made on mercantile accounts which are past due, it is customary to compute interest on the whole debt from the time it became due, and on each payment from the time it was made, until the time of settlement, and to deduct the amount of all the payments, including interest, from the amount of the debt and interest. The labor of computing interest on each item may be avoided, by multiplying the amount due at first, and the balance due after each payment, by the number of days that they are severally at interest, adding all the products, and dividing the amount by 6000. The quotient will be the interest at 6 pe cent.b EXAMPLE FOR ILLUSTRATION. A debt of $630.25 became due March 15th, on which the following payments were made: April 3d, $170; May 20, $245.30; June 17th, $87.50. How much was due Sept. 5, when the account was settled? a The amount received for the note after it is discounted. b It may be readily seen that the same result will be obtained whichever method is adopted, but no accountant who is familiar with both methods, will hesitate to adopt the latter.

Page  219 ~ 67.] LEGAL INTEREST. 219 $ Days. Products. March 15, Amount due... 630.25 X 19 = 11974.75 April 3, 1st payment.. 170. Balance. 460.2.5 X 47 - 21631.75 May 20, 2d payment.... 245.30 Balance.. 214.95 X 28 = 6018.60 June 17, 3d payment... 87.50 Balance 1. 127.45 X 80 10196.00 6000) 49821.10 8.30351 Ans. $127.45 + $8.30 $= 135.75. The whole debt is on interest from March 15 to April 3, 19 days. The principal is then reduced by a payment, to $460.25, which is on interest from April 3 to AMay 20, 47 days. In like manner we find that the 2d balance is on interest 28 days, and the 3d, 80 days. The amount due at settlement, is the unpaid balance of the debt, $127.45, together with the interest, $8.30. EXAMPLES. 1. Sold goods to the value of $650.39, to be paid Jan. 27, 1848. Required the amount due Aug. 19, the following payments having been made on account: Feb. 23, $100; March 15, $150.39; May 20, $200; July 31, $125. Ans. $86.89. 2. A Georgia merchant bought goods to the amount of $575, and obtained credit till Jan. 31, 1849. But being unable to pay the whole debt at one time, he remitted $100 when it became due, and subsequently paid $75 March 3, $100 March 27, 8150 April 17, and the balance June 7. What was the amount of the last payment, the legal rate of interest being 8 per cent.? Ans. $158.51. 67. LEGAL INTEREST. The rate of interest varies in different States of the Union. In the examples given in most American works on Arithmetic, 6 per cent. is understood, unless some other rate is specified.

Page  220 220 THE COUNTING-HOUSE. [ART. XV. In each of the New England States, in New Jersey, Pennsylvania, Delaware, Maryland, Virginia, North Carolina, Tennessee, Kentucky, Ohio, Indiana, Illinois, Missouri, Arkansas, and the District of Columlbia, and on U. S. notes, the rate is 6 per cent. In New York, South Carolina, Michigan, Wisconsin, and Iowa, it is 7 per cent. In Georgia, Alabama, Mississippi, Texas, and Floridcla, 8 per cent. In Louisiana, 5 per cent., though the bank interest is.06, and conventional interest may be as high as.10. In Maryland, the interest on tobacco contracts is.08. In Mississippi, Missouri, and Arkansas, the interest by agreement may be as high as.1.0, and in Illinois, Wisconsin, and Iowa, as high as.12. In the mercantile method of computing interest, compound interest is sometimes charged. But as the courts do not generally allow compound interest, the following rule is recommended in computing interest on notes and bonds: If any payment exceeds the interest due at the time it is made, deduct it from the AMOUNT,a and compute the subsequent interest on the balance. If the p2ayment is less than the interest, deduct it from the INTEREST, reserve the excess of interest to be added to the succeeding interest, and continue the interest on the former pri2ncipal.b a The amount of principal and interest. b " In casting interest on notes, bonds, &c., upon which partial payments have been made, every payment is to be first appropriated to keep down the interest; but the interest is never allowed to form a part of the principal, so as to carry interest. 27 Mass. R. 417; 1 Dall. 378. " When a partial payment exceeds the amount of interest due when it is made, it is correct to compute the interest to the time of the first payment, add it to the principal, subtract the payment, cast interest on the remainder to the time of the second payment, add it to the remainder, and subtract the second payment, and in like manner from one payment to another, until the time of judgment. 1 Pick. 194; 4 Hen. &. Munf. 431; 8 Scrg. & Rawle, 458; 2 Wash. C. C. R. 167; see 3 Wash. C. C. R. 350; Ibid. 376.'" When a partial payment is made before the debt is due, it cannot

Page  221 ~ 67.] LEGAL INTEREST. 221 EXAMPLE FOR ILLUSTRATION. $1000.00 Philadelphia, March 4th, 1841. For value received, I promise to pay John Smith, or order, one thousand dollars on demand, without defalcation.a WILLIAm BROWN. December 1st, 1841, received $75.00. July 17th, 1842, received $15.50. August 18th, 1843, received $30.50. December lth, 1843, received $500.00. January 3d, 1844, received $150.00. What was due on the note, Aug. 18, 1844? First principal, on interest from March 4, 1841. $1000.00 Interest to Dec. 1, 1841 (8mo. 27dy.). 44.50 Amount.. $1044.50 First payment, exceeding the interest due.. 75.00 Balance for a new principal... 969.50 Interest from Dec. 1, 1841, to July 17, 1842 (7mo. 16dy.).... 36.52 Second payment, less than interest due. 15.50 Excess of interest... X 21.02 Interest from July 17, 1842, to Aug. 18, 1843 (13mo. idy.).... 63.18 Interest due Aug. 18, 1843... 84.20 Third payment, less than interest due. 30.50 Excess of interest.. 53.70 Interest from Aug. 18, 1843, to Dec. 11, 1843 (3mo. 23dy.).... 18.26 71.96 Amount due Dec. 11, 1843... 1041.46 Fourth payment, exceeding the interest due. 500.00 Balance for a new principal.... 641.46 be apportioned, part to the debt and part to the interest. As if there be a bond for one hundred dollars, payable in one year, and at the expiration of six months fifty dollars be paid in. This payment shall not be apportioned, part to the principal and part to the interest, but at the end of the year interest shall be charged on the whole sum, and the obligor shall receive credit for the interest of fifty dollars for six months. 1 Dall. 124."-Bouvier' s Laow Dict., vol. 1, p. 717. a The laws of Pennsylvania require the insertion of the words " without defalcation." In other places, they are usually omitted.

Page  222 222 THE COUNTING-HOUSE. [ART. xv. New Principal.... 541.46 Interest from Dec. 11, 1843, to Jan. 3, 1844 (23dy.) 2.07 Amount due Jan. 3, 1844. v... 543.53 Fifth payment, exceeding the interest due. 150.00 Balance for a new principal.... 393.53 Interest from Jan. 3, 1844, to Aug. 18, 1844 (7mo. 15dy.) 14.76 Amount due Aug. 18, 1844... $408.29 EXAMPLES.a 1. Worcester, July 4th, 1840. For value received, I promise to pay Thomas Jackson, or order, six hundred and thirty-nine dollars, on demand.b $639.00 JOHN WINTER. Endorsements. Sept. 5, 1840, received $13.25. Jan. 1, 1841, received $1.560. March 17, 1841, received $72.00. Oct. 3, 1841, received $29.50. July 3, 1842, received $9.00. What was due, Jan. 1, 1843? Ans. $601.83. 2. Portland, May 13th, 1841. For value received, I promise to pay George Appleton, or order, nine hundred dollars, on demand. $900.00 WILLIAM MASON. Endorsements. Aug. 28, 1843, received $175.00. Dec. 13, 1843, received $10.00. April 13, 1844, received $10.00. May 1, 1844, received $500.00. What was due Sept. 13, 1844? Ans. $371.11. 3. New York, Oct. 5th, 1823. For value received, I promise to pay Brown & Oliver, or order, one thousand dollars on demand. $1000.00- JAMES THOMAS. Endorsements. March 4, 1824, received $100.00. July 27, 1825, received $50.00. Oct. 25, 1825, received $100.00. April a For the rate of interest, see p. 220. b In some of the States, a note payable on demand, does not draw interest until demand is made.

Page  223 ~ 68.] EQUATION OF PAYMENTS. 223 13, 1826, received $15.00. Nov. 13, 1826, received $10.00. Dec. 1; 1826. received $500.00. What was due on the note, May 16, 1830? Ans. $532.76. 4. Charleston, Aug. 18th, 1840. For value received, I promise to pay Nathan J. Wilson, or order, four hundred and thirty-one dollars in six months, with interest afterward. EDWARD ELLIS. $431.00 Endorsements. Feb. 18, 1841, received $31.00. Sept. 15, 1841, received $10.00. Nov. 11, 1841, received $5.00. March 29, 1842, received $100.00. May 13, 1843, received $200.00. Dec. 31, 1843, received $2.50. What was due June 16, 1844? Ans. $149.30. 5. Cincinnati, Nov. 1st, 1841. For value received, we promise to pay Samuel Jones, or order, seven hundred and seventy-five dollars and fifty cents, in two months from date. HENRY THOMPSON & CO. $775.50 Endorsements. Feb. 27, 1842, received $15.00. August 20, 1842, received $15.00. Dec. 18, 1842, received $15.00. Jan. 1, 1843, received $15.00. April 1, 1843, received $200.00. Feb. 19, 1844, received $21.00. What was the balance due Oct. 1, 1844? A.ns. $603.64. 68. EQUATION OF PAYMENTS. The ordinary rule for equation of payments, is founded on the supposition that the interest of the money which is not paid until after it is due, is equal to the discount of that which is paid before it is due. This is not strictly correct, but it corresponds to the usual method of computing discount among merchants. The labor of equating may be abbreviated by disregarding the cents if they are less than 50, and counting them as an additional dollar if they are more than 50. When the sums are all large, the units of dollars may be disregarded in a similar manner.

Page  224 224 THE COUNTING-HOUSE. [A RT. XV, To find the equated time for the payment of several debts. u2lt'ipy each charge by the time which has elapsed from the date of the first bill, and divide the sum of the products by the sum of the bills. To find the equated time for the settlement of an account, in which there are both debits and credits: 1st. Find the equated time for each side of the account. 2d. Multtiply the least side of the account by the time between the dates found for each side, and divide the product by the balance of the account. 7zhe quotient will be the time between the date found for the larger side of the account, and the equated time for the settlement of the balance. If the date Jbund for the larger amount is the earliest, count back, but qfj it is the latest, count forwoard from that date. EXAMPLE FOR ILLUSTRATION. Required the time when the balance of the following account became due: Dr. Mlartin Acres in Account with Simon Gray. C(r. 1849 1849 June 14 To Balance $875 30 July 10 By Cash $600 00 Aug. 18 " Mdse. 519 40 Sept. 16 Mdse. 427 90 Sept.22 Bill due 289137 Nov. 18" Draft 350 00 Dec. 18 " Do. 318 01 Dec. 20 "Cash 275 00 Drs. Crs. 875.30 X 0 = 0 600.00 X 26 = 15600. 519.40 X 65 _ 33761. 427.90 X 94 = 40222.60 289.37 X 100 - 28937. 350. X 157 = 54950. 318. X 187 = 59466. 275. X 189 - 51975. 2002.07 122164. 1652.90 162747.60 122164.. 2002.07 — 61. 162747.60 -- 1652.90 = 98. Debits clue 61 days after June 14. Credits due 98 days after June 14. 1652.90 X37- 349.17 -= 175. The balance was therefore due 175 days before the date found for the Dr. side, or 114 days before June 14, which was Feb. 20.

Page  225 ~ 68.] EQUATION OF PAYMENTS. 225 The example may be equated as follows, by disregarding the cents and units of dollars. If the dollars amount to 5 or more, 1 should be added to the number of eagles. 88 X 0 = 0 60 X 26 = 1560 52 X 65 = 3380 43 X 94 = 4042 29 X 100 = 2900 35 X 157 =5495 32 X 187 - 5984 28 X 189 = 5292 201 12264 166 16389 12264 - 201 = 61. 16389~ 166 = 98. 1652.90 X 37 - 349.17 - 175, as before. EXAMPLES FOR THE PUPIL. 1. When was the balance of the following, account due? Dr. Jamzes Day inT Accountt with William Knight. Cr. 184 1848 Jan. 6 To Mdse. ]427 20 Feb. 20 By Cash $750 00 ar. 0',' D. 316 19 Apl. 16 a Mdse. 95 87 Ilay 18' Do. 284 72 {July 3 Diraft 100 00 Ans. Nov. 29, 1847. 2. Find the equated time for the settlement of the following account: Dr. North, ~airrison & Co. in Acc. with J. E. Oliver. (Jr. 1846 1846 Jan. 1 To Balance $283 94 Jan. 12 By Cash $42500 May 15 "Mdse. 217833 Apl. 3 Mdse. 693 13 July 5. Note 500 00 June29 3759 Aug. 30 " iMdse. 894 60 Ans. Dec. 21, 1846. The average price of a mixture consisting of several ingredients, is found in nearly the same manner as the average time for the payment of several debts. The method of finding the average price; is usually called MEDIAL ALLIo GATION. For Examples, see Sect. 74. 15

Page  226 226 THE COUNTING-HOUSE. [ART. XV. 69. ACCOUNTS CURRENT. An ACCOUNT CURRENT contains a statement of the mercantile transactions of one person with another. On the Dr. side are placed all the payments made, and the amounts of merchandise sold, to the merchant who is furnished with the account; on the Cr. side are entered all sums received, and the amounts of merchandise purchased from the said merchant. To facilitate the settlement of interest, it is customary to place against each sum the number of days that elapse from the time that each entry becomes due, until the time of rendering the account, and calculate the interest on each item. The balance of interest is entered on the side which has the greatest amount of interest. EXAMPLE FOR ILLUSTRATION. Abbott 4' Clark, Philadelphia, in Account Cur~rent with Cha/rles Goodhue 4 Co., Boston. DR. 1844 Dol. Cts. No. days. Interest. June 1 To balance clue from former acct.. 153.50 92 2.354 " 13 To amount due on note for goods... 400.00 80 5.333 " 25 To merchandise.. 275.00 68 3.116 July 19 To 90 bbls. flour, at $4.7 393.75 44 2.887 Sept. 1 To balance of interest 3.28 13.690 $1225.53 10.412 3.278S 1844 CR. Dol. Cts. No. days. Interest. June 16 By cash... 375.00 77 4.812. " 30 -By bill ofArnold & Brown 250.00 63 2.625 July 29 By cash 525.00 34 2.975 Sept. 1 By balance to your debit on a new account. 75.53 10.412 $225.53

Page  227 ~ 69.] ACCOUNTs CURRENT. 227 1. Oliver Marriott, Savannah, in /ccount Current wig Daniel Clark, Molbile. Dr. l 8 4 4 Dol. cts. No. of days. Interest. Feb. 16 To balance due from old account, 91.75 " 25 To merchandise, 163.50 Apl. 13 To merchandise, 219.25 May 16 To balance of interest,, 8 44 Cr Dol. cts. No. of days. Interest. Mar. 30 By cash, 400.00 May 16 By balance to new acct. 2. Joseph lMason, Co., Saint Louis, in Sccount Current with Thompson 4 Brother, Lexington. Dr. I 8 4 4 Dol. cts. No, of days. Interest. Jan. 13 To sheeting, 131.50 Feb. 3 To duck, 87.75 " 25 To cambric, 240.00 Mar. 29 To sundries, 300.00 Apl. 1 To balance of interest,. 18 44 Cr. Dol. cts. No. of days. Interost. Jan. 29 By furs, 175.00 Mar. 3 By bill, 200.00 Mar. 18 By cash, 380.75 Apl. 1 By balance to new acct. 3. Henry Chatham, Nashville, in.ccount Current with George Hapgood ~ Sons, Baltimore. Dr. 1 8 4 4 Dol. cts. No. of days. Interest. Apl. 5 To amount due on note for goods, 500.00 May 17 To ditto, 119.50'28 To merchandise, 87.25 June 16 To sundries, 63.00 July 1 To balance to new acct. 1 8 4 4 Cr. Do]. cts. No. of days. Interest Apl. 9 By cash, 350.00 "11 By tobacco, 289.50 " 23:By bill on Atkins&Jones, 200.00 July 1 By balance of interest, -- ---- 0 —— P

Page  228 228 THE COUNTING-HOUSE. [ART. Xt. It is more convenient to enter the product of each entry by the number of days, as in ordinary Equation of Payments, instead of computing the interest on each amount. By dividing the balance of products by 6000, we at once obtain the balance of interest at 6 per cent. For any other rate than 6 per cent., the result may be increased or diminished, as in computing Bank Interest. EXAMPLE. Wilson, Survilliers ~ Co. in Account Current with James N. Martin. DR. CR. 1850 $ ct dys. 1850 $ ctldys.| Jan. 9 Paid for 9001. remitted June 1 By proceeds of sales rento Brown, Shipley &'dered this day. Value Co., at 1.09. 4360 00 142 619120 d.ue Nov. 6, 1850. 15141 751 157 23775 6s 20 Freight on 30 cases per St. Nicholas... 17 10 131 2227 "e C. H., charge for warehousing... 6 60 " 917 Feb. 6 Paid duties, and C. H. storage..... 327 76 114 37302 "8 " Marine Insurance 160 03 112 17920 1 per cent., (instead of usual charges,) per agreement... 151 42 5 per cent. commission and guarantee on $15141.75.... 757 09 Numbers from credit side...... 2377'294 3054850 Amount of products, 3054870 div. hy 60 - 509 15 Balance dueW. S. & Co. 8852 58 15141 753 " " By balance.... 882 5 As the amount of sales in the preceding account is not due, Wilson, Survilliers & Co. should be charged with interest until it becomes due; therefore the product 2377294 is added to the amount of the Dr. products. In forming the products, the cents are disregarded, unless they exceed 50, in which case the number of dollars is increased by 1. EXAMPLES. 1. A. B. in account with C. D. Debits, 1849, July 3, $5000; July 28, $40; Aug. 16, $800; commission and charges, 6~1o on $18000. Credits, 1850, March 2, $9000; Iay 15, $9000. What was the balance of account rendered Jan. 1, 1850. Ans. $10617.89.

Page  229 ~ 70.] PRACTICE. 229 2. E. F. in acct. with G. H. Debits, 1850, Jan. 20, $3300; Jan. 31, $25; Feb. 19, $400; June 8, $900; commission and charges, 7V~o on $13842.68. Credit, proceeds of sales, due Oct. 29, $13842.68. Required the balance due July 1. Ans. $7869.86. The pupil may solve by this method the examples given on page 225. 70. PRACTICE. In PRACTICE many questions arise that can be solved more readily than by adopting any of the ordinary rules. Many of the operations of business, in which compound numbers are concerned, may be abbreviated by first finding values for the highest denomination, and considering the lower denominations as aliqdot parts of the higher.a TABLE OF ALIQUOT PARTS. Of a dol. Of a X. Ofa shi. I Of a ton. Of a cwt. Of a year-. cts. $ s. d. ~ d. 1. cwt. qr. ton. ql. Ib. czvt. in. d../ 50 10 1 6 1 0 22 6 8331 31 6 8 A3 4 -1 5 1 1 4 25 1 5 = 3 4 16 1 3 20 1 4 = 2 = 2 1 2 = 14 2 212 — 16'= 3 4 1- 11 2 =l 8 1 2 12' 2 6 1 1 1 1 1 7 115 10 1=10oi 2 = 4-10 -4= 4 16 -.V. 1 = 10 1 8 12 1 2 22 24 40 1 1 6= 3 12 Zo a 24 4 61 — = 10 214 4 4 8 82 1 4 = 424 5 o 38 0 9 12 = 20= a 7 6= - 4 4 8 216 4 2 8 4 9 18= 4,.,1 ~ 3 5 7 5 3 the number of articles s not very large, and the e 24 con- 4 2 _1 2 l 41= s 7 2= 3 12= ( 715_ " 1 When the number of articles is not very large, and the price COiisists of several denominations, the answer can generally be obtained niost readily by Compound Multiplication. When the number of articles is large, and the price such as to require but few parts to be taken, the readiest mode of solution is by Practice. When the quantity and price, consist each of several denominations, the best mode of solution is by Proportion, or by Fractional Analysis.

Page  230 230 THE COUNTING-HOUSE. [ART. XV. Similar tables may be made to any required extent, but these are sufficient to show their application. One of the following rules may be adopted in nearly all questions that admit of abbreviation by Practice: 1. Assume the price at some unit higher than the given price, and take aliquot parts of the assumed price for the answer. 2. Multiply the price by the integers of the highest denomination, and take aliquot parts for the lower denominations. In the application of either of these rules, there is great room for the exercise of judgment, in determining what parts should be taken to determine the answer most readily. EXAMPLES ILLUSTRATING THE FIRST RULE. 1. At 434 cents a yard, what is the price of 874 yards of muslin? The price at $1 per yd. would be $87.50 at 25ets. = -4 21.875 at 122" = 4 of $4 10.9375 at 6*1" = 4 of $4 5.46875 at 434 $38.28125 Or, as $0.434 = $-s-;'the answer might have been obtained by multiplying $87.50 by -6. 2. What is the value of 961 lb. of tea, at 3s. 10'id. per lb.? The price at ~1 per lb. would be ~96 10s. at 3s. 4d. =-~ 16 1 8 at 5d. = of 3s. 4d. 2 0 21 at ld. = of 5d. 10 0at dcl. - of 5d. 4 A1 at3. 1d.~ 1 1 1d4 at 3s. 103d. ~18 15 11-d. 4 8

Page  231 ~ 70.] PRACTICE. 231 EXAMPLE FOR ILLUSTRATING THE SECOND RULE. What is the value of 11cwt. 3 qr. 17 lb. of sugar, at ~1 3s. 6d. per cwt.? ~ s. d. 2 qr. = -cwt. 1 3 6 price of 1cwt. 161 lb. = Icwt. 11 12 18 6 price of 11cwt. 1qr. = -2 qr. 11 9 " ". 2 qr. 5 10' " " 1 qr. 1lb. = 16 lb. 3 4- ".. 161b. 2-2 " ".. lb. 5 9 13 19 8la price of 11cwt. 3 qr. 17 lb. The following rule is probably as convenient as any that could be given, for computing interest by Practice. 3Multiply 1 per cent. of the principal by - the even number of months, and if there is an odd month, add 30 to the number of days. Divide the days by 6, and multiply 1 of 1 per cent. by, the quotient. If there are any remaining days,b take as many 60ths of 1 per cent. Add the numbers so obtained, and their sum will be the interest at 6 per cent. For any other rate, increase or diminish the result, as in the Bank Rule. EXAMPLE FOR ILLUSTRATION. What is the interest of $9763.25 for 2yr. 9mo. 8dy.? 1 per cent., or the interest 1 per cent. is 97.6325 for 2 months, is $97.6325. of 2yr. 8mo. is 16 Multiplying by 16, we obtain the interest for 2yr. 8mo. The 585.7950 976.325 1mo. remaining, added to the 976.325 mo. remaining, added to the 36dy. is.6 of 2mo. 58.5795 8 days, gives 38 days, in which 2dy. is -- of 2mo. 3.2544 6 is contained 6 times, with 2 3 remainder. As the interest $1623.9539 a The fraction is - 7, but as nothing is reckoned less than -d., it is called 4. b When the remainder is 4, it is more convenient to diminish the quotient by 1, and call the remainder 10, taking 1 or 1 of 1 per pent for the interest.

Page  232 232 THIE COUNTING-HOUSE. [ART. XV. for 6 days is.1 of 1 per cent.,or $9.763+, the interest for 36 days is 6 times as much. The two remaining days are -31 of 60 days, we therefore add ~- of $97.63+. There are a variety of other contractions that may frequently be adopted in practice. A few are given below, which will often be found useful. (1.) When the multiplier consists of any number of 9's, increase it by 1, and subtract the multiplicand from the product. Thus, 18473X 9999 — =184730000-18473 184711527. (2.) To multiply by 5, divide the multiplicand by.2. Thus, 187 X 5 = 187 -.2 = 935. To divide by 5, multiply the dividend by.2. (3.) To multiply by 25, divide the multiplicand by.04. Thus, 1289 X 25 = 1289.04 = 32225. To divide by 25, multiply the dividend by.04. (4.) To multiply by 75, multiply by 100, and subtract - of the product. Thus, 18645 X 75 = 1864500 — 466125 = 1398375. To divide by 75, divide by 100, and add 1 of the quotient. (5.) To multiply by 125, divide the multiplicand by.008. Thus, 1641 X 125 = 1641 *..008 = 205125. To divide by 125, multiply the dividend by.008. (6.) To multiply by 375, divide by.008, and multiply the quotient by 3. Thus, 294 X 375 =294 *-.008 X 3 -= 110250. To divide by 375, multiply by.008 and divide by 3. (7.) To multiply by 625, divide the multiplicand by.0016. Thus, 4812 X 625 = 4812.0016 = 3007500. To divide by 625, multiply the dividend by.0016. (8.) To multiply by 875, multiply by 1000, and subtract ~ of the product. Thus, 735 X 875= 735000 —91875= 643125. To divide by 875, divide by 1000 and add 1 of the quotient. (9.) To multiply by any number within 12 of 100, 1000, &c., annex to the multiplicand as many zeros as there are figures in the multiplier, and subtract as many times the multiplicand as are equivalent to the excess of 100, 1000, &c., over the multiplier. Thus, 24796 X 99989 = 2479600000 - (11 X 24796) = 2479327244. (10.) To square a number ending in 5, multiply the number of tens by one more than itself, and place 25 at the right of the product. Thus, 3 X 4=12, and 35 X 35 =1225; 12 X 13-156, and 125 X 125 = 15625; 6 X 7 =42, and 65 X 65 = 4225.

Page  233 ~ 70.] PRACTICE. 233 (11.) When the tens in two numbers are alike, and the sum of the units is 10, to obtain the product multiply the number of tens by one more than itself for the hundreds, and place the product of the units at the right of this product, for the tens and units. Thus, 4 X 5 =-20, and 43 X47 =2021; 42 X 48 =2016; 44 X 46 -2024; 7 X 8 = 56, and 72 X 78 = 5616; 71 X 79 =5609, &c. (12.) The sum of two numbers multiplied by their difference, is equal to the difference of their squares. HIence we may readily find the product of two numbers, one of which is as much above as the other is below, a certain number of tens. Thus, 87 X 73 =(80 + 7) X (80 - 7) 802 72= 6400 - 49 = 6351. (13.) To square any number between 50 and 60, add the units of the given number to 25 for the hundreds, and annex the square of the units for the tens and units. Thus, for the square of 51; 25 + 1 = 26 7,hundreds, and 1 X 1 = —1; hence 51 X 51 = 2601. In like manner 53 X 53 = 2809; 59 X 59 = 3481. (14.) When one figure of the multiplier is an aliquot part of one or more of the remaining figures, the work may be abbreviated as in the following example:Multiply 489.137 by 7261.8. We see at once that 18 489.137 is a multiple of 6, and 72 2934822 = prod. by 6. is a multiple 8804466 -prod. by 3 X 6 =prod. by 18. of 18. There- 35217864 -prod. by 4 X 18-prod. by 72. fore, multi- 3552015.0666 plying first by 6, we take 3 times the product for the product by 18, and 4 times the product by 18, for the product by 72. (15.) In the ordinary mode of determining the greatest common divisor of two numbers, any prime factor or square number, contained in one number but not in the other, or any prime factor or square number in a remainder that is not in the preceding divisor, may be rejected, and the work thus abbreviated. For example, let the greatest common measure of 689 and 2279 be required.

Page  234 234 THE COUNTING-HOUSE. [ART. XV. Here the square number 4 is a factor of 212, and not of 689. We 689) 2279 (3 2067 therefore divide 212 by 4, and immediately obtain the greatest com- 4) 212 mon measure. In the application of this principle to the reduction of G. C. Meas. 53)689(13 fractions, we observe that 53 divides 689 13 times, and it divides 212, 4 159 times. It therefore divides 3 X 689 159 + 212 or 2279, 3 X 13 + -4 or 43 times. Therefore 689 1_3 Reduce -4 -7 to its lowest terms. 457)563(1 457 Neither 2 nor 53 being factors of 457, 106 2 X 53 the fraction is already in its lowest terms. EXAMPLES. 1. Find the cost of 151b. 10oz. of tea, at $.37y- per lb.; at $.25; at $.314; at $.433; at $.50; at $.56i-; at $.662; at $.871I. 2. What is the price of 891yds. of broadcloth, at $4.75 per yd.? Ans. $426.31~. 3. What is the value of 49A. 3R. 15r. of land, at $125 per acre? Ans. $6230.47. 4. What is the value of 96yds. 3qr. 3na. of broadcloth, at ~1 2s. 6d. per yard? Ans. ~109 is. 1-d. 5: What will 17T. llewt. 2qr. 21lb. of iron cost, at $19.75 per ton? Ans. $347.29. 6. What is the cost of 163A. 2R. 25r. of land, at $15.75 per acre? Ans. $2577.59. 7. What is the value of 364yds. 3qr. Ina. of sheeting, at, 12lcts. a yard? Ans. $45.60. 8. Bought 76bu. 3pk. of potatoes, at 37'cts. a bushel; 19bu. 2pk. of wheat, at $1.10 a bushel; 37bu. lpk. of

Page  235 ~ 70.] PRACTICE. 235 barley, at 622-cts. a bushel; and 10T. 15cwt. of hay, at $16.00 a ton. What was the amount of the whole? Ans. $245.51. 9. What is the interest of $2375 for 5yr. 11mo. 23dy., at.055 per year? Ants. $781.21. 10. What is the interest of $4814.25 for 3yr. 7mo. 14dy., at.06 per year? At.075 per year? Ans. $1046.297; $1307o87. 11. Multiply 576.3 by 99; by 999000. 12. Multiply 7.894 by 5; by 25; by 7500. 13. Multiply 48.302 by 1250; by 375000. 14. Divide 1879.4 by 5; by 250; by 75. 15. Divide 4449.17 by 125; by 375. 16. Multiply 3.0872 by 525125, by adding three partial products. 17. Multiply 41909 by 999625125, in the most expeditious manner. 18. Multiply 89443 by 625; by 875. 19. Divide 141.982 by 625; by 875. 20. Multiply 89443 by 625875. 21. Multiply 283172 by 9992; by 991. 22. What is the square of 15? of 85? of 115? 23. What is the product of 73 X 77? 12 x 18? 44 x 46? 24. What is the product of 81 X 89? 75 X 75? 34 x 36? 25. What is the product of 16 x 24? 19 X 21? 35 x 45? 89 x 71? 67 x 53? 78 x 82? 96 x 84? 113 x 107? 112 x 128? 26. What is the square of 58? of 56? 527 55? 57? 54?

Page  236 236 THE COUNTING-IIOUSE. [ART. XV 27. Find the greatest common divisor of 804 and 938; of 741 and 1083; of 1343 and 1817. 28. Reduce each of the following fractions to its lowest terms 30 781 fiFi7 1147 941 lS -3 12-9 071 899!u' 19333; 1TYT 29. Multiply 476384 by 9995125625. 30. Required the interest of $1729.50 for 3yr. 7mo. 16dy., at 7 per cent. 71. CAUSE AND EFFECT. In all questions which can be solved by ratio or proportion, if the multiplying terms are written in one column, and the dividing terms in another, the factors common to both columns may be cancelled, and the answer obtained by dividing the product of the remaining factors in the column of multipliers, by the product of the remaining factors in the column of divisors. The terms of a proportion may be distinguished into causes and effects, and the alternate products of either cause by the other effect, are equal. For example, if 8 men build 6 rods of wall in a day, 4 men will build 3 rods in the same time. Stating the proportion, we have men. men. rod. rod. 8: 4::6: 3 cause. cause. effect. effect. The produt of the extremes, and the product of the The product of the extremes, and the product of the means, give us the product of each cause by the effect of the other cause. If then we write each effect opposite to its cause, our multipliers and divisors will be obtained without difficulty. Men, animals, and times, are evidently causes, because the increase of either of them, will increase the effect produced.

Page  237 ~ 71.] CAUSE AND EFFECT. 237 In questions of freight, we may regard distances and bulk as causes, producing money for their effect. The principal of a sum of money at interest, is a cause, and the interest is an effect. A little practice will give great facility in distinguishing between causes and effects, in all cases of common occurrence, and by acquiring readiness in making this distinction, a vast amount of labor may be saved. In arranging the terms according to this rule, when we reach the required term, or term ofj demancl, its place may be supplied with a dash. Then, as the product of all the factors on each side mlust be equal, the missing term may be found by cancelling, and dividing the product of all the numbers on the side opposite to the dash, by the product of all the numbers on the side in which the dash is included. EXAMIPLES FOR ILLUSTRATION. 1. If 18 men, in 6 days of 8 hours, build a wall 150 feet long, 2 feet wide, and 4 feet high, in how many days of 12 hours will 24 men -build a wall 200 feet long, 3 feet wide, and 6 feet high? Commencing our statements, men Men. we write 18 men, 6 days of 8 days - ays. hours, as cause, and the effect, C hours b /i hours which is a wall 150ft. long, 2ft.. long 0 long. wide, and 4 ft. high, on the op- ~ wide 3 A wide. posite side. Opposite to each of high high. ptsdhigh 3 h wide, these terms, we write cl days, 12 hours, 24 men, as cause, and 200 long, 3 wide,.6 high, as effect. Cancelling the like factors, we have but 3 X 3 on the side of the multipliers, and 1 on that of the divisors. 3 X 3 * 1 is therefore the missing term, or number of days required. If either term were fractional, the denominator, representing a divisor, should be transposed to the opposite side. By proceeding in this manner, a statement may be made as soon as the question can be proposed, 2. If $27- buy 4.- yards of cloth, that is -yd. wide, how

Page  238 238 THE COUNTING-HOUSE. [ART. XV. many yards of like quality, that is alyd. wide, may be bought for $13-? 8 m3 STATEMENT. Reducing the mixed numbers 4 4 27-X 133- to improper fractions, we trans- 2 4 - - pose the denominators, writing 4- them above the causes,-then 19 - cancel and divide as before. Ans. 1-9 -3-yd. EXAMPLES FOR THE PUPIL. 1. How many men in 18 months, will build a wall that 108 men can build in 16 months? a N. B. The effect in this example is 1 wall. 2. How many bushels of meal will serve 54 persons 12 months, if 15 persons consume 12. bushels in 2 months?b 3. If 27 men build a cistern 30ft. long, 16ft. wide and 10ft. high, in 4 weeks, by working 5 days in a week, and 9 hours a day, how many men, working 6 days in a week, and 12 hours a day, will build a cistern 48ft. long, 12ft. wide, and 20ft. high, in 9 weeks? 4. What is the interest of $360 for. 3y. 4mo., at 6 per cent.?d 5. If $16 gain $3 in 5 mo., how much ought $24 to gain in lOmo.?e 6. If the freight of 1800 lb. for 56 miles, is $1.50, how far may IT. 4cwt. 12 lb. be carried for $6.75?f 7. How much wheat, at $1.20 a bushel, must be S b 54 15 C 27 - 1360 6 24 f 1800 2700 18 16 12 2 4 6 2 1 40 5 10 56 1l 1 12 1- 5 12 --.06 -13 6.751 1.50 48 30 12 16 20 10

Page  239 ~ 71.] CAUSE AND EFFECT. 239 given in exchange for 90 barrels of flour, at $4.75 per barrel? a 8. If the rent of 19A. 3R. of land is 4 lO10s., what will be the rent of 73~A.?b 9. If the expenses of a family of 8 persons are $40 in 10 weeks, how many persons can be supported 121 weeks for $100?c 10. The shadow of a stick that is 5ft. 6in. high, measures 3ft. 4in. What is the height of a tree whose shadow measures 75ft. at the same time? d 11. If 29-1bu. of wheat yield 1760bu. in 5 years, how much will 45~bu. yield in 6 years at the same rate? 12. If 10 compositors, in 2 days of 10 hours, set 66W pages of types, each page containing 45 lines of 50 letters, how many compositors will set 94 pages, each page containing 35 lines of 40 letters,. in 2- days of 8 hours? 13. If 19 men, in 71-3 days of 10~ hours, dig a trench 41-yd. long, 57ft. deep, and 74ft. wide, how long a trench, that is 82ft. deep, and 4 7 ft. wide, will 11 men dig in 291 days of 47 hours? Ans. 5002 4. 3 y 14. If 18 men in 9- months consume flour worth $78.75, when wheat is $1.124 per bushel, how many months will $145 supply 35 men with flour, when wheat is $1.00 per bushel? Ans. 10 590 mo. 15. If 2500 slates, each 8 inches long and 5 inches wide, will cover a roof, how many will be required that are 6 inches long and 4 inches wide? Ants. 4166g. 16. A pile of wood 60ft. long, 10ft. high, and Sft. thick, ~a _- 90 b 2j4 c 2 d 3 2 79 147 8 119 —(9, 100 40 75 10

Page  240 240 TIHE COUNTING-HOUSE. [ART. XV. was sold for $23881. What would be the price of a pile LOft. long, 8ft. high, and 4ft. thick, at the same rate? Ans. $31-. 17. If 5 men, by working 8 hours a day for 12 days, can build a wall 40 rods long, 2 feet thick, and 6 feet high, how many men, working 9 hours a day, will build a similar wall, 30 rods long) 3 feet thick, and 8 feet high, in 4 days? Ans. 20 men. 18. If 7 men, in 84 days, by working 9 hours a day, can build i of a wall that is to be raised 12 feet, how many days must 11 men work, when the days are 8 hours long, to raise the same wall 5 feet? Ans. 74.2 days. 19. If $1000 gain $11- in 80 days, how much will $2500 gain in 12Q days, at the same rate? 20. If a man, by walking 3 miles an hour, for 6 hours a day, can accomplish a journey in 12 days, in how many days would a man walk the same distance, at the rate of 21 miles an hour, for 9 hours a day? 21. If 421 bushels of corn, that weighs 511 pounds a bushel, can be bought with 23 bushels of wheat, that weighs 564 pounds a bushel, how much corn, weighing 60 pounds a bushel, would be equivalent to 100 bushels of wheat that weighs 54 pounds a bushel? Ans. 150 —7TLbu. 22. If a man travels 240 miles in 8 days, when the days are 12 hours long, how many miles will he travel in 24 days, when the days are 16 hours long? 23. If the freight of 2T. Ocwt. for 28 miles, is $14.50, what will be the freight of 9T. 4cwt. for 96 miles? 24. If 4 men, in 3 days of 8 hours, build 40 rods of wall, how many rods will 18 men build in 5 days of 9 hours? Ans. 337' rods. 25. How many men, in 24 days of 16 hours, will do three times as much work as 18 men can perform in 32 days of 12 hours?

