Conic sections and analytical geometry; theoretically and practically illustrated. By Horatio N. Robinson.
Robinson, Horatio N. (Horatio Nelson), 1806-1867.

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Page  I ROBINSON'S MATHEMATICAL SERIES. CONIC SECTIONS AND ANALYTICAL GEOMETRY; THEORETICALLY AND PRACTICALLY ILLUSTRATED. BY HORATIO N. ROBINSON, LL. D., LATE PROFESSOR OF MATHEMATICS IN THE U. S. NAVY, AND AUTMOR OF A Fnu COUml OF MATHEMATICS. NEW YORK: IVISON, PHINNEY & CO., 48 AND 50 WALKER ST. CHICAGO-: S. C. GRIGGS & CO., 89 AND 41 LAKE ST. BOSTON: BROWN, TAGGARD & CHASE. PHILADELPHIA: SOWER, BARNES & ao., AND J. B. LIPPINCOTT & CO. CINCINNATI: MOORE, WILSTACH, KEYES & CO. ST. LOUIS: KEITH & WOODS. DETROIT: RAYMOND & ADAMS. 1862.

Page  II ROBINSON'S SERIES OF MATHEMATICS, The most COMPLETE, most PRACTICAL, and most SCIENTIFIC SERIES of MATHEMATICAL TEXT BOOKS ever issued in this country, (IN TWENTY VOLJUMES.) e. eo 044 ------ 1. Robinsons Progressive Primary Arithmetic............. $ 15 2. Robinson9s Progressive Intellectual Arithmetice........ 25 3. Robinson's Rudiments of Written Arithmetlci.......... 25 4. Robinson9s Progressive Practical Arithmetic,.......... 56 5. *Robinson9s Key to Practical Arithmetic,.................. 50 6. Robinson's Progressive Higher Arithmetic,.............. 75 7. Robinsons Key to Higher Arithmetics.................. 75 8. IRobinson's New Elementary Algebras...................... 75 9. Robinson9s Key to Elementary Algebra.................. 75 10. Robinson's University Algebra,............................... 1 25 11. Robinsons Key to Uni.versity Algebra,...............1.... 1 00 12. REobinsons New University Algebra, (in press,)............. 13. Robinson's New Geometry and Trigonometry,.......... 1 50 14. BRobinson's Surveying and Navigation,......................1 50 15. Robinsons Analytical Geometry and Conie Sections,.. 1 50 16. Robinson's Diff. and Integ. Calculus, (in preparation)...... 1 50 17. Rlobinson's Elementary Astronomy,......................... 75 18. Robinson's University Astronomy,..........................1 75 19. IRobinson's athemahematical Operations,...................... 2 25 20. Rcbinson, s Key to Geometry and Trigonometry, Conie 8ections and Analytical Geometry,.................... 1 00 Entered, according to Act of Congress, in the year 1860, by H. N. ROBINSON, LL. D., in the Clerk's office of the District Court of the United States for the Northern District of New York. J. G. K. TRUAIR & CO., STEREOTYPBR & PRINTERS, SYRACUSE, N. Y.

Page  III PREFACE. In the preparation of the following work the object has been to bring within the compass of one volume of convenient size an elementary treatise on both Conic Sections and Analytical Geometry. In the first part, the properties of the curves known as the Conic Sections are demonstrated, principally by geometrical methods; that is, in the investigations, the curves and parts connected with them are constantly kept before the mind by their graphic representations, and we reason directly upon them. In the purely Analytical Geometry the process is quite different. Here the geometrical magnitudes, themselves, or those having certain relations to them, are represented by algebraic symbols, and we seek to express properties and imposed conditions by means of these symbols. The mind is thus relieved, in a great measure, of the necessity of holding in view the often-times complex figures required in the intermediate steps of the first method. It is, mainly, at the beginning and end of our investigations that we have to deal with concrete quantity. That is, after we have expressed known and imposed conditions, analytically, our reasoning is independent of the kind of quantity involved, until the conclusion is reached in the form of an algebraic expression, which must then receive its geometrical interpretation. Much of the value of Analytical Geometry, as a disciplinary study, will be derived from a careful consideration, in each case, of this process of passing from the concrete to the abstract and the (iii)

Page  IV iv PREFACE. converse, and both teacher and student are earnestly recommended to give it a large share of their attention. In both divisions of the work the object has been to present the subjects in the simplest manner possible, and hence, in the first, analytical methods have been employed in several propositions when results could be thereby much more easily obtained; and for the same reason, in the second division, a few of the demonstrations are almost entirely geometrical. The analytical part terminates, with the exception of some examples, with the Chapter on Planes. Three others might have been added; one on the transformation of Co-ordinates in Space, another on Curves in Space, and a third on Surfaces of Revolution and curved surfaces in general: but the work, as it is, covers more ground than is generally gone over in Schools and Colleges, and is sufficiently extensive for the wants of elementary education. Numerous examples are given under the several divisions in the second part to illustrate and impress the principles. The Author has great pleasure in acknowledging his obligations to Prof. I. F. Quinby, A. M., of the University of Rochester, N. Y., formerly Ass't Professor of Mathematics in the United States Military Academy, at West Point, for valuable services rendered in the preparation of this treatise, as well as for the contribution to it of much that is valuable both in matter and arrangement. His thorough scholarship, as well as his long and successful experience as an instructor in the class-room, preeminently qualified him to perform such labor. December, 1861.

Page  V CONTENTS. CONIC SECTIONS. DEFINITIONS, Conical Surfaces.......................... PAGeE 9 Conic Sections,........................... 10 THE ELLIPSE. Definitions and Explanations,.......................... 11 Propositions relating to the Ellipse,...................... 13 THE PARABOLA. Definitions and Explanations,................ 41 Propositions relating to the Parabola,.................... THE HYPERBOLA. Definitions and Explanations,.......................... 65 Propositions relating to the Hyperbola,................... 67 ASY M P TOTES. Definition,.......................... 91 Propositions establishing relations between the Hyperbola and its Asymptotes,................................ 91 1* (v)

Page  VI vi CONTENTS. ANALYTICAL GEO METRY. General Definitions and Remarks,....................... 96 GENERAL PROPERTIES OF GEOMETRICAL MAGNITUDES. CHAPTER I. OF POSITIONS AND STRAIGHT LINES IN A PLANE AND THE TRANSFORMATION OF CO-ORDINATES. Definitions and Explanations,........................... 97 Propositions relating to Straight Lines in a Plane,......... 100 Transformation of Co-ordinates,......................... 119 Polar Co-ordinates,....................... 122 CHAPTER II. THE CIRCLE. LINES OF THE SECOND ORDER. Propositions relating to the Circle....................... 124 Polar equation of the Circle............................ 132 Application in the solution of Equations of the second degree, 134 Examples,.......................................... 139 CHAPTER III. THE ELLIPSE. The description of the Ellipse and Propositions establishing its properties,................................... 140 Example,........................................... 167 CHAPTER IV. THE PARABOLA. The description of the Parabola and propositions establishing its properties................................... 169 Polar equation of the Parabola,....................... 183 Application in the solution of equations of the second degree, 185 Examples...........1........................... 187

Page  VII C ONT ENTS. vii CHAPTER V. THE HYPERBOLA. The Description of the Curve, and Propositions Establishing its Properties,................................... 188 ASYMPTOTES OF THE HYPERBOLA. Definition and Explanation,............................ 201 The Equation of the Hyperbola referred to its Asymptotes, and Properties deduced therefrom,..................... 202 CHAPTER VI. ON THE GEOMETRICAL REPRESENTATION OF EQUATIONS OF THE SECOND DEGREE BETWEEN TWO VARIABLES. Object of the Discussion............................... 210 Solution and Discussion of the General Equation,.......... 211 Criteria for the Interpretation of any Equation of the Second Degree between two Variables,...................... 221 APPLICATIONS. First, B —4A C<O0, the Ellipse........................ 222 Second, BE-4A C> 0, the Hyperbola..................... 226 Third, B2-4A C-O, the Parabola,...................... 231 Examples,.................................. 233 CHAPTER VII. On the Intersection of Lines, and the Geometrical Solution of Equations,.................. 237 Remarks on the Interpretation of Equations,.............. 244

Page  VIII vimi CONTENTS. CHAP TER VIII. STRAIGHT LINES IN SPACE. Co-ordinate Planes and Axes,.................. 249 The Equations and Relations of Straight Lines in Space,.... 250 CHAPTER IX. ON THE EQUATION OF A PLANE. The Equations and Relations of Planes,.................. 258 Examples Relating to Straight Lines in Space and to Planes,. 269 Miscellaneous Examples,................................. 273

Page  9 CONIC SECTIONS. DEFINITIONS. 1. A Conical Surface, or a Cone is, in its general acceptation, the surface that is generated by the motion of a straight line of indefinite extent, which in its different positions constantly passes through a fixed point and touches a given curve. The moving line is called the generatrix, the curve that it touches the directrix, the fixed point the vertex, and the generatrix in any of its positions an element, of the cone. The generatrix in all its positions extending without limit beyond the vertex on either side, will by its motion generate two similar surfaces seperated by the vertex, called the nappes of the cone. 2. The Axis of a cone is the indefinite line passing through the vertex and the center of the directrix. 3. The intersection of the cone by any plane not passing through its vertex, that cuts all its elements, may be taken as the directrix; and when we regard the cone as limited by such intersection, it is called the base of the cone. If the axis is perpendicular to the plane of the base, the cone is said to be right; and if in addition the base is a circle, we have a right cone with a circular base. This is the same as the cone defined in Geometry, (Book VII, Def. 16), and in the following pages it is to be understood that all references are made to it, unless otherwise stated. (9)

Page  10 10 CONIC SECTIONS. 4. Conic Sections are the figures made by a plane cutting a cone. 5. There are five different figures that can be made by a plane cutting a cone, namely: a triangle, a circle, an ellipse, a parabola, and an hyperbola. REMARK. The three last mentioned are commonly regarded as embracing the whole of conic sections; but with equal propriety the triangle and the circle might be admitted into the same family. On the other hand we may examine the properties of the ellipse, the parabola, and the hyperbola, in like manner as we do a triangle or a circle, without any reference whatever to a cone. It is important to study these curves, on account of their extensive application to astronomy and other sciences. 6. If a plane cut a cone through its vertex, and terminate in any part of its base, the section will evidently be a triangle. 7. If a plane cut a cone parallel to its base, the section will be a circle. 8. If a plane cut a cone obliquely through all of the elements, the section will represent a curve called an ellipse. 9. If a plane cut a cone parallel to one of its elements, or what is the smtne thing, if the cutting plane and an element of the cone make equal angles with the base, then the section will represent a parabola. 10. If a plane cut a cone, making a greater angle with the base than the element of the cone makes, then the section is an hyperbola. 11. And if the plane be continued to cut the other nappe of the cone, this latter intersection will be the opposite hyperbola to the.E former. 12. The Vertices of any section are the points where the cutting plane meets the opposite elements of the cone, or the sides of the vertical triangular section, as A and B.

Page  11 THE ELLIPSE. 11 Hence, the ellipse and the opposite hyperbolas have each two vertices; but the parabola has only one, unless we consider the other as \ at an infinite distance. 13. The Axis, or Transverse Diameter of a conic section, is the line or distance AB between the.... |. vertices. Hence, the axis of a parabola is infinite in length, AB being only a part of it. The properties of the three curves known as the Conic Sections will first be investigated without any reference to the cone whatever; and afterward it will be shown that these curves are the several intersections of a cone by a plane. THE ELLIPSE. DEFINITIONS. 1. The Ellipse is a plane curve described by the motion of a point subjected to the condition that the sum of its distances from two fixed points shall be constantly the same. 2. The two fixed points are called E D the foci. 3. The Center is the point C, the At middle point between the foci. - 4. A Diameter is a straight line through the center, and terminated both ways by the curve. 5. The extremities of a diameter are called its vertices. Thus, DD' is a diameter, and D and D' are its vertices. 6. The Major, or Transverse Axis, is the diameter which passes through the foci. Thus, AA' is the major axis. 7. The Minor, or Conjugate Axis is the diameter at right

Page  12 12 CONIC SECTIONS. angles to the major axis. Thus, CE is the semi minor axis. 8. The distance between the center and either focus is called the eccentricity when the semi major axis is unity. That is, the eccentricity is the ratio between CA and CF; or it is -CA; hence, it is always less than unity. The less the eccentricity, the nearer the ellipse approaches the circle. 9. A Tangent is a straight line which meets the curve in one point only; and, being produced, does not cut it. 10. A Normal to a curve at any point is a perpendicular to the tangent at that point. 11. An Ordinate to a Diameter is a straight line drawn from any point of the curve to the diameter, parallel to a tangent passing through one of the vertices of that diameter. REMARK. —A diameter and its ordinate are not at right angles, unless the diameter be either the major or minor axis. 12. The parts into which a diameter is divided by an ordinate, are called abscissas. 13. Two diameters are said to be conjugate, when either is parallel to the tangent lines at the vertices of the other. 14. The Parameter of a diameter is a third proportional to that diameter and its conjugate. 15. The paramater of the major axis is called the principal parameter, or latus rectum; and, as will be proved, is equal to the double ordinate through the focus. Thus F' G is one half of the principal parameter. 16. A Sub-tangent is that part of the axis produced, which is included between a tangent and the ordinate, drawn from the point of contact. 17. A Sub-normal is that part of the axis which is included between the normal and the ordinate, drawn from the point of contact.

Page  13 THi.' ELLiPSE. 18 PROPOSITION I. PROBLEM. To describe an Ellipse. Assume any two points, as F and G / F' and take a thread longer than d the distance between these points, A' fastening one of its extremities at the point F and the other at the ID point F'. Now if the point of a pencil be placed in the loop and moved entirely around the points F and F', the thread being constantly kept tense, it will describe a curve as represented in the adjoining figure, and, by definition 1, this curve is an ellipse. PROPOSITION II.-THEOREM. The major axis of an ellipse is equal to the sum of the two lines drawn from any point in the curve to the foci. Suppose the point of a pencil at p D to move along in the loop, holding the threads F'D and iFD at A' F' A equal tension; when D) arrives at A, there will be two lines of threads between F and A. He-nce, the entire length of the threads will be measured by F'F~-2FA. Also, when D arrives at A', the length of the threads is measured by FF'+ 2F'A'. Therefore,.'FF'+2FA=FF'+2F'A' Hence,.... FA=F'A' From the expression FF'+2FA, take away FA, and add F'A', and the sum will not be changed, and we have F'F+2FA=A' F' +FF' +FA=A'A Therefore,. F'D+FD= A'A Hence the theorem; the major axis of an ellipse, etc. 2

Page  14 14 CONIC SECTIONS. PROPOSITION III.-THEOREM. An ellipse is bisected by either of its axes. Let FZP be the foci, AA' the ma- p:.P jor and BB' the minor axis of an ellipse; then will either of these, A axis divide the ellipse into equal. i parts. Take any point, as P in the el- I' S lipse, and from this point draw ordinates, one to the major and another to the minor axis, and produce these ordinates, the first to P', the second to P", making the parts produced equal to the ordinates themselves. It is evident that the proposition will be established when we have proved that P' and P' are points of the curve. First. F is a point in the perpendicular to PP' at its middle point; therefore FP'- FP (Scho. 1, Th. 18, B. 1 Geom.) for the same reason F'P'=F'P. Whence, by addition, FP' + F' P' =-FP+ F /P. That is, the sum of the distances from P' to the foci is equal to the sum of the distances from P to the foci; bout by hypothesis P is a point of the ellipse; therefore P' is also a point of the ellipse, (Def. 1). Second. The trapezoids P"d CFt, PdCF are equal, because F' C=FC, dP='-dP by construction, and the angles at d and C in each are equal, being right angles; these figures will therefore coincide when applied, and we have P'F' equal to PF and the angle /PF'F equal to the angle PFF'. Hence the triangles P"'F/, PFF' are equal having the two sides P'F', F'Fand the included angle P"F'F in the one equal, each to each to the two sides PF, FFY' and the included angle PFFT in the other. Therefore, P'F'+ P"F= PFt + F P That is, the sum of the distances from P" to the foci is

Page  15 THE ELLIPSE. 15 equal to the sum of the distances from P to the foci, and since P is a point of the ellipse P" must also be found on the ellipse. Hence the theorem; an ellipse is bisected, etc. PROPOSITION IV.-T HEORE M. The distance from either focus of an ellipse to the extremity of the minor axis is equal to the semi-major axis. Let AA' be the major axis, F'and D F' the foci, and CD the semi-minor axis of an ellipse; then will D= AD C F A F',D be equal to CA. Because F' C= CF and CD is at right angles to F'F, we have F'D=FD. But, F'D+ FD=A'A Or, 2FD=A'A Therefore, FD-=A'A, or CA. Hence the theorem; the distance from either focus, etc. SCHOLIUM.-The half of the minor axis is a mean proportional between the distance from either focus to the principal vertices. In the right-angled triangle FCD we have 2 2 2 ~D_ =-FD __F But, ED=AC Therefore, - - 2-A -2 EU2 = (A C+OC) (A C-FC) =AP'XAE Or, AF: UD= CD: FA' PROPOSITION V. -T HEOREM Every diameter of an ellipse is bisected at the center. Let D be any point in the curve, and C the center. Draw D C, and produce it. From F' draw F'l' parallel

Page  16 16 CONIC SECTIONS. to ED; and from F draw rD' par- D allel to'D. The figure'DFD'P' is aparallelogram by construction; and F' F therefore its opposite sides are equal. Hence, the sum of the two sides P'D' and )D'Fis equal to F'D and DF; therefore, by definition 1, the point D' is in the ellipse. But the two diagonals of a parallelogram bisect each other; therefore, DC= CD', and the diameter DD' is bisected at the center, C, and DD' represents any diameter whatever. Hence the theorem; every diameter, etc. Cor. The quadrilateral formed by drawing lines from the extremities of a diameter to the foci of an ellipse, is a parallelogram. PROPOSITION VI.-THE OREM. A tangent to the ellipse makes equal angles with the two straight lines drawn from the point of contact to the foci. Let F and F' be the foci and t I D any point in the curve. Draw / P'D and FiD, and produce F'D to H, making.DH=DF, and draw PH. Bisect FHin T. Draw TD and produce it to t. Now, (by Cor. 2, Th. 18, B. I, Geom.), the angle FDT= the angle HD T, and HTD=its vertical angle F'Dt. Therefore, FD T=-F'D t. It now remains to be shown that Tt meets the curve only at the point D, and is, therefore, a tangent. If possible, let it meet the curve in some other point, as t, and draw Ft, tH, and F't. (By Scholium 1, Th. 18, B. I, Geom.) P=tH. To each of these add PFt; Then, F't+ tH= F't+Ft

Page  17 THE ELLIPSE. 17 But F't and tH are, together, greater than F' H, because a straight line is the shortest distance between two points; that is, F' t and Ft, the two lines from the foci, are, together, greater than FH, or greater than F'D+ FD; therefore, the point t is without the ellipse, and t is any point in the line Tt, except D. Therefore, DT is a tangent, touching the ellipse at D; and it makes equal angles with the lines drawn from the point of contact to the foci. Hence the theorem; a tangent, etc. Cor. The tangents at the vertices of either axis are perpendicular to that axis; and, as the ordinates are parallel to the tangents, it follows that all ordinates to either axis must cut that axis at right angles, and be parallel to the other axis. SCHOLIUM 1.-From this proposition we derive the following simple rule for drawing a tangent line to an ellipse at any point: Through the given point draw a line bisecting the angle included between the line connecting this point with one of the foci and the line produced connecting it with the other focus. SCEOLIUM 2. Any point in the curve maybe considered as a point in a tangent to the curve at that point. It is found by experiment that rays of light, heat and sound are incident upon, and reflected from surfaces under equal angles; that is, for a ray of either of these principles the angles of incidence and reflection are equal. Therefore, if a reflecting surface be formed by turning an ellipse about its major axis, the light, heat, or sound which proceeds from one of the foci of this surface will be concentrated in the other focus. Whispering galleries are made on this principle, and all theaters and large assembly rooms should more or less approximate this figure. The concentration of the rays of heat from one of these points to the other, is the reason why they are called the foci or burning points. 2* B

Page  18 18 CONIC SECTIONS. P R O P OSITION V I I.-T'H E O R E 1M. Tangents to the ellipse, at the vertices of a diameter, are parallel to each other. Let DD' be the diameter, and F' T and F the foci. Draw F'D, F'D', FD, and FlY. Draw the tangents, Tt and Ss, one through the point D, the other through the point 1Y. These tan- s gents will be parallel. D' By Cor. Prop. 5, F'D'FD is a parallelogram, and the angle F'D'F is equal to its opposite angle, F'DF. But the sum of all the angles that can be made on one side of a line is equal to two right angles. Therefore, by leaving out the equal angles which form the opposite angles of the parallelogram, we have sD'F'+SDlF- tDF' TDF But (by Prop. 6) sD'F'-=SD'F; and also tDF'= TDF; therefore, the sum of the two angles in either member of this equation is double either of the angles, and the above equation may be changed to 2SD'.P —2tDF' or SD'_F-=tDF' But DE' and D'F are parallel; therefore SD'F and tDF are, in effect, alternate angles, showing that T and Ss are parallel. Gor. If tangents be drawn through the vertices of any two conjugate diameters, they will form a parallelogram circumscribing the ellipse. PROPOSITION VIII. -THEOREM. If, from the vertex of any diameter of an ellipse, straight lines are drawn through the foci, meeting the conjugate diameter, the part of either line intercepted by the conjugate, is equal to one half the major axis.

Page  19 THE ELLIPSE. 19 Let DD' be the diameter, and I Dt the tangent. Through thecenter T draw EE' parallel to D. Draw I'D and DF, and produce DF to K; and' from F draw FG parallel to EE' E' or iY. Now, by reason of the parallels, we have the following equations among the angles: tDG= DGF Also, tDG = DK TDF=DFG J TDF= DKH But (Prop. 6) tDG= TD.F; Therefore, D GF — DFG; And, DHK=DKH Hence, the triangles DGF'and DHK are isosceles. Whence, DG=DF, and DII=DK. Because HC is parallel to FG, and F' C= CF, therefore, _F'H=H- G Add, D-FDG and we have F'HE+DF=DIH But the sum of the lines in both members of this equation is _FID+DF, which is equal to the major axis of the ellipse; therefore, either member is one half the major axis; that is, DH, and its equal, DK, are each equal to one half the major axis. Hence the theorem; if from the vertex of any diameter, etc. PROPOSITION I X.-T H E O R E M. Perpendiculars from the foci of an ellipse upon a tangent, meet the tangent in the circumference of a circle whose diameter is the major axis. Let F,F be the foci, C the center of the ellipse, and D a point through which passes the tangent Tt. Draw F' D

Page  20 20 CONIC SECTIONS. and FD, produce F'D to H, mak- G ing DH=FED, and produce FD to G, making DG-F'D. Then F'H t and FG are each equal to the major axis, A'A. Draw FH meeting the tangent in A' F. C T and F'G meeting it in t. Draw the dotted lines, CT and Ct. S By Prop. 6, the angle ED T=the angle P` Dt; and since opposite or vertical angles are equal, it follows that the four angles formed by the lines intersecting at D, are all equal. The triangles DP G and DHEF are isosceles by construction; and as their vertical angles at D are bisected by the line AT, therefore F' t-tG, FT= TH, and PT and "' t are perpendicular to the tangent DT. Comparing the triangles F' GP and F' Ct, we find that F'Cis equal to the half of F'F, and Pi't, the half of FG; therefore, Ct is the half of FG; but A'A=FG; hence, Ct=-A'A= CA. Comparing the triangles iFF' and FCT, we find the sides PFH and FF' cut proportionally in T and C; therefore, they are equi-angular and similar, and CT is parallel to'1H, and equal to one half of it. That is, CT is equal to CA; and CA, CT, and Ct are all equal; and hence a circumference described from the center C, with the radius CA, will pass through the points T and t. Hence the theorem: perpendiculars from the foci, etc. PROPOSITION X.-T HEORE M. The product of the perpendiculars from the foci of an ellipse upon a tangent, is equal to the square of one half the minor axis. Produce TC and GF', and they will mleet in the circumference at S; for FT and F't are both perpendicular to

Page  21 THE ELLIPSE. 21 the same line Tt, they are there- G fore parallel; and the two triangles, CFT and CF'S, having a side, FC, t T of the one, equal to the side, CF', of the other, and their angles equal, A each to each, are themselves equal. F Therefore, CS= CT, S is in the circumference, and SF'-=FT. S Now, since A'A and St are two lines that intersect each other in a circle, therefore (Th. 17, B. III, Geom.), SF' x F't-A F x F' A Or, FTx F' t=A'F' x F'A. But, by the Scholium to Prop. 4, it is shown that A'F' x F'A= the square of one half the minor axis. Therefore, FTx F' t= the square of one half the minor axis. Hence the theorem; The product of the perpendiculars, etc. Cor. The two triangles, FTD and F'tD, are similar, and from them we have TF: F't-=FD:: DF'; that is, perpendiculars let fall from the foci upon a tangent, are to each other as the distances of the point of contact from the foci. PROPOSITION XI.-THEOREM. If a tangent, drawn to an ellipse at any point, be produced until it meets either axis, and from the point of tangency an ordinate be drawn to the same axis, one half of the axis will be a mean proportional between the distances from the center to the intersections of these lines with the axis. Let T be a tangent at any B point in the ellipse, as P. Draw F'P and FP, F and F' being the foci, and produce A, F' G F F T F'P to Q, making PQ=PF; join T,Q, and draw PG perpendicular to the axis AA'.

Page  22 22 CONIC SECTIONS. The triangles PET and PTQ are equal, because PT is common, PQ-=P' by construction, and the L TPF= the angle L TPQ (Th. 6). Therefore, TP bisects the angle FTQ, and QT=-FT. As the angle at T is bisected by TP, the sides about this angle in the triangle F'TQ are to each other, as the segments of the third side, (Th. 24, B. II, Geom.) That is, F'T: TQ::F'P: PQ Or, F'T: FT:: F'P: PF From this last proportion we have (Th. 9, B. II, Geom.), Ji' T+FT:' T- FT: F'P+PF: F'P —PF Or, since.F'T+FT=2CT and F'P+PF=2 CA, by substitution we have 2C: F'F:: 2CA: F'P -PF (1) Again, because PG is drawn perpendicular to the base of the triangle F'PF, the base is to the sum of the two sides, as the difference of the sides is to the difference of the segments of the base, (Prop. 6, P1. Trig.) Whence, F'F: F'P+PF:: F'P —PF: 2CG (2) If we multiply proportions (1) and (2), term by term, omitting in the resulting proportion the factor F'F, common to the terms of the first couplet, and the factor F'P-PF, common to the terms of the second couplet, we shall have 2CT:2CA::2CA: 2CG Or, CT: CA:: CA: CG In like manner it may be proved that Ct: CB:: CB: Cg Hence the theorem; If a tangent, drawn to an ellipse, etc. PROPOSITION XII.-T HEOREM. The sub-tangent on either axis of an ellipse is equal to the corresponding sub-tangent of the circle described on that axis as a diameter.

Page  23 THE ELLIPSE. 23 Let P be the point of tan- *' gency of the tangent line Tt to the ellipse, of which AA' is the B major axis and C the center. Draw the ordinate P G to this axis, and produce it to meet Al C tC A T the circumference of the circle described on AA' as a diameter, at B, and draw BC and BT, T being the intersection of the tangent with the major axis; then will the line BT be a tangent to the circumference, at the point B. By the preceding theorem we have CT: CA:: CA: CG And since CA= CB, this proportion becomes CT: CB:: CB: CG Hence, the triangles CB T and CBG have the common angle C, and the sides about this angle proportional; they are therefore similar(Cor. 2 Th. 17, B. II, Geom.). But CBG is a right-angled triangle; therefore, CBT is also right-angled, the right angle being at B. Now, since the line BT is perpendicular to the radius CB at its extremity, it is tangent to the circumference, and GT is therefore a common sub-tangent to the ellipse and circle. If a circumference be described on the minor axis as a diameter, it may be proved in like manner that the corresponding sub-tangents of the ellipse and circle are equal. Hence, the theorem; The sub-tangent on either axis, etc. SCHOLIUM 1. —This proposition furnishes another easy rule for drawing a tangent line to an ellipse, at any point. RULE. On the major axis as a diameter, describe a semi-circumference, and from the given point on the ellipse draw an ordinate to the major axis; draw a tangent to the semi-circumference at the point in which the ordinate produced meets it. The line that connects the point in which this tangent intersects the major axis with the given point on the ellipse, will be the required tangent.

Page  24 24 CONIC SECTIONS. SCHOLIUM 2.-Because CB T is a right-angled triangle, CG GT=BG; but A' GA G=BG2 Therefore, CG GT=A' GA G PROPOSITION XIII.-THEOREM. The square of either semi-axis of an ellipse is to the square of the other semi-axis, as the rectangle of any two abscissas of the former axis is to the square of the corresponding ordinate. From any point, as P, of the - ellipse of which C is the center, g- h i.. AA' the major, and BB the minor axis, draw the ordinate A C G T PG to the major axis; then it is to be proved that B' CA::1:: AG GA':PG Through P draw a tangent line intersecting the axes at Tand t; then, by Prop. 11, we have CT:: CA:: CA: CG Whence, C07 CG= CA2 and by multiplying both members of this equation by CG, it becomes CT CG -CA CG which may be resolved into the proportion CA2: CGT:: CT: CG From this we find, (Cor. Th. 8, B. II, Geom.), CA2: CA -C:: CT: GT (1) Again, drawing the ordinate Pg to the minor axis, we have Ct: CB:: CB: Cg or PG Whence, Ct PG=- CB2 Multiplying both members of this equation by PG, it becomes

Page  25 THE ELLIPSE. 25 Ct PG 2- CB PG from which we have the proportion CB-: PG:: C: PG By similar triangles we have Ct: PG:: CT: GT And, since the first couplet in this proportion is the same as the second couplet in the preceding, the terms of the other couplets are proportional. That is, CBS: PG":: CT: GT (2) By comparing proportions (1) and (2), we obtain -of2 A-G: C CGA2-_CG2 (3) But CA — C-= (CA+ CG) (A- CG)=A' G A G; Whence, by inverting the means in proportion (3) and substituting the values of CA-_CG2, we have finally CB: CX2:: PG2 A A' G vt G or, CA2: IB2:: AG A GA: P-2 By a process in all respects similar to the above, we will fnd that 2 2 2 CB::CA:: Bg B'g: (pg) Hence the theorem; the square of either semi-axis, etc. SCHOLIUM 1.-From the theorem just demonstrated is readily deduced what is called, in Analytical Geometry, the equation of the ellipse referred to its center and axes. If we take any point, as P, on the curve, and can find a general relation between A G and PG, or between CG and PG, the equation expressing such relation will be the equation of the curve. Let us represent CA, one half of the major axis, by A, and CB, one half of the minor axis, by B; that is, the symbols A and B denote the numerical values of these semi-axes, respectively. Also, denote the CG by x, and PG by y, then A' G-A -+x, and A G=A —x; and by the theorem we have A': B2:: (A+x) (A-x):y' Whence, A2'y'=-A'B'-B'x' Or, A'ay2+B2x'= A9'B 3

Page  26 26 CONIC SECTIONS. This is the required equation in which the variable quantities, x and y, are called the co-ordinates of the curve, the first, x, being the abscissa, and the second, y, the ordinate; the center C from which these variable distances are estimated, is called the origin of co-ordinates, and the major and minor axes are the axes of co-ordinates. Had we donoted A' G by x, without changing y, then we should have A G —2A —x, And A': B':: (2A-x) x: y Whence, y= —rA(2Ax-x2), which is the equation of the ellipse when the origin of co-ordinates is on the curve at A'. SCHoLIUM 2. —If a circle be described on either axis of an ellipse as a diameter, then any ordinate of the circle to this axis is to the corresponding ordinate of the ellipse, as one half of this axis is to one half of the other axis. Retaining the notation in Scholium 1, and producing the ordinate PG to meet the circumference described on A'A as a diameter, at P, we have, by the theorem, A2: B2:: (A+x) (A-x): y2 But (A+x) (A —x) - GP'2 Whence, A2: B2: GP' y2 Or, A B:: GP':y That is, GP': y:: A: B By describing a circle on BB' as a diameter, we may in like manner prove that pg: Pg:: B: A PROPOSITIO'N XIV. THEOREM. The squares of the ordinate to either axis of an ellipse are to each other, as the rectangles of the corresponding abscissas. B V Let AA' be the major, and BB' g p the minor axis of the ellipse, and I _ TEG, P'G' any two ordinates to A' A C G G-A the first axis. Denoting CG by by x, CG' by x', PG byy and 3 gP' G' by y', we have, by Scho. 1,

Page  27 THE ELLIPSE. 27 Prop. 13, A2y2 + B 2x 2= A2B2 and A2y'2 +~ B2x'2=A2B2 J32 B2 Whence y2 =B (A2-x2)=-A (A+xz) (A-x) (1) B2 B2 and y12 _3-(A2 -A(A+xf) (A-x') (2) Dividing equation (1) by equation (2), member by member, and omitting the common factors in the numerator and denominator of the second member of the resulting equation, it becomes y2 (A+x) (A-x) y', (A+x') (A-x') By simply inspecting the figure, we perceive that A+x and A-x represent the abscissas of the axis AA', corresponding to the ordinate y; and A+x', and A-x' those corresponding to the ordinate y'. By placing the two equations first written above, under the form A2 X2 =2 (B2_y2) A2 X'-B (B -2) and proceeding as before, we should find X2 (B+y)(B-y) X'- (B+y') (B-y') in which B+y, B-y are the abscessas of the axis BB', corresponding to the ordinate x=CG=Pg; and B+y', B-y' are those corresponding to the ordinate x'= CG'= P'g'. Hence the theorem; the squares of the ordinates, etc. PROPOSITION XV.-THEOREM. The parameter of the transverse axis of an ellipse, or, the latus rectum, is the double ordinate to this axis through the focus.

Page  28 28 CONIC SECTIONS. Let F and F' be the foci of an ellipse of which AA' and BB' re- spectively are the major and mi- A' A nor axes. Through the focus F draw the double ordinate PP'. Then will B' PP be the parameter of the major axis. We will denote the semi-major axis by A, the semiminor axis by B, the semi-ordinate through the focus by P, and the distance from the center to the focus by c. The equation of the curve referred to the center and axis, is A2y2+ B2x2=A2B2. If in this equation we substitute c for x, y will become P, and we have A2P2+2C- 2A2=B2. Transposing the term B2C2, and factoring the second member of the resulting equation, it becomes A2P2=B2 (A2-c2) (1) In the right-angled triangle BCF, since BF=A (Prop. 4) and Bc=B, we have A2-c2=B2. Replacing A2 — in eq. (1) by its value, that equation becomes A2 p2=32 -B2 Or, by taking the square roots of both members, A.P=B.B Whence, A: B:: B: P Or, 2A:2B::2B:2P 2P is therefore a third proportional to the major and minor axes, and (Def. 14) it is the parameter of the former axis. Hence the theorem; the parameter, etc.

Page  29 THE ELLIPSE. 29 PROPOSITION XVI.-THE ORE M. The area of an ellipse is a mean proportional between two circles described, the one on the major, and the other on the minor axis as diameters. On the major axis AA' of the G' G" ellipse represented in the figure, G describe a circle, and suppose this axis to be divided into'any number of equal parts. A' Il A Through the points of division draw ordinates to the circle, and join the extremities of these consecutive ordinates, and also those of the corresponding ordinates of the ellipse, by straight lines. We shall thus form in the semi-circle a number of trapezoids, and a like number in the semiellipse. Let GH, G'H' be two adjacent ordinates of the circle, and gH g'I1' those of the ellipse answering to them; and let us denote GH by Y, G'H' by Y', gHfby y, g'H' by y', and the part HH1' of the axis by x. The trapezoidal areas, GIH7 G1', gHff'g', are respectively measured by y+ y7 y+y, +2 -x andY x (Th. 34, B. I, Geom.) But (Prop. 13, Scho. 2) A:B:: Y:y:: Y7:yl Hence (Th. 7, B. II, Geom.) A: B:: Y+ 3:Y+y:: Y+ I y+y 2 2 Y+ Y, y+y' or, A: B::.2 2' If the ordinates following YI, y' in order, be represented by Y", y", etc., we shall also have 3*

Page  30 80 CONIC SECTIONS. rV+ Y"t y'+y." A:B:: 2 x: 2 2 2 That is, any trapezoid in the circle will be to the corresponding trapezoid in the ellipse, constantly in the ratio of A to B; and therefore the sum of the trapezoids in the circle will be to the sum of the trapezoids in the ellipse as A is to B; and this will hold true, however great the number of trapezoids in each. Calling the first sum S, and the second s, we shall then have A: B:: S: s But, when the number of equal parts into which the axis AA' is divided, is increased without limit, S becomes the area of the semi-circle and s that of the semi-ellipse. Therefore, A: B:: area semi-circle: area semi-ellipse. Or, A: B:: area circle: area ellipse. By substituting in this last proportion for area circle, its value 7rA2, it becomes A: B::'rA2: area ellipse. Whence area ellipse=rAB, which is a mean proportional between -wA2 and lrB2. Hence the theorem; the area of an ellipse, etc. SCHOLIUM.-This theorem leads to the following rule in mensuration for finding the area of an ellipse. RuLE.=-Multiply the product of the semi-major and semi-minor axes by 3.1416. PROPOSITION XVII.-THEOREM. If a cone be cut by a plane making an angle with the base less than that made by an element of the cone, the section is an ellipse. Let Vbe the vertex of a cone, and suppose it to be cut by a plane at right-angles to the plane of the opposite

Page  31 THE ELLIPSE. 31 elements, VN VB, these elements v being cut by the first plane at A and B. Then, if the secant plane be not parallel to the base of the cone, the section- will be an ellipse, K of which AB is the major axis. Through any two points, F and -—..... H/, on AB, draw the lines KL, MN, / parallel to the base of the cone, and through these lines conceive planes to be passed also parallel to this base. The sections of the cone made by these planes will be circles, of which KGL and -IIN are the semi-circumferences, passing the first through G, and the second through I, the extremities of the perpendiculars to BA, lying in the section made by the oblique plane. The triangles AEL, AHN, are similar; so also are the triangles BMff, BKF; and from them we derive the following proportions: AF: IFL":: AH: IN BF: KF.:: BH: HT By multiplication, ATFBF: FLKF:: AXHBH: HLN JJM Because KL is a diameter of a circle, and FG an ordinate to this diameter, we have KF FL= EG2, andt for a like reason, HAI-'HN=-HI Therefore, AFBBF: _:G2: Af B:. HI 2 or, AF'BF: AHlHB:: FG2: H12 This proportion expresses the property of the ellipse proved in (Prop. 14); and the section A GIB is, therefore, an ellipse. Hence the theorem; if a cone be cut, etc. ScTIoLIuM.-The proportion AF' BE: AH' HB:: FG: HI' would still hold true, were the line AB parallel to the base of the cone, and the section a circle; the ratios would then become equal

Page  32 32 CONIC SECTIONS. to unity. The circle may therefore be regarded as a particular case of the ellipse; PROP OSITION XVIII.-THE OREM. If, from one of the vertices of each of two conjugate diameters of an ellipse, ordinates be drawn to either axis, the sum of the squares of these ordinates will be equal to the square of the other semi-axis. LetAPP'A' QQ' be t an ellipse, of which AA' is the major and BB' the minor axis; r' also let PQ, P'Q' be T G' G any two conjugate Q diameters. Through the vertices of these ~ diameters draw the tangents to the ellipse and the ordinates to the axes, as represented in the figure. Then we are to prove that cA2= (Pg)2+(P g)2 Cg2 + CSy2 and CB2=(PG)2+(P G')2=()2+(6/)2 Now (by Prop. 11) we have CT: CA:: CA: CG, also, C: CA:: CA: Cn Whence, CA2= CT- CG, (1) and CA2- Ct' C-n. Therefore, CT' CG= Ct( Cn, which, resolved into a proportion, gives Ct': CT:: G: n (2) By the construction, it is evident that the triangles CPT, CQft', are similar, as are also the triangles P G T and CQ'n.