Page  241 ~71.] CAUSE AND EFFECT. 241 26. If 14 men, in 53 days, by working 8 hours a day, reap 38y' acres of grain, how many men will reap 37~ acres in 63 zdays, by working 9 hours a day? 27. How much wheat, that weighs 60 lb. per bushel, would be required to supply a garrison of 1400 men 9 months, if 2800 bushels, weighing 58 lb. per bushel, supply 800 men 3-a months? Ans. 12180bu. 28. How many hours a day must 15 men work, to dig a trench 400ft. long, Gft. wide, and 3ft. deep, in 187~ days, if 72 men can dig a trench 250ft. long, 8ft. wide, and 4ft. deep, in 31J days, by working 7 hours a day? 29. How many men can be furnished with 4 suits each, by 1140 yards of cloth that is 1'yd. wide, if 2016 yards, 4yd. wide, furnish 112 men 3 suits apiece? 30. If 16 compositors set 150 pages of types, each page containing 48 lines, and each line 50 letters, in 3 days of 10 hours, how many compositors will be required to set 500 pages of 72 lines each, and 45 letters in a line, in 6 days of 8 hours? Ans. 45. 31. If $1700, at 6 per cent., yield an interest of $350 in a given time, what will be the interest of $3900 at 7 per cent. for one half the time? 32. If 30 reams are required for 1500 pamphlets of 10 sheets each, how many reams will be required for 740 pamphlets of 121 sheets each? 33. If 17 yards of serge, that is i wide, are required to line a cloak containing 9.75 yards of cloth that is 51 quarters wide, how many yards of a yard wide, would line a cloak containing 10.5 yards of cloth 61 quarters wide? Ans. 1 5259-O yd. 34. If a cistern discharges 831 gallons of water in 1.3 hours, how much will it discharge in 6- hours? 16

Page  242 242 THE COUNTING-HOUSE. [ART. XV. 35. When exchange on London is at a premium of 9~ per cent., what is the value of $1863.50, in English money, the par of exchange being ~1 — $4.444-? 72. EXCHANGE. The term EXCHANGE, in commerce, is generally employed to designate that species of mercantile transactions, by which the debts of individuals residing at a distance from their creditors, are cancelled without the transmission of money. A BILL OF EXCHANGE is an order addressed to some person at a distance, directing him to pay a certain sum to the person in whose favor the bill is drawn, or to his order. The person who draws the bill is called the drtawer; the person in whose favor it is drawn, the remitter or payee; the person on whom it is drawn, the ctrawee. The drawee is also called the acceptor, when he has accepted, or engaged to pay the bills. Though bills of Exchange are originally drawn by creditors on their debtors, they are very rarely transmitted directly, but pass from hand to hand like any other circulating medium, and are bought and sold in the market. When the remitter disposes of a bill, he writes his name on the back, and is termed the endorser. If he endorses in favor of any particular individual, he gives a special endorsement, and such endorsee must also endorse the bill if he negotiates it. But if the endorsement is blank, the bill may be passed at pleasure from hand to hand. Every endorser, as well as the acceptor, is held responsible for the payment of the bill, and may be sued for its recovery. INLAND, or DOMESTIC EXCHANGE, includes the commer cial transactions within the limits of one country. FOREIGY EXCItANGE relates to the transactions of one country with another. The TRUE PAR OF EXCHIANGE is the value of the cur

Page  243 ~ 72.] EXCHANGE. 243 rency of one country estimated in the currency of another, by comparing the quantity of gold and silver in their respective coins. The exchange with England apparently furnishes an exception to this rule, the nominal par being $4.444- per ~, while the actual value of the pound sterling, which is the real par, is about $4.87. Hence, exchange on England is generally said to be from 8 to 10 per cent. above par. The CouRSE OF EXCHANGE, or the fluctuation above or below par, depends generally on the amounts due between different countries. Thus, when the debts and credits between two countries are equal, the real exchange is at par. But if New York owes London more than London owes New York, there will be a greater demand for bills on London, and this demand will produce a rise in the price, or cause the bills to be at a premium. The premium, however, can never exceed the cost of transporting specie; for if it did, all debts would be paid'in money or merchandise, instead of bills of exchange. The nonzinal premium, however, may exceed the cost of remitting coin, when the nominal par is above the real par. The operation of Bills of Exchange, may be explained by a single example. If A. of Boston owes B. of Paris, and C. of Paris owes D. of Boston, A. purchases in the market a bill upon Paris; that is, he buys of D. an order on his' debtor C., to pay A. or his order the amount desired. A. endorses the bill, and sends it to B., who receives payment from C. Thus the two debts are cancelled by a single remittance; the inconvenience of exporting and re-importing coin is removed, and all danger of loss is obviated by sending three bills (called the First, Second, and Third of Exchange), either of which being paid, the others are void. An ACCEPTANCE is an engagement to pay the amount of the bill, and may be either absolute or qualified. An abso

Page  244 244 THE COUNTING-HOUSE. [ART. XV\ lute acceptance binds the drawee when the bill becomes due, and in making it the drawee usually writes "Accepted," and subscribes his name at the bottom, or across the body of the bill. A qualified acceptance implies some condition, as the sale of merchandise, &c., and does not bind the acceptor until the condition is complied with. If a bill is made payable at a certain time after sight, the acceptance should be dated. A bill should be presented for payment during the regular hours of business, on the day it becomes due. When acceptance or payment has been refused, the holder should give immediate notice to all the parties whom he intends to hold responsible for the payment of the bill. This notice is usually accompanied with a PROTEST, which is an instrument prepared by a public notary, stating that acceptance or payment has been demanded and refused, and that the holder of the bill intends to recover any damages which he may sustain in consequence. In some places on the continent of Europe, banks of deposit are established, and exchanges are frequently made by transferring the amounts credited on the books of the bank, from one person to another. The deposits on which these credits are based, are called banco, and they usually bear a premium above the ordinary currency of the country. This premium is called the agio. The comparative market value of gold and silver is constantly varying, and the mint value is differently estimated by different governments. Thus, in England the relative worth of the two metals is as 1 to 14.29; in France as 1 to 15.52, and in the United States as 1 to i5.99. In England, silver is so much overvalued, that it would banish the gold coins from circulation, were there not a statute providing that only gold shall be legal tender in all payments of more than 40 shillings. The relative value of the precious metals should always be considered, in estimating the true par of exchange with any country.

Page  245 ~72.] EXCHANGE. 245 DOMESTIC EXCHANGE. Inland Exchange is usually effected by checks or DRAFTS, similar in form to the following:$1275.25 Philadelphia, June 3, 1850. Sixty days from date, pay to James N. Lewis, or order, Twelve Hundred and Seventy-Five Dollars and Twenty-Five Cents, and charge the same to my account. WILLIAM MORRIS. To Markham & Jones, Merchants, Cincinnati. The premium or discount on drafts, may be owing either to a difference in the value of the circulating medium, or to fluctuations in the demand. The English denominations of shillings and pence, are still retained in this country to some extent. At the formation of the Constitution, the continental currency had suffered a greater depreciation in some of the colonies than in others. Thus, while a pound in New England was worth $3.33~, in Pennsylvania it was but $2.664, and in New York but $2.50. The value in Federal Money, of the old currencies of the different States, is as follows:A shilling of New England, Virginia, Kentucky, or Tennessee, is 16- cents. A shilling of New York or North Carolina, is 12~ cents. A shilling of New Jersey, Pennsylvania, Delaware, or Maryland, is 13M cents. A shilling of South Carolina or Georgia, is 21 cents. We cannot remind teachers too often of the signal benefits they may confer upon their pupils, by communicating collateral know

Page  246 246 THE COUNTING-HOUSE. [ART. XVE. ledge to them;a that is, such knowledge as is directly connected with the subject of their lessons, though rarely, if ever, found in a text-book. This practice should be commenced with a child the first day he enters the school-room, and should never be discontinued until the day when, for the last time, he leaves it. If teachers would make themselves familiar with such books as 3Miss Mayo's Lessons on Objects; Mrs. Hamilton's Questions; the first fifty pages. of Wilmsen's Children's Friend, and similar works, it would be impossible for them to keep school, or even to hear a recitation, without overflowing with information, both instructive and delightful. The school-room would then cease to be a place so far out of the world; and the gulf which has so long separated it from actual life would be bridged over. When it was our fortune to be a teacher of the Greek and Latin classics, we used to think it as much a part of our daily duty to be prepared with illustrative anecdotes and historical facts, drawn from the manners and customs of other nations and times, in order to render each lesson more useful and interesting, as to be prepared for translation or syntax. The whole business'of the school-room, from morning till night, should, in this way, be made attractive and profitable. Children do love information which is adapted to their capacities, and they will desire to go where it can be found, as naturally as bees to flowers. An absurd objection is sometimes urged against such a course; namely, that it will only amuse children, turn what should be toil into, pastime, and create a disrelish for close, pains-taking, solitary application. This objection is theoretic merely. It is never made by those who have tried the experiment. It is urged only by such as are too ignorant or too indolent to make the necessary preparation. Not only reason, but experience, proves that it is the best possible means of kindling a desire for knowledge in the bosoms of the young; and when this desire is once kindled, the teacher has only to direct the car, instead of dragging it. We propose, on the present occasion, to give a specimen of the kind of instruction we mean; to show, by an example, how collateral knowledge may be appropriately introduced to illustrate and enrich the matters contained in the text-book. And we may remark, in passing, that it is strange how any teacher can ever use the term text-book, without being reminded that it is only a collection of texts, which it is his duty to explain and illustrate. In our cities, every mexchant and most business men have a The remainder of this Section was originally published in the December numbers of the Common School Journal, for 1847.

Page  247 ~ 72.] EXCHIANGE. 247 much to do with bills of exchange and promissory notes. In the country, too, almost every man has something to do with notes of hand, either as promisor or payee, endorser or endorsee. If a man borrows money, he makes these instruments; if he lends money, he receives them. Every respectable man is liable to be on a jury, where questions respecting this class of securities are tried; and no man is so poor, so ignorant, or so far outside of all society, as not to hear conversations about them. Suppose,. then, a class of advanced scholars, whose minds have been previously awakened by a proper course of instruction, to be asked in what way they suppose that commercial transactions between the merchants of different nations are carried on. The citizens of the United States, for instance, send abroad their productions to different quarters of the world, to the amount of a hundred millions of dollars or more annually. In what manner do they receive their pay? In money, or otherwise? Should any one say in money; then explain to him the immense trouble, risk, and expense, of bringing a hundred millions of dollars from other countries, across the ocean to this, which amount must soon be sent abroad again to pay for foreign productions which we want. Here the historical fact may be stated, that we learn from the Pandects,a that, when a Roman capitalist had lent money to a foreigner, the common mode of collecting the debt was, to send a trusty slave to the foreign country to receive the debt and its interest, and to bring them home. But this was necessarily both expensive and perilous. Some may suppose that money may not be remitted for the settlement of each transaction, but that the traffic may, be carried on by barter. One merchant may send flour, and receive his pay in cutlery; another beef, and receive broadcloth, &c. It will be easy to answer any suggestions of this kind by showing, that, on such a plan, each man would have to trade in everything, or, at least, in a great variety of things, and with a great number of men. But the same man could never trade in books, leather, jewelry, iron-ware, silks, pork, tea, fish, fruits, logwood, flour, cotton, rice, oil, feathers, coal, hemp, molasses, indigo, otter skins, &c. &c.; or if, by any possibility, one man could trade in all these things, he never could trade with all parts of the world from which they come. Barter, therefore, must always be confined to a small number of articles, and to the same place. a Pandects, the digest of the civil law published by Justinian. Ex plain who Justinian was, and when and where he lived.

Page  248 248 THE COUNTING-HOUSE. [ART. XV. Here the subject may be dismissed, for the first day, and the children sent away to devise or to ascertain in what way commercial transactions between different nations may be made expeditious, safe, and cheap. We suggest the suspension of the subject at this point, because we deem the proper course, in regard to all instruction, to be, first, to awaken the child's mind to a sense of the necessity or desirableness of knowledge, and to put it into an inquiring or receptive state; and then, secondly, to rectify the views which his unaided judgment may suggest, or to impart, when necessary, the precise knowledge he needs. At subsequent recitations, let the subject be taken up again, and either the pupils or the teacher will explain it as follows:A., in Boston, is about to ship flour for the Liverpool market. B., in Liverpool, is a coma merchant, and will buy A.'s flour. At the time of this transaction, C., also a merchant in Boston, wants cloths from D., a manufacturer in Manchester. When A. ships his flour, he draws a bill of exchange on B., in Liverpool, in which he requests B. to pay to himself, or to some other person named on the face of the bill, and to the order of whoever is so named, a sum stated, supposed to be the value of the flour when it shall reach Liverpool. But C., in Boston, who wants cloths from D., in Manchester, has no money in England to pay for them; C. therefore buys A.'s bill on B., and pays for it in our money. If the bill be made payable to A.'s order, A. endorses it to C.; and C. then endorses it to D.,-or he endorses it in blank, as it is called, which phrase the teacher must explain,-and sends it to his agent or correspondent in England. When the bill arrives in England, C.'s agent or correspondent presents it to B.; and, the flour having arrived, B. accepts, that is, promises to pay it, according to its terms. It is then taken to the manufacturer D., and, the cloths having been bbought, it is delivered to him. D. therefore becomes B.'s creditor,, and receives payment from him in English money, as A. had received his pay from C. in Boston money. Thus the transaction is completed without the trouble, expense, or risk of sending a cent of money across the ocean, to be sunk by storms, or plundered by pirates. But suppose, in the above case, that C., after having bought a Let it be explained that the word " corn," in England, never has the same signification as with us. Here, it is commonly used for Indian corn or maize; but in England, it is a generic term, and means wheat, barley, and other cereal grains.

Page  249 ~ 72.] EXCHANGE. 249 A.'s bill of exchange, does not wish to use it for three months; but his neighbor E. wants it, or one like it, to be used immediately. Is there any way by which C. can transfer this bill to E., receive his money, and so have the use of it for the three months,or must he go back to A., have the bargain rescinded, the bill cancelled, and a new one drawn in favor of E.? There is such a way. When a bill or note is drawn payable to the order of any one, it is payable to whomsoever that one shall order it to be paid. In the case supposed, therefore, C. has only to write his name on the bill, with the words, "Pay to E.," and E. receives C.'s whole interest in it. If he says, "Pay to E., or his order," then E. may order it to be paid to any one else; and so on. It is this transferability, or quality of being transferable from hand to hand, that makes bills of exchange and promissory notes negotiable instruments. This word negotiable is an important one, and the meaning of it should be precisely understood. By the civil law of the European continent, bills of exchange and promissory notes were early recognised as mercantile instruments, and, from their nature, negotiable. But in England there was a strong prejudice against the assignment or transfer of debts, because of the abuses liable to be practised, if one man could buy up a debt against another, and sue and imprison him on it. There are still laws, both in England and in this country, against buying up debts for purposes of oppression. Anciently, the common law of England forbade the assignment or negotiation of promissory notes. But the statutes of 3 and 4 Anne gave negotiability to notes, placing them, as mercantile contracts, on the same footing as inland bills of exchange. These English statutes have been generally adopted in the United States, as a part of our common law. At the present time, therefore, bills of exchange and promissory notes, by their quality of negotiability, are the means by which debts and credits are transferred from one person to another, with safety, despatch, and economy. They afford means of circulation for all the property they represent, and thus they enlarge in every country its stock of circulating wealth, or its means of trade. Promissory notes are of two kinds;-negotiable, and not negotiable. A negotiable note expresses on its face that it is payable, not only to the person named in it, but to any other person who shall acquire the legal interest in it. If it be made payable to John Stiles or order, it is then negotiable by endorsement; if to John Stiles or bearer, it is then negotiable by delivery.

Page  250 250 THE COUNTING-IOUSE. [ART. XV. A note not negotiable, expresses on its face that it is payable to the particular person named in it,-as to John Stiles. Such a note is payable only to the party named. All valid promissory notes import a valuable consideration; that is, an action at law may be sustained upon them without specially setting forth or proving a consideration for the note. In this they differ from other unsealed contracts. In a promissory note, there are two original parties,-the maker of the note, who is called the promisor; and the party to whom it is made payable, who is called the promisee or payee. A valid negotiable promissory note is a written promise for the payment of money, at all events. The promise must be in writing, but it may be in ink or pencil; and all but the signature may be in printed letters. The signature gives efficacy to the note, and must be in the handwriting of the promisor or of his authorized agent. The form of words used is not material, provided the note contains a written promise to pay. A mere acknowledgment of indebtedness is not sufficient; Thus, "I owe you $300," though in writing, is only a due bill; it is not a promissory note. The note must be for the payment of money. Therefore, a written promise to pay in goods or labor, is not a negotiable promissory note, although put into the form of a note, and payable "to order" or "bearer." A negotiable note must, on its face, fix and make certain the amount of money to be paid, either in words or figures. Hence, a written promise to pay " all that shall be due on final settlement," or " all that shall be realized from the growing crop," or "" a11 that shall be received from John Stiles," is not a negotiable promissory note. Even though a part of the sum to be paid should be made certain, on the face of the note, it is yet not a negotiable promissory note, even for the part which is so made certain. The money, by the note itself, must be made payable at all events, and independently of any contingency. Therefore, a written promise to pay "when certain goods are sold," or "when a certain ship arrives," is not a negotiable promissory note. So, if the note is made payable out of a particular fund, as "my next month's wages," it is not a negotiable promissory note. And the promise must be to pay money on a day certain,-a y fixed by the note itself. Therefore, a promise to pay $100,

Page  251 ~ 72.] EXCHANGE. 251 "when A, shall come of age," is not a negotiable promissory note; for A. may never come of age. But a note promising to pay $100 when A. shall die, is valid; for A.'s death on some day is certain; and the note, by its own terms, fixes that day for payment. A note must contain no uncertainty as to the person to whom it is payable. Hence, a written promise to pay to "A. or B.," is not a negotiable promissory note. But a written promise to pay to "A. or bearer," is good; for, in legal effect, such a note is payable to "bearer;" and any person who has legal possession of the note, and presents it for payment, is the "bearer" intended. A note may be issued with a blank for the payee's name; and, in such case, any bona fide holder may fill iup the blank with his own or any other name, and the note will then be treated as though it had been valid in all respects from its date. It is indispensable that the maker's'name should appear on the note as promisor. The name, however, may be written in ink or pencil, and at the top, or bottom, or in the margin of the paper. It is not indispensable to the validity of a note that it should be dated, because it is allowable to show the time when it was made, by evidence extrinsic to the note itself; but this is always expensive, often difficult, and sometimes impossible. If a note be postdated or antedated, the time of its actual issue may always be shown, when required for substantial justice. It is not necessary that a note should specify any place of payment; but when it is the intent of the parties that it shall be paid at a particular place, the place must be specified in the body of the note. A memorandum at the bottom of the note, or on its margin, is not sufficient. Nor is it essential that a note should be attested. An attestation, however, in Massachusetts, takes a note out of the statute of limitations, as to the payee, his executor or administrator. A promissory note may be made by one person, or by two or more persons. When made by two or more persons, it may be joint, or joint and several. When two or more persons sign a note written thus: "We promise to pay," &c., it is a joint note only. If they sign a note written thus: "I promise to pay," &c., it is a joint and several note. When a note is joint, all the promisors must be jointly sued; if joint and several, either promisor may be sued alone. When a note written thus, "We promise," &c., is signed thus,

Page  252 252 THE COUNTING-HOUSE. [ART. XV "A. B. as principal, and C. D. as surety," it is still the joint note of A. B. and C. D. Had it been written, "I promise," &c., and signed in the same way, it would be a joint and several note. The words "principal" and "surety" only show the relation of the makers to each other; they do not affect other parties. By the phrase "negotiable promissory note," is meant an instrument, negotiable and possessing all the privileges of a promissory note in commerce. A note not negotiable is nevertheless a binding contract between the parties to it. It is in reference to the transfer of a note from hand to hand, — like a bank bill or a Bank of England note,-that the question of its negotiability becomes material. THE TRANSFER OF NOTES. A note may be transferred by delivery, or by endorsement. As TO TRANSFER BY DELIVERY.-The rule is, that no person whose name is not on the note, as a party thereto, is liable on the note. Therefore, when a note payable to bearer, or endorsed in blank, is transferred by the holder, by delivery only, the party transferring it is not liable upon it. By.not endorsing it, he is understood to mean that he will not be responsible on it; and such, therefore, is the contract between him and the party receiving it. But if, in such case, the note is received, by the party to whom it is delivered, as a conditional payment of a debt previously due him, or as a conditional satisfaction of any other valuable consideration then given, the party transferring it, if the note is dishonored, (that is, if not paid,) on legal presentment and notice, will be responsible for the debt, or consideration, though not directly suable on the note. And though a party transferring a note by delivery only, is not liable on the note, he is not exempt from all obligations or responsibilities. In the first place, by legal implication, he warrants his own title to the note, and his right to transfer it by delivery. Then he warrants that the riote is genuine, and not forged or fictitious. And he warrants, moreover, that he has no knowledge of any facts which make the note worthless.; for instance, if the note be a bank note, and the party transferring it knows the bank has failed, and conceals this knowledge, his act is a fraud, and the

Page  253 ~ 72.] EXCHANGE. 253 consideration he received may be recovered back. The fraud makes void the contract. And even if the failure of the bank, at the time of the transfer, was unknown to either of the parties to it, it is the better opinion that the transferrer must bear the loss, because it is implied in the transaction that the note would be paid on due presentment. As TO TRANSFER BY ENDORSEMENT. —When a note is payable to a person, or his order, it is properly transferable only by endorsement, as nothing else will give to the holder a legal title, so that he can, at law, hold the parties to the note directly liable to him. By a mere assignment of a negotiable note, the holder acquires only the same rights that the assignment would give him, if the note were not negotiable. No particular form of words is required to make an endorsement legal; generally it is enough if the signature of the endorser is on the note, without any words at all; and this is the usual mode of endorsing notes. The endorsement may be on either side, or any part of the note, or on a paper annexed to it, and in ink or in pencil. A note transferable by delivery only, may be endorsed; and then the endorser incurs the same obligations and liabilities as if the note had been originally made transferable by endorsement only. The time of endorsing a note may be material, for if a person, (not the payee of a negotiable note,) endorses it when it is made, he will be liable at all events, not as endorser, but as guarantor. If he endorses it afterwards, (not being a regular endorser,) he will be liable if his act is founded on any legal consideration, but not otherwise. Every endorser, by his endorsement, contracts with every subsequent holder of the note,1. That the instrument itself, and the signatures antecedent to his, are genuine. 2. That he, (the endorser,) has a good title to the note. 3. That he is competent to bind himself as endorser by his endorsement. 4. That the maker is competent to bind himself as maker, and will, on presentment, pay the note. 5. That if, when duly presented, it is not paid by the maker, the endorser, on due notice, will pay it. An endorsement may be in "blank," or " in full," or " restrictive," or "general," or " qualified," or "conditional." A " blank" endorsement is merely the name of the endorser written on the note,

Page  254 254 THE COUNTING-HOUSE. [ART. XV. After such an endorsement, a note may be transferred by delivery only, and be circulated like a bank note; and any holder may write out, over the endorser's name, the contract implied b&y law on the part of the endorser, and sue upon it. An endorsement is said to be "in fillu," when it mentions the name of the person in whose favor itis made, and then the endorsee can transfer his interest in it only by writing his own endorsement on it. In order to make an endorsement "restrictive," there must be express words, showing that intent; as, " Pay to John Stiles only." An endorsement is said to be " general," when it is in blank, or payable to the endorsee or order. A " qrualified" endorsement is one which affects the liability of the endorser, but not the negotiability of the note; as when to the endorsement is added, " without recourse," or " at endorsee's own risk," &c. A " conditional" endorsement limits the validity of the endorsement to some future event, and may be either precedent or subsequent; as, 1st, "Pay John Stiles the within on my marriage;" or, 2d, "Pay John Stiles, or order, the within in six months, unless he sooner receives it from my agent." Whoever receives an endorsed note, contracts with the endorser, (and if there are many, with each of them,) that the note shall be presented to the promisor for payment at the proper time; that no extra time for payment shall be allowed; and that notice of non-payment shall be immediately given to the endorser; and a default in any of these particulars discharges the endorser. Due presentment for payment requires that the note should be presented as soon as it becomes clue. If the holder could delay a day, he might two days, or a year; but any delay may injuriously affect the endorser, and his remedy against other persons. Therefore, if the holder of the note does not present it to the promisor on the day it becomes due, the endorsers are discharged. And the rule is so, although the holder received the note so near the time of its maturity as to make the demand in legal time impossible. And such demand for payment is required though it is known that the maker is dead or an insolvent. Where a note is made payable on demand, the time at which payment must be demanded, depends on the circumstances of the case, the rule being that payment must be demanded in rea8onable time.

Page  255 ~ 73.] ARBITRATION OF EXCHANGE. 255 And in Massachusetts, by statute, the endorser is excused, if the demand for, payment on the maker is not made within sixty days from the date of the note. If a note is payable generally, that is, without any place being designated, it may be presented at the maker's counting-house or dwelling-house. If it is presented at the counting-house, it must be within the hours in which, by the usage of the city or place, counting-houses are kept open; if at the dwelling-house, then at hours while the family are up, and the maker may be presumed not to have gone to bed. And where a note is made payable at a particular place, the demand must be made at the place fixed, as well as at the proper time; otherwise the endorser is discharged. Where a note is payable by'a partnership, presentment to either of the partners is sufficient. Where the promisors are only joint contractors, and not partners, demand must be made on each. The demand must be made witli the note; and if any particular bank or place is fixed for payment, the note must be there, in order to make the demand valid. On the failure -of the maker to pay, the holder must give due notice of it to each party liable to him; and if he fails to do so to any party, such party is discharged. And when the endorser lives in the same place with the holder, notice may be given on the day when the demand was made, or the day after, but not later. When the endorser and holder live in different towns, the notice may be by mail, by special messenger, or by private hand. And notice by the mail on the day, or the day after, is good, but not later. Where there are numerous endorsers, each is entitled to notice, and each is to give notice to all parties prior to himself; and each endorser has the next day after receiving notice, in which to give notice to any prior party whom he seeks to hold liable to himself. 73. ARBITRATION OF EXCHANGE. Merchants often find an advantage in remitting bills circuitously, rather than directly to the' place where they are due. The determination of the value of such remittances

Page  256 o256 THE COUNTING-HOUSE. [ART. XV. is called ARBITRATION OF EXCHANGE, and is best determined by the CHAIN RULE. EXAMPLE FOR ILLUSTRATION. A French merchant wishes to pay in London a bill of ~1500. How many francs must he pay to procure remittances through Russia, Hamburg, and Spain, allowing ~13 = —75 roubles, 5 roubles = 9 mares of Hamburg; 3 mares = I Spanish dollar; and 9 dollars = 50 francs, We write the quantities which ~13 5 roubles. are equivalent to each other, s. rou. ) = p mares. as antecedent and consequent, marcs A - 1 dollar. mnakigq each consequent of the same denomination as the next antece- dol. dent. The like factors on op- francs - 150 ~ posite sides are cancelled, and 13) 375000 the products divided as in ~ 71, 28846fr. 155-c. to obtain the answer. 13 The question may be otherwise stated in the following manner: If ~13 produce 75 ~13 1500~ roubles, 5 roubles produce 9 marcs, 3 marcs mar. 3 produce $1.00, and $9.00 produce 50 francs, $ 9 how many francs will ~1500 produce? The i5 rou second set, or set of demand, contains but 9 mar a single cause and effect. The first, or 1$ given set, contains a number of causes and fr. -- 50 fr. effects,, but they are so connected, that all the terms may be multiplied together, as a single compound term. Thus, if ~13 produce 75 roubles, and 5 roubles produce 9 mares, ~18 will produce Y-5- of 9 mares, and ~13 X 5 will produce 75 X 9 marcs. In the same way, it may be shown that ~13 X 5 X 3 =$75 X 9 X 1, and ~13 X 5 X 3 X 9 =75 X X 1 X 50 fr. Then how many francs will ~1500 produce? EXAMPLES FOR THE PUPIL. 1. A London merchant wishing to pay 1000 milrees in Lisbon, remits as follows: To Amsterdam, at 36 schillings 7 groats per ~; thence to Cadiz, at 17 groats for 2 rials of

Page  257 ~ 73.] ARBITRATION OF EXCHANGE. 257 plate; thence to Leghorn, at 17 pezze for 100 rials; thence to Lisbon, at 1497 rees for 2 pezze. How many pounds did he remit? Ans. ~152 3s. 3Md. 2. If a merchant of New York remits $5000 to Havre, at 5fr. 35c. for $1; thence to London, at 49fr. for ~2; thence to Hamburg, at 1 mare for is. 6d.; and thence to St. Petersburg, at 8 roubles for 17 mares, how many roubles can he pay with his remittance?a Ans. 6850rou. 74cop. 3. If 33 copecks are equal to 5 English pence, 11 English pence are equal to 3 piastres, 13 piastres are equal to I florin, and 5 florins are equal to 29 francs, how many francs are equal to 9000 copecks? Ans. 165f. 92c. 4. If a man receives $30 for building 8 rods of wall, and he can purchase 3 barrels of four for $14, and 3cwt. of sugar for 4 barrels of flour, and 21 lb. of tea for 2cwt. of sugar, how many pounds of tea could he purchase by building 17 rods of wall? Ans. 1071b. 9{oz. 5. If 13 days' work will purchase 1 hogshead of molasses, and 2 hogsheads of molasses are worth 5 tons of hay, and 3 tons of hay are worth 4 bags of coffee, how many bags of coffee can be bought with 39 days' labor? Ans. 10 bags. 6. If 70 braces of Venice are equal to 75 braces of Leghorn, and 7 braces of Leghorn are equal to 4 yards, how many yards are there in 79.375 braces of Venice? Ans. 48-117yd. 7. A merchant in New York orders ~500 sterling, due him in London, to be sent by the following circuit: To HaIllburg, at 15 mares banco per ~; thence to Copenhagen, at 100 mares banco for 33 rix-dollars; thence to Bordeaux, a The pupil must carefully observe the rule, and make each conse. quent of the same denomination as the next antecedent. 17

Page  258 258 THE COUNTING-HOUSE. [ART. XV. at 3 rix-dollars for 18 francs; thence to Lisbon, at 125 francs for 18 milrees; and thence to New York. at $1.25 per milree. What was the arbitrated value of a dollar by this remittance?a Ans. 3s. 8.89 + d. 8. Amsterdam exchanges with London, at 34 schillings 4 pfennings per ~, and with Lisbon at 52 pfennings for 400 reas. What is the arbitrated exchange between London and Lisbon, by way of Amsterdam? Ans. 1- = 3 C 169-3. 9. The exchange between New York and London is $4.84 per pound; between London and Amsterdam, 35 schillings per pound; between Amsterdam and Paris, 58 groats for 6 francs; between Paris and Venice, 10 francs per ducat; and between Venice and Cadiz, 360 maravedis per ducat. HI-ow many maravedis will be equivalent to $4500, by this circuitous remittance? Ans. 14542601 6-0 mar. 10. If 100 lb. of Amsterdam are equal to 105 lb. of Antwerp, 1001b. of Antwerp to 1421b. of Genoa, 1001b. of Genoa to 701b. of Leipsic, and 1001b. of Leipsic to 104llb. of America, how many lb. of Amsterdam are equal to 14911b. of America? Ans. 1373-7lb. 74. ALLIGATION. When it is desired to make a mixture of a given value, with a variety of ingredients, the following method is usually adopted: — 1. Having written the values of the ingredients in a perpendicular column, connect by a line each value that is less than the required average with one or more that is greater, and each value that is greater with one or more that is less. 2. WTrite the difference between each value and the average, a After finding the number of dollars which are equivalent to ~50/D, tie itrbitrated value of $1 is found by dividing ~500 by that number.

Page  259 ~ 74.] ALLIGATION. 259 opposite the ingredient with which that value is connected, and the difference, (or the sum of the differences, if there be more than one,) opposite each ingredient, will be the quantity of that ingredient required. EXAMPLES FOt ILLUSTRATION. 1. How much sugar at 5cts., 7cts., 8cts., lOcts., and 12cts., must be mixed together, that the mixture may be worth 9cts. a pound? 5 1+ 5 1 9 180 1 + l 9 8- } 9 8 -— 3 — _ - 4+ 1 12 4+2 L12 2+-1 Ist Ans. 4 lb. at 5, 1 t. at 7, 4 b. 2d Ans. 3 b. at 5, S3b. at 7, 1 b. S Ans. I lb. at 5, 3 b. at 7, 3b at 8, 71b. at 10, 5 lb. at 12. at 8, lb. at 10, 6 b. at 12. at 8 4 lb. at 10, 3 b. at 12. We may obtain as many answers as there are different ways of connecting the numbers above, with those below the average. To prove the rule correct, let us examine the second of the above answers. If we were mixing sugars at 5 and 12cts. to sell the mixture at 9cts., we should gain 4cts. on every pound of the former, and lose 3cts. on every pound of the latter. Then, on 3 lb. of the former we should gain 12cts. and on 4 lb. of the latter we should lose 12cts.; therefore, if we mix these quantities, we shall neither gain nor lose by selling the mixture at 9cts. In the same way it may be shown that 3 lb. at 7cts. and 2 lb. at 12cts., 1 lb. at 8cts. and 1 lb. at 10cts. may be sold at the average of 9cts., and the same reasoning will prove the truth of each of the other answers. 2. A farmer wishes to mix 10 bushels of barley at 50cts., 4 bushels of oats at 40cts., and 16 bushels of rye at 75cts. with wheat at $1.25, and corn at 90cts. a bushel, so that the mixture may be worth $1.00 per bushel. We may regard the limited quantities as a single ingredient of 30 bushels; worth 62cts. a bushel. Proceeding in the usual way, we find that 25 bushels at 62cts., 25 at 62 25 90cts., and 48 at $1 25, would give us 1.00 90 125 a mixture of the desired average value. 1.25- _ 38 + 10 But as we have 30 bushels at 62cts., we must take 2 or 5 of these proportionate quantities, and we have 30 bushels at 90cts., and 570bu. at $1.25, for the answer.

Page  260 260 THE COUNTING-IIOUSE. [ART. XV. In most questions in Alligation, ain infinite number of answers may be obtained, but it will readily be perceived that the preceding method gives only a few of those answers. The following rule is not only more general, but also more analytical in its character. Assume any quantity you please of each ingredient, find the cost of the whole, and also the cost of the same quantity at the mean rate proposed. If the assumed quantities cost TOO MUCH, take such additional quantities of the lower priced, or such diminished quantities of the higher priced ingredients, as' will exactly counterbalance the excess. If TOO LITTLE, take such additional quantities of the higher priced, or such diminished quantities of the lower priced ingredients, as will exactly counterbalance the deficiency. EXAMPLE FOR ILLUSTRATION. In what proportion should I mix sugars at 5ets., 7cts., 8cts., l0cts., and 12cts., in order that the mixture may be worth 9cts. a pound? We may commence by taking any 31b. at ets. = 15 quantity we please of each ingredient, 1 lb. at 7cts..07 and finding the cost of the whole. If, 2lb. at 8cts. —.16 for example, we take 3 lb. at 5cts., 1 lb. 4 lb. at l0cts. =.40 at 7cts., 2 lb. at 8cts., 4 lb. at lOcts., and 5 lb. at 12cts..60 5 lb. at 12cts., (making 15 lb. in all,) the 15 lb. cost 1.38 whole cost will be $1.38. But 15 lb. at 15 lb. at 9cts. - 1.35 9cts. would cost only $1.35, therefore our Excess.03 estimated quantities give an excess of 3 cents above the required cost. To balance this excess, we must either add some of the sugar that costs less than the average price proposed, or take out some of that which costs more than the average. For every pound that we add at 5 cents, there will be a deficiency of 4 cents from the mean rate; for every pound at 7cts., a deficiency of 2cts.; for every pound at 8cts., a deficiency of lct. Then we may either add a of a lb. more at 5cts., or ] lb. more at 7cts., or 3 lb. more at 8cts., or 1 lb. more at 8cts. and 1 lb. at 7cts., or any other quantity that will make a deficiency of 3cts. If the quantities first assumed had f;iven a deficiency instead of

Page  261 ~ 74.] ALLIGATION. 261 an excess, we should have been obliged to take some additional quantity of one or more of the ingredients whose value is greater than the proposed average, or to take out some of the ingredients whose value is 3 lb. at 7cts..21 less than the average. The cost of the 21b. at 8cts. =.16 quantities assumed in the margin, would 1 lb. at l0cts. =.10 be only 9lcts., but 12 lb. at 9ets. would 2 lb. at 12cts. =.24 cost $1.08. There is, therefore, a defi- 121lb. cost.91 ciency of 17cts., which may be balanced 12lb. at 9cts. 1.08 by taking enough of the sugar that costs Deficiency.17 more than 9cts., to make an excess of 17cts. Thus, 5 lb. additional at 12cts., and 2 lb. at 10cts., would answer the required conditions. The pupil may determine other values for himself. The following are therefore some of the answers to the proposed question:-(1.) 38- lb. at 5cts., 1 lb. at 7cts., 2 lb. at 8cts., 4 lb. at l0cts., and 5 lb. at 12cts. (2.) 31b. at 5cts., 2 lb. at 7cts., 2 lb. at 8cts., 4 lb. at 10cts., and 5 lb. at 12cts. (3.) 3 lb. at 5cts., 1 lb. at 7cts., 5 lb. at 8cts., 4 lb. at 10cts., and 5 lb. at 12cts. (4.) 3 lb. at 5ets., 2 lb. at 7cts., 3 lb. at 8cts., 4 lb. at l0cts., 5 lb. at 12cts. (5.) 4 lb. at 5cts., 3 lb. at 7cts., 2 lb. at 8cts., 3 lb. at 10cts., 7 lb. at 12cts. Let the pupil perform the above examples by taking out some of the quantities originally assumed. EXAMPLES FOR THE PUPIL. 1. A mixture is made of 24bu. of grain at $1.10 per bu., 28bu. at $.90, and 44bu. at $.60. How must the mixture be sold, in order to gain $ 3 per bushel? a.Ans. $1 per bushel. 2. What is the fineness of a composition, consisting of 3 lb. 6oz. of gold, 23 carats fine, 41b. 8oz. of 21 carats, 3 lb. 9oz. of 20 carats, and 2 lb. 2oz. of alloy? Ans. 18 carats. 3. A grocer mixes 651b. of sugar at $.08, 30 lb. at $.07, a The first three examples are not, strictly speaking, examples in Alligation, but, in accordance with general custom, they are inserteA in this place. See Remark at the close of Section 68.

Page  262 262 THE COUNTING-HOUSE. [ART. XV. 25 lb. at $.05, and 20 lb. at $.061. How must the mixture be sold, in order to gain 25 per cent.? Ans. 8-4cts. per lb. 4. In what proportions may I mix teas at GOcts., 70cts., $1.10, and $1.20, per pound, so as to gain 121 per cent. by selling at $.90 per pound? a Ans. 4 lb. at 60; 3 lb. at 70; 1 lb. at $1.10; 2 lb. at $1.20. 5. A corn merchant bought wheat at $1.20, at $1.10, at $.90, and at $.70 per bushel; but the markets having fallen, he is desirous to sell at $.80 per bushel, and is willing to lose 20 per cent. In what proportions may a mixture be made, to answer the conditions of the question? AAns. 3bu. at $1.20, lbu. at $1.10, lbu. at $.90, 2bu. at $.70. 6. A cask contains 50 gallons of acid, worth 70 cents per gallon; how much will it contain after the value of the liquor has been reduced to 60 cents per gallon, by pouring in water? A2ns. 581 gallons. 7. A silversmith mixes 20oz. of silver, at 5s. 9d. per ounce, with two other kinds, at 5s. 6d., and 5s. 3d., and as much alloy as reduces the mass to the value of 4s. lOd. per ounce. Required the weight of the whole composition. Ans. 688oZ. 8. A refiner melts 14lb. of gold, 22 carats fine, with 16 lb. of 20 carats fine. How much alloy must be added, in order to make the composition 18 carats fine? Ans. 48 lb. 9. A New York merchant shipped 8000 bushels of grain, consisting of wheat, barley, and rye, which he sold in London at 7s. 4d. per bushel. The prime cost of the white wheat was 6s. 6d., the red wheat 5s. 9d., the barley 3s. 4d., and the rye 4s. 3d. per bushel. The port charges and other a An infinite number of answers can be obtained to some of the Kemaining examples in this section, besides the ones that are given.

Page  263 ~ 75.] GENERAL AVERAGE. 263 incidental expenses, amounted to ~58 6s. 8d.; the freight was 1Od. per bushel, and he gained 91d. per bushel by the transaction. How many bushels of each kind of grain would answer the conditions of the question? Ans. 22855bu. white wheat; 38579-bu. red wheat; 2857bu. barley; 1571-bu. rye. 75. GENERAL AVERAGE.a Whenever any sacrifice of property is made, or any expense necessarily incurred, for the preservation of a ship and cargo, the loss is divided among all the parties interested, either as owners of the vessel, or of the property on board. " Thus, where the goods of a particular merchant are thrown overboard in a storm, to save the ship from sinking; or where the masts, cables, anchors, or other furniture of the ship are cut away or destroyed for the preservation of the whole; or money or goods are given as a composition to pirates to save the rest; or an expense is incurred in reclaiming the ship, or defending a suit in a foreign court of admiralty, and obtaining her discharge from an unjust capture or detention; in these and the like cases, where any sacrifice is deliberately and voluntarily made, or any expense fairly and bone fide incurred, to prevent a total loss, such sacrifice or expense is the proper subject of a general contribution, and ought to be ratably borne by the owners of the ship, freight, and cargo, so that the loss may fall equally on all, according to the equitable maxim of the civil law:-No one ought to be enriched by another's loss."b The loss is distributed in such cases, by GENERAL AVERAGE. But if any sacrifice is made for the sake of the ship only, or of the cargo only, the loss must be borne by the parties immediately interested, and is consequently defrayed by a particculcar average. In New York, only - of the value of the freight is contributory; in the other United States ports, *- of the value is taken. The remainder of the freight is reserved for seamen's wages, because if the seamen were laid under this a McCulloch, Hunt's Mer. Mag. b Mr. Serjeant Marshall.