Page  33 THE ELLIPSE, 33 From these-triangles we derive the proportions Ct' CT:: CQ': PT C: PT:: Cn:GT Whence, Ct': CT:: Cn: G T Comparing the last proportion with proportion (2) above, we have CG: Cn:: C: GT Whence, (Cn)2= CG' GT But GT= CT-CG; then (Cn)2= CG (CT-CG), from which we get (Cn)2+ CG2-= CG CT= (See eq. 1.) Substituting, in this equation, for (Cn)2, its equalCG', it becomes CA2= 2 + CG'1 In a similar manner it may be proved that B7_ pGj 2+p' yi2 Hence the theorem; if from one of the vertices of each, etc. PROPOSITION XIX.-THEOREM. The sum of the squares of any two conjugate diameters of an ellipse is a constant quantity, and equal to the sum of the squares of the axes. t The annexed figare, being the same r as that employed in,T G the preceding proposition, by that proposition we have CA == C + CG" and CB'-PG2+PT G By addition, CA'+ =C — CG:+PG2+C+P' G~+Y

Page  34 34 CONIC SECTIONS. But CG and PG are the two sides of the right-angled triangle C'PG, and CG' and P'G' are the two sides of the right-angled triangle CP' G'; Therefore, GA2 _+ CB2= P2+,6_,P2 Whence, 4 ICA2+4 CB'-=4 CP2+4 CP'2 The first member of this equation expresses the sum of the squares of the axes, and the second member the sum of the squares of the two conjugate diameters. Hence the theorem; the sum of the squares of any two, etc. PROPOSITION XX.- THEORE M. The parallelogram formed by drawing tangents through the vertices of any two conjugate diameters of an ellipse, is equal to the rectangle of the axes. Employing the figure of the last two propositions, we r have, from proposi- A.t tion 18, 2 --- CA = CG + C G from-which, by trans- o position and factoring the second member, we get CG =(CA + CG') ( CA-CG')=A G'A' G' But CA2: CB2:: AG'A'G': P'G'2; (Prop. 13.) Whence, CA2: CB2:: CG2: PG'2 Or, CA: CB:: CG: P'G'-Qn (1) But, CT: CA:: CA: CG (2) (Prop. 11.) Multiplying proportions (1) and (2), term by term, omitting, in the first couplet of the resulting proportion, the common factor CA, and in the second couplet the common factor CG, we find CT: CB:: CA: Q'n

Page  35 THE ELLIPSE. 35 Whence, CT' Q'n= CA- CB Or, 4 CTQ'n=4 CACB The first member of this equation measures eight times the area of the triangle CQ' T, and this triangle is equivalent to one half of the parallelogram CQ'mP, because it has the same base, CQ', as the parallelogram, and its vertex is in the side opposite the base. This parallelogram is obviously one fourth of that formed by the tangent lines through the vertices of the conjugate diameters; 4CT.Q'n therefore, measures the area of this parallelogram. Also, 4 CAG CB is the measure of the rectangle that would be formed by drawing tangent lines through the vertices of the major and minor axes of the ellipse. Hence, the theorem; the parallelogram formed, etc. PROPOSITION X X I.-T H E O R E M. If a normal line be drawn to an ellipse at any point, and also an ordinate to the major axis from the same point, then will the square of the semi-major axis be to the square of the semi-minor axis, as the distance from the center to the foot of the ordinate is to the sub-normal on the major axis. Let P be the assumed point in the ellipse, and through this point draw the tangent PT, the normal PD), and the ordinate A,' D G T PG, to the major axis; then C being the center of the ellipse, and A denoting the semi-major, and B the semi-minor axis, it is to be proved that A2: B2:: CG': DG By (Prop. 13) we have A2: B2:: A G AG: PG (1) and because DPT is a right-angled triangle, and PG is a

Page  36 36 CONIC SECTIONS. perpendicular let fall from the vertex of the right-angle upon the hypotenuse, we also have (Th. 25, B. II, Geom.) PG2 =DG.GT But A' GA G= G- GT (Scho. 2, Prop. 12) Substituting in proportion (1), for the terms of the second couplet, their values, it becomes A2: B2:: CG'GT: 1DG'GT or A2: B2::CG': DG. Hence the theorem; if a normal line be drawn, etc. Cor. If CG=x, then this theorem will give for the subnormal, D G, the value B x, which is its analytical expression. PROPOSITION XXII. —THEOREM. If two tangents be drawn to an ellipse, the one through the vertex of the major axis and the other through the vertex of any other diameter, each meeting the diameter of the other produced, the two tangential triangles thus formed will be equivalent. Let PP' be any diameter of N the ellipse whose major axis r 1 M E is AA'. Draw the tangents V AN and PT, the first meeting A'' C A S T the diameter produced at N, and the second the axis pro- vw;p duced at T; the triangles CAN and CPT thus formed are equivalent. Draw the ordinate PD; then by similar triangles we have CD: CA:: CP: CN But CD: CA:: CA: CT (Prop. 11) Whence CP: CN:: CA: CT Therefore, CP' CT= CN CA

Page  37 THE ELLIPSE. 37 Multiplying both numbers of this equation by sin. C, it becomes CP' CT sin. C= CN' CA sin. C or, CT' CP sin. C=~ CA' CN sin. C (1) But CP' sin. C=PD, and CN sin. C=AN; therefore the first member of equation (1) measures the area of the triangle CPT, and the the second member measures that of the triangle CAN. Hence the theorem; if two tangents be drawn to an, etc. Cor. 1. Taking the common area CAEP, from each triangle, and there is left APEN=AAET. Cor. 2. Taking the common A CDP, from each triangle, and there is left APDT=trapezoidal area PDAN. PROPOSITION XXIII.-THEOREM. The supposition of Proposition 22 being retained, then, if a secant line be drawn parallel to the second tangent, and ordinates to the major axis be drawn from the points of intersection of the secant with the curve, thus forming two other triangles, these triangles will be equivalent each to each to the corresponding trapezoids cut off, by the ordinates, from the triangle determined by the tangent through the vertex of the major axis. Draw the secant QnS par- Q allel to the tangent PT, and rE also the ordinates QR, ng, producing the latter to p. Then A\ in' / - A S T is ASQR=trapezoid ANVR,, and ASng=trapezoid ANpg. vV"The three triangles, CVR, CPD, CNA are similar, by construction; therefore,' A CNA CED:: CA2:: CD2 Whence, trapezoid A.NPD: A CNA:: CA2 —iD2 CA2(1) (Th. 8, B. II, Geom.) 4

Page  38 38 CONIC SECTIONS. In like manner, trapezoid ANVYR: ACNA:: CA-CR2: CA2 (2) Dividing proportion (1) by (2), term by term, we get trapezoid ANPD CA2_67f2 trapezoid ANVR 1: A2Whence, trapez. ANPD: trapez. ANVR:: CA2 C — 2: CA2_-CR2 But PDb -R2:: A'DDA: A'R4RA, (Prop. 14); and since A'D= CA+ CD, AR-= CA+ CR, DA= CA- CD and RA= CA —CR, we have D2 Q R2:: (CA + CD) (CA- CD) (CA+ CR) (CA-CR):: CA2_ (JB2 CA — CR2 Therefore, trapezoid ANPD: trapezoid ANVR:: PD2: QR2, But the trapezoid ANPD=A TPD, (Cor. 2, Prop. 22); whence, A TPD: trapezoid ANVR:: PD: QR2 (3) and since the triangles TPD and SQR are similar, we have ATPD: ASQR::PD2 Q 2 (4) By comparing proportions (3) and (4) we find A TPD: trapezoid ANVR:: ATPD: ASQtR Whence, trapezoid ANVR =ASQR; and by a similar process we should find that trapezoid APg-= ASng. Hence the theorem; if a secant line be drawn parallel, etc. Cor. 1. Taking the trapezoid AJApg from the trapezoid ANVR, and the ASng from the ASQR, we have trapezoid gp VR=trapezoid gnQR. Cor. 2. The spaces ANVR, TP VR, and SQR are equivalent, one to another. Cor. 3. Conceive QR and QS to move parallel to their present positions, until R coincides with C; then QR

Page  39 THE ELLIPSE. 39 becomes the semi-minor axis, the space ANVR the triangle ANC, and the A QRS equivalent to the A CPT. PROPOSITION XXIV.-THEOREM. Any diameter of the ellipse bisects all of the chords of the ellipse drawn parallel to the tangent through the vertex of the diameter. By Cor. 1 to the preceding N proposition we have - J 1 trapez. gp VR=trapez. gnQIR. If from each of these equals R'. A S T we subtract the common area gnm VR, there will remain the vPA Amnp, equivalent to the AQm V; and as these triangles are also equi-angular, they are absolutely equal. Therefore, Qm=inn. Hence the theorem; any diameter of the ellipse bisects, etc. REMARK.-The property of the ellipse demonstrated in this proposition is merely a generalization of that previously proved in Prop. 3. PROPOSIT ION XX V.-TH E O R E M. The square of any semi-diameter of an ellipse is to the square of its semi-conjugate, as the rectangle of any two abscissas of the former diameter is to the square of the corresponding ordinate. Let AA' be the major axis N of the ellipse, CP any semi-? diameter and CFP its semiconjugate. Draw the tan- L R' R CA ST gents TP and AN, the ordinate Qm, producing it to meet v, _ the axis at S; and P' V', parallel to AN, and in other

Page  40 40 CONIC SECTIONS. respects make the construction as indicated in the figure. It is then to be proved that CP2: -P':2: Pmm nP": m Now in the present construction, the triangles CP'R' and CV'R' take the place of the triangles SQR and CVR respectively, in Prop. 23; and hence by that proposition, the triangles CP' V', CAN, and CPT are equivalent one to another. The triangles CPT and CmiS are similar; therefore, ACPT: aCnS:: CP: Cm WWhence, A CPT: ACPT — ACmS:: CP: CP - m Or, ACPT: trapez. mPTS:: CP OP -O — (1) From the similar triangles, CP' V' and m Q V, we have ACP'V': AmQV::'P'2 mQ2 But area Sm VR+ A CV~R+ AmQ V= area Sm VR+ OCVR+trapez. mPTS, (Prop. 23.); therefore, AmQ V= trapez. m PTS; also ACP' V'= CPT. Substituting these values in the preceding proportion, it becomes A CPT: trapez. mPTS:: CP'2: mQ (2) By comparing proportions (1) and (2), we get CP2: CP2- m:: CP mQ2 Or, CP2: CP'2:: - -: ~ Whence, CP2: CP'2:: (CP+ Cm) (CP-Cm): mQ Or, 0P2 C2P': Ptm'mP~: m Hence the theorem; the square of any semi-diameter, etc. REMARK. The property of the ellipse relating to conjugate diameters, established by this proposition, is but the generalization of that before demonstrated in reference to the axes, in Prop. 13.

Page  41 THE PARABOLA. 41 THE PARABOLA. DEFINITIONS. L The Parabola is a plane curve, generated by the motion of a point subjected to the condition that its distances from a fixed point and a fixed straight line shall be constantly equal. 2. The fixed point is called the B a focus of the parabola, and the fixed line the directrix. A Thus, in the figure, F is the focus B, and BB" the directrix of the para-,, bola PVP'P", etc. B 3. A Diameter of the parabola is a line drawn through any point of the curve, in a direction from the directrix, and at right-angles to it, 4. The Vertex of a diameter is the point of the curve through which the diameter is drawn. 5. The Principal Diameter, or the Axis, of the parabola is the diameter passing through the focus. The vertex of the axis is called the principal vertex, or simply the vertex of the parabola. The vertex of the parabola bisects the perpendicular distance from the focus to the directrix, and all the diamleters of the parabola are parallel lines. 6. An Ordinate to a diameter is a straight line drawn from any point of the curve to the diameter, parallel to the 4*

Page  42 42 CONIC SECTIONS. tangent line through its vertex. Thus, PD, drawn parallel to the tangent V' T, / A is an ordinate to the diameter V'D. It will be shown that DP=DG; and hence / PG is called a double ordinate. 7. An Abscissa is the part of the diameter between the vertex and an ordinate. B G Thus, V'D is the abscissa corresponding to the ordinate PD. 8. The Parameter of any diameter of the parabola is one of the extremes of a proportion, of which any ordinate to the diameter is the mean, and the corresponding abscissa the other extreme. 9. The parameter of the axis of the parabola is called the principal parameter, or simply the parameter of the parabola. It will be shown to be equal to the double ordinate to the axis through the focus. Thus, BB', the chord drawn through the focus at right-angles to the axis, is the parameter of the parabola. The principal parameter is sometimes called the latusrectum. 10. A Sub-tangent, on any diameter, is the distance from the point of intersection of a tangent line with the diameter produced to the foot of that ordinate to this diameter that is drawn from the point of contact. 11. A Sub-normal, on any diameter, is the part of the diameter intercepted between the normal to the curve, at any point, T~ and the ordinate from the same point to N the diameter. Thus, in the figure, V'N F being any diameter, PT a- tangent, and PYN a normal at the point P, and PQ an ordinate to the diameter; then TQ is a sub-tangent and QN a sub-normal on this diameter.

Page  43 THE PARABOLA. 43 When the terms, sub-tangent and sub-normal, are used without reference to the diameter on which they are taken, the axis will always be understood. PROPOSITION I.-PROBLEM. To describe a parabola mechanically. Let CD be the given line, and F the D given point. Take a square, as DBG, and to one side of it, GB, attach a thread, B P G and let the thread be of the same length In v as the side GB of the square. Fasten one c end of the thread at the point G, the other end at'. Put the other side of the square against the given line, CD, and with the point of a pencil, in the thread, bring the thread up to the side of the square. Slide the side BD of the square along the line CD, and at the same time keep the thread close against the other side, permitting the thread to slide round the point of the pencil. As the side BD of the square is moved along the line CD, the pencil will describe the curve represented as passing through the points V and P. For GP+PF=the length of the thread, and GP+PB=-the length of the thread. By subtraction, PF-PB=O, or P-F=PB. This result is true at any and every position of the point P; that is, it is true for every point on the curve corresponding to definition 1. Hence, FV= V-H. If the square be turned over and moved in the opposite direction, the other part of the parabola, on the other side of the line PFH, may be described. Cor. It is obvious that chords of the curve which are perpendicular to the axis, are bisected by it.

Page  44 44 CONIC SECTIONS. PROPOSITION II.-THEOREM. Any point within the parabola, or on the concave side of the curve, is nearer to the focus than to the directrix; and any point without the parabola, or on the convex side of the curve, is nearer to the directrix than to the focus. Let F be the focus and HB' the directrix B' A' of a parabola. B i, A.First.-TakeA, any point within the curve. From A draw AFto the focus, and AB per- iF - pendicular to the directrix; then will AXF be less than AB. Since A is within the curve, and B is without it, the line AB must cut the curve at some point, as P. Draw PF. By the definition of the parabola, PB=PF; adding PA to each member of this equation, we have PB+PA BA =PA+P But PA and PF being two sides of the triangle APF, are together greater than the third side AF; therefore their equal, BA, is greater than AF. Second. —Now let us take any point, as A', without the curve, and from this point draw A'F to the focus, and A'B' perpendicular to the directrix. Because A' is without the curve and PF is within it, A'F must cut the curve at some point, as P. From this point let fall the perpendicular, BP, upon the directrix, and draw A'B. As before, PB=PPF; adding A'P to each member of this equation, and we have A'.P+ PB=A'P+ PF-A'F. But A'P and PB being two sides of the triangle A'PB, are together greater than the third side, A'B; therefore their equal, A'F, is greater than A'B. Now A'B, the hypotenuse of the right-angled triangle A'BB' is greater than either side; hence, A'B is greater than A'B'; much more then is A'F greater than A'B'. Hence the theorem; any point within the parabola, etc.

Page  45 THE PARABOLA. 45 Cor. Conversely: If the distance of any point from the directrix is less than the distance from the same point to the focus, such point is without the parabola; and, if the distance from any point to the directrix is greater than the distance from the same point to the focus, such point is within the parabola. First.-Let A' be a point so taken that A'B'<A'F. Now A' is not a point on the curve, since the distances A'B' and A':F are unequal; and A' is not within the curve, for in that case A'B' would be greater than A'F according to the proposition, which is contrary to the hypothesis. Therefore A' being neither on nor within the parabola, must be without it. Second.-Let A be a point so taken that AB>.AF. Then, as before, A is not on the curve, since AF and AB are unequal; and A is not without the curve, for in that case AB would be less than AF, which is contrary to the hypothesis. Therefore, since A is neither on nor without the parabola, it must be within it. PROPOSITION III.-THEOREM. If a line be drawn from the focus of a parabola to any point of the directrix, the perpendicular that bisects this line will be a tangent to the curve. Let F be the focus, and HD the di- A rectrix of a parabola. D Assume any point whatever, as B, in b the directrix, and join this point to the / focus jy the line BF; then will tA, the H V F perpendicular to BiF through its middle point t, be a tangent to the parabola. Through B draw BL perpendicular to the directrix, and join P, its intersection with tP, to the focus. Then, since P is a point in the perpendicular to BF at its middle point, it is equally distant froimithe extremities of BF; that is, PB=PF. P is there

Page  46 46 CONIC SECTIONS. fore a point in the parabola, (Def. 1). Hence, the line tP meets the curve at the point P. We will now prove that all other points in the line IP are without the parabola. Take A, any point except P in the line tP, and draw AF, AB; also draw AD perpendicular to the directrix. AF is equal to AB, because A is a point in the perpendicular to B]? at its middle point; but AB, the hypotenuse of the right-angled triangle ABD, is greater than the side AD; therefore AD is less than AE, and the point A is without the parabola. (Cor., Prop. 2). The line tA and the parabola have then no point in common except the point P. This line is therefore tangent to the parabola. SCHOLIUM 1.-The triangles BPt and FPt are equal; therefore the angles FEt and BPt are equal. Hence, to draw a tangent to the parabola at a given point, we have the following RULE.-From the given point draw a line to the focus, and another perpendicular to the directrix, and through the given point draw a line bisecting the angle formed by these two lines. The bisecting line will be the required tangent. SCHOLIUM 2.-Just at the point P the tangent and the curve coincide with each other; and the same is true at every point of the curve. Now, because the angles BPt and FPt are equal, and the angles BPt and LPA are vertical, it follows that the angles LPA and FPt are equal. Hence it follows, from the law of reflection, that if rays of light parallel to the axis VF be incident upon the curve, they will all be reflected to the focus F. If therefore a reflecting surface were formed, by turning a parabola about its axis, all the rays of light that meet it parallel with the axis, will be reflected to the focus; and for this reason many attempts have been made to form perfect parabolic mirrors for reflecting telescopes. If a light be placed at the focus of such a mirror, it will reflect all its rays in one direction; hence, in certain situations, parabolic mirrors have been made for lighthouses, for the purpose of throwing all the light seaward. Cor. 1. The angle BPF continually increases, as the

Page  47 THE PARABOLA. 47 pencil P moves toward V, and at V it becomes equal to two right angles; and the tangent at V is perpendicular to the axis, which is called the vertical tangent. C(or. 2. The vertical tangent bisects all the lines drawn from the focus of a parabola to the directrix. Let Vt be the vertical tangent; then because the two right-angled triangles FVt and PFHB are similar, and VF= VHY, we have Ft= tB. PROPOSITION IV.-THE OREM. The distance from the focus of a parabola to the point of contact of any tangent line to the curve, is equal to the distance from the focus to the intersection of the tangent with the axis. Through the point P of the parabola By of which P is the focus and BII the directrix, draw the tangent line PT, meeting the axis produced at the point T Ii v F D c T; then will PP be equal to FT Draw PB perpendicular to the directrix, and join F,B. The angles BPT and TPF are equal, (Scho. 1, Prop. 3); and since PB is parallel to IC, the alternate angles BPT, and PTC are also equal. Hence the angle TPPF is equal to the angle PTE, and the triangle PET is isosceles; therefore FP=FT. Hence the theorem; the distance from the focus to, etc. SCHOLIUM.-To draw a tangent line to a parabola at a given point, we have the following RULE.-Produce the axis, and lay off on it from the focus a distance equal to the distance from the focus to the point of contact. The line drawn through the point thus determined and the given point will be the required tangent.

Page  48 48 CONIC SECTIONS. PROPOSITION V.-THEOREM. The perpendicular distance from the focus of a parabola to any tangent to the curve, is a mean proportional between the distance from the focus to the vertex and the distance from the focus to the point of contact. In the figure of the preceding proposi- B tion draw in addition the vertical tangent Vt; then we are to prove that F-t - VFPFP. Because TtF and VFt are TIR V F D C similar right-angled triangles, we have TF: Ft:: Pt: VF. But TF=PF, (Prop.4); therefore, PF: Fi:: Ft: VP Whence, P-=PF. V.F Hence, the theorem; the perpendicular distance from,etc. PROPOSITION VI.-TTHEOREM. The sub-tangent on the axis of the parabola is bisected at the vertex. In the figure which is constructed as B in the two preceding propositions, draw in addition the ordinate PD, from the point of contact to the axis; then we T H F C are to prove that TD is bisected at the vertex V. The two right-angled triangles TRt and tEP have the side Ft common, and the angle FTt equal to the angle FPt; hence the remaining angles are equal, and the triangles themselves are equal; therefore t T=tP. From the similar triangles TDP, TVt, we have the proportion Tt: tP:: TV: VD But tT=tP; whence TV- VD Hence the theorem; the sub-tangent on the axis, etc.

Page  49 THE PARABOLA. 49 Cor. Since TV=-? TD, it follows that VIt=PD. That is, The part of the vertical tangent included between thle vertex and any tangent line to the parabola, is equal to one half of the ordinate to the axis from the point of contact. PROPOSITION VII.-THEOREM. The sub-normal is equal to twice the distance from the focus to the vertex of the parabola. In the figure (which is the same as that D P of the last three propositions), PC is a normal to the parabola at the point C, and DC is the sub-normal; it is to be T It V F I) C proved that DC=2F7V. Because BH and PD are parallel lines included between the parallel lines BP and HD, they are equal. BF and PC are also parallel, since each is perpendicular to the tangent PT; hence BF=PC, and also the two triangles HBF and DPC are equal. Therefore JHF=DC; but HF= 2FTV; whence DC-=FV. Hence the theorem; the sub-nornmal is equal to twice, etc. ScHOLIuM.-This proposition suggests another easy process for constructing a tangent to a parabola at a given point. RULE. —Draw an ordinate to the axis from a given point, and from the foot of this ordirate lay off on the axis, in the opposite direction of the vertex, twice the distance from the focus to the vertex. Through the point thus determined and the given point draw a line, and it will be the regquired tangent. PROPOSITION VIII.-THEOREM. Any ordinate to the axis of a parabola is a mean proportional between the correspondibq sub-tanqtenWt and sutb-norm.al. 5 D

Page  50 50 CONIC SECTIONS Assume anypoi It, as P, in the parabo- B p I of' whiCh F7 is the focus and JIB the \ directrix. Through this point draw the tangent PT, the normal PC, and the or- T ii v F D C dinate PD to the axis. Then in reference to the point P, TD is the sub-tangent, and -DC the sub-normal on the axis; and we are to prove that TD: PD:: PD: DC The triangle TIC is right-angled at P, and PD is a perpendicular let fall from the vertex of this angle upon the hypotenuse. Therefore, PD is a mean proportional between the segments of the hypotenuse, (Th. 25, B. II, Geom.) Hence the theorem; any ordinate to the axis, etc. SCIOLIUM 1.-For a given parabola, the fourth term of the proportion, TD: PD:: PD: DC, is a constant quantity, and equal to twice the distance from the focus to the vertex, (Prop. 7). By placing the product of the means of this proportion equal to the product of the extremes, we have P-D2= TD'D C= TD'2D C, which may be again resolved into the proportion TD: PD:: PD: 2DC Or, VD: PD:: P:2D C But VD is the abscissa, and PD is the ordinate of the point P; hence (Def. 8) 2D C is the parameter of the parabola, and is equal to four times the distance from the focus to the vertex, or to twice the distance from the focus to the directrix. SCHOLIUM 2.-If we designate the ordinate PD by y, the abscissa VD by x, and the parameter by 2p, the above proportion becomes x: y: y: y 2p Whence, Y -2px. This equation expresses the general relation between the abscissa and ordinate of any point of the curve, and is called, in Analytical Geometry, the equation of the parabola referred to its principal vertex as an origin. Cor. The sub-normal in the parabola is equal to one-half of the parameter.

Page  51 THE PARABOLA. 51 PROPOSITION IX. —T HEORE M. The parameter, or latus rectum, of the parabola is equal to twice that ordinate to the axis which passes through the focus. Let F be the focus, and BB' the directrix of a parabola; and through the focus B draw a perpendicular to the axis intersecting H the curve at P and P'. From P and P' let fall \ the perpendiculars PB, P'B', on the direc- B trix. Then will 2PF be equal to 2E11, or r to the parameter of the parabola. By the definition of the parabola, PF=PB; and because PP' and BB' are parallel, and the parallels PB and EH are included between them, we have PB=FH. Hence P.F=:F1, or 2PF=2FH= the parameter. Cor. Since the axis bisects those chords of the parabola which are perpendicular to it, FP=FP'. That is, FP'; therefore PP'=2FH. That is, The parameter of the parabola is equal to the double ordinate through the focus. PROPOSITION X.-THEOREM. The squares of any two ordinates to the axis of a parabola are to each other as their corresponding abscissas. Let y and y' denote the ordinates, and x and x' the abscissas of any two points of the parabola; then, by Scho. 2, Prop. 8, we have the two following equations: y2 2px and y_2 -2px' Dividing the first of these equations by the second, member by member, we have y2 2px x y'2 2px' x' Whence y2: y'2:: x x Hence the theorem; the squares of any two ordinates, etc.

Page  52 52 CONIC SECTIONS. PROPOSITION XI.-THEOREM. If a perpendicular be drawn from the focus of a parabola to any tangent line to the curve, the intersection of the perpendicular with the tangent will be on the vertical tangent. Let F be the focus, and BH the di- B rectrix of the parabola, and PT a tangent to the curve at the point P. From Fdraw FB perpendicular to the tangent, T R1 v F D c intersecting it at t, and the directrix at B. We will now prove that-the point t is also the intersection of the vertical tangent with the tangent PT. Because the triangle TFP is isosceles, the perpendicular Ft bisects the base PT; therefore tP=tT. Again, since Vt and DP are both perpendicular to the axis, they are parallel, and the vertical tangent divides the sides of the triangle TDP proportionally. Hence, TV: VD:: Tt: tP; but TV= VD (Prop. 6) therefore, Tt-=tP. That is, the tangent PT is bisected by both the perpendicular let fall upon it from the focus, and the vertical tangent. Therefore the tangent PT, the vertical tangent and the perpendicular FB, meet in the common point t. Hence the theorem; if a perpendicular be drawn, etc. PROPOSITION XII.-THEOREM. The parameter of the parabola is to the sum of any two ordinates to the axis, as the difference of those ordinates is to the difference of the corresponding abscissas. Take any two points, as P and Q, in the parabola represented in the following figure, and through these points draw the double ordinates Pp and Qq. VD and VE are the corresponding abscissas. Draw PS' and pt parallel to the axis. Ther since

Page  53 THE PARABOLA. 53 PD=-Dp and QE=Eq, we have QE+_PD Q = Qt, equal to the sum of the two ordinates; S and QE —PD= QS, equal to their difference; also VE —VD=DE, equal to the v E — difference of the corresponding abscissas. We are now to prove that 2p: Qt:: QS: DE in which 2p denotes the parameter of the parabola. Because PD and QE are ordinates to the axis, we have (Scho. 2, Prop. 8) PD2 =2p VD (1) and QE2=2p~ VE (2) Whence Q-E2 —p2=2p (VE- VD)-2p'DE (3) But QE2-PD2=(Q+E+PD) (QE-PD)= Qt QS, therefore Qt. QS=2p..DE (4) Whence 2p: Qt:: QS: DE Hence the theorem; the parameter of the parabola, etc. Cor. By dividing eq. (4) by eq. (2), member by member, we obtain Qt QS DE QE2 VE Whence VE: DE:: QE2: Qt QS PROPOSITItON XIII.-THEO E M. If a tangent line be drawn to a parabola at any point, and from any point of the tangent a line be drawn parallel to the axis terminating in the double ordinate from the point of contact, this line will be cut by the curve into parts having to each other the same ratio as the segments into which it divides the double ordinate. 5*

Page  54 54 CONIC SECTIONS. Take any point as P in the parabo- nI la represented in the figure, and of which VD is the axis, and through this point draw the tangent PT to the curve, and the double ordinate PQ to the axis. Assume a point in the tangent at pleasure, as A, and through it P -D L draw A C parallel to the axis, cutting \ the curve at B and the double ordinate at C. Then we are to prove that AB: BC-.: PC: CQ By similar triangles we have PC: CA:: PD:D T; but DT=2D V (Prop. 6) therefore. PC: CA.:: P D: 2D)V (1) But DV: PD: PD: 2p (Scho. 2, Prop. 8) or 2D V: PD:. 2PD: 2p. Inverting terms, PD 2D V.: 2p: 2PD=PQ (2) By comparing proportions (1) and (2), we get PC: CA:: 2p: PQ But 2p: CQ:: PC: BC (Prop. 12) Multiplying the last two proportions, term by term, we have 2p PC: CACQ:: 2p.PC: BCPQ The first and third terms of this proportion are equal; therefore the second and fourth are also equal. Hence we have the proportion CA-.BC:: PQ: CQ Whence by division, CA —BC:B C:: PQ —CQ: CQ or AB: BC::. PC: CQ If we take any other point, HA, on the tangent, and through it draw the line HL parallel to the axis, intersecting the curve at K and the ordinate at L, we will have, in like manner, HK: KL:: PL: LQ Hence the theorem; if a tangent be drawn, etc.

Page  55 THE PARABOLA. 55 PROPOSITION X1V.-T HEOREM. If any two points be taken on a tangent line to a parabola, and ";'oq'l these points lines parallel to the axis be drawn to meet ile curve, such lines will be to cach other as the squares of the d&stances of the points from the point of contact. The figure and construction being l! the same as in the foregoing proposition, we are to prove that AB: IK:: PA2: P2 A We have B f AB: BC:: PC: CQ (1) (Prop. 13.) Multiplying the terms of the second PxQ- D L couplet of this proportion by PC, it// \ becomes AB: BC:. PC2: PCQ (2) But, (Cor. Prop. 12) VDI): BC:: PD: PC CQ (3) Dividing proportion (2) by proportion (3), term by term, we have AB 1P 1 Whence, AB: VD:: P C2 PD (4) From the similar triangles, APC and TPD, we get the proportion PA: PT2:: PC2 ID (5) By comparing proportions (4) and (5) we find AB: D: PA2: PT2 (6) In like manner we can prove that IK: VD: PH2:' 2 (7) Dividing proportion (6) by proportion (7), term by term, we have 2 AB 1I PA Whence, AB: HK:: PA: Pi Hence the theorem; if any two points be taken, etc.

Page  56 56 CONIC SECTIONS. APPLICATION.-Conceive PH to be the direction in which a body thrown from the surface of the earth, would move, if it were undisturbed by the resistance of the air and by the force of gravity. It would then move along the line PH, passing over equal spaces in equal times. When a body falls under the action of gravity, one of the laws of its motion is, that the spaces are proportional to the squares of the times of descent; hence, if we suppose gravity to act upon the body in the direction A C, the lines AB, TV, HK, etc., must be to each other as the squares PA, PT2, PH', etc.; that is, the real path of a projectile in vacuo, possesses the property of the parabola that has been demonstrated in this proposition. In other words, The path of a projectile, undisturbed by the resistance of the air, is a parabola, more or less curved, depending upon the direction and intensity of the projectile force. PROPOSITION XV. -THEOREM. The abscissas of any diameter of the parabola are to each other as the squares of their corresponding ordinates. Let P be any point on a parabola, PL a tangent line, and PF a diame- T ter through this point. From the points B, V,K, etc., assumed at pleas- X_ ure on the curve, draw ordinates and / parallels to the diameter, forming the quadrilaterals PCBA, PD VT, etc. / Now, since the ordinates to any di- E ameter of the parabola are parallel to F the tangent line through the vertex of that diameter, these quadrilaterals are parallelograms and their opposite sides are equal. But, by the preceding proposition, we have AB: TV: HK, etc.,: PA2 PT2 PH etc. or PC: PD: PE, etc., BC: -VD KE, etc.

Page  57 THE PARABOLA. 57 By definition 6, PC is the ordinate and BC the abscissa of the point B, and so on. Hence the theorem; the abscissas of any diameter, etc. PROPOSITION XVI.-THE ORE M. If a secant line be drawn parallel to any tangent line to the parabola, and ordinates to the axis be drawnfrom the point of contact and the two intersections of the secant with the curve, these three ordinates will be in arithmetical progression. Let CT be the tangent line to the parabola, and EH the parallel secant. Draw the ordinates EG, CD, and K HI, to the axis VI, and through E -T VI DN draw EK parallel to VI We are now to prove that EG +HI= 2 CD L The similar triangles, HKE and CDT, give the prop;rtion ITK::KE CD: D T=-2 VD and, by proposition 12, we have 2p: KL.: K: KBE. Therefore 2p: KL:: CD: 2 VD, (1) and from the equation, y2=2px, we get, by making y= CD and x= VD, 2p: 2CD:: CD: 2 VD (2) By dividing proportion (1) by (2), term by term, we shall have KL Whence KL — 2CD But KL HI+ KI= HI+ E G; therefore HI+ EG= 2 CD Hience the theorem; if a secant line be drawn, etc.

Page  58 58 CONIC SECTIONS. SCHOLIUM 1.-If we draw CM parallel, and MN perpendicular to VI, then 2 CD=2MN=EG+-HI; and since MNis parallel to each of the lines EG and HI, the point M bisects the line EH. That is, the diameter through C bisects its ordinate EH; and as HE is any ordinate to this diameter, it follows that A diameter of the parabola divides into equalparts all chords of the curve parallel to the tangent through the vertex of the diameter. SCHOLIUM 2. —Hence, as the abscissas of any diameter of the parabola and their ordinates have the same relations as those of the axis, namely; that the ordinates are bisected by the diameter, and their squares are proportional to the abscissas; so all the other properties of this curve, before demonstrated in reference to the abscissas and ordinates of the axis, will likewise hold good in reference to the abscissas and ordinates of any diameter. PROPOSITION XVII.-THEOREM. The square of an ordinate to any diameter of the parabola is equal to four times the product of the corresponding abscissa and the distance from the vertex of that diameter to the focus. Let VXbe the axis of a paraola, and through any point, as P, of the D W curve, draw the tangent PT, and the diameter PW; also draw the A NEM secant Qq, parallel PT, and produce the ordinate QN, and the diameter I W, to meet at D. From the focus let fall the perpendicular PFY upon the tangent, and draw -F P and VY. We are now to prove that Qv2=4PF-Pv Because FY is perpendicular to PT, Qv parallel to PT and DQ parallel to each of the lines PM and VY, the triangles DQ V, PMT, TYVand TYF are all similar. Whence Qv2: QD2 TI-y2 (1) But TF =PF2 and Y-F2=P.F- VF. (Prop. 5)

Page  59 THE PARABOLA. 59 Substituting these values in proportion (1) and dividing the third and fourth terms of the result by PE1, it becomes Qv: QD2: PPF: VF (2) Again, from the triangles QDv and PMT we get QD: Dv:: PM: MT=2VM:: PM2: 2PM- VM But (Scho. 2, Prop. 8) PM'-4 VF VM Whence QD: Dv:: 4 VF' VM: 2PM VJ;:: 4 VF: 2PM therefore 2PM- QD=4 VF Dv C3) By subtracting the equation QN-2= 4 VP- VN from the equation PM2=4 VF' VM, member from member, we have PM -QN =4 VF (VM- VN) =4 VF PNM =4 VEKDP Whence (PM+ QN) (PM —QN)=(PM+ QN) D DQ=4 VF. DP (4) Subtracting eq. (4) from eq. (3), member from member, we obtain (PM —QN) DQ=4 VF (Dv-DP)-=4 VP- Pv and because PM-QN=DQ, this last equation becomes DQ-=4 V" Piv Substituting this value of DIN in proportion (2), we have Qv2: 4 VF Pov:: PF: VF or Qv2: 4Pv:: PF: 1 Whence Qv2=4P- Pv Hence the theorem; the square of an ordinate, etc. Cor. If, in the course of this demonstration, we had used the triangle vdq in the place of vDQ, to which it is similar, we would have found that qv2=4PF.Pv; whence Qv=qv. And since the same may be proved for any ordinate, it follows that

Page  60 60 CONIC SECTIONS. All the ordinates of the parabola to any of its diameters are bisected by that diameter. SCHOLIuM. —The parameter of any diameter of the parabola has been defined (Def. 8) to be one of the extremes of a proportion, of which any ordinate to the diameter is the mean and the corresponding abscissa the other extreme. Now, we have just shown that Qv —qv2 4PF'Pv. Whence, Pv: Qv:: Qv: 4PF. 4PF, which remains constant for the same diameter, is therefore the parameter of the diameter PTV. And as the same may be shown for any other diameter, we conclude that The parameter of any diameter of the parabola is equal to four times the distance from the vertex of that diameter to the focus. PROPOSITION XVIII.-T IEOREM. The parameter of any diameter of the parabola is equal to the double ordinate to this diameter that passes through the focus. Through any point, as P, of the pa- / - rabola of which F is the focus and V the vertex, draw the diameter PW, the tangent PT, and, through the focus the r double ordinate BD), to the diameter. n It is now to be proved that 4PF, or the parameter to this diameter, is equal to BD. Because PW is parallel to TX, and BD to TP, TPvF is a parallelogram, and Pv= TIF. But PF=FT (Prop. 4), hence Pv-= P. By the preceding proposition, Bv2=4PF Pv =-4PF.P_ F Whence, Bv=2PF; therefore, 2Bv —BD=4PF. hIence the theorem; the parameter of any diameter, etc. PROPOSITION XIX. —T H E O RE M. The area of the portion of the parabola included between the curve, the ordinate from any of its points to the axis, and

Page  61 THE PARABOLA. 61 the corresponding abscissa, is equivalent to two thirds of the rectangle contained by the abscissa and ordinate. Let VD be the axis of a parabo- Axr.. la, and VIP any portion of the. curve. Draw the extreme ordinate PD to the axis, and complete the FUG D rectangle VAPD; then will the area included between the curve VIP, the ordinate PD, and the abscissa VD, be equlva~ lent to two thirds of the rectangle VAPD. Take any point I, between P and the vertex, and draw PI, producing it to meet the axis produced at E. Now, from the similar triangles, PQI and PD1E, we get the proportion P: QI:: PD: DE. Whence PQ - DE= QI PD= GD' PD. (1) If we suppose the point Ito approach P, the secant line PE will, at the same time, approach the tangent PT; and finally, when I comes indefinitely near to P, the secant will sensibly coincide with the tangent PT, and DE may then be replaced by DT=2D V=2PA. Under this supposition, eq. (1) becomes 2PQ' PA=PD' GD. That is, when the rectangles GDPH and. APQ C become indefinitely small, we shall have Rect. G DP=-2 Rect. APQC. We will call Rect. GDPH the interior rectangle, and Rect. APQ C the exterior rectangle. If another point be taken very near to 1, and between it and the vertex, and with reference to it the interior and exterior rectangles be constructed as before, we should again have the interior equivalent to twice the exterior rectangle. Let us conceive this process to be continued until all possible interior and exterior rectangles are constructed; then would we have Sum interior rectangles=2 sum exterior rectangles.

Page  62 62 CONIC SECTIONS. But, under the supposition that these rectangles are indefinitely small, the sum of the interior rectangles becomes the interior curvilinear area, and the sum of the exterior rectangles the exterior curvilinear area, and the two sums make up the rectangle APD V. Therefore, if this rectangle were divided into three equal parts, the interior area would contain two of these parts. Hence the theorem; the area of the portion of the, etc. PROPOSITION X X.-THEORE M. If a parabola be revolved on its axis, the solid generated will be equivalent to one half of its circumscribing cylinder. Conceive the parabola in the fig- A ure, which is constructed as in the last proposition, to revolve on its axis V-D. We are then to find the....y'". measure of the volume generated. T - V G D The rectangle ID will generate a cylinder having DQ for the radius of its base, and D G for its axis; and the rectangle AI will generate a cylindical band, whose length is CI, and thickness PQ. The solidity of the cylinder =rDQ2' DG The solidity of the band =_r(3-D2 Q-2) G — 4r[PD —(PD-PQ)']~ VG=r[2PD' PQ-PQ2] VG Now, under the supposition that the point 1 is indefinitely near to P, D Q may be replaced by PD, VG by VD, and PQ may be neglected as insensible in comparison with 2PD PQ. These conditions being introduced in the above expressions for the solidities of the cylinder and band, they become The solidity of the cylinder=-r-P-2 ~ DG The solidity of the band =27rPD' PQ * VD

Page  63 THE PARABOLA. 63. Whence, sol. of cylinder: sol. of band:: PD DG: 2PD PQ~ VD (1) But, when I and P are sensibly the same point, PQ: GD:: PD: 2VD therefore, 2 VD' PQ=PD- GD, or 2 V.D PQ PD=PD'DG The terms in the last couplet of proportion (1) are therefore equal, and we have sol. of cylinder: sol. of band:: 1: 1 or sol. of cylinder=sol. of band. In the same manner we may prove that any other interior cylinder is equivalent to the corresponding exterior band. Hence the sum of all the possible interior solids is equivalent to the sum of the exterior solids. But the two sums make up the cylinder generated by the rectangle VDPA; therefore either sum is equivalent to one half of the cylinder. Hence the theorem; if a parabola be revolved, etc. REMARK.-The body generated by the revolution of a parabola about its axis is called a Paraboloid of Revolution. PROPOSITION X X I. -T HEOREE M. If a cone be cut by a plane parallel to one of its elements, the section will be a parabola. Let MVN be a section of a cone by a plane passing through its axis, and in this A section draw AHparallel to the element VM. L Through AlX conceive a plane to be passed perpendicular to the plane MVN; then will':"a - N the section DA GI of the cone made by this last plane, be a parabola. In the plane MVN, draw MN and KL perpendicular to the axis of the cone, and through them, pass planes perpendicular to this axis. The sections of the cone, by these planes, will be circles,

Page  64 64 CONIC SECTIONS. of which ]MN and KL, respectively, are the diameters. Through the points P and H, in which All meets KL and ]IN, draw in the section DA GI the lines FPG and HI, perpendicular to AHl. Because the planes DA1 and IMVN are at right angles to each other, PG is perpendicular to KL, and HI is perpendicular to JWN. Now, from the similar triangles AEL, AHN, we have AF: AHf:: L: HN (1) By reason of the parallels, KF= MJI; multiplying the first term of the second couplet of proportion (1) by KP, and the second term by 2fH, it becomes AF: At::PFL KPF: HtN' 2XE (2) But PG is an ordinate of the circle of which KL is the diameter, and HI an ordinate of the circle of which lN is the diameter: therefore FL.KFP=PG2, and HN31tH — f-I2 (Cor., Th. 17, B. III, Geom.) Substituting, for the terms of the second couplet, in proportion (2), these values, it becomes AF: AH:: PG2: Hi2 This proportion expresses the property that was demonstrated in proposition 15 to belong to the parabola. Hence the theorem; if a cone be cut by a plane, etc. Cor. From the proportion, AP: Al:: PG: 72HI we _TG_ - 2 PW_ 2 get -; that is, or which is a third AF AHf' AP AH') proportional to any abscissa and the corresponding ordinate of the section, is constant, and (by Def. 8) is the parameter of the section.

Page  65 THE HYPERBOLA. 65 THE HYPERBOLA. DEFINITIONS. 1. The Hyperbola is a plane curve, generated by the motion of a point subjected to the condition that the difference of its distances from two fixed points shall be constantly equal to a given line. REMARK 1. —The distance between the foci is also supposed to be known, and the given line must be less than the distance between the fixed points; that is, less than the distance between the foci. REMARK 2.-The ellipse is a curve confined by two fixed points called the foci; and the sum of two lines drawn from any point in the curve is constantly equal to a given line. In the hyperbola, the difference of two lines drawn from any point in the curve, to the fixed points, is equal to the given line. The ellipse is but a single curve, and the foci are within it; but it will be shown in the course of our investigation, that The hyperbola consists of two equal and opposite branches, and the least distance between them is the given line. 2. The Center of the hyperbola is the middle point of the straight line joining the foci. 3. The Eccentricity of the hyperbola is the distance from the center to either focus. 4. A Diameter of the hyperbola is a straight line passing through the center, and terminating' in the opposite branches of the curve. The extremities of a diameter called its vertices. 6* E

Page  66 66 CONIC SECTIONS. 5. The Major, or Transverse Axis, of the hyperbola is the diameter that, produced, passes through the foci. 6. T1he MIinor, or Colnjugate Axis, of the hyperbola bisects the major axis at right-angles; and its half is a mean proportional between the distances from either focus to the vertices of the major axis. 7. An Ordinate to a diameter of the hyperbola is a straight line, drawn from any point of the curve to meet the diameter produced, and is parallel to the tangent at the vertex of the diameter. 8. An Abscissa is the part of the diameter produced that is included between its vertex and the ordinate. 9. Conjugate Hyperbolas are two hyperbolas so related that the major and minor axes of the one are, respectively, the minor and major axes of the other. 10. Two diameters of the hyperbola are conjugate, when either is parallel to the tangent lines drawn through the vertices of the other. The conjugate to a diameter of one hyperbola will terminate in the branches of the conjugate hyperbola. 11. The Parameter of any diameter of the hyperbola is a third proportional to that diameter and its conjugate. 12. The parameter of the major axis of the hyperbola is called the principal parameter, the latus-rectum, or simply the parameter; and it will be proved to be equal to the chord of the hyperbola through the focus and at rightangles to the major axis. EXPLANATORY REMARRKS.-Thus, let F'F be two fixed points. Draw a line between them, and bisect it in C. Take CA, CA', each equal to one IF half the given line, and CA may be any distance i C less than CF; A'A is the given line, and is called the major axis of the hyperbola. Now, let us suppose the curve already found and represented by ADP. Take any point, as P, and join P, F and P, F'; then, by Def. 1, the difference between PF'

Page  67 THE HYPERBOLA. 67 and PF must be equal to the given line A'A; and conversely, if PF'-PFP=A'A, then P is a point in the curve. By taking any point, P, in the curve, and joining P, F and P, F' a triangle PFF' is always formed, having F'P for its base, and A'A for the difference of the sides; and these are all the conditions necessary to define the curve. As a triangle can be formed directly opposite PF'F, which shall be in all respects exactly equal to it, the two triangles having FP'P for a common side; the difference of the other two sides of this opposite triangle will be equal to A'A, and correspond with the condition of the curve. Hence, a curve can be formed about the focus F', exactly similar and equal to the curve about the focus F. We perceive, then, that the hyperbola cG' Ei is composed of two equal curves called G branches, the one on the right of the center and curving around the right-hand focus, and the other on the left of the' F-' AF center and curving around the left-hand focus. In like manner, by making CB equal to a mean proportional between JJ' E' FPA and FA', and constructing above and below the center the branches of the hyperbola of which BB'=_2 CB is the major, and AA' the minor axis, we have the hyperbola which is conjugate to the first. PlP is a diameter of the hyperbola, PT a tangent line through the vertex of the diameter, and Q Q', parallel to PT and terminating in the branches of the conjugate hyperbola, is conjugate to the diameter PP'. HD) is the ordinate from the point H to the diameter CP, and PD is the corresponding abscissa. PROPOSITION I.-PROBLEM. To describe an hyperbola mechanically. Take a ruler, F'H, and fasten one end at the point F, on which the ruler may turn as a hinge. At the other end, attach a thread, the length of which is less than that of the

Page  68 68 CONIC SECTIONS. ruler by the given line A'A. Fasten the other end of the thread at F. With the pencil, P, press the thread against the ruler, and keep it at: D equal tension between the points H and F. Let the ruler turn on the point F', keeping the pencil close to the ruler and letting the thread slide round the pencil; the pencil will thus describe a curve on the paper. If the ruler be changed, and made to revolve about the other focus as a fixed point, the opposite branch of the curve can be described. In all positions of P, except when at A or A', PP and PF will be two sides of a triangle, and the difference of these two sides is constantly equal to the difference between the ruler and the thread; but that difference was made equal to the given line A'A; hence, by Definition 1, the curve thus described must be an hyperbola. Cor. From any point, as P, of the hyperbola, draw the ordinate PD to the major axis, and produce this ordinate to P', making DP' equal to PD; and draw FP, FP', PFP and F'P'. Then, because 1)D is a perpendicular to PP at its middle point, we have FP=-P', and F'P=F'P'; whence FP'P-FP=F'P'-FP', and P' is a point of the hyperbola. Therefore, PP' is a chord of the hyperbola at right angles to the major axis, and is bisected by this axis; and as the same may be proved for any other chord drawn at right angles to the major axis, we conclude that All chords of the hyperbola which are drawn at right angles to the major axis are bisected by that axis. It may be proved, in like manner, that All chords of the hyperbola which are drawn at right angles to the conjugate axis are bisected by that axis.

Page  69 THE HYPERBOLA. 69 PROPOSITION II.-THEOREM. If a point be taken within either branch of the hyperbola, or on the concave side of the curve, the difference of its distances from the foci will be greater than the major axis; and if a point be taken without both branches, or on the convex side of both curves, the difference of its distances from the foci will be less than the major axis. Let AA' be the major axis, and x F and F' the foci of an hyperbola. x Q Within the branch APX take any \ / point, Q, and draw FQ and' Q; 2ii then we are to prove F'A' A F First.-That FiF Q —FQ is greater than AA'. Since Q is within the branch APX, the line F'Q must cut the curve at some point, as P. Draw PF and FQ. By the definition of the hyperbola, F'_P-PF=AA'. Adding PQ+PF to both members of this equation, it becomes F'P-PF+PQ+PF=AA'+PQ+PF or, F'Q=AA'+PQ+PF. But PQ and PF being two sides of the triangle FPQ, are together greater than the third side FQ. Therefore F' Q>AA'+FQ; and, by taking FQ from both members of this inequality, we have F'Q-FQ>AA'. Second. —Take any point, q, without both branches of the hyperbola, and join this point to either focus, as F. Then since q is without the branch APF, the line qF must cut the curve at some point, P. Draw qF, qF', and PF'. Because P is a point on the curve, we have P' —PF =AA'. Adding Pq+PF to the members of this equation it becomes PF —PF+ Pq+ PF=AA' + PF+Pq or, PF + Pq=AA'+PPF+Pq=AA'+qF.

Page  70 70 CONIC SECTIONS. But PP and Pq, being two sides of the triangle F'Pq, are together greater than the third side qE'. Whence qF'<AA'+q-F; and by taking qF from both members of this inequality, we have qF'-qF<AA'. Hence the theorem; if a point be taken, etc. Cor. Conversely: If the difference of the distances from any point to the foci of an hyperbola be greater than the major axis, the point will be within one of the branches of the curve; and if this difference be less than the major axis, the point will be without both branches. For, let the point Q be so taken that P Q —FQ>AA'; then the point Q cannot be on the curve; for in that case we should have F'Q-FQ=AA'. And it cannot bewithout both branches of the curve, for then we should have F' Q-FQ<AA', from what is proved above. But it is contrary to the hypothesis that F' Q —FQ is either equal to or less than AA'; hence the point Q must be within one of the branches of the hyperbola. In like manner we prove that, if the point q be so chosen that qF'-qF<AA', this point must be without both branches of the hyperbola. PROPOSITION III.-THEOREM. A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci. Let F', F be the foci, and P T any point on the curve; draw PF', PF and bisect the angle\ F'PF bythe line TT'; this line will be a tangent at P. If TT' be a tangent at P, ev- R TAF ery other point on this line will be without the curve.

Page  71 THE HYPERBOLA. 71 Take PG=PF and draw GF; TT' bisects GF, and any point in the line TT' is at equal distances from F and G (Scho. 1, Th. 18, B. I, Geom). By the definition of the curve, F' G-A'A the given line. Now take any other point than P in TT', as E, and draw EF', EF and EG. Because EF is equal to EG we have EF-' -EF=B- E'- EG. But EF' —EG, is less than FIG, because the difference of any two sides of a triangle is less than the third side. That is, BEF'-E- is less than A'A; consequently the point E is without the curve (Prop. 2), and as E is any point on the line TT', except P, therefore, the line TT', which bisects the angle at P, is a tangent to the curve at that point. Hence the theorem; a tangent to the hyperbola, etc. SCHOLIUM.-It should be observed that by joining the variable point, P, in the curve, to the two invariable points, F' and F, we form a triangle; and that the tangent to the curve at the point P, bisects the angle of that triangle at P. But when any angle of a triangle is bisected, the bisecting line cuts the base into segments proportional to the other sides. (Th. 24, B. II, Geom). Therefore, F'P: PF=F'T: T F Represent _PP by r' and PF by r; then r': r=FT: TF But as r' must be greater than r by a given quantity, a, therefore, ra: r=F'T: T F a Or, l+-: =FT: TF Let it be observed that a is a constant quantity, and r a variable one which can increase without limit; and when r is immensely great in respect to a, the fraction - is extremely minute, and the first term of the above proportion would not in any practical sense differ from the second; therefore, in that case, the third term would not essen

Page  72 72 CONIC SECTIONS. tially differ from the fourth; that is, F' T does not essentially differ from FT when r, or the distance of P from F' is immensely great. Hence, the tangent at any point P, of the hyperbola, can never cross the line EF' at its middle point, but it may approach within the least imaginable distance to that point. If, however, we conceive the point P to be removed to an infinite distance on the curve, the tangent at that point would cut AA' at its middle point C, and the tangent itself is then called an asymptote. PROPOSITION IV.-T HEORE M. Every diameter of the hyperbola is bisected at the center. Let F and A' be the foci, and T AA' the major axis of an hyperbola. Take any point, as P, in one v k of the branches of the curve; draw C PF and PF', and complete the,'\ parallelogram PFP'.F'. We will now prove that P' is a point in the opposite branch of the hyperbola, and that PP' passes through, and is bisected at, the center, C. Because PFP'F' is a parallelogram, the opposite sides are equal; therefore FP-PI'-EFP' -P'F'; but since I is, by hypothesis, a point of the hyperbola, F'P —PF AA'; hence FP'-P'FT=AA', and P' is also a point of the hvly[erbolu. Again, the diagonals, F'F, P'P of the parallelogram, mutually bisect each other; hence C is the middle point of the line joining the foci, and (Def. 2) is the center of the hyperbola. P P' is therefore a diameter, and is bisected at the center, C. Hence, the theorem; every diameter of the hyperbola, etc. PROPOSITION V. -THEOREM. Tcanqents to the hyplerbola at the vertices of a diameter are pa2rallel to each other.

Page  73 THE HYPERBOLA. 73 At the extremities of the diameter, PP', of the hyperbola represented in the figure, draw the tan- \ v gents TT' and VV'. We are now F AF to prove that these tangents are parallel. By proposition (Prop. 3) TT' bisects the angle FPE', and VV' also bisects the angle F'P'F. But these angles being the opposite angles of the parallelogram FPF'P', are equal; therefore the L T'PF-the LPT'F=the L VP'F. But the L's PT'_F VP'F, formed by the line fi'P' meeting the tangents, are opposite exterior and interior angles. The tangents are therefore parallel (Cor. 1, Th. 7, B. I, Geom). Hence the theorem; tangents to the hyperbola, etc. PROPOSITION VI.-THEOREM. The perpendiculars let fall from the foci of an hyperbola on any tangent line to the curve, intersect the tangent on the circumference of the circle described on the major axis as a diameter. In the hyperbola of which AA' T7 is the major axis, F and F' the foci, and C the center, take ally point in one of the branches, as P, and through it draw the tan- F' f A.x gent line TH'. From the foci let fall on the tangent the perpendic- / ulars Ff1, F'H', draw PP and PF', and produce FH1 to intersect PF' in G. We are now to prove that H and 11' are in the circumference of a circle of which AA' is the diameter. Draw CH, producing it to meet FH_' in Q. Then, because PN- is a tangent to the curve, it bisects the angle FPF'; ther'efore tle right-angled triangles, FPH and 7

Page  74 74 CONIC- SECTIONS. HTPG, being mutually equiangular, and having the side PH common, are equal. Whence, Fi=-tHG and PF1= PG. But, by the definition of the hyperbola, F'P-PF — AA'; hence F'P-PG=F'G-=AA'. Since CHf bisects the sides F'F and FG of the triangle FG.F', we have F'.F:FC::F'G: CH but 1F'F=2FC; therefore iF' G=2 CH=AA' If then with C as a center and CA as a radius, a circumference be described, it will pass through the point H. Again; the triangles EFHC and F' CQ are in all respects equal; hence CQ= CH, and Q is also a point in the circumference of the circle of which AA' is the diameter. Therefore, the right-angled triangle Q1i'H, having for its hypotenuse a diameter IQ of this circle, must have the vertex, -' of its right angle at some point in the circumference. Hence the theorem; the perpendiculars let fall, etc. PROPOSITION VII.-THEOREM. The product of the perpendiculars let fall from the foci of an hyperbola upon a tangent to the curve at any point, is equal to the square of the semi-minor axis. Resuming the figure of the pre- T ceding proposition; then, since the semi-minor axis, which we will represent by B, is a mean proportional between the distances from pF'..A; F either focus to the extremities of the~ major axis, we are to prove that B23= -A x FA'= FHfx F'l' By the construction, the triangles PFHC and CQE are equal; therefore FH=F'tQ (1)

Page  75 THE HYPERtBOLA. 75 Multiplying both members of eq. (1) by PH'tw it becomes F. t' HI'H PF Q' F'H' (2) Again, it was proved in the last proposition that the points H, Hi' and Q were in the circumference: of the circle described on AA' as a diameter; therefore F'IH' and PIA are secants to this circumference, and we have F'" Q: FAI':: PA: FH' (Cor., Th. 18, B. III, Geom). Whence, F' Q. EHIJI' =FA' F _'FA (3) But.A'-=FA, =F'AFA', and F'Q=E-. Making these substitutions in eq. (3) it becomes FHiE -FrH'= FA * FA'=-B2. Hence the theorem: the product of the perpendiculars, etc. Cor. 1. The triangles PFPH, PF'1_' are similar; therefore, PE: PE':: FT: F'H' That is: The distances from any point on the hyperbola to the foci, are, to each other, as the perpendiculars let fall from the foci upon the tangent at that point. Cor. 2. From the proportion in corrollary 1, we get PF*E' -,lH' PP' F'Hf.' F'H _1= —- P; whence H2 = — P PE',? P.F But by the proposition, F'/H' ~ F=132; therefore, E —2 B because' G- P1 -2 CA +PbecauseEG AA'=2 CA, and PG=PP. In like manner it may be proved that 2 2(2CA+PF) PPF PFS PROPOSITION VIII.-THEOREM. If a tangent be drawn to the hyperbola at any point, and also an ordinate to the major axis from the point of contact, then will the semi-major axis be a mean proportional between the

Page  76 76 CONIC SECTIONS. distance from the center to the foot of the ordinate, and the distance from the center to the intersection of the tangent with this axis. Let AA' be the major axis, TF' the foci and C the center of the G hyperbola. Through any point, \ as P, taken on one of the branches, draw the tangent PT intersect- F - F C M \ ing the axis at T; also draw PF, PF' to the foci, and the ordinate P1M to the axis. We are now to prove that CT: CA.:: CA: CM. Because PT bisects the vertical angle of the triangle FPF' (Prop. 3), it divides the base into segments proportional to the adjacent sides (Th. 24, B. II, Geom.) Therefore, F'T: TF:: F'P: PF. Whence, F' T- TF-: T+ TF:: F'P-PF:'P P+ PF That is, 2CT: F'F:: AA' —2CA: F'P+PTF Or, by inverting the means, 2CT: 2CA::: F:F'P+PF (P) Now, making ITF"-=MF, and drawing PF", we have, from the triangle FI'PF",'l": F'P+ PT":: Fl'P-PF": TF'1'M —IF" (Prop 6, PI. Trig.) But, because the triangle FPI "is isosceles, and PM is a perpendicular from the vertical angle upon the base, PF= PF", TF'TF"=F'F+2JM= 2 CTF+2FTM=2 CM; therefore the preceding proportion becomes 2CM: F'P+PF:: 2 CA: 7'F or, 2CM: 2CA'::i'P+PT: lVl'F (2) Multiplying proportions (1) and (2), term by term, observing that the terms of the second couplet of the resulting proportion are equal, we have

Page  77 THE HYPERBOLA. 77 4CT. CMH: 4A2::1:1 Whence, CT' C02= CA2; which, resolved into a proportion, becomes CT: CA:: CA: CM. Hence the theorem; if a tangent be drawn, etc. SCHOLIvM.-The property of the hyperbola demonstrated in this proposition is not restricted to the major axis, but also holds true in reference to the minor axis. The tangent intersects the minor axis at the point t1, and PG is an ordinate to this axis from the point of contact. Now, the similar triangles tCT, TifF, give the proportion Ct: FH C: T: T (1) and from the similar triangles PMT, TF'1T, we also have PM: F H:: MT: H'T (2) Multiplying proportions (1) and (2), term by term, we get Ct.PM: FHF'H':: CTMT: TH:H7 T (3) But FHffF'H'=B (Prop. 7). Moreover, drawing the ordinate TV, and the radius CVof the circle, and the line VAf1, we have by the proposition CT: CA:: CA: CM or, CT: CV:: CV: CM Therefore, the triangles VCT and MCV, having the angle C common and the sides about this angle proportional, are similar (Cor. 2, Th. 17, B. II, Geom.); and because the first is right-angled, the second is also right-angled, the right angle being at V; hence VT = CT-MT (Th. 25, B. II, Geom). Also, AA' and HH' are two chords of a circle intersecting each other at T; hence HT TH'=A T TA'= V=T (Th. 17, B. III, Geom). Substituting for the terms of proportion (3) these several values, it becomes Ct'PM: B:: VT: VT2::: 1 Whence, Ct-PM= By Therefore, Ct: B:: B: PM= CG 7*

Page  78 '78 CONIC SECTIONS. Cor. It has been proved that the triangle CVM is rightangled at V; therefore, VM is a tangent at the point V to the circumference on AA' as a diameter, and TMl is its sub-tangent. But Ti is also the sub-tangent on the major axis of the hyperbola answering to the tangent PT; hence If a tangent be drawn to the hyperbola at any point, and through the point in which the tangent intersects the major axis an ordinate be drawn to the circle of which this axis is a diameter, the sub-tangent on the major axis corresponding to the tangent through the extremity of this ordinate will be the same as that of the tangent to the hyperbola. PROPOSITION IX.-THEOREM. In any hyperbola the square of the semi-major axis is to the square of the semi-minor axis, as the rectangle of the distances from the foot of any ordinate to the major axis, to the vertices of this axis, is to the square of the ordinate. Resuming the figure to Proposition 8, the construction of which G needs no further explanation, we are to prove that CA: CB2:: At'M. AM: P M2, F assuming CB to represent the semi-minor axis. From the similar triangles PMT, THF and TI'F', we derive the proportions PM: FH:::MT: TH PM: F'I': MT: THI Whence (1) PM:IbF-'I'::]JIMT: TH TB' (1) But F FH.'t' is equal to the square of the semi-minor axis (Prop. 7); and because the chords, Hf' and AA', of the circle intersect each other at T, we have

Page  79 THE HYPERBOLA. 79 T ffTH'=A T'TA'= VT2 (Th. 17, B. III, Geom.) These values of the consequents of proportion (1) being substituted, it becomes PM: -C2:: MT: VT2 (2) The triangles CVT and TVM are similar, and give the proportion MT2: VT- V.M2: =CV=CA2 (3) Comparing proportions (2) and (3), we find that PM.2 BC2: VM2:CA (4) Because MV is a tangent and MA' a secant to the circle A VA'H', we have 2 VM — A'M' AM (Th. 18, B. III, Geom.) Placing this value of VM2 in proportion (4) and inverting the means of the resulting proportion, it becomes PM2: A'M AM:: BC2: CA2 or, CA: BC:: A'M AM: PM2 Hence the theorem; in any hyperbola the square of the, etc. Cor. Proportion (4) above may be put under the form CA: BC:: VM2: PM2 (a) and from the right-angled triangle CVM we have C-2+ V-M2= CM2 from which, because CV= CA, we get VM=7 CM2 —-CA2. Also, the right-angled triangles CVM, VTMare similar; therefore, CM': VM:: VM: MT Whence VM 2= CMJ MT. Now, if in proportion (a) we place for VM2 these values, successively, we shall have the two proportions CA2:'B-C2:: C]' MT: PM2 (b) and - -i. -2 -C22 -CA2: PM2 (c)

Page  80 80 CONIC SECTIONS. SCHOLIUM 1. —Let us denote CA by a, CB by b, CM by x, and PMbyy; then A'M==x-+a and AM=x-a. Because CUM- CA ( CM+ CA) ( CM- CA)=AM' AM, proportion (c), by substitution, now becomes a' b:: (x+a) (x-a)'y. (a') Whence a~y2=b-x 2-a 2b2 or, ay2- bsx2= -a2b. This equation is called, in analytical geometry, the equation of the hyperbola referred to its center and axes, in which x, the distance from the center to the foot of any ordinate to the major axis, is called the abscissa. The equation a2y'2-b2x2= —a2b2 therefore expresses the relation between the abscissa and ordinate of any point of the curve. SCHOLIUM 2.-Let y' denote the ordinate and x' the abscissa of a second point of the hyperbola; then we shall have a2 b:: (x'+a) (x'-a): y'2 Comparing this proportion with proportion (a'), scholium 1, we find Y2 -: y. (x+a) (x-a): (x'+a) (x' —a) That is: In any hyperbola the squares of any two ordinates to the major axis are to each other, as the rectangles of the corresponding distances from the feet of these ordinates to the vertices of the axis. A similar property was proved for the ellipse and the parabola. PROPOSITION X.-T H EO R E M. The parameter of the major axis, or the latus-rectum, of the hyperbola is equal to the double ordinate to this axis through the focus. Through the focus F of the hyperbola, of which AA' is the major and BB' r\ p the minor axis, draw the chord PP' at right angles to the major axis; then de- A F\ noting the parameter by P, we are to / prove that /' AA': BB': BB': PP'=P (Def. 11.)

Page  81 THE HYPERBOLA. 81 By definition 6, BC2 =FA' *'FA, and by proposition 9 we have AC.: BC2:: FA' FA: pF2 =(pp)2 (Cor. Prop. 1.) Whence AC2: BC:: B C- (1pp')2 Therefore AC: BC:.BC: IPP' (Th. 10, B. II, Geom.) Multiplying all the terms of this last proportion by 2, it becomes 2AC: 2BC:: 2BC: PP' or, AA': BB':: BB': PP' Hence the theorem; the parameter of the major axis, etc. PROPOSITION XI.-THEOREM. If from the vertices of any two conjugate diameters of the h Tperbola ordinates be drawn to either axis, the difference of the s /pares of these ordinates will be equal to the square of one half the other axis. Let AA', BB' be the axes, and PP', QQ' any two conjugate diameters of the conjugate hyperbolas represented in the figure. Then, 1 4M drawing the ordinates QV, PM, to the major axes, and the ordinates PS=]WC, QD= VC, to the minor axis, it is to be proved that 0A"=MAC2- VC' and that -CB= Q V2_M2 Draw the tangents PT and Qt, the first intersecting the major axis at T and the minor axis at T', and the second intersecting the minor axis at t' and the major axis at t. Now, by proposition 8, we have, with reference to the tangent PT, CT: CA:: CA: CM, F

Page  82 82 CONIC SECTIONS. and by the scholium to the same proposition, we also have, with reference to the tangent Qt to the conjugate hyperbola, Ct: CA'- CA:: CA: CV The first proportion gives CA2= CT' CM, and the second CA2= Ct - CV, Whence CT CM-= Ct CV, which, in the form of a proportion, becomes CM': CV:: Ct: CT (1) From the similar triangles tCQ, CTP, we get Ct: CT:: QC: PT (2) and from the triangles CQ V, TPM QC: PT:: CV: TM (3) Comparing proportions (1), (2) and (3), it is seen that CM: CV:: CV: Ti Whence CV2= CM' TM; but TM= CM —CT; Therefore CV'= C-M — CT' CM. And because CT' CM= CA (Prop. 8), we have CV'=C C -CA or, CA-Cnm e-CV Again we have CT': CB:: CB: PM (Scho., Prop. 8) and Ct': CB:: CB: CD=-QV (Prop. 8) Whence CT' P1M= Ct'. Q V, which, resolved into a proportion, becomes PM: Q V:: Ct': CT' (4) From the similar triangles, T' CP, Ct' Q, we get Ct': CTt: t'Q: CP (5) And from the triangles t'DQ, CPM, we also get t'Q: CP:: t'D: PM (6) From proportions (4), (5) and (6) we deduce

Page  83 THE HYPERBOLA. 83 PM: QV: t'D: PM Whence PMl2- Q V' t'D; but t'D= Q V-Ct'; therefore, PM2= QV2 — C' QV= QV2- Ct CD. And because Ct' * CD= CB2 (Prop. 8) we have PM2= Q V2 CB2 or CB2= Q -2 -P-MI Hence the theorem; from the vertices of any two, etc. Cor. By corollary to proposition 9 we have CA:CB 2::CM2 CA2: PM2 In like manner, in reference to the conjugate hyperbola, we shall have CB:CA2::CD2_-CB2 QD.:- QV2-CB2:.CV - or, CB2: QV-CB CA2: CV2-ABy composition, CB2: QV2:.C2.: CA-2+-CVBut by this proposition we have CA2= CM2 —CV2; hence CA2 +-CV= — CM therefore -CB2-: Q V2:: CA-: CM2 Whence CB: QV:: CA: CM or, CA: CB:: CM: QV PROPOSITION XII. -THEOREM. The difference of the squares of any two conjugate diameters of an hyperbola is constantly equal to the difference of the squares of the axes. In the figure, which is the same as that of the preceding proposi- \ tion, PP' and Q Q' are any two conjugate diameters (Def. 10). It is to be proved that PP -QQ'2-AA'2-BB'2 v T-, By proposition 11 we have By proposition 11 we have

Page  84 84 CONIC SECTIONS. CA2 CM22 and CB= Q V2_PM2 therefore CA2-CB2 CM2+P 1M2_-(CV2 Q2) or, CA- CB== -0P2 Multiplying each member of this equation by 4, observing that 4CA2=AA'2 &c., it becomes AA2 -BB'-P'2 p2- jQ Q'2 Hence the theorem; the difference of the squares, etc. PROPOSITION XIII.-THEOREM. The parallelogram formed by drawing tangent lines through the vertices of any two conjugate diameters of the hyperbola is equivalent to the rectangle contained by the axes. Let LMNO be a parallelogram formed by drawing tangent lines Q through the vertices of the two con- P jugate diameters PP', QQ' of the A conjugate hyperbolas represented in / the figure. It is to be proved that /' B area LMNO=f AA' x BB'. We have CA: CB:: CS: QV (1) (Cor. Prop 11.) Also, CT: CA:: CA: CS (2) (Prop. 8.) Multiplying proportions (1) and (2), term by term, omitting in the first couplet of the result the common factor CA, and in the second the common factor CS, we find CT: CB:: CA: QV Whence CT Q V-=CA' CB But CT' Q V measures twice the area of the triangle CQT, and this triangle is equivalent to the half of the parallelogram Q CPL, because they have the common base QC and are between the same parallels QC, LT (Th. 30, B. I, Geom.)

Page  85 THE HYPERBOLA. 85 Now the parallelogram QCPL is one-fourth of the parallelogram LMNO, and CAG CB measures one fourth of the rectangle contained by the axes; therefore the parallelogram and rectangle are equivalent. Hence the theorem; the parallelogram formed, etc. PROPOSITION X1V.-THEOREM. If a tangent to the hyperbola be drawn through the vertex of the major axis, and an ordinate to any diameter be drawn from the same point, the semi-diameter will be a mean proportional between the distances, on the diameter, from the center to the tangent, and from the center to the ordinate. Let CA be the semi-major axis and CP any semi-diameter of this hyperbola. Draw the tangents At, PT, the i ordinate AHto the diameter, and the c ordinate PM to the major axis. It is T A -M now to be proved that CP2- Ct CH. We have CT: CA:: CA: CM, (Prop. 8) also CA: Ct:: CM: CP from the similar A's CAt, CMP Multiplying these proportions term by term, omitting in the result the common factor in the first couplet, and also that in the second, we find CT: Ct:: CA: CP (1) Again we have CP: CT:: CH: CA from the similar A's CPT, CIA. Proceeding with these last proportions as with those above, we find CP: Ct:: CH: CP W: ence, CP2= Ct' CH. Hence the theorem; if a tangent to the hyperbola, etc. Cor. 1. From proportion (1) we get CT' CP= Ct CA; but the triangles CTP, CAt, having a common angle, C, are 8

Page  86 86 CONIC SECTIONS. to each other as the rectangles of the sides about this angle (Th. 23, B. II, Geom.) Therefore A CTP=- ICtA. Cor. 2. If from the equivalent areas A CTP, A CtA we take the common area CTVt there will remain A TA V-= At VP. Cor. 3. If we add to each of the triangles TA V, t VP, the trapezoid VAMP, we shall have area TAMP= area tAMP. PROPOSITION XV.-THEOREM. If through any point of an hyperbola there be drawn a tangent, and an ordinate to any diameter, the semi-diameter will be a mean proportional between the distances on the diameter from the center to the tangent, and from the center to the ordinate. Take any point as D on the hyperbola of which CA is the semi- D major axis, and through this point B draw the tangent D Tand the semidiameter CD, also take any other point, as P, on the curve, and draw C Lt m G the tangent Pt, the ordinate PHto the diameter through D, and the ordinates PQ and DG to the axis. The semi-diameter CD and the tangent Pt intersect each other at t'. We will now prove that CD-' — Cf' CH Let CB represent the semi-conjugate axis, then by corollary to proposition 9 (proportion (b)) we have 2 >2 s CA: CB:: CG TG: D6 and Z2: C2:: CQttQ: P7 Whence CG' TG: CQ tQ: D G2: PQ ~ ~ but DliG: P Q:: TVG: LQ2, from the similar A's TGD, LQP;

Page  87 THEI HYPERiBOLA. 87 therefore CG TG: CQ tQ:: TG: LQ (1) Drawing Dm parallel to Pt we have the similar A's mGD, tQP which give the proportion DG: PQ:: Gm: Qt. (2) The A's TGD, LQP also give DG: PQ:: TG: LQ (3) From proportions (2) and (3) we get TG: LQ:: Gm: Qt (4) Multiplying proportions (1) and (4) term by term, there results, CG.TG2: CQtQ LQ:: TGWGm:nLQQt Dividing the first and third terms of this proportion by TG2and the second and fourth terms by Qt LQ it becomes CG: CQ:: Gm: LQ or CG: Gm:: CQ: LQ (5) Whence CG: CG-Gm:: CQ: CQ-LQ That is CG: Cm:: CQ: CL (6) Again CT CG-= CA - CQ Ct, (Prop. 8.) therefore CG: Ct:: CQ: CT The antecedents in this last proportion and in proportion (6) are the same, the consequents are therefore proportional, and we have Ct: CT:: Cm: CL We have also, Cm: CD:: Ct: Ct' from the similar A's CmD, Ctt' And CT: CD:: CL: CH from the similar A's CTD CLIH By the multiplication of the last three proportions term by term we find Ct Cm CT:CD' CT:: Cnm Ct CL: CL Ct' C CH Whence CT: C-D CT:: CL: CL Ct' CH or 1: CD2:: 1: Ct' CCH therefore Of'- Ct' CIH

Page  88 88 C'ONIC SECTIONS. Hence the theorem; if through any point of an, etc. REMARK. —The property of the hyperbola just established is the generalization of that demonstrated in the preceding proposition. PROPOSITION XVI. —THEOREM. The square of any semi-diameter of the hyperbola is to the square of its semi-conjugate as the rectangle of the distances from the foot of any ordinate to the first diameter, to the vertices of that diameter, is to the square of the ordinate. Let PP' and QQ' be any two conjugate diameters of the conju- K gate hyperbolas represented in \ the figure. Through any point as p, Th G draw the tangent GT' inter- H secting the first diameter at T and the second at T', and from // \ the same point draw the ordinates GE/, GK, to these diameters. We will now prove that, CP: CQ:: PH'P'H: GH'_ By the preceding proposition we have CP = CT' CH and multiplying each member of this equation by Cf it becomes CP2 C CH= CT CH 2 Whence CP2.Gl:: CT: CHfrom which by division we get CP-2: GC2- -P2:: C'T: C —CT= TH, (1) Again we have CQ2 — CT' CK (Prop. 15) and multiplying each member of this equation by CK it becomes CQ- CK= CT' CK.2 Whence CQ2: CK2:: CT': CK=-GH (2) The similar A's TCT', TfG give the proportion CT': GH:: CT: TH (3) Comparing proportions (2) and (3) we obtain CQ2: CK2::CT: TH (4)

Page  89 THE HYPERBOLA. 89 And by comparing proportions (1) and (4) we obtain -2 Cic2 -Cp2 - Cp2 or cP2: c- CQK2GH2 But because CF= CP' and C-2-CP2-=(C- CP) (CUH+ CP) = PH. ( SH+ CP) the last proportion above becomes P2 Q2:: PHP'.H: GIL2 Hence the theorem; The square of any semi-diameter, etc. REMARK.-The property of the hyperbola with reference to any two conjugate diameters just demonstrated is the same as that with reference to the axes established in proposition 9. Cor. If the ordinate GHi be produced to intersect the curve at Gt and the above construction and demonstration be supposed made for the point G' instead of G, we should finally get the same proportion as before, except the fourth term, which would be G'_2; therefore, G'H= GIl. HIence we conclude that Any diameter of the hyperbola bisects all the chords drawn parallel to a tangent line through the vertex of that diameter. P R OPOSITION X V II.-T H E O R E M. The squares of the ordinates to any diameter of the hyperbola are to one another as the rectangles of the corresponding distances from the feet of these ordinates to the vertices of the diameter. Resuming the figure to the G proposition which precedes and drawing any other ordinate gh to the diameter PP', it is to be / H proved that G-H:.2.PH. P'I. Ph.P'h By the foregoing proposition /' we have two proportions following, vlz: — P-: CQ2:Ph P. pH: G2 UP2: U(2:: Ph P'h: gh2 8*

Page  90 90 CONIC SECTIONS. Since the ratio CP2: CQ2 is common to these proportions the remaining terms are proportional. That is GH2: gh2:: PH P':t Ph P'h Hence the theorem- The squares of the ordinates, etc. PROPOSITION XVI I I.-T HE O RE M. If a cone be cut by a plane making an angle with its base greater than that made by an element of the cone, the section will be an hyperbola. Let the A's MVN, BVR be the R\- B sections of two opposite cones by a \ plane through the common axis, and y BH a line in this section not passing through the vertex, and making K with MN the LBIHN> the LBMN. / Through this line pass a plane at /I right angles to the first plane, mak- M. N ing in the lower cone the section 1 IGAG'I'; then will this section be one of the branches of an hyperbola. Let KL and MN be the diameters of two circular sections made by planes at right angles to the axis of the cone, and at F and H, the intersections of these lines with BI, erect the perpendiculars FG, HI to the plane MVN. lFG is the intersection of the plane of the section IGA G'I' with the plane of the circle of which KL is the diameter and is a common ordinate of the section and ol the circle; so likewise is HI a common ordinate of the section and of the circle of which MN is the diameter. Now by the similar A's AFL, AHN, and BFK, BIM we have AF: AH:::FL: iN (1) and B F: BH:: FK: HM (2) Multiplying proportions (1) and (2), term by term, we get

Page  91 THE HYPERBOLA. 91 AFB.F: AH'BHE: FL-FK: -N E.HNHIM (3) But because LGK and NIM are semi-circles, PG2 2 EFLFIK and H-I =HNHM. Substituting these values for the terms of the last couplet of proportion (3) it becomes AF.BF: AlHBHI:: PG2:: 2 If we denote any two ordinates of the corresponding section of the opposite cone by fg and hi we should have in like manner Af.Bf: Ah Bh: (fg)2: (hi)2 If, therefore, AB be taken as a diameter of the curves cut out of the opposite cones by a plane through AH, at right angles to the plane VMN, we have proved that these curves possess the property which was demonstrated in the preceding proposition to belong to the hyperbola. Hence the theorem; if a curve be cut by a plane, etc. ASYMPTOTES. DEFINITION.-An Asymptote to a curve is a straight line which continually approaches the curve without ever meeting it, or, which meets it only at an infinite distance. We shall for the present assume, what will be afterwards proved, that the diagonals of the rectangle constructed by drawing tangent lines through the vertices of the axis of the hyperbola possess the property of asymptotes, and they are therefore called the asymptotes, of the hyperbola. PROPOSITION XIX.- THEOREM. If an ordinate to the major axis of an hyperbola be produced to meet the asymptotes, the rectangle of the segments into which it is divided by either of its intersections with the curve will be equivalent to the square of the semi-conjugate axis.

Page  92 92 CONIC SECTIONS. Let CA, CB be the semi-axes and Ct, n Ct' the asymptotes of an hyperbola.Through any point, as P, of the curve, B' draw the ordinate PQ to the major axis and produce it to meet the asymptotes at n c Q and n'. By the enunciation we are required to prove that CB2 Pn'Pn'p By Cor. proposition 9 we have v CA2: CB2:: Q2- A2: PQ2 (1) And from the similar triangles CAB', CQn CA2 AB'2 CB2 2 2 (2) Comparing proportions (1) and (2) we find CQ2: G-Q2- A2: Qn: PQ2 which gives by division A2: CQ2:: Qn2 PQ: Qn2 or GA': Qn2 —PQ::Q.::: (3) From proportions (2) and (3) we get (CA2: CB2:::2 n2 —Q2 In this proportion the antecedents are the same the consequents are therefore equal; that is CB2 Qn2- __ 2 = (Qn+PQ) (Qn-PQ)=Pn- Pn Itence the theorem; if an ordinate to the major axis, etc. Cor. Let us take another point p in the curve and from it draw the ordinate p Q' to the major axis; then, as before, we shall have CB2e pt pt'; t and t' being the intersections of the ordinate, produced, with the asymptotes. Whence Pn Pn'=pt'pt', which in the form of a proportion becomes Pn: Pt:: pt': Pn' PROPOSITION XX.-TIHEOREM. The parallelograms formed by drawing through the different points of the hyperbola lines parallel to and meeting the asymptotes are equivalent one to another, and any one is equivalent to one half of the rectangle contained by the semi-axes.

Page  93 THE HYPERBOLA. 93 Let CA, CB be the semi-axes and Cn, Cn' the asymptotes of an hyperbola. From any point, as P, of the curve draw the or- B dinate PQ to the major axis, producing it c to meet the asymptotes at n, n', and through c P and the vertex A draw parallels to the bX asymptotes, forming the parallelograms PmCt, A-ECD. This last is a rhombus because its adjacent sides CE, CD are equal, being the semi-diagonals of equal rectangles. It will now be proved that Area Pm Ct = area AECD=- Rect. AB'BC. By the proposition which precedes we have CB-2=Pn Pn' (1) And from the similar triangles AB'E, Pnm, and the similar triangles ADb', Ptn' we also have AE: AB'= CB: mP: Pn AD: Ab=CB:: Pt: Pn' Multiplying these proportions, term by term, we find AE'AD: CB 2:: mP m Pt: Pn Pn' By equation (1) the consequents of this proportion are equal, therefore the antecedents are also equal. That is, AB - AD=mP Pt If the first member of this equation be multiplied by sin. LDAE, and the second member by the sine of the equal LmPt it becomes AE' AD * sin. DAE=mP _Pt' sin mPt But AE AD * sin DAE measures the area of the rhombus AECD and mP' Pt sin. mPt measures the area of the parallelogram Pm Ct; therefore the parallelogram and the rhombus are equivalent. Moreover, because the A's AEC, ADC are equal, and the A's A-EC, AEB' are equivalent, it follows that the rhombus AECD is equiva

Page  94 941 CONIC SECTIONS. lent to the AAB' C, or, to one half of the rectangle contained by the semi-axes. Hence the theorem; the parallelograms formed, etc. Cor. 1. If from the rhombus AECD and the parallelogram Pm Ct the common part be taken, there will remain the parallelogram AKtD, equivalent to the parallelogram PmEK, and if to each of these the curvilinear area AKP be added, we shall have Area APmE= area APtD. Had we proceeded in the same way with the parallelogram PmCt and any parallelogram other than AECD we should have had a like result; therefore If from any two points in the hyperbola parallels be drawn to each asymptote, the area bounded' by the parallels to one asymptote, the other asymptote, and the curve will be equivalent to the other area like bounded. SCHOLIUM. —If one half of the rectangle contained by the semiaxes of the hyperbola be denoted by a, the distance Cm by x, and the distance mp= Ct by y, then, by this proposition, we shall have the equation xy=a, which, in analytical geometry, is called the equation of the hyperbola referred to its center and asymptotes. Cor. 2. In the equation xy=a, y expresses the distance of any point of the curve from the asymptote on which a x is estimated. From this equation we get y —-. Now it is evident that as x increases y decreases, and finally when x becomes infinite, y becomes zero. That is, the asymptote continually approaches the hyperbola without ever meeting it, or without meeting it within a finite distance. We were, therefore, justified in assuming that the diagonals of the rectangle formed by the tangents through the vertices of the axes were asymptotes to the hyperbola.

Page  95 ANALYTICAL GEOMETRY.