Page  264 264 THE COUNTING-HOUSE. [ART. XV. obligation, they might be tempted to oppose a sacrifice necessary for the general safety. When the loss of masts, or ship furniture, is compensated by general average, as the new articles are supposed to be of more value than those that were lost, only i- of the value is contributed. EXAMPLES. 1. It became necessary, in the Downs, to cut the cable of a ship destined for Hull; the ship afterwards struck on a bar, and the captain was compelled to cut away the mast, and throw overboard part of the cargo; in which operation another part was injured. The ship, being cleared from the sands, was forced to take refuge in Ramsgate harbor, to avoid further injury from the storm. Required, from the following statement, the proper adjustment of the loss. AMOUNT OFr LOSSES. CONTRIBUTORY VALUES. Goods of A., cast over- Goods of A., cast overboard... $2000 board... $2000 Damage of B.'s goods. 800 Sound value of B.'s goods 4000 Freight of A.'s goods. 400 Goods of C.... 2000 i of new cable, anchor, " "D... 8000 and mast... 800 " " E.. 20000 Pilotage, port-duties, and Value of the ship.. 8000 expenses of bringing i of freight.. 3200 ship off the sands. 700 Adjusting average. 16 Postage... 4 Total loss. $4720 Tot. contributory values $47200 Ans. The owners of the vessel receive $800; C. pays $200. A. receives... $1800; D. " $800. B. "... $400; E. " $2000. 2. In a storm, goods belonging to A., worth $500, were thrown overboard, and the losses of the owners of the vessel amounted to 81500. Adjust the generil average, the total

Page  265 ~ 76.] PRODUCT OF MINES. 265 value of A.'s goods being $800; B.'s goods, $1200; C.'s goods, $3000; value of the ship, $10000; total -freight, $1500..Ans. A. receives $400; B. pays $150. Owners receive $125; C. pays $375. XVI. STATISTICS. 76. PRODUCT OF MINES IN THE UNITED STATES, ACCORDING TO THE CENSUS OF 1810.a States and Territo- C ar. Lead. Gld Other Coa. Salt aite, re. ast&ar Pouuds. Colde. Metals. Toss of 25 uRshel Marble,&c.. ons. Value. bushels. Value. ~Maine......... 6122............... 00.... 50000 $10750 N. Hampshire.. 145 1000........ 10300 1069 1200 16(038 Masschulse tts. et 15336...2500....... 376596 790855 Rhode Island.. 4126....... 17800 Connecticut... 10118... 1357 1500 31346G9 Vermont - 730.... 70500........... 33855 New York... 82781 670000........ 84564.. 2867881 1541480 New Jersey..........85..... 39550. 500 35721 Pennsylvania.. 185639.. 100200 1274709 549478 238831 Delaware.......... 460... 110 16000 Mryland...... 16776............. 28800 929 1200 22750 Virginia....... 24697 8786148 $51758...... 379569' 1745618 84489 N. Carolina.... 1931 10000 255618 1000 53 4493 3350 South Carolina 2415.......... 37418....... 2250 3000 Gcorgia....... 494........ 121881.............. 51990 A labna...... 105........ 61230........ 845.. 13700 Mississippi............................. Louisinlla..... 2766.............................. Teln essee.. 25 802....... 1500........ 498...... 30100 IKentucky...... 32853.............. 23131 210195 19592 Olio..O... 42702............... 16000 125775 297350 195831 Istdlia -a........ 830....................... 86514 6-100 35021 Illinois........ 158 8755000 200........ 15282 20000 74228 Misso i....... 298 5295455........ 1500 8901 13150 28110 Arkansas...................................9 8700 15500 Michigan...... 601........ 2700 Florida.......... 12090 2650 Wisconsin 3 15129350............ Iowa.................. 500090............ 357350 Total, _ EXAMPLES. 1. Find the total products embraced in the above table. 2. Estimating the average value of the iron at $22 per ton, lead at 41- cents per lb., coal at $3.75 per ton, and salt at $.40 per bushel, what was the entire value of the above products? a Tucker's Progress.

Page  266 266 STATISTICS. [ART. XVI. 17 7 TABLE OF THE AGRICULTURAL PRODUCTS OF THE UNITED STATES, ACCORDING TO THEI CENSUS OF 1840.a States and. Territo I Bushels o Bushels o f Blo Bushels of Bush els of ushelsof Bushels of Bushels of ries. WheIat, Barley. Oats. Rye. Buckwheat Ind. Coln. Po tatoes. Maine.......... 848166 355161 1076409 137941 51513 05)528 10392280 N. Hampshire.. 422124 121899 1296114 308148 1051.03 1162572 6206606 Massachuseetts. 157923 165319 1319680 536014 87000 1809192 5385652 Rhode Island.. 3098 66490 171517 34521 2979 450-1098 911.973 Connecticut... 87009 33759 14531262 737424 303013 15004411 3114238 Vermont...... 495800 511781 2222581 230993 228416] 1119678 869(751 New York.... 12286418 2520068 20675847 2979323 22S78S5 10972286 301236141 New Jersey... 774203 12501 3083524 1].665820 856117l 4361975 2072069 Pennsylvania.. 13213077 209893 20641.819 6613873 2113742 14240022'95356(33 Delaware.... 315165 5260 27405 331546 1.1299| 2099359 200712 Marvland l... 3..83 3194 3534211 723577 736061 8233086G 10364133 Virginia.... 10109716 87430 13451.062 1482799 2413822 3415775911 29-1GG 660 N. Carolina... 1960855 3574 3193941 21.3971. 15391238937631 2609239 S. Carolina.. 968354 3967 1486208 44731 72S1472280() 5; 269831.3 Georgia... 180.. S30 1297 1610030 61)03 1411209051221 1291(3(6 Alabama...... 828052 7692 1406353 51.008 58 2()09-70041 17()8356 Mississippi.... 196626 1 54 668624 11444 61 13161237 16.30100 Louisiana 60........ 107353 1815........ 5952912. 8l34311 Tennessee...... 4569692 4809 7035678 3041320 1 7118 44 961 1.88 19)0:1370 Kentucky 4S03152 17491. 7155974 13213173 8169 398547120 1.055085 Ohio.......... 1.6571661 212440 14393103 81.4205 6331.3913.3G681.41 58()5021 Indiana........ 4049375 2015 5816)5 129621 190)192815587 1.52579 Illinois........ 3335393 82251 4980808 88197 578S-I'263-4211 202552)0 Missouri...... 1037386 9801 2234947 68608 1531 1173325 4 783768 Arkansas...... 105878 760 189553 6219 88 481632 29360()8 Michigan.. 21571.08 127802 2114051 34236 113592 2277039 21()9205 Florida........ 412 30 13829 305...... 898971 26461.7 WVisconsin..... 212116 11062 406514 1965 10654 379359 41960(8 Iowrta..,.. - 154693 728 216385 3792 212 1406241 234063 Dist. of Col.... 12147 294 15751 5081. 27' 391485 12035 Total........ EXAMPLES. 1. Find the total products of each article embraced in the above table. 2. Find the value of the entire product of the wheat, at $1.10 per bushel; of the barley, at $.65 per bu.; of the oats, at 8.40 per bu.; of the rye, at $.70 per bu.; of the buckwheat, at $.75 per bu.; of the corn, at $.80 per bu.; of the potatoes, at $.37~ per bu. 3. What percentage of the entire wheat harvest was produced in 1840, by the four principal wheat-growing States? a Tucker' s Progress.

Page  267 ~78.] AGRICULTURAL PRODUCTS. 267 7 S. TABLE OF AGRICULTURAL PRODUCTS. —CONTINUED.a; Ip..ay Tobacco. Rice. Cotton. W5ool. Sugar. Dairy States, &c. Toos. Pounds. Pounds. Pounds. Pounds. Pounds. Products. Maine...... 691353 30............. 1465551 257464 $14196902 N. lamp.... 496107 1151................ 1260512 7 11.62368 163851:3 Mass........ 569395 640955................. 941900 579227 2373299 1 R. Island... 63449 317............. 183830 50 223229 Connecticut 42670-1 4716573.......7........ 889870 51764- 13765334 1 Vermont.... 836739 585................ 3699235 461-793-14 2008737 New York.3127047 744................ 9845295 100-18109 10-196021 New Jersey 334861 1.922........3......... 397207 56 1328032 Pennsylv'a. 13116G13 32501.8 304..........30 8561G 2265755 3187292 Delaware.. 22.183 272........ 33 641..04..... 113828 M1:ryland... 10667 2816012........ 5673 48201 36266 45746G irglnlia.... 3G41708. 75317106I 2956 31494483 2538374 154-1833 14SO-18 N. Carolina 101369 167723591 2S20388 51926190 625044 7163 6743-19 S. Carolina. 24618 51519160590861 61710274 299170 30000 577810 Georgia..... 1G696,9i 162894112381732 163392396 371303 329744 605 1.72 A1labama.... 12718 273302 14901.9 1173l38823 2'20353 10143 265200 Mississippi. 171 83-171 777195 193401 577 1751.96 77 359585 Louisiana... 24651 119821 3604534 15255536SS 49283 119947720 153069 Tennessee.. 31:233 29550432 7977 27701 277 1060332 258073 472141 Kentucky.... S8306 53436909 16376 6914561786847 1377835 931363 Olhio........ 1022037 5942275........6......... 36853 15 6363386 1848869 In liana... 178029 1820306........ 180 1237919 37277915 742261 Illinois..... 164932 564326 460 200947 650007 399813 428175 Missouri... 49083 9067913 50 121122 562265 2748S53 100-132 Arkansas.. 586 148-439 5454 6028642 64943 15;2 59205) Michiglan.... 13080.5 1602................. 153375 1329784 301052 ltlorilda..... 11.97 75274 481420 1211.0533 7285 275317 231)01 WVisconsin.. 30938 115................. 6777 135288 35677 Iowa....... 17953 S076................. 230391 41450 23609 Dist. of Col. 1331 55550............... 707.5566 Total.... -. EXAMPLES. 1. Which of the States yielded the heaviest product of each of the foregoing articles, and what percentage of the entire product of each article, was the heaviest product? 2. Find the average amount yielded by each of the producing States, of each article. 3. Find the total product of each article, and the value of the cotton, at 12~ cents per pound. 4. Find what percentage of each of the foregoing articles was produced by the New England States; by the Middle States; by the Southern States; by the South-western States; by the North-western States. a Tucker's Progress.

Page  268 268 STATISTICS. [ART. XVI. 17 j TABLE OF OCCUPATIONS IN THE UNITED STATES, ACCORDING TO THE CENSUS OF 1810.a States and Territo- Mining. Agricul- Conl- Mau- Ocean Interna Learned Total ries. ture. menrce. factures. Navig. aNavig. Plofesions. Total. Maine......... 36 101630 2921 21879 10091 539 1889 N. Hampshire.. 13 77949 1379 17826 452 198 1640 Vermont....... 77 73150 1303 13174 41 146 1563 Massachusetts. 499 87837 8063 85176 27153 372 3804 Rhode Island.. 35 16617 1348 21271 1717 228 457 Connecticut.... 151 56955 2743 27932 2700 431 1697 N. Eng. States.075082 N. Engr. States................................................. 675082 New York..... 1898 455954 28468 173193 5511 101.67 14111 New Jersey... 266 56701 2283 27004 1143 1.625 1627 Pennsylvania.. 4603 207533 15338 105883 181.5 3951 6706 Delaware...... 5 16015 467 4060 401 235 1900 Mlaryland...... 320 72046 3281 21529 717 1528 1666 Dist. of' Col.......... 384 240 2278 126H 80 203 Middle States............................................ 1251560 Virginia..9905 318771 6361 54147 582 2952 3866 N. CaroliasL..... 589 217095 1734 14322 3271 370 1086 South Carolina, 51 198363 1958 10325 381 34S 1481 Georgia.... 574 209383 2428 7984 262 352 1250 74 62 35 2 1250 Florida......... 1 12117 481 1177 435 214 South. States............................................1738 Sootla. S tales..1073870 Alabanma...... 96 177439 2212 71.95 256 758 1514 Mississippi.... 14 139721 1303 4151 33 100 1506 Louisiana...... I 7928 854 7565 13221 662 1018 Arknsas...... 41 26355 215 11.73 31 39 301 Tennessee..... 103 227739 2217 17815 55 302 2042 Southw.States.............................................. 713107 Missouri.. 742 92418 2522 11100 39 1885 1469 Kentucky... 331 197738 344,18 23217 44 968 2487 Ohio....... 7041 272579 9201 66265 21.2 3323 5663 Indiana........ 233 148806 3076 20590 89 627 2257 Illinois.. 782 105337 2506 13185 63 310 2021 Michignan 40 56521 728 6890 24 166 904 Wisconsin 794 7047 479 1814 14 209 259 Iowa.......... 217 10469 355 1629 131 78 365 Northw.States...................................... 085242 Total....... 1521113719951t 117607 791749 56021. 33076 65255 EXAMPLES. 1. Fill all the blanks in the above table. 2. Find the average number emlployed in each occupation, in each of the New England States; in each of the Middle States; in each of the Southern States; in each of the South-western States; in each of the North-western States; in the entire Union. a Tucker's Progress.

Page  269 ~80.] EDUCATION. 269 S9o TABLE SI-IOWING THE STATE OF EDUCATION IN THE UNITED STATES, AND THE NUMBER OF WTI-IITE PERSONS OVER 20 YEARS OF AGE WHO COULD NOT READ OR WRITE, ACCORDING TO THE CENSUS OF 1840.a States and Territories. fO tY _ o Maine............. 4 266 86 8477 3385 16'1477 60212 3241 New H ampshire.... 2 433 68 5799 2127 83632 7715 942 Vermont........ 3 233 46 4113 2402 82817 14701 2276 Massachusetts..... 4 769 251 167,16 3362 160257 158351 4448 Rhode Island...... 2 324 52 3664 434 17355 10749 1614 Connecticut....... 4 -832 127 4865 1619 65739 10912 526 New Engsland States Neow ork......... 12 1285 505 3471.5 10593 502367 27075 41452 New Jersey........ 3 443 66 3027 1207 52583 7128 6385 Pennsylvania....... 20 2034 29() 15970 4978 179I989 73908 33940 Delaware.......... 1 23 20 764 152 6924 1571 4832 Maryland.......... 12 813 133 4'289 565 16 l51 6624 11817 District of Columbia 2 224 26 1389 29 851 482 1033 Middle States...... Virginia.......... 13 1097 382 110833 1561 35331 9791 58787 North Carolina..... 2 158 1il 41398 632 14937 124 56609 South Carolina..... 1 168 117 4326 566 12520 3524 20615 Georgia...........1 622 176 7878 601 15561 1333 30717 Florida..................... 18 732 51. 925 14 1303 Southern States.... Alabama........... 2 152 ml 5018 639 16243 3213 22592 Mississippi......... 7 454 71 2553 382 8236 107 8360 Louisiana.......... 12 989 52 1995 179 3573 1190 4861 Arkansas............. 8 300 113 261.4........ 6567 Tennessee......... 8 402 152 5539 983 25090 6907 58531 South-west'n States Missouri........... 6 495 4,7 1.926 612 J6788 526 19457 IKentucky.......... 10 1419 1.16 4906 952 24641 429 40018 Ohio............... 18 1717 73 4310 5186 218609 51812 35394 Indiana............ 4 322 54 2946 1521 481.89 6929 38101( Illinois............ 5 311 42 1967 1241 34876 1683 27502 Michigan.......... 5 158 12 485 975 29701 998 2173 Wisconsin................ 2 65 77 1937 315 1701 Iowa......................... 1 25 63 1500........ 1118 North-west'n States Total............. EXAMPLE. 1. Fill the blanks in the above table, and find the percentage of the entire populationb embraced in each class, in each section of the Union. a Tucker's Progress. b See Table of Population, page 36.

Page  270 270 STATISTICS. [ART. XVI. SUMMJARY OF THE ANNUAL PRODUCTS OF INDUSTRY IN THE SEVERAL STATES, AS ESTIMATED IN THE CENSUS OF 1840.a VALUE OF ANNUAL PRODUCTS FROM States anl lerrsitories. Agriculture. tuIafes. Commerce. Mising. Forest. Fisheries. Total. Ma.inse.. $1.5856270 $656153303 $1505380 $327376 $1877663 $1280713 _ N.Hamp. 11377752 6545811 1001533 88373 440801 02S11 Vermont 17879155 5685425 758S899 389488 430224......... Mass.... 16065627 4351.8057 7004691 2020572 377354 6483996 R. Islalnd 2199309 S640626 1.294956 1624110 44610 659312 Coanec't 113 1776 12778963 1963281 820410 181575 907723 N. E. S. 74740889............ -................... 1 17..... 187657294 NI.Yorlk.. 1082 752S1 47454514 24311.715 7408070 504()781 1310072 N. Jeo rsey 1209853 10696257 1206929 1073921 3161326 124140 Peunshl.. 681809241 33354 279 10593368 17666146( 1203578 35360 DI)el w'o. 3109-140 1538879 266257 54555 131.19 181285 [Maryland] 175a86720 6212677 3199087 1056210 241194 225773 D. of' Col. l176942 900526 802725............. 87400.iddlle S.......................................... 390558, 03 Vilroiit. 590(85821 83-19218 52099451 33216290 617760 95173 N. Caro'a 29 7 58311 2053697 13222841 3724S6 1446108 251792 [S. Caro'a 21.553691 2248915 2632421 187608 549626 1275 Georgia.. 31.468271 1953950 2248488 191.631 117439 584 1 lo id 18314237 434544 464637 2700 27350 213219'S. S.tLtes.............................................. 175321836 Al aolma 2l4896513 1732770 2273267 S1310 1774G65........ Mi ssissi'26: 94565 1585790 1453686........ 205297......... Louisi'a. 22S51375 4087655 7868898 165280 71751......... tArkas's. 5086757 1145309 420635 18225 217469........ Teocoss.. 31660180 2477103 223947 1371331 2251791......... IS. W. S................... I........'............1..... 138607378'issouoroi. 10-184263 236070S 234-925 187669 448559........ Ke atour y. 292265451 5092353 2580575 15399t 9 184799.........,Ohio.... 37802001 14588091 8050316 2442682 1013063 10525 Indianaa.. 1.7247743 3676705 18661.55 660836 80(00 1192 Illiois.. 13701466 3213981 1493125 2932721 4981.......0.. 1 1Iic ligan 41502889 1376249 6298292 56790 467540......... Wiscon. 5(68105 304692 189957 384603 4305S0 27663 loM.,. 69295 170087 136525 13250 309......... N. W S.............170080925 - Tot l..............................-..-........... 10:313 7 j E XAMPLE. 1. Fill all the blanks in the above table, and find what per centage of the total products was derived from each source. Find also the proportion to each inhabitant of the total products of industry in each State. a Tucker's Progress.

Page  271 ~ 82.] PERMUTATION. 271 XVII. PERMUTATION AND COMBINATION. 8?~. PERMUTATION. PERMUTATION shows the number of changes that can be made in the order of a given number of things. PROBLEM I. To find the number of changes that can be made of any given number of things, all different from each other. HIow many changes may be made in the position of 4 persons at table? If there were but two persons, a and 6, they could sit in but two positions, ab and be. If there were three, the third could sit at the head, in the middle, or at the foot, in each of the two changes, and there could then be 1 X 2 X 3 = 6 changes. If there were 4, the fourth could sit as the 1st, 2d, 3d, or 4th, in each of these 6 changes, and there would then be 1 X 2 X 3 X 4=24 changes. RULE. Multiply together the series of numbers 1, 2, 3, &c., up to the given number, and the product will be the number sought. 1. How many variations may be made in the order of the 9 digits? Ans. 362880. 2. How many changes may be made in the position of the letters of the alphabet? Ans. 403291461126605635584000000. 3. How long a time will be required for 8 persons to seat themselyes at table in every possible order, if they eat 3 meals a day? PROBLEM II. Any number of different things being given, to find how mlany changes can be made out of them by taking a given number of the things at a time. If we have five things, each one of the 5 may be placed before each of the others, and we thus have 5 X 4 permutations of 2 out

Page  272 272 PERMUTATION AND COMBINATION. [ART. XVII of 5. If we take 3 at a time, the third thing may be placed as 1st, 2d, and 3d, in each of these permutations, and we have 5 X 4 X 3 permutations of 3 out of 5. For a similar reason we have 5 X 4 X I X 2 permutations of 4 out of 5, &c. RULE. Take a series of numbers, commencing with the number of things given, and decreasing by 1, until the number of terms is equal to the number of things to be taken at a time. The product of all the terms will be the answer required. 4. How many changes can be rung with 8 bells, taking 5 at a time? Ans. 6720. 5. H-ow many numbers of 4 different figures each, can be expressed by the 9 digits? 6. In how many different ways may 10 letters of the alphabet be arranged? Anis. 19275223968000. PROBLEM III. To find the number of permutations in any given number of things, among which there are several of a kind. How many permutations can be made of the letters in the word terrier? If the letters were all different, the permutations, according to Problem I., would be 1 X 2 X 3 X 4 X 5 X 6 X 7 =5040. But the permutations of the three r's would, if they were all different, be 1 X 2 X 3, which could be combined with each of the other changes; the number must therefore be divided by 1 X 2 X 3. For the same reason, it must also be divided by 1 X 2, on account of the 2 e's. Then the true number sought is 1X2X 3X< 4X5 X 7 -~ 420. 1X2 X3X1X2 RULE. Take the natural series, from 1 up to the number of things of the first kind, and the same series up to the number'of things of each succeeding kind, and form the continued product of all the series. By the continued product, divide the number of permutations

Page  273 ~ 83.] COMBINATION. 273 of which the given things would be capable, if they were all different, and the quotient will be the number sought. 7. How many changes can be made in the order of the letters, in the word Philadelphia? Ans. 14968800. 8. How many different numbers can be made, that will employ all the figures in the number 119089907343? 9. How many permutations can be made with the letters in the word Cincinnati? Ans. 201600. 3. COMBINATION. COMBINATION shows in how many ways a less number of things may be chosen from a greater. If we have ten articles, each may be combined with every one of the nine remaining ones, and therefore we may have 10 X 9 permutations of 2 out of 10. But each combination will evidently be repeated; thus, we have ab and ba, ac and ca, &c. Therefore, the number of combinations will be 10 X 9 2 If now we add an eleventh article, each of the eleven may be joined to each of the combinations of the remaining ten, and we shall have 11 X 10 X 9 permutations. But each combination will 1X2 be three times repeated; thus we shall have abc, bac, and cab; abd, bad, and dab, &c. The number of combinations of 3 out of 11 will therefore be 1i X 10 X 9 Hence we obtain the following RULE. Write for a numerator the descending series, commencing with the number from which the combinations are to be made, and for a denominator the ascending series, commencing with 1, giving to each series as many terms as are equivalent to the number in one combination. Cancel the like factors in the numerator and denominator, and divide. 10. How many combinations of 4 letters, can be made from the alphabet? Ans. 14950. 18

Page  274 274 INVOLUTION AND EVOLUTION. [ART. XVIII. 11. How many combinations of 7 can be made from 18 apples? Ans. 31824. 12. How many ranks of 10 men, may be made in a company of 80? 13. How many locks of different wards, may be unlocked with a key of 6 wards? [Find the number of combinations of 1I 2, 3, 4, 5, and 6 in 6, and the sum of all the combinations will be the number required.] Alns. 63. XVIII. INVOLUTION AND EVOLUTION. S 4. INVOLUTION. INVOLUTION is the repeated multiplication of a number by itself. The product obtained by Involution is called a power. The root is the number involved, or the first power. If the root be multiplied by itself, or employed twice as a factor, the product is the second power. If the root is employed three times as a factor, it is raised to the 3d power; if 5 times, to the 5th power, &c. Thus, 2 is the 1st power of 2 or 2'. 2 x 2or 4,is the2dpower of 2, or 22. 2 x 2 X 2 or 8, is the 3d power of 2, or 23. 2x2x2X2X2 or 32, is the 5th power of 2, or 25. The power is usually denoted by a small figure over the right of the root, called the expo. nent, or in4dex. When there is no exponent, the number is regarded as the 1st power. The second power is often called the square, because the number of square feet in any square surface, is obtained by multiplying the number of feet in one side by itself. The third power is often called the cube, because the number of cubic feet in any cubical block, may be obtained by raising the number of feet in one side to the 3d power.

Page  275 ~ 84.] INVOLUTION. 275 The 4th power is sometimes called the bi-quadrate, or the square squared; the 5th power, the first sursolid; the 6th power, the square cubed, or the cube squared; the 7th power, the second sursolid; the 8th power, the bi-quadrate squared; the 9th power, the cube cubed; the 10th power, the 1st sursolid squared, &c. If the exponents of any two powers of the same number be added, we shall obtain the exponent of their product. Thus 63 x 65 = 6 X 6 x 6 x 6 x 6 x 6 6668; 42x 43-4 x 4 x 4x 4 X 4= 45. In any two powers of the same number, if we subtract the smaller exponent from the larger, we shall obtain the exponent of their quotient. Thus 68 65 = 6x6x6x6x6x6x6x x 6x6 6 6X 6= 6 6 x 66 x 6x 6 = 63. 6x6x6x6x6 We may represent any power of a number by multiplying its exponent. Thus, the 7th power of 5 is 5'; the 3d power of 2' is 26, because 22 X 22 X 22 = 26. These properties form the basis of the system of Logarithms. 1. What is the 2d power of 6? the 3d power? 2. Find the value of.94; 123; (~)5; 29. 3. Find the value of 164; 1.64;.164; (1)3. 4. What is the square of 13.68? of 9? 5. What is the difference between 34 and 43? 6. What is the value of 1"7; 37; 23 x 22? 7. What power of 9 is equivalent to 95 X 93; 92 X 910~ 94x 96; 9 X 97 x 98? 8. Multiply 1279 by 1277, and divide the product by 127's. 9. Divide 319 by 319; 178 by 175; 427 by 426. 10. What is the sixth power of 41? P~o ~ uctu113 UL~ ~aUI1 VW~~ + U

Page  276 276' INVOLUTION AND EVOLUTION. [ART. XVIII. 11. What is the 9th power of 53? the 12th power of 185? the 24th power of 172? 85. EVOLUTION. EVOLUTION iS the process by which we discover the root of any given power. Thus, 3 is the 2d root of 9, the 3d root of 27, the 5th root of 243, because 9 = 32; 27 = 33; 243 = 35. So the 2d or square root of 49 is 7; the 3d or cube root of 125 is 5; the 4th root of 16 is 2; the 5th root of 1024 is 4, &c. We may denote a root by a radical sign, or by afractional exponent. The radical sign is V/, and when employed by itself denotes the square root. If we wish to denote the 3d, 5th, 7th, &c. root, the index of the root is written above the radical sign, thus, V-, 5-, &c. In fractional exponents, the numerator expresses the power of the number, and the denominator expresses the root. Thus, (27)- = -_ 3 4 _ /27; (16)4 -= 4163; (32)5 = 324, &c. The product, or the quotient, of two second, third, or other roots, is the 2d, 3d, &c., root of the product or quotient. Thus, V27 x V/125 = 3/27 X 125 or ~/3375. For 27 = 33= 3 X 3 3, and 125=5 x 5 X 5. Then 27 X 125 = 3 x 3 x 3 X 5 x 5 x 5 =3 X 5 3 x 5 X 3 x 5 =153. Therefore V27 x 125 =15. In a similar manner it may be shown that 3375 -- /125 = V27. The power of any root may be obtained by multiplying the fractional exponent. Thus, the 4th power of 273= 27-. For by the last proposition, V/272 X V/27'2 X /272 X 7 =272 = /278-27-. The root of any power or root may be obtained by di. viding the exponent by the index of the desired root. Thus, 3 3 1 f/35=3} + =3T.

Page  277 ~ 85.] EVOLUTION. 277 This is the converse of the last proposition. For if the 1 3 1 1 3d power of 31 is 315, or 35, the 3d root of 3-5 must be 1 315 If the numerator and denominator of fractional indices be multiplied or divided by the same number, the value of 4 8 2 the quantity is not altered. Thus, 3" 6 3 1 = 33. For the multiplication of the numerator involves the number to a certain power, and the multiplication of the denominator extracts the corresponding root. Then the 3d root of the 3d power, the 5th root of the 5th power, &c., is the Ist power. We may multiply or divide any two roots of the same number, by adding or subtracting the fractional exponents. Thus, /2 x V 23 -+ = 2; v 5 -/ 5-3 4_ 512. For by the last proposition we have V/2 x /2 = /22-x V/23, which is equivalent to I/2- or 23. Also I/5 * t 5 ='~/ 54 1 53= V 5 or 51'. When the exact root of a number can be obtained, it is called a rational number. An irrational number, or surd, is one whose exact root cannot be obtained. Thus, V/16, V 27, V 64, /81, are rational numbers, equivalent to 4, 3, 4, 3, respectively. But /5, V19,?/16, are allsurds, and their roots can only be obtained approximately. A number which has a rational root, is called a perfect power. Thus, 16 is a perfect 2d power, and a perfect 4th power, but an imperfect power of any other degree. But 5, 7, 12, &c., are imperfect powers of any degree. 1. What is the square root of 9? the cube root of 8? 2. What is the 4th root of 81? the 5th root of 32? 3. What is the value of 9/1; 252; V/ 64; 64-1-u?

Page  278 278 INVOLUTION AND EVOLUTION. [ART. XVIII. 4. What is the product of /8 by'12; of /9 by 74? 5. Multiply /3 by / 3; 7 by /-7; 43 by'/42. 1/ 1 _-1 6. Divide 63 by 65; /5 by 4/Z; 17-3 by V/17. 4 7. Find the 4th power of V/9; the 6th power of 85-. 8. What is the cube root of 76; the 5th root of 11? SO6. ROOTS OF ALL PowERS.a When the exponent of a power can be resolved into two or more factors, by successively extracting the roots denoted by those factors, we may obtain the root desired. Thus, as 12 - 3 X 2 X 2, the cube root of the square root of the square root of a number, is equal to the 12th root. So the square root of the square root is the 4th root; the cube root of the cube root is the 9th root; the cube root of the square root is the 6th root, &c. The following rule will, however, be generally found more convenient for determining the roots of all powers greater than the cube. a It-does not seem necessary to give the ordinary rules for the extraction of the square and cube roots, as the pupil is supposed to be already familiar with them. But the following formulas will probably be found useful, in explaining the usual method of finding the trial and complete divisors. The square of any number consisting of tens and units The square of the tens + (2 X the tens + the units) X the units. The cube of any number consisting of tens and units The cube of the tens -i3 X the square of the tens+ ) 3 X the tens X the units + X the units. the square of the units. The entire portion of the root which has been found, at any step, may be considered as the tens, and the next root figure will then represent the units.

Page  279 ~ 86.] ROOTS OF ALL POWERS. 279 GENERAL RULE. At the left of the number whose root is required, arrange as many columns as are equal to the index of the root, writing I at the head of the first or left hand column, and zero at the head of each of the others. Divide the number into periods of as many figures as the index of the root requires. Write the root of the left hand period as the first figure of the true root. Multiply the number in the first column by the root figure, and add the product to the second column; add the product of this sum by the root figure to the third column, and so proceed, subtracting the product of the last column from the given number. Repeat this process, stopping at the last column, and thus proceed, stopping one column sooner each time, until the last sum falls in the second column. To determine the second root figure, consider the number in the last column as a trial divisor, and proceed with the second root figure thus obtained,a precisely as with the first. Continue the operation until the root is completed, or the approximation carried as far as is desired. In order to avoid error, observe carefully the value of each root figure and each product. Thus, if the first root figure is hundreds, the number in the second column will be hundreds,-in the third, ten thousands,-in the fourth, millions, &c. EXAMPLES FOR ILLUSTRATION. 1. What is the third root of 205692449327? a If the root figure thus found proves too large, erase it and try a smaller number.

Page  280 280 INVOLUTION AND EVOLUTION. [ART. XVIII. 1 0 0 205692448927s(5903 5 thous. 25 mill. 125 bill. 5 25 cd..806,92 5 50 80379 mill. 10 (1) 75 t. d. 3134,49327 5 1431 ten thous. 313449827 un. (1) 159 hun. 8931 c. d. 9 1512 168 (2) 10443 t. d. 9 53109 un. (2) 17703 un. 104483109 c. d. The complete divisors are marked c. d., the trial divisors, t. d. The figures at which the new additions commence are marked (1), (2). The partial dividends by which each root figure is determined, are distinguished by a comma. They always terminate with the first figure of the period that is annexed. The abbreviations, thous., mill., &c., show the value of the figures against which they are placed. 2. Extract the 5th root of 858533232.56832. 1 0 o 0 0 858533232.56832(61.2 6 tens. 36 hund. 216 thous. 1296 ten thou. 7776 hund. thous. 6 36 216 1296 c. d. 8093,3232 6 72 618 5184 66996301 12 108 864 (1) 6480 t. d. 139369315,6832 6 108 1296 2196301 un. 13936931.56832 18 216 (1) 2160 66996301 c. d. 6 144 36301 un. 2232904 24 (1) 360 2196301 (2) 69229205 t. d. 6 301 un 36603 4554528416 ten thous. (1) 301 un. 36301 22329'304 696846578416 c. d. 1 302 36906 302 36603 (2) 2269810 1 303 7454208 thous. 303 36906 2277264208 1 304 304 (2) 37210 1 6104 hund, (2) 3052 tenths 37271.04 The additions to the left hand column may be made mentally, and thus shorten the labor. There are other abbreviations, for which the student is referred to the Chapter on Approximations.

Page  281 C1GO TABLE OF ROOTS AND POWERS. 1st power, 1 2 _3-. 4.. 7 8 2d power, 1 4 9 16 25 36 49 64 81 3d power, 1 8 27 64 125 216 343 512 729 4th power, 1 16 81 256 625 1296 2401 4096 6561 O 5th power, 1 32 243 1024 3125 7776 16807 32768 59049 k 6th power, 1 64 729 4096 15625 46656 117649 262144 531441 7th power, 1 128 2187 16384 78125 279936 823543 2037152 4782969 8th power, 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 O 9th plower, 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 10th power, 1-1024 59049 1048576 9765625 60466176 282475249 1073741.824 3486784401 11lth power, 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 12th power, 1 4096 531441 167772t6 244140625 2176782336 13841287201 68719476736 282429536481 13th power, 1 8192 1594323 67108864 12207031-25 13060694016 96889010407 549755813888 2541865828329 14th power, 1 16384 4782969 268435456 6103515625 78364164096 678223072849 4398046511104 22876792454961'7 15th power, 1 32768 14348907 1073741824 30517578125 470184984576 4747561509943 35184372088832 205891132094649 eOD COO

Page  282 282 INVOLUTION AND EVOLUTION. [ART. XVIIY. The first root figure in each of the following examples may be found by the table of Powers and Roots. 1. Extract the square root of 350026681. 2. Extract the square root of 3; 5; 6.5. 3. Extract the cube root of 2924207. 4. Extract the cube root of 13; 12.5. 5. Extract the fifth root of 65.7748550151. 6. Extract the 7th root of 1.246688292353624506368. 7. APPLICATION OF THE SQUARE ROOT. The areas of any similar figures are proportioned to the squares of their like dimensions. The area of any circle is equal to the square of its diameter multiplied by.7854. The circumference of a circle is equal to its diameter multiplied by 3.1416.a The area of a triangle is equal to the base multiplied by half the height. In any right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides. The distance through which bodies fall, when falling freely, are as the squares of the times. In a vacuum, a body would fall 161-1ft. in 1 second. Then we have the proportion, letting n represent any number of seconds, sec. sec. ft. ft. (1)2: n2:: 16-1: distance in n seconds. Any three terms of this proportion being given, the fourth may be readily founi. But it should be remarked, that in consequence of the resistance of the air, the space a The more exact ratio is, 3.14159265358979323846264338328.

Page  283 ~ 87.] APPLICATION OF THE SQUARE ROOT. 283 actually fallen through is somewhat less than that given by the formula. If b represents the base of a right-angled triangle, p the perpendicular, and lb the hypothenuse, h= p2q_ b2; b - / h2-p p2 -; = h2 2- b. The square root of the area of any surface will give the side of a square, equal in area to the given surface. EXAMPLES. 1. What is the diameter of a circle that is 16 times as large as one whose diameter is 13 feet? Ans. 52ft. 2. The area of a circle is 7632 feet; what is the diameter? Ans. 98.5ft. 3. A horse is fastened to a post in the centre of a field. What is the length of a rope that will allow him to graze: an acre? Ans. 7.136 rods. 4. A ladder 75 feet long, rests against the trunk of a tree at a point 50 feet from the ground. How far is the foot of the ladder from the root of the tree? An~s. 55.9ft. 5. The length of a room is 18 feet, and the width 12 feet. WVhat is the distance between the opposite corners? What length of rope would reach from an upper corner to, the opposite lower corner, the height being 10 feet? Ans. 21.6ft.; 23.832ft. 6. The circumference of a circle is 29 rods. What is the side of a square having an equal area? Ans. 8.18 rods. 7. Two ships left the same port; one sailed 125 miles north, the other 100 miles east. How far were they then apart? Ans. 160m. 24.96r. 8. A kite accidentally lodged in the top of a tree, but the line breaking, I measure its length, which is 210 feet.

Page  284 284 INVOLUTION AND EVOLUTION. [ART. XVIII. What is the height of the tree, the foot being 189 feet from my standing place? Ans. 91.53ft. 9. Desiring to know the height of a precipice, I drop a stone from the summit, and observe by my watch that it strikes the ground in 32- seconds. What is the height? Ans. 197.02ft. 10. A bag of sand is dropped from a balloon 14 miles above the surface of the earth. How long will it be in falling? Ans. 20.25sec. When one number bears the same ratio to a second as the second does to a third, the second number is called a mnean proportional between the other two. Thus, in the proportion 3: 6:: 6: 12, 6 is a mean proportional between 3 and 12. The mean proportional between andy two numbers is equal t the square rloot of their product. 11. Find a mean proportional between 7 and 252. 12. Find a mean proportional between.75 and 12. 13. Find a mean proportional between A- and -I5. 14. Find a mean proportional between'2o and.875. 15. Find mean proportionals between -1- and 16; 5 and 6; 25 and 13; 2 and A. S S. APPLICATION OF THE CUBE ROOT. The solid contents and the weights of similar bodies are to each other as the cubes of their diameters, or of their similar sides. The solid contents of a sphere may be found by multiplying the cube of the diameter by.5236. The cube root of the solid contents of any body, will give the side of a cube, equal in solidity to the given body. a No allowance is made for resistance of the air, in the answers that are given.

Page  285 ~ 88.] APPLICATION OF THE CUBE ROOT. 285 EXAMPLES. 1. What are the solid contents of the earth, supposing it a perfect sphere, whose diameter is 7920 miles? Ans. 260120860876.8 cubic miles. 2. If a ball 2 inches in diameter, weighs 1~ pounds, what would be the weight of a similar ball 6 inches in diameter? Ans. 40 lb. 3. What is the side of a cubical box that will hold 1 bushel? Ans. 12.907in. 4. What is the side of a cubical pile that contains 256 cords of wood? Ans. 32ft. 5. If a tree 1 foot in diameter, yields 2 cords of wood, how much wood is there in a similar tree that is 3ft. 6in. in diameter? Ans. 85` cords. 6. If a pound avoirdupois of gold is worth $200, and a cubic inch weighs 11~ oz., what would be the value of a gold ball 1 foot in diameter? Ans. $243000. 7. What is the diameter of a ball that weighs 27 times as much as one 3 feet 6 inches in diameter? Ans. 10ft. 6in. 8. If a hollow sphere 3 feet in diameter and 21 inches thick, weighs 12 tons, what would be the dimensions of a similar sphere that would weigh 324 tons? Ans. Diameter 9ft.; thickness 7 inches. 9. What is the side of a cubical block of wood, that weighs as much as a sphere of the same material, 15 inches in diameter? Ans. 12.09 inches. 10. The length of a ship's keel is 70ft., the breadth of beam 25ft., and the depth of the hold 12~ft. Required the dimensions of another vessel, built on the same model, but of twice the tonnage.

Page  286 286 PROGRESSION) OR SERIES. [ART. XIX. 11. If a ship whose keel measures 90 feet, carries 420 tons, what will be the tonnage of a similar vessel with a keel 60ft. long? XIX. PROGRESSION, OR SERIES. 89. ARITHMETICAL AND GEOMETRICAL PROGRESSION. LET a represent the less extreme of a series, 1 the greater extreme, n the number of terms, s the sum of all the terms in an arithmetical series, p the product of all the terms in a geometrical series, d the arithmetical difference, and r the geometrical ratio. Then In Arithmnetical Progression. lt Geometrical Progression. I=a + (n -a) c (1) I = a Xn q (5) (a + (2) P V(a/ )' (6) 2 i =-O-)a =(-(s1) d r_, a=(7) i= a (4) a =-n -r (8) In comparing these tables, we see that addclition corresponds to multiplication; subtraction " " division; multtilication " ifnvolution; division " " evolution. If, therefore, we had a series of numbers bearing the same ratio to the natural series, as an Arithmetical to a Geometrical Progression, the labor of multiplication would be reduced to that of simple addition, and involution to simple multiplication. Such a series constitutes a TABLE OF LOGARITHMS.

Page  287 ~ 89.] ARITHMETICAL PROGRESSION. 287 EXAMPLES. 1. Determine the value of n, when a, d, and 1 are given, by the 2d method of analysis, stated in Section 57. By formula (1) we perceive that if 1 be subtracted from n, the remainder multiplied by d, and a added to the product, the sum will be 1. Reversing the operation, if we subtract a from 1, divide the remainder by d, and add 1 to the quotient, the sum will be n. Ans. n- +., 2. From formula (2) find the value of 1, when a, n, and s are given, and the value of a, when 4 n, and s are given. Ans. 2s 2s 3. Determine the value of n, when the values of a, i, and s are known. Ans 2s Ans. n — 2s 4. From formulas (1) and (2) determine the value of a, when d, t, and s are known. Substituting for I in formula (2), its value in formula (1), we n(2an-1 d) have s = n(2a — -d) 2 Reversing all the operations that must be performed on a to produce this result, we find that a = (2 (n -1) d). 2. 5. From formulas (2) and (3), how may we find the value of 1, when dc, a, and s are known? a 2s Ans. I= (25 J n 1 d) *- 2. a There are two formulas, which the pupil could hardly be expected ro obtain, without considerable knowledge of Algebra. They are, therefore, inserted here, in order that he may have all the formulas that are necessary to solve any question in Arithmetical Progression that can possibly occur. When a, d, and s are given, d - 2a + / (d -2)2 + 8ds 2d

Page  288 288 PROGRESSION, OR SERIES. [ART. XIx. 6. A laborer agreed to dig a well 39 yards deep, for which he was to be paid as follows: 75 cents for the first yard, $1.25 for the second yard, and so on, increasing 50 cents for each subsequent yard. What would the last yard cost, and what would he receive for the whole job? 2d Ans. $399.75. 7. The formula for determining the sum of any geomet1x r —a rical series is s = — 1 Determine from this formula, the value of r when s, a, and 1 are given. s-I 8. When one of the extremes, the ratio, and the sum of the terms are given, how would you find the other extreme? Give separate answers for each extreme. Ans. a= X r —(r —) X s. -=(r -1) s + a r 9. What is the sum of the series 2, 1, ~, +, &c., to infinity? (The last term in any infinite decreasing series is 0.) Ans. 4. 10. If a man commences at 21 years of age, and annually puts $500 at compound interest, how much will he be worth when he is 50 years old? Ans. $36819.90. 11. Insert 2 mean proportionals between I and 343. (As When d, 1, and s are given, 21+ d ~ V/(21 + d)2 - 8ds 2d The sign following d in the second formula, is sometimes +, and sometimes -. The proper sign can easily be determined by trial. a If ( X — a). (r —1) =s, r X s- s -l X r- a. Then r X s=l X r -s- a. Subtracting 1 X r from r X s, we have r s - I X r-s - a. But r X s —r X I =r X (s —I); and therefore, dividing by s - 1, we obtain the answer, r = (s - a) -- (s1). This analysis will be more difficult to follow than any of those required in arithmetical progression; but the pupil should pass nothing over until he understands it perfectly.

Page  289 ~ 90.] IHARMONICAL PROGRESSION. 289 there are to be 2 means, the number of terms is 4, and the extremes 1 and 343.) Ans. 7, 49. 12. Insert 5 mean proportionals between 4 and 2916. Ans. 12,: 36, 108, 324, 972. 13. Every oviparous fish deposits annually,'at the spawning season, many thousands of ova. If we estimate the average number deposited by each pair of herrings to be only 2000, to what number would the offspring of a single pair amount in the eighth year, supposing that every egg produced a fish? Ans. 2 septillion, a number which would make a mass larger than the whole globe. 14. According to some experiments, it has been found that one stem of the hyoscyamus sometimes produces more than 50000 seeds. At this rate, if every seed should produce a fertile plant, what number of plants would be contained in the fourth crop from a! single seed? Ans. 6250 quadrillion, a number that the whole surface of the earth would not be sufficient to contain. 15. If the human race, after making a proper deduction for those who died, had doubled every twenty years, how many of the descendants of Adam would have been living when he was 500 years old? Ans. 33554430. 90. HARMONICAL PROGRESSION.a When three numbers are such that the first is to the third, as the difference between the first and second is to the difference between the second and third, they are said to be in HARMONICAAL PROPORTION; and a series of numbers, in continued harmonical proportion, constitutes a HARMON1CAL PROGRESSION. The reciprocal of a number, is the quotient of 1 by the a So called, because if a musical string be divided in harmonical proportion, the different parts will vibrate in unison. 19

Page  290 290 PROGRESSION, OR SERIES. [ART. XIX, number. Thus, is the reciprocal of 2; 4 is the reciprocal of —; 3 is the reciprocal of, &c. The reciprocals of any equidierent series form a harmonical proportion. I. Two numbers being given, to find a third in harmonical proportion. Consider the reciprocals of the numbers as two terms of an equidifferent series. The thircd term will be the reciprocal of the number sought. Find a third harmonical proportional to 120 and 40. The reciprocals are -i —, and or, or ~-. The third term of the equidifferent series is 5-0 and its reciprocal 24 is the harmonical proportional sought. II. To insert any number of harlmonical means between two numbers. 7Find as many arithmetical means between the reciprocals of the given numbers.; These means will be the reciprocals of the harmonical means. Insert 4 harmonical means between 20 and 120. The reciprocals are - and y1i or y-6 and ~ 2*. The four arithmetical means are l 4 o and~ and ~n whose reciprocals are 24, 30, 40, and 60,-the desired harmonical means. EXAMPLES. 1. The first two terms of a harmonical progression are 60 and 30. Required the ten succeeding terms. 2. The first two terms of a harmonical proportion are 348075 and 69615. Find the six succeeding terms. 3. Insert 6 harmonical means between 630 and 50410 4. Insert 8 harmonical means between 10 and 60. 5. Insert 2 harmonical means between ~ and -. 6. Insert 4 harmonical means between 1 and -.

Page  291 ~ 91.] COMPOUND INTEREST. 291 91. COMPOUND INTEREST. Compound Interest may be computed by Geometrical Progression; a = the amount of $1 for the time that should elapse between two successive payments of interest; r = a; it-n the number of payments. The labor of computing Compound Interest, may be abridged by a table in which the amount of $1 is computed at different rates, and for a number of years. (See Table I., p. 293.) I. To find the amount of any sum by the Table, multiply the given sum by the amount of $1 for the given time. EXAMPLE. What will be the amount, at 7 per cent. compound interest, of $200 for 15yr.? $1 in 15yr. at 7 per cent. amounts to 2.759031, and 2.759031 X $200 = $551.8062. II. To compute compound discount, or to find the present worth at compound interest, of any sum due at a future time, divide the given sum by the amount of $1 for the given time. EXAMPLE.-When money is worth 5 per cent. compound interest, what is the present worth of $5000 due in 19yr. 4mo. 24dy.? $1 at 5 per cent. would amount in 19yr. 4mo. 24dy. to $2.577489, and $5000 - 2.577489 -$1939.87. III. To find the time in which any principal will amount to a given sum, divide the amount by the principal, and look for the guotient in the Table, under the given rate. EXAMPLE.-In what time, at 6 per cent. compound interest, will $25 amount to $48? 4- 1= 1.92. 1 would amount to 1.898299 in 11 years, or to 2.012196 in 12yr. 1.92 exceeds 1.898299 by.021701, and 2.012196 exceeds 1.898299 by.113897. Then if the gain in 12 months is.113897, in what time would there be a gain of.021701?.113897:.021701:: 12mo.: 2mo. 8dy. very nearly. IV. To find the rate at which any principal will amount to a given sum in a given time, divide the amount by the principal, and look for the quotient in the Table, opposite the given time. EXAMPLE.-At what rate of compound interest, will $250 amount to $550 in 18 years?

Page  292 292 PROGRESSION, OR SERIES. [ART. XIX. 5 - =: 2.2. In the line of 18 years, we find 2.2 under 41 per cent. EXAMPLES. 1. Find the amount of $637.25, at 5 per cent. compound interest, for 16yr. 3mo. 15dy. Ans. $1411.32. 2. Allowing 7 per cent. compound interest, what is the present worth of $1000, due in 35yr. 5mo. 6dy.? Ans. $90.91. 3. At 6 per cent. compound interest, in what time will $250 amount to $1000? Ans. 23yr. 9mo. 13dy. 4. At what rate of compound interest will $127.75 amount to $201.22, in 10yr. 3mo. 24dy.? Ans. 4~ per cent. 9 2. ANNUITIES. Any sum of money to be paid regularly, at stated periods, is called an ANNUITY. The payment may be stipulated for a given number of years, in which case it is called an annuity certain, or it may be dependent upon some particular circumstance, as the life of one or more individuals. The latter is called a contingent annuitty. A pepetual annuity, is one which can only be terminated by the grantor, on the payment of a sum whose interest will be equivalent to the annuity. Of this character is the consolidated debt of England. An annuity in possession, is one on which there is a present claim; an annuity in reversion, or deferred annuity, is one that does not commence until the lapse of a stated time, or the occurrence ofsome uncertain event, as the death of an individual. The present.worth of an annuity, is the sum which, at compound interest for the time of its duration, would amount to the sum of all the payments, each being placed at compound interest as it became due.

Page  293 ~92.] ANNUITIES. 293 TABLE I. SHOWING THE AMOUNT OF $1.00, AT COMPOUND INTEREST, tROll 1 YEAR TO 60. Year. 3 p. cent., 3 p. cent. 4 p. cent. 4p. cnt.. cent. 6p.cent. 7 p. cent, 1 1.030000 1.035000 1.040000 1.045000 1.050000 1.060000 1.070000 2 1.060900 1.071225 1.081600 1.092025 1.102500 1.123600 1.144900 3 1.092727 1.108718 1.124864 1.141166 1.157625 1.191016 1.125043 4 1.125509 1.147523 1.169859 1.192519 1.215506 1.262477 1..:10796 5 1.159274 1.187686 1.216653 1.246182 1.276282 1.338226 1.401552 6 1.194052 1.22-255 1.265319 1.302260 1.340096 1.418519 1.5(07.30 7 1.229874 1.272279 1.315932 1.360862 1.407100 1.503630 1.605781 8 1.266770 1.316809 1.368569 1.422101 1.477455 1.593848 1.718186 9 1.304773 1.362897 1.423312 1.486095 1.551328 1.689479'3 1.838459 102 1-.3439116 1.410599 1.480244 1.552969 1.628895 1.790848 1.967151 11 1.384234 1.459970 1.539454 1.622853 1.710339 1.898299 2.104852 12 1.425761 1.511069 1.601032 1'.695881 1.795856 2.012196 2.252192 13 1.468534 1.563956 1.665073 1.772196 1.885649 2'.132928 2.409845 14 1.512590 1.618694 1.731676 1.851945 1.979932 2.260904 2.578534 15 1.557967 1.675349'3 1.800943 1.935282 2.078928 2.396558 2.759031 _ 1.607067......_.... _ _.1...... —_ 16.604 0 1.733986 1.872981 2.022370 2.182875 2.540352 2.952164 1 17 1.652848 1.794675 1.947900 2.113377 2.292018 2.692773 3.158815 18 1.702433 1.857489 2.025816 2.208479 2.406619 2.854339 3.37!1)31 19 1.753506 1.922501 2.106849 2.:307860 2.526950 3.025599 3.61C521 20 1.806111 1.989789 2.191123 2.411714 2.653298 3.207135 3.86: 683 21 1.860295 2.059431 2.278768 2.520241 2.785963 3.399564 4.140o5(;1 22 1.916103 2.131512 2.369(J19/ 2.63:3652 2.925261 3.6035:37 4.4:20.1(30 23 1.973586 2.206114 2.4464715 2.752166 3.071524 3.81!)750 4.746(528 24 2.032794 2.283328 2.5t3304 2.87l6014 3.2251(00 4.0483!35 5.0 I2:t651 25 2.093778 2.363245 2.665836 3.005434 3.386355 4.291871 5.42-l1 i 26 2.156591 2.445959 2.772470 3.140679 3.555673 4.549383 5.807351 27 2.~2128'3 2.5:31567 2.883369 3.282009 3.733456 4.822346 6.2138(i6 28 2.287928 2.620177 2.998703 3.4297 00 3.920129 5.11 1687 ]6.648836 i 29 2.356565 2.711878 3.118651 3.584036 4.116136 5.418388 7.114255 30 2.427262 2.806794 3.243397 3.745318 4.321942.5.743491 7.(i12253 31 2.500080 2.905031 3.373133 3.913857 4.538039 6.088101 8.145110 32 2.575083 3.006708 3.508059' 4.089981 4.764941 6.453:387 8.715268 33 2.652335 3.111942 3.648381 4.274030 5.003188 6.8405910 ).3253373 34 2.731905 3.220860 3.794316 4.466361 5.253348 7.2510)25 9.978110 35 2.813862 3.333590 3.946089 4.667348 5.516015 7.686087 10.G76G578 36 2.898278 3.450266 4.103932 4.877378 5.791816 8.147252 11.42:3)339 37 2.985227 3.57.102.5 4.268090 5.096860 6.081407 8.636087 12.2238i'14 38 3.074783 3.693011 4.438813 5.326219 6.385477 9.1,54252 13.07,i277 39 3.167027 3.82.5372 4.616366 5.565899 6.704751 9.703507 13.194827 1 40 3.262038 3.959260 4.801021 5.816364 7.039989 10.285718 14.!)71465 41 3.359899 4.097834 4.993061 6.078101 7.391988 10.902861 16.022677 42 3.460696 4.241258 5.192784 6.351615 7.761587' 11.5570:33 17.144265 43 3.564517 4.38t3702 5.400495 6.637438 8.149667 12.250455 18.344:363| 44 3.671452 4.543342 5.616515 6.936123 8.557150 12.985482 19.6284(69 45 3.781596 4.702358 5.841176 7.248248 8.985008 13.764611 21.002461 4 3.895044 4.866941 6.074823 7.574420 9.434258 14.590487 22.472634 47 4.011895 5.037284 6.317816( 7.915268 9.905971 15.4653(17 24.045718 48 4.1322252 5.213589 6.570528 8.271455 10.401267 16.393872 25.728918 49 4 | 56429! 5.3)G(6065 6.83:349 8.643(71 10.92133 17.377504 27.529943 50 1 4.383906 5.58)3927 7.106683 9.032636 11.467400 18.420154 29.415703.9j

Page  294 294 PROGRESSION3 OR SERIES. LART. XIX. TABLE II. THE AMIOUNT OF AN ANNUITY OF $1.00, FROM 1 YEAR TO 50. Year. 3 p. cent. 3L p. cent. 4 p. cent. 4Q p. cent. 5 p. cent. 5 p. cent. 6 p. cent. 1 1.00 1.00 00000 1. ooo000000 1.000000.000000 2 2.o00000 2.035000 2.040000 2.045000 2.050000 2.055000 2.060000 3 3.030900 3.1065325 3.121600 3.137025 3.152500 3.168025 3.183600 4 4.1;83627 4.214543 4.246464 4.278191 4.310125 4.342266 4.3746161 5 5.30913 5.362466 5.416322 5.47'0710 5.525631 5.5810911 5.637093 6 6.468410 6.550152 6.632975 6.716892 6.801913 6.888051 6.975319 7 7.662462 7.7739408 7.898294 8.019152 8.142008 8.266894 8.393838 8 8.892336 9.051687 9.214226 9.380014 9.549109 9.721573 9.897468 9 10.159106 10.368496 10.582795 10.802114 11.026564 11.25625'3 11.491316 10 11.463879 11.731393 12.006107 12.288210 12.577893 12.8753541 13.18079f5 11 12.807796 13.141992 13.486351 13.841179 14.206787 14.583498 14.971643 12 14.192029 14.601962 15.025805 15.464032 15.917127 16.385500 16.869941 13 15.617790 16.113030 16.626838 17.159913 17.712983 18.286798 18.882138 14 17.086324 17.676986 18.291911 18.932109 19.598632 20.292572 21.015066 15 18.598914 19.295681 20.023588 20.784054 21.578564 22.408663 23.275971 16 20.156881 20.971030 21.824531 22.719337 23.657492 24.641139 25.6725281 17 21.761588 22.705016 23.697512 24.741707 25.840366 26.9'36402 28.212880, 18 23.414436 24.499691 25.645413 26.855084 28.132385 29.481205 30.905653 19 25.116868 26.357180 27.671229) 29.063562 30.539604 32.102671 33.759992i 20 26.870374 28.279682 2'.778078 31.371423 33.065954 34.868318 36.785592' 21 128.676486 30.269471 31.969202 33.783137 35.719252 37.78G6075 39.992727 22 30.536780 32.328902 34.247970 36.303378 38.505214 40.864309 43.392290) 23 32.452884 34.460414 36.617888 38.9370:30 41.430475 44.111846( 46.9958281 24 34.426470. 36.666528 39.082604 41.689196 44.501999 47.537998 50.815577 225 36.459264 38.949857 41.645908 44.565210 47.720(39 51.552588 54.864512 26 38.553042 41.313102 44.311745 47.570645 51.113454 54.965979 59156383 27 40.709634 43.759060 47.084214 50.711324 54.669126 58.989109 63.7057(6 28 42.950923 46.290627 49.967583 53.993333 58.402583 63.233510 68.528116 29 45.218850 48.910799 52.966286 57.423033 62.352712 67.711353 73.63798 30 47.575416 51.622677.56.084938 61.007070 66.438847 72.435478 79.058186 31 50.002678 54.429471 59.328335 64.752388 70.760790 77.419429 84.801677] 32 52.502759 57.334502 62.701469 68.6662451 75.298829! 82.677498 Y0.889778 3:3 55.077841 60.341210| 66.205527 72.756226[ 80.063770 88.2247601 97.343165 34 57.730177 63.453152 69.857909 77.0302561 85.066959 94.077122 104.183755 35 60.462082 66.674013 73.652225 81.496618 90.320307 100.251363 111.434780 36 63.275944 70.007603 77.598314 86.163966 95.836323 106.765188 119.120867 37 66.174223 73.457869 81.702246| 91.041344 101.628139 113.6372745 127.268119 38 69.159449 77.0288951 85.970336 96.138205 107.709546 120.8873241 35.9!04206 39 72.23423'3 80.724906 90.409150 101.464424 114.095023 128.5361271 145.0584581 410 75.401260 84.550278| 95.0125516 107.003023 1520.799774 136.605614 154.761966 41 78.6632 88.509537 99.826536 112.846688 127.839763 145.1189236 165.047684 42 82.023196 92.6073711104.819598 118.924789 135.231751 154.100464 175.950545 43 85.483892 96.848629 110.012382 125.2764041 142.9933339 163.57589! 187.5075771 44 89.048409 101.238331 115.412877 131.913842 151.143006 173.5726691 199.758032 45 92.719861 105.781673 121.029392 138.849965 155.700156 184.119165212.743514 46 96.501457 110.484031 126.870568 146.098214 168.685164 195.245720 226.508125 47 100.396501 115.350973 132.945390 153.672633 178.119422 206.984234 241.098612 48 104.408396 120.388257 139.263206i 161.587902 188.0'5193! 219.368367 256.564529, 49 108.540648 125.60146, 145.83:3734 lG6.8593571 188.42663i3 232.4353627, 272.95840 1! | 501 112.796867| 130.997910] 152.6670384 178.5030281 209.:3479?!'(i! 246.217477; 12;0.335t!05[

Page  295 ~ 92.] ANNUITIES. 295 TABLE III. THHE PRESENT WORTI- OF AN ANNUITY OF $1.00, FROMi 1 YEAR TO 50. Yea.r./ 3 P. ce ent. 4 p. cent..4 p. cent. 6 p. cent. Year. 1 0.97087 0.96618 0.96154 0.95694 0.95238 0.94786 0.94339 1 2 1.931347 1.89969 1.88609 1.87267 1.85941 1.84631 1.83339 2 3 2.82861 2.80164 2.77509 2.74896 2.72325 2.69793 2.67301 3 4 3.71710 3.67308 3.62990 3.58753 3.54595 3.50514 3.46511 4 5 4.57971 4.51505 4.451821 4.38998 4.32918 4.27028 4.'21236 5 6 5.41719 5.32855 5.24214 5.15787 5.07569 4.99553 4.91732 6 7 6.23028 6.11454 6.00205 5.89270 5.78637 5.68297 5.58238 7 8 7.01969 6.87396 6.73274 6.59589 6.46321 6.33457 6.20979 8 9 7.78611 7.60769 7.43533 7.26879 7'.10782 6.95220 6.80169 9 10 8.53020 8.31661 8.11090 7.91272 7.72173 7.53762 7.36009 10 11 9.25262 9.00155 8.76048 8.52892 8.30641 8.09254 7.88687 11 12 9.95400 9.66333 9.38507 9.11858 8.86325 8.61852 8.38384 12 13 10.63495 10.30274 9.985651 9.68285 9.39357 9.11708 8.852681 13 14 lt.29607 10.92053 10.56312 10.22283 9.898641 9.58965 9.29498 14 15 11.93794 11.51741 11.11839 10.73955 10.37966 10.03 59 9.71225 15 16 12.56110l 12.09412 11.65230 11.23401 10.83777 10.46216 10.10589 16 17 13.16612 12.65132 12.16567 11.70719 11.27407 10.86461 10.47726 17 18 13.75351 13.18968 12.65930 12.15999 11.68959 11.24607 10).82760 18 19 14.32380 13.-70984 13.13394 12.5.329 12.08532 11.60765 11.15812 19 20 14.87747 14.21240 13.59033 13.00794 12.46221 11.95034 11.46992 20 21 15.41502 11.69797 14.02916 13.40472 12'82115 12.27524 11.76408 21 ~22 15.93692( 15.16712 14.45112 13.78442 13.16300 12.58317 12.04158 22 23 G16.44361 15.62041 14.85684 14.14777 13.48857 12.87504 12.30338 23 24 16.93554 16.05337 15.24696 14.49548 13.79864 13.15170 12.55036 24 25 17.41315 16.48151 15.62208 14.82821 14.09394 13.41391 12.78336 25 26/ 17-87684 16.89035 15.98277 15.14661 14.37518 13.66250 13.00317 26 27 18.327031 17.285336 16.329591 15.45130 14.64303 13.89810 13.21053 27 28 18.76411 17.66702| 16.663081 15.74287 14.89813 14.12142 13.40616 28 29 19.188453 18.03577| 16.98371 16.02189 15,14107 14.33310 13.59072 29 30 19.60044| 18.392053 17.29203 16.28889 15.37245 14.53375 13.7648:3 30 31 20.00043 1.8.73628 17.58849 16.54439 15.59281 14.72393 13.92909 31 321 20.388771 19.06887 17.87355 16.78889 15.80268 14.90420 14.08404 32 3:3 20.765791 19.39021 18.14765 17.02286 16.00255 15.075077 14.23023[ 33 34 21.13184 19.70068 18.411201 17.246761 16.19290 15.237031 14.368141 35 2.48722 20.00066 18.66461| 17.46101 16.37419 15.390553 14.49825 35.3(6 21.83225| 20.290491 18.90828 17.66604 16.54685 15.53607 14.62099 36 37 22.16724[ 20.57053 19.14258 17.86224 16.71129) 15.674001 14.7:3678 37 38 22.49246 20.84109 19.36786 18.04999 16.86789 15.80474 14.84602 38 3! 22.80822 21.10250 19-58448| 18.2291i5 17.01704 15.92866i( 14.94'07 39 40 23.11477 21.35507 19.79277 18.40158 17.15909 16.04612 15.04630 40 41 123.41241 21.599101 19.99305 18.56611 17.29437 16.15746 15.13802 41 42 2:3.70136 21.83488 20.18563 18.72355 17.42321 16.2299 15.2;2454 42 43 23.98190 22.06269 20.37079 18.87421 17.545!91 16.36303 15.30617 43 44 24.25427 22.28279 20.54884 19.01838 17.66277 16.45785 15.38318 44 45 24.51871 22.49545 20.720041 19.1535 6 17.77407 16.54772 15.45 3 45 46 24.77515 22.70092 20.88465 19.28837 17.88007 16.63283 15'2437 46 47 25.02471 22.89943 21.04294 19.41471 17.98102 16.71357 15.58903 47 48 25.2166711 23.09124 21.19513 1,.5:3561 18.07716 19.79011 15.65003 48 49 25.50166 23.27656 21.34147 19.65130 18,.687P2 16.8626li 15.70757 49 50 25.729761 23.45562 21.48218 19.762011 18.255931 16.93143 15.6186 50

Page  296 296 PRoGRESSION, oR SERIES. [ART. XIX. Nearly all the questions that occur in annuities, can be easily solved by the first method of Analysis, given in Section 57, p. 190. This will be readily seen, by comparing the following RULES. 1. To find the amount due on an annuity thaft has remained unpaid for a given time. Multiply the amount of an annuity of ONE dollar for the given time,a by the NUMBER of dollars in the given annuity. 2. To find the present worth of an annuity certain. Multiply the present worth of an annuity of ONE dollar for the given time,b by the NUMBER of dollars in the given annuity. 3. To find the present worth of a c.-erpetual annuity. Divide the annuity by the interest that ONE dollar would yield, in the time that elapses between the several payments of the annuity. 4. To find the annuity, whden tle present worth, or thle amount is given. If the present worth is given, divide by the present worth of an annuity of ONE dollar for the time the annuity is to continue. If the amount is given, divide by the amount of an annuity of ONE dollar. 5. To find the present worth of an annuity in reversion. Multiply the present worth in reversion, of an annuity a This amount may be taken from Table II., p. 294, or it may be determined by finding the sum of a geometrical progression, (Ex. 7, sect. 89,) in which a- =the annuity, r- the amount of $1 for the time that should elapse between two successive payments, and n — the number of payments due. b The present worth may be taken from Table III. p. 295, or it may be found by dividing the amount of an annuity of $1 for the given time, by the amount of $1 at compound interest, for the same time.

Page  297 ~ 92.] ANNUITIES. 297 of ONE dollar,a by the NUMBER Of dollars in the given annuity. EXAMPLES. 1. An estate that yields an annual income of $2000, is offered for sale for the amount of 10 years' income at 6 per cent. compound interest. What is the price of the estate? Anqs. $26361.59. 2. A gentleman wishes to present his estate to his children, reserving enough to yield $700 per annum for 15 years. How much must he reserve, allowing 5 per cent. compound interest? Anzs. $7265.76. 3. For how much should an estate that rents for $175 per year, be sold, to allow the purchaser 6 per cent. interest on his investment? Ans. $29163-. 4. What sum of money must a man lay up annually, to amount to $10000 in 20 years, the investments being all made at 6 per cent. compound interest? Ans. $271.85. 5. A father leaves an annual rent of $400 to his eldest child for 5 years, and the reversion of it for the 8 succeeding years to his youngest child. What is the present worth of each legacy, at 6 per cent.? Ans. $1684.94; $1856.13. 6. If a person saves $250 per annum, and invests it at 6 per cent. compound interest, how much will he be worth at the end of 25 years? Ans. $13716.13. 7. What sum invested at 6 per cent. compound interest, will yield me an income of $1600 per annum for 25 years? Ans. $20453.38. a This value may be found from Table III., p. 295, by subtracting the present worth of an annuity continuing until the reversion comnmences, from the present worth of' an annuity, continuing until the reversion terminates. Thus, the present worth of an annuity of $1, to commence in 3 years, and continue 8 years, computing interest at 6 per cent., is $7.88687 - $2.67301 = $5.21386.

Page  298 298 PRocGRESSIOi OR SERIES. [ART. XIX. 8. What sum will build a wall worth $1000, and renew it every 15 years, at 5 per cent. compound interest? N. B. At 5 per cent. compound interest, $1 in 15 years will amount to $2.078928. The interest is therefore $1.078928. The amount necessary to renew the wall, is 51000 — 1.078928, to which must be added the $1000 expended for building the wall at first. Ans. $1926.85. 9. A builder takes a lease of a lot of ground for 25 years, and erects buildings on it which cost him $20000. Allowing money to be worth 6 per cent. compound interest, what cleara annual rent must he receive from the buildings to reimburse his expenditure, at the termination of the lease,the rent commencing one year after the lease is given? Ans. $1593.58. 10. What sum must be paid, allowing 6 per cent. conmpound interest, to extend a lease 7 years,-the clear annuz-al rent being $500, and the lease having 4 years to run? Anzs. $2210.88. 11. What is the amount of a pension of $400 a year, payable semi-annually, for 3 years and 6 months, at 7 per cent. per annum? Ans. $1555.88. 12. What is the par value of an annual income of ~500 in the 4 per cent. consols?b Ans. ~12500. 13. A railroad has been constructed through a farm, in consequence of which, the owner of the estate is obliged to expend $400 in fencing, that must be renewed at the expiration of every 12 years. WVhat sum should he now receive, to compensate him for the required expenditure, money being worth 6 per cent. compound interest? Ans. $795.18. a The clear annual rent, is the amount received after deducting ground-rent, taxes, and other expenses. I CoNsoLs is an abbreviation for the consolidated annuities of the British National Debt.

Page  299 ~ 92.] A.N NUITIES. 299 14. The executors of an estate wish to dispose of an unexpired lease that has 8 years to run, for a premium of $1500. What amount must be added to the annual rent, for that purpose? Ans. q$241. 55. 15. What is the present worth of a reversion of $700 per annum, to commence in 20 years, and continue 30 years thereafter, allowing 6 per cent. compound interest? Ans. $3004.36. 16. The British National Debt is about ~800000000. If ~8000000 were applied annually to the reduction of this debt,a in what time would it be paid off, calculating compound interest at the rate of 5 per cent.? Ans. 86yr. 10mo. 13dy. 17. There are two adjoining farms, each renting for $400 per annuim, but the rent of one is payable semi-annually, the rent of the other quarterly. What will be the difference in the income from the two farins, at the expiration of 20 years, provided all the rent is invested as fast as it becomes due, at 6 per cent. compound interest?b Ans. 8190.67. 18. An estate is sold for $50000, of which $5000 is to be paid in cash, and the rest in semi-annual instalments of 6$2250. But the purchaser proposing to discharge the whole debt at once, he wishes to know what sum of money will be required, allowing discount at the rate of 7 per cent. compound interest. Ans. $836977.90. 19. A gentleman takes a lease for ten years, at $450 per annum. At the expiration of. two years, he wishes to give up the lease, but the landlord will not consent, unless the tenant will either pay down a year's rent in advance, or 560 a The interest is supposed to be regularly paid, in addition to the sinkinug fiund for the reduction of the debt. b Tlie amount of 80 quarterly payments of $1 each, would be $152.7092.

Page  300 300 POSITION. [ART. XX. per annum, during the whole term of the contract. Which proposal is the more favorable, and how much will he save by accepting it, money being worth 6 per cent. per annum? 2d Ans. He will save 877.41. 20. What is the difference in value, between the present worth of a lease, for 100 years, of an estate that rents for $1500 per annum, and the perpetuity of the same estate, computing interest at 7 per cent.? Ants. $24.69, or less than one week's rent. N. B. The present worth of an annuity of $1, to continue 100 years, at 7 per cent., is $14.269251. XX. POSITION. THE answers to many difficult analytical questions, can be obtained by assuming one or more numbers, and working with them as if they were the true numbers sought. The method of obtaining the correct result in such cases, is called POSITION. SINGLE POSITION requires only one assumed number. It is used in solving questions in which the required number is increased or diminished by any of its parts or multiples, either by addition, subtraction, multiplication, or division. DOUBLE POSITION requires two assumed numbers. It is applicable to all questions that can be solved by Single Position, and to nearly all questions that can be solved by algebraical equations. 93. SINGLE POSITION. Questions in Single Position may be solved by the following rule: Assume any convenient number, and proceed with it

Page  301 ~94,] DOUBLE POSITION. 301 according to the conditions of the question. Multiply the assumed number by the number given in the question, and divide the product by the result obtained with the assumed number. EXAMPLE FOR ILLUSTRATION. Divide $1584 among three persons, in such manner that ~ of the first share, 1 of the second share, and I of the third share, shall all be equal. Suppose I- the first share 250. Then the first share will be 500 250 X 1584 — 2250 =176, second share 750 is ~ the first share. third share 1000 Ans. 1st share $352 2d " 528 Result 2250 ad " 704. Solve by the above rule, Examples 8, 15, 22, 23, 25, 28, 29, 30, 32, 33, 42, 46, 47, 58, 59, 64, 71, 73, 74, 79, 81, Section 60. 941. DOUBLE POSITION. RULE I. Assume two convenient numbers, proceed with them separately, according to the conditions of the question, and note the result obtained from each operation. Multiply the error of either result by the difference of the assumed numbers, and divide the product by the difference of the results. The quotient will be a correction to be added to the assumed number, 11i tf gives a result too small, or to be subtracted from it, if the result is too large. In many cases the correct result is obtained at the first trial, but if greater accuracy is required, take the number obtained by the first trial, and the nearer of the two numbers first assumed, or any other that appears more nearly correct, as new assumed numbers. Repeat the operation as before, and you will obtain a new answer, more accurate than the former. This process may be repeated until you obtain the true answer, or a number sufficiently correct for your purpose. This Rule fails in those questions in which the result of the operations to be performed is not a known number, but the re

Page  302 302 POSITION. [ART. XX. quired number, or one depending upon it, such as some multiple, or some part of it. Such cases may usually be solved by the follo wing rule. RULE II. Assume two convenient numbers, and perform on each of them the operations indicated by the question. Note the errors of the results, and mark each of them with the sign + or -, according as it is in excess or defect. Then form the following proportion — The difference of the errors, when they are caiice,a or their sum, when they are uwlikee: the difference of the assumed numbers:: either error: the correction of the assumed number which produced that error. EXAMPLES FOR ILLUSTRATION. 1. "A hunter wishes to measure the width of Z ravine, and having no other means of doing it, he fires a rifle ball at a small knot on a tree, which stands on the opposite bank. On going over, he finds that the ball struck 6 inches below the knot. By previous experiments, he knows that the ball drops 2 inches in going 50 rods, and its deflection is proportional to the square of the distane added to - of its cube. What is the width of the ravine?" Suppose 100 I Le)2 + 3 of (100)3 X 2 -131 fil'st resutlt. 50 3 50/ 3 Subtract 6 Error in excess 17 Suppose 75 5)2 + 3 Of (U7 )3 X 2 = 61 second result. Subtract 6 Error in excess 4 Then by Pule I, 7-1- X 25 - 6 — = 27. -, correction for first supposition. 100 -27 72 9 rods, approximate width. Working with this result, we obtain for the deflection 6.168 inches, which is so near the true deflection that'72 rods may be assunmed as the true width. a The errors are said to be alike, when both are too great, or both too small; and unlike, when one is too great, and the other too small.

Page  303 ~ 94.] DOUBLE POSITION. 303 2. A person being asked the time, replied: " The sun now rises at 5 and sets at 7. Now if you add I of the hours that have passed since sunrise, to'-!1 of those which must elapse before sunset, you will have the exact time of the day." To find how many hours have elapsed since sunrise, by Rule II. Suppose 6. Suppose 3. 6 + J-2 of 8 = 6U1. -a + -6of 11 =11 81-.` The result should be 11. The result should be 8. 1 st Error -4:-.2 2d Error + 1 — 41: 3:: 4-'-: 2- correction to be subtracted from 6. 6 -2'- = 3-2 5 + 31 = 8~ the true time. Proof. - of 31- + 1_i f of 1 8. EXAMPLES FOR THE PUPIL. 1. A farmer engaged a servant, agreeing to pay him $1.50 for every day he should work, and to charge him $.50 per day for his board, every day he should be idle. At the end of 13 weeks, the man received $86.50. How many days did he work? Ans. 66 days. 2. Two men have the same income. A. saves - of his, but B. spends $325 per year more than A., and at the end of 5 years finds himself $625 in debt. What is the annual income of each? Ans. $1000. 3. A person distributed in charity 2d. apiece among several poor children, and had 4d. left. He would have given them 3d. apiece, but had not enough money by 10d. What was the number of children? Ans, 14. 4. A man has a chaise worth $130, and two horses. If the first horse be harnessed to the chaise, their joint value will be 3 times that of the second horse; but if the second horse be harnessed to the chaise, their joint value will be twice that of the first horse. Required the value of each horse. Ans. PFirst horse $104; second $78.

Page  304 304 POSITION. [ART. XX. 5. Find two numbers, whose difference is 29, and their product 546. Ans. 13 and 42. 6. A man sold a horse for $144, and thereby gained as much per cent. as was equivalent to the number of dollars that the horse cost him. How much did he give for the horse? Ans. $80. 7. The fourth power of a certain number, diminished by 7 times the number, and increased by 3 times its cube, equals 3381. What is the number? Ans. 7. 8. Find a number to which if 7 times its square be added, the sum will be 500. Ans. 8.3804 +. 9. John is now three times as old as Charles, but five years ago he was four times as old. Required the age of each. Ans. John 45; Charles 15. 10. Five times a certain number, increased by 12, is equivalent to 7 times the number, diminished by 20. What is the number? Ans. 16. 11. What number is that whose half is as much less than 75, as its double is greater than 94? Ans. 675. 12. A farmer purchased a number of geese for ~6 5s. l-e retained 5, and sold the remainder for Is. 3d. apiece more than he paid, thus receiving what he paid for the whole. How many did he buy? Ans. 25. 13. The area of a certain field is 187 square rods, and the length exceeds the breadth by 6 rods. What are the dimensions? 14. There is a fish, whose head weighs 9 pounds; his tail weighs as much as his head and half his body; and his body weighs as much as his head and tail both. What is the weight of the fish? Ans. 72 lb. 15. What would have been the width of the ravine, in the example given on p. 302, if the ball had struck 8 inches below the knot? Ans. 80- rods, nearly.

Page  305 ~95.] MULTIPLICATION. 305 XXI. APPROXIMATIONS. 95. 3MULTIPLICATION. In MUITrTIPLICATION, if only a certain degree of accuracy is desired, the product may be obtained by writing the units' figure of the multiplier under that figure of the multiplicand whose place we would reserve in the product, and inverting the order of the remaining figures. In multiplying, we commence, for each partial product, with the figure of the multiplicand immediately above the multiplying figure, carrying the tens which would arise from the multiplication of the two rejected figures at the right. EXAMPLE. Required the product of 287.613952 by 15.98421, correct to the fourth decimal place. 287.613952 287.613952 12489.51 15.98421 2876.1395 28 7613952 1438.0698 575 227904 258.8525 11504 55808 23.0091 230091 1616 1.1505 2588525 568 575 14380697 60 29 28761395 2 4597.2818 4597.28180769792 The units' figure of the multiplier being placed under the 4th decimal of the multiplicand, and the whole multiplier reversed, the product of each figure by the one above it will be ten-thousandths. Therefore, the right-hand figure of each partial product will fall in the column of ten-thousandths. In the second product, multiplying 52 by 5, we obtain 260, which, being nearer 300 than 200, we carry 3 to the product of 9 by 5. The multiplication has also been performed in the usual way, the vertical line showing the figures that are rejected. If the multiplicand does not contain enough decimal figures to correspond with the inverted multiplier, the deficiency should be supplied by annexing zeros. The sa-me contraction may be applied to integers, if we wish only to obtain the thousands, millions, &c., of the product. 20

Page  306 306 APPROXIMATIONS. [ART. xxr. 9 6. DIVISION. In DIVISION, a similar contraction may be made when the divisor is large, a contraction which is also applicable in the extraction of roots. The first quotient figure is of the same numerical value as the figure of the dividend which stands immediately over the units of the divisor, at the first step of the division. After the first remainder has been obtained, instead of bringing down the remaining figures of the dividend, we may cut off the right-hand figure of the divisor at each step, as in the following example. 842.15) 28417.95255 (83.057 342.15 ) 28417.9 5255 ( 83.057 27372 0 27372 0 0459 104595 10264 10264 5 195 195 025 171 171 075 24 2319505 24 2319505 In the complete division, the contraction is indicated by the vertical line. In each multiplication, the tens arising from the product of the quotient figure by the suppressed figure of the divisor, must always be carried as in contracted multiplication. The right-hand figure of the quotient thus obtained, cannot always be relied upon. If greater accuracy is desired, the division may be extended further before commencing the contraction. IN DIVISION or CIRCULATING DECIBIALS, we may adopt the following rule. Make the repetends of the divisor and dividend similar and conterminous, and from the result, considered as whole numbers, subtract the finite part of each. Perform the division with the remainders as with whole numbers, and the true quotients will be obtained. EXAMPLE. Divide 36.91 by 5.273.

Page  307 ~97.] CONTINUED FRACTIONS. 307 The example is here solved 5.7273 ) 36.9191 by contracted decimal division. 5 36 The exact fractional quotient is 5273268) 36919155(7.001191 7562279 or 71263 The ef- 36912876 feet of subtracting the' finite 6279 parts of the divisor and dividend 5273 is the same as reducing the two 1006 numbers to improper fractions, 527 and dividing the numerators. 479 474 5 97. CONTINUED FRACTIONS. CONTINUED FRACTIONS arise from the approximate valuation of fractions whose terms are large, and prime to each other. If, for example, we desire approximate values for the fraction 89 we may commence by dividing both terms of the fraction by the I 1 numerator, which gives us 54. Disregarding the 4, we have for a first approximate value, which is greater than the true value, because the approximate denominator is less than the true denominator. But as the denominator is between 5 and 6, the fraction is between - and 6. If we desire greater accuracy, we may divide 42 in the same manner as the first fraction, which gives us 1 for the value of, or 24 1 2for a second value of the origina fraction. Disregarding th for a second value of the original fraction. Disregarding the 5, the continued fraction becomes, 1 or -, which is less than the true value, because the supposed denominator is greater than the true denominator. We therefore know that the fraction is between -a and A2.