Page  96 ANALYTICAL GEOMETRY. GENERAL DEFINITIONS AND REMARKS, Analytical Geometry, as the terms imply, proposes to investigate geometrical truths by means of analysis. In it the magnitudes under consideration are represent by simbols, such as letters, terms, simple or combined, and equations; and problems are then solved and the properties and relations of magnitude established by processes purely algebraic. A sigle letter, without an exponent, will alwoys be understood as denoting the length of a line; and in general, any expression of the first degree denotes the length of a line and is, for this reason, said to be linear; so likewise, an equation all of whose terms are of the first degree is called a linear equation. An expression of the second degree will represent the measure of a surface, and an expression of the third degree will represent the measure of a volume. When a term is of a higher degree than the third, a sufficient number of its literal factors, to reduce it to this degree, must be regarded as numerical or abstract. The subject of Analytical Geometry naturally resolves itself into two parts. First. That which relates to the solution of determinate problems; that is, problems in which it is required to determine certain unknown magnitudes from the relations which they bear to others that are known. In this case we must be able to express the relations between the known and unknown magnitudes by independent equnations equal in number to the required magnitudes. (96)

Page  97 GENERAL PROPERTIES. 97 After having obtained, by a solution of the equations of the problem, the algebraic expressions for the quantities sought, it may be necessary, or, at least desirable, to construct their values, by which we mean, to draw a geometrical figure in which the parts represent the given and determined magnitudes, and have to each other the relations imposed by the conditions of the problem. This is called the construction of the expression. This branch of analytical geometry, which may bb termed Determinate Geometry, being of the least importance, relatively, will be omitted, after this reference, in the present treatise, and we shall pass at once to division. Second. That which has for its object to discover and discuss the general properties of geometrical magnitudes. In this the magnitudes are represented by equations expressing relations between constant quantities, and, either two or three indeterminate or variable quantities, and for this reason it is sometimes called Indeterminate Geometry. GENERAL PROPERTIES OF GEOMETRICAL MAGNITUDES, CHAPTER I. OF POSITIONS AND STRAIGHT LINES IN A PLANE,AND THE TRANSFORMATION OF CO-ORDINATES. DEFINITIONS. 1. Co-ordinate Axes are two straight lines drawn in a plane through any assumed point and making with each other any given angle. One of these lines is the axis of abscissas or the axis of X; the other is the axis of ordinates, or the axis of Y, and their intersection is the origin of coordinates. 2. Abscissas are distances estimated from the axis of Y on liiles parallel to the axis of X; ordinates are distances 9

Page  98 98 ANALYTICAL GEOMETRY. estimated from the axis of X on lines parallel to the axis of Y. 3. The abscissa and ordinate of a point together are called the co-ordinates of the point. 4. The co-ordinate axes are said to be rectangular mwhen they are at right angles to each other, otherwise they are oblique. 5. The two different directions in which distances may'be estimated from either axis, on lines parallel to the other, are distinguished by the signs plus and minus. 6. Abscissas are designated by the letter x and ordinates by the letter y, and when unaccented they are called general co-ordinates, because they refer to no particular one of the points under consideration. When particular points are to be considered the co-ordinates of one are denoted by x' and y'; of another by x" and y", etc., which are read x prime, y prime, x second, y second, etc. ILLvSTRATIoNs.-Through any point A draw the lines XX', YY' making withY each other any given angle. Call XX' the axis of abscissas and YY' the axis of ordinates. A is the origin of co-or- x-p A P dinates, or zero point. The four angular spaces into which the plane is divi- D — /;P ded are named, respectively, first, second, third, and fourth angles. YAX is the first angle, YAX' is the second angle, Y'AX' is the third angle, and Y'A' is the fourth angle. Take any point, as P, in the first angle, and from it draw Pp parallel to the axis of Y and Pp' parallel to the axis of X, the first meeting the axis of X at p, and the second the axis of Y at p'; then p'P=Ap is the abscissa, and pP=Ap' is the ordinate of the point P. Now produce Pp' to P' making p'P'=p'P, and from P' draw a parallel to the axis of Y meeting the axis of SX at p"; then the point P' is in the second angle, and p' P'

Page  99 GENERAL PROPERTIES. 99 =-Ap" is its abscissa, and p"P'=Ap' is the ordinate. By like constructions we determine the position of the point I' in the third angle, and that of the point P"' in the fourth angle. It is evident that the abscissas of these four points are numerically equal, as are likewise their ordinates; but if we have reference to the algebraic signs of the co-ordinates, each point will be assigned to its appropriate angle and will be completely distinguished from the others. Abscissas estimated to the right of the axis of Y are positive and those estimated to the left are negative. Ordinates estimated from the axis of X upwards are positive, those estimated downwards are negative. We shall therefore have for points In the 1st angle, x positive, y positive. " " 2d " x negative, y positive. 3d " x negative y negative. 4th " x positive y negative. From what precedes we see that the position of a point in the plane of the co-ordinate axis is fully determined by its co-ordinates. To construct this position we lay off on the axis of X the given abscissa, to the right, or to the left of the origin, according to the sign; also lay off on the axis of Y the given ordinate, upwards from the origin if the sign be plus, downwards if it be minus. The lines drawn through the points thus found, parallel to the coordinate axes, will intersect at the required point and fix its position. As rectangular co-ordinates are more readily apprehended than oblique, and as discussions and algebraic expressions are generally less complicated where references are made to the former, than when made to the latter, rectangular co-ordinates will be habitually employed in the following pages. When we have occasion to use others it will be so stated.

Page  100 100 ANALYTICAL GEOMETRY. PROPOSITION I. To find the equation of a straight line, Let XX', YY' be two rectangu- Y tar co-ordinate axes. A being the L" L origin draw any line as L'L through this point, and designate the natu- p pn ral taigenltof the angle LAXby a. X' p iP x Then take any distance on AX as AP, and represent it by x, and L', y the perpendicular distance PMy. Then by trigonometry we have Rad: tan. MAP:: AP: P3M or 1: a:: x: y Whence y=ax (1) Now this equation is general; that is, it applies to any point M on the line AL, because we can make x greater or less, and PM will be greater or less in like proportion and 1 will move along on the line AL as we move P on the line AXY. Because the point 3Mwill continue on the line AL through all changes of x and y, we say that y=ax is the equation of the line AL. Now let us diminish x to 0, and the equation reduces to y=O at the same time, which brings M1 to the point A. Let x pass the line YY', then AP' becomes-x, and the corresponding value of y will be P'M', and,being below the line X'X, will, therefore,beminus. Therefore y=ax. is the general equation of the line LL', extending indefinitely in either direction. If the tangent a becomes less, the line will incline more towards the line X'X. When a= O the line will coincide with YY'. Now let AP"' be +x, and a become -a, then P"'3M"' i;ll correspond to y, and becomes minus y, because it is

Page  101 STRAIGHT LIN ES. 101 below the axis XX'. Or, algebraically y=-ax, indicating some point MIi" below the horizontal axis. It is, therefore, obvious that y=ax may represent any line, as LL', passing through A. from the 1st into the 3d quadrant, and that y=-ax may be made to represent any line, as L"L"', passing through A from the 2d into the 4th quadrant. Therefore y=-4-ax may be made to represent any straight line passing throuyh the zero point. In case we have -a and -x, that is, both a and x minus at the same time, their product will be +ax, showing that y must be plus by the rules of algebra. As an exercise, let the learner examine these lines and see whether they correspond to the equation. When we have -a we must draw the line from A to the right and below AX; then XAL"' is the angle whose natural tangent is -a. But the opposite angle XIAL" is the same in value. When we have -x we must take the distance as AP' to the left of the axis YY', and the corresponding line P'M" is above XX', and therefore plus, as it ought to be. But the equation of a straight Y line passing through the zero point is not sufficiently general B/ for practical application; we will therefore suppose a line to pass L A _ P X in any direction across the axis Q YY', cutting it at the distance AB or AD (+b) or b distance above or below the zero point A, L' Y' and find its equation. Through the zero point A draw a line, AN, parallel to ML. Take any point on the line AX and through P draw 9*

Page  102 102 ANALYTICAL GEOMETRY. PM parallel to A Y, then ABMNwill be a parallelogram. Put AP= x. PM=y. The tangent of the angle NAP=a. Then will NP-=ax. To each of these equals add NM= b, then we shall have y-=ax+b for the relation between the values of x and y corresponding to the point M, and as M is any variable point on the line ML corresponding to the variations of x, this equation is said to be the equation of the line ML. When b is minus the line is then QL', and cuts the axis YY' in D, a point as far below A as B is above A. Hence we perceive that the equation y=- ax-+-b may represent the equation of any line in the plane YAX. If we give to a, x, and b, their proper signs, in each case of application we may write y=ax+b for the equation of any straight line in a plane. Cor. Since the equation y=ax+b truly expresses the relation between the co-ordinates of any point of the line, it follows that if the co-ordinates x' and y' of any particular point of the line be substituted for the variables x and y the equation must hold true; but if the co-ordinates x" and y", of any point out of the line be substituted for the variables, the equation cannot be true. What appears in the particular case of a straight line are general principles which we thus enunciate, viz: 1st. If the co-ordinates of a particular point, in any line whatever, be substituted for the variables in the equation of the line, the equation must be satisfied; but if the co-ordinates of a point out the line, be substituted for the variables in its equation, the equation cannot be satisfied. 2d. If the co-ordinates of anypoint be substituted for the variables in the equation of a line, and the equation be satisfied, the

Page  103 STRAIGHIT LINES. 103 point must be on the line; but if the equation be not satisfied by the substitution, the point cannot be on the line. These are principles of the highest importance in analytical geometry, and should be thoroughly committed and fully understood by the student. SCHoLIUM.-Instead of rectangular, let us as- y sume the oblique co-ordinate axes AX and AY, making with each other an angle denoted by m. Through the origin draw the line AP making with the axis of x the angle PAD-n; then the angle PAD'-m-n. Take any point as P in the line and from it draw PD' and PD parallel, respectively, - D X to the axes of Xand Y; From the triangle APD we have (Prop. 4, Sec. 1, Plane Trig.) PD: AD:: Sin. PAD=Sin. PAD' or y:x::Sin. n: Sin. (m —n.) Whence Y= sin. n sin. m —n sin n But i n s constant for the same line and may be represin. (m —n sented by a. Therefore, for any straight line passing through the origin of a system of oblique co-ordinate axes we have, as before, the equation y — ax. And if we denote by b the distance from the origin to the point at which a parallel line cuts the axis of Yabove or below the origin we shall also have for the equation of this line y —ax+ b, in which it must be remembered that a denotes the sine of the angle that the line makes with axis of x divided by the sine of the angle it makes with the axis of Y. To fix in the minds of learners a complete comprehension of the equation of a straight line, we give the following practical EXAMPLES. 1. Draw the line whose equation is y —2x+3. (1) Then draw the line represented by y= -x+2 (2) and determine where these two lines intersect.

Page  104 104 ANALYTICAL GEOMETRY. Draw YY' and XX' at right angles, and taking any convenient unit of measure lay it off on each of the axes from the origin in both positive and negative M 2 directions a sufficient number of times. Equation (1) is true for all values of L Q x and y. It is true then when x=0. 1 2 But when x_0 the point on the line 1 must be on the axis YPY. S When x=O. y=3. y' This shows that the line sought for must cut YY' at the point +3. The equation is equally true when y —0. But when y=O, the corresponding point on the line sought must be on the axis XX', and on the same supposition the equation becomes 0-z2x+3, Or x= —i. That is, midway between -1 and -2 is another point in the line which is represented by y=2x+-3, but two points in any right line must define the line; therefore ML is the line sought. Taking equation (2) and making x=O will give y=2, and making y= —0 will give x=2; therefore MQ must be the line whose equation is y —-x+2, and these two lines with the axis XX' form the triangle LM1Q, whose base is a3 and altitude about 2j. But let the equations decide, (not about,) but exactly the position of the point MI of intersection. This point being in both lines, the co-ordinates x and y corresponding to this point are the same, therefore we may subtract one equation from the other, and the result will be a true equation, giving 3x+1=0. Or x= ——. Eliminating x from the two equations we find y-2i. 2. For another example we 7 equire the projection of the line representecl by the equation Y — 2. 420 Making x —O, then y. —2. Making y=O, then x=-840. Using the last figure, we perceive that the line sought for must

Page  105 STRAIGHT LINES. 105 pass through S two units below the zero point, and it must take such a direction S V as to meet the axis XX7 at the distance of 840 units to the left of zero. Hence its absolute projection is practically impossible. 3. Coilstrutct the line whose equation is y=2x+5. 4. Construct the line whose equation is y=-3x-3. 5. Construct the line represented by 2y3x-+5. 6. Construct the line represented by y= —4x-3. The lines represented by equations 4 and 6 will intersect the axis of Y at the same point. Why? 7. Construct the line whose equation is y=2x+3. 8. Construct the line whose equation is y=2x —3. The last two lines intercept a portion of the axis of Ywhich is the base of an isosceles triangle of which the two lines are the sides. What are the base and perpendicular, and where the vertex of the triangle? ANS. The base is 6, the perpendicular 1i, vertex on the axis of X. Construct the lines represented by the following equations. 9. 3x+5y-15-0 10. 2x-6y+7 —0 11. x+y+2=0 12. — x+y+3=0 13. 2x-y4-=0 PROPOSITION II To find the distance between two given points in the plane of the co-ordinate axis. Also, tofind the angle made by the line joining the two given points, and the axis of X. Let the two given points be P Y and Q, and because the point P is said to be given, we know the two distances Q AN=x', NP=y'. r And because the point Q is given we know the two distances. AM-x" and MQ=y". A N M X

Page  106 106 ANALYTICAL GE OMETRY. Then, AM-AN=NM=PR=x" —x'; and MQ -MR= QR=y"-y'. In the right angled triangle PRQ we have (PR)2+( RQ)2=(PQ)2. But D=PQ. That is D2=(x1"-x')2+ (yl"-y)2, Or D-V (x" -x' )2+(y"-y/) Thus we discover that the distance between any two given points is equal to the square root of the sum of the squares of the differences of their abscissas and ordinates. If one of these points be the origin or zero point, then x'=0 and y'=O, and we have D-='(X")2+ (y")2, a result obviously true. Tofind the angle between PQ and AX. PR is drawn parallel to AX, therefore the angle sought is the same in value as the angle QPR. Designate the tangent of this angle by a, then by trigonometry we have PR.: RQ:: radius tan. QPR. That is, x"-x': y"-y':: 1: a. Whence a -- xt In case y"=y', PQ will coincide with PR, and be parallel to AX, and the tangent of the angle will then be 0, and this is shown by the equation, for then 0 In case x"-x', then PQ will coincide with RQ and be parallel to A Y, and tangent a will be infinite, and this too the equation shows, for if we make x"-x' or x"-x' -0, the equation will become 0_-y'_o

Page  107 STRAIGHT LINES. 107 PROPOSITION III. To find the equation of a line drawn through any given point. Let P be the given point: The equation must be in the form y==ax+b (1) Because the line must pass through the given point whose co-ordinates are x' and y', we must have y'=ax/+b. (2) Subtracting equation (2) from equation (1) member from member, we have y —'=a(x-x') (3) for the equation sought. In this equation a is indeterminate, as it ought to be, because an infinite number of straight lines can be drawn through the point P. We may give to y' and x' their numerical values, and give any value whatever to a, then we can construct a particular line that will run through the given point P. Suppose x'=2, y'=3, and make a=4. Then the equation will become y-3=4(x,-2). Or y=4x-5. This equation is obviously that of a straight line, hence equation (3) is of the required form. PROPOSITION IV. To find the equation of a line which passes through two given points. Let AX and A Y be the co-ordinate axes, and P and Q the given points. Denote the co-ordinates of P by x', y' and of Q by x", y". The required equation must be of the form y=ax+b (1)

Page  108 108 ANALYTICAL GEOMETRY. -We will now determine such Y values for a and b as will cause the ]Ene represented by this equation to pass through the given points. As the line is to pass through the point P, the co-ordinates x', y' of this point when substituted N for the variables x, y must satisfy A x the equation, and we shall have y'-ax'+b (2) And because the line is to pass through the point Q, whose co-ordinates are xy" we will also have y"=ax"+b (3) Subtracting eq. (2) from eq. (3) member from member, we get y" —y'=a (x"-x') Whence a-Y"-y (4) From eqs. (1) and (2) we obtain in like manner y-y'=a(x-x') (5) Substituting for a in eq. (5) its value in eq. (4) we find Y-Y Y. (x-X') (6) for the equation sought. If we subtract eq. (3) from eq. (1) member from member, and substitute for a in the resulting equation its value in eq. (4) we find Y-Y-y _ (x-x") (7) for the required equation. By simply clearing eqs. (6) and (7) of fractions and reducing, it may be shown that they are in fact but different forms of the same equation. To prove that either of these equations is that of a line passing through the points P and Q, we have but to sub

Page  109 STRAIGHT LINES. 109 stitute -in it, for x and y, the co-ordinates of these points. It will be found that when these substitutions are made for either point, the equation will be satisfied. We will illustrate the use of these equations by the following EXAMPLES. 1. The co-ordinates of P are x'-3, y'- 4, and of Q, x"=-1, y"=3. What is the equation of the line that passes through these points? Here a=_y" —y 3x"-X' -1- 3 And the equation y-y (x —x-') becomes y -4=(x 3) or y= x+3} By substituting in the equation y —y"-y —' (xwe get y-3= —(x+1) or y= x+3, the same as that above. 2. Find the equation of the straight line that is determined by the points whose co-ordinates are x' -4, y'= -1 and x"=42, y"- l4 Ans. y=- 4 X-11-6. 3. The co-ordinates of one point are x'-6, y'-5, and of another they are x"=-3, y"=3. What is the equation of the straight line that passes through these points? Ans. y=-x+32. PROPOSITION V. To find the equation of a straight line which shall pass through a given point and make, with a givwn litne, a giren angle. The equation of the given line must be in the form Y=ax+b. (1) 10

Page  110 110 ANALYTICAL GEOMETRY. Because the other line must pass through a given point its equation must be (Prop. III.) y —y'=a'(x-x'). (2) We have now to determine the value of a'. When a and a' are equal, the two lines must be parallel, and the inclination of the two lines will be greater or less according to the relative values of a and a'. Let PQ be the given line, y making with the axis of Xan angle whose tangent is a and PR the other line which shall pass through the given point P and make with PQ, a given an- Q/ R x gle QPR. The tangent of the / A angle PRX is equal to a'. Because PRX=PQR+ QPR. QPR=PRX-PQR Tan. QPR=tan. (PRX-PQR.) As the angle QPR is supposed to be known or given, we may designate its tangent by m, and m is a known quantity. Now by trigonometry we have m=tan. PRB)-PQR)=a1-a, (3) 1+aa' Whence a'= a+m 1 -ma This value of a' put in eq. (2) gives (?= (a+m (xx) (4) Y-Y - \1-mal' () for the equation sought. Cor. 1. When the given inclination is 900, m its tangent is infinite, and then a'= —. We decide this in the a following manner. An infinite quantity cannot be increased or diminished

Page  111 STRAIGHT LINES. 111 relatively, by the addition or subtraction of finite quantities, therefore, on that supposition, a+m becomes m or 1-ma -ma a APPLICATION.-TO make sure that we comprehend this proposition and its resulting equation, we give the following example: The equation of a given line is y=2x+6. Draw another line that will in- 10Y\/M tersect this at an angle of 45~ and pass through a given point P, whose co-ordinates are x'=31,y'=2. Draw the line MN correspond- P ing to the equation y=2x+6. Locate the point P from its given co- I /N \ I x ordinates. Because the angle of intersection is to be 450, m=l, and a=2. Substituting these values in eq. (4) we have Or y=-3x+1242. Constructing the line MR corresponding to this equa. tion, we perceive it must pass through P and make the angle NMR 45~, as was required. The teacher can propose any number of like examples. Cor. Equation (3) gives the tangent of the angle of the inclination of any two lines which make with the axis of X angles whose tangents are a and a'. That is, we have in general terms al'-a l+aa' In case the two lines are parallel m=-0. Whence a'=a, an obvious result.

Page  112 112 ANALYTICAL GEOMETRY. In case the two lines are perpendicular to each other, m must be infinite, and therefore we must put 1+aa'= O to correspond with this hypothesis, and this gives _. 1 a' a a result found in Cor. 1. To show the practical value of this equation we require the angle of inclination of the two lines represented by the equations y=3x-6, and y=-x+2. Here a —3 and a'=- 1. Whence m =2. 1-3 This is the natural tangent of the angle sought, and if we have not a table of natural tanfgents at hand, we will take the log. of the number and add 10 to the index, then we shall have in the present example 10.301030 for the log. tangent which corresponds to 630 26' 6" nearly. The minus sign merely indicates the position of the angle; it is below the angular point. 2. What is the inclination of the two lines whose equation are 2y=5x+8 and 3y=-2x+6? Ans. The tangent of their inclination is 4j Log. 4.75 plus 10=10.676694. The inclination of the lines is therefore 780 6' 5". 3. Find the equation of a line which will make an angle of 56~ in the line whose equation is 2y=56x+4. As the required line is to pass through no particular point, but is merely to make a given angle with the known line, we may assume it to pass through the origin of co-ordinates. Its equation will then be of the form

Page  113 STRAIGHT LINES. 113 y=a'x. We must now determine such a value for a' that the two lines will make with each other an angle of 56~. Represent the tangent of the given angle by t; then by corollary (2) a'5 t=al+-5a' In the tables we find that log. tangent of 56~ to be 10. 171013, from which subtracting 10 to reduce it to the log. of the natural tangent and we have 0.171013 for the log. of 1. The number corresponding to this is 1.483. a 5 Whence a - =1.483 1+2-a' From which we find a'=-1,473 nearly and the equation of the line making with the given line, an angle of 56~ is therefore y=-1.473x. PROPOSITION VI. To find the co-ordinates which will locate the point of intersection of two straight lines given by their equations. We have already done this in a particular example in Prop. I, and now we propose to deduce general expressions for the same thing. Let y=ax+b be the first line. And y=a'x+b' be the second line. For their point of intersection y and x in one equation will become the same as in the other. Therefore we may subtract one equation from the other, and the result will be a true equation. For the sake of perspicuity, let x, and y, represent the co-ordinates of the common point, then by subtraction (a-a')x +b —b=O Whence x = (b-b') and Y, a=b-ab (a-a') a' —a 10'*

Page  114 114 ANALYTICAL GEOMETRY. EXAMPLE. At what point will the lines represented by the two equations y= —2x+1 and y=-5x+10 intersect each other. Here a= —2, a'=5, b=1, b'=10. Whence x= —, y34. If we take another line not parallel to either of these, the three will form a triangle. Then if we locate the three points of intersection and join them, we shall have the triangle. PROPOSITION VII. To draw a perpendicular from a given point to a given straight line and to find its length. Let y-=ax+b be the equation of the given straight line, and x', y' the co-ordinates of the given point. The equation of the line which passes through the given point must take the form y-y' =a' (x-x'). (Prop. 3.) And as this must be perpendicular to the given line, we must have a' —-. Therefore the equations for the a two lines must be y=ax+b for the given line; (1) and yy' y —1 ( x —'); Or y —x+(-+y') for the perpendicular line (2) a a Let x, and y, represent the co-ordinates of the point of intersection of these two lines. Then by Prop. 6,

Page  115 STRAIGHT LINES. 115 xi- 1. and y = + x aandj aOrxl( —-(ab-x-ZaY')) and y,- b+ax'+a2y' \21 a/+l Or we may conceive x and y to represent the co-ordinates of the point of intersection, and eliminating y from eqs. (1) and (2) we shall find x as above, and afterwards we can eliminate y. Now the length of the perpendicular is represented by j(x,-x')'"+(y1, —y)2 =-D. (Prop. II.) Whence -abb -ay —-a2x' 2 (b+ax' —-Y')2 \ a2+1 / + a2+1 the perpendicular. If we put u=b+ax'-y', the quantities under the radical will become a2u+2 U t(a.2 + -1)2 U 4(a2+1)2 (a2+1)2 4(a2+1)2 Ua2-1 Whence the perpendicular = ~b+ ax.-y,%/a~+ EXAMPLES. 1. The equation of a given line is y=3x-10, and the co-ordinates of a given point are x'-4 and y'=5. What is the length of the perpendicular from this given point to the given straight line? Ans. -1~/90. 2. The equation of a line is y=-5x-15, and the coordinates of a given point are x'=4 and y'=-5. What is the length of the perpendicular from the given point to the straight line? Ans. 7.84+.

Page  116 116 ANALYTICAL GEOMETRY. PROPOSITION VIII. To find the equation of a straight line which will bisect the angle contained by two other straight lines. Let y-=ax+b (1) and y=a'x+b' (2) be the equations of two straight lines which intersect; the co-ordinates of the point of intersection are X1=(b-b) 1alb-ab (Prop. VI. We now require a third line which shall pass through the same point of intersection and form such an angle with the axis of Z (the tangent of which may be represented by m) that this line will bisect the angle included between the other two lines. Whence by (Prop. V.) the equation of the line sought must be y-y,1=m(x-x) (3) in which we are to find the value of m. Let PNrepresent the line cor- r M responding to equation (1) PMthe R line whose equation is (2), and PR N the line required. Now the position or inclination of PN to AX depends entirely on the value of a, and the inclination of PM depends on a' and both are A independent of the position of the point P. Now RPNBRPX' —NP.X and MPR —IMP=PX-RPX'. Whence by the application of a well known equation in plane trigonometry, (Equation (29), p. 253 Plane Trig.) we have tan. RPN=tan. (RP' —NRXP')= mr-a 1+am And tan. MPR=tan. (MPX —RPX — a -m 1+alm

Page  117 STRAIGHT LINES. 117 But by hypothesis these two angles RPN and MPR are to be equal to each other. Therefore mn-a a'-m 1+am-l+a'm Whence m2+ 2(1-aa') m=1. (4) a'+a This equation will give two values of m; one will correspond to the line PR, and the other to a line at right angles to PR. If the proper value m be taken from this equation and put in eq. (3), we shall have the equation required. Practically we had better let the equations stand as they are, and substitute the values of a, a' x, and y, corresponding to any particular case. To illustrate the foregoing proposition we propose the following EXAMPLES. Two lines intersect each other: y=-2x+5 is the equation of one line. (1) y=4x+6 is that of the other line. (2) Find the equation of the line which bisects the angle contained by these two lines: Here a=-2, a'=4, b=5, b'=6. Whence x =-A, and y,1= 6. Thus (3) becomes Y-16 6m(x+ ). And eq. (4) becomes m2+9m —1. Whence m=-0.1097 or m —9.1097. y-1 =O.1097(x+*). Or y — 6 = —9.1097(x+ A).

Page  118 118 ANALYTICAL GEOMETRY. Equation (4) is that of the line required; (3) that of the line at right angles to the line required. All will be obvious if we construct the lines represented by the eqs. (1), (2), (3), and (4). For another example, find the equation of a line which bisects the angle contained by the two lines whose equations are y=x+12, y=-20x+2. Here a-l, a'=-20. Whence (4) becomes m2 —~m-=1. Therefore m —0.385, or +2.6. NOTE. —Two straight lines whose equations are y=ax+b and y'-a+b' will always intersect at a point (unless a-a') and with the axis of Y form a triangle. The area of such triangle is expressed by (a —ax (2) From the given equations we find the co-ordinates of the intersection of the lines to be x1 =-_!Y 24 For the line bisecting the angle included between the given lines we have either y-_242 =-0.385(x+ ~-) (1) or, y —242?=2.6(x+ l o) (2) By transposition and reduction (1) becomes y=-0.385x+11.75 (3) and (2) becomes y-=2.6x+12.76 (4) The lines represented by eqs. (3) and (4) are at right angles to each other. The latter line bisects the angle included between the given lines, and the former the adjacent or supplemental angle. 3. From the intersection of two lines whose equations are

Page  119 STRAIGHT LINES. 119 3y+5x=4 (1) and 2y-=3x + 4 (2) A third line is drawn making, with the axis of X, an angle of 30~. Find the intersection of the given lines and the equation of the third line. ( The co-ordinates of the points of intersection Ans. are xl = —-4, Y1 —, and the required equation is y —— =0,5773(x+ 4 ). 4. Two lines are represented by the equations 2y-3x= —1 and 2y+3x=3 What kind of a triangle do these lines form with the intercepted portion of the axis of Y, and what are its sides and its area? Ans. The triangle is isosceles; its base on the axis Ansi. of Y is 2, the other sides are each 1.201+, and its area 0.66+. 5. Two lines are given by the equations -2-y+ 3yx= —2 and 2{y-2-x=4 Required the equation of the line drawn from the point whose co-ordinates are x"=3, y", =0 to the intersection of the given lines and the distance between these two points. Ans. { The equation sought is y= —0.717x+2.1523 and the distance is'/(1.8)2+(2.52)2. TRANSFORMATION OF CO-ORDINATES. It is often desirable to change the reference of points from one system of co-ordinate axes to another differing from the first either in respect to the origin or the direction of the axes, or both. The operation by which this is done is called the transformation of co-ordinates. The

Page  120 120 ANALYTICAL GEOMET RY. system of co-ordinate axes from which we pass is the primitive system and that to which we pass is the new system. Let AX and A Y be the primitive axes. Take any point, as A', the co-ordinates of which referred to AX and A Y are x=a, y=b and through it draw the new axes A. A__ X A'X', and A' Y' parallel to the primative axes. Then denoting. X the co-ordinates of any point, as M, referred to the primitive axes by x and y, and the coordinates of the same point referred to the new axes by x' and y', it is apparent that x-a+x' y=b+y' By giving to a and b suitable signs and values we may place the new origin at any point in the plane of the primitive axes and the above formulas are those for passing from one system of axes to a system of parallel axes having a different origin. The formulas for the transformation of co-ordinates must express the values of the primitive co-ordinates of points in terms of the new co-ordinates and those quantities which fix the position of the new in respect to the primitive axes. PROPOSITION IX. To find the formulas for passing from a system of rectangular to a system of oblique co-ordinates from a diferent origin. Let AX, A Y be the primitive axes and A'X7, A' Y' the new axes. Through any point as M draw MP' parallel to A' Y' and MP perpendicular to A'X. Then A'P' is the new abscissa, P'M the new ordinate of the point M, and AP and PM are respectively the primitive abscissa and ordinate of the same point.

Page  121 STRAIGHT LINES. Let AB=a, BA'=b, AP=x, Y Y' P]J=y, A'P'-x', P'M-y' the an- gle X'A''X"=m, and the angle Y'A'X"=n. Now by trigonome- H try we have. A'K=x'cos.m KP'= LH-x' sin. n P'H=KL-y' cos. n. B p -X And MB=-Iy' sin. n. Whence x=-x' cos.m+y' cos.n, y=b+x' sin. m+y' sin.n, the fbrmulas required. SCHOLIUM. —In case the two systems have the same origin, we merely suppress a and b, and then the formulas required are x=x' cos. mq-y' cos. n y=x' sin. m+y' sin. n. PROPOSITION X. To find the formulas for passing from a system of oblique coordinates to a system of rectangular co-ordinates, the origin being the same. Take the formulas of the last problem x=x' cos. m+y' cos. n, y-x' sin. m+y' sin. n. We now regard the oblique as the primitive axes, and require the corresponding values on the rectangular axes. That is, we require the values of x' and y'. If we multiply the first by sin. n, and the second by cos. n, and subtract their products, y' will be eliminated, and if x' be eliminated in a similar manner, we shall obtain x sin. n-y cos. n y_ cos. m-x sin n sin. (n —m) sin.(n-m) SCHOLIUM.-If the zero point be changed at the same time in reference to the oblique system, we shall have x'=a+ x sin. n-cos. n y' b y cos.m —x sin. m sin. (a —m) sin.(r,-m We will close this subject by the following 11

Page  122 122 ANALYTICAL GEOMETRY. EXAMPLE. The equation of a line referred to rectangular co-ordinates is y-a'x+Vb. Change it to a system of oblique co-ordinates having the same zero point. Substituting for x and y their values as above, we have x' sin. m+y' sin. n-a'(x cos. m+y' cos. n)+b'. Reducing,_(a' cos. m-sin. m')x'+ sin. n-a' cos. m sin. n-a' cos. m POLAR CO-ORDINATES. There are other methods by which the relative positions of points in a plane may be analytically established than that of referring them to two rectilinear axes intersecting each other under a given angle. For example, suppose the line Y' y AB to revolve in a plane about the point A. If the angle that this line makes with a fixed line / H passing through A be known, and also the length of AB, it is evident A that the extremity B of this line will be determined, and that there iX X is no point whatever in the plane the position of which may not be assigned by giving to AB and the angle which it makes in the fixed line appropriate values. The variable distance AB is called the radius vector, the angle thatitmakes with the fixed linethe variable angle and the point A about which the radius vector turns, the pole. The radius vector and the variable angle together constitute a system of polar co-ordinates.

Page  123 STRAIGHT LINES. 123 Denote variable angle BAD by v, the radius vector by r and by x and y, the co-ordinates of B referred to the rectangular axes AX, A Y; then by trigonometry we have x=r cos. v and y=r sin. v. Now from the first of these we have r- (v may reCOS. v volve all the way round the pole), and as x and cos. v are both positive and both negative at the same time, that is, both positive in the first and fourth quadrants, and both negative in the second and third quadrants, therefore r will always be positive. Consequently, should a negative radius appear in any equation, we must infer that some incompatible conditions have been admitted into the equation. PROPOSITION XI. To find theformulasfor changing the reference of pointsfrom a system of rectangular co-ordinate axes to a system of polar co-ordinates. Let A'X, A' Y be the co- Y ordinate axes, A the pole AB the radius vector of any point, B and AD parallel to A'X the H fixed line from which the variable angleis estimated. De- A b D note the co-ordinates A'E, AE of the pole by a and b and.4 E C X the radius vector AB by r. Draw BC perpendicular to A'X; then is A' C=x the abscissa, and B C=y the ordinate of the point B. From the figure we have A' C-A'E+EC=A'E+AF=A'E+AB cos.v and BC=BF+FC=BF~+AE=AE+A AB sin. v

Page  124 124 AN ALYTICAL GEOMETRY. Whence x-a+r cos. v y=b+r sin. v. SCHOLIUM.-If instead of estimating the variable angle from the line AD, which is parallel to the axis A'X, we estimate it from the line AH which makes with the axis the given angle HAD=m we shall have x=a+r cos. (v+m) y=b+r sin. (x-+m) CHAPTER II. THE CIRCLE. LINES OF THE SECOND ORDER. Straight lines can be represented by equations of the first degree, and they are therefore called lines of the first order. The circumference of a circle, and all the conic sections, are lines of the second order, because the equations which represent them are of the second degree. PROPOSITION I. To find the equation of a circle. yt y Let the origin be the center of \ the circle. Draw AM to any point in the circumference, and let X A / x fall MP perpendicular to the axis ofX. Put AP=-x, PM-=y and AM-R. Then the right angled triangle APM gives 2 +y2-2 (1) and this is the equation of the circle when the zero point is the center.

Page  125 THE CIRCLE. 125 When y=O, xZ-R2, or ix=R, that is, P is at X or A'. When x=O, y2=R2, or ~y=-R, showing that M on the circumference is then at Y or Y". When x is positive, then P is on the right of the axis of Y, and when negative, P is on the left of that axis, or between A and A'. When we make radius unity, as we often do in trigonometry, then x2+yz-=l, and then giving to x or y any value plus or minus within the limit of unity, the equation will give us the corresponding value of the other letter. In trigonometry y is called the sine of the are XM, and x its cosine. Hence in trigonometry we have sin.2+cos.2=l. Now if we remove the origin to A' and call the distance A'P —x, the AP=x-R, and the triangle APM gives (x — R)2+y21=R2. Whence y2= 2RBxx2. This is the equation of the circle, when the origin is on the circumference. When x-O,y=O at the same time. When x is greater than 2R, y becomes imaginary, showing that such an hypothesis is inconsistent with the existence of a point in the eircumference of the circle. There is still a more general equation of the circle when the zero point is neither at the center nor in the circumference. The figure will fully illustrate. Let AB=c, BC=b. Put AP y =x, or AP'=x, and PM or P/ M M"'-y, CM, CM', &c. each=-R. In the circle we observefour I D equal right angled triangles. The numerical expression is the same for each. Signs only indicate positions. A PE B i X 11*

Page  126 126 ANALYTICAL GEOMETRY. Now in case CDM is the triangle wefix upon, We put AP=x, then BP= CD-=(x —c), PM-:y, MD=y- CB=(y — b). Whence (x-c)2+(y-b)2=.2 (1) In case CDM' is the triangle, we put AP=x and PM' =Y. Then (x-c)2+(b-y)2=_Rs (2) In case CD'IM"' is the triangle, we put AP'=x, P'M"' =Y. Then (c-x)'+ (y-b)2= R (3) If CD'IM" is the triangle, we put P'M"=y. Then (C-x)2+b y)2= —_ (4) Equations (1), (2), (3), and (4), are in all respects numerically the same, for (c-x)2 (x —c)2, and (b-y)2=(y-b)2. Hence we may take equation (1) to represent the general equation of the circle referred to rectangular co-ordinates. The equation (x —c)2+(y-b)2=-R2 (1) includes all the others by attributing proper values and signs to c and b. If we suppose both c and b equal 0, it transfers the zero point to the center of the circle, and the equation becomes x2+y2=_t2 To find where the circle cuts the axis of X we must make y=0. This reduces the general equation (1) to (x-c)2+ b2-=2. Or (x-_)2=B2_-b2. Now if b is numerically greater than B, the first member being a square, (and therefore positive,) must be equal to a negative quantity, which is impossible,-showir'g that in that case the circle does not meet or cut the axis of X, and this is obvious from the figure. In case b=B, then (x-c)2-0, or x-=c, showing that the

Page  127 THfE CIRCLE. 127 circle would then touch the axis of X. If we make x=O, eq. (1) becomes c2+ (y-b)2= R2. Or (y-b)2=_R2_C2. This equation shows that if c is greater than R, the circle does not cut the axis of Y, and this is also obvious from the figure. If c be less than B, the second member is positive in value, and y=bt::/R2'c_2, showing that if the circumference cut the axis at all, it must be in two points, as at M", M"'. PROPOSITION II. The supplementary chords in the circle are perpendicular to each other. DEFINITION.-Two lines drawn, one through each extremity of any diameter of a curve, and which intersect the curve in the same point, are called supplementary chords. That is, the chord of an arc, and the chord of its supplement. In common geometry this proposition is enunciated thus: All angles in a semi-circle are right angles. The equation of a straight y line which will pass through D the given point B, must be of the form (Prop. III. Chap. I.) F y —y'=a(x-x'). (1) a The equation of a straight B A A' line which will pass through the given point X, must l:e of the form Y-y'aI='(x-x'). (2)

Page  128 128 ANALYTICAL GEOMETRY. -At the point B, y'-O, and x'= —R, or — x'=R. Therefore eq. (1) becomes y=a(x+R?). (3) And for like reason eq. (2) becomes y=a'(x-R). (4) For the point ill whlice: these lineAs intersect x and y in eq. (3) are the same as x and y ill eq. (4); hence, these equations mlay be multiplied together under this supposition, and the result will be a true equation. That is, y-"=aa'(x2-R2). (5) But as the point of intersection must be on the curve, by hypothesis, therefore, x and y must conform to the following equation: y2+x2=R2. Or y2=-1(x2-R2). (6) Whence aa'= —1, or aa'+1+O. This last equation shows that the two lines are perpendicular to each other, as proved by (Cor. 2, Prop. 5., Chap. 1.) Because a and a' are indeterminate, we conclude that an infinite number of supplemental chords may be drawn in the semi-circle, which is obviously true. PROPOSTION III. To find the equation of a line tangent to the circumference of a circle at a given point. Let C be the center of the circle, P the point of tangency, and.....Q Q a point assumed at pleasure in r — the circumference. Denote the co-ordinates of P by x', y', and those of Q by x, y. C The equation of a line passing through two points whose co-or

Page  129 THE CIRCLE. 129 dinates are x', y' and x", y" is of the form (Prop. 4, Chap. 1). YIY- -xY, (x-x"). (1) We are to introduce in this equation, first, the condition that the points P and Q are in the circumference of the circle, which will make the line a secant line, and then the further condition that the point Q shall coincide with the point P, which will cause the secant line to become the required tangent line. Because the points P and Q are in the circumference of the circle, we have x'2+y'2=_X and 2x"+y"2.=2 Whence by subtraction and factoring, (X'+) (x'-x")+(Y'+Y) +(y'-y (2) from which we find y'-y" x'+e' X'_x"- - -y'-y" This value of "Y-,substituted in equation (1) gives us for the equation of the secant line, y y xf <(x x'X).(3) Now, if we suppose this line to turn about the point P until Q unites with P, we shall have x"=x' and y"=y', and the secant line will become a tangent to the circumference at the point P. Under this supposition eq. (3) becomes y-y'=-_, (x —x, (4) in which y- is the value of'the tangent of the angle which the tangent line makes with axis of X. I

Page  130 130 ANAAL GEOMETRY. By clearing this equation of fractions, and substituting for x'2+y'2 its value, _R2, we have finally for the equation of the tangent line, yy'+xx'-=2. (5) This is the general equation of a tangent line; x',y', are the co-ordinates of the tangent point, and x, y, the co-ordinates of any other point in the line. SCHOLIUM 1.-For the point in which Q the tangent line cuts the axis of X, we make y'= 0, then R2 X=_= AT. For the point in which it meets the A axis of Y, we make x'= ~ and y=_,=AQ. y:I SCHOLIUM 2.-A line is said to be normal to a curve when it is perpendicular to the tangent line at the point of contact. Join A, P, and if APT is a right angle, then AP is a normal, and AB, a portion of the axis of X under it, is called the subnormal. The line BT under the tangent is called the subtangent. Let us now discover whether APT is or is not a right angle. Put a'= the tangent of the angle PA T, then by trigonometry a'. XI But a=-_. Eq. (6) Whence aa'- 1. Or a'a Therefore AP is at right angles to PT. (Prop. 5. Chap. 1.) That is, a tangent line to the circumference of a circle at any point is perpendicular to the radius drawn to thatpoint. SCHOLIUM 3.-Admitting the principle, which is a well-known truth of elementary geometry, demonstrated in the preceding scholium, we would not, in getting the equation of a tangent line to the

Page  131 THE CIRCLE. 181 circle, draw a line cutting the curve in two points, but would draw the tangent line PT at once, and admit that the angle APT was a right angle. Then it is clear that the angle APB= the angle PTB. Now to find the equation of the line, we let x' and y' represent the co-ordinates A B of the point P, and x and y the general co-ordinates of the line, and a the tangent of its angle with the axis of X, then (by Prop III, Chap. I,) we have y — y- a(x'-x). Now the triangle APEB gives us the following expression for the tangent of the angle APB, or its equal PTB, a= — Iy' This value of a put in the preceding equation, will give us y' —y=-(x'- x). Or yI2_yy_=_ x + xSx Whence yy'+xx'=R2,the same as before. PROPOSITION IV. To.find the equation of a line tangent to the circumference of a circle, which shall pass through a given point without the circle. Let H (see last figure to the preceding proposition) be the given point, and x" and y" its co-ordinates, and x' and y' the co-ordinates of the point of tangency P. The equation of the line passing through the two points HE and P must be of the form y-y"=-a(x-x") (1) in which a- YY Since PH is supposed to be tangent at the point P,

Page  132 132 ANALYTICAL GEOMETRY. and x' and y' are the co-ordinates of this point, equation (6) Prop. 3, gives us xi a=-. Placing this value of a in equation (1) we have y- y"= —(x-x') for the equation sought. This equation combined with X'2 +y 12= which fixes the point P on the circumference will determine the values of x' and y', and as there will -be two real values for each, it shows that two tangents can be drawn from H, or from any point without the circle, which is obviously true. SCHOLIUM. We can find the value of the tangent PT by means of the similar triangles ABP, PBT, which give': R:: y': PT. PT=RY. More general and elegant formulas, applicable to all the conic sections, will be found in the calculus for the normals, subnormals, tangents and subtangents. OF THE POLAR EQUATION OF THE CIRCLE. The polar equation of a curve is the equation of the curve expressed in terms of polar co-ordinates. The variable distance from the pole to any point in the curve is called the radius vector, and the angle which the radius vector makes with a given straight line is called the variable angle.