Page  308 808 APPROXIMATIONS. [ART. XXI. Still greater accuracy may be obtained by reducing 5, which gives us 1+ 5 — I Rejecting the 5-, we have, 1 1 17 +5 - 1,or -1X or o1 21 5- 93 for a third approximate value, greater than the true value. The fraction is, therefore, between -1- and 2-. After one farther approximation, we should obtain the original fraction. In fractions whose terms are very large, as in the ratio of the diameter to the circumference of a circle, these approximate values are often very useful. They, moreover, have the advantage of admitting any required degree of accuracy, for the error in adopting any approximation, is always less than the difference between the fraction taken and the one following. Thus, in the present example, if we had adopted -1- as the true value of q8g9q the error would have been less than 1 - 1, or -1. For forming the successive approximations, we have the following RULE. Divide the greater term by the less, and the divisor by the remainder, &c., as in finding the greatest common measure. Assume 1 for the numerator, and the first quotient for the denominator of the first approximate value. Multiply the terms of this fraction by the second quotient, and add 1 to the product of the denominator, for the second approximate value. For each succeeding approximation, multiply the terms of the last approximate fraction by the following quotient, and add the corresponding terms of the preceding fraction. If the fraction given is improper, the reciprocals of the fractions thus obtained, will'be the approximations desired. EXAMPLE. Required less approximate values for the ratio of the circum

Page  309 ~ 97.] CONTINUED FRACTIONS. 309 ference to the diameter of a circle, one approximate ratio being 3141 59 1 00000)31459(3 100000)3140159o(3 Equivalent Continued Fraction. 300000 14159)100000(7 3+' 99113 7 + 1 15 +- 1 887)14159(15 1 1 887 25+ 25.-. 1 5289 1 + 1 4435 7-i 854)887(1 854 For convenience, the frac33)854(25 tion may be written, 66 3+ 1 1 1 1 1 - 7+ 15+ 1+ 25+ 1+ 7+ 4' 194 165 29)33(1 29 4)29(7 28 1)4(4 1 ~3 1st approximate value. I X 7 -- 7 3 X 7 + 1- 22 2d approximate value. 7 X 15 - 1=-106 22 X 15 + 3 = 333 3d approximate value. 106 X 1+- 7-113 4th approximate value. 833 X 1 - 22 —355. &c., &c. The reciprocals of these values are, 22 333 355' 106 113 The second ratio is the one given by Archimedes. The fourth is that of Adrian Metius, and is even more exact than the ratio 3.14159, from which we have derived it. An infinite continued fraction may be made equivalent to any given number by the following rule: Assume any number you please for a denominator; add the assumed

Page  310 810 APPROXIMATIONS. [ART. XXI. slumber to the given number, and multiply the sum by the given number, for a numerator. 40 40 40 6_ 6 6 EXAMPLES. 5- 3+ 3+ 3+; 2- - 1- i- 1+ The value of any infinite continued fraction, with but one numerator and denominator, may be found, by the following rule: To 4 times the numerator add the square of the denominator. Fromn the square root of the sumn, subtract the denominator, and divide the remainder by 2. The quotient is the value sought. EXAMPLES. -4- (s= 160 _- 9 -3) 2 =5; 4 4 = (12 + 16 - 4) 2 =.6457+ The SQUARE ROOT of any number may be expressed in the form of a continued fraction, after part of the root is found,-by making each numerator equal to the remainder, and each denominator equal to twice the root found. Thus, in extracting the square root of 17, the first root figure is 4, and the remainder 1. Then the true root is 4 + the continued fraction 1 I 1 1 1 1 0 8+ s8- s8- 8+ 8-+~ &C. In like manner, the square root of 14, is + 5 5 5 5 &3 q- 6-+ 6-+ 6%+ 6~ c. Reducing the fraction, we have, first, —,3 z5 = y~ or 4) nearly, giving the first approximate root 34. Second, 5 205 17 630 -26 or 23 41 nearly, giving a second approximate root 3 7. Third, 1350 23 or 23 6205 = - 8 31 nearly, giving a third approximate root 38 2. This approximation is of use in affording convenient fractional expressions for those roots which are of most frequent occurrence. Thus, the diagonal of a square is to its side as v/ 2is to 1. By the rule just given, we obtain successively for approximate values of /2, 11 12 L5 1 12 129 2 5Y 12' 2Q9 70' The last of these values, 10 or., is a very convenient one.

Page  311 ~ 98.] EVOLUTION 3811 9S. EVOLUTION. In the Extraction of Roots, we may commence with any complete divisor, cutting off the right hand figLure at each step, as in contracted division. At whatever place this contraction is conmmenced, as many additional root figures will be obtained-as are equal to the number of figures in the divisor less 1, but the last figure so obtained cannot be relied upon. To illustrate this mode of contraction, we will extract the 5th root of 69. 0 0 0 0 69.(2.3323285 2 4 8 16 32 2 4 8 16 37 2 8 24 64 32.36343 4 12 32 80 4.63657 2 12 48 27.8781 4.29498 6 24 80 107.8781 34159 2 16 12.927 32.0424 29334 8 40 92.927 139.9205 4825 2 3.09 13.881 3.246- 4407 10.3 43.09 106.808 143.166 418 3 3.18 1.4' 3.288 294 10.6 46.27 108.2 146.454 124 1.4 22- 117 109.6 146.67 7 22 7 146.89 After obtaining the third trial divisor, we commence rejecting one figure from the trial divisor, two from the number at the foot of the preceding column, thiree from the third column, &c., and proceed in a similar way with each subsequent trial divisor, until the figures from the preceding columns are entirely cancelled. But in every instance, allowance must be made for the product of the figures rejected, as in simple contracted division. The following is a general rule for the approximation of ANY RooT desired. RULE. Call the first two figures of the root found in the usual way, the ASCERTAINED ROOT.

Page  312 812 APPROXIMATIONS. [ART. XXI. Involve the ascertained root to the given power, and multiply by the index of the root for a dividend. Subtract the power of the ascertained root from the corresponding periods of the given number, for a divisor. Divide, and reserve the quotient. To 6 times the reserved quotient, add the index of the root, plus 1, for a second dividend. To 6 times the reserved quotient, add 4 times the index of the root, subtract 2 from the sum, and multiply by the reserved quotient for a second divisor. Divide, add 1 to the quotient, and multiply by the ascertained root for the true root nearly. If greater accuracy is desired, repeat the process with the root thus found. By this rule, the number of figures in surd roots, may generally be tripled at each operation. EXAMPLE. The following is the application of the rule, in extracting the 5th root of 659901. Ascertained 1root 14. 145 = 537824 given no. 659901 index 5 145 - 537824 dividend 2689120 1st divisor 122077 2689120 - 122077 = 22.02806, reserved quotient. reserved quotient 22.02806 6 132.16836 index +- 1 6. second dividend 138.16836 6 X reserved quotient = 132.16836 4 X 5 —2-18. 150.16836 Multiply by 22.02806 2d divisor 3307.91764 138.16836 3307.91764 -.041768 1.041768 X 14 = 14.584752, approximate root, correct to the fourth decimal place. This contraction is of use in extracting the higher roots. Any root below the 10th may be obtained in the usual way, nearly as readily, and with much greater accuracy.

Page  313 ~ 100.] PROPERTIES OF SQUARES AND CUBES. 313 99. EXAMPLES IN APPROXIMATION. 1. Multiply 11817.93642 by 2581.36, and reserve two decimal places. 2. Divide 2704.1583 by 361.8901. 3. Divide 4.3097 by 18.6i5843. 4. What are the approximate values of.785398, which is nearly the ratio of the area of a circle, to that of its circumscribing square? 4AS.7 I; - 1 7 23 5 42 Allzs. 1; 4 5 Yj 1; 4 3 4; 5 &; 5. Form infinite continued fractions, equivalent to 15; to 7; to 20; to)5~. 6. Determine the value of the infinite fraction 3.75 3.75 3.75 Ans..(18) 7. Express in the form of a continued fraction /19; (18)-; 8. Find the 5th root of 729. Ans. 3.73719. 9. Extract the 17th root of 1.004. Ans. 1.00023. XXII. PROPERTIES OF NUMBERS.a 109. PROPERTIES OF SQUARES AND CUBES. 1. EVERY square number terminates in 1, 4, 5, 6, or 9, or in an even number of ciphers preceded by one of these figures. If a square number ends in 1, 4, 5, or 9, the last figure but one will be even, but if it ends in 6, the preceding figure will be odd. If a square ends in 5, it will end in 25, and the figure preceding 25 must be even. No square number can end in two even digits, except two ciphers, or two fours. No square number can end in three equal digits, except three fours; nor in more than three equal digits, unless they are ciphers. a Hutton, Barlow, and private sources.

Page  314 314 PROPERTIES OF NUMBERS. [ART. XXIIT. 2. Every square number is divisible by 3, or becomes so when diminished by 1. The same remark may be made of 4. If 1 be deducted from any odd square number, the remainder will be divisible by 8. If a square be either multiplied or divided by a square, the product or the quotient will be a square. 3. Every number is either a square, or is divisible into two, three, or four squares. 4. Every power of 5, or of any number terminating in 5, necessarily ends in 6. A similar remark may be made of 1, and 6. 5. Assume any two numbers whatever; then one of them, or their sum, or their difference, must be divisible by 3. 6. If the sum of two squares forms a square, the product of their square roots will be divisible by 6. 7. If 1 be added to the product of two numbers whose difference is 2, the sum will be the square of the intermediate number. 8. If a cube be divisible by 6, its root will also be divisible by 6. And if a cube, when divided by 6, has any remainder, its root divided by 6 will have the same remainder. 9. All exact cubes are divisible by 4, or can be made so by adding or subtracting 1. The same remark may be made of 7, and 9. 10. The cube root of any exact cube, consisting of not more than six figures, may be determined by inspection. Divide the number into periods, (as in the usual mode of extracting the cube root,) and to the root of the greatest cube contained in the first period, affix the root of the cube that terminates in the right hand figure of the second period. 11. If a cube terminates in ciphers, the number of ciphers must be divisible by 3. 12. Any square may be divided into two other squares, in the following manner: Assume any two numbers at pleasure, and by their product multiply double the root of the given square, for a numerator. Take the sum of the squares of the assumed numbers for a denominator. The resulting fraction will be the root of one of the squares sought. Subtract the square of this root from the given square, and the remainder will be the other square required.

Page  315 ~ 100.] PROPERTIES OF SQUARES AND CUBES. 315 13. If we add the cubes of the series, 1, 2, 3, 4, commencing at the beginning, and taking any number of terms whatever, the sum will always be a square. Thus, 1 + 8 = 9; 1 + 8 + 27 36; 1+8+-t-27 + 64 =100. 14. If we write down the series of squares of the natural numbers, and take the difference between the successive terms, and the difference of these differences, the second differences will always be 2, as may be seen below: Squares. 1 4 9 16 25 36 Ist Diff. 3 5 7 9 11 2d Diff. 2 2 2 2 15. If the successive differences of the series of cubes be taken, the third differences are always 6, =1 X 2 X 3, as may be seen below: Cubes. 1 8 27 64 125 216 1st Diff. 7 19 37 61 91 2d Diff. 12 18 24 30 3d Diff. 6 6 6 16. The fourth differences of the series of fourth powers are always equal to 1 X 2 X 3 X 4 = 24; the fifth differences of the series of fifth powers are always equal to 1 X 2 X 3 X4 X 5120; and so on. EXABIPLES. 1. By which of the foregoing rules do you know that neither of the following numbers can be a square? 952; 827; 1814; 2795; 3725; 308; 711; 866; 299; 25000; 334; 779; 426; 47800. 2. Divide 24 into three squares. Ans. 16 + 4 + 4. 3. Find four square numbers, whose sum will make 30. 4. If you divide the cube of 8709512863 by 6, what will be the remainder? 5. By what rules do you know that neither of the following numbers is an exact cube? 87042; 14284; 730176; 4080000; 51858. 6. Determine by inspection, the cube root of each of the following exact cubes. 12167; 21952; 103823; 39304; 42875; 97336: 79507; 132651; 24389. Ans. 23; 28; 47, &c.

Page  316 816 PROPERTIES OF NUMBERS. [ART. XXII 7. Divide 49 into two other squares. (6 X 14) 7056 49 7056 122s5 Ans. Assuming 2 and 3; (4- 169 - 169 4+9 1_T69 169 169 j Assuming and 8 (8 X 14) 12544. 49 12544 194481 1+64 225 4225 4225 45 42 &c. &c. &c. X1 0X. PRINME AND COMPOSITE NUMBERS. Every number that cannot be divided by any other number, (except 1,) without a remainder, is called a PRIME NUMBIER. Two or more numbers that have no common divisor, are said to be prime to each other. Every prime number is prime to all other numbers except its own multiples. There are no known means of determining at once whether a proposed number is a prime; but the following properties and rules will enable us to determine all the divisors of any number. 1. 2 is a factor of all numbers terminated by 0, 2, 4, 6, or 8. For, as 2 will divide 10, it will also divide any number of tens, or any number of tens plus 2, 4, 6, or 8. Numbers divisible by 2 are called EVEN; all others, ODD numbers. 2. 5 is a factor of all numbers terminated by 0 or 5. For, as 5 will divide 10, it will also divide any number of tens, or any number of tens phis 5. 3. 3, or 9, is a factor of all numbers in which the sum of the figures is exactly divisible by 3, or 9. For, if front any power of 10, as 10, 100, 1000, &c., we subtract 1, the remainder consists entirely of 9's, and is, therefore, divisible by both 3 and 9. Hence, any power of 10 is divisible by 3 and 9 with 1 remainder; therefore, any number of tens, hundreds, thousands, &c., diminished by as many units, will be divisible by 3 and by 9. Let us, then, examine the number 34794. 3 ten thousands - 3; 4 thousands —4; 7 hundreds —7; 9 tens- 9; and 4 units-4; each divided by 3 or 9, give no remainder. Therefore, 34794 - 3 - 4 -7-9-4, is divisible by 3 and by 9, and if the sum of the numbers subtracted, or in other words, the sum of the digits, is similarly divisible, the number itself will be so. 4. 11 is a factor of all numbers in which the sum of the odd digits, (the 1st, 3d, 5th, &c.,) and the sum of the even digits, (the 2d, 4th, 6th, &c.,) are equal, or their difference is some multiple of 11. For any number of tens, thousands, hundred thousands,

Page  317 ~ 101.] PRIME AND COMJPOSITE NUMBERS. 317 &c., (which represent the even digits,) increased by as many units, will be divisible by 11. Any number of hundreds, ten thousands, millions, &c., (which represent the odd digits,) diminished by as many units, will also be divisible by 11. Take, then, the number 635173. 6 hundred thousands + 6; 3 ten thousands 3; 5 thousands-+5; 1 hundred —1; 70+7; and 3-3; each divided by 11 give no remainder. Therefore, 635173 -18 + 7 or 635173 - 11, is divisible by 11, and 635173 itself must be so. 5. 4 is a factor of all numbers, in which the two terminating figures are divisible by 4. For, as 4 will divide 100, it will also divide any number of hundreds, or any number of hundreds plus any number of units divisible by 4. 6. 25 is a factor of all numbers terminating in 25, 50, 75, or two zeros. For, as 25 will divide 100, it will also divide any number of hundreds, or any number of hundreds plus 25, 50, or 75. 7. Every number that is divisible by two or more numbers prime to each other, is divisible by their product. Take, for example, 105, which is divisible by both 3 and 5. This number may be resolved into the factors 5 X 21; 5 X 21, must therefore be divisible by 3. But as 3 will not divide 5, it must divide the other factor 21, and the number may be resolved into the factors 5 X 3 X 7 or 15 X 7. Hence we deduce the following additional properties. 8. Every even number that is divisible by 3 is also divisible by 6; and every even number that is divisible by 9 is also divisible by 18., 9. Every number divisible by 3 or 9, in which the two terminating figures are divisible by 4, is divisible by 12 or 36. 10. Every number divisible by 3 or 9, whose terminating digit is 0 or 5, is divisible by 15 or 45. 11. Every prime number greater than 2, is one greater or one less than some multiple of 4. 12. Every prime number greater than 3, is one greater or one less than some multiple of 6. 13. Every number that has no prime factor, equal to or less than its square root, is itself a prime number. For the product of any two factors, each greater than the square root of a number, -would evidently be greater than the number itself. Therefore, if we attempt the division of any supposed prime, by all the primes less

Page  318 318 PROPERTIES OF NUMBERS. [ART. XXIT than its square root, and discover no factor, the number is itself a prime. TO FIND ALL THE DIVISORS OF A NUMBER. What numbers will divide 5940 without a remainder? We first resolve the number into all its prime factors, 215940 by commencing with 2 and dividing as often as possible, 2 2970 by each of the prime numbers in succession. We thus 3 1485 find that 5940 =22 X 33 X 5 X 11, or2 X 2 X 3 X 3 X 3 495 3 X 5 X 11. It may, therefore, have as many compos- 3 165 5 55 ite divisors as we can form distinct products of these 11 11 prime factors. In order to determine all the possible 1 products, we arrange 1, with the powers of the factor that is employed the greatest number of times, in a horizontal line. We then multiply each of the numbers in the first line, by each of the powers of another factor; each of the numbers of the preceding lines,, by each of the powers of a third factor, &c., as in the following table. 1 3 9 27 -33 2 6 18 54 = 33 X 2 4 12 36 108-33 X 2S 5 15 45 135 33X 5 10 30 90 270 = 33 X 2 X 5 20 60 180 540-= 33X 92 X 5 11 33 99 297 - 33 X 11 22 66 198 594 = 33 X 2 X 11 44 132 396 1188 = 33 X 22 X 11 55 165 495 1485 = 33 X 5 X 11 110 330 990 2970= 33 X 2 X 5 X 11 220 660 1980 5940 = 33 X 22 X 5 X 11 Tilhe numbers of the first line having been arranged as directed, we multiply them separately by 2 and 22. All the numbers of these three lines, are multiplied by 5, which gives us three new lines of divisors. All the numbers of these six lines are multiplied by 11, which gives us six new lines of divisors. We thus obtain 48 numbers that will divide 5940 without a remainder, and an examination of the table will show that these are all the divisors, since the prime factors are combined in every possible way. We are able to determine without actual trial, the number of exact divisors of any given number. By the foregoing table we

Page  319 ~101.] PRIME AND COMPOSITE NUMBERS. 319 perceive that 33 had 4, or 3+1 divisors. 33 \< 22 has 12, or 3+1 X 2+1. 33 X22 X Shas24or3 + 1 X 2 +- 1X 1 - 1. In like manner each new factor can be multiplied by all the preceding divisors, as many times as are equivalent to the exponent of its power, thus forming so many new divisors to be added to the preceding. Hence, for finding the number of divisors of any given number, we have the following RULE. Add 1 to the exponent of each of the prime factors of the given number, and multiply together the exponents thus increased. The product thus obtained, is the number of divisors sought. If any other number than 10 were adopted as the base of a system of niUmeration, the number preceding the base would have the same properties as the figure 9 in our present system. For example, 1183, expressed by a scale of 8 would be 2237. The sum of the digits 2 + 2 + 3 + 7 -14 being divisible by 7, the number itself is so divisible. A pCefect number is one that is equal to the sum of all its aliquot parts. Thus, 6, the aliquot parts of which are 1, 2, and 3, is a perfect number, because 1 + 2 + 3 = 6. The following are the only perfect numbers known: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2417851639228158837784576, 9903520314282971830448816128. Every perfect number must terminate either in 6 or in 28. Two numbers are said to be amicable, when each is equivalent to the sum of all the aliquot parts of the other. Thus, 220 and 284 are amicable numbers, because 220 = 1 + 2 + 4 + 71 + 142, which are the aliquot parts of 284, and 284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110, which are the aliquot parts of 220. There are very few amicable numbers known. EXAnIPLE S. 1. Which of the following numbers are prime? 733; 949; 917; 619; 1009; 1001; 989; 11571. 2. Find all the prime factors of 780; 468; 3944; 6972; 1849; 2899; 883; 15664. 3. Find all the divisors of 94800; 21100; 6922.

Page  320 320 PROPERTIES OF NUMBERS. [ART. XXI. 4. How many divisors has 20736? 44100? 29930? 5940? 16384? 15309? 5. Show that if 7 were adopted as the base of a numerical scale, 548712 would be expressed in such a manner that the sum of its digits would be divisible by 6. 6. Prove that 137438691328 is a perfect number. 7. Prove that 17296 and 18416 are amicable numbers. 8. Prove that 9363584 and 9437056 are amicable numbers. 9. Show that 120 and 672 are each equal to half the sum of their aliquot parts. 1 O2. FIGURATE NUMBERS. Figurate, or polygonal numbers, are formed by adding the successive terms of an arithmetical series. Thus, if we add the successive terms of the natural series, 1, 2, 3, 4, 5, 6, 7, &c., we obtain the figurate series 1, 3, 6, 10, 15, 21, 28, &c., which are called triangular numbers, because they can always be arranged in the form of an equilateral triangle. If we take the arithmetical progression 1, 3, 5, 7, 9, &c., in which the common difference is 2, we obtain the figurate series, 1, 4, 9, 16, 25, &c., which... are called square nunzbers, because they can always be arranged in the form of a square. The arithmetical progression 1, 4, 7, 10, &c., in which the common difference is 3, furnishes the figurate series 1, 5, 12, 22, &c., which are called pentagonal numbers, because they can always be arranged in the form of a polygon with five sides. In a similar manner are produced the hexagonal, heptagonal, octagonal, and other figurate series, the number of sides of the polygon in which the numbers can be arranged, being always two greater than the common difference of the arithmetical progression from which they are derived. EXAMPLES. 1. Find the first 20 triangular numbers. 2; What are the first 20 pentagonal numbers? 3. What are the first 10 hexagonal numbers?

Page  321 ~ 103.] THE FUNDAMENTAL RULES. 321 4. Find the first 10 dodecagonal numbers. ~5. Find the first five 17-gonal numbers..0@3. THE FUNDAMENTAL RULES. It may readily be perceived that the rules of Arithmetic merely indicate convenient modes of obtaining a desired result, and that, in many. cases, a variety of processes will suggest themselves, either one of which will serve our purpose. But it may at first seem incredible, that the sum of any number of quantities can be obtained without addition; the difference of two numbers, without subtraction; the product of two numbers, without multiplication; and the quotient of two numbers, without division. Yet such is the case, and the following rules are not only interesting from their curiosity, but from the connexion which they show between the several operations that may be performed upon numbers. I. To obtain the sum of a series of numbers, by subtraction. Assume any number larger than the required sum, and from the assumed number subtract in succession each of the given numbers. Subtract the final remainder from the number first assumed, and the result will be the sum required. EXAMPLE.-Find the sum of 69, 93, and 237. Assume 1000 1000 69 601 PROOr. 931 399 Ans. 1000o- (1000- 69 - 93 -287) 93 69- +93- +237. 838 237 601 II. To find the difference of two numbers, by multiplication and division. Write nine times the subtrahend under the minuend, and add each figure of the upper number to the figures in the same place and all the inferior places of the lower number, carrying as in ordinary addition. Proceed in this manner, stopping at the figure that falls immediately under the left hand figure of the minuend, and the result will be the difference sought. ExAmPLE.-Required the difference between 874 and 10757. 21

Page  322 322 PROPERTIES OF NUMBERS. [ART. XXII. 7+- 6= 13; 1 to carry + 5 + 6+6 = 18; 10757 1 to carry - 7 + 8 + 6 + 6 = 28; 2 to carry 7866 = 9 X 874 + - + 7 8 + 6 + 6 =29; 2 to carry 1 -I 09883 Ans. 7 + 8 6 + 6=-30, but as the 0 falls under the left hand figure of the minuend, we stop there, and find the true remiinder to be 9883. III. To find the product of two numnbers by addition, subtraction and division. Resolve either factor into a number of submultiples of some power of ten, which submultipleb, when combined, either by addition or subtraction, will reproduce the original factor. Write 1 as a common numerator, and the submultiples as denominators of a series of fractions, and divide the other factor by the decimal expression for each of the fractions thus formed. The several quotients, combined in the same manner as the original submultiples, will give the product desired. EXAMPLE. What is the product of 5769 X 2841? 5769 = 5000 + 500 + 250 + 20 —1. 5~ ~ - + - +- I -1- 1 -.0002 +.002 +.004q.05-1.0-00' o 803-. - + - - 16389729 Ans. IV. To find the quotient of two numbers, by addition and multiplaication. 1. Employ as a multiplier, the difference between the given divisor, and the least power of 10 which is larger than the divisor. 2. Write the first figure of the dividend in the quotient, and add the product of the first dividend figure by the employed multiplier, to as many of the succeeding figures as are equivalent to the number of figures in the divisor. If the sum has a greater number of figures than the divisor, write the left hand figure under the figure in the quotient, and proceed as before, until a sum is obtained having the same number of figures as the divisor. 3. To this sum annex the remaining figures of the dividend. Place the left hand figure of the result as the second quotient figure, and proceed as in paragraph (2). Continue this process until all the figures of the dividend have been employed. 4. Add the several figures in the quotient, and cut off from the right hand as many figures as there are in the divisor. The figures so cut off will represent a remainder; the remaining figures are the quotient.

Page  323 ~ 103.] THE FUNDAMENTAL RULES. 323 [If the remainder is larger than the divisor, subtract the divisor as often as possible, and increase the quotient by a number equivalent to the number of subtractions. If the divisor is contained two or more times in the next larger power of 10, the quotient may be obtained more readily, by employing as a multiplier the difference between some power of 10 and the greatest multiple of the divisor contained in it. The quotient thus found, should be multipled in the same manner as the original divisor.] EXAMPLE 1. Required the quotient of 284175 by 89. 100 - 89 = 11, the employed multiplier. 284175 (2182 11X 2- 22 1 1 106 3192 11 X I= 11 17175 11 X 1= 11 8275 Ans. 3192w-'. 11 X 8 88 115 11 X 1= 11 265 11 X 2== 22 87 The several multiplications and additions may be made mentally, and the quotient thus obtained by employing very few figures. If the divisor is very near some power of ten, or a submultiple of any number which is but little smaller than some power of ten, the quotient can be obtained in this way with great facility. EXAMPLE 2. Divide 573612 by 9. 57361 2 53634.6 11 Ans. 637346 EXAMPLE 8. Divide 47281591 by 997. 47281 591 Ans. 474238 o. 47423.860 When the divisor consists entirely of 9's, the following rule for obtaining the quotient will be found more convenient:

Page  324 324 PROPERTIES OF' NUMBERS. [ART. XXII. RULE FOR DIVIDING BY 9's. When the divisor consists of any number of 9's, increase it by 1, for a new divisor. Divide the dividend by this new divisor. By the same divisor, divide the integers of the quotient, and proceed in a similar manner, until a quotient is obtained less than the divisor. Add all the quotients together, observing the number of units carried from decimals to integers. Add this number to the right hand decimal figure, and the integers will represent the quotient, and the decimals the remainder. When all but the units figure of the divisor are 9's, increase the divisor by the difference between the units figure and 10, and divide as above directed, multiplying each quotient after the first, by the number added to the divisor. Multiply the number carried from decimals to integers, by the number added to the divisor, and add the product to the decimals for the true remainder. If this increased remainder exceeds the divisor, increase the quotient by 1, and subtract the divisor from the remainder for the true remainder. EXAMPLES FOR ILLUSTRATION. Divide 8905473 by 999. The divisor, increased by 1, is 1000. Dividing by the 8905.473 rule, and adding the quotients, we obtain 8.905 8914.386. There being 1 unit to carry 8 from decimals, we add 1 to the right-hand 8914.386 decimal figure, and find the quotient is 1 8914.387. 8914.387 quotient. This rule is founded on the decimal value of g and it will be easily seen that the process is nearly the same as in the multiplication by.Oi00i +. In dividing the numerators of fractions obtained by the multiplication of circulating decimals, the rule will often be of use. Divide 1549638144 by 9991. The divisor, increased by 9, is 10000. Di- 154963.8144 viding first by this number, we multi- 139.4667 ply the in'egers of the first quotient by 9, writing the first figure of the 155103.4062 product under the right hand decimal 1 X 9 9 figure, which is equivalent to multi-.4071 remainder. plying by 9, and dividing by 10000.

Page  325 ~ 104.] CURIOUS PROBLEMS. 325 We multiply the integers of this second number, and write them in the same manner, and add the several numbers together. There being 1 unit to carry from decimals, we multiply it by 9, and add the product to 4062, which gives 4071 for the true remainder. The foregoing rules furnish us with a general method for determining whether any number is divisible by any other given number, without actually performing the division. EXAMPLES. 1. Find by subtraction, the amount of 1690, 84, 207, 168, and 4493. 2. Find by multiplication and addition, the difference between 847 and 10082; between 19804 and 2973. 3. Multiply 764 by 1972, by Case III. 4. Divide 21709 by 837, by Case IV. 5. Divide 684291708 by 9; by 97; by 99; by 995; by 990; by 999; by 9999. 1 @4. CURIOUS PROBLEMS. 1. To add a column of numbers at a glance. The numbers to be added should be arranged in pairs, each member of the pair being the arithmetical complement of the other. The key line may be written in the middle, and the sum of the whole may be found by prefixing the figure which represents the number of pairs, to the key line. For example, if we first write the three numbers, 4719082, 3604227, 4719082 1518729, and reserve the third as a key line, then 3604227 take each figure of the second number from 9 except 1518729 the right hand figure, which we take from 10, we 6395773 obtain a fourth number, 6395773. Proceeding in a 5280918 similar manner with the first number, we obtain 21518729 5280918 for our fifth number, thus completing two pairs besides the key line. Then prefixing 2 to the key line, we obtain 21518729 for the sum of the five numbers. It will be more difficult to detect the method pursued, if we subtract each figure of one of the numbers from 8, except the right hand figure, which we take from 9. Each figure of the key line must then be diminished by 1. Other variations may be made

Page  326 326 PROPERTIES OF NUMBERS. [ART. XXII. in the process, and the position of the key line in the column may be altered at pleasure. 2. To tell two or olre znumbers which a person has thought qf, neither number being greater than 9. Direct the person to double the first number thought of, add 1 to the product, multiply the sum by 5, and add to the product the second number. If there be a third, let him double the first sum, and add 1 to it, multiply by 5, and add the third number. Thus proceed for each additional number thought of. Finally, if there were but two numbers thought of, direct him to subtract 5 from the result; if three, 55; if four, 555, and so on. The remainder will be composed of the figures thought of, in their proper order. For example, suppose the numbers thought of to be 2, 2, 8, 9. 2 X 2+ 1 — =5; 5 X 5-=25; 25-+2=27; 27 X 2 + 1 =55; 55 X 5-275; 275+ 8 =283; 283 X 2+ 1 =567; 567 X 5 2835; 2835 + 9 = 2844; 2844 - 555 =2289, the figures of which indicate, in their order, the four numbers thought of. 3. What is the product of ~11 11s. 11d. by ~11 1s. lid.? Questions similar to this are to be found in many of the old arithmetics, and different answers have been given, according to the different views of the proposers. But in reality the problem is absurd, the error consisting in the supposition that applicate, or concrete numbers, are capable of being multiplied together. The thorough arithmetician should never lose sight of the fact, that all the operations of arithmetic are performed on abstract numbers. We say, indeed, that the area of any rectangular surface is found by multiplying the length by the breadth; but this is merely a convenient expression, adopted to avoid circumlocution. Our meaning is, that if we multiply the NUMSBER of feet in the length, by the NUMBER of feet in the breadth, the product will represent the NUMBERx of square feet in the area. So in geometry, when we say AB X CD c the rectangle AD, we mean that if the line AB were repeated for every point of the line CD, we should have the surface AD. If it were possible to form a product of concrete numbers, we should be obliged to call the product of 1~ by 1~, 1 square pound; Is. X Is., would then be 1 square shilling; Id. X ld. — 1 square d.; lqr. X lqr., =1 square qr. 1 sq. ~ would then be equal to 400sq. s.; 1 sq. s. = 144 s. d.; l sq. d.=16 sg. qr.; and the product of ~11 lls. Ild. by ~11 11s. lid., would be 134 sq. ~

Page  327 ~ 105.] CHRItONOLOGY. 327 185 sq. s. 49 sq. d. The impossibility of conceiving of a square pound, a sqcuare shilling, or a square pennzy, shows at once the absurdity of the original question. But the investigation is a useful one, both because it furnishes exercise in an intricate kind of multiplication, and because it shows the error of the very prevalent idea, that dollars, cents, and mills, can be multiplied by dollars, cents, and mills. The number of curious problems might be extended indefinitely, if our limits would allow. The few here given may suffice to excite an interest in such investigations, and lead the pupil tc exercise his own ingenuity, and to consult the works of authors who have treated the subject more fully. XXIII. MISCELLANEOUS PROBLEMS. 105. CHRONOLOGY. AcCORDING to the Julian Calendar or OLD STYLE, the solar year was considered as being 365 days and 6 hours. The 6 hours in 4 years amounted to a day, therefore every fourth year was called a Leap Year, and consisted of 366 days. But the true solar year is about 11 minutes less than the Julian year, and on this account, in 1582, it was found that spring commenced 10 days later than at the establishment of the Julian Calendar. Pope Gregory XIII., therefore, caused ten days to be taken out of the month of October in that year, and to prevent the recurrence of a similar variation, he ordered that the centurial years should not be regarded as leap years, unless the number of centuries were divisible by 4. This computation, which is called the Gregorian or NEW STYLE, was soon adopted in the greater part of Europe; but in England and America, the change was not made until 1752, when the error had amounted tc eleven days.

Page  328 828 MISCELLANEOUS PROBLEMS. [ART. XXIII. It was then ordered that the 3d of September should be called the 14th, and the Gregorian calendar adopted for the future. In Russia, the Old Style was retained until the year 1830. One of the first seven letters, A, B, C, D, E, F, G, is attached to every day in the year; thus A is applied to Jan. Ist, 8th, 15th, &c.; B, to Jan. 2d, 9th, 16th, &c.; C, to Jan. 3d, 10th, 17th, &c. In this manner all days in any year which have the same letter, fall on the same day of the week. The DOMINICAL LETTER for any year is the letter that falls against all the Sundays. Thus, the 6th of January, 1850, fell on Sunday, and the dominical letter was, therefore, the 6th letter, or F. But in leap year there are two dominical letters, the first for January and February, the second for the remainder of the year. PROBLEMS. I. To find the domini2ical etter for any year, accorcling to the Julian or OLD STYLE. To the given year add one fourth of itself, plus 4, and divide the sum by 7. If there is no remainder, the dominical letter is G; if 1 remainder, F; and so on in inverse order. If the given year be leap year, the letter thus found will be the dominical letter for the last 10 months, and the next following letter, for the remainder of the year. What was the dominical letter for A. D. 1531? To the given year, we add Given year 1531 one fourth of itself, (rejecting one-fourth t. 382 4 the fraction,) and 4. Dividing this sum by 7, we have a re- 7 ) 1917 mainder 6, which indicates 273 -- 6 remainder. that the dominical letter sought is the 6th from G, counting in retrograde order, which is A. What were the dominical letters for A. D. 564?

Page  329 ~ 105.] CHRONOLOGY. 329 The remainder 2 indicates that the dominical 564 letter is the 2d from G, or E. But the year being 141 leap year, the dominical letter for January and 4 February, will be the next following, or F. The 7 ) 709 two letters sought are therefore F, E. 1i 4 2 If the given year were before the Christian era, the remainder would indicate the direct order of the letters. Thus, 1 denotes A; 2 denotes B; 5, E, &c. II. To find the domninical letter for any year, according to the Gregorian or NEW STYLE. Divide the centuries by 4, and take the remainder from 3. Add twice this remainder to 4 of the odd years, and divide the sum by 7. If there is no remainder, the dominical letter is G; if 1 remainder, F, &c., as in the preceding rule. What is the dominical letter for 1895? 4)18 cent. Dividing 18 centuries by 4, the remainder 4 + 2 is 2. Taking this remainder from 3, we.3-2= 1 have a remainder of 1. Twice 1 added to 2 X 1 = 2 95 years plus I of 95, (rejecting the fraction,) Odd years 95 4 V IJUL VIIUVILI23 gives 120, which, divided by 7, gives a reimainder 1, indicating that the dominical 7)120 letter is the Ist before G, which is F. 17 + 1 III. To find, the day of the week corresponding to any given day of the qmonth. The dominical letter found by one of the preceding rules, will indicate the day on which the first Sunday in January will fall. The day of the week for the corresponding day of each succeeding month, may be found by the initials of the following couplet:At Dover Dwell George Brown, Esquire, Good Captain French, And David Friar. On what day of the week was the Declaration of Independence signed?

Page  330 330 nMISCELLANEOUS PROBLEMS. [ART. XXIII. The dominical letters for 1776 were G, F. Therefore, the first Sunday in January was the 7th of the month. Then A representing the 7th Jan., D would represent the 7th Feb.; D the 7tl March; G the 7th April; B the 7th May; E the 7th June, and G the 7th July. But 1776 being a Leap Year, the dominical letter after February is one clay earlier in the month, and a day of the month which would otherwise be represented by G, will be represented by A or Sunday. The 7th July, therefore, came on Sunday, and the 4th on Thursday. The initials 0. S. denote the Old Style. In all cases not thus marked, the New Style is understood. EXAMPLES, 1. Washington was born on the 22d Feb. 1732. What was the day of the week? Ans. Friday. 2. The pilgrims landed at Plymouth, Dec. 11, 1620. 0. S. What was the day of the week? Ans. 3Monday. 3. The Battle of Waterloo was fought June 18, 1815. Is it probable that a letter, purporting to have been written at the time, and dated Friday, June 18, is authentic? Ans. No; because the battle was fought on Susnday. 4. Constantine Paleologus was the last Christian Emperor of Constantinople. On the 29th of 3May, 1453, the city was taken by the Turks, the Emperor Constantine killed, and Mohammed II. ascended the throne, thus founding the present empire of Turkey in Europe. What was the day of the week? Ans. Tuesday. CONTEMPORARY PRINCES, FRO1I EGBERT TO QUEEN VICTORIA. The following table is compiled principally from Wade's British History. The teacher will find in it the materials for a great variety of Chronological questions. In order to determine the date of the commencement of each reign, add the number following the name to the number at the head of the column. Ex.: Ethelwolf, 836; Athelstan, 925.