Page  133 THE CIRCLE. 188 PROPOSITION V. To find the polar equation of the circle. When the center is the pole or the fixed point, the equation is r2=X2+y2=R2 (1) and the radius vector B is then constant. Now let P be the pole, and the co-ordinates of that point referred to the center and rectangular axes / XI be a and b. Make P2M=r, and tIfPX'=v the variable angle; AN -x and N21-=y. Then (Prop. 11, Chap. 1.) we have x=a+r cos. v, and y=b+r sin v. These values of x and y substituted in eq. (1), (observing that cos.2v+sin.2v-=1,) will give r2+2(a cos. v+b sin. v)r+a2+b2 —i=O which is the polar equation sought. SConLIUM 1.-P may be at any point on the plane. Suppose it at B'. Then a < -R and b=O0. Substituting these values in the equation, and it reduces to r' 2Rrcos. v=O. As there is no absolute term, r=O will satisfy the equation and correspond to one point in the curve, and this is true, as P is supposed to be in the curve. Dividing by r, and r=2R cos. v. This value of r will be positive when cos. v. is positive, and negative when cos. v is negative; but r being a radius vector can never be negative, and the figure shows this, as r never passes to the left of B, but runs into zero at that point. When v=0, cos. v=l, then r=BB'. When v-=90, cos. v=0, and r becomes 0 at B', and the variations of v from 0 to 90, determine all the points in the semi-circumference BDB'. 12

Page  134 134 ANALYTICAL GEOMETRY. SCHOLIUM 2.-If the pole be placed at B, then a=+R and b=O, which reduces the general equation to r= -2R cos. v. Here it is necessary that cos. v should be negative to make r positive, therefore v must commence at 900 and vary to 270~; that is, be on the left of the axis of Y drawn through B, and this corresponds with the figure. APPLICATION. The polar equation of the circle in its most general form is r'+2(a cos. v+b sin v)r+a'+b'=R. (1) If we make b=O, it puts the polar point somewhere on the axis of X, and reduces the equation to r'+2a cos. v.r+a'=R'. (2) Now if we make v=0, then will cos. M v-1, and the lines represented by ~r would refer to the points X, X', in the circle. A This hypothesis reduces the last equa- tion to r'+2ar= (R2-a') (3) and this equation is the same in form as the common quadratic ins algebra, or in the same form as x' ~px=_q. Whence x=r, 2a= ~p, and R' —a'q= a= ~ ip, R=/V+a / =Vq+ a-'. These results show us that if we describe a circle with the radius V/q+ p', and place P on the axis of X at a distance from the center equal to to 4p, then PX represents one value of x, and PX7 the other. That is, x=- p +Vg+iP2=PX. Or x= - p —_z -/+ Ip2= Pa, and this is the common solution. When p is negative, the polar point is laid off to the left from the center at P'. The operation refers to the right angled triangle APM.

Page  135 THE CIRCLE. 135 AP=ip, PM= q, and AM4I+ip'. Let the form of the quadratic be X2 ~px= —q. Then comparing this with the polar equation of the circle, we have 2a=i~p. R2'-a=-q. x=~+p. R=~ 4p~2-q. Take AX=R and describe a semi- y circle. Take AP=ip and AP'= — ip. From P and J' draw the lines PM, and P'M7 to touch the circle; and draw AM, AM. Here AP is the hypotenuse of a P X' A X P right angled triangle. In the first case AP was a side. In this figure as in the other, PM= 4/g; but here it is inclined to the axis of X; in the first figure it was perpendicular to it. The figure thus drawn, we have PX for one value of x, and PX' is the other, which may be determined geometrically. If _ x +px= — x=-ip+ 41p' —q=PX, or x= —ip- /ip —g=PX'. Observe that the first part of the value of x, is minus, corresponding to a position from P to the left. If x'-px= —-q, we take P' for one extremity of the line x. x=Ip+ /p2- q=P'X, or x=@- /- p'2-g= P'X. Here the first part of the value of x, (ip), is plus, because it is laid off to the right of the point P. Because R= -/~p — g R or AM becomes less and less as the numerical value of q approaches the value of p'2. When these two are equal, R=O, and the circle becomes a point. When q is greater than ip', the circle has more than vanished, giving no real existence to any of these lines, and the values of x are said to be imaginary. We have found another method of geometrizing quadratic equations, which we consider well worthy of notice, although it is of but little practical utility.

Page  136 136 ANALYTICAL GEOMETRY. It will be remembered that the equation of a straight line passing through the origin of co-ordinates is y=ax, (1) and that the general equation of the circle is (XTc)2+ (yJ=b)2= R2. (2) If we make b= 0, the center of the circle must be somewhere on the axis of X. Let AM represent a line, the M equation of which is y=ax, and if we take a=1, AM will incline 45~ from either axis, as rep- p A PI B X resented in the figure. Hence y=x, and making b=O, if these / MA two values be substituted in eq. (2) and that equation reduced, we shall find Y2=cy= C (3) This equation has the common quadratic form. Equation (1) responds to any point in the straight line NMM. Equation (2) responds to any point in the circumference BMXM. Therefore equation (3) which results from the combination of eqs. (1) and i 2), must respond to the points M and 2M', the points in which the circle cuts the line. That is, PM and'P'M' are the two roots of equation (3), and when one is above the axis of X, as in this figure, it is the positive root, and P'M' being below the axis of X, it is the negative root. When both roots of equation (3) are positive, the circle will cut the line in two points above the axis of X. When the two roots are minus, the circle will cut the line in two points below the axis of X. When the two roots of any equation in the form of eq. (3) are equal and positive, the circle will touch the line above the axis of X. If the roots are equal and negative,

Page  137 TIIE CIRCLE. 137the circle will touch the line below the axis of X. In case the roots of eq. (3) are imaginary, the circle will not meet the line. We give the following examples for illustration: y2 —2y=5. To determine the values of y by a geometrical construct tion of this kind, we must make c=-2, and 2c 5. 2 Whence R=3.74, the radius of the circle. Take any distance on the axes for the unit of measure, and set off the distance c on the axis of X from the origin, for the center of the circle; to the right, if c is negative, and to the left, if c is positive. Then from the center, with a radius equal to R= V2q+c2, describe a circumference cutting the line drawn midway between the two axes, as in the figure. In this example the center of the circle is at C, the distance of two units from the origin A, to the right. Then, with the radius 8.74 we described the circumference, cutting the line in M and M', and we find by measure (when the construction is accurate) that MP=4.44, the positive root, and M'P' —1.44, the negative root. For another example we require the roots of the following equation by construction: y2+6y=27. N. B. When the numerals are too large in any equation for convenience, we can always reduce them in the following manner: Put y=nz, then the equation becomes n2z2+6nz-27. Or z2+-6z= —2. n n2 12'

Page  138 138 ANALYTICAL GEOMETRY. Now let n= any number whatever. If n=3, then z2+2z=3. D P' R2__20 HIere c=2.... — 3. 2 2 Whence R —/10 —3.16. At the distance of two units to the left of the origin, is the center of the circle. We see by the figure that 1 is the positive root, and -3 the negative root. But y=nz, n-3, z=1, y=3 or-9. We give one more example. Construct the equation y2+4y=-6. Here c=4, and 12 —6. Whence R-=2. 2 Using the same figure as before, the center of the circle to this example is at D, and as the radius is only 2, the circumference does not cut the line M'M, showing that the equation has no real roots. We have said that this method of finding the roots of a quadratic was of little practical value. The reason of this conclusion is based on the fact that it requires more labor to obtain the value of the radius of the circle than it does to find the roots themselves. Nevertheless this method is an interesting and instructive application of geometry in the solution of equations. When we find the polar equation of the parabola, we shall then have another method of constructing the roots of quadratics which will not require the extraction of the square root. To facilitate the geometrical solution of quadratic equations which we have thus indicated, the operator should provide himself with an accurately constructed scale, which is represented in the following figure. It

Page  139 THE CIRCLE. 189 consists of two lines, or axes, 6 at right angles to each other, 4 and another line drawn 5 through their intersection and 3 making with them an angle, of 45~. Ontheaxes,anycon- 6 5 4 3 g/22 3 4 6 venient unit, as the inch, the 3 half, or the fourth of an inch, 4 etc., is laid off a sufficient 5 number of times, to the right 6 and the left, above and below the origin, from which the divisions are numbered 1, 2, 3, etc., or 10, 20, 30, etc., or.1,.2,..3, etc. To use this scale, a piece of thin, transparent paper, through which the numbers may be distinctly seen, is fastened over it, and with the proper center and radius the circumference of a circle is described. The distances from the axis of X of the intersections of this circumference, with the inclined line through the origin, will be the roots of the equation, and their numerical values may be determined by the scale. By removing one piece of paper from the scale and substituting another, we are prepared for the solution of another equation, and so on. EXAMPLES. 1. Given x2+11x-=80, to find x. Ans. x=5, or —16. 2. Given x2-3x=28, to find x. Ans. x=7, or-4. 3. Given x2_x —2, to find x. Ans. x=2, or-1. 4. Given x2 —12x=-32, to find x. Ans. x=4, or 8. 5. Given x2 —12x —36, to find x. Ans. Each value is 6. 6. Given x2-12x —38, to find x. Both values imaginary. T. Given x2+6x= —10, to find x. Both values imaginary. 8. Given x2=81, to find x. Ans. x=9, or-9.

Page  140 140 ANALY TIC.AL GEOMETRY. For example 8, c=o and 2 -- 81; Whence, R=9V2. This method may therefore be used for extracting the square root of numbers. In such cases, the center of the circle is at the zero point. CHAPTER III. THE ELLIPSE. We have already developed the properties of the Etlipse, Parabola and Hyperbola by geometrical processes, and it is now proposed to re-examine these curves, and develop their properties by analysis. As he proceeds, the student cannot fail to perceive the superior beauty and simplicity of the analytical methods of investigation; and, even if a knowledge of the conic sections were not, as it is, of the highest practical value, the mental discipline to be acquired by this study would, of itself, be a sufficient compensation for the time and labor given to it. As all needful definitions relating to these curves have been given in the CONIC SECTIONS, we shall not repeat them here, but will refer those to whom such reference may be necessary to the appropriate heads in that division of the work. PROPOSITION I. To find the equation of the ellipse referred to its axes as the axes of co-ordinates, the major axis and the distance from the center to the focus being given. Let AA' be the major axis, FE the foci, and C the center of an ellipse. Make CF-=c CA=A. Take any

Page  141 THE ELLIPSE. 141 point on the curve, and from it l D let fall the perpendicular Pt on' the major axis; then, by our A A conventional notation, is Ct=x, -E' C i tA tP=y. As F'P+PF=2A, we may put F'P=A+z, and PF=A —z. Then the two right angled triangles F'Pt, FPt, give us (c+x)2+y2=(A +Z)2 (1) (x-c)2+y2=(A z)2 (2) For the points in the curve which cause t to fall between C and F, we would have (c-x)2+y2=(A-z)2 (3) But when expanded, there is no difference between eqs. (2) and (3), and by giving proper values and signs to x and y, eqs. (1) and (2) will respond to any point in the curve as well as to the point P. Subtracting eq. (2) from eq. (1), member from member, and dividing the resulting equation by 4, we find Cx=Az, or z=e x (4) This last equation shows that F'P, the radius vector, varies as the abscissa x. Add eqs. (1) and (2), member to member, and divide the result by 2, and we have c2+x2+y2 A2+Z2 Substituting the value of z2 from eq. (4), and clearing of fractions, we have 2A2 + A2X2 + A2= A4+ C2X2. Or, A2y2+ (A2 —2)x2A2(A2_-c2). (5) Now conceive the point P to move along describing the curve, and when it comes to the point D, so that DC makes a right angle with the axis of X, the two triangles DC(F and DCF' are right angled and equal. DF and

Page  142 142 ANALYTICAL GEOMETRY. DEF each is equal to A, and as CP, CF', each is equal to c, we have D C2= A2 —2. It is customary to denote D C half the minor axis of the ellipse by B, as well as half the major axis by A, and adhering to this notation B2=A2-c2. (6) Substituting this in eq. (5), we have for the equation of the ellipse A2y2+ 12x2= A21B2, referred to its center for the origin of co-ordinates. If we wish to transfer the origin of co-ordinates from the center of the ellipse to the extremity A' of its major axis, we must put x=-A+x', and y=y'. Substituting these values of x and y in the last equation, and reducing, we have y'2-_2 (2Ax' —x'2) A2 Or without the primes, we have B2 y_,(2Ax-e), for the equation of the ellipse when the origin is at the extremity of the major axis. Cor. 1. If it were possible for B to be equal to A, then c must be equal to 0, as shown by eq. (6). Or, while c has a value, it is impossible for B to equal A. If B=A, then c=O, and the equation becomes A2y2 +A2x2= A2A2. Or y2+x2=A2, the equation of the circle. Therefore the circle may be called an ellipse, whose eccentricity is zero, or whose eccentricity is infinitely small.

Page  143 T H E EILL.rPS E. 143 Cor. 2. To find where the curve cuts the axis of 1, make y=O in the equation, then x= -A, showing that it extends to equal distances from the center. To find where the curve cuts the axis of Y, make x=O, and then y=- B. Plus B refers to the point D, -B indicates the point directly opposite to D, on the lower side of the axis of X. Finally, let x have any value whatever, less than A, then Y=-(A2_x2). an equation showing two values of y, numerically equal, indicating that the curve is symmetrical in respect to the axis of X. If we give to y any value less than B, the general equation gives B Showing that the curve is symmetrical in respect to the axis of Y. SCHOLIUM.-The ordinate which passes through one of the foci, corresponds to x-c. But A' —B'=c. Hence A'-c' or A'-x- =B'. Or (A2-x')-=B, and this value substituted in the last equation, gives y= t. Whence 2B is the measure of A A the parameter of any ellipse. PROPOSITION II. Every diameter of the ellipse is bisected in the center. Through the center draw the line DD'. Let x, and y, denote the co-ordinates of the point D, and x', y', the co-ordinates of the point D'.

Page  144 144 ANALYTICAL GEOMETRY. The equation of the curve is D A2y2+.B2x2= A2B2. The equation of a line passing, /c through the center, must be of the form y=ax. This equation combined with the D equation of the curve, gives x AB aAB V/a2A2+B2 V/a2A2+1B xI_= AB y- aAB..a2A2+ B2 v/(a2A2 +Bz These equations show that the co-ordinates of the point D, are the same as those of the point D', except opposite in signs. Hence DD' is bisected at the center. PROPOSITION III. The squares of the ordinates to either axis of an ellipse are to one another as the rectangles of their corresponding abscissas. Let y be any ordinate, and x - its corresponding abscissa. Then, by the first proposition, y y\ we shall have.-X A y2= -B(22A —x)x. A2 Let y' be any other ordinate, and x' its corresponding abscissa, and by the same proposition we must have y'1= A2(2A-x') x'. Dividing one of these equations by the other, omitting common factors in the numerator and denominator of the second member of the new equation, we shall have

Page  145 THE ELLIPSE. 145 2 _ (2A-x)x y'2 (2A-x')xf Hence, y9: y'2=(2A-x)x: (2A-x')x'. () By simply inspecting the figure, we cannot fail to perceive that (2A-x), and x, are the abscissas corresponding to the ordinate y, and (2A-x') and x' are those corresponding to y'. If we transfer the origin to the lower extremity of the conjugate axis, the equation of the ellipse may be put under the form x2 AL(2B )y, and by a process in all respects similar to the above, we rove that x2' x'2: (2B-y)y: (2B —y')y'. Therefore, the squares of the ordinates, etc. SCHOLIUM. —-Suppose one of these ordinates, as y' to represent half the minor axis, that is, y'=B. Then the corresponding value of x' will be A and (2A-x',) will be A, also. Whence proportion (1) will become y' B' —(2A —x)x A'. In respect to the third term we perceive that if A'His represented by x, AH will be (2A-x), and if G is a point in the circle, whose diameter is A'A, and GH the ordinate, then (2A —x)x= G=H, and the proportion becomes y' B= GHI: Ag. Or y: GH=B: A. Or A: B= GH: y=DH. If a circumference be described on the conjugate axis as a diameter, and an ordinate of the circle to this diameter be denoted by X and the corresponding ordinate of the ellipse by x, it may be shown in like manner that A: B::x: X. 18 K

Page  146 146 ANALYTICAL GEnOMETRY. PROPOSITION IV. The area of an ellipse is a mean proportional between the areas of two circles, the diameter of the one being the major axis, and of the other the minor axis. On the major axis A'A of the ellipse as a diameter describe a B circle, and in the semicircle A'D A ifscribe a polygon of any number of sides. From the verti- E' A ces of the angles of this polygon draw ordinates to the major axis, and join the points in which they intersect the ellipse by straight lines, thus constructing a polygon of the same number of sides in the semi-ellipse A'D'A. Take.the origin of co-ordinates at A', and denote the ordinates BE, CF, -etc., of the circle by Y, Y', etc., the ordinates B'E, C'F, etc., of the ellipse by y, y', etc., and the corresponding abscissas, which are common to ellipse and circle, by x, x', etc. Then by the scholium to Prop. 3, we have Y: y:::A: B and Y': y': A: B, whence Y: Y'::y:y' from which, by composition, we get Y+ Y' y y: Y: y:: A: B But the area of the trapezoid BEFC is measured by (r+r Y)(x'-x) or (Y+YI)( 2 x) and that of the trapezoid B'EFC' by Y+Y' )(x' —x) or (y+y,)(x'-x therefore, trapez. BEFC Y+ Y' A trapez.B'EFC' y+y' -B

Page  147 THE ELLIPSE. 147 That is, trapez. BEFC: trapez. B'EFC': A: B; or, in words, any trapezoid of the semi-circle is to the corresponding trapezoid of the semi-ellipse as A is to B. From this we conclude that the sum of the trapezoids in the semi-circle is to the sum of the trapezoids in the semi-ellipse as A is to B. But by making these trapezoids indefinitely small, and their number, therefore, indefinitely great, the first sum will become the area of the semi-circle and the second, the area of the semi-ellipse. Hence, Area semi-circle: area semi-ellipse:: A: B or, area circle: area ellipse:: A: B That is, 7rA2: area ellipse:: A: B Whence, area ellipse= 7A2- B 7rA.B A But rA.B is a mean proportional between 7rA2 and rpB2. Hence; The area of an ellipse is a mean proportional, etc. SCHOLIUM.-Hence the common rule in mensuration to find the area of an ellipse. RULE.-Multiply the semi-major and semi-minor axes together, and multiply that product by 3.1416. PROPOSITION V. To find the product of the tangents of the angles that two supplementary chords through the vertices of the transverse axis of an ellipse make with that axis, on the same side. Let x, y, be the co-ordinates of any point, as P, and x', y', the coordinates of the point A A'. Then the equation of a line which passes through the two points A' and P, (Prop. 3, Chap. 1,) will be

Page  148 148 ANALYTICAL GEOMETRY. y —y'=a(x-x'). (1) The eqiuation of the line which passes through the points A and.P, will be of the form y-y"-a'(x-x"). (2) For the given point A', we have y-O0, and x' —-A. Whence eq. (1) becomes y=a(x+A). (3) For the given point A we have y"=O, and xi-=A, which values substituted in eq. (2) give y=a'(x- A). (4) As y and x are the co-ordinates of the same point P in both lines, we may combine eqs. (3) and (4) in any manner we please. Multiplying them member by member, we have y2= aa'(x2-A2). (5) Because e is a point in the ellipse, the equation of the curve gives e"" p (A2,e~C ) -B(2A2)z (6) A2 / A2 Comparing eqs. (5) and (6), we find aa___B A2 for the equation sought. SCHOLIUM 1. —In case the ellipse becomes a circle, that is, in case A=B, (a'+1=0, showing that the angle A'PA would then be a right angle, as it ought to be, by (Prop. II, Chap. II.) Because A is less than unity, or aa' less than 1,* or radius; A2 the two angles PA'A and PAA' are together less than 90~; therefore, the angle at P is obtuse, or greater than 900. SCHOLIUM 2.-Since aa' has a constant value, the sum of the two, a+a', will be least when a —a'. * In trigonometry we learn that tan. x cot. x-R2-1. That is, the product of two tangents, the sum of whose arcs is 900, is equal to 1. When the sum is less than 900, the product will be a fraction.

Page  149 THE ELLIPSE. 149 Hence the angle at P will be greatest when P is at the vertex of the minor axis, and the supplementary chords equal; and the angle at P will become nearer a right angle as P approaches A or At. PROPOSITION V1. To.find the equation of a straight line which shall be tangent to an ellipse. Assume any two points, as Q P' and Q, on the ellipse, and denote the co-ordinates of the K first by x', y', and of the second by x", y". Through these points draw a line, the equation of which (Prop. 4, Chap. 1,) is y —y- a(x —x), (1) in which a:y yN We must now determine the value of a when this line becomes a tangent line to the ellipse. Because the points P and Q are in the curve, the coordinates of those points must satisfy the following equa. tions: A2y'2+ s2. A2IB A2y'2+B2x'2-A2B2. By subtraction A 2y 2_(y,"T2) + B2 (x,2x"2)-O0. Or A2(y'+y")(y'-y")= —-B2(x'+x)(x'-x)., (2) Whence a= — y _B2(x +x") x' —x" A2(y'+y") Now conceive the line to revolve on the point P until Q coincides with P, then PR will be tangent to the curve. But when Q coincides with P, we shall have yt=y", and x'=x". 13*

Page  150 150 ANALYTICAL GEOMETRY. Under this supposition, we have B2xt A2y' The value of a put in eq. (1), gives Reducing A2yy+B2 A2y2 X Reducing A2yy'+B2xx —A~y'2 +B2x12. Or A2yy'+B2xx- A2B2. This is the equation sought, x and y being the general co-ordinates of the line. SCHOLIUM. —To find where the tangent meets the axis of X, we must make y= O. This gives x="= CT.. In case the ellipse becomes a circle, B=A, and then the equation will be-,, come yy'+xPA2, \ I T the equation for a tangent line to a circle; and to find where this tangent meets the axis of'X, we make y=O, and x=A -CT, as before. In short, as these results are both independent of B, the minor axis, it follows that the circle and all ellipses on the major axis AB have tangents terminating at the same point T on the axis of X, if drawn from the same ordinate, as shown in the figure. SCHOLIUM 2.-To find the point in which the tangent to an ellipse meets the axis of Y, we make x=O0, then the equation for the tangent becomes B2 As this equation is independent of A, it shows that all ellipses having the same minor axis, have tangents terminating in the same point on the axis of Y, if drawn from the same abscissa. SCHOLIUM 3. If from CT we subtract C'R, we shall have RT,

Page  151 THIE ELLIPSE. 151 a common subtangent to a circle, and all ellipses which have 2A for a major diameter. That is R T=41-x'= As-X'2 Xt X= xf xi We can also find RT by the triangle P2T, as we have the tangent of the angle at iT, (Ba2 ) to the radius 1. Whence we have the following proportion: 1: _B2 RT y' A2y' RT —-- Ay. B2 XI' The minus sign indicates that the measure from T is towards the left. PROPOSITION VII. To find the equation of a normal line to the ellipse. Since the normal passes through the point of tangency, its equation will be in the form y-y' a'(x —x). (1) Because PIN is at right angles to the tangent, aa'+1=O. t C N R But by the last proposition aA2x' A2y' Whence a'- Ay',, and this value of a' put in eq. (1) gives Y-Y' (x — ), for the equation sought. SCHOLIUM 1.-To find where the normal cuts the axis of X, we must make y=O, then we shall have

Page  152 152 ANALYTICAL GEOMETRY. m- [A2_B2 ~ — A2 / APPLICATION.-Meridians on the earth are ellipses; the semimajor axis through the equator is A=3963. miles, and the semiminor axis from the center to the pole is B —3949.5. A plumb line is everywhere at right angles to the surface, and of course its prolongation would be a normal line like PN. In latitude 420, what is the deviation of a plumb line from the center of the earth? In other words, how far from the center of the earth would a plumb line meet the plane of the equator? Or, what would be the value of CN? As this ellipse differs but little from a circle, we may take CR for the cosine of 420, which must be represented by x'. This being assumed, we have x'=2945. A Bj )2945.=20,+mies=CON. Ans. SCHOLIUM 2.-To find NR, the subnormal, we simply subtract CN from CR, whence:R=,_( A' — ): ~,._ B A' A2 A2 We can also find the subnormal from the similar triangles PR T, PNR, thus: TR: RP::RP: RN. y2:y'::y' -NR. Whence 1N'RPROPOSITION VIII. Lines drawn from the foci to any point in the ellipse make equal angles with the tangent line drawn through the same point. Let C be the center of H the ellipse, PT the tangent line, and PF, PF, the two lines drawn to the foci. Denote the distance CF=T-/A2 —B2 by c, CPF

Page  153 THE ELLI P SE. 153 by —c, the angle FPTby V, and the tangents of the angles PTJ, PFT, by a and a'. Now FPT= P TX-PFT. By trigonometry, (Eq. 29, p. 253, Robinson's Geometry), we have Tan. FPT=tan. (PTX-PFT). That is, tan. V= a —a,. (1) 1+aa Prop. 6f, gives us t- B2'x x', y', being the co-ordinates of the point P. Let x, y, be the co-ordinates of the point E, then from Prop. 4, Chap. 1, we have atY.'-Y But at the point F, y=O and x=c. Whence a Y These values of a and a' substituted in eq. (1) give -B2x' yI Tan. V= A2y' x'-c B2z2+B2cx-A,2y12.Bx = A,2y'(x'. —c)-B2x'y' A2(x'-C) B2Cex, —'A2B2 B2(cx'.-A2) B2 Tan. V=-(A B,2)xy.A2e' cy'(c'x A,2) - cy' observing that A2y'2+B2xt2 =A2B2, andA2- -B2-c. The equation of the line PF will become the equation of the line PP by simply changing +c to -c, for then we shall have the co-ordinates of the other focus. We now have B2 tan. FPT-_:B cy, But if c is made -c, then tan. PIPT B CIy'

Page  154 154 ANALYTICAL GEOMETRY. As these two tangents are numerically the same, differing only in signs, the lines are equally inclined to the straight lines from which the angles are measured, or the angles are supplements of each other. Whence FPT+P PT=-180. But F'PPH+F'PT=-180. Therefore FPT=-F'PH. Cor. The normal being perpendicular to the tangent, it must bisect the angle made by the two lines drawn from the tangent point to the foci. SCHOLIUM.-Any point in the curve may be considered as a point in a tangent to the curve at that point. It is found by experiment that light, heat and sound, after they approach to, are reflected off, from any reflecting surface at equal angles; that is, for any ray, the angle of reflection is equal to the angle of incidence. Therefore, if a light be placed at one focus of an ellipsoidal reflecting surface, such as we may conceive to be generated by revolving an ellipse about its major axis, the reflected rays will be concentrated at the other focus. If the sides of a room be ellipsoidal, and a stove is placed at one focus, the heat will be concentrated at the other. Whispering galleries are made on this principle, and all theaters and large assembly rooms should more or less approximate to this figure. The concentration of the rays of heat from one of these points to the other, is the reason why they are called the foci, or burning points. PROPOSITION IX. The product of the tangents of the angles that a tangent line to the ellipse and a diameter through the point of contact, make with the major axis on the same side, is equal to minus the square of the semi-minor divided by the square of the semimajor axis.

Page  155 THE ELLIPSE. 155 Let PT be the tangent D line and PP' the diameter through the point of contact X P, and denote the co-ordi- C A nates of P by x', y'. The equation of the diameter is y=alx, in which a' is the tangent of the angle POT. Since this line passes through the point P, we must have y'- a'x' Whence a'-y ( For the tangent of the angle PTXwe have B2x' Ba - — x (2) A2y' Multiplying eqs. (1) and (2), member by member, we find aa' = B2 A2 SCHOLIUM.-The product of the tangents of the angles that a diameter and a tangent line through its vertex make with the major axis of an ellipse is the same (Prop. 5) as that of the tangents of the angles that supplementary chords drawn through the vertices of the major axis make with it. Hence, if a=a, then a'=a'. That is, if the diameter is parallel to one of the chords, the tangent line will be parallel to the other chord, and conversely. This suggests an easy rule for drawing a tangent line to an ellipse at a given point, or parallel to a given line. OF THE ELLIPSE REFERRED TO CONJUGATE DIAMETERS. Two diameters of an ellipse are conjugate when either is parallel to the tangent lines drawn through the vertices of the other.

Page  156 156 ANALYTICAL GEOMETRY. Since a diameter and the tangent line through its vertex make, with the major axis, angles whose tangents satisfy the equation B2 aa A2 it follows that the tangents of the angles which any two conjugate diameters make with the major axis must also satisfy the same equation. Now let m be the angle whose tangent is a, and n be the angle whose tangent is a', then a-sin m, and a,-sin. n. cos. m~ cos. n Substituting these values in the last equation, and reducing, we obtain A2 sin. m sin. n+B2 cos. m cos. n-=O, which expresses the relation which must exist between A, B, m, and n, to fix the position of any two conjugate diameters in respect to the major axis, and this equation is called the equation of condition for conjugate diameters. In this equation of condition, m and n are undetermined, showing that an infinite number of conjugate diameters might be drawn, but whenever any value is assigned to one of these angles, that value must be put in the equation, and then a deduction made for the value of the other angle. PROPOSITION X. To find the equation of the ellipse referred to its center and conjugate diameters. The equation of the ellipse referred to its major and minor axes, is A2y2 +B2X2=A 2B2. The formulas for changing rectangular eo6,'odinatas

Page  157 THE ELLIPSE..t into oblique, the origin being the same, are (Prop. 9, Chap. 1,) x=x' cos. m+y' cos. n. y=x' sin. m+y' sin. n. Squaring these, and substituting the values of x2 and y2 in the equation of the ellipse above, we have (A2sin2n+B2cos2n)y2+ (A2sin2m +B2cos2m)x'2 +2(A2sin.m sin.n+PB2cos.m cos.n)y'x' But if we now assume the condition that the new axes shall be conjugate diameters, then A2sin. m sin. n+B2cos. m cos. n=-0, which reduces the preceding equation to (F) (A2sin.2n+3B2cos. n)y'+ (A2sin. m+ cos.2 m)'2=A2B which is the equation required. But it can be simplified as follows: The equation refers to the two diameters B"B' and ]Y'D' as co-ordinate axes. For the point B' we must make y'= —O, then A2B2A A2sin.(m+ B2cos.2m (CB')2 A/2. (P) Designating CB' by A', and CD' by B'. For the point D' we must make x'=O. Then A2B2 Y1 A2sin.2n+B2cos2n-( CD')2=B". (Q) From (P) we have (A2sin.2m+B2cos.1m)-=AB/ A212 From (Q) (A28in.'n+B2cos.2n)-= B-, These values put in (F) give A2B2 A2B2 y/2+ _X/2 =A22. A2t A/2 Whence A2y'2+B12X12= Aa2B/n. 14

Page  158 158 ANALYTICAL GEOMETRY. We may omit the accents to x' and y', as they are general variables, and then we have A'2y2+B12x2-A'2B". for the equation of the ellipse referred to its center and conjugate diameters. ScHoLIUM.-In this equation, if we assign any value to x less than A', there will result two values of y, numerically equal, and to every assumed value of y less than B', there will result two corresponding values of x, numerically equal, differing only in signs, showing that the curve is symmetrical in respect to its conjugate diameters, and that each diameter bisects all chords which are parallel to the other. OBSERVATION.-As this equation is of the same form as that of the general equation referred to rectangular co-ordinates on the major and minor axis, we may infer at once that we can find equations for ordinates, tangent lines, etc., referred to conjugate diameters, which will be in the same form as those already found, which refer to the axes. But as a general thing, it will not do to draw summary conclusions. PROPOSITION XI. As the square of any diameter of the ellipse is to the square of its conjugate, so is the rectangle of any two segments of the diameter to the square of the corresponding ordinate. Let CD be represented by A', and CE by B', CH by x, and GH by y, B K D then by the last proposition we have A 2y2+B'2x2.-A'2BV2. Which may be put under the form D A 2y2-B,2(A2 _ X2). Whence A'2: B': (A'2 — x2) y2. Or (2A')': (2B')2: (A'+x)(A'-x): y. Now 2A' and 2B' represent the conjugate diameters D'D, E'E, and since C.H represents x, A'+x-D'H, and

Page  159 THE ELLIPSE. 159 A' —x-HED. Also y-=GH. Hence the above proportions correspond to (D'D)2: (E'E)2:: D'HJx D: (GH)2. SCHOLIUM.-As x is no particular distance from C, CF may represent x, then LF will represent y, and the proportion then becomes (D'D)' (E'E)':: D'FXFD: (LF)'. Comparing the two proportions, we perceive that D'H-HD: D'FFD:: Gb' ~ LF'. That is, The rectangle of the abscissas are to one another as tIh squares of the corresponding ordinates. The same property as was demonstrated in respect to rectangular co-ordinates in Prop. 3. In the same manner we may prove that Eh-hE': EffE':: (hg)': (fe)' PROPOSITION XII. To find the equation of a tangent line to an ellipse referred to its conjugate diameters. Conceive a line to cut the curve in two points, whose co-ordinates are x', y', and x", y", x and y being the coordinates of any point on the line. The equation of a line passing through two points is of the form y-y'=-a(x —x), (1) an equation in which a is to be determined when the line touches the curve. From the equation of the ellipse referred to its conjugate axes we have A'y'f2+ B'2X12=A/2B12. Ay''2+ B2x2= A,'B'2. Subtracting one of these equations from the other, and operating as in Prop. 6, we shall find Bt2Xt

Page  160 160 A.NALYTI AL GEOMETRY. This value of a put in eq. (1) will give Y -B'2x'X) Yl= A12~tl(x-x-VReducing, and A'2y'y+B'2x'x-A'2B'2, which is the equation sought, and it is in the same form as that in Prop. 6, agreeably to the observation made at the close of Prop. 10, PROPOSITION XIII, To transform the equation of the ellipse in reference to conjugate diameters to its equation in reference to the axes. The equation of the ellipse in reference to its conjugate diameter is A12y22+ B'2x'2- A/2B2 (1) And the formulas for passing from oblique to rectangular axes are (Prop. 10, Chap. 1,) _x= sin. n-y cos. n y y cos. m —z sin. m sin.(n —m)' sin. (n —m) These values of x' and y' substituted in eq. (1) give (A'2 cos.2 m+ B2 cos.2 n)y2+ (A'2 sin.2 m+B'2 sin.2 n)x2 _ -2(A'2 sin. m cos. mr+B'2 sin. n cos. n)xy A'2 B12 sin.2(n-m). This equation must be true for any point in the curve, x being measured on the major axis, and y the corresponding ordinate at right angles. This being the case, such values of A', B', m, and n, must be taken as will reduce the preceding equation to the well known form A2y2+B2x2_A2B2. Therefore we must assume A2 cos.-2 m+B'2 cos. 2 n=A2. C1) A'2 sin.2 m+B'2 sin. 2 n-=B. (2) Al2sin.m cos.m+~B2sin.n cos.n — 0. (3) A'2 B'2 sin. (n —m) =A2B2. (4)

Page  161 THE ELLIPSE. 161 The values of m and n must be taken so as to respond to the following equation, because the axes are in fact conjugate diameters. A2sin.m sin.n+B2cos.m cos.n —O. (5) These equations unfold two very interesting properties. SCHOLIUM 1.-By adding eqs. (1) and (2) we find A'2 +B'2 -=A A2+B2. Or 4A'2+4B'2.4A2 +4B2. That is, the sum of the squares of any two coftjugate diamneters is equal to the sum of the sguares of the axes. SCHOLIUM 2. —Equation eq. (3) or (5) will give us m when n is given; or give us n when m is given. SCHOLIUM 3.-The square root of eq. (4) gives A'B' sin.(n — m)AB, ihich shows the equality of two surfaces, one of which is obviously the rectangle of the two axes. Let us examine the other. Let n represent the angle NCB, M and m the angle PCB. Then the angle NCP will be represented by (n —m). Since the angle MNK is the supplement of NCP, the two angles have the same sine and M= A'. In the rightangled triangle NKM, we have 1: A':: sin.(n-m): MK. MK=A' sin.(n —m). But NC-=B'. Whenee MK.NC=A'B' sin.(n-m —) =the parallelogram NCPM. Four times this parallelogram if the parallelogram ML, and four times the parallelogram DCB.H, which is measured by A XB, is equal to the parallelogram HF. Hence eq. (4) reveals this general truth: The rectangle which is formed by drawing tangent lines through 14* L

Page  162 162 ANALYTICAL GEOMETRY. the vertices of the axes of an ellipse, is equivalent to any parallelogram which can be formed by drawing tangents through the vertices of conjugate diameters. NoTE.-The student had better test his knowledge in respect to the truths embraced in scholiums 1 and 3, by an example: Suppose the 8emi-major axis of an ellipse is 10, and the semi-minor axis 6, and the inclination of one of the conjugate diameters to the axis of X is taken at 800 and designated by m. We are required to find A'2 and B'2, which together should equal A2 +B2, or 136, and the area NCPM, which should equal AB, or 60, if the foregoing theory is true. Equation (5) will give us the value of n as follows: 100' tan.n+36'i/3=0. Or tan.n=-36_3 100 Log. 36+- log. 3-log. 100 plus 10 added to the index to correspond with the tables, gives 9.794863 for the log. tangent of the angle n, which gives 310 56' 42", and the sign being negative, shows that 31~ 56' 421 must be taken below the axis of X, or we must take the supplement of it, NCB, for n, whence n=1480~ 3' 18", and (n —m)=1180~ 3' 18". To find A'2 and B's, we take the formulas from Prop. 10. A'2 A2B2 100'36 360069.23 A2sin.230+B2cos.230 100'i+361 52 B'2 - A2B B' 3600 A2sin.231~56'42"+B2cos.2(31o56'42") 27 99+25 92 66'77. And their sum=136. This agrees with scholium 1. As radius 10.000000 Is to A'~(log.69.23) 0.920147 So is sine (n-rm) 61~ 56' 42" 9.945713 log. MKE 0.865860 Log. B'=i log. (66.77)........ 0.912290 AB=60. log. 60= 1.778150

Page  163 THE ELLIPSE. 168 PROPOSITION XIV. To find the general polar equation of an ellipse. If we designate the co-ordinates of the pole P, by a and b, and es- x timate the angles v from the line \ PX' parallel to the transverse axis, we shall have the following formulas: x-=a+r cos.v. y=b+r sin v. These values of x and y substituted in the general equation A2y2+B2x2A2B2, will produce A2 sin.2v I r+2A2b sin.v r+A2b2+BVa2=A2'B,' 3B2 cos.2v + 22a cos.v~ for the general polar equation of the ellipse. SCHOLIUM i.-When P is at the center, a=0, and b=0, and then the general polar equation reduces to A2B' A2sin.2v+B2cos.'v a result corresponding to equations (P) and ( Q) in Prop. 10. SCHOLIUM 2.-When P is on the curve A'b'-+B'a'=A'B, therefore A'sin.'v r'+2A'b sin. Vr- 0 B'cos.2vv +2B2a cos.vI This equation will give two values of r, one of which is 0, as it should be. The other value will correspond to a chord, according to the values assigned to a, b, and v. Dividing the last equation by the equation r=Oj and we have A'sin. vlr+2A'b sin.v =0. B'cos.2vl +2B2acos.v The value of r in this equation is the value of a chord. When the chord becomes 0, the value of r in the last equation becomes 0 also, and then A'b sin.v+B'a cos.v = 0.

Page  164 864 ANALYTICAL GEOMETRY. Or tan.v=- Ba A2b a result corresponding to Prop. 6, as it ought to do, because the radius vector then becomes tangent to the curve. ScriotiuM 3. —When P is placed at the extremity of the major axis on the right, and if v=0, then sin. v=O, and cos. v-1 a=A, and b=0; these values substituted in the general equation will reduce it to B'r2+2B2Ar=O, which gives r=O, and r= -2A, obviously true results. When P is placed at either focus, then a- VA'-B2=c, and b=-0. These values substituted, and we shall have (A' sin.2v+-Bcos..v)r' +2B'a cos.vr=_B4. It is difficult to deduce the values of r from this equation, therefore we adopt a more simple method. Let F be the focus, and FP any radius, and put the angle PFI:D=v. By Prop. 1, of the ellipse, we learn that C D (1) FP=r=A+A 7 an equation in which c= V/A-B%, and x any variably distance CD. Take the triangle PDF, and by trigonometry we have r:: cos.v: c+x. Whence x=r cos.v-c. This value of x placed in (1), will give cr. cos.v-C' r=A+ A Whence (A —c cos.v)r-=A' —_ Or - _ A-c cos.v This equation will correspond to all points in the curve by giving to cos.v all possible values from 1 to -1. Hence, the greatest value of r is (A+c), and the least value (A-c), obvious results when the polar point is at F.

Page  165 THE ELLIPSE. 10 The above equation may be simplified a little by introducing the eccentricity. The eccentricity of an ellipse is the distance from the center to either focus, when the semi-major axis is taken as unity. Designate the eccentricity by e, then 1: e=A: c. Whence c=eA. Substituting this value of c in the preceding equation, we have A2-eA _ A(1-e') A-eA cos. v9 1-e cos. v This equation is much used in astronomy. PROPOSITION XV.-PROBLEM, Given the relative values of three different radii, drawn from the focus of an ellipse, together with the angles between them, to find the relative major axis of the ellipse, the eccentricity, and the position of the major axis, or its angle from one of the given radii. Let r, r', and r", represent the r three given radii, m the angle between r and r', and n that between r and r". The angle between the radius r and the major axis is supposed to be unknown, and we therefore, call it x. From the last proposition, we have A(1 —-e2) ( 1 -e cs. x A (1-e) (2)'-'e cos.(n) (2) A(1 —e2) X1-e cos. (x+n) (3) Equating the value of A(1 —e2) obtained from eqs. (1) and (2), and we have r-re cos. x=r' —re cos. (x+m)

Page  166 166 ANALYTICAL GEOMETRY. Or, e- -- (4) Or, e r cos. x-r' cos. (x+m). In like manner frbm eqs. (1) and (3), we have r-re cos. x=-r"-r"e cos. (x+n). Or, e= -(5 r cos. x-r cos. (x+n) (5) Equating the second members of eqs. (4) and (5), we have r-r' r-' r cos.x-r' cos.(x+m)-r cos.x-r" cos.(x+n) Whence, r-r' r cos. x-r' cos. (x+m) r-r" r cos. x-r" cos.(x+n) r cos. x-r' cos. x cos. m+r' sin. x sin. m r cos. x —r cos. x cos. n+r" sin. x sin. n r-r' cos. rn+r' sin. m tan. x r —r' cos. n-+r" sin. n tan. x For the sake of brevity, put r-r'=d, r —r-d, the known quantity r-r' cos. m-a, and r-r"cos.n-b. Then the preceding equation becomes d a+r'sin.m tan.x d' b+r"sin.n tan.x From which we get successively db+dr" sin. n tan. x= —ad'+d'r' sin. m tan. x (dr" sin. n-d' r' sin. m) tan. x=ad'-db, ad'-db tan. X —drF sin.n —d'r' sin.nt The value of x from this equation determines the position of the major axis with respect to that of r, which is supposed to be known, as it may be by observation. Having x, eq. (4) or (5) will give e the eccentricity. If the values of e found from these equations do not agree, the discrepancy is due to errors of observation, and in such cases the mean result is taken for the eccentricity.

Page  167 THE ELLIPSE. 167 Equations (1), (2) and (3) contain A, the semi-major axis, as a common factor in their second members. This factor, therefore, does not affect the relative values of r, r' and r", and as it disappears in the subsequent part of the investigation, it shows that the angle x and the eccentricity are entirely independent of the magnitude of the ellipse. To apply the preceding formulas, we propose the following EXAMPLE. On the first day of August, 1846, an astronomer observed the sun's longitude to be 1280 47' 31", and by comparing this observation with observations made on the previous and subsequent days, he found its motion in longitude was then at the rate of 57' 24".9 per day. By like observations made on the first of September, he determined the sun's longitude to be 1580 37' 46", and its mean daily motion for that time 58' 6". 6; and at a third time, on the 10th of October, the observed longitude was 1960 48' 4", and mean daily motion 59' 22".9. From these data are required the longitude of the solar apogee, and the eceentricity of the apparent solar orbit. It is demonstrated in astronomy that.the relative distances to the sun, when the earth is in different parts of its orbit, must be to each other inversely as the square root of the sun's apparent angular motion at the several points; therefore, (r)2, (r')2, and (r2, must be in the proportion of 1 1, and 1 57' 24" 9 58' 6" 6 59' 22" 9 Or as the numbers 1 1 1 1 and 1 3444.9' 3486.6 3562.9 Multiply by 3562.9 and the proportion will not be changed, and we may put r= (3562.9i r= (3562.9 1. and 3444.9/ 3486.6'

Page  168 Is68 8ANALYTICAL.GEOMETRY. By the aid of logarithms we soon find r=1.016982 r' -1.010857 and r —=1. ~Hence r —r-'=d=-0.006125, r-r"=-d'= 0.016982. 1580 37' 46" 1960 481 4" 128 47 31 128 47 31 m-29 50 15 n 68 0 33 To substitute in our formulas, we must have the natural sine and cosine of m and n. sin. m=sin. 29~ 50'15"-0.497542, cos.=0.867440. sin. n=-sin. 68~ 0' 33"-0.927238, cos.=0.374472. r —r' cos..m- a- 0.140124. r —r" cos. n= b - 0. 642510. ad'=0.0023695, db=0.00393537. d'r' sin. m=-0.008538616. dr" sin. n=0.005679332. These values substituted in the formula tan.x= ad —db db —d' dr"sin.n-d'r'sin.m d'r'sin.m —drs'in.n' give tan. x.00156586 15.6586.00285928 28.5928 Log. 15.6586 plus 10 to the index-11.194746 ].og. 28.5928 1.456224 Log. tan. -280 42' 45 9.738522 Long. of r 128~ 47' 1"' Long. apogee 1000 4' 46" According to observation, the longitude of the solar apogee on the 1st of January, 1800, was 990 30' 8"39, and it increases at the rate of 61"9 per annum. This would give, for the longitude of the apogee on the 1st of January, 1861, 1000 33' 03"54. To find e, the eccentricity, we employ eq. (5), which is

Page  169 THE PARABOLA. 169 e — r cos.x-r"' cos.(x+n)' Whence, by substituting the values of r, r", cos. x, etc., we find 0.016982.016982 rcos. 28042'45"-cos. 96~43q8".891891+.11694.016982 0.016838 1.0088 CHAPTER IV. THE PARABOLA. To describe a parabola. Let CD be the directrix, and F the D focus. Take a square, as DBG, and B U - to one side of it, GB, attach a thread, if and let the thread be of the same HIV, length as the side GB of the square. c Fasten one end of the thread at the point G, the other end at F. Put the other side of the square against CD, and with a pencil, P, in the thread, bring the thread up to the side of the square. Slide one end of the square along the line CD, and at the same time keep the thread close against the other side, permitting the thread to slide round the pencil P. As the side of the square, BD, is moved along the line CD, the pencil will describe the curve represented as passing through the points V and P. GP+ PF= the thread. GP+-PB= the thread. By subtraction PF-PB=0, or PF= PB. This result is true at any and every position of the point P; that is, it is true for every point on the curve. Hence, _ FV= -VH. 15

Page  170 170 ANALYTICAL GEOMETRY. If the square be turned over and moved in the opposite direction, the other part of the parabola, on the other side of the line ~FH may be described. PROPOSITION I. To find the equation of the parabola. Take the axis of the parabola for the axis of abscissas and the line at right angles to it through the vertex for the axis of ordinates. The perpendicular distance from the HI v[ F I C focus F to the directrix BH, is called p, a constant quantity, and when this constant is large, we have a parabola on a large scale, and when small, we have a parabola on a small scale. By the definition of the curve, V is midway between F and the line BBH, and PF=-PB. Put VD=x and PD=y, and operate on the right angled triangle PDF..FD=x-21p, PB=x+21p=PF. (FD)2+(PD)2=( PF)2 That is, (x —~p+y)+-( +~p)2. Whence y2=2px, the equation sought. Cor. 1. If we make x=-O, we have y=O at the same time, showing that the curve passes through the point V, corresponding to the definition of the curve. As y= —/2pz, it follows that for every value of x there are two values of y, numerically equal, one +, the other -, which shows that the curve is symmetrical in respect to the axis of X. Cor. 2. If we convert the equation y2 2px into a proportion, we shall have x:y: y: 2p,

Page  171 THE PARABOLA. 171 a proportion showing that the parameter of the axis is a third proportional to any abscissa and its corresponding ordinate. Cor. 3. If we substitute jp for x in the equation y2=2px we get y-p or 2y=2p. That is the parameter of the axis of the parabola is equal to the double ordinate through the focus, or, it is equal to four times the distance from the vertex to the directrix. PROPOSITION II. The squares of ordinates to the axis of the parabola are to one another as their corresponding abscissas. Let x, y, be the co-ordinates of any point P, and x',y', the co-ordinates of any other point in the curve. Then by the equation of the curve we must have y2-2px. (1) y2- 2px', (2) y2 _ X By division Whence y2: y2::x-:. PROPOSITION III. To find the equation of a tangent line to theparabola. Draw the line SPQ intersecting the parabola in the two points P and Q. Denote the co-ordinates of the first point by x', y', and of the sec- S ond, by x", y". The equation of the straight line T passing through these points is y-y' a(x-x') ()

Page  172 172 ANALYTICAL GEOMETRY. y'-y" in which a is equal to,, It is now required to find the value of a when the point Q unites with P, or, when the secant line becomes a tangent line at the point P. Since P and Q are on the parabola we must have yl'22px' And y"2-2px" Whence y' ~ —2- 2p(x' x") Or (y'-y")(y' +y)-2p(x' —e) y_-y" 2p —x Therefore a-.Ypx,, Substituting this value of a in eq. (1) we have for the equation of the secant line. 2p (2) Now if this line be turned about P until Q coincides with P we shall have y"-y' and the line becomes tangent to the curve at the point P. Under this supposition the value of a becomes P and equation (2) reduces to y- y' -(x- ) Or y y' -y' 2=pX —px' But y'2=-2px'; substituting this value y'2 in the last equation, transposing and reducing, we have finally y y'-p(x+x') (3) for the equation of the tangent line. Cor. To find the point in which the tangent meets the axis of -X, we must make y=-0, this makes makes B p(x+x')=O. Or XI=-x. T vF C

Page  173 THE PARABOLA. 173 That is, VD= VT, or the sub-tangent is bisected by the vertex. Hence, to draw a tangent line from any given point, as P, we draw the ordinate PD, then make TV= VD, and from the point T draw the line TP, and it will be tangent at P, as required. PROPOSITION IV. Tofind the equation of a normal line in the parabola. The equation of a straight line passing through the point P is y-y' =a(x-x'). (1) Let xl, yl, be the general co-ordinates of another line passing through the same point, and a' the tangent of the angle it makes with the axis of the parabola, its equation will then be y1-y'=a'(x1 —x'). (2) But if these two lines are perpendicular to each other, we must have aa' —1. (3) But since the first line is a tangent, p — _ fy This value substituted in eq. (3) gives a' Y And this value put in eq, (2) will give y,- 1 y' —Y (x -x') for the equation required. 15*

Page  174 174 ANALYTICAL GEOMETRY. Cor. 1. To find the point in which the normal meets the axis of., we must make y,=0. Then by B a little reduction we shall have But VC=xl, and VD- x'. Therefore DC=p, that is, The sub-normal is a constant quantity, double the distance between the vertex and focus. Cor. 2. Since TV=-VD, and VF= —DC, TF=FC. Therefore, if the point F be the center of a circle of which the radius is FC, the circumference of that circle will pass through the point P, because TPC is a right angle. Hence the triangle PFTis isosceles. Therefore, If from the point of contact of a tangent line to the parabola a line be drawn to the focus it will make an angle with the tangent equal to that made by the tangent with the axis. Cor. 8. Now as Vbisects TD and VB is, parallel to PD, the point B bisects TP. Draw PB, and that line bisects the base of an isosceles triangle, it is therefore perpendicular to the base. Hence, we have this general truth: Iffrom the focus of a parabola a pcrpendicular be tdrawn to any tangent to the curve, it will meet the tangent on the axis of Y. Also, from the two similar right-angled triangles, FB V and FB T, we have TPF::FB: FB: $FV. Whence BP-2= TF FV. But FV is constant, therefore (BF)2 varies as TF, or as its equal PF. SCHOLIUM. —Conceive a line drawn par- A allel to the axis of the parabola to meet p the curve at P; that line will make an angle with the tangent equal to the angle FTP. But the angle FTP is equal to A F the angle FPT; hence the L LPA-=the

Page  175 THE PARABOLA. 175 L FPT. Now, since light is incident upon and reflected from surfaces under equal angles, if we suppose LP to be a ray of light incident at P, the reflected ray will pass through the focus F,. and this will be true for rays incident on every point in the curve; hence, if a reflecting mirror have a parabolic surface, all the rays of light that meet it parallel with the axis will be reflected to the focus; and for this reason many attempts have been made to form perfect parabolic mirrors for reflecting telescopes. If a light be placed at the focus of such a mirror, it will reflect all its rays in one direction; hence, in certain situations, parabolic mirrors have been made for lighthouses for the purpose of throwing all the light seaward. PROPOSITION V. If two tangents be drawn to a parabola at the extremities of any chord passing through the focus, these tangents will be perpendicular to each other, and their point of intersection will be on the directrix. Let PP' be any chord through the focus of the parabola, and PT, P' T the tangents B drawn through its extremities. Through T T, their intersection, draw BB' perpendicular to the axis HF, and from the focus let H F fall the perpendiculars Ft,'t' upon the B P tangents producing them to intersect BB' at B and B'. Draw, also, the lines PB, P'B', and It'. _,irst.-The equation of the chord is y a (xP2 (1) and of the parabola y2=-2px (2) Combining eqs. (1) and (2) and eliminating x, we find that the ordinates of the extremities of the chord are the roots of the equation y22y=pa a

Page  176 176 ANALYTICAL GEOMETRY. Whence y,_p+pVai+1 and y", p —p/a-+1 ~~a a Therefore the tangents of the angles that the tangent lines at the extremities of the chord make with the axis are p a and P a Y' — +/a+l y" 1 -an+1 The product of these tangents is a a X -1 1+- x-+_ 1__'/ 1 Whence we conclude that the tangent lines are perpendicular to each other. Second. —Because the AtFt' is right-angled and FV is a perpendicular let fall from the vertex of the right angle upon the hypothenuse, we have (The. 25, B. II, Geom.) Et:_F'2:: Vt: Vt' and because itt and BB' are parallel, (Cor. 3, Prop. 4), we also have ~.B: ~HB' But (Cor. 3, Prop. 4,) i?'-2 ~p FP-' Therefore FP: FP'::HB: HB' H]ence the lines PB, P'B' are parallel to the axis of the parabola, and (Cor. 2, Prop. 4,) the angles BPt and tPF are equal. Therefore the right-angled triangles BPI and tPF are equal, and PB=PF'. In the same way we prove that P'B'-P'F. The line BB' is therefore the directrix of the parabola. Cor. Conversely: If two tangents to the parabola are perpendicular to each other, the chord joining the points of contact passes through the focus.

Page  177 THE PARABOLA. 177 For, if not, draw a chord from one of the points of contact through the focus, and at the extremity of this chord draw a third tangent. Then the second and third tangents being both perpendicular to the first, must be parallel. But a tangent line to a parabola, at a point whose ordinate is y', makes with the axis an angle having P for its tangent; and as no two ordinates of the parabola are algebraically equal, it is impossible that the curve should have parallel tangent lines. PROPOSITION VI. Toj ind the equation of the parabola referred to a tangent line and the diameter passing through the point of contact as the co-ordinate axes. Let V be the vertex and VX the axis of the parabola. Through / M any point of the curve, as P, draw the tangent P Y and the diameter p PR, and take these lines for a sys- R S tem of oblique co-ordinate axes.'. From a point M, assumed at plea- R' sX sure, on the parabola, draw MR X parallel to P Y and MS perpendicular to VX; also, draw PQ perpendicular to VX. Let our notation be VQ=c, PQ-=b, VS'-x, 2MS'=y, PR=x', MR-y' and LMRS=LMR'S'=m; then the formulas for changing the reference of points from a system of rectangular to a system of oblique co-ordinate axes having a different origin, give, by making Ln=O, VS'=x=c+x'+y'cos.m MS'-y-b+ y'sin.m M

Page  178 178 ANALYTICAL GEOMETRY. These values of x and y substituted in the equation of the parabola referred to V as the origin which is y2=2px (1) will give b2 +2by'sin.m+y'2sin. 2m=- 2pc+2px'+2py'cos.m (2) Because P is on the curve, b2-2pc, and because RM is parallel to the tangent P Y, we also have (Prop. 3,) sin.m p cos.m b Whence 2by'sin.m=2py' cos.m By means of these relations we can reduce eq. (2) to yl2sin.2m=2px' Or _y,= 2p____ Or yJ"'sin.2m If we denote 2p sin.2m by 2p' the equation of the curve referred to the origin P and the oblique axes PX, PY, becomes y': =2p'x' an. equation of the same form as that before found when the vertex V was the origin and the axes rectangular. Cor. 1. Since the equation gives y'-=4-2p'x', that is for every value of x' two values of y', numerically equal, it follows that every diameter of the parabola bisects all chords of the curve drawn parallel to a tangent through the vertex of the diameter. Cor. 2. The squares of the ordinates to any diameter of the parabola are to each other as their corresponding abscissas. Let x, y and x', y' be the co-ordinates of any two points in the curve, then y2-2p'x y'2=2p'z' Whence Y2 X yt2 XI

Page  179 THE PARABOLA. 179 Or y2.: y2. x: x' Cor. 3. By a process in no respect differing from that followed in proposition 3 we shall find yy'- p'(x+x') for the equation of a tangent line to the parabola when referred to any diameter and the tangent drawn through its vertex as the co-ordinates axes. If, in this equation, we make y=O we get x+x'-O or x=-x'. That is, the subtangent on any diameter of the parabola is bisected at the vertex of that diameter. SCHOLIUM.-Projectiles, if not disturbed by the resistance of the atmosphere, would describe parabolas. A Let P be the point from which a projec- v tile is thrown in any direction PHr. Undisturbed by the atmosphere and by gravity, it would continue to move in that line, describ- D ing equal spaces in equal times. But grav- L ity causes bodies to fall through spaces proportional to the squares of the times. From P draw PL in the direction of a plumb line, the direction in which bodies fall when acted upon by gravity alone, and draw from A, T, H, etc., points taken at pleasure on PH, lines parallel to PL. Make AB equal to the distance through which a body starting from rest, would fall while the undisturbed projectile would move through the space PA, and lay off TV to correspond to the proportion PA: PT":: AB: TV (1) Also lay off ZK to correspond to the proportion PA2 P- "::AB: H (2) In the same way we may construct other distances on lines drawn from points of PH parallel to PL. Now through the points B, V, K, etc., draw parallels to PH, intersecting PL in C, D, L, etc., and through the points B, V,

Page  180 180 ANALYTICAL GEOMETRY. K, etc., trace a course. This curve will represent the path described by a projectile in vacuo, and will be a parabola. Because AB is parallel to PC, and PA parallel to BC, the figure PABCis a parallelogram, and so are each of the other figures,.PTVD, PHKL, etc. Let PA=y, PT=y', P-H=y" etc. and PC=x, PD=x', PL=x" etc. Then proportions (1) and (2) become respectively y2' y2::x' X y2: y" "2:: xit But by corollary 2 of this proposition, the curve that possesses the property expressed by these propositions is the parabola, and we therefore conclude that the path described by a projectile in vacuo is that curve. PROPOSITION VII. The parameter of any diameter of the parabola is four times the distance from the vertex of that diameter to the focus. We are to prove that 2p'-4PF..Y Q Let the angle YPR=m as before. Then by (Prop. 3,) sin._ p (1) cos. m b The co-ordinates of the point P being c, b, as in the last proposition, we have 62=2pc. (2) From eq. (1) b2sin.2m-p2cos2m. -p2(1 —sin.2m)=p2 —-p2sin.2m. Or sin.2m= p2 _ p2 p b2+p2 2pc+p2 2c+p But in the last proposition 2P =2p Whence sin.2M sin. 2m-P. p

Page  181 THE PARABOLA. 181 Therefore p'2-2c+p. Or 2p'=4( c-) But ( c+) =PF. (Prop. 1.) Hence 2p', the parameter of the diameter PR, is four times the distance of the vertex of the diameter from the focus. ScHoLITUM.-Through the focus F draw a line parallel to the tangent P Y. Designate PR by x, and RQ by y. Then, by (Prop. 6), y2 =2p'x. But PF=FT, (Prop. 4, Cor. 2.) And PR= TF, because TFRP is a parallelogram. Whence PR=PF; and, since PR=x, and PF=c+E 2''therefore 4x=-4(c+P) =2p', or x-P This value of x put in the equation of the curve gives y=p', or 2y=2p'. That is, the quantity 2p', which has been called the parameter of the diameter PR, is equal to the double ordinate passing through the focus. PROPOSITION VIII. If an ordinate be drawn to any diameter of the parabola, the area included between the curve, the ordinate and the corresponding abscissa, is two-thirds of the parallelogram constructed upon these co-ordinates. Let V'P'PQ be a portion of a p parabola included between the arc S VTP'P, and the co-ordinates V'Q, Si PQ of the extreme point P, re- o! ferred to the diameter V' Q and the T TVQ' L Q tangent through its vertex. 16

Page  182 182 ANALYTICAL GEOMETRY Take a point, P', on the curve between P and V'; draw the chord PP' and the ordinates PQ, P' Q'. Through N, the middle point of PP', draw the diameter NS, and at P and P' draw tangents to the parabola intersecting each other at M and the diameter V' Q produced at T and T'. The tangents at the points P and P' have a common subtangent on the diameter VS, because these points, when referred to this diameter and the tangent at its vertex, have the same abscissa, VA, (Cor. 3, Prop. 6). The point 1M is therefore common to the two tangents and the diameter VS produced. By this construction we have formed the trapezoid PQQ'P' within, and the triangle TMT' without, the parabola, and we will now compare the areas of these figures. From N draw NL parallel to PQ, and from Q draw QO perpendicular to P' Q', and let us denote the angle YV' Q that the tangent at V' makes with the diameter V' Q by m. By the corollary just referred to we have V' T- V'Q and V'T'= V'I Q'. Whence T' T- Q' Q; and because N is the middle point of PP' we also have NL PQ+P Qe 2 Therefore (Th. 34, B. I, Geom.,) the area of the trapezoid PQQ P is measured by NLx QO= NLx Q' Q sin.m= Q' Q x NL sin.m. But NL sin.m is equal to the perpendicular let fall from N upon Q' Q which is equal to that from M upon the same line. Hence the area of the triangle TMT' is measured by i T' Tx NL sin.m —= Q Q x NL sin.m. The area of the trapezoid is, therefore, twice that of the triangle. If another point be taken between P' and V', and we make with reference to it and P the construction that

Page  183 THE PARABOLA. 183 has just been made with reference to P' and P, we shall have another trapezoid within, and triangle without, the parabola, and the area of the trapezoid will be twice that of the triangle. Let us suppose this process continued until we have inscribed a polygon in the parabola between the limits P and V'; then, if the distance of the consecutive points P, P', etc., be supposed indefinitely small, it is evident that the sum of the trapezoids will become the interior curvilinear area PP' V Q, and the sum of the triangles the exterior curvilinear area TP V' V. Since any one of these trapezoids is to the corresponding triangle as two is to one, the sum of the trapezoids will be to the sum of the triangles in the same proportion. But the interior and exterior area together make up the triangle PQT. Therefore interior area=-APQ T, and APQT= TQx PQsin.m= V' Qx PQsin.m. But V' Qx PQ sin. ma measures the area of the parallelogram constructed upon the abscissa V' Q and the ordinate PQ. We will denote V' Q by x and PQ by y. Then the expression for the area in question becomes Ixy.sin.m Cor. When the diameter is the axis of the Q P parabola, then m=900, and sin. m=1, and the expression for the area becomes Wxy. That is, every segment of a parabola at right angles v D with the axis is two-thirdsaits circumscribing rectangle. P R O P OSITION IX. Tofind the general polar equation of the parabola. Let P be the polar point whose co-ordinates'referred to the principal vertex, V, are c and b. Put VD=x, and DM

Page  184 184 ANALYTICAL GEOMETRY. -y; then by the equation of the curve we M have y2=2px. (1) Put PM-=R, the angle MPX=-m, then V Dy7E we shall have VD=x=c+B cos. m. DDM-=y=b+R sin. m. These values of x and y substituted in eq. (1) will give (b+R1 sin. m)2= 2p(c+R cos. m). (2) Expanding and reducing this equation, (R being the variable quantity), we find B2 sin.2m+ 2R(b sin. m-p cos. m)=2pc —b2 for the general polar equation of the parabola required. Cor. 1. When P is on the curve, b2=2pc, and the general equation becomes R2sin.2m+2R(b sin.m-p cos.m)=O. Here one value of B is 0, as it should be, and the other value is.R_2(p cos. m-b sin. m) sin.2m When mn=90, cos.m-=0 and sin.m=1. Then this last equation becomes R=-2b, a result obviously true. Cor. 2. When the pole is at the focus F, then b=0, and c=P, and these values reduce the general equation to R2sin.2m —2Rp cos.m-=p2. But sin.2m-=1cos.2m. Whence R2 —R2cos.2m —2Rp cos.m-p2. Or R2=p2 + 2Rp cos.m+R2cos.'2m. Or R=p+.R cos.rn. Whence R= P 1 cos. wme and this is the polar equation when the focus is the pole.

Page  185 THE PARA.BOLAo 185 When m=O, cos.m=1, and then the equation becomes R-= P or Br-P= infinite, 1-1' O showing that there is no termination of the curve at the right of the focus on the axis. When m=90~, cos.m=O, then RBp, as it ought to be, because p is the ordinate passing through the focus. When m=1800, cos.m=-1-, then R-=p; that is, the distance from the focus to the vertex is Jp. As m can be taken both above and below the axis and the cos. m is the same to the same arc above and below, it follows that the curve must be symmetrical in respect to the axis. SCHoLIUM 1.-If we take p for the unit of measure, that is, assume p=l, then the general polar equation will become R2sin.2m+2R(b sin.m —cos.m)=2c —b2. Now if we suppose m=900, then sin.m=l, cos. m=0, and R would be represented by the line PM7, and the equation would become R2+2bR=(2c —b2), and this equation is in the common form of a quadratic. Hence, a parabola in whichp=1 will solve any quadratic equation by making c= VB, BP=b, then PM' will give one value of the unknown quantity. To apply this to the solution of equations, we construct a parabola as here represented. Now, suppose we require the value of +- M' y, by construction, in the following equa- + tion,.1 P y-2+2y=8. 0X1F 2 3 4 5 Here 2b=2, and 2c-b2=8. Whence b==, and c=4.5. \ Lay off c on the axis, and from the ex- M tremity lay off b at right angles, above the axis if b is plus, and below if minus. This being done, we find P is the polar point corresponding to16*

Page  186 186 ANALYTICAL GEOMETRY. this example, and PM'-=2 is the plus value of y, and PM=-4 is the minus value. Had the equation been y2 —2y=8, then P' would have been the polar point, and P'.M=4 the plus value, and P'M=-2 the minus value. For another example let us construct the roots of the following equation: y2-6y= — 7. Here b=-3, and 2c —b2=-7. Whence c=1. From 1 on the axis take 3 downward, to find the polar point P'. Now the roots are P"m and P"rm', both plus. P'm=1.58, and P'm'= —4.414. Equations having two minus roots will have their polar points above the curve. When c comes out negative, the ordinates cinnot meet the curve, showing that the roots would not be real but imaginary. The roots of equations having large numerals cannot be constructed unless the numerals are first reduced. To reduce the numerals in any equation, as y2+72y= 146, we proceed as follows: Put y=nz, then n2z -+72nz=146 z +72 z 146 Now we can assign any value to n that we please. Suppose n=10, then the equation becomes z2 +72z —1.46. The roots of this equation can be constructed, and the values of y are ten times those of z. SCHOLIUM 2.-The method of solving quadratic equations employed in Scholium 1 may be easily applied to the construction of the square roots of numbers. Thus, if the square root of 20 were required, and we represent it by y, we shall have,2.20.

Page  187 THE PARABOLA. 187 an incomplete quadratic equation; but it may be put under the form of a complete quadratic by introducing in the first number the term ~ 0 xy, and we shall then have ys ~ 0 xy=20. Here 2b=0, and 2c —b =20; whence c-10; which shows that the ordinate corresponding to the abscissa 10 on the axis of the parabola will represent the square root of 20. In the same way the square roots of other numbers may be determined EXAMPLES. 1. What is the square root of 50 7 Let each unit of the scale represent 10, then 50 will be represented by 5. The half of 5 is 2J. An ordinate drawn from 2~ on the axis of X will be about 2.24, and the square root of 10 will be represented by an ordinate drawn from 5, which will be about 3,16. Hence, the square root of 50 cannot differ much from (2.24) (3.16) =7,0786. ANOTHER SOLUTION. 50=25 x2, V/50=5v2. From 1 on the axis of X draw an ordinate; it will measure 1.4+. Hence, /50=5(1.4+)=7,+.'W hat is the square root of 175? 175=25 x 7, /175=5x 7. An ordinate drawn from 3.5 the half of 7 will measure 2.65. Therefore /Vi75=5(2.65)-13.25 nearly. 3. Given xa —lxz=8 to find T. Ans. x=2.9.+ 4. Given X-2 +=x-T tofind x. Ans. x-0.60+. 5. Given ~y2s —-y=2 to find y. Ans. y=3.17, or-2.5+.

Page  188 188 ANALYTICAL GEOMETRY. CHAPTER V. THE HYPERBOLA. To describe an hyperbola. The definition of this curve suggests the following method of describing it mechanically: Take a ruler EI', and fasten one end at the point F', on which the ruler may turn as a hinge. At the \ IP other end of the ruler attach a thread, and let its length be less than that of'' As the ruler by the given line A'A. Fasten the other end of the thread at F. With a pencil, P, press the thread against the ruler and keep it at' equal tension between the points H and F. Let the ruler turn on the point E', keeping the pencil close to the ruler and letting the thread slide round the pencil; the pencil will thus describe a curve on the paper. If the ruler be changed and made to revolve about the other focus as a fixed point, the opposite branch of the curve can be described. In all positions of P, except when at A or A', PF' and PF will be two sides of a triangle, and the difference of these two sides is constantly equal to the difference between the ruler and the thread; but that difference was made equal to the given line A'A; hence, by definition, the curve thus described must be an hyperbola. PROPOSITION I. To find the equation of the hyperbola referred to its center and axes.

Page  189 THE HY PERBOLA. 189 Let C be the center, F and.F' the foci, and AA' the transverse axis of an hyperbola. Draw CC' at right angles to AA', and take these lines for the co-ordinate axes. From P, C any point of the curve, draw PF, PF' to the foci, and PH perpendicular to AA'. Make CF-c, CA=A, CH-=x, and PH=y; then the equation which expresses the relation between the variables x and y, and the constances c and A, will be the equation of a hyperbola. By the definition of the curve we have r —r=2A. () The right-angled, PEHP gives r2=(xc)2+y2. (2) The right-angled APPF' gives r'2=(x+c)2+y2. (3) Subtracting eq. (2) from eq. (3) we get r2-r2-=-4cx. (4) Dividing eq. (4) by eq. (1) we have 2cx Combining eqs. (1) and (5) we find r'=A+C, and r=-A+. This value of r substituted in eq. (2) gives ~2,2 A2+2cx+9X2 x2_2cx c2+Y2. A2 Reducing, we find A2y2+ (A2_C2)X2=A 2(A2__cC for the equation sought. SCHOLIUM. —AS is greater than A, it follows that (A2-c2) must be negative; therefore we may assume this value equal to -B2. Then the equation becomes A2y2-B2 x2= A2B2.

Page  190 190 ANALYTI CAL GEOMET'RY. This form is preferred to the former one on account of its similarity to the equation of the ellipse, the difference being only in the negative value of B2. Because A -c2= —-B2, A2+.B=-cg Now to show the geometrical magnitude of B, take C as a center, and CF as a radius, and describe the circle TEF'. From A draw AH at right,. A angles to CF. Now CH=c, CA=A, and if we put AH=B, we shall have A2+B2=c2, as above. Whence AH must equal B. PROPOSITION II. To determine the figure of the hyperbola from its equation. Resuming the equation A2y2-B2X2=-A2B2, and solving it in respect to y, we find Y- ma,ix2_A2, If we make x=O, or assign to it any value less than A, the corresponding value of y will be imaginary, showing that the curve does not exist above or below the line A'A. If we make x=A, then y=-0, showing two points in the curve, both at A. If we give to x any value greater A C A than A, we shall have two values of y, numerically equal, showing that the curve is symmetrically divided by the axis A'A produced. If we now assign the same value to x taken negatively, that is, make x (-x), we shall have two other values of.y, the same as before, corresponding to the left branch of the curve. Therefore, the two branches of the curve are

Page  191 THE HYPERBOLA. 191 equal inmagnitude, and are in all respects symmetrical but opposite in position. Hence every diameter, as DD', is bisected in the center, for any other hypothesis would be absurd. SCHOLIUM 1. —If through the center, C, we draw CD, CD', at right angles to A'A, \ D and each equal to B, we can have two opposite A branches of an hyperbola passing through D -f and D' above and below C. as the two others which pass through the points A' and A, at / I the right and left of C. The hyperbola which passes through D and D' is said to be conjugate to that which passes through A and A', or the two are conjugate to each other. DD' is the conjugate diameter to A'A, and DD' may be less than, equal to, or greater than A'A, according to the relative values of c and A in Prop. 1. When B is numerically equal to A, the equation of the curve becomes y2-x2 =-A2, and DD'=AA'. In this case the hyperbola is said to be equilateral. SCHOLIUM 2.-To find the value of the double ordinate which passes through the focus, we must take the equation of the curve A2y2-B2x2 =-A2B2, and make x=c, then Ay2=B-2(c2-A2). But we have shown that A2 +B2 —c2, or B2=c2 2... Whence A2y2 —B4 Or Ay=B', or 2y 2B That is, 2A: 2B ~: 2B: 2y, showing that the parameter of the hyperbola is equal to the double ordinate, to the major axis, that passes through the focus. SCHOLIUM 3.-To find the equation for the conjugate hyperbola which passes through the points D, D', we take the general equation A2y2 — Bx2 =- -A2B2,

Page  192 192 ANALYTICAL GEOMETRY. and change A into B and x into y, the equation then becomes B2x2-A2y2 =-A2B2, which is the equation for conjugate hyperbola. PROPOSITION III. To find the equation of the hyperbola when the origin is at the vertex of the transverse axis. When the origin is at the center, the equation is A2y2_B2X2_ _A2_B2. And now, if we move the origin, to the vertex at the right, we must put x-=A+x'. Substituting this value of x in the equation of the hyperbola referred to its center and axes, we have A2y2-B2x'2-2B2Ax'=0. We may now omit the accents, and put the equation under the following form: which is the equation of the hyperbol+2Ax), which is the equation of the hyperbola when the origmn is the vertex and the co-ordinates rectangular. PROPOSITION IV. To find the equation of a tangent line to the hyperbola, the origin being the center. In the first place, conceive a line cutting the curve in two points, P and Q. Let x and y be co-ordinates of any point on the line, as S, x' p and y' co-ordinates of the point P on the curve, and x" and y" the co- s/ ordinates of the point Q on the.. cuirve.

Page  193 THE HYPERBOLA. 193 The student can now work through the proposition in precisely the same manner as Prop. 6, of the ellipse was worked, using the equation for the hyperbola in place of that of the ellipse, and in conclusion he will find A2yy' — B2xx' =-A2B2, for the equation sought. Cor. To find the point in which a tangent line cuts the axis of X, we must make y=O, in the equation for the tangent; then A2 x= CT. P' CT FD N If we subtract this from CD (x') we shall have the subtangent TD-_'-A2 x2_A2 PROPOSITION V. To find the equation of a normal to the hyperbola. Let a be the tangent of the angle that the line TP makes with the transverse axis, (see last figure), and a' the same with reference to the line PN. Then if PN is a normal, it must be at right angles to PT, and hence we must have aa'+1=O. (1) Let x' and y' be the cor-ordinates of the point P on the curve, and x, y, the co-ordinates of any point on the line PN, then we must have y-y'=a'(x —x'). (2) In working the last proposition, for the tangent line PTwe should have found B2x' a A2y' This value of a put in eq. (1) will show us that a'- A2Y' B2x 17 N

Page  194 194 ANALYTICAL GEOMETRY. And this value of a' put in eq. (2) will give us y-y =- Ay' (X-x')) for the equation of the normal required. Cor. To find the point in which the normal cuts the axis of X, we must make y=0. This reduces the equation to 1= m_X-x',). A2 If we subtract C-D, (x'), from CON, we shall have DN, the sub-normal. That is, (A2+B) )x' —x':B,' the sub-normal. \A,,]*2 A2 PROPOSITION VI. A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci. If we can prove that F'P: PF:: FIT:TF, (1) it will then follow (Th. 24, B. II, Geom.,) that the angle P'PT=the angle TPF. In Prop. 1, of the hyperbola, we find that ~F'P r'=A+Cex and PF r= —A+CX, A and by corollary to Prop. 4 A 2 A 2 1' T=F' C+ CT=c+_, and TF=e —_, x x We will now assume the proportion /A+C. -Aex + (2,

Page  195 THiE HYPERBOLA. 195 Multiply the terms of the first couplet by A, and those of the last couplet by x, then we shall have (A2 +cx): ( A2 +cx):: (cx+A 2) XZ. Observing that the first and third terms of this proportion are equal, therefore xz=cx —A2. Or z=c-_ A TP. Now the first three terms of proportion (2) were taken equal to the first three terms of proportion (1), and we have proved that the fourth term of proportion (2) must be equal to the fourth term of proportion (1), therefore proportion (1) is true, and consequently F' PT= TPF. Cor. 1. As TT' is a tangent, and PN its normal. it follows that the angle TPN= the angle T'PN, for each is a right angle. From these equals take away the equals TPF, T'PQ, and the remainder EPN must equal the remainder QPN. That is, the normal line at any point of the hyperbola bisects the exterior angle formed by two lines drawn from the foci to that point. A2 Cor. 2. The value of CT we have found to be A, and the value of CD is x, and it is obvious that A2 A.:A: x, is a true proportion. Therefore (A) is a mean proportional between CT and CD. A tangent line can never meet the axis in the center, because the above proportion must always exist, and to make the first term zero in value, we must suppose x to be infinite. Therefore a tangent line passing through the center cannot meet the hyperbola short of an infinite distance therefrom. Such a line is called an asymptote.

Page  196 196 ANALYTICAL GEOMETRY. OF THE CONJUGATE DIAMETERS OF THE HYPERBOLA. DEFINITION.- Two diameters of an hyperbola are said to be conjugate when each is parallel to a tangent line drawn through the vertex of the other. According to this definition, GG' and IH' in the adjoining figure are conjugate diameters. EXPLANATION. 1. —The tangent line which passes through the point H is parallel to CG. Hence CG makes the same \ D angle with the axis as that tangent line A does. H C If we designate the co-ordinates of the point H, in reference to the center and axes by x' and y', and by a the tangent of the angle made by the inclination of CG with the axis, then in the investigation (Prop. 6,) we find BYx' a _ _-.. (1) A'y' Now if we designate the tangent of the angle which CI makes with the axis by a', the equation of CH must be of the form because the line passes through the center. because the line passes through the center. Whence a' —. (2) (2) Multiplying eqs. (1) and (2) together member by member, and we find aa Bto which equation all conjugate diameters must correspond. EXPLANATION 2. —If we designate the angle GCB by n, and HCB by m, we shall have sin. m sin. n -.-a-=a,.=a. COS. m cos. n B2 And tan. m tan. n=-. A'

Page  197 THE HYPERBOLA. 197 PROPOSITION VII. To find the equation of the hyperbola referred to its center and conjugate diameters. The equation of the curve referred to the center and axes is A2y- tB2X2=_A2_B. Now, to change rectangular co-ordinates into oblique, the origin being the same, we must put x-x' cos. m+y' cos. n } Chap. 1, Prop. 9. And y-x' sin. m+y' sin. nj These values of x and y, substituted in the above general equation, will produce (sin. 2n A2-cos. 2n 2)y'2+(sin.2 mA2 —cos. 2mB)x'2 1 { + 2(sin.m sin.nA2-cos.m cos.n B2)x'y' — A2B2. (1) Because the diameters are conjugate, we must have sin. m sin. n 32 cos. m cos. n A2 Whence (sin. m sin. n A2 —cos. m cos. n B2)=0 (k) This last equation reduces eq. (1) to (sin.2nA2 —cos.2nB2)y'2 + (sin.2mA 2 —cos.2nB2)x'2=-A2B2 (2) which is the equation of the hyperbola referred to the center and conjugate diameters. If we make y'=O, we shall have 12= _A2B-2 2 2 (3) (sin.2mA2-cos. mB2) If we make x'=O, we shall have C \ y'2 -A2B2 (4) (sin.2nA2 —os.2nB2)=CG2 (4). D' If we put A'2 to represent CH2, and regard it as positive, the denominator in eq. (3) must be negative, the nu17*

Page  198 198 ANALYTIC AL GEOMETRY. merator being negative. That is, sin.2mA2 must be less than cos.2mB2. That is, sin.2mA2<cos.2m2 m. Or tan. m<A. But tan. m tan. n=. A2& Whence tan. n> A, or, sin.2 nA2>B2 Cos.2 n. Therefore the denominator in eq. (4) is positive, but the numerator being negative, therefore CG2 must be negative. Put it equal to —B'2. Now equations (3) and (4) become Al _ A~ _2_B2 -A2B2 (sin.2 mA2 —-cos.2mB2)' (sin.2 nA2-cos.2 nB2)? Or (sin. 2mA - cos.2mB2)= A,,'2 2 (sin.2nA 2-cos.2nB2) 2B2 Comparing these equations with eq. (2) we perceive that eq. (2) may be written thus: A2B2 Ar2_ B2al2 A2B2. s/2 A/2 Whence A'2y'2 — B12X2= —A 2B12. Omitting the accents of x' and y', since they are general variables, we have A 2y2 _B2X2=-A 2.B2, for the equation of the hyperbola referred to its center and conjugate diameters. SCHOLIUM 1.-As this equation is precisely similar to that referred to the center and axes, it follows that all results hitherto determined in respect to the latter will apply to conjugate diameters by changing A to A' and B to B', For instance, the equation for a tangent line in respect to the center and axes has been found to be Avy'- B'xx'= -ABa.