Page  331 ~ 105.] CHRONOLOGY. 331 800. 900. 1000. ENGLAND.-Eghert; E- ENG. - Edward the El- ENG.-Edmund Ironside, thelwolf, 36; Ethel- dler, 1; Athelstain, 25; 16; Canute the Great, bald, 57;Ethelbert, 60; Edmund, 41; Edred, 17; Harold I-larefoot, Ethelredl I_ 66; Alfred 46; Edw, 55; Ed gar, 136; Iar dicanUte, 309 the Great, 72. 59; Edw. the Iartyr. Edwardl the Confessor, 75; Ethelred II., 78. 41, Harold II., 65 Vnm. I., the Conqueror, 66; W\rn II. Rut'fus, 87. SCOTLAND. - Aehaius, ScoT.-Constant. III., 1;SCOT.-ITIalcoln II., 4; 787; Congale III.. 19; MIalcolm I.38, Indul- Dunean,34; 31 tacbeth, Alpin, 31; Kennetlh II., phts, 58; Duphus, 68, 40, Malcolm III., 57; 34; Donald V.. 54; Cullenus, 72; Kenneth Donald VI., 93; DunConstantine II., 58; IIl., 73; Const. IV., can II., 90; Edgar, 6. Ethus, 74; Gregory the 91;'Grimus, 97. Greet, 75; Donald'VI., Fr.-Itenry I., 31; Phi92. lip I., 60. FRANCE.-Charlemagne; Fr.-Robert, 22; Ralph, GER.-Heniry II., the Louis I., 14; Charles 23; Louis IV., 36; Lo- Saint, 2; Conrad the the Bald, 43; Louis tharius, 54; Louis V., Salic, 2;, Henry III., II., the Stammerer, 77; 86; Hugh Capet, 87, 39; Henry IV., 56. Louis III. and Carlo- Robert the Pious 7 S.-Sancho III., the man, 79; Charles the Great; Ferdinand I Fat, 84; Hugh, 88; in Castile, 33; Garcia Charles the Simple, 98. GER. - Conrad I., It IV. in Navarre, RamiHenry I., 19; Otho the rez I. in Arragon, 35; G ERMANY. - Charle- Great, 36; Otho II., Sncho IV, Nayv., 51 magne; Louis I., 14; 73 Otho III.. 83. Sanchlo I., Arr., 63; Louis II., 43; Carlo- S-neho I., Cast., 65; man, Louis III., the Alfonzo I., Cast., 72; Younger, and Charles SP.-Sancho I., 5; Gar- Sancho V, Nay. and the Fat, 76; Arnoldl, cia 11., 26; Sancho II., Arr., 76; Peter I., N. 87; Louis IV., the In- 70; Garcia III., 94. and A., 91. fant, 99. PA. ST.-Romanus For- PA. ST. -Jno. XVII.,nd SPAIN.-Garcia I., 58; mnosus and John IX.,: XVIII. 3; SergiislV., Fortunio, 80. Benedict IV., 5; Leo 9; Benedlict VIII.. 1i2; V. and Christopher, 6; Jno. XIX., 21, Bened. PAPAL STATES. - Leo Sergias III, 7; Arias- IX., 33 Gregory VI., III., 795; Stephen V., tasius, 10; Lando arnd 44; Clement II., 47; 16; Pasehal I., 1.7; John X., 12; Leo VI., Da1cmtsia 11., 48; Leo Eugene II., 20; Va- 28; Stephen VIII., 29; IX., 49; Victor II.,55; lentine,'24; Gregory John XI. 31; LeoVIi., StephenX., 57; NichoIV., 27;, Sergius II., 36; Stephen IX., 40; 1-ls II., 58; Alexrnder 43; Leo IV., 47; Be- Martin II., 43; Aga- II.,; Gregory VII. nedlict III., 55; Nicho- pet II., 46; John XII., 73; Victor III., 85 las I., 5S; Adrian II., 56; Benedict V., 65; Urban II., 87; Pascal 66; John VIII., 73; Jolh XIII., 66; Donus II., 99. Martin I,.85; Adriais It. and Benedict VI. Roe -Savtopolk 1.,15, Ill", 841; Sieplien VI., 73; Benedi t VIL-, 74;, rsod e I, of' Kiew, 85; Forsuosus, 91; Ste- John XIV., 81 Polii IS; Iii isv I. 5ol phen VII., 97. XV., 85; John XVI., Swu stosl iw I., 7 86; Gregoiry V, 963 Wiewolod I. 78, Swac Silvester Ii. 9. tpolk II., 9") RUSIA.. RUSe-IghoT 1. 13;Sw-er Co-s-Count. IX., 11clne, tosl~aw I 45, Jaro- 25 R om,, niis Ill.,2 polk I., 72; Waldimer Michael IV., 94, Mithe Great, 80. cliael V., 36; Zoe and COIN ST A NT I N OP L is- CON. - Ale. andei, 11, Theodroil. aniid Const -Ir-ene, 797; Nicepho- Rominiis, 19; Constan- X., 40; TIheodlori,5; rus I., 2; Michael I., tineVIl.,45; Romanus Mich. VI., 5b, Isaac 11; Leo V., 1i3; Mi- II., 59; Nicephiorus I11 I, 157, Coiist XI.,59 ehsuel It., 20; Thieophii- 63; Joth Z imices, 69; F tidociaoacur Ruin i'mic ins, 29; Michael Ill., Basil l. and Con.VIII., Ill., 67, Mi'ichi VN 42; Basil, 67; Leo VI., 76. 7, Nicepliorus1III,7 86. Alexius I., jt

Page  332 332 MISCELLANEOUS PROBLEMS. [ART. XXII. 1100. 1200. 1300. ENGLAND.-HlIeny I.; ENG.-Henry III., 16; ENG.-Edward II., 7; Stephen,36;HelnryII., Edward I., 72. Edw. III.,27; Richard 54; Richard I., Cceur II., 77; I-Ienry IV., 99. de Lion, 89; John, 99. SCOTLAND. - Alexander SCOT. —AlexanderlI.,14; ScoT. —Robert Bruce, 6; I., 7; David I., 24; Alex. III., 45; Marga- David II., 29; Edwsard Malcolm IV., 53; WVil- ret, 86; John Baliol, Baliol, 32; Robert II., liam I., 65. 88; Interregnum, 96. 70; Robert III., 90. FrxANcE. —Looui si VI., the FR. - Louis VIII., 23; F. —LouisX., K. of NaGross, 8; Louis VII., Louis IX., (St. Louis,) varre, 14; Philip tie 37; Philip II., 77; Au- 26; Philip III., the Tall, K. of Nav., 16; gustus, 80. Bold, 70; Philip IV., Chas. IV. (the Fail), the Fair, 85. K. of Nav., 22; Philip GERMANY.-HenryV., 6; GER.-Frederic II., 12; VI. (the Fortunate), Lotharius II., the Sax- Conrad IV., 50; Wm. 28; John T., (thle Good,) on, 25; Conrad III., of Holland.54-; Rich)d, 50; Chas.V.(the\Vise), 38; Frederick I., Bar- Duke of Cornwall, 57; 64; Chas. VI., S0. barossa, 52; HenryVI., Rodlolph ofHapsburggh GER. — enry VII., 8; 90; Philip and Otho 73; Adolphus of Nas- Louis of Bavl-ria, and IV., 98. sal, 92; Albert of Aus- Fred. of Austria, 14; tria, 98. Chas. IV., 46;, NVinceslaus, 78. SPAIN.-AlphonSO I., N. SP.-James I., Ar., 13; SP.-Alphlonso V., C.,14; and Arr., 4; Urraca, Henry I., C., 14; Fer- Alph. 1V., Ar., 27; JoCas., 9; Alphonso II., din. III., C., 17; Theo- anna II., N., 28; l'eter Cas., 26; Garciaz V., baldI,N., 34; Alphon- II., Ar. 36; Chas. I., Nav., RamirezII.,Ar., so IV., C., 52; Theob. N., 49; Peter I.,C.,50 34; Petronilla and Ray- It., N., 53; Henry I., Henrv II., C.,69; Johii mondo, Arr., 37; San- N., 70; Joanna I., N., C., 79; Charles Ill., cho VI., the WVise, N. 74; Peter III., Ar., 76; N., 86; John I.,A., 87; 50; Sancho II., Cast., Sancho IV., C. 84; A1- Henry III.,C., 90; Mar57; Alphonso II., Arr., phonso III., Ar., 85; tin, A., 95. 62; Sancho VII., N., Jtas. II., Ar., 91; Ferd. 94; Peter II.. Arr., 96. IV., C., 95. PAPAL STATES.- Ielas PA. ST.-Honorius IIT., PA. ST.-Benedict X., IT., 18; Calixtus II., 17; Gregory IX., 27; Clem.V., 5; Jno X TXI. 19; Honorius II., 25; Celestine IV., 41; In- 16; Alex. II., 27; BeInnocent II., 30; Ce- noc. IV.,43; Alex. IV., ned. XI., 34; Clem VI., lestine II., 43; Lucius 54; Urb.IV.,62; Greg. 42; Innocent VI., 53; II., 44; Eugene III., X., 64; Clem. IV., 65; Urban V., 63; Oreg. 45; Anastasius IV., 54; Innoc. V., Adriin V., XI., 71; Itrban VI., 78; Adrian IV., 55; Alex. and John XX., 76; Ni- Boniface IX., 90. III., 59; Lucius 111., cholas II., 77; Mirtin 81.; Urban III., 85; IV., 81; Honorius IV., Gregory VII., 87; Cle- 85; Nich. IV., 88; Cement lII., 88; Celestine lestineV., 91; Boniface III., 91; Innocent III., VIII., 95. 98. Rrss rA.-WValdimir II., Rius.-Jurje ITI., 13; Con- R u ss. -- ichailow, 5; 13; Mistislaw, 25; Ja- stantine, 17; Jaroslaw Jurje IIl., 17; lrau I., ropolk II., 32; Wse- II., 38; Alexander, 45; of Moscow, 28; Semenei, wolod II., 38; Isaslaw Newskoi50; Jaroslaw 40; IwanlI., 53; DimiII., 46; Jurje I., 49; III., 62; Wasilej I., 70; trej Il., 59; Dimlit. II., Andrej, 57; Michel 1., Dirnitrej, 75; Andrej, 63; ~Wasilej II., 89. 75; -;Vsewolod III., 77. 81; Danilo, 94. CONSTANTINOPLE.-Jno. CoN.-Isaac IT. restored, CoN.-Johin Canta, 41; Comnenus, 18; Manuel 3; MIourzouffe,4; Bald- John Pa.leologusn 55; Com..43; Alex'sCom., win, 4; Henry, 6; Pe- Manuel Pal., 91. 80; iAndronicus, 83; ter, 17; Robert, 19 Isaac II., 85; Alexius John,with BaldwrinII., Angelus, 95. 28; Baldwin IT., alone 37; Michael Paleolo_ us 61; Andronicus,82. L~~,,

Page  333 ~105.] CHRONOLOGY. 333 1400. 1500. 1600. FNGLAND.-.Hen. V., 13; ENG. —Henry VIII., 9; GREAT BRITAIN. —Jas. Hen. VI., 22; Edward Edward VI., 47; Mary, I., 3; Charles I., 25; IV., 61; Edw. V. and 53; Elizabeth, 58. Cromwell, 53; Charles Richard III., 83; Hen. II., 60; James II., 85; VlI., 85. WVm. and Mary, 89. ScoTLArND.-James I.,5; ScoT.-James V., 12; Jamnes II., 37; James Mary, 42; James VI., III., 60; JamesIV., 87. 67. FRANcE.-Charles VII., FR.-Francis I., 15; F. —Louis XIII., 10 the Victor, 22; Louis Henry II., 47; Francis Louis XIV., the Great, XI., the Prudent, 61; 11.,59; CharlesIX.,60; 43. CharlesVIII., the Affa- Henry III., 74; Henry ble, 83; Louis XII., 98. IV., the Great, 89. GEaRMaANY.-Robert; Si- GsER. —Charles V, 19; GER.-Mialthias,12; Fergismund, 11; Albert II, Ferdinand I., 58; Sax- dinand II., 19; Ferd. 37; Frederic III., 40; imilian II., 64; Ro- III., 37; Leopold I., Maximilian I., 93. dolph II., 76. 58. SPAIN. —John II., Cast., SP.-Charles I., 16; Phi- SP.-Philip IV., 21; 6; Ferdinand I., Arr., lip II., 56; Philip III., Charles II., 65. 12; Alphonso V., Arr., 98. 16; Blanche, N., and John I., A., 25; Henry IV., C., 54; Ferdin. II. and Isabella, C., 74; Ferd. II., the Catholic, A., 79; Eleanor and Francis Phoebus,Nav., 79; Catharine, N., 83. PAPAL STATES. -Inno- PA. ST. —Pius III. and PA. ST.-Leo XI. and cent VII., 4; Gregory Julius 1I., 3; Leo X., Paul V., 5; Greg. XV., XII., 6; Alexander V., 13; Adrian VI., 22; 21; Urban VIII., 23; 9; John XXII., 10; ClementVII., 23; Paul Innoc.X., 44; AlexanMart-;17 7; Eugene II., 34; JuliusII., 50; der VII., 55; Clement IV., lt; Nicholas V., Marcellinus II., 55; IX., 67; Clem. X, 70; 47; Calixtus III., 55; Paul IV., 56; Pius Innoc. XI., 76; Alex. Pius I[.,58; Paul lI., IV., 59; Pius V., 66; VIII., 89; Innoc.XII., 64; Sixtus IV., 71; Gregory XIII., 72; Six- 91. Innoc.VIII., 84; Alex. tus V., 85; Urban VII. VI., 92. and Gregory XIV., 90; Innocent IX., 91; Clement VIII., 92. RUSSIA. —WTasilej III., RTrs. —Vasilej IV., 5; Rus. —Wasilej Schuis25; Iwan Wasilej III., Iwan Wasilej IV., 33; koi, 6; Mich. Fedrow62. Feodor I., 84; Boris itsch, 13; Alexej, 45; Godunow, 98. Feodor lI., 76; Iwan V., 82; Pet. the Great, 85. CONSTANTINOPLE..-J no. CoN.-Selim I., 12; So- CoN.- Achmed I., 4; Paleologus, 25; Con- lyman, 20; Selim II., Mustapha, 17; Osman stantine Paleol., 48; 66; Amurath, 74; Mo- I., 18; Mustapha reMohammed II., 53; Ba- hammed III., 95. storedl, 2-2; Amurath jazet, 81. IV., 23; Ibrahim, 40; Mohammied IV., 48; Solyman 1I., 87; Achmed II., 91; Mustapha II., 95.

Page  334 834 MISCELLANEOUS PROBLEMS. [ART. XXIII 1700. 180( GREAT BIITAIN. —Anne, 2; George GT. BRIT.-George IV., 20 William I., 141; Geo. II., 27; Geo. III., 60. IV., 30; Victoria, 37. FRANCE.-Louis XV., 15; Louis FR.-Napoleon Emperor, 4; Louis XVI., 74; Republic, 92. XVIII., 14:; Charles X., 2- i Louis Philippe, 30; Republic, 48. GERxMAv-.-Joseph I, 5; Chas. VI., AvsTRIA.-Francis I., 6; Ferdinand 11; Chas. VII., 42; Francis I. and I., 35; Francis Joseph I., 48. M1aria Theresa, 45; Joseph II., 05; Leopold II., 90; Francis II., 92. SPAIN.-Philip V.; Ferdinand VI., Sp.-Ferd.VII. and Joseph Napoleon, 51; Charles II1., 59; Chas. IV., 88. 8; Ferd. VII., 14; Isabella II., 33. PAPAL STATEs.-Clement XI.; Inno- PA. ST. —Pius VII., Leo XII., 23; cent XIIt., 21; Benedict XIII., 24; Pius VIII., 29; Gregory XVI., 31; Clerm. XII., 30; Bened. XIV., 40; Pius IX., 46. Clem. XIII., U8; Clem. XIV., 00; Pius VI., 75. RussIA.-Catharine I., 25; Peter II., Rus. —Alexander I., 1; Nicholas I., 27; Anne, 30; IwanVI., 40; Eliza- 25. beth, 41; Peter I1I. and Catharine II., 62; Paul 1., 06. CONSTANTINOPLE.-Achmed III., 3; CON.-Mohammed VI., 8; AbdulMohamnned V., 30; Osmran II., 54; Medjid, 39. Mustnpha III., 57; Abdul-Hanet, 74; Selim III., 89. TIHE SMALLER EUROPEAN STATES, FROM 1700. 1700. 1800. SwEDEIN.-Charles XII., 1697; Ulrica Sw:E:.-Charles XIII., 9; Charles Eleanora, 19; Fredleric, 20: Adol- John XIV., 1S; Oscar I., 44. plihus Frederic, 59; Gustavus III., 71; Gustavus IV., 92. DENMARK.-YFred. IV., 1699; Chris- DE\N.-Frederic VI., 8; Frederic i tian VI., 30; Fred. V., 46; Chris- VII., 30; Christian VIII., 39; Fretian VII., 6(. deric VIII., 48. NAPLEs.-Charles II., 13; Charles NAP.-Joseph Napoleon, 8; Joachimn III., 35; Ferdinand IV., 59. Murat, 15; Ferdinand I. (of thie two Sicilies), 21; Francis, 26; Ferdiand lII., 30. PoRTUGAL.-Joohn V., 6; Joseph PORT.-Pedro IV., 26; Maria da Emanurlel 50; Maria, 77; JohnVI., Gloria, 20. 99. PRUssIA.-Frederic I., 1; Frederic PRUs.-Frederic William IV., 40. William I., 13; Frederic II., thei Great, 40; Frederic William II., 80; Frederic WVilliam III., 97.

Page  335 ~ 106.] THE IMOON'S AGE AND SOUTIIING. 335 BIRTHS AND DEATI-IS OF CELEBRATED PERSONS. Born. I)ied. Born. I)ied. Solon........ B...... C 650 B.C. 559 Oliver Cr(nmwell.... A 0,1 () A. ). 1657 Confucius............. -- 479 liltm............. t..l 1(74 Socrates........... " 470' 400 Algelno Sdnietl v...' 1117' 16S3 Plato............... 430 " 317 Moliere........... I -2 1i" 1673 Dernosthenes....... 3SI31 322 Pascal 16;d 3 16 (S2 Alexander the Great 356 3'23 Georlge ox....... i 24 169(0 IIannibal....... " 24-17 183 Christopher eren.. 16:392 1723 ScipioAfricanus.. " 235'- 181 Isaac Newuon......, 1364' 1727 Cicero............ " 106 ~ 43 I,eilinitz............ 1646 1.716 Virgil.........., 70 " 10 Fenelon...........1. 651l 1715 Constf;ntine........ A.D. 274 A.D. 337 P.eter the Great..... 1h672 1725 Mallholnet........... 570 " 32 Handel............ c 1680 17(60 Charlemnag ne o....... a 74.2 Haroun Alraschid. " 765 809 Belrklev...........'; 1S 1753 Canute............. 0: --- 1036 Voltailre........... 16!)94 177S Jenghis IZhan.......... 1160 1227 Jolln W'esley......: 17051' 179() Dante............. 12655 1" 1321 Euler.............. 11 183 Petrarch............ 1304 1374 Lin nmus........... 17 i' 171 1177 Gutenherg........... " 1300 9 1.468 Rousseau............ t' 71~.'. 1 177 Jo;in of Are........ 0. 1410' 1431 Ht-ume; 1717 1776 Columbus.......... 1 1.442 " 1506 Blackstone....... 3 I 17S(I Copernices........ " 1473 " 151:3 Ada(m Smith........ 3 1790 Michael Ang elo.. 1 1.474 1564.Josllhta Reynoldis... l 17B3 i 1792 Pizarro........... 1 1475 - 1541 Cowper............ " 31 10 Iuther............". 1.83 " 1546. Gibbon............ 1737 1791 Raphael............ 1483 " 1520 Robespierre........ 1759 194 Ioyola.............. I.)1 "1 1556 Npol. " 1135 18 1 Napier............ 1550 1617 Fourier......... 1772 " 1837 EXAMPLES. 1. Columbus discovered America Oct. 11, 1492, O. S. What was the day of the week? Ans. Thursday. 2. On what day of the week was the commencement of the year in which Henry III. ascended the English throne?a Als. Friday. 106. THE 3MOON'S AGE AND SOUTHING. I. The Golden Numknber. The mean time of the moon's revolution around the earth, is 29.53 days. In 19 years, or 228 solar months, there are 235 lunar months. This period of 19 years, is called a lunar cycle, and all the changes and eclipses of the moon, are the same for each cycle. a In England, until the year 1752, the year was considered as beginning on the 25th of Miarch.

Page  336 336 MISCELLANEOUS PROBLEMS. [ART. XXIII. The year of the lunar cycle corresponding to any given year, is called the GOLDEN NUMBER for that year. It is found by the following rule: Add 1 to the given year, and divide by 19. Thle remnainder vwi be the golden number. (If the remainder is 0, the golden zumber is 19.) Required the golden number for A. D. 1853. 1-84 = 971}. Ans. 11. II. The Ecpaet. The Epact is the moon's age at the beginning of the year. It generally either increases by 11, or diminishes by 19, every year, and never exceeds 29. It is found by the following rule: Divide the given year by 19, and nmultiply the remainder by 11. TlZeproduct will be the epact, if it does got exceed 29. I-f the product exceeds 29, divide it by 30, and the reaindcler will be the epact. The epact for 1845: 1 845 = 97-; 2 x 11 = 22. Ans. 22. The epact for 1857: 18597 19714; 14 X 11 = 154; 1'?4 5_4. Ans. 4. The epact for 1862: jl 8 = 98. Ans. 0. III. The Number or0 Epact for the MJonth. The Numlber for any month, shows what the moon's age would be at the beginning of that month, provided it was new moon on the first of January. It is found by the following rule: Divide the nulmber of dclays in the precedingz months, (reckoning fronm the beyinning of the year,) by 29.53, and the 9nearest whole nlumber to the remainder, is the epact or lumnber for the 9mntol, required. 31 ~ 28+ O31 The epact for April. In common years, 29.3 - 3 Ans. 0. 29.53

Page  337 ~106.] THE MOON'S AGE AND SOUTHING. 337 The epact for April. In leap years, 3129+53 -= 1.41 29.53 329.53. Ans. 1. IV. The Moon's Age. The Moon's Age, is the number of days that have elapsed since new moon. It never exceeds 30. It is found by the following rule: To the epact for the year, add the number for the month, and the day of the month, and divide by 30. The remainder will be the moon's age. What was the moon's age, Aug. 18th, 1849? 6+5+18 30 29 — 3s0' A.Ans. 29 days. V. The Moon's Southing. The SOUTHING is the time when the moon is on the meridian.. It is found by the following rule: Mzultiply the moon's age by.8, and the product will be the hours past noon. If the hours exceed 12, subtract 12, and the remainder wt7l represent the time after midnight. Find the time of the moon's southing, for July 4th, 1776. Epact 9; number for the month 5; moon's age 18. 18 x.8 = 14.4h. = 2h. 24m. after midnight, moon south. EXAMPLES. 1. Find the Golden Number for the years of accessions of each of the monarchs of Great Britain. Ans. Jas. I. 8; Chas. I. 11; Cromwell, 1; Chas. II. 8; &c. 2. Find the epact for the years of accession of William the Conqueror, and of each of the ten following monarchs of England. Ans. Wmin. I. 22; Wm. II. 14; Hen. I. 7; Stephen 15, &c. a See Table of Contemporary Princes. 22

Page  338 338 MISCELLANEOUS PROBLEMS. [ART. XXIIL 3. Find the moon's number for each month of the year, both for common years, and for leap years. Ans. common years; Jan. 0; Feb. 1; Mar. 29; &c. leap years; " 0; " 1; " 1;&c. 4. What was the moon's age on the day of the battle of Bunker Hill, June 17th, 1775? 5. The French army commenced its retreat from Moscow, Oct. 18th, 1812. At what time on that day was the moon on the meridian? 1 07~. MENSURATION. In the following formulas, a represents the area; al the altitude; b the base; br the breadth; c the circumference; ca the conjugate axis, (of an ellipse); cs the convex surface; d the diameter; fa the filred axis, (of a solid of revolution); h the hypothenuse; he the height; Z the length; lb the lower base, (of a frustum); p the perpendicular; ra the revolving axis; s the solidity; sh the slant height, (of a cone, frustum, &c.); ub the upper base. 1. THE PARALLELOGRAM. a = b X al. 2. THE TRIANGLE. a = - x a To find the area of a triangle, when the three sides are given: I. Add the three sides together, and take half their sum. II. From this half sum take each side separately. III. Form the continued product of the half sum and the three remainders, and extract the square root of that product. 3. TEIE RIGHT ANGLED TRIANGLE. h = / b2 + p2 4. THE CIRCLE. c = 3.1416 X d; a =.7854 X d2; a =.07958 x c2. 5. THE ELLIPSE. a -- ta X ca X.7854.

Page  339 ~1:07.] MENSURATION. 339 6. THE PARALLELOPIPEDON. s = 1 X br X he. 7. THE PRISM. s = al x a of b. 8. THE CYLINDER. Cs = al X c of b; s = al X a of b. sh X c of b 9. THE CONE,AND PYRAMID. CS = S of b al X a of b 10. THE FR-USTUM. CS = (C of Ub + C of lb) X sh 2 (a of ub +a of lb + /a of ub x a of lb) X al S-3 11. THE SPHERE. Cs = cX d; S-.5236 X d3; s=.01689 X C3. 12. TIIE ELLIPSOID. s =fa X ra2 X.5236. 13. THE POLYGON. To find the area of a regular polygon when one of its sides is given: Multiply the square of the side, by the multiplier standing opposite the name of the polygon in the following table. The radius of the inscribed circle, is found by multiplying the side by the proper number in the right hand column. No. of Radius of InSides Names Multipliers. scribed Circle. 3 Triangle.433013.288675 4 Square 1.000000.500000 5 Pentagon 1.720477.688191 6 Hexagon 2.598076.866025 7 Heptagon 3.633912 1.088262 8 Octagon 4.828427 1.207107 9 Nonagon 6.181824 1.373739 10 Decagon 7.694209 1.538842 11 Hendecagon 9.365640 1.702844 12 Dodecagon 11.196152 1.866025 14. THE POLYEDRON. To find the surface of a regular polyedron: Multiply the square of the linear edge, by the tabular number in the column of surfaces.

Page  340 340 MISOCELLANEOUS PROBLEMS. [ART. XXIII. To find the solidity of a regular polyedron: Multiply the cube of the linear edge, by the tabular number in the column of solidities. Sides. 4 Tetraedron 1.73205.11785 6 Hexaedron 6.00000 1.00000 8 Octaedron 3.46410.47140 12 Dodecaedron 20.64578 7.66312 20 Icosaedron 8.66025 2.18169 EXAMPLES. 1. By the second method of Analysis, (~~ 57 and 59,) determine the formulas for finding b, and al, in a parallelogram. Ans. b = a -- al; al = a - b. 2. What are the formulas for b and at, in a triangle? 3. Required the formulas for b and p, in a right-angled triangle. Ans.' b = h2 -- P22; p= / h2 - b2. 4. Find two formulas for d, and an additional formula for c, in a circle. Ans. d=c * 3.1416 d = a.78 54; c= /a --.07958. 5. Obtain formulas for ta and ca in an ellipse. 6. What are the formulas for i, br, and he in a parallelopipedon? 7. Find formulas for al and a of 6 in a prism. 8. Obtain formulas for c of b and a of b, and two formulas for al in a cylinder. 9. Obtain formulas for sh, al, c of b, and a of b in a cone or pyramid. 10. Find two formulas for c and for d in a sphere. Partial Ans. c = /s --.01689.

Page  341 ~ 107.] MENSURATION. 341 11. Required formulas for c of ub, c of Ib, sh, and aL, in the frustum of a pyramid, or of a cone. 2cs 2cs Ans. c of ub =2cs - c of; c of 6b = c of u; sh sh sh = 2cs - (c of ub + c of lb); al = 3s -- (a of zub + a of lb +,Va of Bub x a of lb). 12. What are the formulas forfa and ra in an ellipsoid? Ans. fa = s (.5236 X ra); ra = /s (.5236 x fa.) 13. The solidity of a prism is 516 c. ft., and the altitude 27-Ift. What is the area of its base? Ans. 184 —! sq. ft. 14. The convex surface of a pyramid contains 47 sq. ft., and the circumference of its base is 31ft. Required its slant height? Ans. 28 ft. 15. The solidity of a frustum is 1447~ c, ft.; the area of the upper base 49 sq. ft.; the area of the lower base 81 sq. ft. What is its altitude? Ans. 22-ft. 16. The solidity of a cylinder is 28 c. ft., and the altitude 14ft. What is the diameter of the base? Ans. 1.59ft. 17. Required the radius of a sphere whose solidity is 75 c. ft. Ans. 2.616ft. 18. The fixed axis of an ellipsoid measures 10ft.; and its solidity is 80 c. ft. What is the length of its revolving axis? Ans. 3.9088ft. 19. The area of a regular dodecagon is 47ft. Required the length of each side, and the radius of the inscribed circle. Ans. 2.0488ft.; 3.823ft. 20. What is the area of a triangle, whose sides measure respectively 16 rods, 27 rods, and 19 rods? Ans. 3R, 29.398r,

Page  342 342 MISCELLANEOUS PROBLEMS. [ART. XXIIr. 21. A regular icosaedron contains 81 c. ft. What is the length of each of its linear edges?a Ans. 3.336ft. tiA. TONNAGE OF VESSELS. The accurate determination of the tonnage of any vessel, is a very difficult problem. The best method probably, is, to divide the vessel into a number of sections, and to determine by numerous measurements, the average length, breadth, and depth of each section. The solid contents can thus be found in cubic feet, and by dividing the number of cubic feet by 40,b the answer will be obtained in tons. The accuracy of the result, will depend on the number of measurements that are taken. This process is, however, a very tedious one, and numerous rules have been framed, in order to abridge the labor of computation. Of these rules, the following are the most frequently employed. I. Carpenters' Rule. Measure the length, breadth, and depth, all in feet. Divide the continued product of the three dimensions by 95, and the quotient will be the tonnage. If the vessel is double-decked, half the breadth is taken as the depth. II. United States Government Rule. If the vessel is single-decked, take the length from the fore part of the main stem to the after part of the stern post above the upper deck, the breadth-at the broadest part above the main wales, and the depth from the under side of the a Questions similar to the foregoing, may be multiplied by the teacher, to any extent that may seem desirable. b Dividing by 40 will give the actual tonnage. Each of the following rules gives a result considerably less than the true tonnage. In freighting by the ton, the owner may charge either by the ton of weight, or the ton of measure, whichever will yield the largest tonnage.

Page  343 ~108.] TONNAGE OF VESSELS. 343 deck to the ceiling, in the hold. From the length deduct a of the breadth, multiply the remainder by the breadth, and that product by the depth. Divide the last product by 95, and the quotient will be the government tonnage. If the vessel is double-decked, substitute half of the breadth for the depth, and proceed as directed for singledecked vessels. III. -British Government Rule. Divide the length of the upper deck between the after part of the stem and the fore part of the stern-post, into 6 equal parts. At the foremost, the middle, and the aftermost of these points of division, measure in feet, and decimals of a foot, the depths from the under side of the upper deck to the ceiling at the limber strake. (In the case of a break in the upper deck, the depths are to be measured from a line stretched in a continuation of the deck.) Divide each of the three depths into 5 equal parts, and measure the inside breadths at the following points, viz: at - and 4 below the upper deck of each of the extreme depths, and at 5 and - below the upper deck of the niidship depth. At half the midship depth measure the length from the after part of the stent to the fore part of the stern-post. To twice the midship depth add the two extreme depths, for the sum of the depths. Add the upper and lower breadths at the foremost division, 3 times the upper breadth and the lower breadth at the midship division, and the upper and twice the lower breadth at the after division, for the sum of the breadths. Multiply the sum of the depths by the sum of the breadths, and the product by the length, and divide the final product by 3500, which will give the number of tons for register. If the vessel has a halfdeck, or a break in the upper deck, measure the inside mean length, breadth, and height of such part thereof as may be included within the bulk-head. Divide the continued product of these three measurements by 92.4, and the quotient will be the number of tons to be added to the result as above found.a Previous to Jan. 1st, 1846, the British rules for estimating tonnage were similar to our own. - But this rule led to the building of ships of forms improper for the purpose of safe navigation, in order that, by measuring less than their real burden, they might evade a part of the

Page  344 344 MISCELLANEOUS PROBLEMS. [ART. XXIfI. If the vessel is laden, measure the length on the upper deck between the after part of the stem and the fore part of the sternpost, the inside breadth on the under side of the upper deck at the middle point of the length, and the depth from the under side of the upper deck down the pump-well to the skin. Divide the continued product of these three dimensions by 130, and the quotient will be the register tonnage. In open vessels, the depths are to be measured from the upper edge of the upper strake. EXAMPLES. 1. Find by rule 1, the tonnage of a double-decked vessel 163 feet long, and 31 feet wide. Ans. 824~ tons. 2. Find by rule 2, the government tonnage of a doubledecked vessel 180 ft. long, and 32ft. wide. Ans. 866.6 tons. 3. Required the British government tonnage of a vessel with the following measurements: length at half the midship depth 175ft.; depths, at the foremost point of division 29ft., at the middle point 30ft., at the aftermost point 28ft.; breadths, at the. and 4 depths of the foremost division 33 and 24ft., at the 2 and -4 depths of the midship division 39 and 36ft., at the - and 4 depths of the aftermost division, 35 and 29ft.; half-deck, length 75ft., mean breadth 35ft., height 7ft. Ans. 1971.41 tons. 109. GAUGING. The easiest way of finding the contents of casks, is by the diagonal rod. The contents given by the rod are only correct for casks of the most common forms. The same result as the rod would give, may be found in the following manner: duties. The method now employed, gives the tonnage of all ships, however built, with tolerable accuracy, and therefore removes the temptation to build vessels of an unsuitable form.

Page  345 ~ 109.] GAUGING. 845 Take any rod, and putting it in at the bung, extend it to the opposite corner of either head, and measure the distance in inches. Extend it in a similar manner to the other head, and measure the distance. Half the sum of these two distances, will be the diagonal. Then, Multiply the cube of the diagonal by -0 =44 9 The quotient will be the contents in imperial gallons. For wine gallons, substitute 173; for beer gallons, 141.6; for dry gallons, 148.5, in the place of 144. In order to determine the contents with accuracy, casks are usually divided into four varieties, according to the degree of their curvature. In each of the following formulas for those four varieties, D denotes the bung diameter, d the head diameter, and a the length of the cask. I. WhVen the casks is formed ilce the middle zone of a spheroid..0009442 x a X (2 D2 + d2) = contents in imperial gallons. For the contents in wine gallons, substitute.0011333; for beer gallons,.0009284; for dry gallons,.000974, in the place of.0009442. II. When the cask is formned like the middle zone of a parabolic spindle..0001888 X a X (8 D2 + 4 D X d + 3dA2) = contents in imperial gallons. For wine, beer, or dry measure, substitute.0002267,.0001856,.000195, in the place of.0001888. III. - WhAen the cask is formed like two equal frustums of a parabolic conoid..0014163 x a X (D2 + d') = contents in imperial gallons. For wine, beer, or dry measure, substitute.0017,.0013926,.001461, in the place of.0014163.

Page  346 346 MISCELLANEOUS PROBLEMS. [ART. XXIII. IV. When the cask is formed like two eqaal frustums of a cone..0009442 X a X (D2 - D x d +d2) = contents in imperial gallons. For the other measures, substitute as in Case I. The following method is more accurate than either of the foregoing, when the diameter midway between the head and the bung, can be accurately determined:Add the squares of the head diameter, of the bung diameter, and of twice the middle diameter. The sum, multiplied by.0004721 times the length, gives the contents in imperial gallons. For wine gallons, substitute.0005667; for beer gallons,.0004642; for dry gallons,.000487, in the place of.0004721. If the staves are of uniform thickness, the middle diameter may be found by measuring the circumference, dividing by 3.1416, and subtracting twice the thickness of the staves from the result. EXAMPLES. 1. The diagonal of a cask is 32 inches. Required its contents in wine gallons.'Ans. (32)3 x 173 - 64000 = 88.576 gallons. 2. Find the contents in imperial gallons, of casks of each of the four varieties, the length of each cask being 40 inches, and the diameters 30 and 36 inches. Ans. 1st var..0009442 X 40 x (2 x 362 + 302) = 131.886 gallons. 2d var..0001888 X 40 x (8 x 362+4 x 36 x 30 +3 x 302) = 131.314 gallons. 3d var..0014163 X 40 x (362 302)=124.408 gallons. 4th var..0009442 x 40 X (362- 36 X 30+ 302) - 123.628 gallons.

Page  347 ~ 110.] MISCELLANEOUS EXAMPLES. 347 3. The head diameter of a cask is 25 in., the bung diameter 30in., the middle diameter 28 in., and the length 36 in. Required the contents in beer measure. Ancs. (252 + 302 + 562) x.0004721 X 36 = 79.216 gallons. 4. The circumference of a cask at the bung is 113 in., at the head 91 in., and at a point midway between the head and the bung 106 in. Required the contents in each measure, the length being 40in., and the thickness of the staves e in.a Ans. 119.16 imperial gallons. 143.03 wine gallons. 117.16 beer gallons. 122.92 dry gallons. 1 10. MISCELLANEOUS EXAMPLES. 1. If the multiplicand is 7, and the product 2, what is the multiplier? 2. The dividend is 1, and the quotient 8; what is the divisor? 3. The dividend is 6, and the quotient 182; what is the divisor? 4. The sum of two numbers is 7-, and one of the numbers is 4.759; what is the other? 5. The difference of two numbers is 13-, and the greater number is 29.43; what is the less? 6. The suni of two numbers is 7 and their difference is 6-33T; what are the numbers? a When the circumferences are given, it is not necessary to find each diameter separately. We may proceed with the circumferences precisely as with the diameters, until we obtain the sum of the squares. This sum should be divided by the square of 3.1416 (9.8696), and the quotient will be the sum of the squares of the diameters. A deduction should -be made from each outside circumference, of six times the thickness of the staves, and the remainder will be nearly equivalent to the inside circumference.

Page  348 848 MISCELLANEOUS PROBLEMS. [ART. XXIrI. 7. What number must be subtracted from 39- to leave 17k?9 25 8. What number must be added to 2375r to make 47.432? 9. What number must be multiplied by 38, and the product divided by 7.365 to give 87- as a quotient? 10. What is the difference between -436 and p- of 7T. 15cwt. 3qr.? 11. What is the difference between 28 miles, and 27m. 7fur. 39r. 5yd. 2ft. 11.9in? 12. Reduce X of 9m. 7fur. 7fu 9r. 5yd. 2ft. to inches. 13. If from a purse containing ~35 7s. 11id., I pay to each of 15 laborers, ~1 9s. 83d., how much will be left? 14. Find the sum, the difference, and the product of 874.91 and 42. 15. In 1284 lb. of water, there is 142 lb. of hydrogen. How much hydrogen is there in 273- lb. of water? 16. A grocer sold 17cwt. 3qr. 17 lb. of sugar, at 6 cents a pound, receiving in exchange 20 barrels of flour, at $4J per barrel, and the balance in money. How much money did he receive? 17. From 5 of 3T. 17cwt., subtract 1 of 7T. 3cwt. lqr. 181b. 18. What will be the freight of, 17cewt. for 89- miles, if $7.63 be paid for carrying 11-3 tons 9- miles? 19. How many raisins, at 8-2 cents a pound, must be given in exchange for 163gal. 2qt. 3gi. of molasses, at $.374per gallon? 20. Bought 16cwt. 3qr. 16 lb. of rice, at $4.00 per cwt., and 9cwt. 2qr. 5 lb. of pearl barley, at $4,.374 per cwt. How much would be gained on the whole, by selling each at 4j cents a pound?

Page  349 ~ 110.] MISCELLANEOUS EXAMPLES. 349 21. If 635yd. of broadcloth cost $255, at what price must it be sold per yd., in order to gain $25.50? 22. A hogshead of sugar at $7.00 per cwt. cost $43.75. What did it weigh? 23. How much shalloon that is {yd. wide, will line 145 yards of cloth, that is 13yd. wide? 24. What is the price of 16 boxes of raisins, each holding 92 lb., at 91 cents per lb.? 25. How much money, that is 9 per cent. below par, will pay a debt of $187.50? 26. If 9 men mow 10 acres of grass in a day, how much will 11 men mow in 2- days? 27. In what time will $150 gain $6.37~-, at 6 per cent., simple interest? 28. When molasses is 31- cents a gallon, how many hogsheads, each holding 97ga1. 3qt., can I buy for $366.561? 29. What will be the price of 7 bales of sheeting, each bale containing 9i pieces, and each piece measuring 3043yd., if 26yd. cost $2.924? 30. Bought 391 bushels of potatoes for $12.871. At what price per bushel must they be sold, in order to gain 15 per cent.? 31. What is the interest of $9431 for 3yr. 7mo. 13dy., at 7 per cent.? 32. How much may a man spend per day, whose income is $500 a year, after deducting -5 per cent. for taxes? 33. The words in Johnson's Dictionary have been classifled as follows:a Articles, 3; nouns, 20409; adjectives, a Gregory.