Page  199 THE HYPERBOLA. 199 Therefore, in respect to conjugate diameters it must be A'" yy' —B'2xx' — A'2B'2, and so on for normals, sub-normals, tangents and sub-tangents. SCHoLIUM 2.-If we take the equation A'2y2_ B2X2 — _A'2B1', and resolve it in relation to y, we shall find Y I that for every value of x greater than A' we shall find two values of y numerically equal, N which shows that ON bisects JIM and every X X line drawn parallel to MM, or parallel to a M tangent drawn through L, the vertex of the diameter LL'. Let the student observe that these several geometrical truths were discovered by changing rectangular to oblique co-ordinates. We will now take the reverse operation, in the hope of discovering other geometrical truths. Hence the following: PROPOSITION VIII. To change the equation of the hyperbola in reference to any system of conjugate diameters, to its equation in reference to the axes. The equation of the hyperbola referred to conjugate diameters is A'2y'2-_B/2X/2=-Al' 2B/2 To change oblique to rectangular co-ordinates, the formulas are (Chap. 1, Prop. 10,)'=_x sin. n-y cos. n,_y cos. m-x sin. m sin. (n-m) sin. (n —m) Substituting these values of x' and y' in the equation, we shall have 2(y COS. m-x sin. r)2 3'2(x Sin. n-y cos. n)2= A'2B'2 sin.2 (n —m) sin.2 (n-rm) By expanding and reducing, we shall have

Page  200 200 ANALYTICAL GEOMETRY. (A cos.2m-B'2cos.2n)y2+(A'2sin.2m-B'2sin.2n)x2 2(-A'2sin.m cos.m+B'2sin.n cos.n)xy — A'2B'2 sin.2(n —m). which, to be the equation of the hyperbola when referred to the center and axes, must take the well known form, A2y2 —B2x2=-A2B2. Or in other words, these two equations must be, in fact, identical, and we shall therefore have A'2 cos.2m Br2 cos.2n=A2. (1) A'2 sin.2 mB2 sin..2 n-B2. (2) -A'2 sin. m cos. m+ B'2 sin. n cos. n= 0. (3) A1'2 B'2 sin.2(n-m)= -A2B2. (4) By adding eqs. (1) and (2), observing that (cos.2m+ sin.2 m)- =l, we shall have A"2_B2 —A2 B2 Or 4A'2 —4B'2=4A —4,B2 which equation shows this general geometrical truth: That the difference of the squares of any two conjugate diameters is equal to the difference of the squares of the axes. Hence, there can be no equal conjugate diameters unless A=B, and then every diameter will be equal to its conjugate: that is, A'=B'. Equation (3) corresponds to tan.mtan.n= B2 the equaA2' tion of condition for conjugate diameters. Equation (4) reduces to D G A'B' sin. (n-m)=AB. A The first member is the measure C of the parallelogram GCffT, and it being equal to Ax B, shows this geometrical truth: That the parallelogram formed by drawing tangent lines through the vertices of any system of conjugate diameters of

Page  201 THE HYPERBOLA. 201 the hyperbola, is equivalent to the rectangle formed by drawing tangent lines through the vertices of the axes. REMARK.-The reader should observe that this proposition is similar to (Prop. 13,) of the ellipse, and the general equation here found, and the incidental equations (1), (2), (3), and (4), might have been directly deduced from the ellipse by changing B into B 4V1, and B' into B'V,1. OB THE ASYMPTOTES OF THE HYPERBOLA. DEFINITION.-If tangent lines be drawn through the vertices of the axes of a system of conjugate hyperbolas, the diagonals of the rectangle so formed, produced indefinitely, are called asymptotes of the hyperbolas. Let AA', 1B3B', be the axes of I1 D conjugate hyperbolas, and through the vertices A, A', B, B', let tangents to the curves be drawvn form- X ing the rectangle, as seen in the figure. The diagonals of this rectangle produced, that is, DD' and D BE', are the asymptotes to the curves corresponding to the definition. If we represent the angle D)CX by m, E' CX will be m also, for these two angles are equal because CB= CB'. It is obvious that tan. m —. sin..2 m B2'Whence2 -- COS2. m A2 But cos.2 m=l —sin.2m. Therefore sin. 2m X1 1 -sin.m A2

Page  202 202 ANALYTICAL GEOMETRY. B2 and c2 =A2 Consequently sin.2 m= A2+B2' and cos.2 m= which equations furnish the value of the angle which the asymptotes form with the transverse axis. PROPOSITION IX. To find the equation of the hyperbola, referred to its center and asymptotes. Let CM=x, and PM-y. Then the equation of the curve referred to its center and axes is A2Uy2B2X2=-A2B2. (1) From P draw PH parallel to CE, and PQ parallel to CM. Let CH=x', and HP=-y. Now the object of this proposition is to find the values of x and y in terms of C < A hM x' and y', to substitute them in eq. (1). The resulting equation reduced to its most simple form will be the equation sought. The angle IfCM is designated by m, and because HP is parallel to CE, and PQ parallel to CM, the angle JHPQ is also equal to m. Now in the right angled triangle CGHh we have Hh =x' sin. Dn, and Chex' cos. m. In the right angled triangle PQH we have HQ y' sin. m, and PQ=y' cos. m. Whence HIh-HfQ-Qh-=PM=y=x' sin. m-y' sin. m. Or y-(x'- y) sin. m. (2) Ch+ QP= CMG-x=x' cos. m+y' cos. m. Or x=(x'+y') cos. m. (3) These values of y and x found in eqs. (2) and (3) substituted in eq. (1) will give

Page  203 THE HYPERBOLA. 203 A2(x'-_y)2 sin.2 m —B2 (x'+y')2 cos.2 m=-A2P2. Placing in this equation the values of sin.2m and cos.2m, previously determined, we have A2B2 A A2B2 2_ 2B- _ __ (XB2_ -B2 A2-+-(x' +B2 Dividing through by A2B2, and at the same time multiplying by (A2+B2), we get (~, —y,)2_(X +y)2 -(A 2+ 2). Or -4x'y'-= (A2+B2). Or x'yt='A2 —2+' 4 - which is the equation of the hyperbola referred to its center and asymptotes. Cor. As x' and y' are general variables, we may omit the accents, and as the second member is a constant quantity, we may represent it by 1/2. Then 3f2 xy=M2, or xM y This last equation shows that x increases as y decreases; that is, the curve approaches nearer and nearer the asymptote as the distance from the center becomes greater and greater. But x can never become infinite until y becomes 0; that is, the asymptote meets the curve at an infinite distance, corresponding to Cor. 2, Prop. 6. PROPOSITION X. All parallelograms constructed upon the abscissas, and ordinates of the hyperbola referred to its asymptotes are equivalent, each to each, and each equivalent to'AB. Let x and y be the co-ordinates corresponding to any point in the curve, as P. Then by the equation of the curve in relation to the center and asymptotes, we have xy=M2. (1)

Page  204 204 ANALYTICAL GEOMETRY. Also let x', y', represent the co-ordinates r of the point Q. Then q iP Xly'=M2. (2) - /Q The angle p CD between the asymptotes c iA we will represent by 2m. Now multiply D\ both members of equations (1) and (2) by sin. 2m. Then we shall have xy sin. 2m=M2 sin. 2m. (3) x'y' sin. 2m=-M2 sin. 2m. (4) The first member of eq. (3) represents the parallelogram CP, and the first member of eq. (4) represents the parallelogram CQ; and as each of these parallelograms is equivalent to the same constant quantity, they are equivalent to each other. Now A is another point in the curve, and therefore the parallelogram AHCD is equal to (M2 sin. 2m), and therefore equal to CQ, or CP. Hence all parallelograms bounded by the asymptotes and terminating in a point in the curve, are equivalent to one another, and each equivalent to the parallelogram AHCD, which has for one of its diagonals half of the transverse axis of A. We have now to find the analytical expression for this parallelogram. The angle HCA=m, A CD-m, and because AH is parallel to CD, CAH-=m. Hence, the triangle CAH is isosceles, and C= —HA. The angle AHq=2m. Now by trigonometry sin. 2m: A: sin. m: Hff. But sin. 2m=2 sin. m cos. m. Whence 2 sin. m cos. m: A: sin. m-: fH. C-f A. 2 cos.m Multiply each member of this equation by CA=A, and sin. m, then

Page  205 THE HYPERBOLA. 205 A2 sin. m A2 A.(CH)sin. m=.m tatn.m. 2 cos. m 2 The first member of this equation represents the area of the parallelogram CHAD, and the tan. m=. Hence, the parallelogram is equal A.B —AB, which is the value 2A also of all the other parallelograms, as CQ, CP, etc. PROPOSITION XI. To find the equation of a tangent line to the hyperbola referred to its center and asymptotes. Let P and Q be any two points on the curve, and denote the co-ordinates of the first by x', y', and of the second by x", y". The equation of a straight line pass- P ing through these points will be of the form y-y'=a(x-x') (1) in which a= I —Y" We are now to find the value of a when the line becomes a tangent at the point P. Because P and Q are points in the curve, we have x'y'- =Xy". From each member of this last equation subtract x'y", then ty' -xfy"- x"y"-xty". Or x'(y' —")=-y"(x' —x). Whence a=-y"- Y" x - X' This value of a put in eq. (1) gives y (x-x). () 18

Page  206 206 ANALYTICAL GEOMETRY. Now if we suppose the line to revolve on the point P as a center until Q coincides with P, then the line will be a tangent, and x'=x",-and y'=y", and eq. (2) will become which is the equation sought. Cor. To find the point in which the G tangent line meets the axis of X, we must make y=O; then x=2x'. That is, Ct is twice CR, and as BRP and CT are parallel, tP=PT. \ A tangent line included between the asym.p- T. \a totes is bisected by the point of tangency. D. ScHroLIUM.-From any point on the asymptote, as D, draw DG parallel to Tt, and from C draw CP, and produce it to S. By scholium 2 to Prop. 7 we learn that CP produced will bisect all lines parallel to t and within the curve; hence gd is bisected in S. But as CP bisects tT, it bisects all lines parallel to tT within the asymptotes, and D G is also bisected in S; hence dD= Gg. In the same manner we might prove dh=kv, because hk is parallel to some tangent which might be drawn to the curve, the same as D G is parallel to the particular tangent t T. Hence, If any line be drawn cutting the hyperbola, the parts between the asymptotes and the curve are equal. This property enables us to describe the hyperbola by points, when the asymptotes and one point in the curve are given. Through the given point d, draw any line, as DG, and from G set off Gg=dD, and then g will be a point in the curve. Draw any other line, as hk, and set off kv=dh; then v is another point in the curve. And thus we might find other points between ^ and g, or on either side of v and g.

Page  207 THE HYPERBOLA. 207 PROPOSITION XII. To find the polar equation of the hyperbola, the pole being at either focus. Take any point P in the hyperbola, P and let its distance from the nearest C G focus be represented by r, and its distance from the other focus be repre- A A c A< F H sented by r'. Put CH=x, CF=c, and CA=A. Then, by Prop. 1, we have r= -A+czA (1) r'-A+ cx (2) A' Now the problem requires us to replace the symbol x, in these formulas, by its value, expressed in terms of r and r', and some function of the angle that these lines make with the transverse axis. _First.-In the right-angled triangle PFH, if we designate the angle PIEH by v, we shall have 1: r:: cos. v: FT=Ir cos. v. CH-1=CF+FPH. That is, x=c+r cos. v. The value of x put in eq. (1) gives C2+Acr cos. v r-A A' Whence r — c2-A2 (<) A-c cos. v Second.-In the right-angled triangle F'PIH, if we designate the angle PF'" by v', we shall have 1: r':: cos. v': FI-=r' cos. v'. But F'l-=F' C+ C. That is, r' cos. v'=c+x. Or x-r' cos. v' —c,

Page  208 208 ANALYTICAL GEOMETRY. and this value of x put in eq. (2) gives crtcos. v' c2 r-A+ A Whence r A22 (4) A - cos. v' Equations (3) and (4) are the polar equations required. Let us examine eq. (3). Suppose v=0, then cos. v —l, and r C2_A2 r-A -— A-c. A-c But a radius vector can never be a minus quantity, therefore there is no portion of the curve on the axis to the right of F. To find the length of r when it first strikes the curve, we find the value of the denominator when its value first becomes positive, which must be when A becomes equal to c cos. v; that is, when the denominator is 0. the value of r will be real and infinite. If A-c cos. v=0, then cos. v —A This equation shows that when r first meets the curve it is parallel to the asymptote, and infinite. When v=90~, cos. v=0, and then r is perpendicular at c2_.A2.B2 the point F, and equal to, or B. half the parameA A' ter of the curve, as it ought to be. When v=180~, then cos. v —1, and -c cos. v=c; then r= _< —-- c-A = IA c+A a result obviously true..As v increases, the value of r will remain positive, and, consequently, give points of the hyperbola until cos. v again becomes equal to A, which will be when the radius C

Page  209 THE HYPERBOLA. 209 vector makes with the transverse axis an angle equal to 3600 minus that whose cosine is A. Equation (3) will therefore determine all points in the right hand branch of the hyperbola. Now let us examine equation (4). If we make v'=O, then r,= A -=A+c=F'A, A-c as it ought to be. To find when r' will have the greatest possible value, we must put A-c cos. v'-O. Whence cos.v'2 A. This shows that v' is then of such a value as to make r' parallel to the asymptote and infinite in length. If we increase the value of v' from this point, the denominator will become positive, while the numerator is negative, which shows that then r' will become negative, indicating that it will not meet the curve. The value of r will continue negative until the radius vector falls below the transverse axis, and makes with it an angle having + A for its cosine. Values of v between this and 360~ will render r positive and give points of the hyperbola. Equation (4) will, therefore, also determine all the points in the right hand branch of the hyperbola; By changing the sign of c, we change the pole to the focus F', and eqs. (3) and (4), which then determine the left hand branch of the hyperbola, become C2 _A2 A+c cos. v' and'i A2-cc2 (4) A+c cos. v' 10* o

Page  210 210 ANALYTICAL GEOMETRY. GENERAL REMARKS.-When the origin of co-ordinates is at the circumference of a circle, its equation is y- = 2Rx-x2. When the origin of a parabola is at its vertex, its equation is y2=-2px. When the origin of co-ordinates of the ellipse is at the vertex of the major axis, the equation of the curve is B2 y2 = (2Ax- -X2). When the origin of co-ordinates is at the vertex of the hyperbola, the equation for that curve is 2 = 2(2Ax+-2). A2 But all of these are comprised in the general equation y2=2px+qx2. In the circle and the ellipse, q is negative; in the hyperbola it is positive, and in the parabola it is 0. CHAPTER VI. ON THE GEOMETRICAL REPRESENTATION OF EQUATIONS OF THE SECOND DEGREE BETWEEN TWO VARIABLES. 1.-It has been shown in Chap. 1, that every equation of the first degree between two variables may be represented by a straight line. It has also been shown that the equations of the circle, the ellipse, the parabola and the hyperbola were all some of the different forms of an equation of the second degree between two variables. It is now proposed to prove' that, when an equation of the second degree between two variables represents any geometrical magnitude, it is some one of these curves. The limits assigned to this work compel us to be as brief in this investigation as is consistent with clearness. We shall, therefore, restrict ourselves to a demonstration

Page  211 INTERPRETATION OF EQUATIONS. 211 of this proposition; the determination of the criteria by which it may be decided in every case presented, to which of the conic sections the curve represented by the equation belongs, and the indication of the processes by which the curve may be constructed. 2.-The equation of the second degree between two variables, in its most general form, is Ay2+Bxy+ Cx2 +.Dy+Ex+F=O, for, by giving suitable values to the arbitrary constants, A, B, C, etc., every particular case of such equation may be deduced from it. The formulas for the transformation of co-ordinates being of the first degree in respect to the variables, the degree of an equation will not be changed by changing the reference of the equation from one system of co-ordinate axes to another. We may therefore assume that our co-ordinate axes are rectangular without impairing the generality of our investigation. The resolution, in respect to y, of the general equation gives y B A _D B1 X2 x2+2BD x+D2 2A 2A 2A4 C -4AE-4AF Now let AX, A Y be the co-ordinate axes, and draw the straight line MQ, whose equation is B D 2A 2A A D E X For any value, AD, of x, the or- Mel dinate, DC, of this line, is expressed by | x B x P'x 2A 2A' and this ordinate, increased and diminished successively by what the radical part, when real, of the general value of y becomes for the same substitution for x, will give

Page  212 212 ANALYTICAL GEOMETRY. two ordinates, DP, DP', corresponding to the abscissa AD. Since P and P are two points whose co-ordinates, when substituted for x and y, will satisfy the equation, Ay2+Bxy+Cx2+, etc., =0, they are points in the line that this equation represents. By thus constructing the values of y answering to assumed values of x, we may determine any number of points in the curve. In getting the points P and P', we laid off, on a parallel to the axis of y, equal distances above and below the point C; PP' is, therefore, a chord of the curve parallel to that axis, and is bisected at the point C. The solution of the general equation in respect to x, gives x=B- Ei 1 1e l3 y2+2BEly+E2 2C 2C 2 4A C -4CD — 4CF The equation B E X a, —-y —-, is that of a straight line, making, with the axis of y, an angle whose tangent is B and intersecting the axis of X at a distance from the origin equal to — 2C' As above, it may be shown that any value of y that makes the radical part of the general value of x real, responds to two points of the curve, and that the straight line whose equation is B _E 2 0y20C bisects the chord, parallel to the axis of X, that joins these points. By placing the quantity under the radical sign in the value of y equal to 0, we have an equation of the second degree in respect to x, which will give two values for x.

Page  213 INTERPRETATION OF EQUATIONS. 213 If these values are real the corresponding points of the curve are on the line 1MQ; that is, they are the intersections of this line with the curve, since, for each of these values, there will be but one value of y, which, in connection with that of x, will satisfy the general equation, and these values also satisfy the equation, B DI 2A 2A In like manner, placing the quantity under the radical sign in the value of x equal to 0, we shall find two values of y, to each of which there will respond a single value of x, and the points of the curve answering to these values of y will be the intersections of the curve with the line whose equation is B E 2C 2C A diameter of a curve is defined to be any straight line that bisects a system of parallel chords of the curve. From the preceding discussion we therefore conclude, 1. That if an equation of the second degree between two variables be resolved in respect to either variable, the equation that resultsfrom placing this variable equal to that part of its value which is independent of the radical sign will be the equation of that diameter of the curve which bisects the system of chords parallel to the axis of the variable. 2. The values of the other variable found from the equation which results from placing the quantity under the radical sign equal to zero, in connection with the corresponding values of the first variable, will be the co-ordinates of the vertices of the diameter. 3. The formulas for changing the reference of points from a system of rectangular co-ordinate axes to any other system having a different origin are x=a+x'cos. m+y'cos. n. y=b+x'sin. m+y'sin. n.

Page  214 214 ANALYTICAL GEOMETRY. Substituting these values of x and y in the equation Ay2 -Bxy+ Cx2+.Dy+Ex+F -=O developing, and arranging the terms of the resulting equation with reference to the powers of y' and x' and their product, we find (A sin.2 n+B sin. n cos. n+ Ccos.2 n) y' 2 +(A sin.2 m+B sin. m cos. m+ C CoS2 m) X'2 +[2A sin. m sin. n+B (sin. m cos. n +sin n cos. m) +2 C os. m cos. n]x y' +[(2Ab+Ba+D) sin. n+(2 Ca+Bb+E) =0 (1) Cos. nly' +[2Ab+Bb+D) sin. m+(2Ca+Bb+.E) cos. mix' +Ab2+Bab+ Ca2+Db+Ea+F. Since we have four arbitrary quantities, a, b, nm, and n entering this equation we may cause them to satisfy any four reasonable conditions. Let us see if, by means of them, it be possible to reduce the coefficientfs of the first powers, and of the product of the variables, separately to zero. We should then have )2A sin. m sin. n+B (sin. m cos. n+sin. n {l2A cos. m) +2C cos. m cos. n. (2Ab+Ba+D) sin. n+(2Ca+Bb+-E) cos. n=O (3) (2Ab+Ba+D) sin. nz+(2Ca+Bb+-E) cos. m=0 (4) These equations may be put under the form 2A tan. m tan. n+B (tan. m+tan. n)+2C=0 (2) (2Ab+Ba+D) tan. n+2Ca+Bb+E-0 (3') (2Ab+Ba+~D) tan. m+2Ca+Bb+E=O (4') Now, since it is necessary that m and n should differ in value, it is evident that, in order to satisfy eqs. (3') and (4'), we must have 2Ab+_Ba+D=O (5) And 2 Ca+Bb+.E=O (6)

Page  215 INTERPRfETATION OF EQUATIONS. 215 2A.E-_BD Whence a=2-BD B2 —4.A C 2 CD-BE And B2-4AC These values of a and b are infinite when B2-4A C=O, and it will then be impossible to satisfy both eqs. (3') and (4'), and consequently to destroy the co-efficients of the first powers of the two variables in eq. (1); we shall, for the present, assume that B2 —4A C is either greater or less than zero. By transposition and division eqs. (5) and (6) become b, B a D 2A 2A And a — b - 2C 2C the first of which, if a and b be regarded as variables, is the equation of the diameter that bisects the chords of the curve which are parallel to the axis of y, and the second, that of the diameter which bisects the chords which are parallel to the axis of X. The values of a and b, given above, are, therefore, the co-ordinates of the intersection of these diameters. Since eq. (2') contains both of the undetermined quantities, m and n, we are at liberty to assume the value of either, and the equation will then give the value of the other. Let us take for the new axis of X the diameter whose equation is yBxD 2A 2A then tan. m-_. This value of tan. m substituted in 2A eq. (2') gives (B-B) tan. m=B2 —4AC, Or tan. n= B2-A C o

Page  216 216 ANALYTICAL GEOMETRY. That is, the new axis of y is at right angles to the primitive axis of 2. The values of a, b, and tan. n which we have thus found, in connection with the assumed value of tan. m, will reduce the co-efficients of the first powers and of the product of the variables in eq. (1) to zero. To find what the co-efficients of y'2 and x'2 become, we must first get the values of the sines and cosines of the angles m and n from the values of tan. m and tan. n. Since tan. m- = and n-90~ we have 2A sin. m=4- B cos. m= 2A.4A2+B2 —,/4A2+B2 sin. n —1 cos. n=O. The sign i is written before the value of sin. m, and the sign = before that of cos. m, because if the essential sign of tan. m is minus, which will be the case when A and B have the same sign, sin. m and cos. rn must have opposite signs; but if the essential sign of tan. m is plus, then A and B have opposite signs, and sin. m and cos. m must have like signs. Making these substitutions in eq. (1) it will become, whether the signs of A and B are like or unlike, Ay'2-A( B2( —4A4 C) x'2=- (Ab2+Bab+ Ca2+Db+Ea 4A2+B2 +F. (1') Now, since the first term of the general equation may always be supposed positive, the two terms in the first member of equation (1') will have like signs when B24A C<O, and unlike signs when B2 —4A C>O. In the first case the form of the equation is that of the equation of the ellipse, and in the second, the form is that of the equation of the hyperbola, referred in either case, to the center and conjugate diameters.

Page  217 INTERPRETATI N OF EQUATIONS. 217 Hence, when the transformation by which eq. (1') was derived from the general equation Ay2+B.cy+ Cx2+Dy+Ex+F= 0 is possible, we conclude that the latter equation will represent either the ellipse, or hyperbola, according as B2-4A C<O, or B2 —4A C> 0. 4.-Let us now examine the case in which B2 —4A C=O. Since, under this hypothesis, the co-efficients of the first powers of both variables in eq. (1) cannot be destroyed, we will see if it be possible to destroy the absolute term of the equation, and the co-efficients of the product of the variables, the second power of one variable and the first power of the other variable. Then the equations to be satisfied are Ab2 +Bab+ 0Ca2 +~Db+Ea+F=0. (7) 2Asin.msin.nBin. nB(sin.m cos.n+sin.ncos.m) =0. (2) +2Ccos.m cos.n Asin.2m+B sin.m cos.m+ Ccos. 2m=0. (8) (2Ab+Ba+D)sin.n+(2 Ca+Bb+E)cos.n=-O. (3) when it is required that the co-efficients of x'2 and y' should reduce to zero in connection with the absolute term and the co-officient of x'y', in eq. (1). To reduce the co-efficients of y'2 and x' to zero, instead of those of.X2 and y', it would be necessary to replace eqs. (8) and (3) by A sin.2n+B sin.n cos.n+ Ccos. 2- 0. (9) (2Ab+Ba+]D)sin.m+(2Ca+Bb+~E)cos.m=O. (4) Equations (2) and (8) may be written 2A tan.m tan.n+B(tan.m+tan.n)+2C=O. (2') A tan.2m+B tan.m+ C=0. (8') From eq. (8') we find tan. B -1 — 1 B tan.m= 2A2-A% B24A C 2A' 18

Page  218 218 ANALYTICAL GEOMETRY. and this value of tan. m substituted in eq. (2') gives 2A(B-B) tan.n=B-2-4A C, or tan. n — That is, when tan. m is equal to B eq. (2') and, therefore, eq. (2), will be satisfied independently of the angle n. Equation (7), being what the general equation becomes when a and b take the place of x and y respectively, shows that the new origin of co-ordinates must be on the curve. Solving this equation with reference to b, and introducing the condition B2-4A C0O, we find B D 1 I b=- a- 24 - 2(BD-2AiE)a+D24AF Now, because the imposed conditions require that the transformed equation shall be of the form fMy 2= Nx', it follows that every value of x' must give two numerically equal values of y'; hence, the new axis of Y must be parallel to the system of chords bisected by the new axis of X. That is, n must be equal to 900, and, consequently, sin. n= 1, cos. n=0. Equation (3) will therefore become 2Ab+Ba+D- 0. Whence b — Ba — aD and the radical part of the 2A 2A' value of b will disappear, or we shall have 2(BD-2AE)a+_D2- 4AF=O. From which we get D2- 4AF 2(BD —2AE) These values of a and b place the new origin at the vertex of the diameter whose equation is B D Y 2- ---- 2A 2A'

Page  219 INTERPRETATION OF EQUATIONS. 219 and make the new axis of Y a tangent line to the curve at the vertex of this diameter. The values of a, b, m and n which we have now found, substituted in eq. (1), will reduce it to Ay' 2 +(2 Ca+.Bb+ E)cos. m x'-= 0. Or y+ 1 (2 Ca+Bb+ E)cos. m x'=0. Denoting the co-efficient of x' by -2p', this last equation becomes yt2-2p'x', (10) which is of the form of the equation of the parabola referred to a tangent line and the diameter passing through the point of contact. The transformation by which eq. (10) was derived from the general equation is always possible when B2 —4A C =0, unless we also have BD —2AE=O. If we suppose that both of these conditions are satisfied, the general value of y, which is B D 1 Y A-2A2A'- (B'-4A C)x'+2(BD-2AE)x+D'-4AF reduces to B D 11 - Y 2A 2A.2A44D2-4AF whence B D II 2A 2A 2A B 4 A F B _D 1 and Y= —2A X2A 2AD2-4AF, which are the equations of two parallel straight lines. Under the suppositions just made, the general equation may be written under the form (2Ay+Bx +-)+I-.Dt —4AF) (2Ay+Bx+D — f Ds 4Ao )-0, which may be satisfied by making, first one, then the other factor of the first member, equal to zero. Each of

Page  220 220 ANALYTICAL GEOMETRY. the equations thus obtained, being of the first degree in respect to x and y, will represent a right line. If the further condition, D2 —4A.F<0, be imposed, the right lines will have no existence, and the general equation can be satisfied by no real values of x and y. The value of 2p', the parameter of the diameter which becomes the new axis of X, will be found by substituting in the expression 1(2 Ca +-Bb+E)cos. m, the values of a, b and cos. m. These values are D2 4AF 4ADE-4ABF BD2 2(BD-2AE) 4A(BD-2AE) 2A cos. m —4A. 44A2 +B2 To reduce eq. (1) to the form x'2= 2p"y' (11) we must satisfy equations (7), (2), (9) and (4). From eq. (4) we find tan. n=- and this value of tan. n substituted in eq. (2') gives tan. m=~; results which might have been anticipated, since eqs. (3) and (4) are the same, except that m in the former takes the place of n in the latter. Because eq. (11) will give two numerically equal values of x' for every value of y', the new axis of X must be parallel to the system of chords bisected by the new axis of y; hence m=00, sin. m=O, cos. m=l, and equation (4) therefore reduces to 2 Ca+Bb+E=0 Whence a- B b —. 2C 2C Solving eq. (7) with reference to a we have

Page  221 INTERPRETATION OF EQUATIONS. 221 2Cb 2 2-2(BE-20 )b+,] CF By comparing this value of a with that which precedes we find 2(BE-2 CD)b+E2-4 CF= 0, Whence b= E2- 4 C 2(BE-2 CD) These values of a and b place the new origin at the vertex of the diameter whose equation is - B E 2C 2C 20 B Or x The transformation by which eq. (4) is derived from eq. (1) will be impossible when b is infinite; that is when BE-2 CD= 0. It may be easily proved that when B2-4AC= 0, the condition BD-2AE=-O necessarily includes the condition BE-2CD= 0; that is, when we cannot transform eq. (1) into eq. (10), it will also be impossible to transform it into eq. (11). For BD-2AE=0 gives2A D- And B2 —4AC=0 gives 2A B 2C E Whence - =0, or BE-2CD=0. 5.-We have now established the following criteria for the interpretation of any equation of the second degree between two variables, viz: For the ellipse, B2 —4A C<O. For the hyperbola, -B2-4A &>0. For the parabola, B2-4A 6=0. It remains for us to indicate the construction of any of these curves from its equation, and in doing this, we 19*

Page  222 222 ANALYTICAL GEOMETRY. shall follow the order in which the conditions are given above. First, B2 — 4A C<O, the ellipse. 6.-Let us resume the formulas. 2AE —BD aB2 -4AC b=2CD-BE, tan. m=-_B B2-4A C 2A ay2- A B2-4AC)x12=-(A b 2+Bab+ Ca +Db+Ea 4A2+_B2 +F,~ (1') and suppose, for a particular case, B=O, and A=C. We shall then have a =- b= _AA 2A And y12+ X2= D2+ E2 —4AF 4A2 That is, the general equation, under the suppositions E Dth made, represents a circle having a= —2 forthe co-ordinates of its center, and D2+E -- 4AF for its radius. Draw AX, A Y for the primitive y co-ordinate axes, lay off AB= --,AD=- D and through the B x points B and D draw the parallels ct D BC and DC to the axes. Their intersection, C, is the center of the circle, and the circumference described with CE=IjD2~+E2-4A f as a radius, will be 4A2 that represented by the given equation. The general equation gives

Page  223 INTERPRETATION OF EQUATIONS. 223 B D 1 =-2A-2-A 2A (B2 —4A C)x2+2(BD -2AE)x +D2-4AF. Placing the quantity under the radical sign, in this value of y, equal to zero, we have x 2+2(BD-2A-E) +D2 —4AF' B2-4A C xB —4AC=O, and denoting the roots of this equation by x' and x", the value of y may be written B D 1 1y 2A 2A 2A (B2')-x( ). Now x' and x" are the abscissas of the vertices of the diameter whose equation is B D 2A 2A The corresponding values of y are _ Bx'+D 2A Bx"+ D 2A Substituting these values of x', x" and y', y" in the formula we have 2xX2A MB2+4A2 for the length of the diameter. The diameter which is conjugate to this is that which is parallel to the axis of y. We find the ordinates of its vertices by substituting a= +x" for x in eq. (q), which 2 then becomes -B(-' +')D= 4A4 f- 4A C-B29 4A 2A 4A Denoting these two values of y by y,, y2, their difference, which is the length of the conjugate diameter, is Y1y~ —Y — -x 4AC-B

Page  224 224 ANALYTICAL GEOMETRY. To find the angle that the conjugate diameters make with each A B D E X other, let VV' be the first diameter \ At and QQ' the second. The angle that V V' makes with the axis of X is equal to V' VR, and its cosine is VR x__" x 2A VV'- x" —x'! ~/'B2+ 4A2' - B2+4A2 and the LQCV'=the LBVV'=900+the LV'VR. When the roots of eq. (p) are equal, the vertices of the first diameter, and also those of its conjugate, coincide, and the ellipse reduces to a point. Equation (q) may then be put under the form Bx+D.x —x' 2A 2A B —4AC. Because B2 —4A C is negative, this value of y will be imaginary for every value of x except the particular one, x=x', which causes the radical to disappear. When the roots of eq. (p) are real and unequal, that one of the factors (x-x'), (x-x") under the radical in eq. (q), which corresponds to the root which is algebraically the greater, will be negative, while the other will be positive, for all values of x included between the limits of the smaller and greater roots. The quantity under the radical, being then composed of the product of three factors, two of which are negative and one positive, will itself be positive and the corresponding values of y will therefore be real. All values of x which exceed the greater, and, also, all talues of x which are less than the smaller, of these roots, will render the quantity under the radical negative and the corresponding values of y imaginary. The roots x' and x" are therefore the limits within which we would

Page  225 INTERPRETATION OF EQUATIONS. 225 select values of x to substitute in the equation to get the co-ordinates of points of the curve. When the roots of eq. (p) are imaginary, the product of the factors (x-x'), (x-x") under the radical in eq. (q) will remain positive for all real values of x; and because the other factor is B2 —4A C<0, the radical will always be imaginary: that is, no real value of x which will give a real value for y. There is, then, in this case, no point in the plane of the co-ordinate axes whose co-ordinates will satisfy eq. (q), and, consequently, the equation from which it was derived, and the curve, has no existence, or it is imaginary. By the solution of eq. (p) it will be found that when the expression (B)D-2AE)2 —(B-4A C)(D —4AF) is positive, the roots of the equation are real and unequal; when the expression is zero the roots are real and equal, and when negative the roots are imaginary. If we solve the general equation with reference to x instead of y, and place the quantity under the radical sign equal to zero, we shall find that when the expression (B —2 CD)2-(B2-4A C) (E2 —4 CF) is positive, the roots of the resulting equation are real and unequal; when zero, these roots are real and equal, and when negative they are imaginary. It might be inferred that if these roots are real and unequal, equal, or imaginary when the general equation is resolved with reference to one variable, they would be like characterized when it is resolved with reference to the other. To prove this, we develope the first of the above expressions and find that it becomes 4A (A(E)2+ C(D)+ F(B) —BDE-4A CF.) The development of the second is

Page  226 226 ANALYTICAL GEOMETRY. 4CA(Ey)2+ C(D)2+F(B)2-BDE-4A CF.) The only difference in these developments is that the coefficient of the parenthesis in the first is 4A, and in the second it is 4C; but when B2 —4A C<O, A and C must have the same sign, hence these expressions must be positive, negative, or zero at the same time. Second, B2-4A C>0, the hyperbola. 7.-We will begin by supposing B=O, and A=-C. The formulas for a, b and tan. m will then give a= E b — D tan.m=O0 2A' 2A' and eq. (1') will become y1 X12 P-E2 —-4AF 4A2 This is the equation of an equilateral hyperbola whose semi-axis is the square root of the numerical value of the expression D —-4A F. Since tan. m=O, mO-0, and one of the axes of the hyperbola is parallel and the other perpendicular to the primitive axis of X. If the sign of A is negative, the transverse is the parallel axis; if negative, it is the perpendicular axis. To construct the curve, let AX and A Ybe the primitive co-ordinate \ It. axes. Lay off the positive abscissa..X / AD - and the negative ordinate F' F AE=- f; the parallels to the axes drawn through D and E will be the axes of the hyperbola, and C will be its center. On these axes, lay off from the center, the distances CV, CV', CR, CR', each

Page  227 INTERPRETATION OF EQUATIONS. 227 equal to 4 ---- -4AF, and we have the axes of conjugate equilateral hyperbolas. The foci may be found by describing a circumference with C as a center and CH, the hypothenuse of the isosceles right-angled triangle CV.E, as a radius; the circumference will intersect the axes at the foci. For another case, let us suppose A=O and C=O; then the value — which was assumed for tan. m becomes 2A infinite, or the new axis of X is perpendicular to the primitive axis of X, and since tan. n is also infinite, the new co-ordinates axes would coincide; in other words, with this value of tan. m, it would be impossible, under the hypothesis, to transform the original equation into eq. (1'). But if A=O, and C=O, the co-efficient of x'y' in eq. (1) becomes B(sin. m cos. n+sin. n cos. m). Placing this equal to zero, and dividing through by B cos. m cos. n, we have tan. m+tan. n=O, Or tan. m= —tan. n. Since we are at liberty to select a value for either m or n, let us make n=450; then m= —450. The values of a and b, which will destroy the co-efficients of x' and y' D E are, a= — b=.E Substituting these values in eq. (1), reducing and transposing, we have y /2. x2=2(DE —BF) B2 which is also the equation of the equilateral hyperbola, the co-ordinates of whose center are a= —-, b —---- B B

Page  228 228 ANALYTICAL GEOMETRY. and whose semi-axis is the square root of the numerical value of 2(DE-B) The asymptotes of this hyperbola B2 are parallel to the primitive axes, and if 2( BF) is negative, the transverse axis makes a negative angle with the primitive axis of 1X, if positive, it makes a positive angle with that axis. There is another case in which the transformation by which eq. (1') was obtained, cannot be made with the value- B for tan m. It is that in which A becomes zero, 2A and C does not. We then assume for tan. m the tangent of the angle that the diameter whose equation is B /E 2 CY 2C makes with the axis of X. That is, we make tan. m=-2C Proceeding with this as with the value —B, we shall find for the transformed equation 2 C( 12o4A C )X- _(Ab2+Bab+ Ca2+Db+ Ea+F V40C2~1 2 By making A=O, this equation becomes Cy, 2__ CB2 2=-(Bab+ Ca2+.Db+.Ea++F) V/4 C2+13 which is that of an hyperbola referred to a system of conjugate diameters, one of which bisects the chords which are parallel to the primitive axis of X. In the general case the course to be pursued for the hyperbola differs so little from that already indicated for the ellipse, that it is unnecessary to dwell upon it at length.

Page  229 INTERPRETATION OF EQUATIONS. 229 The quantity under the radical in the general value of y placed equal to zero gives the equation X2$+2(BD-2AE) D2 —4AF_ B2 —4AC + B2 — 4A.C - 0 The roots of this equation are the abscissas of the vertices of the diameter, whose equation is B 1D 2A 2A When these roots are real and unequal, the diameter terminates in the hyperbola; when imaginary, it terminates in the conjugate hyperbola. Denoting these abscissas, when real, by x' and x", and the corresponding ordinates by y' and y", we have y__Bx'+D 2A,_ _Bx"+D 2A By placing these values of x', x" and y', y" in the formula _/(x' —X")2+(y,-_y") we shall have the length of the diameter, and the angle included between it and its conjugate will be found precisely as in the ellipse. If x' be the smaller and x" the greater abscissa, then all values of x between x' and x" will give imaginary values'for y, and will answer to no points of the curve; but all values of x less than x', and also all values of x greater than x" will give real values for y', and such values of x with the corresponding values of y will be the co-ordinates of points of the hyperbola. When the roots x', x" are imaginary, the diameter whose equation is B _D 2A 2A 20

Page  230 230 ANALYTICAL GEOMETRY. terminates in the hyperbola which is conjugated to that represented by the given equation, and the diameter which is conjugate to this diameter will terminate in the given hyperbola. The conjugate diameter may be found in the case of both the ellipse and hyperbola by making first y'=0 in eq. (1'), and taking the square root of the corresponding numerical value of x'2, and then x'=0, and taking the square root of the corresponding numerical value of y'2. 8. —In the transformation of co-ordinates by which the original equation was changed into eq. (1) had the condition, that the new co-ordinate axes should be rectangular, been imposed, as it might, we would have had n-m=90~, n=90~+m. Sin. n=cos. m, cos. n=-sin. m. These values being substituted in eq. (2) will give 2A sin. m cos. m —B sin.2 m+B cos.2m-2 Csin. m cos. m=0, which, by dividing through by cos.2m, and denoting sin. m by t, becomes cos.?z 2At-BI+ _B-2 Ct- 0. Whence t_ A- 1iB 2+(A 2 B -A B 4 C)2. Since the product of these two values of t is equal to 1, they are the tangents of the angles that two straight lines at right angles to each other make with the axis of X. Now, if eqs. (5) and (6) are satisfied at the same time; that is, if the new origin be placed at the point of which the co-ordinates are aAE —BD b 2 CD —BE B2 4A C' B2-4A C' the values of t just found will be the tangents of the angles that the axes of the ellipse, or hyperbola, as the case may be, make with the primitive axis of X. Denoting these tangents by t' and t', we shall have

Page  231 INTERPRETATION OF EQUATIONS. 281 y —b t(x-a), y-b= t"(x-a), for the equations of the axes, and by combining the equations of the axes with the original equation, we may find the co-ordinates of their vertices, and, consequently, their length. 9.-When the roots x' and x" become equal, the value of y may be written Bx+D x —x' C. 2A 2A 4A For the hyperbola, B2-4A C>0, and these values of y are real. We therefore have B D x —x' Y — X + IP4Af (r) 2A 2A 2A 4AC and y: B D x-x' and y=- 2A _ B2 1A C. (B) 2A 2A 2 A These equations represent two right lines, and, since the co-efficients of x, when the second members are arranged with reference to it, are different, these lines will intersect. We see that by making x=x', the two equations will give the same value for y. Hence, x=x', and y=___Bx'+D are the co-ordinates of the intersection of 2A the lines. The line BE, whose equation is A 2A 2A c still has the property of bisecting all lines drawn parallel to the axis of Y, which are limited by the lines D BC and BD, whose equations are eqs. (r) and (s). Third, B —4AC=0, theparabola. 10.-The equation of the diameter that bisects the chords of the curve which are parallel to the axis of Y is

Page  232 232 ANALYTICAL GEOMETRY. B D 2A 2A' and that of the diameter which bisects the chords parallel to the axis of X is B E 2 X y 2C 2_ or Y 2 BC Since a tangent line drawn through the vertex of a diameter is parallel to the chords that the diameter bisects, it follows that the diameters represented by the above equations are perpendicular to each other, and, therefore, (Prop. 5, Chap. 4), their intersection, in the case of the parabola, is on the directrix. The abscissa of the vertex of the first diameter is the value of x given by the equation 2(BD-2AE)x+D2 —4AF=O, the first member of which is the quantity under the radical in the general value of y, after we have made B2 —4A C=O. Denoting this abscissa by x' we have,_ -2 _ 4A.F 2(BD-2AE)' and. y, Bx'+D 2A If we denote the co-ordinates of the vertex of the second diameter by x" and y", we have,,_' E2-4 CF 2(BE-2 CD)' X"= -By t+E 2C Let P and P' be the two vertices thus found. Through the first draw PT parallel to the axis of Y, and through the second, P' T parallel to the axis of X. These lines will be tangent to the parabola at P and P' respectively,

Page  233 INTERPRETAT'I'()ON OF EQUATIONS. 233 and their intersection, T, will be Y a point of the directrix. The a lines CM, BN, drawn through D A P and P', making, with the axis c of X, angles having for their common tangent \N _ 2C 2A-' are diameters of the curve, and BC drawn through T perpendicular to these diameters, is the directrix. With P as a center and PC as a radius, or with P' as a center and P'B as a radius, describe an arc of a circle. This arc will cut the chord PP' at the focus F. The perpendicular Et), drawn through F to the di.ectrix, is the axis, and the middle point, V, of EID, is the vertex of the parabola. EXAMPLES. It will aid in the construction of the curve represented by any equation to find the points in which it is intersected by the co-ordinate axes. If we make either variable equal to zero in the equation, the values of the other variable given by the resulting equation will be the distances from the origin to the intersections of the curve, with axis of the latter variable. When the roots of the equation which we solve are real and unequal, there will be two intersections, where real and equal, the axis will be tangent to the curve at the point thus determined, and when imaginary, the curve and the axis will have no common points.. —Construct the curve represented by the equation y2+ 2xy+ 3x2-4x=-. Whence y-X —xv'.-2x(x- 2). Here A=1, B=2, C=3; therefore B2 —4AC<O, and 20*

Page  234 234 ANALYTICAL GEOMETRY. the curve is an ellipse which passes through the origin of co-ordinates, since the equation has no absolute term. y= —x L Y is the equation of a diameter of the r K -K' curve and the co-ordinates of its ver- R\ R G ticesarex'= O,y'=Oandx" 2,y"=-2. A E By making x=l1 in the original equation, we find y=+. 41+, or -2.41 L for the ordinates of the vertices of the diameter conjugate to the first. F The length of the first diameter is'! equal to 4V8=2.82+, and the length of the second is +.41+2.41=2.82. 2.-Determine the curve that corresponds to the equation y +2xy+x2-6y+9 = O. Here A=1, B=2, C=1, hence B2-4AC=O, and the curve is a parabola. We find y= —x +3 / —6x, And x= —y/6y — 9. The diameter whose equation is y= —x+3 has x'=-O, and y'=3 for the co-ordinates of its vertex. The axis of y is therefore tangent to the curve. The co-ordinates of the vertex of the diameter whose equation is x=-y are, x"= —1~, and y"=l, and a line drawn through this point parallel to the axis of X will be tangent to the curve. Let P' be the vertex of the first diameter and P that of the second. The chord PP' passes through the focus. P'S', PSmaking with the S axis of X, on the negative side, angles of 450 are diameters of the - curve, and B T a perpendicular to T PSis the directrix. B A

Page  235 INTERPRETATION OF EQUATIONS. 235 3. —Determine the curve of which the equation is y2+2xy-2x-4y-x+10= 0.. In this case A=1, B=2, C —2; hence B —4A C>O, and the curve is an hyperbola. The equation gives y= —x+ 2tV3x2 — x —6. The abscissas of the vertices of the diameter whose equation is y=-x+2 are the roots of the equation 3x2-3x —6 —=0. Whence x'= —1, and x"=2, and the corresponding values of y are y'=3 and y"=O. The diameter which is parallel to the axis of y is conjugate to PP', D and terminates in the conjugate by- P' perbola. The co-ordinates of its vertices are imaginary and may be found by making x-= in the original B equation. We would thus find / 5.2,/1l y- 3 -t 2 ~ 2 The conjugate diameter will therefore be about 5.2. The point E in which the curve intersects the axis of X is on the left of the origin and at a distance from it equal to 21 units. 4.-Determine the curve represented by the equation y2 + 6xy+9x2 -2y-6x-15- 0. In this, the condition B2 —4A C= 0 is satisfied, and the curve is the parabola; but it answers to the case in which the parabola reduces to two parallel lines. In fact the equation may be put under the form (y+ 3x)2-2(y+ 3x)=15. Whence y+3x=141- 6, Or y+3x-=5 or -3.