Page  350 350 MISCELLANEOUS PROBLEMS. [ART. XXIII. 9053; pronouns, 41; verbs active 5445, neuter 2425, passive 1, defective 5, auxiliary 1, impersonal 3; verbal noun, 1; participles, 38; participial adjectives, 125; participial nouns, 3; adverbs in ly, 2096; other adverbs, 496; prepositions, 69; conjunctions, 19; interjections, 68. What per-centage of the entire number belongs to each part of speech? 34. Owing to the curvature of the earth, the difference of level in 1 mile is 8 inches. What would be the difference in - mile, the level varying as the square of the distance? in 5,- miles? Ans. 2in.; 22 —4ft. 35. Suppose a school of 180 boys to breathe 20 times each per minute, requiring for each respiration 30 c. in. of air, what amount of carbonic acid would they produce in two sessions, of 3 hours each, estimating that 5 per cent. of the air inhaled is changed into an equal volume of carbonic acid? Ans. 1125 c. ft. 36. A vessel of 400 tons has a keel 48 feet long. What length of keel has a vessel of 750 tons, that is built on the same model? Aurs. 59.19ft. 37. There are two cannon balls, one weighing 28 pounds, and the other 9 pounds. What is the diameter of the greater, that of the less being 5 inches? Ans. 7.3in., nearly. 38. In an establishment in Lowell 50 cows were kept, of which the average number giving milk was 35. Each cow consumed annually 4.18 tons of hay at $18.50 per ton, and green vegetables worth $20.36. What was the loss in two years, exclusive of the amount expended for attendance, the whole amount of milk obtained being 99705 quarts, worth five cents a quart? 39. Bought 6 bales of cinnamon, weighing gross 4cwt. 3qr. 2 lb., tare 9 lb., at $11 per cwt., and paid charges for

Page  351 ~ 110.] MISCELLANEOUS EXAMPLES. 351 freight, duties, &c. $11.438. What rate per cent. would be gained or lost, by selling the whole at $.15 per lb.? Ans. 25 per cent. gained. 40. Find the value of 4 cases of gum tragacanth, at ~20 8s. per cwt., duty 25 per cent. ad valorem, the cases weighing as follows, viz: cwt. qr. lb. No. 15 gross 2 1 7 tare 411b. No. 16 " 2 1 11 " 42" No. 17 " 2 1 7 "1 40" Draft 2 lb. per case. No. 18 " 2 0 27 " 39' J Ans. ~196 5s. 21d. 41. Required the amount of commission at 2} per cent., and brokerage at - per cent., and the net proceeds of 50 bags of cotton, weighing gross 115cwt. lqr. 11 lb., draft i lb. per bag, tare 4 lb. per cwt., the whole being sold at 7~cts. per lb. Ans. Commission and brokerage $27.92. Net proceeds $902.68. 42. In the British Sanitary Report of 1843, it is stated that the proportionate numbers of the population at different ages, in the United States, and in England and Wales, were as follows: — United States. England & Wales. Under 5 years 1744 1324 5 and under 10 1417 1197 10 " " 15 1210 1089 15 ". c 20 1091 997 20 " 830 1816 1780 30 CC" 40 1160 1289 40" " 50 732 959 50 " " 60 436 645 60 " "c 70 245 440 70 "' " 80 113 216 80 " " 90 32 59 90 and upwards 4 5 Required the average age, both in England and in

Page  352 352 MISCELLANEOUS PROBLEMS. [ART. XXIII. America, of all the inhabitants;-of all above 15;-above 20;-above 50. 43. The commissioners of a certain county are about building a new court house, which will cost $75000. They hire money for the purpose at an annual interest of 5 per cent., and propose to pay the debt thus incurred by 50 equal annual instalments. What amount must be paid each year? Ans. $4108.25. 44. Three men bought a grindstone 50 inches in diameter, for which A. paid 75 cents, B. $1.50, and C. $2.00. What part of the diameter ought each to wear away, allowing the diameter of the axle to be 2 inches? Ans. A. 4.62in.; B. 11.05in.; C. 32.33in.a 45. An estate of $20000 is to be divided between two sons in the following manner: the elder is to receive $100 the first month, $300 the second month, &c., in arithmetical progression, and the younger is to receive $1000 per month, until the whole is paid. What is the share of each, and how long will they be in receiving it? Ans. $10000; lOmo. 46. A ladder standing upright against a wall reaches the top, but the foot being removed 12 feet from the wall, it reaches to a point 6 feet from the top. Required the length of the ladder, and the height of the wall. Ans. 15ft. 47. A block of stone 12ft. long, 3ft. wide, and 2ft. thick, is to be floated on a pine raft, which is 20ft. long, and 6ft. wide. What must be the depth of the raft, in order that it may float 4 inches above the water, the specific gravity of the pine being 575, and the sp. grav. of the stone 2500? Ans. 4ft. 3.76 +-in. 48. Wishing to estimate the height of a hill which is 5 miles distant, I hold a foot rule at the distance of 2ft. from my eye, and find that 1 inch on the rule intercepts the rays a This answer is obtained by supposing that A. uses the stone first, B. second, and C. last. The question admits of five other solutions.

Page  353 ~ 110.] MISCELLANEOUS EXAMPLES. 353 from the top of the hill and from the horizon. What is the height of the hill? Ans. 1100ft. -49. What is the distance of a thunder cloud, if three seconds elapse between the flash and report? Ans. 3270ft. 50. If the travel over a hill, by friction and gravity causes 8000 days' work of a horse, at 75 cents per day, which can be partially avoided by a road along the'base of the hill, the travel over which would require only 4000 days' work, and if the new road will require an extra annual outlay of $600 for repairs, how much can be saved by expending $12000 in making the improvement, computing interest at 5010? Ans. $36000. 51. If an engine has sufficient force to draw 100 tons over level ground, what additional power must be exerted on an ascending grade of 40ft. per mile? Ans. 15cwt. 1632 lb. 52. What is the power of a steam engine with a cylinder 40 inches in diameter, making the usual estimate of the effective force of the steam and the stroke of the piston? Ans. 64 horse power. 53. What is the amount of pressure on a dam 150ft. by 18ft., the average depth of water being Gft.? Ans. 1012500 lb. 54. My expenditure having exceeded my income by 15 per cent., I find that by saving - of my income for the succeeding year, I can supply the deficiency with interest, and have $4.60 left. What is my income? AIns. $600. 55. Find 11 terms of a harmonical progression, two of the terms being 4 and 8. Ans. 8, 4, 22, 2, 13, 11, 14, 1, 8, 4, 56. At the breaking up of the ice in a river, a tree is cut down by a block of ice, which has a surface of 10000 23

Page  354 354 MISCELLANEOUS PROBLEMIS. [ART. XXIII. square feet, and is 1 foot thick. How many axes, each weighing 10lb., and moving with a velocity of 20ft. per second, would have the same momentum, the velocity of the ice being 3-ft. per second, and its specific gravity 930? Ans. 8687 —. 57. Gregory King, in 1695, made the following estimate of the expense of England, France, and Holland, in diet:a England. France. Holland. Total. In bread-stuffs.. ~4300000 ~10600000 ~1400000 In meats... 3300000 5600000 800000 In butter, cheese, 2300000 4200000 600000 and milk In malt liquors. 5800000 100000 1200000 In spirituous drinks 1300000 9000000 400000 In fish, fowls, and eggs.... 1700000 3900000 1100000 In fruits and garden produce... 1200000 3600000 400000 In groceries and sweetmeats.. 1100000 3000000 300000 Total 58. Estimating the population of England, in 1695, at 51 million, France at 132 million, and Holland at 2s million, what was the average amount annually expended for each article of diet by each individual, in each nation? What was the average annual expense of each individual in the three nations? 2d Ans. ~3 3s. 4.7d. 59. A man spends 25 cents a day for wine and cigars; how much will he lose by the expenditure in 48 years, supposing money to be worth 6 per cent.? Ans. $23427.55. 60. A road has been constructed through a farmer's land, taking 1~ acres, worth $75 per acre. In addition to the loss of his land, he is obliged to expend $100 in a Wade.

Page  355 ~ 110.] MISCELLANEOUS EXAMPLES. 355 building a fence, which must be renewed every 12 years. What damages should he receive, money being worth 6 per cent. compound interest? Ans. $292.54. 61. Find 6 weights with which any number of pounds, fronm I to 364, can be weighed. 62. What is the difference between the area of a circle whose circumference is 157-25ft., and the area of the greatest square that can be inscribed in it? Ans. 713.5ft. 63. What number is that which is divisible by 11, but if divided by any number less than 11, leaves 1 remainder? Ans. 25201. 64. The average efect of a bushel of coals, weighing 60 lb., when consumed by an engine now working at Wheal Towan, in Cornwall, is sufficient to raise 70 million pounds one foot high.a How many pounds of the same coal would furnish sufficient power to raise a man, weighing 150 lb., to the summit of Mlont Blanec, an elevation of 15680ft.? Ans. 214 lb. 65. The Menai bridge consists of a mass of iron about 4 million pounds in weight, suspended at an average height of 170ft. above the sea.a How many bushels of coal would have sufficed to raise it to its present position? 66. Estimating the quantity of granite in the great pyramid at 75614816 c. ft., the specific gravity at 2700, and the average height to which the materials were raised at 125ft., how many chaldrons of coal would have furnished the power necessary for its erection? Ans. 632 - -o4- 9 chaldrons, a quantity consumed in some foundries in a week. 67. The entire expense of the Revolutionary War was estimated by the Register of the Treasury, in 1790, a Working Man's Friend,

Page  356 356 MISCELLANEOUS PROBLEMS. [ART. XXIIL at $135193700, to meet which an issue was made of $359547027-7 in continental money.a What was the average loss per cent. by the depreciation of the continental currency? Ans. 62.398 + per cent. 68. Four men bought a grindstone, 40 inches in diameter, each contributing an equal amount. How much of the diameter ought each to grind away? Ans. 1st. 5.359in.; 2d. 6.3568in.; 3d. 8.2842in.; 4th. 20in. 69. A. and B. are on opposite sides of a circular field that is 120 rods in diameter, and commence travelling around it in the same direction. How many times will each go round the field before the slower is overtaken, A. going 39 rods in 3 minutes, and B. 66M rods in 5 minutes? Ans. A. 191 times; B. 20 times. 70. A man sold a horse for $65.25, thereby gaining as much per cent. as the horse cost him. What did he give for the horse? Ans. $45. 71. Professor Wheatstone endeavored to determine the velocity of electricity by using a revolving mirror, and observing the relative position of the images made by the sparks at the two extremities of a long wire. What interval of time would be indicated by an angular deviation of I~ in the appearance of the two sparks, supposing the mirror to make 800 revolutions in 1 second? Ans. 8 sec. 72. It has been estimated that the average quantity of air contaminated by respiration, insensible perspiration, and lights, is 4 c. ft. per minute for each individual; the average amount cooled by the draught from each door or window, 11 c. ft. per minute; the number of c. ft. cooled by radiation through the windows, 1' times the number of sq. ft. of the glass exposed to the external air.' According to this estimate, how much fresh air should be supplied per a Encyclopaedia Americana. bTredgold.

Page  357 ~ 110.] MISCELLANEOUS EXAMPLES. 357 minute, in summer, for a hall containing 2000 people? How much heated fresh air, in winter, for a church with a congregation of 600, there being 28 windows and doors, and 1000 sq. ft. of glass? 1st Ans. 8000 c. ft. per nain. 2d Ans. 4208 c. ft. per min. 73. A. and B. own adjoining farms. A. and his family do no labor in winter, except to take the necessary care of his stock and household; but the family of B., by shoemaking, braiding straw, and other similar employmlents, make $125. To how much will B.'s winter labor amount in 30 years, if the proceeds are all invested at 6 per cent. compound interest? Ans. $9882.27. 74. A farmer has a lane leading through his pasture, to the public highway, and instead of fencing in the lane, he has a gate at each extremity. If 1 minutes' delay is occasioned by opening and closing each gate, and if he is obliged to pass through the lane 12 times a day, how much will he lose by the gates in a year, supposing his time to be worth 20 cents an hour? Ans. $36.50. 75. A pump in which the water is to be raised to the height of lOft., has a bore 6in. in diameter. What should be the diameter of the bore of a pump, which is to raise the water 25 feet, in order that the two pumps may be worked with equal ease? Ans. 3.79in. 76. Suppose a man whose average weight is 160 lbs., to drink three half-pint cups of coffee per day for 40 years, the average specific gravity of the coffee being 1100, to how many times his own weight will the whole amount? Ans. 24106~ lb., or 150.665625 times his own weight. 77. It has been stated by one of the most careful and successful manufacturers, that on substituting, in one of his cotton mills, a better for a poorer educated class of operatives, he was able to add 12 per cent. to the speed of his machinery, without any increase of damage or danger from the acceleration. What amount would be saved by the

Page  358 358 MISCELLANEOUS PROBLEMS. [ART. XXTL employment of educated labor, from this source alone, in a business of $500000? 78. The number of females engaged in the various manufactures of Massachusetts, has been estimated at 4000, and their average annual wages at $100 eachl. The superintendent of the Merrimack Mills, in 1841, estimated the average wages of the best educated operatives, at i79 per cent. above the general average wages of the mills, and the average wages of the least educated, at 18~ per cent. below the general average. According to this estimate, how much would be gained by elevating the whole 40000 to the highest standard, and how much would be lost by degrading them to the lowest standard? 79. The researches and discoveries of M. Meneville, in regard to the fly which was lately so destructive to the olive in the south' of France, are said to have increased the value of the annual product of this fruit, 81000000. What amount of profit will France derive in 50 years, from the education which led to such a discovery, supposing the entire increase of value to be invested annually at 5 per cent. compound interest? 80. The aggregate quantity of water annually discharged by the Mississippi river, has been estimated at 14883360636880 c. ftR. To how many cubic miles is this equivalent? Ans. 101.1 c. miles. 81. Estimating the area of the Mississippi valley at 1400000 sq. miles,a and the average depth of rain water that falls annually in the valley, at 52 inches,a how many cubic feet of water fall in the whole valley during the year, and what part of it passes off by evaporation? Ans. 169128960000000 c. ft., of which about gpasses off by evaporation, and 2 is discharged by the river. a Proceedings of Amer. Association, 1848.

Page  359 ~ 110.] MISCELLANEOUS EXAMPLES. 359 82. The removal of the forests from the valley of the Mississippi has increased evaporation to such an extent, that the inundations of the river have become much less frequent and less formidable than they were at the first settlement of the country.a What amount of water was annually discharged by the Mississippi, 30 years ago, supposing that it has since decreased 25 per cent.? Ans. 19844480849173 c. ft. 83. If 23232 c. in. of the Mississippi water, deposit 44 c. in. of sediment;, how long would it require to deposit the present delta, which is estimated to contain 13600 sq. m. of surface, and to be of the average depth of - mile? Ans. 142034 years. 84. What is the approximate weight of the earth, estimating its mean diameter at 7912 miles? Ans. 13527679878424215242145792 lb. 85. If 12 oxen eat 31 acres of grass in 4 weeks, and 21 oxen eat 10 acres of the like pasture in 9 weeks, how many oxen will eat 24 acres in 18 weeks, the grass being at first equal on every acre, and growing uniformly? This example is taken from Newton's Universal Arithmetic. It can be solved most readily by making three distinct questions. (1.) If 12 oxen eat I3o acres of grass, with the growth, in 4 weeks, how many oxen will eat 24 acres, with 4 weeks' growth, in 18 weeks? Stating the question by analysis, or by proportion, g1 we obtain 12 4 X 2=191- oxen, for the answer. 4 (2.) If 21 oxen eat 10 acres, with the growth, in 9 weeks, how many oxen will eat 24 acres, with 9 weeks' growth, in 18 weeks? a Proc. of Amer. Association, 1848.

Page  360 360 MISCELLANEOUS IROBLEMS. [ART. XXIII. The answer, found as before, is 21 21 X 621 02 5 6,4 405 Now, if 19- oxen in 18 weeks, eat 24 acres with 4 weeks' growth, and 25A- oxen in the same time, eat the same number of acres with 9 weeks' growth, 5 weeks' growth on 24 acres will support 6 oxen 18 weeks. Then, (3.) If 6 oxen in 18 weeks eat 5 weeks' growth, how many oxen in the same time will eat 9 weeks' growth? The answer is 104. We have already found that 251 6oxen in 18 weeks, will eat 24 acres with 9 weeks' growth, 915 and if we add the number which would eat the growth of the remaining 9 weeks, we obtain 36 oxen for the answer sought. 86. If 15 oxen eat 41z- acres of grass in 2 weeks, and 24 oxen eat 142 acres in 5 weeks, how many oxen will eat 48 acres in 8 weeks, the grass being at first equal on every acre, and growing uniformly? Ans. 60 oxen. 87. If 11 oxen eat 244- acres of grass in 5 weeks, and 10 oxen eat 26W acres in 4 weeks, how many acres of similar pasture will 42 oxen eat in 7 weeks, the grass growing uniformly? Ans. 782 A. 88. What number is that which is 169 greater than the greatest square number below, and 114 less than the least square number above itself? Ans. 20050. 89. A. and B. can do o-l of a piece of work in a day, B. and C. can do 1 of it in 2~ days, and A. and C. can do 1 of it in 4-1T days. In what time would each do it alone, and in what time would it be done if they all worked together? Ans. A. in 15 days; B. in 30 days; C. in 18 days; all together in 6-: days. 90. There are some monads not exceeding -oin. in diameter, in which 6 spots have been observed, separated by membranous partitions not thicker than - of the diameter

Page  361 ~ 110.] MISCELLANEOUS EXAIMPLES. 361 of the spots. If these estimates are correct, what is the thickness of the partitions? Ans. x1 -~in. 91. What amount can be recovered from the underwriters, upon the following transaction? —~5000 was insured upon goods from New York to London, the goods being valued in the invoice, at ~7200. In a storm, a part of this property valued at ~1107 5s. was thrown overboard, and the remainder was so much damaged, that it sold for only ~5407 15s. 6d., whereas if it had arrived safe, it would have sold for ~6723 lOs.; besides which, the owner of the goods was obliged to contribute towards a general average, at the rate of 2.225 per cent. on the invoice value of the whole. Ans. ~1025. 92. Exported 45 hogsheads of sugar from New York to Amsterdam, weighing gross 454cwt. 2qr. 18 lb., tare 53 lb. per hhd., which were sold at 12} groats per lb., subject to a discount of 5 per cent. The original cost and charges in New York, were $2437.50, the amount of charges in Amsterdam, 1149 florins 7 groats, and the agio of the bank of Holland was 4~ per cent. How much was gained or lost by the adventure, the net proceeds being remitted at the exchange of 40cts. per florin? Ans. $2638.19 gained. 93. How far can the top of Bunker Hill Monument, which is 282ft. above the level of the sea, be seen from the deck of a vessel, the spectator's eye being 15ft. above the water?a Ans. 25.3 miles. 94. The average power of draft of a horse, moving 5 a The distance at which bodies may be seen, is found by the following rule:To the earth's diameter (41815224 feet), add the height of the eye, and multiply the sum by the height of the eye. The square root of the product is the distance at which an object oN THr, SURFACE of the earth or water can be seen. Work in the same way with the height of the object, and the sum of the two results is the distance at which the object may be seen.

Page  362 362 MISCELLANEOUS PROBLEMIS. [ART. XXIII. miles per hour for 8 hours a day, being 75 lb., what will be the annual cost of transportation over a road 40 miles long, on which the average friction is -1- of the weight, estimating the amount transported at 90000 tons, and the value of a horse's labor at 62- cents a day? Ans. 867200. 95. If the road, in the preceding example, could be improved by macadamizing or otherwise, so that the friction would be reduced to 6- of the weight, how much might profitably be expended in making the improvement, money being worth 6 per cent.? Ans. $6533331. 96. The shadow cast by a Drummond light, at the distance of 80 rods, was observed to be of the same intensity as that cast by the full moon, which was shining at the time. To how many such lights was the moon's light equivalent, her mean distance being 240000 miles? Ans. 921600000000. 97. Find the least 3 integers, such that 3 of the first, -15 of the second, and 27 of the third, shall be equal. Ans. 140, 147, 150. 98. For the purchase of a certain estate, A. offers $150 premium, and $300 rent per annum; B. offers $400 premium, and $250 per annum; C. offers $650 premium, and $200 per annum, and D. offers $1800 in ready money. Whose offer is the best, and what is the difference between them, computing 5 per cent. compound interest? Partial Ans. A.'s offer is the best. 99. Which is of the greater value, the income of an estate of $500 a year for 15 years to come, or the reversion of the same estate for ever, at the expiration of the 15 years, interest at 6 per cent.? Ans. The income for 15 years. 100. If a ball were put in motion by a force which

Page  363 ~ 110.] MISCELLANEOUS EXAMPLES. 363 would drive it 12 miles the first hour, 10 miles the second, and so on in geometrical progression, what distance would it go in the whole? Ans. 72 miles. 101. What is the least number which, if divided by 2, will leave 1 remainder; by 3 will leave 2; by 4 will leave 3; by 5 will leave 4; by 6 will leave 5; but by 7 will leave no remainder? Ans. 119. 102. Required the least three numbers, which, divided by 20, will leave 19 remainder; if divided by 19 will leave 18, and so on, (always leaving one less than the divisor), to unity. Ans. 232792559; 465585119; 698377679. 103. A trader offers to receive a young man as partner, proposing, if he will advance $500, to allow him $200 per annum; if he will advance $1000, to allow $275 per annum; and if he advances $1500, he will allow $350 per annum. What per cent. is offered for the use of the money, and how much for the young man's time? Ans. 15 per cent.; $125 per annum. 104. A shepherd sold to one man, half his flock and half a sheep; to a second, half the remainder and half a sheep; and to a third, half the remainder and half a sheep, when he had 20 left. How many had he at first? Ans. 167. 105. The annual cost of transportation on a road 40 miles long, being estimated at $30000 per mile, what amount of saving can be effected by expending $50000 to shorten the road 3 miles, and 82000000 to reduce the friction to - its present amount, the annual cost of repairs being the same in both cases, and the rate of interest being 5y per cent.? Ans. $9677272fs. 106. A city of 50000 inhabitants is to be supplied with water, from a river 300 feet below the proposed reservoir. Estimating the average daily consumption at 10 ale gallons

Page  364 364 MISCELLANEOUS PROBLEMLS. [ART. XXIII. for each individual, what must be the power of an engine working 12 hours a day, to raise the requisite supply? Ans. 48.53 horse power. 107. A commission sale of 60 bags of coffee was effected at Rotterdam, at 22 stivers per lb., with an allowance of 2 per cent., and of 1 per cent. on the remainder; weight gro. 50cwt. lqr. 11llb., draft 1 per cent., tare 6 lb. per bag; commission and guarantee, 3 per cent.; other charges, 269 florins 10 stivers. Required the net proceeds of the sale, a bill being remitted at the exchange of 39 cents per forin. Ans. $2003.58. 108. Bought in Cadiz four chests of Peruvian bark, at 25 rials of plate per lb., weighing gro. 6cwt. 3qr. 16 lb., tare 25 lb. per chest. The export duty was 150 rials vellon per quintal of 100 lb., and the amount of other charges was 1210 rials of plate. What was the whole amount of the purchase? Ans. $1851.40. 109. At a time when bills upon the treasury bore at Jamaica a premium of 7~ per cent., and dollars a premium of 4 per cent., 14000 dollars were purchased and consigned to London, to be disposed of there; they weighed 1010 lb. 3oz. troy, and were sold to the Bank of England at 5s. 3d. per oz. The charges were, freight 1~ per cent.; 14 bags, at 6d. each; weighing, 2s. 6d.; brokerage, A per cent.; commission, - per cent.; and insurance on ~3000, at 4 per cent. It is required to find the rate per cent. of the amount of the charges, estimated upon the cost of the dollars, exclusive of the premium, and to find what would have been gained or lost, if in preference to dollars, their actual cost had been laid out in government bills. Ans. Value of the dollars, ~3182 5s. 9d.; net proceeds of the sales, ~2994 3s. lid; rate of the charges, 6 1 per cent.; gain by government bills, ~84 9s. 8d.

Page  365 ~ 110.] MISCELLANEOUS EXAIPLES. 36:5 110. "' One evening I chanced with a tinker to sit, Whose tongue ran a great deal too fast for his wit; He talked of his art with abundance of mettle, So I asked him to make me a flat-bottomed kettle. Let the top and the bottom diameters be In just such proportion as five is to three: Twelve inches the depth I proposed, and no more, And to hold in ale gallons seven less than a score. lie promised to do it, and straight to work went, But when he had done it he found it too scant. IHe altered it then, but too big he had made it; And when it held right, the diameters failed it. Thus making it often too big and too little, The tinker at last had quite spoiled his kettle, But declared-he would bring his said promise to pass, Or else that he'd spoil every ounce of his brass. Now to keep him from ruin, I pray find him out The diameters' length, for he'll ne'er do't without." Ans. 24.4in.; 14.64in. 11. Two vessels are 30 miles apart, and are sailing, the first with, and the second against a current of 21 miles per hour. In still water, each would sail 7 miles per hour. In what time will they meet, and what will be the distance of each from its present position? 112. At what time between half past 7 and 8 o'clock, are the hour and minute hands exactly 13 minutes apart? Ans. 52-41m. past 7. 113. The annual loss in the United States, in consequence of intemperance, has been estimated at $108000000.a If this amount were saved, how much would be left after purchasing 4000000 sheep at $2.50, 400000 head of cattle at $25, 200000 cows at $20, 40000 horses at $100, 500000 suits of men's clothes at $20, 1000000 suits of boys' clothes a Amer. Temp. Soc. Reports.

Page  366 8MISCELLANEOUS PROBLEMS. [ART. XXIII. at $10, 500000 suits of womens' clothes at $10, 1000000 suits of girls' clothes at $3, 1200000 bbl. flour at $5, 800000 bbl. beef at $10, 800000 bbl. pork at $12.50, 3000000bu. corn at $.50, 2000000bu. potatoes at $.25, 100000001b. sugar at $.10, 400000lb. rice at 5cts., 2000000gal. molasses at 40cts., building 1000 churches at $5000, 8000 school houses at $500, and supporting 50000 families at $300? Ans. $180000. 114. What is the difference between 10000 years according to our present calendar, and 10000 solar years, estimating the solar year at 365d. 5h. 48m. 49'sec.?a Ans. 2d. 14h. 30m. 115. In 1840, Prof. Bessel, from the corrected parallax of the star 61 Cygni, estimated its distance from the earth at 592200 times the mean distance of the sun. According to this estimate, how long would the star's light be in reaching us? Ans. 3391dy. 9h. 13m. 45s. 116. A micrometer screw is made with threads 5o of an inch apart, and an index is affixed to the head, which points to the degrees on a graduated circle. What thickness would be measured by turning the index 1~ 15'? Ans. 1 4in. 117. A. and B. are carrying a weight of 150 lb., which is suspended on a pole, at the distance of 2ft. 6in. from A., and lft. 9in. from B. What amount does each one support? Ans. A. 61lb. 12-4Loz.; B. 88 lb. 3 —oz. 118. Archimedes boasted, that if he had a place to stand, he could move the world. If he weighed 150lb., how far would he be obliged to move, in order to move the earth 1 inch? [See Ex. 84.] Anzs. 1423366990574938472.448 miles. 119. At Bilin, in Germany, is a bed of tripoli, composed a Carpenter.

Page  367 ~110.] MISCELLANEOUS EXAMIPLES. 367 almost entirely of the sheaths of a kind of animalcule, the length of each sheath being about y3t10 of an inch.a How many such sheaths would there be in a cubic inch? Ans. 42875000000. 120. A grain of copper dissolved in nitric acid, and mixed with three pints of water, (wine measure,) gives a blue color to the whole. What is the weight of the quantity contained in 1 0 O-( of a cubic inch, which is sufficiently large to be visible to the naked eye?a Ans. a-c2n5 of a grain. 121. In the manufacture of embroidery wire, a cylinder of silver, weighing 360oz., is covered with 2oz. of gold. This is drawn into wire, of which 4000ft. weigh loz. A foot of this wire may be divided into 1200 parts visible to the naked eye. What would be the weight of a particle of the gold upon this wire, that would be visible under a microscope magnifying 500 times each way? Ans., I GOUUOz. 122. The squares of the times of revolutions of the planets around the sun, are proportioned to the cubes of their mean distances. The mean distance of the earth from the sun, is 95 million miles, and it revolves round the sun in 365d. 5h. 48m. 50s. What is the mean distance of Venus, her time of revolution being 224d. 16ah.?b Ans. 69 million miles, nearly. 123. What do I gain per cent. by purchasing goods at 8 months' credit, and selling them immediately for cash, at the same price, money being worth 6 per cent. per annum? Ants. 3-1- per cent. 124. A. and B. enter into partnership, A. advancing $4800 to carry on the business. B. has no money, but being thoroughly acquainted with the trade, agrees to be manager. B.'s annual salary, before he engaged in the coparta Carpenter. b Nesbit.

Page  368 16i8 MISCELLANEOUS PROBLEMS. [ART. XXIIIr nership, was $250. By the terms of the contract, A. is to be allowed 7 per cent. for the interest of his money and the risk of,the business, and the net profits, after making this deduction, are to be divided in the proportion which 6 per cent. on A.'s capital bears to B.'s former salary. The profits at the end of the year, amounted to $5000. Eow should they be divided? Ans. A. $2832.71; B. $2167.29. 125. What quantity of each of the following ingredients, would be required to make a ton of flint glass, estimating the waste in melting at 2 per cent.? White sand, 9 parts; red lead, or litharge, 6.5 parts; pearlash with a little nitre, 4.5 parts.a Ants. Sand 1028.6 lb.; litharge 742.9 lb.; pearlash 514.3 lb. 126. The dust of the puff ball consists of seeds, which vary fronim 10 to -3G- of an inch in diameter. How many seeds, of the average diameter of — oin., would there be in 1i cubic inches? Ants. 284375000000000. 127. Estimating the entire wealth of the world at $1407374883553.28, in what time would 1 cent absorb all the property on the globe, if it were placed at compound interest, so as to double every 11 years? Ans. 517 years. 128. A merchant failing in business, offers to settle with his creditors by paying them 80 cents on a dollar in 6 months, without interest, or to give security for the payment of the whole in 4- years, without interest. Which proposal is the more advantageous, supposing that the composition first proposed can be invested at 6 per cent. compound interest? 129. Bought flour for $3.50 per barrel, on 6 months. At what price must it be sold to gain 20 per cent., and allow 3 months' credit, money being worth 8 per cent. per annum? Anzs. $4.11-.2 a Parnell.

Page  369 ~ 110.] MISCELLANEOUS EXAMPLES. 369 130. A. has 70cwt. of sugar, which he would sell for 87 per cwt. for cash, but in barter he values it at $8.50 per cwt. B. has wheat worth in ready money $1.12 per bu., which he wishes to exchange with A. At what price should it be valued in barter? Ans. $1.36. 131. Insured 150 hogsheads of sugar, valued at $125 per hogshead, from Jamaica to Boston, at 8 per cent., to return 4 per cent. for convoy and arrival without loss. Of the quantity insured, 30hhds. were shipped on board a vessel that was totally lost, 64hhds. arrived safe by a vessel that sailed with convoy the whole of her voyage, 45hhds. arrived with convoy, but in consequence of a mast being cut away, and some goods being thrown overboard to lighten the vessel in a storm, the merchandise shipped was obliged to contribute to the loss, at the rate of 2 per cent. on its value, and the rest of the sugar was not shipped. Required the amount to be recovered from the underwriters, and the amount of return premium for short interest, and for convoy and arrival.a Ans. Underwriters to pay $3862.50. Return premium, $419.00. 132. What is the product of $18.75 by $.061-? 133. James Davis, of Waverly, Ross county, Ohio, cultivated, in the year 1849, 1800 acres exclusively in Indian corn. With the crop he filled a corn-crib 3 miles long, 10ft. high, and 6ft. wide.b What was the average yield per acre, and how many bushels would the whole make, when shelled? Ans. 422.4bu. in the ear, per acre. 396000bu., when shelled. a The custom varies as to the amount of return premium, when part of the risk is avoided. Some companies reserve, per cent. on the amount insured; some reserve 10 per cent. of the premium paid in; on "open policies," no premium is usually paid, except for the amount actually at risk. In the present instance, 10 per cent. of the premium paid on the molasses which was not shipped, is supposed to be reserved by the company. b Cincinnati Gazette. 24

Page  370 370 MISCELLANEOUS PROBLEMS. [ART. XXIII. 134. Sysla, the reputed inventor of the game of chess, is said to have asked as a reward, one grain of wheat for the first square on the chess-board, two for the second, and so on in geometrical progression. What would have been the amount of his reward, there being 64 squares on the board, and 9200 grains of wheat in a pint? What would be the height of a cubical bin that would contain it, supposing the base to be 10 miles square? Ans. 18446744073709551615 grains; 31329388712142.58 bushels; height of bin, 2.65 miles. 135. If the cost of a railroad is $40000 per mile, and the annual repairs and expenses are estimated at $2500 per mile, how much may be profitably expended at the outset, in order to shorten the proposed route 22 miles, provided the stock pays an annual dividend of 8 per cent.? Ans. $178125. 136. What is the weight of a round iron rod, 10 feet long, and 4 inches in diameter? Ans. 538 lb. 10*-oz. 137. A mule and an ass travelling together, the ass began to complain that her burthen was too heavy. "Lazy animal," said the mule, "you have little reason to complain; for if I take two of your bags, I shall have three times as many as you, but if I give you two of mine, we shall have only an equal number." With how many bags was each loaded? 138. How would you distribute among 3 persons 21 bags, 7 of which are full of corn, 7 half full, and 7 empty, so as to give to each the same quantity of corn and the same number of bags? This question admits of two solutions. 139. What is the 7th root of 63? Ans. 1.80737+. 140. A farmer has a stack of hay, from which he sells a

Page  371 ~ 110.] MISCELLANEOUS EXAMPLES. 371 quantity which is to the quantity remaining, as 4 to 5. He then uses 15 loads, and. finds that the quantity left is to the quantity sold, as 1 to 2. Required the number of loads at first in the stack. Ans. 45 loads. 141. At what points would the numbering on Fahrenheit's and on Delisle's thermometer-scale be the same, the only difference being in the sign?a Ans. + 96-4~ Fah.; - 96-4~ Del. 142. If the temperature below 0~ on the Russian scale is marked +, and the temperature above 00 is marked b at what points will the numbering be the same, on the Russian and Centigrade scales? Ans. 600. 143. The prime cost of 109cwt. 3qr. 18 lb. of sugar was $769.375; the freight, insurance, and other expenses, amounted to $92.31. What did it cost per cwt., and at what price must it be sold per lb. to gain 20 per cent., supposing 4 per cent. to be lost by overweight in retailing? Ans. $7 per cwt.; 8 cts. per lb. 144. A merchant receives an invoice of goods, which will probably sell for $16000, by which he will realize a profit of 33~ per cent. above the prime cost; but preferring to sacrifice a portion of his profit, rather than to risk the entire loss of the goods, he determines to insure them. The premium is 3- per cent., the policy $2.00, the agent's commission e per cent., and if there is any loss, the broker will charge 1 per cent. for procuring a settlement with the underwriters. What amount should the merchant insure, a The pupil should observe that Reaumur's and the Centigrade scales correspond only at 00, but all the other scales have the same numbering at two points of equal temperature, differing at one of the points only by the sign. b The signs are sometimes applied in this way on the Russian scale, but the notation given on p. 197, appears more philosophical, on accoun' of its uniformity.

Page  372 372 MISCELLANEOUS PROBLEMS. [ART. XXnI in order to recover the prime cost of the goods, and all expenses attending the insurance, in case of a total loss? Ans. $12633.68. 145. A trader wishes to sell his merchandise at wholesale, so as to make a profit of 20 per cent., after deducting 30~Io from the retail price. He therefore adds 50 per cent. to the cost of the articles in cash, and deducts 30 per cent. firom the amount, for the wholesale price, allowing a credit of 8 months. Computing interest at 6 per cent. per annunm, what is his actual loss on all sales at this rate, supposing that his expenses and bad debts amount to 15 per cent. of his total sales? Ans. 143- 9 per cent. 146. What percentage should have been added to the prime cost of the articles in the foregoing example, to yield a net profit of 10 per cent. after making the discount proposed? Ans. 923139 per cent. 147. A merchant received on consignment three bales of sheeting, marked A., B., and C. A. contained 420 yards of a quality 15 per cent. better than B., B. contained 380 yards of a quality 10 per cent. poorer than C., and C. contained 450 yards. The whole were sold together at 12-i cents per yard; how much should be credited to each, after deducting 2~ per cent. commission? An/s. A. $53.98; B. $42.47; C. $55.89. 148. An estate was offered for sale for $12000, but the price appearing too high, the tenant took a lease for 25 years at $800 per annum. How much did he gain or lose, estimating compound interest at 6 per cent.? Ans. $1022.67. 149. If 32 oxen in 3 weeks consume all the grass on 23 acres of pasture, and if 22 oxen in 5 weeks consume the grass on 25 acres, in what time will 88 oxen consume 115 acres of similar pasturage, the grass growing uniformly? Ans. 5.88 weeks.

Page  373 ~ 110.] MISCELLANEOUS EXAMRPLES. 373 150. The steam power at present employed in Great Britain and Ireland, is estimated as equivalent to the power of 8000000 men.a If the power of a horse is 5 times as great as that of a man, and if a horse requires 8 times the quantity of soil for producing food that a human being does, what population could be supported by the additional food which would be required, if horse power were substituted for steam? Ans. 12800000. 151. The hydraulic press of Branmah can, by the exertion of a single man, produce a pressure of 1500 atmospheres.b Estimating the mean height of the mercury in the barometer at 29.53 inches, what would be the pressure exerted on a surface 2ft. 3in. long, and ift. 9in. wide? Ans. 5502T. 5cwt. lqr. 151b. 12oz. 152. A species of lace is made by covering an inclined flat surface with a paste made of leaves, and drawing with a camel's hair pencil, the pattern which is to be left open. A number of caterpillars, of a species which spins a strong web, are then placed at the bottom, and they commence eating and spinning their way to the top, carefully avoiding every part touched by the oil, but devouring every other part of the paste.b I-low many square yards of the lace thus made, would there be in 1 lb. avoirdupois, the average weight being 48- grains troy per sq. yd.? Ans. 1615- sq. yd. 153. Estimating the weight of a globe of air ift. in diamleter, at - lb. avoirdupois, and the weight of an equal globe of hydrogen, allowing for impurities, at I as much, what weight would be sustained by a balloon 1Sft. in diameter, if fully inflated with hydrogen gas? Ans. 194.4 lb. 154. Having observed that the shadow of a cloud is 15 seconds in passing from one point to another, I measure a Chambers' Mechanics. b Babbage.

Page  374 374 MISCELLANEOUS PROBLEMS. [ART. XXIII. the distance between the two points, and find that it is 314 rods. What is the velocity of the wind per hour? Ans. 23m. 200r. 155. A., B. and C. have a loaf of sugar, weighing 48 lb., which they wish to divide equally between them, but having only a 4 lb. weight and a 7 lb. weight, it is required to find how the division can be made. 156. A hundred hurdles may be so placed, as to enclose 200 sheep, and with 2 more the fold may be so made as to hold 400. How can this be done? 157. The animalcules of iron ochre, are about 1a7 of an inch in diameter. How many such animalcules would occupy 1 c. ft.? Ans. 2985984000000000. 158. An ounce of gold forms a cube about -5 of an inch thick, but by hammering, it may be extended so as to cover a surface of 146 sq. ft. How many leaves formed in this manner, would equal in thickness a leaf of writing paper 1 — of an inch thick? Ains. 1938. 159. Bought wheat for cash, at $.90, at $.95, and at $1.10 per bushel. In what proportions may the three kinds be mixed, so as to gain 20 per cent. by selling at $1.25 per bushel, on 6 months' credit, money being worth 7 per cent. per annum? Ans. 11.62bu. at $.90; 11.62bu. at $.95; 20.13bu. at $1.10. 160. A farmer has paid in cash $4000 for the lease of a farm for 8 years. If money is worth 5 per cent. compound interest, what income ought he to receive from the farm each year, in order to recover his outlay, and lay up $200 a year, his family expenses being $350? Ains. $1168.89. 161. A family of 10 persons hired a house for 6 months, at a rent of $780 per annum. At the expiration of 14 weeks they received 4 new boarders, and at the expiration of every 3 weeks, during the remainder of the term, they

Page  375 ~ 110.] MISCELLANEOUS EXAMIPLES. 375 received 4 more. How much of the rent should be paid by one of each class? Ans. 1st class $30.49T-512d class $9.49T-~-1T; 3d class $6.27 1;-9; 4th class $3.77J-s9; 5th class $1.73,1. 162. Estimate of the cost of manufacturing pins, "' Elevens," of which 5546 weigh I lb.:aTime of Cost of ma. NAME OF THE PROCESS. andsmakin kin Handearns of pins. of pins. per day.. a Ilours. Pence. s. d. 1. Drawing wire............ lan.3636 1.2500 3 3 225 2; Straightening wire....... Wo7Tlnan.3000.2810 1 ~ 51 Sragenn e Girl.3000.1420 0 6 26 3. Pointing................ M{an.3000 1.7750 5 3 319 4. Twisting and cutting ~ Boy.0400.0147 0 4 3 the heads........... Man.0400.2103 5 4 38 5. Headingr......... Wo........ o man 4.0000 5.1000 1 3 901 6. Tinning, or Whiteni Man.1071.6666 6 0 121 6. Tinning orWiteni Voman.1071.3333 3 0 60 1 7. Papering. WVoman 2.1314 3.1973 1 6 576 Estimating the premium on sterling money at 9- per cent., what is the cost of each pin, in our currency? At what price should the pins be sold per ounce, after paying 15 per cent. profit to the manufacturer, 30 per cent. ad valorem for duties, 18 per cent. to the American importer, and 25 per cent. to the retailer, allowing 10 per cent. of the retail price for loss of interest? Ans. a 75 of a cent; 3.9837+ cents. 163. If a powerful Drummond light, at the distance of 30 feet, casts a shadow of the same intensity as that cast by the sun, to how many such lights is the sun's light equivalent, estimating the distance of the sun at 95 million miles? Ans. 279558400000000000000. 164. It has been supposed that 10 turns of Babbage's calculating machine may be made in a minute.a At this rate, how many places of figures would the machine reach a Babbage.

Page  376 376 MISCELLANEOUS PROBLE MS. [ART. XXIII. in a million centuries, supposing it to be so regulated as to commence with 1, and give all the following numbers in their natural order? Ans. 15 places. 165. It has been estimated" that a man in a properly ventilated room can work 12 hours a day, with no greater inconvenience than would be occasioned by 10 hours' work in a room badly ventilated, and that where there is proper ventilation, a man may gain 10 years' good labor on account of unimpaired health. According to this estimate, what is the loss, in 30 years, to each individual in a badly ventilated workshop, valuing the labor at 10 cents per hour? Ans. $5008. 166. In the town of Bury, England, with an estimated population of 25000, the expenditure for beer and spirits, in the year 1836, was estimated at ~54190." If this sum was 24 per cent. of the entire loss, resulting from the waste of money, ill health, loss of labor, and the other evils attendant upon intoxication, what was the average loss from intemperance, for each man, woman, and child, in the place, estimating the pound sterling at $4.80? Ans. $43.352. X 1. TABLE OF PRIME AND COMPOSITE NuMnBERS. The following table contains all the prime numbers, and the factors of all odd composite numbers, below 12700, the prime numbers being indicated by a dash. For numbers below 1000, all the factors are given. For the odd numbers above 1000, one or more factors will be found in the table, which will reduce each number to a prime, or to some number less than 1000, and under the latter number the remaining factors may be found. The hundreds are placed at the head of the table, and the tens and units at the left hand. a British Sanitary Reports.

Page  377 ~111.1 TABLE OF PRII:ME AND COMIPOITE NUMABERS. "i77 0 1 2 3 1 7 8 00 _ 252.52 23.52 22.3,52 24.52 22.53 23.3.52 22.52.7 25.522.32.352 0l1 - 3.67 7.4:3 3.167 32.89 17.53 02 - 2.3.17 2.101 2.151 2.3.67 2.251 2.7.43 2.33.13 2.401 2.11,41 03 - 7.2 3.101 13.31 32.67 19.37 11.73 3.7.43 04 22 23.13 21.3.17 24.19 22.101 23.7.9 22.151 26.11 22.36.7 23.1.1: 01 - 3.5.7 5,41 5.f61 34.5 5.101 5.112 3.5.47 5.7.23 5. 181 06 2.3 2.53 2.103 2.32.17 2.7.29 2.11.23 2.3.101 2.353 2.13.31 2.3.11l1 07 - 32.23 11.37 3.132 --- 7.1(1 3.269 ~0 23 22.33 24.13 2a.7.11 23.:3.17 22.127 26.19 22.3.59 23.101 2. "7 09 32 11.15~ 3.1013 3.7.29 -- 32.101 10 2.5 2.5.11 2.3.5.7 2.5.31 2.5.41 2.3.5.17 2.5.61 2.5.71 2.34.5 2.5.7.13!11 - 3.37 1. 3.137 7.73 13.47 3,.79 -- 12 22.3 24.7 22.531 23.3.13 22.103 29 22.32.17 23.89 22,7.2 9 24.3 19 13 - 3.71 7.59 353.1 —.- 23.31 3.271 11.83 014 2.7 2.3.19S 2.107.1157 2.3.23 2.257 2.307 2.3.7.17 2.11.37 2.427 15 3.5 5.23 5.43 32,5,7 5.83 5.103 3.5.41 5.11.13 5.163 3.5.61 16 24 22.29 23.33 22.791 25.13 22.3.4:3 23.7.11 22.179 24.3.17 22.229 17 - 32. 13 7.31 3.139 11.47 3.23.) 19.43 7.131 18,2.32 2.59 2.10i.) 2.3.53 2.11.19 2.7.37 2.3.103 2.359 2.4091 2.33.17 19 -_ 7.17 3.73 11.29 3.173 -- --- 32.7.13 - 120 22.5 23.3.5 22.5.11 26.5 22.3.5.7 23.5.13 22.5.31 24.32.5 22.5.41 23.5.23,21 3.7 112 13.17 3,107 33.23 7.103 — _ 3.307 2-5 2~.11 2.61 2.3.37 2.7.23 2.211 2.32.29 2.311 2.192 2.3.137 2.461 23| - 3.41 17.19 32.47 7.19 3.241 -_ 13.71 241 23.3 22.31 2.7 2234 23.3 22.131 2.3.13 22.181 23.103 2.3,7.11 5 52 53 32,52 52.131 52.17 3.52.7 54 52.29 3.52.11 52.37 26 2.13 2.32.7 2.113 2.163 2.3.71 2.263 2.313 2.3.112 2.7.59 2.463, 27 331 I 3.1091 7.61 17.31 3.11.19 _ -- 132,10:3 28'~2.7 2 22.3.19 23.41 22.107 24.3.11 2.157 23.7.13 23.23 25, 2.:3.5 2.513 2.5.23 2.3.5.11 2.5.43 2.5.53 2,32.5.7 2.5.73 2.5.8312.3.5.31'31 - 3.7.11 312,5 9 - 17.43 3.27l 72,19:32 2 5 22.3.11 23.29 22.83 24.33 22.7.19 23.79 22.3.61 2;.131 2,5;33'331 3.11 7 7.1(93l 32,371 13.411 3.211 — _ 72.171 3.311 134'2.17 2.67 2.32.13 2.167 2.7.31 2.3.89 2.317 2.367 2.3.1391 2.4671 35 5.7 33.5 5.47 5.67 3.5.29 5.107 5.127 3,5,72 5.167 5.11.17 3G2,632- 23.17 22.59 24.3.7 22.109.5 23.67 22.3.53 25.235 22.51.1 123.32.13 37- - 3,79 19'.23 3.179 72.13 11.7 332.31 38 2'.191 2.3.3 2.7.317 2.132 2.3.73 2.269 2.11. 2.32.41 2.4191 2.7,t67 39 3.13 3.113/ 72.131 32.7(1 --- 3.313 o0 23.5, 22.5.71 24.3.5 2'2.5.174 23.5.31 22.33.5 2'1.5 22.5.37 23.3.5.71 2,25.47 411 -_ 3.47 -- 11.31 32,72 - - 3.13.19 292 --'42 2.3.7 2.71 2.1125329 2.13 2.17 2271 2.3.107 2 1 7.53 2.42.1 2.3.157 13 - 11.131 35 731 -- 33.181 1 ~ — ---- 3.281- 23.41 i4422.11 24.32 22.61 23.43 22.3.37 25.17| 22.7.23 23.3.31 22.211 24.59 14,5 32-5 5.29 5,72,3.5.23 5.89 5.109 3.5.43 5.149 5.132 33.5.7 46 2.23 2.73 2.3.41 2.173 2.223 2.3.7.13 2.17.19 2.373 2.32.471 2.11.43 147 - 3.72 13:19 3.149 - I — 32.83 7.112 -- 2181 2.3. 2.2.371 23.31 22.3.29/ 28.7j 22.1537 2334 22.11.17 24.53 22.3.,79 149 72 — J 3.83 3261 11.59 7.107 3.283 13.73 56[3.5.29/ _.107 __._ _

Page  378 378 MIMISCELLANEOUS IrrcOBLEMs. [ART. XXIII.' —i3 — -- 4 5 ]7 1 _ _ _ _ _ _ _ _ _ oO 2.1 2.3.52 2.53 2.52.7 2.32.52 2.52.11 25213 23.53 252.17 2.52.19 51 3.17 - 33.13 11.41 19.29 3.7:31 23.37, 3.317 52 22.13 23.19 22.79 23.13 22.113 22.3.23' 2=.163 24.47 2.3.71 23.7.17 53 - 32.17 11.23, 3.151 7.9 3 251 54 2.33 2.7.1.1 22,127 2.3.59 2.227 222 7: 2.3.109: _ 2.13.21 2.7.61 232531 K 5.11 5.31 3.5.17 5.71 5.7.13 3.5.37i 5.131 5.151. 3 2.5.19 5.1911 569!2"7 22.3.13 28 22.89 23.3.19 23.139 24.41 2.33.7i 23,1070 2.239) 3.19 ----- 3.7.17 32,73 3[ 1. —3 - 3.7-17 T2. ~~~~~~~3.31 ~ -- 3.11.291 56 2.29 2,79 2.3.43 2.179 2.229 2.32.31i 2.7.47 2.3792.3.11.13 2.479 5~ - 3.53 7.37 - 33.17 13.43 --- 3.11.23 7.137 60 22.3.5 25.5 22.5.11 23.32.5 22.5.23 24.5.7 2.3.5.11 23.5.19 2=.5.431 0,6.3.5! 6i1 7.23 32.2) 192 ----- 3.11.17 ~ 3.7.41 31a2 62 2.31 2.34 2.131 2.181 2.3.7.11 2.281 2.331 2.3.127 2.4311 2.13.37 6:3 32.7 -- 3.11 3.13.17 7.109 -- 1 323.10 64 2 2.41 23.3.11 22.7.13 22.29 2.3.47 2,.83 22.191 25.33 2 2_31 65 5.13 3.5.11 3.553 5.73 3.5.31 5.113:5.7.19 32.5.171 5.1731 5.193 66 2.3.11 2.83 2.7.109 2.3.61 2.233 2.283 2.33.37 2.383 2.4332.3.7.223 67 - - 3.89 - - 34,7 23.29 13.591 37.12: - 68 22.17 23.3.7 22.67 24.23 2.32.13 23.71 22.167 28.3 2.7.31 23.1121 69 3.23 13- 1 32.41 7.67 3.223 --- 11.79 19 76 2.5.7 2.5.17 2.33.5 2.5.37 2.5.47 2.3.5.19 2.5.67 2.5.7.1112.3.5.19 2.5.97 71 - 32.119 1 7.53 3.1571 11.61 3.257 13.67 - 72 23 22.43 24,17 22.3.31 23.59 22.11.13 25.3.7 2%.193 23.109 22,.3 73 -- - 3.7.13 11.43 3.191 --- - 32.7 7.139 74 2.37 2.3.29 2.137 2.11.177 2.3.379 2.7.41 2.337 232.43 2.19.23 2.487 75 3.5 2 5.7 52.11 353 52.39 52.23 33.52 52.33 53.7j 3,12,3 761 22.19 28.11 2.3.23 23.47 22.7.17 22.32 22.13 2.97 22.3 9493 77 7.11 3.59 13.021 32.53 - - -- - 3.7.37 -— I 78 2.3.13 2.89 2.139 2.3.7 2.239 2172 2.3.113 2.389 2.4391 2.3.163 - 32.3 - - 3.193 7.97 19.41 3.293 11.89 29 24.5 22.32-.5 23.5.7 22.5.19 25.3.5 22.5.29 23.5.17 22.3.5.13 24.5.11 12.5.72 34 _ I _____ )3.127 13.37 7.83 3.227 11.71 -- 32.109 i2 2.41 2.7.1:1 2.3.47 2.191 2.241 2.3.97 2.11.31 2.17.23 2.3272 2.4911 3:3 [ 3.61. -- 3.7.23 1153 --- 33.29 -- -- 84 2~2.3.7 23.231 22.71 2".31 22.1112 23.73~3 22,32.19 24.72122.13.17i 23.3.41 85 5.171 5.37 3.5.11 5.7.11 5.97 32.5.133 5.137 5.1571 3.5.59 3.197 86 2.43 2.3.31 2.11.13 2.193 2.33 2.213 2.73 2.3.131 2.443 2.17.29j 87 3.29 11.17 7.41 3243 3.229 --- 3.7.47j 188 23.11 22.47 28.32 22.97 O2.61 22.3.72 24.43 22.197 23.3.37 22.13. 9 89 3,71 1723 3.163 19.31 13.53 3.263 7.127 23)3 9 2.32.5 2.5.19 2.5.29 2.3.5.13 2.5.72- 2.5.59 2.3.5.23 2.5.79 2.5.89 2.32.5.11 191 7.13 - 3.97 17.23 - 3.197 7.113 34.11 22.23 26.3 22.73 23.72 2.3.41 24.37 22.173 2.3.11 2.223 2.31 $4] 2=1312.321 3.31 - 3.131 17.29 32.7. 1 1 13.61 19.47 3.331 9 2.47 2.97 2.3.72 2.197 2.13.19 2.3.11 2.347 2.397 12.3.149 2.7.71 5 5.19 3.5.13 5.59 5.79 32.5.11 5.7.17.139 3.5.53 5.179 5.199 196 28,3 22,72 23.37 22.32.11 24.31 22.149 23.3.29 2,219 9 22.7 22.3.83 j97 - - 33.11 7.7 3.199 17.41 --- 3.13.23 -- 8 2.72 2.32.11 2.143 2.199 2.3.83 2.13.23 2.349 2.3.7.19 2.449 2.499 99 32.1 - 13.23 3.7.191 -7 2 - 3.233 17.47 29.31 33.37

Page  379 111] TABLE OP PRIME A~ND COI-roTOFIT'- u- NUMJiBEl.S. 379 10 11112 13 14 15 10117 133 19 20 521 22 0:131'25 2 627K2 29 I~~~ —...... i............ 017-7 — 3 19 - 3- -31131 741 3 03 17- 3 - 23 3 7 1' 3 a - - 7 3 -3 9 3 - 05 3 51 5 35 5 3 5 3 3 3 5 5 07l19 3 17 - 3 11 - 3 13 - 3 7- 3 29 23 3 7 3 09- I- 3 7 - 3- 2 3 3 472 3 13- 3 5332 11 3 11 7 3 17 - 3 21- 3 3 7 4j 13 3 - 13 3 17 3. 7- 3 3 19 7 3 2- 31 15 5 0 3 5 5 3 5 50 3 5 0 31 5 3 5 5 0 5 5 173 — 31337317233-2137- 3 19 323 - 3 7 311719 3 131 7 3i4111 3 3' 21 - 19 3 - 3 3 17 473 - -II 37-4 7 231 23 3 - - 3 — 3 3 75Ji3 23 343 7 3 371 20 0 3 5 5 3 5 5 5 5 3 0 0 3 0 0 3 5 5 3 2713 73 - 3 3 41 - 3 1713 3 7 37 311 - _~ K -3 17 29 -- 3 11 31 1 31 - -I 17 7 3 11.1 371- 9, 31-3- 1133 7K31 -3 - 23 3 11 3 19:31 33 -11 3 31 - 3 23 - 319 37 3 17-'35 3 5 0 3 5 5 3 5 3 5 5 3 5 5 3 5 3 5 37 17 3 - 7 3 29 3 11 13 3 - 3- 43 3 7 - 339! -17 3 13- 3 11 37 3 7 3- - 3- 7:3 171 37 17 3 11 23 3 - 7 3 13 3 - 319 3 17 453 7 33. 11 11 31 - 31 9 29 3 37- 3.1 3 45 5 0 3 0 5 3 51 5 5 3 5 5 03 50 3 5 5 3 ii~~~~~~~~~~~~ 47 3 31 2U 3 4 1 -73 3 23 19 3 -4 37 49 3- 1 3 17 432 3 7 3 1 7 3 ~ ~ ~3 51 - 3 7 1317 3 II 3 13' 53 3 7 3 3- 17 31 3 131 1 3 7 3 - 55 5 3 5 33 5 53 51 i 3 5 3 5 5 3 5 5 7 7 133 3313- 7 191 1 1 33 3 I 31 ) 93 1-1 3 3- 313 2917 7 3 31 3 1 1 61 313 371 3 7 323 13 3 11 3 (3:- 29 41 3 13 3 31 17! 3 11 3 7 1 i1 3 5 5 3 5 5 3 0 6 3 5 3 5 3 5 067 11 3 7 - 3 - 3 7 3 Ill - 3 17 3 47 3.(39 - 3 3 3 -23 3 3 233 3 7 / 3 h r71 3 31 3 - 3 7 - 3 1 13 3 - 7 3 - 1 31 73 _( 3 19- 3 11 7 3 3 411 3 31 3 41 3 71. 5 5 3 0 0 3 5 5 35 5 3 5 5 3 5 77 3 11 3719 3 3 31 7 3 3 1 31 79 13 3 - 73- 23 3 3- 43 3 37- 3 3 81 23 3 3' 41 13 3 - 3 29 7 3 43 11 1:337 3 - 3- 73 3731 - 11 7 83 3 3 3 - 3 3 0 5 3 85 3/ 5 5 3 5 5 33 337 3 01 3 7 - 3 37i 7 3 13 - 689 3 29 - 3 - 7 3 - - 3 Il 3- 19 3 - 7'390IJ - 1 3 2134 - 37 19 3 31 1113 7729 3 71 -5~~~~~~~~~~~~~~~~~~. 93 3 7 3 -I1 3 1 1 3 3- 3 - 3 11 41 95 3 2 1 3 7 - 3 5 5 3 5' 3 5 3 1 3 1 97 11 3 - 3 7 3 13 3-11 7 3 1 -99 73 11 - 3 7 3 3, - j 3 23 13

Page  380 380 MISCELLANEOUS PROBLEnMS. [ART. XXIII, L 30 31 32 33 34 35 36 37K38 39 40 41 42 43 44 4546 4731 48 49] 01- 7 1 - 19 3 13 47I- 3 11 32 7 43 3 13 03 7 2 3 32 4131 3 - 3- 113213 7 19- 3 - 05 5 5 5 5 5 5 5 5 5 1 5 5 5 1 5 5 5 5 5 5 07 31 13 3 - - 7 11 32- 3 7 59 13 - 17 32 11 7 MU17 - - 3 7 11 3 - 13 319 7 3 31 - 32 11 17 I - 17 13 73 3 37 7 -3 11 13 29 7 -' 13 23 11 7 — 3 - 47i 31 7- 32 1 19 3 - 7 3 - 173 15 5 5 5 51 5 1 5,' 5 5 533 531 3 5 5 1 0 J7 7 3- 31 17- - 7 11- 13 23- 3 7 - 32 533 11 119 29 - 1 317 - 19 - - 3' - 7 32 - 31 11 61 2119- - 3q117 17612- 3a- 13729- 11- 3 % 23 - 3= 11 - 7 13 - 17 - - 32 74111- - 23 - 7 3: 25 5 5 5/.5 5/ 5/5 5 5 51 51 5t 5.5 5 5 5/ 5/ 5' 51 51 231359355~~~~~~~~~~~~3= /9 31,1 13315 27 3 53 73 3231- 13- 43 7 - - 193 7 29 3 131 29 13 7- - 32 1911I 7- 17 - -2 43 7 3 11311 31 7 31 32 - 47 11 - 713- 29 17 61 7 23111 19 —33 32 13 53 11 - - 7 -- 19 37 - 17 7 1 1 3 41 3:135 5 5 5 5 5 51 51 5 5 5 5 5 5 5 5 5 5 37. — 13 47 7 32- 37 3 3111 7 19- 1713 3 7 3 43141 7 19- 3- 11 13 7 - 3 - 2317 - 7 3 11 1414 31327 13 31 11 29 23 73 41 3 19 711 473'4 1 721 - 1 ) 7 1 1 47 3 5 5' 2 i43 17 723 - 1/ 3 -19 7 13 3 - 43 3 7 3 22) - A45 51 51 5' 5! 5;47 11 3 17- 32 7 3 — - 19 11 7 - 3 47 37 17{ 49- 47 19 17- 741 23 3 11 - 32 7 - 3 - - 3! 13 7 51 32 23 3 753 311-3 2- 7 319-3753 43[1- 7 3 l11 13 32a -]59 7/ 1- 3 61 29f11 7;231]31 fj 33 7 3 11 3 i6 nI 7' 23 t5 5 5 5 5 5 5 5 5 5 5 5 51 5 51 5 5 51 5[ 5 37 - 3 — 13 3 — 11- -.7 -367 3 59 713- - I- - 7173711 — 3 747 3 43 61 -293- - 3 7 - 11 17131 191 - 7 3 - 59 3- 11;63 3 - 1319-7 11 53 - 3 17 23; 7 32/ - [l- 3 7 65 5 5 5 5/ 53 5 51 I 51 5! 5 5 5 5 51 5 67 - Il 7 - 29 191- 3 71 32 17 11 3- 13 7 31 - 6911 3 -43 3 -3713113 17 41 3 7 19 3271377- 13 32 7 1123 43'- 31 17 732 3 - 3 13 719 3 - 23[32 - 7 329- 13- 1- - 7 17 - 37 11-.75 51 5 5 5 51 5 515' 5 5 5 / 1 1 55.7717 32=29 11 19 7- 31- 4132 7 31 23 3 17- 7 179- 11 331 7 3 13 - 233 71129 3 19- 32 7 13 81 13- 17 7592- 39 - 3219 7 37 3 13- 32 31 7 3 17 83 3 7 17 32- 29/13 1 7 3 47i- 32 - - 7- 11 85 5 5 51 3 51 5 5 5 5 5 5 5 5 5' 5 5 87 7 - 19 3 11 17 3 713 361 53 3 41 7 11 43- 32 - 89- 3 11 3 37 732- -29 59 7 672 13332 - 3 911 1-3- 3 - -17 3 131- 111 7{- 32 - 3 677 933 3 3137 13 7- 3- 17 - 71 32 23 - 3 13- 7931'1 1 1 1 1 1 1 1I' 1 a 1.5 5 1 61 1 95 5 5 5 5 5 i5 5 5 5 5 5 5 5 5 5 97 19 23 7 43 13 11 - - 32 7 17 31 - 1 71 32 519 19 9 7- I/ -1 59 32 29l 7 31- 131 3 5 11 7 371- 23

Page  381 ~111.1 TABLE OF PRIME AND COMPOSITE NUMBERS. 381 390 3 3 33 54 56 57358 59 6061 62 63 64 63 66 67 68 69 01 3- 7 32 11 3 7 171 - 32 - 371 1 7 - 3 67j 03 3211 13 3 1 - 32'17 - 11 91 7 31 I - 32 07 3- 41 29 3213 11 31 3 7 43 32 - 19' 3 13- - 3 7 711 11 37 19 3 41 32 13 23 3 - 11 7 19 392 47 7 l11 31 3 231 32 33 7 11 37 - 1 2-13[ 3 323 - 32; 19 53 19 3 17 7 39 31 35 1U 71 7 3 3.5 5 3 2 5 713.5 5 1313.5 5 3 32 17 429 3.3 17'9 7 37 13 - 32 411 7 61 111 3 - 32 7 13 3 17 - I 173 7 17 ~32 -~ 3731 1 3 13 29 3 71 7 41 ] 31 32033 33 73 31 332. - 7 - 23 - 47 -3 I1 7 -159, 32- 31 1: 3 7 - 31 3 1.1 37 332 - 29 2 ~ 41 i I 25 5o~ 59 51 53 53133 32' 5,3 5 52 5532 53 52 57 52 53 53. 52 1 27 11 3 27 73 33-117 3 4 1 3- - 2 32 -179 3' 37 711 13 32 73 41 4 23 32 73 617 3 1 3' 1 7 2372 1 1 31 2 32 - 1 ~~~~~~~11.1 7 -1 I:31 322 7 -- 3 - - 3 TY 3711- 31 13 59 7 11) 53i 32 29 43 37 19- 3 2 43 32' 1 7 7 3 3- 1 7 347 3 1 - - 3 35 19 13'3.5 11 5 33 73 3153.5 5 1713.5 29 7 32 5 5 3,5 5 19 49' 9 9 221 37 3 6'~ 32 ( - 032 3 1 3- 17329 - 32 37 37 231 11 33 7 3 13 3 - 17 3- 41 3 - - 43 7 3 - 3 133 19 7 29 3 - - 32 7 17 3 47 13 3 23 7 3 341 713215i 3 7 3 3 13 7i23792 17 19 31 29 3 -2 11 43 3 37 7 13 - 23 3 - 7 - 3 - 1 3 7 11 3 53 45 5 7/5 5 32 5 5i3.5, 7 29 3.5 5 5 32 5 7 3.5 19 / 3.5 7 - 3 - 13 3 7 3 19 1 - 3 - 11 7 - 7 3 4149 TAa 19 2~ 3 -31 7- 32 2 11 3 7 7137 611 17 32 -3' 51. - 1-59-5J/-23 7I - 31 -y 11- 3 - 17 29- /- - 32 43 17/1613331i 532 11171135 16743-r..3 2231 3- -11 53 31 - 17 53 7 32 3 3 - 32 1 93. 3 7 5 32 5 ~193.5 5 325 71 37 31. 35 7 3.5 13 57 13 3 7 11 1 31 a 47 -1 13 11 79 7 29 - 3 59 3 3 53 17 13 31 591 7 3 11 - 3 3 - 19 - 173 713- 3 431 31 23 3 7 - 11 - 3 7 19 311 3[19/ 111 7 3 3324359 29] -[ 717-[ 31[ 32[ 23[-/ 63 3I 3 3'717 11 - 4 3 7 - 32 - - 3 7 81 11 32- 13 41 - - 37311332.- - 17l- 7 13 (35 5 57 3. 2 59 513.5 13 5 3.5/3 2 17 5 3.5 71 3 3 35 11J 177 1123237 3 37 1 3 2 113 3 24 7 — - 3 3 - 29 13 5 7 33 1! l 32 1113 i 7 3 -7 4331 - l 1 3:3 2 7- - 2 33343132 3753 2373 73 13 32 1 - 23 3243 - 3-,93 I 3 711 - 3 13- 23 32 711 - 3 2 19 i937 -331 3 733175332431233.3 3 31 32 7 3I8j1 ~ 31 1 - J 3I 171 7 1-32- 1341 3! 1 81~~~~~~ { 3' 67[ - I7 3 1 71 32 7 3 37 3 7 32 61 13 3 29 4137 - - 95 3 1 7 3.5 53345 13 7 11 3% 5 3.5 5 3.5 7 23 32 1 IS7 7 17 - 3 37 11 32 231 - 3 3 131 7 3a I 1 71 17 189 7 - 41 1'7 11 3,2 - 7 13 53i 3 193 - 32 ] 31 8 3 2 91 3 291 11 32 17- 7-43 3! 41j 32, 71 - 13 93 1`1 312 7 - 7 -~ 3 71 13 3S 11 7 3 43 19 23 - 61 3a-,95 5 53 13 73.5 1719 3a 11 23 3- 5- 5 5 3.5 5 1 3 3 2 7'99r 31 - 3 i 2 11 4 13 1 7 7 19i - - 3,21 671 -1 7 13 3 ~~7 2 7 i - I3 3 7 7 1

Page  382 382 MISCELLANEOUS PROBLEMS. [ART. XXIII. 10 71 72 73 74 75 76 77 78 79 8 8283184 85 86 87 88 89 01. - 19 7 3 13 11 17 29 - 32 - 59331 - 477.11- 13 32 03 471 386711 41- - 32 7 53 37131 19311 7 32 - 9: 05 3.515.7 11 13. 5 1932 23 5.7 3.5 5 3.5 11 5 32 5 5 3.5 13 07 1 3 - - 3 - 3.7 37 - 1767 9 32 7 47 19 - 3 09 43 32 31 7 13 19 11i- 32 - 7 3 67 3 23 -- I a3 70 3 a.~9 11 32 13 3 - 29943 113 73 32 37 133 79313267 113 - 3 - 7113.7 11 2, 32 13 41 319434 17, 47- 32 - 7 3: 150,3 53.557 5 325 3.5 5 5.7 3.5 3 1 32 13 5 3.541 5 / 3. 11 7 7 3 -/ -37 - - 32 -9_ 317 72 3 3 3, 321 11 3 73 19 31 7 320 47 - 7 13 - 32.21 17 -1 2 33 324 892, 13 3 53 3.7 - 37 3- 11,2 3 17 31 3 - 32 - 71 3 29 - 3 7 13 1 1 012 52 5 3 52 52- 32 50 2 52 02 52 52. 52'5 52 52 2 5 52 52 521 127 - 32 17 7; 1 3 2 9, 3 - 2:3 32 19 Il - I""3 i3::359332399193 - 13i 3 29 32 - - 13.7 7 3' 59 - 31 1 1 1 3 - 3 - W 32 7l,__l —,-l- ___ -- 2) 32':3l 4 73 3 17 13 324711 3 471 3 - 19 32 - 3 13 7 3 - 1- 321 17 19 3.71 -2 31 - 13 3 ] 3 223 89 41 73 - 35 3.55 532 5 11 3.5 5.7 a 3.5' 5 5 32 5 5.7 3.5 1 5 3.5:37 -31 32 - 33 7 3 13- 3 - 32 3! 59 99 43 43 3 7 71 32 17 3 7 318 2, 53 32 7 41 3 37 13 3 7 2 3 17 7 41 19 23 32 - 3.7 - 43 - 3 7 3 1 - 2 ea/23 13 3.7 11 7- 3 - - 43 71 37 11 5 5 32 13 5 3o5 1 1 5 35 57 5 17 5 3.5 5.7 11' 29 5 47 327 - 1 311 3- 3 61 19 32 13 - 3 17 - 11 - 32123 49 19 3 11 - 13 - 47 - 3 20 731l 11 8 3 32 13 - 13 9 531 -3 - 3 7 23 3 - 11 37- 7 3 17 41 3i 53 153 3 23- 32 299 13 3 - - 11 - 31.32 7 37 17 -_ 13 F 1732. 5 513.5 5 5 111 5 37 35.7 133.5: 19l 29 3.5 17' 5.7 32 57 - 17 41 7 - 11 13 2 731 7 3 23 6! 343 11 329 17 13 99 13 - 1711 - - 2 3 41 3l 13. 11 32 I 19 3 17 [o~~~~~i5~~~3 l'l 3=',{61 23 11 53 17 3 - 47 13 7 19 3- 11 32 7 3- - 29 63 13 3237 7 3 73 7 3 - 11 32- -3.7 - -23 32 5 513.5 o l' 3.5 5 11 32 5 23/3.53 5.7, 5L3.5 5 5 32 11 67 373 13 53I9 23 17 31 - 31 3 37 33 1 32 11 - 3.7 7) 3'-7 32 - 17 43 13, - 37 32 4 - 37 771 3 711 11 32, 31 61 3 19 11 31 - 32! I 43393 7 37r3 17 3 7-734 = 311 433 3 13 17 32 173 11[ 3si 7] 73[ 47 - - 3[ -} 171 3~/ 1 1[-[ 3 37[- 3,"[ 3 1! 19[ 3~!75 s5252 5 52 5,2 5s25= 52 3 l 52is 52 52 5252 52 52 5 52 5 2: j 5 73.7 - 1 31- 32 7.11 314 113 31 7 - 7 11 471 79 3 23!47 32 13 1 3{- 79 3 - I7 32 61 2:3 11 13 41 81 73 43 32 61 - 3.71- 31. 37 03 32 13' 17 111 3 8:3 7 83':3i 1 5 1 - ]"/7- 13 43 - 32 59 7 11 83 1731 - 13 32sC~ 11 - 2: E3 7 -3 11 3g 1 85 13 3. 315 3.5 31 29 3 5;i 11 5 53.5 5 17 32 5.7 5 3.5 87 )[- 3.7 83[- 32 — I r 1 7 - 3 32 31 17 29- 43 i8!, IT 3 2 1 (~~~~~~~~~~23 3 - 11.7- 17 1.3 89, 8) 17 13 37 3{2 - 11 23 3- 19 3=- 137 17 q 89 7 3 93 1 3 7 13 61 32 3 7 1 3 59 17 3 93 41 -59 37 3 3 - 1 3 2.- 17 9 11 53555732 5 535511 3.5 23 52 37 53557 97 47 3[- 131321 143 31 23 53 1 3 7 7 32 -9 13 191 31 [99 30 2 3 - 1 1 3 -11 13 31 43 371 3- 3.7I 11-;03

Page  383 ~ 111.] TABLE OF PRIME AND COMPOSITE NUMBERS. 383 90 91 92 93 94 95 9 97 98'99 100101102 103 104 105 100 107 109 1091..........' — -- I..... — i-i.........I' 01 19 3 71 17 3 - 89 11 733,71012 - 3 - - 29 7 il i03 3 - - 3.7 -I 13 11 31',- 7 -I 191- 101 33 2g3 11 13 - 0. 51 53.51 5.7 5 3.5 5117 3.5 3715.71 3.5 43' 13 3.5 5 11 3.5 5 5 3.5,0 7t 73 41i'07 - 7 11 41 23 3 13 17.3.7 - - 32 59 11 3 19 - 43 101 13 10911- 29972 373 19 1733- 11 41 13 7 311032 -32 - 2 3 61 3 - 7 -174 1 -3.7 2923 33 - 19 37 33- 337 67- 3.7 - 1 3132 3171 3 7 - 13- - - 3 11 71 15 3.5 5 1913.5 5.7 11 3.5 21i 133.5 515.71 3.5 5 53.5 11 5 3.1 371 71 32 1 7 4 31 5 41 - 47 3.701 1719 11 13 3 7i 21 32 lbK120 131 3 /7 191 - - 3113 43 3 11 17 23067 37 331 3101 I~~~21 31I —i 1 32~~~~~~3'0131 7-13- 3_'2331129 3-173.71371! 307 23 7 3 7 23 - 33 89 3719 13 53 31 717 3 -7 3 ~~~~311 52 5 5 5 5 52 5 52 52 522 52 52 525 52 5252 5252 52 52, 52 52 7 3 3- 3 7 7 1 31 32 37 313.7 23 - 11 - 17 3 7 ~~~~~~~~~~19 -/ 13 - 137 1119 3.7 - 13 23 3 53 17 1 - - 32 - 13 3 l31 11 23 1731- 32 37 291 - 7 1113- 19- - 3.7 - 1 3313 17 - 13 -- 79 - 33 - 331- 03 33 135 13 3.5 5 5 3.5 541 3.5 5.7 513.5 5 233.5 55.7 3.5 19 13 375 37 7 - 3 - - 1123 13 3219 3 21 - 3.741 11 32 23 19 - 11 - 37 93 — 3 13 32 41 11 -- 3 2 3117 3 - 3- 9 1 453 83 732 37 3 43 41 13- 193 - 1761 8313.7 -3 - 31329 3 731 45 3.5 31 533.5 5 233.5 511153. 15.7 5 3.5 5 513. 5 5.7 3.5 11 47 83 3 7 13 47 - 11 139 43 29 17 731 - 3 31 53 3.7 11 - 41 - 3- 11 321 -1- 3j71- 1 17 37 3.7 11 2 3 3 1951 3.7 - 29 3 13 - 3 7 - 31 19 - 17 11 7 3 - 13 3 47 53 1134 13 47 123 41 7 31 1 59 8373 1331 - 7 - 1 531 - 31 31 55 5 3.5 5 31 3.5 53,511 5 3.5 57 19 53.5 5 5 3.5 135.7 3- 7/ 19 29 11/ - 3 1 39 - - 17 - 31 3.753-43477 32 79 1 3 3-233.7- 3 - I 1129- 13 I- -- -- -- - - - - -- -- - -- -11/ —— 41 61 13- 3.72 - 3- 431 7- 32 31 13 1159 7 17 - 971 03 319 17 51 3 - 731 313 733 329 1143 -3.7 - 47 17 191 655.7 3.5 17 51 3.5 51 3. 5 1119 55 3.5 5.7 5 3.5 5 41171. 67 -89 317- 39 7- 11 — - 7 33 - 37 3- 1 69 3 53 13 33217 711-713 — 3.7- 19 1347 89 3 71 —!- - - - - - - 17147 32 73 - 11 9 1 3 -1 13 38 7 3 71 31 3- 7 231 43 - 11 13 - 3 17 29 32 - 7 3 - 23 3 7 13 3.7 83175',252 5 25 2 3 s 5'252 52 5252 52 52 5252 52 52 51 52 52 521 177 37 - - I 131 -| 3 17 11 3 433 - 7 3 13 3 8-37/ 711 39/ -3 37 2313 -67 317 1 -3 1 4 3 8 3 11 3.5 5 5.7 3.5 13 193.5 5 53.51 1 31 3.5 1 5 3.5 7 1 - 13 3713.7 53 - 3 -- 311 61 32 13 - 3 - 23 It - 8 6.9 3'2 741 343- 13! 297 19 231 - 3 17- 3.7 - - 11'1 13 19- - 2311 - 3.7 97- 43' 41- 1 - 11 2I 3 3.7 2 - 31 11 53 3 7 13 3 I- 47 19 7 11 17 43 3 C 1.71.5 11 5 3.5 19{5.7 3.5 5 3.5 5, 29 35 51 13 3.5 17 5 3 97 11 17 321- - 3.7- 97 3 13 2311 737 31 — 1i 59 17 7 393_- 17 13 3 291 53 41 19 11 - 31 3 - 3 131 - 3.7 17

Page  384 38 4 MISCELLANEOUS PROBLEMS. [ART. XXIII. 110 III 112 113 114 115 116 117 1181119 120 121 122 123 124 12 126......~~~~~~~~~~~~~~~11 1.... 01 19 17 23 3 13 3132 - 311 - 3.75- - 3303- 3 178 3.7- 414729- 313 - 379 305 31 5 3 5.7 5 3.5.11 5 3.5 5 5.7 3.5 5 23 3.5 41 5. 1:31 291 57 3 17:375323 - 3.7 3 31 19 3.11 71 0 >01 3!7 11 43 3 17 13 32'7 - 3- 29 3. 11- 7 33: 11 13 41 37 3 17 71 31 43 -3au1 - 13 3.7 11 3- - 3 1012 3.713- 3.11 41 3 7- 4313 5.5 531 3.5 5.7233.5 17 53.5 55.73.5 1 5 3.5 113 - 3- 7 349 - - 13 17 613.7 19 109 3- 311 1 3 - 13 3.7 19 - 32 5 29 17- 3 9 7 11 1321 10:3337 7- 33 4 - 3 13 317 101 32 I 93,7 2 733 7 29 13 - 23 59 19 3.7 - I 33I 3 17 7 13 552 55 5 52 52 52 53~ 59 52 52 52 52 3 52 52 2)7 - 3103 47 13- 7.11 32 - - 19 67 - 3.7 17 - 23 l21 41 31 19 - 11 3.7 29 37 3 791 23 13 7 - 32 17 73, —I,,. l-7; —41 3 21 3- 1132 23 13 3 4153 7 33 19 31 17 33 17 32 47 7 37 19 - 3 - - 3,7 11 13 3 - 83 3 35 5 17 3.5 5 5 3.5 13 5 3.5 5.7 29 3.5 5 5 3.5 23 5,7,137 13 37 17 3 - 83 33 97 19 23 - 53 3 13 - 3.7 - 39 13 47- 23 31 11 10 3.7 - - 3 61- 33 7 - 3.11t 41_ 01 13 7_ 3- 19-41 3.11-_ -_!41 61 13 11 1 7 9 43 33 11 - 19 - 973 3- 3 32 -- 3.7 -23 37 47 4 455 3.5 13 51 3.55 17 35 235 35 5.7 31 3.5 19 13 3,51 47- 71 23 7- 32 191 359 13 7 37 3s-.3 _ 49 20 - 11072-35331 41 3.7 - - 32535947 13 51 43 3.7- -3.11- 61 3 17 91329 23- 7 53 7 19 31- 13 3 4323 33- 17 3- 113.7I 1~~~~~~~~~~~1 55 3.5 23 3,5 29 5 3. 51 5 3.5 55.11 3.5 5.7 47 3.5 5 57 3 41 19 1 371 1 3- 10332 2 3 559 - 3 3 3 89 1 159 j 3 3.7 1 3 1 7 3 19 01 32 - 3.7 73 11 13 19 259 33 7 - 61 47 17 53 1 1u )- 1 7 ~51 - 23 7 3 13 3 343 - 3.7 -- 3i7 i _i5 - 377 --- 71 138 I 3 31177 3 89 3 13 - 73 3- - 177.11 71 32 01! 31 13 - 37 3- -3.11 19 75.2 52 52 52 52 52 512 512! 5~2 512 512 52 52 52 5s 52 5 2 77!19 - 3.731-2:3-17- -37-291-3:1.I5. -_1 — - 79 3 7 - 3 3 3 13 - 17 - 73.11 47 19 3 - - 3.7 31 81 7 3 2 19 43 37 - 3,1092 - 3 13 3i 7 23 3 83 53 3- -- 137- - 17 23 43 31 I71 29 133 7 1 91- 11 53.5 51 37 3,5 5. 5.7 3.5 5 5 17 5 5 3.5 5 5.11 3.5 43 18 l 1A1 J 1337 13 -- 177,113 - 4171 391 3 19 7 3- 07 33 13233.7107 73174- - 3237 93 3.7 23 - 32 - 11 3 7 20 9 891 19 11 313 7 3 i 13' 67 53] 7[- 3/ - 1.- 1 93[ 197 1 53i 3 ~ ~ ~ ~ ~9 19 17 1317 95 5.7 5. 3.5 43 1 3.5 5 5.7 3.5 1 3.5 5 37 17 5.11 5 971 33~ 1:323)- - 3.7 47- 31 -- 3 23 - 139j I3I - I- 3 7 - 19 73 13 37 1 7 3 294 17'9 39 23 - 13 i791 33 - 1 2 — 3.7 47- 31 - j D 11 1 1 2 7

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