Page  236 236 ANALYTICAL GEOMETRY. The first member of the equation may therefore be resolved into the factors y+3x-5, and y+3x+3, which, placed separately equal to zero, give for the parallel lines the equations y=-3x+ 5, And y=-3x-3. 5. —Determine the curve of which the equation is y2-4xy+ 5x2-2y+ 5=0. In this we have B2-4A C<O, and the curve is an ellipse, but it answers to the case in which the curve becomes imaginary. For, resolving the equation in relation to y, we find y=2x+1t-+-/-(x-2)2. The quantity under the radical in this value of y will be negative for every real value of x, hence, all values of y are imaginary; that is, there is no point whose co-ordinates will satisfy the given equation. By inspection we may also discover that the first member of the equation can be placed under the form (y-2x-1)2+ (x-2)2, which is the sum of two squares, and must therefore remain positive for all real values of x and y. 6. —What kind of a curve corresponds to the equation y2 —2xy-x2-2y+2x+3=0? Ans. It is an hyperbola. The axis of Y is midway between the two branches. One branch of the curve cuts the axis of A at the point -1; the other branch cuts the same axis at the point +3. T. —Determine the curve represented by the equation y — 2xy+2x2 2x+4= 0. Resolving, we find (yX)2+(x-1)2+ 3= 0.

Page  237 INTERSECTION OF LINES. 237 The condition for the ellipse is satisfied, but the curve is imaginary. 8.- What kind of a curve corresponds to the equation y2 —2xy+x2+x=O? Ans. It is a parabola passing through the origin and extending without limit, in the direction of x and y negative. 9. —What kind of a curve corresponds to the equation y —2xy+ x2-2y —1=0? Ans. It is a parabola, cutting the axis of X at the distance of-i1 and +1 from the origin, and extending indefinitely in the direction of plus x and plus y. 10. — What kind of a curve corresponds to the equation y2-4xy+4x2=0? Ans. It is a straight line passing through the origin, making an angle of 26~ 34' with the axis of Y. 11.-What kind of a curve corresponds to the equation y2 -2xy+ 2x2 —2y+ 2x=? Ans. It is an ellipse limited by parallels to the axis of Y drawn through the points -1, and +1, on the axis of X. CHAPTER VII. ON THE INTERSECTIONS OF LINES AND THE GEOMETRICAL SOLUTION OF EQUATIONS. We have seen that the equation of a straight line is y=tx+c, And that the general equation of a circle is (xhfa)2+(y+b)2 R2. The first is a simple, the second a quadratic equation,

Page  238 238 ANALYTICAL GEOMETRY. and if the value of x derived from the first be substituted in the second, we shall have a resulting equation of the second degree, in which y cannot correspond to every point in the straight line, nor to every point in the circumference of the circle, but it will correspond to the two points in which the straight line cuts the circumference, and to those points only. And if the straight line should not cut the circumference, the values of y in the resulting equation must necessarily become imaginary. All this has been shown in the application of the polar equation of the circle, in Chap. 2. Let us now extend this principle still further. The equation of the parabola is y2=2px, an equation of the second degree, and the equation of a circle is (x-+a)2+ (y-+-b)2 -RM also an equation of the second degree. But when two equations of the second degree are combined, they will produce an equation of the fourth degree. But this resulting equation of the fourth degree cannot correspond to all points in the parabola, nor to all points in the circumference of the circle, but it must correspond equally to both; hence, it will correspond to the points of intersection, and if the two curves do not intersect, the combination of their equations will produce an equation whose roots are imaginary. Let us take the equation y2=2px, and take p for the unit of measure, (that is, the distance from the directrix to the focus is unity,) then x=Y2, and this value of x substituted in the equation of the circle, will give 2

Page  239 INTERSECTION OF LINES. 239 Let the vertex of the parabola y be the origin of rectangular coordinates..Take AP=x, and let it refer to either the parabola or the circle, and let PM=y, AF=S, AI-=a, C —C=b, and CM= R. Now in the right angle triangle CMD, we have CD=IHP=x-a, MD=y —b, and corresponding to this particular figure, we shall have in lieu of the proceding equation -— a) +(y-b)2R2. Whence y4+(4-4a)y2 —8by=4(R2-a2-b2.) (F) This equation is of the fourth degree, hence it must have four roots, and this corresponds with the figure, for the circle cuts the parabola in four points, M, M', M", and M"'. The second term of the equation is wanting, that is, the co-efficient to y3 is 0, and hence it follows from the theory of equations, that the sum of the four roots must be zero. The sum of two of them, which are above the axis of AX, (the two plus roots,) must be equal to the sum of the two minus roots corresponding to the points M" and MI"'. The values of a and b and R may be such as to place the center C in such a position that the circumference can cut the parabola in only two points, and then the resulting equation will be such as to give two real and two imaginary roots. Indeed, a circumference referred to the same unit of measure and to the same co-ordinates, might not cut the

Page  240 240 ANALYTICAL GEOMETRY. parabola at all, and in that case the resulting equation would have only imaginary roots. In case the circle touches the parabola, the equation will have two equal roots. Now it is plain that if we can construct a figure that will truly represent any equation in this form, that figure will be a solution to the equation. For instance, a figure correctly drawn will show the magnitude of PiM, one of the roots of the equation. We will illustrate by the following EXAMPLES. I.-Find the roots of the equation y4 —11.14y2 —6.74y+ 9.9225-0. This equation is the same in form as our theoretical equation (F), and therefore we can solve it geometrically as follows: Draw rectangular co-ordinates, as in the figure, and take AF=S-, and construct the parabola. To find the center of the circle and the radius, we put 4 —4a —11.14, (1) -8b —6.74, (25 and 4(R2-a2-_b2)=-9.9225. (3) From eq. (1), a=3.78. From eq. (2), b=0.84. And these values of a and b, substituted in eq. (3), give?= 3.34, nearly. Take from the'scale which cor- y responds to AF=-, AH-=a=3.78, fC= 0. 84, and from C as a center, with a radius equal to 3.34, des-'-.. cribe the circumference cutting the Fi { C D parabola in the four points, M, M',, P. X M", and MI". The distance of M from the axis of X is +3.5, of M' it is +0.7, of 31" it is -1.5, and of M"' it is -2.7, anal these are the four roots of the equation.

Page  241 INTERSECTION OF LINES. 241 Their sum is 0, as it ought to be, because the equation contains no third power of y. 2. —Find the roots of the equation y4+y3+6y2+ 12y-72=O. This equation contains the third power of y; therefore this geometrical solution will not apply until that term is removed. But we can remove that term by putting (See theory of transforming equations in algebra). This value of y substituted in the equation, it becomes z4+ 5,z2+ 91z= 74- T, and this equation is in the proper form. Nowput 4 —4a=55, -8b=9, and 4(R2 -a2-b2)=741j. Whence a= — 2, b= — 3, and R=4.485. These values of a and b designate the point C' for the center of the circle. From this center, with a radius =4.485, we strike the circumference, cutting the parabola in the two points m and m'. The point m is 21 units above the axis AX, and the point m' is -2f units from the same line, and these are the two roots of the equation. The other two roots are imaginary, shown by the fact that this circumference can cut the parabola in two points only. If we conceive the circumference of a circle to pass through the vertex of the parabola A, then will a2+ b2-=R2, and this supposition reduces the general equation (F) to y4+(4-4a)y2-8by= 0. Here y= i0 will satisfy the equation, and this is as it should be, for the circumference actually touches the parabola on the axis of X. Now divide this last equation by this value of y, and we have y3+(4-4a)y=8b. (G) 21 Q

Page  242 242 ANALYTICAL GEOMETRY. Here is an equation of the third degree, referring to a parabola and a circle; the circumference cutting the parabola at its vertex for one point, and if it cuts the parabola in any other point, that other point will designate another root in equation (G). It is possible for a circle to touch one side of the parabola within, and cut at the vertex A and at some other point. Therefore it is possible for an equation in the form of eq. (G) to have three real roots, and two of them equal. The circumferences of most circles, however, can cut the parabola in A and in one other point, showing one real root and two imaginary roots. Equation (G) can be used to effect a mechanical solution of all numerical equations of the third degree, in that form.* We will illustrate this by one or two EXAMPLES. 1.-Given y3+4y=39, to find the value of y by construction. (See fig. following page) Put 4-4a=4, and 8b=39. Whence a=O, and b=41. These values of a and b designate the point C on the axis of Y for the center of the circle, CA=4', the radius. The circle again cutsr the parabola in P, and PQ measures three units, the only real root of the equation. 2.-Given y3-75y=250, to find the values of y by construction. When the co-efficients are large, a large figure is required; but to avoid this inconvenience, we reduce the co-efficients, as shown in Chap. 2. * Observe that the second term, or y2, in a regular cubic is wanting. Hence, if any example contains that term, it must be removed before a geometrical solution can be given.

Page  243 INTERSECTION OF LINES. 248 Thus put y=nz. y Then the equation becomes c n3z3-75nz=250. m z75 z=250. n A X Now take n-5, then we have T z3-3z= —2. In this last equation the co-efficients are sufficiently small to apply to a construction. Put 4-4a= —3, and 8b=2. Whence a=l —, and b=-. These values of a and b designate the point D for the center of the circle. DA is the radius. The circle cuts the parabola in t, and touches it in T, showing that one root of the equation is +2, and two others each equal to -1. But y=nz. That is, y=5x2, or-5, -5. Or the roots of the original equation are +10, -5, -5. When an equation contains the second power of the unknown quantity, it must be removed by transformation before this method of solution can be applied. 3. —Given y3-48y=128 to find the values of y by construction. Ans. +8, -4, -4. 4. —Given y3-13y —-12, to find the values of y by construction. Ans. +1, +3, and -4. Conversely we can describe a parobola, and, take any point, as H, at pleasure, and with HA as a radius, describe a circle and find the equation to which it belongs. This circle cuts the parabola in the points m, n and o, indicating an equation whose roots are +1, +2.4, and -3.4. We may also find the particular equation from the general equation y3+(4-4a)y= 8b,

Page  244 244 ANALYTICAL GEOMETRY. observing the locality of H, which corresponds to a=3-3 and b= —l, and taking these values of a and b, we have y3-9.2y= —8, for the equation sought. REMARKS ON THE INTERPRETATION OF EQU3ATIONS. In every science it is important to take an occasional retrospective view of first principles, and the conviction that none demand this more imperatively than geometry will excuse us for reconsidering the following truths so often in substance, if not in words, called to mind before. An equation, geometrically considered, whatever may be its degree, is but the equation of a point, and can only designate a point. Thus, the equation y-=ax+b designates a point, which point is found by measuring any assumed value which may be given to x from the origin of co-ordinates on the axis of X, and from that extremity measuring a distance represented by (ax+b) on a line parallel to the axis of Y. The extremity of the last measure is the point designated by the equation. If we assume another value for x, and measure again in the same way, we shall find the point which now corresponds to the value of x. Again, assume another value for x, and find the designated point. Lastly, if we connect these several points, we shall find them all in the same right line, and in this sense the equation of the first degree, y=ax+b, is the general equation of a right line, but the right line is found by finding points in the line and connecting them. In like manner the equation of the second degree y=-i/V2Rx —x2, only designates a point when we assume any value for x, (not inconsistent with the existence of the equation), and take the plus sign. It will also designate another point

Page  245 INTERPRETATION OF EQUATIONS. 245 when we take the minus sign. Taking another value of x, and thus finding two other points, we shall have four points,-still another value of x and we can find two other points, and so on, we might find any number of points. Lastly, on comparing these points we shall find that they are all in the circumference of the same circle, and hence we say that the preceding equation is the equation of a circle. Yet it can designate only one, or at most, two points at a time. If we assume different values for y, and find the corresponding values of x, the result will be the same circle, because the x and y mutually depend upon each other. Now let us take the last practical example y3 —13y= —12, and, for the sake of perspicuity, change y into x, then we shall have x3-13x+ 12-=0. Now we can suppose y-=O to be another equation; then will y=x3-13x+12 (A) be an independent equation between two variables, and of the third degree. The particular hypothesis that y=O, gives three values to x, (+1, + 3, and -4), that is, three points are designated: the first at the distance of one unit to the right of the axis of Y; the second at the distance of three units on the same side of the axis of Y; and the third point four units on the opposite side of the same axis, and this is all the equation can show until we make another hypothesis. Again, let us assume y=5, then equation (A) becomes 5=x3-13x+12, or x3-13x+7=0, and this is, in effect, changing tBe origin five units on the axis of Y. A solution of this last equation fixes three other points on a line parallel to the axis of X. Again, let us assume y=10, then equation (A) becomes x3-13x+2 0, 21*

Page  246 246 ANALYTICAL GEOMETRY. and a solution of this equation gives three other points. And thus we may proceed, assigning different values to y, and deducing the corresponding values of x, as appears in the following table, commencing at the origin of the co-ordinates, where y=O, and varying each way. y —30.0388 x=-2.2814 +4.1628 -2.0814 y=25. x= —1.1 +4.03 -2.91 y —20. x=-0.40 +3.80 -3.41 y-=15. x=-0.20 +3.70 -3.50 y 10,. x=+0.14 +3.52 -3.66 y-5. x=+0.55 +3.3 -3.85 When y=O. then will x- +. +3. 4. y= —5 x=+1.66 +2.477 -4.14 y=-6.0388 x=+2.0814 +2.0814 -4.1623 Taking y=O, a solution of the equation y=x3 —13x+12, gives the -5 / three points a, a, a, on the axis of X.. Then taking y=5, and a solution C 10 C C gives three points b, b, b, on a line x a a a X parallel to the axis of X, and at the }- in distance of 5 units above said axis. -1 Again, taking y=10, and another solution gives the three points c, c, c. Now joining the three points (a, b, c,) (a, b, c), and (a, b, c), we shall have apparently three curves corresponding to the equation of the third degree, and thus, we might hastily conclude that every equation of the third degree would give three curves, and every equation of the fourth degree four curves, etc., etc., but this is not true. If we continue finding points as before, we shall find that the three curves (a, b, c,) (a, b, c,) and (a, b, c,) are but different portions of the same curve, and we can now venture to draw this general conclusion: That in an equation involving y, the ordinate, to the first power,

Page  247 INTERPRETATION OF EQUATIONS. 247 and the abscissa, x, to the third power, the axis of X, or lines parallel to that axis, may cut the curve in three points. From analogy, we also infer that if we have an equation involving x to the fourth power, the axis of X, or its parallels, will cut the curve in four points; and if we have an equation involving x to the fifth power, that axis or its parallels will cut the curve in five points, and so on. In the equation under consideration, (y=x3-13x+12), if we assume y greater than 30.0388, or less than -6.0388, we shall find that two values of x in each case will become imaginary, and on each side of these limits the parallels to X will cut the curve only in one point. Two points vanish at a time, and this corresponds with the truth demonstrated in algebra, "that imaginary roots enter equations in pairs." The points m, m, the turning points in the curve, are called maximum points, and can be found only by approximation, using the ordinary processes of computation, but the peculiar operation of the calculus gives these points at once. To find the points in the curve we might have assumed different values of x in succession, and deduced the corresponding values of y, but this would have given but one point for each assumption; and to define the curve with sufficient accuracy, many assumptions must be made with very small variations to x. We solved the equations approximately and with great rapidity by means of the circle and parabola as previously shown. We conclude this subject by the following example: Let the equation of a curve be (a2 —2)(x-b)2= x2y2, from which we are required to give a geometrical delineation of the curve. From the equation we have (a2 x2)(x-b)2 x

Page  248 248 ANALYTICAL GEOMETRY. The following figure represents the curve which will be recognized as corresponding to the equation, after a little explanation. If x-=O, then y becomes infinite, and therefore the ordinate at A is an asymptote to the curve. If AB=b, M and P be taken between A and B, E A then /M and Pm will be equal, and lie on different sides of the abscissa m m AP. If x=b, then the two values of y vanish, because x-b=-O; and consequently, the curve passes through B, and has there a duplex point. If AP be taken greater than AB, then there will be two values of y, as before, having contrary signs, that value which was positive before, now becomes negative, and the negative value becomes positive. But if AD be taken =a, and P come to D, then the two values of y vanish, because Va2_x2 0. And if AP is taken greater than AD, then a2 —X2 becomes negative, and the value of y impossible; and therefore, the curve does not extend beyond D. If x now be supposed negative, we shall find y=-ta2 —X2 X (b+x) -x. If x vanish, both these values of y become infinite, and consequently, the curve has two infinite arcs on each side of the asymptote AK. If x increase, it is plain y diminishes, and if x becomes =-a, y vanishes, and consequently the curve passes through B, if AE be taken =AD, on the opposite side. If x be supposed, numerically, greater than -a, then y becomes impossible; and no part of the curve can be found beyond E. This curve is the conchoid of the ancients.

Page  249 STRAIGHT LINES IN SPACE. 249 CHAPTER VIII. STRAIGHT LINES IN SPACE. Straight lines in one and the same plane are referred to two co-ordinate axes in that plane, -but straight lines in space require three co-ordinate axes, made by the intersection of three planes. To take the most simple view of the subject, conceive a horizontal plane cut by a meridian plane, and by a perpendicular east and west plane. The common point of intersection we shall call the origin or zero point, and we might conceive this point to be the center of a sphere, and about it will be eight quadrangular spaces corresponding to the eight quadrants of a sphere, which extended, would comprise all space. The horizontal east and west line of intersection of these planes, we shall call the axis of X. The horizontal intersection in the direction of the meridian, the axis of Y; and that perpendicular to it in the plane of the meridian, the axis of Z. Distances estimated from the zero point horizontally to the right, as we look towards the north, we shall designate as plus, to the left minus. Distances measured on the axis of Y and parallel thereto, towards us from the zero point, we shall call plus; those in the opposite direction will therefore be minus. Perpendicular distances from the horizontal plane upwards are taken as plus, downward minus. The horizontal plane is called the plane of xy, the meridian plane is designated as the plane of yz, and the perpendicular east and west plane the plane of xz. Now let it be observed that x will be plus or minus, according to its direction from the plane of yz, y will be plus or minus, according to its direction from the plane

Page  250 250 ANALYTICAL GEOMETRY. xz, and z will be plus or minus, according as it is above or below the horizontal place xy. PROPOSITION I. To find the equation of a straight line in space. Conceive a straight line passing in any direction through space, and conceive a plane coinciding with it, and perpendicular to the plane xz. The intersection of this plane with the plane xz, will form a line on the plane xz, and this is said to be the projection of the line on the plane xz, and the equation of this projected line will be in the form x=az+7r. (Chap. 1, Prop. 1.) Conceive another plane coinciding with the proposed line, and perpendicular to the plane yz, its intersection with the plane yz is said to be the projection of the line on the plane yx, and the equation of this projected line is in the form y=bz+P. These two equations taken together are said to be equations of the line, because the first equation is a general equation for all lines that can be drawn in the first projecting plane, and the second equation is a general equation for all lines that can be drawn in the second projecting plane; therefore taken together, they express the intersection of the two planes, which is the line itself. For illustration, we give the following example: Construct the line whose equations are x-2z+1 - y=3z-2

Page  251 STRAIGHT LINES IN SPACE. 251 Make z=O, then x= 1, andy=-2. z Now take AP=1, and draw Pm m parallel to the axis of Y, making Pm=-2; then m is the point in A the plane xy, through which the / line must pass. Now take z equal to any number at pleasure, say 1, then we shall Y have x=3 and y=1. Take AP'=3, P'm'=-+1, and from the point m' in the plane xy erect m'n perpendicular to the plane xy, and make it equal to 1, because we took z-=1, then n is another point in the line. Draw n m and produce it, and it will be the line designated by the equations. PROPOSITION II. To fJind the equation of a straight line which shall pass through a given point. Let the co-ordinates of the given point be represented by x, y', z'. The equations sought must satisfy the general equations. x=az+rr. y-bz+. J (1) The equations corresponding to the given point are x' az'+7r. y' -bz'+P. Subtracting eq. (1) from these, respectively, we have x'-x=a(z'-z), and y'-y=b(z'-z), the equations required. PROPOSITION III. To find the equations of a straight line which shall pass through two given points.

Page  252 252 ANALYTICAL GEOMETRY. Let the co-ordinates of the second point be x", y", z". Now by the second proposition, the equations which express the condition that the line passes through the two points, will be x" —x'-a(z" —z'), And y" —y'=b(z"-z'). Whence a= -_ b- y /y z"-__z' zl"_zt Substituting the values of a and b in the equations of a line passing through a single point (Prop. 2,) we have = z"-z' ) (z —z'). (. = z — - z ) for the equations required. PROPOSITION IV. To fJind the condition underwhich two straight lines intersect in space, and the co-ordinates of the point of intersection. Let the equation of the lines be x=az+7r. y-bz+p. x=a'z+7r'. y= b'z+P'. If the two lines intersect, the co-ordinates of the commo-n point, which may be denoted by x, y, z, will satisfy all of these four equations, therefore by subtraction, we have (a —a')Z+ r-' O=, (b-b')z+P-P'=0. Whence, by eliminating z, we find a —a' b-b' which is the condition under which two lines intersect. Now z='-, and this value of z being substituted a-a in the first equations, we obtain x-Z' —a' and ybl-b a —a b-b'

Page  253 STRAIGHT LINES IN SPACE. 253 for the value of the co-ordinates of the point of intersection. Cor. —If a=a', the denominators in the second member will become 0, making x and y infinite; that is, the point of intersection is at an infinite distance from the origin, and the lines are therefore parallel. PROPOSITION V.-PROBLEM. To express analytically the distance of a given point from the origin. Let P be the given point in Z space; it is in the perpendicular at the point N, which is in the plane xy. A W The angle AMN=900. Also, the angle ANP=90~. Let AM=x, MN=y, NP=-z. / Then AN2-x2 +y2. But = AN2+Np -X2+y2+ z2. Now if we designate AP by r, we shall have i-=X2+y2+Z2 for the expression required. PROPOSITION V I.- PROBLEM To express analytically the length of a line in space. Let PP'=D be the line in question. Let the co-ordinates of the point P be x, y, z, and of the point P' be x', P' Y', z' Now MM'=x' —x=NQ. A M w QN'=y' —y. -N-' (x' —x)2+(y'- y)2 — pR2 P'R=z' z. N 22

Page  254 254 ANALYTICAL GEOMETRY. In the triangle PRP' we have PP_2= _ PRj2 PR2=(x' x)2+(y — y)2+(z' —z)2. Or D2-'(x'- x)2+(y-ly)2+(z'_z)2, (1) which is the expression required. SCHOLIUM.-If through one extremity of the line, as P, we draw PA to the origin, and from the other extremity P, we draw P'S parallel and equal to PA, and draw AS, it will be parallel to PP', and equal to it, and this virtually reduces this proposition to the previous one. This also may be drawn from the equation, for if A is one extremity of the line, its co-ordinates x, y, and z are each equal to zero, and D2=X. 2+YI2+ZP PROPOSITION VII.-PROBLEM. Tofind the inclination of any line in space to the three axes. From the origin draw a line I parallel to the given line; then the inclination of this line to the / P axes will be the same as that of,~M X the given line. The equations for the line pass- /' r_ ing from the origin are x=az, and y=bz. (1) Let X represent the inclination of this line with the axis of x, Y its inclination with the axis of y, and Z its inclination with the axis of z. The three points P, N, M, are in a plane which is parallel to the plane zy, and AM is a perpendicular between the two planes. AMP is a right-angled triangle, the right angle being at M. Let AP=r and AM=x. Then, by trigonometry, we have As r: sin. 90~:: x cos. X. Whence x=r cos. Z. Also, as r: sin. 900:: y: cos. Y. Whence y=r cos. Y.

Page  255 STBRAIGHT LINE.S. IN SPACE. 255 Also, as r: sin. 90~:: z cos. Z. Whence z=r cos. Z. From Prop. 5 we have r2= X2+y2+z2. (2) Substituting the values of x, y, and z, as above, we have r2=r2 cos.2X+r2 cos.2 Y+r2 cos.2Z. Dividing by r2 will give cos.2X+cos.2 Y+cos.2Z=1, (3) an equation which is easily called to mind, and one that is useful in the higher mathematics. If in eq. (2) we substitute the values of x2 and y2 taken from eq. (1), we shall have r2=a2z2+b2z2+z2. (4) But we have three other values of r2 as follows: r2 x2 r2= Y_ and r2 z.2 Cos.2' cos. Y' cos. Z Whence. (5) cos. Cos. Y —'zvl+a2'b2. (+) And 1 (7) cos. Z_:V, ~+a2+b2. ( In eq. (5) put the value of x drawn from eq. (1), and in eq. (6) the value of y from eq. (1), and reduce, and we shall obtain cos.X=_ a 4 —ll+a2 + b. ~V~ab +b- The analytical expressions cos. Y- for the inclination of a line ~IV/l+a2+62 in space to the three co-orcos. Z=- dinates. -V/l+a2 + b2 The double sign shows two angles supplemental to each other, the plus sign corresponds to the acute angle, and the minus sign to the obtuse angle.

Page  256 256 ANALYTICAL GEOMETRY. PROPOSITION VIII. To find the inclination of two lines in terms of their separate inclinations to the axes. Through the origin draw two lines respectively parallel to the given lines. An expression for the cosine of the angle between these two lines is the quantity sought. Let AP be parallel to one of the given lines, and A Q parallel to the other. The angle PA Q is the angle sought. Let the equations of one of these lines be x=- az, y=bz, and of the other xt=a/z', yf=b'z'. Let AP=r, A Q=r', PQ-D, and the angle PA Q- V. Now in plane trigonometry (Prop. 8, p. 260, Geom.,) we have cos. V= r2+r 2-D2) 2rr' From Prop. 6 we have D2-(x'-x)2 +(y'-y)2 +(z,-)2. Expanding this, it becomes (D2= (x'2 +y'2 +z/2)+(x2 +yW +z2) -2x'x-2y'y-2z'z. But by Prop. 5 we have X2 +y2 +z2 =r2, and x2 +yl2 +Z2 —r2. Whence 2x'x+2y'y+2z'z=r2+r2 —Da. This equation applied to eq. (1) reduces it to cos. V= lx+yy~+z'z rr' But r and r' may have any values taken at pleasure; their lengths will have no effect on the angle V. Therefore, for convenience, we take each of them equal to unity. Whence cos. V=x'x+y'y+z'z. (2)

Page  257 STRAIGHT LINES IN SPACE. 257 But in Prop. 7 we found that x=r cos.X, y=r cos. Y, etc., and that x'=r' cos.X', y'=r' cos. Y', etc.; and since we have taken r=-1 and r'=l1, x=cos. X, etc., and x'= cos.X', etc. Hence cos. V=cos..Xcos.X' +cos. Ycos. Y'+cos.Zcos.Z'. (3) But by Prop. 7 we have a a' cos._ -_ and cos. X=, etc. 1s/l+a2+b2' 4-/ l+a'2+b'2 Substituting these values in eq. (3) we have I C.= 1+aa'+bb' cos. V- -: /(V1+a2 + b2-)(V/1+a' 2 +b2) for the expression required. The cos. V will be plus or minus, according as we take the signs of the radicals in the denominator alike or unlike. The plus sign corresponds to an acute angle, the minus sign to its supplement. Cor. 1. —If we make V=90~, then cos. V=0, and the equation becomes 1+aa' +bb'-=O, which is the equation of condition to make two lines at right angles in space. Cor. 2.-If we make V=O, the two straight lines will become parallel, and the equation will become 1l+aa'+bb' =t1_ /l+a2+b2 V/l+a'2+b'2 Squaring, clearing of fractions, and reducing, we shall find (a'-a)2+ (b' —b)2+ (ab'-ab)2= 0. Each term being a square, will be positive, and therefore the equation can only be satisfied by making each term separately equal to 0. Whence a'=a, b'=b, and ab'=a'b. The third condition is in consequence of the first two. 22* R

Page  258 258 ANALYTICAL GEOMETRY. CHAPTER IX. ON THE EQUATION OF A PLANE. An equation which can represent any point in a line is said to be the equation of the line. Similarly, an equation which can represent or indicate any point in a plane, is, in the language of analytical geometry, the equation of the plane. PROPOSITION I. To find the equation of a plane. Let us suppose that we have a plane which cuts the axes of. X, Y and Z at the points B, C and D, respectively; then, if these points be connected by the straight lines BC, CD and DB, it is evi- z dent that these lines are the intersections of the plane with the planes of the co-ordinate axes. 1B Now a plane may be conceived as a surface generated by moving a c straight line in such a manner that Y/C in all its positions it shall be parallel to its first position and intersect another fixed straight line. Thus the line DC, so moving that in the several positions, D' C', D"C", etc., it remains parallel to DC and constantly intersects DB, will generate the plane determined by the points D, Cand B. The line DB being in the plane xy, its equations are y=O, z=mx+b, (1) and for the line DC we have x-=O, z=ny+b. (2) The plane passed through the line D' C' parallel to the

Page  259 EQUATION OF A PLANE. 259 plane zy, cuts the axis of X at the point p. Denoting Ap by c, the equations of the line D' C' become x=c, z=ny+b'. (3) It is obvious that eqs. (3) can be made to represent the moving line in all its positions by giving suitable values to c and b', and that, for any one of its positions, the coordinates of its intersection with the line DB must satisfy both eqs. (1) and (3)., That is, c and b', in the first and second of eqs. (3), must be the same as x and z, respectively, in the second of eqs. (1). Hence b'-z-ny, and b'-mx+b. Equating these two values of b', we have z-ny=mx+b, or z-=mx+ny+b, (4) This equation expresses the relation between the co-ordinates x, y and z for any point whatever in the plane generated by the motion of the line DC,. and is, therefore the equation of this plane. Cor. 1. —Every equation 6f the first degree between three variables, by transposition and division, may be reduced to the form of eq. (4), and will, therefore, be the equation of a plane. Cor. 2.-In eq. (4), m is the tangent of the angle which the intersection of the plane with the plane xz makes with the axis of X, n the tangent of the angle that the intersection with the plane yz makes with the axis of Y, and b the distance from the origin to the point in which the plane cuts the axis of Z. Hence, if any equation of the first degree between three variables be solved with respect to one of the variables, the co-efficient of either of the other variables denotes the tangent of the angle that the intersection of the plane represented by the equation,with the plane of the axes of the first and second variables, makes with the axis of the second variable.

Page  260 260 ANALYTICAL GEOMETRY. SCHOLIUM.-If we assume A B bD t=- -, _=-, -X C C and substitute these values in eq. (4), it will become, by reduction and transposition, Ax+By+ Cz+D=0, which is the form under which the equation of the plane is very often presented. From this equation we deduce the following general truths: First. —If we suppose a plane to pass through the origin of the co-ordinates for this point, x=O, y=O, and z=O, and these values substituted in the equation of the plane will give D=O also. Therefore, when a plane passes through the origin of co-ordinates, the general equation for the plane reduces to Ax +By+ Cz=O. Second.- To find the points in which the plane cuts the axes, we reason thus: z R The equation of the plane must respond R to each and every point in the plane; the point P, therefore, in which the plane cuts o X the axis of X, must correspond to y=O 0 and z=0, and these values, substituted in the equation, reduces it to Ax+D=O. Or s= — D OP. A For the point Q we must take x=O and z=O. And Y=D- OQ D For the point R, z= — — OR. C Third.-If we suppose the plane to be perpendicular to the plane XY, PR', its intersection with, or trace on, the plane XZ, must be drawn parallel to OZ, and the plane will meet the axis of Z at the distance infinity. That is, OR, or its equal, D —),must be infinite, which requires that C=O, which reduces the general equation of the plane to

Page  261 EQUATION OF A PLANE. 261 Ax+By+D=O, which is the equation of the trace or line PQ on the plane XY. If the plane were perpendicular to the plane ZX, the line 0 Q, or its equal, B ) must be infinite, which requires that B=O, and this reduces the general equation to Ax+ Cz+D=O, which is the equation for the trace PR, and hence we may conclude in general terms, That when a plane is perpendicular to any one of the co-ordinate planes, its equation is that of its trace on the same plane. PROPOSTION II.-PROBLEM. To find the length of a perpendicular drawn from the origin to a plane, and to find its inclination with the three co-ordinate axes. Let RPQ be the plane, and from the Z origin, 0, draw Op perpendicular to the R plane; this line will be at right-angles to every line drawn in the plane from /x the point p. Whence OpQ=900, OpR=900, and S OpP=90. Let Op=p. Designate the angle pOP by X, pOQ by Y, and pOR by Z. By the preceding scholium we learn that ___D D an OP= OQDA_=B - and OR= _ A' B 3' A, B, C and D being the constants in the equation of a plane. Now, in the right-angled triangle OpP, we have OP: 1:: Op: cos... That is, -_D::: p: cos. X. (1)

Page  262 262 ANALYTICAL GEOMETRY. The right-angled triangle OpQ gives -: l:'p: cos. Y. O The right-angled triangle OpR gives D I: p: cos.Z. () C Proportion (1) gives us cos.2'X= A2, (4) D2 (2) gives cos.2Y=P2 (5) and (3) gives os.2 Z=- 2 C. (6) Adding these three equations, and observing that the sum of the first members is unity, (Prop. 7, Chap. 8), and we have? (A2+B2+ C2)=1. D2 Whence p=- _D /A2+B2+ -C2 This value of p placed in eqs. (4), (5) and (6), by reduction, will give cos. X==- A (8) V/A2+B2+ C2 cos. Y= B4- (9) 4A2+B2+ C2 cos. Z= C (10) VA2+B2+ C2 Expressions (7), (8), (9) and (10) are those sought. PROPOSITION II1.-PROBL EM. To find the analytical expressions for the inclination of a plane to the three co-ordinate planes respectively.

Page  263 EQUA'TION OF A PLANE. 263 Let Ax+IBy+ Cz+D-=O be the equa- R tion of the plane, and let PQ represent its line of intersection with the co-ordinate plane (xy). /o, From the origin, O, draw OS perpendicular to the trace PQ. Draw pS. OpS is a right-angled triangle, right- ~ angled at p, and the angle OSp measures the inclination of the plane with the horizontal plane (xy). Our object is to find the angle OSp. In the right-angled triangle POQ we have found D OD OP= —, OQ=. A' Whence PQ % —S/.A2+B. Now PS, a segment of the hypothenuse made by the perpendicular OS, is a third proportional to PQ and PO. Therefore D D D,/B4 A 2+ B2 --- ----: - PS. r D BD Or A2+B2:: B:: -—:. PS=- The other segment, QS, id a third proportional to PQ and OQ. Therefore B __: __ DQ D AD Or VA2+B2: -A:: -: QS=BvA2+B2But the perpendicular, OS, is a mean proportional between these two segments. Therefore we have OS= D VA2+B2 Now, by simple permutation, we may conclude that the perpendicular from the origin 0 to the trace PR is

Page  264 264 ANALYTICAL GEOMETRY. D and that to the trace QR is D V B2+ C2 We shall designate the angle which the plane makes with the plane of (xy) by (xy), and the angle it makes with (xz) by (xz), and that with (yz) by (yz). Now the triangle OpS gives OS: sin. 90~:: Op: sin. OSp. 1) D That is, DA'+ 1: sin. OSp. VA2+B2:1:BV2+1+ C2si.p Whence sin.2OSp sin.2(xy)- A2 -(XZ)= A2+ C +2C" Similarly, sin.2(xz) A2+ C2 A2++ C2 And sin.2(yz) + C A2+B2+ C2 But by trigonometry we know that cos.2=1 —sin..'Whence cos.2(xy)1- — A2+B2 2 etc. A2+B2+ C2 A2+B2+C2,e Whence cos.(xy)/A2+B2+ C2 cos.(xz)- B Expressions sought. 4A2+2+ 2 C2 cos.(yz) -A /A2+B2+ C2 Squaring, and adding the last three equations, we find cos.2(xy) + cos.2(xz) + cos.(yz)= 1. That is, the sum of the squares of the cosines of the three angles which a plane forms with the three co-ordinate planes, is equal to radius square, or unity.

Page  265 EQUATION OF A PLANE. 265 PROPOSITION IV.-PROBLEM. To find the equation of the intersection of two planes. Let Ax+By+ Cz+D=O, (1) A'x+B'y+ C'z+D'=O, (2) be the equations of the two planes. If the two planes intersect, the values of x, y and z will be the same for any point in the line of intersection. Hence, we may combine the equations for that line. Multiply eq. (1) by C' and eq. (2) by C, and subtract the products, and we shall have (AC'-A' C)x+(B C'-B' C)y+(DC-D' C)=o, for the equation of the line of intersection on the plane (xy). If we eliminate y in a similar manner, we shall have the equation of the line of intersection on the plane (xz); and eliminating x will give us the equation of the line of intersection on the plane (yz). PROPOSITION V.-PROBLEM. To find the equation to a perpendicular let fall from a given point (x', y', z',) upon a given plane. As the perpendicular is to pass through a given point, its equations must be of the form x —x'=a(z —z'), (1) Y —y'=b(z-z'), (2) in which a and b are to be determined. The equation of the plane is Ax+By+ Cz+D=O. The line and the plane being perpendicular to each other, by hypothesis, the projection of the line and the trace of the plane on any one of the co-ordinate planes will be perpendicular to each other. For the traces of the given plane on the planes (xz) and (yz), we have Ax+ Cz+D=O and By+ Cz+D=O. 23

Page  266 266 ANALYTICAL GEOMETRY. From the former x=_CzD (3) A A' From the latter y=-Cz- D (4) W B, Now eqs. (1) and (3) represent lines which are at right angles with each other. Also, eqs. (2) and (4) represent lines at right angles with each other. But when two lines are at right angles, (Prop. 5, Chap. 1), and a and a' are their trigonometrical tangents, we must have (aa'+1-=O). That is, — aC+l0, or aA B Like reasoning gives us b=3, and these values put in eqs. (1) and (2) give x- x-A(Z —') i for the equations aY-YIB(z-z) sought. PROPOSITION VI. -PROBLEM. To find the angle included by two planes given by their equations. Let Ax+By+ Cz+D=O, (1) And A'x+By'+ C'z+D'=O, (2) be the equations of the planes. Conceive lines drawn from the origin perpendicular to each of the planes. Then it is obvious that the angle contained between these two lines is the supplement of the inclination of the planes. But an angle and its supplemrent have numerically the same trigonometrical expression.

Page  267 EQUATION OF A PLANE. 267 D)esgnate the angle between the two planes by V, then Proposition 8, in the last chapter gives cos. V- +aa_+ bb (3) (+(V1+a2+ b2)((/1+a'+ b 2) The equations of the two perpendicular lines from the origin must be in the form x=az, y=bz, x=a-z y=b'z. But because the first line is perpendicular to the first plane, we must have a=A and b-B, (Prop. 5.) C C And to make the second line perpendicular to the second plane requires that a'-A, and b'C, C' These values of a, b, and a', b', substituted in eq. (3) will give, by reduction, AAcos. +BB+ CC' cos. ~= *:/A'2+.B2+. C2VA,_+B,+,12 for the equation required. Cor.-When two planes are at right angles, cos. V=O, which will make AA' +BB' + CC'= O. PROPOSITION VII.-PROBLEM. To find the inclination of a line to a plane. Let MN be the plane given by its equation Ax+By+ Cz+~D=O, and let PQ be the line given by its equations

Page  268 268 ANALYTICAL GEOMETRY. x=az+a. y=bz+P. z p Take any point P in the given line, and let fall PR, the perpendicular, upon the plane; RQ is its projection on A x the plane, and PQR, which we will denote by V, is obviously the least an- /Y gle included between the line and the plane, and it is the angle sought. Let x=a'z+r', and y=b'-z+', be the equation of the perpendicular PR, and because it is perpendicular to the plane, we must have (by the last proposition) a'-A and b,'Because PQ and PR are two lines in space, if we designate the angle included by V, we shall have cos. V=- +aa'+bb' (Prop. 8, Chap. 8.) /l+a2+b2Vl/+a/2+ b/2 But the cos. V is the same as the sin. PQR, or sin. v, as the two angles are complements of each other. Making this change, and substituting the values of a' and b', we have sin. v=~ Aa+Bb+C /l1+a2+b2v C2+ B2+A2 for the required result. Cor. —When v=O, sin. v=0, and this hypothesis gives Aa+ Bb+ C=O, for the equation expressing the condition that the given line is parallel to the given plane. We now conclude this branch of our subject with a few practical examples, by which a student can test his knowledge of the two preceding chapters.

Page  269 EQUATION OF A PLANE. 269 EXAMPLES. 1. —What is the distance between two points in space of which the co-ordinates are x-3, y=5, z=-2, x'=-2, y'=-1-, z=6. Ans. 11.180+. 2.-Of which the co-ordinates are x-=, y=-5, z=-3, x'-4, yl= —4, zl1. Ans. 5-I nearly. 3.-The equations of the projections of a straight line on the co-ordinate planes (xz), (yz), are x=2z+1, y= z-2, required the equation of projection on the plane (xy). Ans. y=Jx —2. 4.-The equations of the projections of a line on the co-ordinate planes (xy) and (yz) are 2y-x-5 and 2y=z-4, required the equation of the projection on the plane (xz). Ans. x=z+l. 5.-Required the equations of the three projections of a straight line which passes through two points whose co-ordinates are x'=2, y'=l, z1=0, and x"=-3, y"=O, z"=-1. What are the projections on the planes (xz) and (yz)? Ans. x=5z+2, y=z+1. And from these equations we find the projection on the plane (xy), that is, 5y=x+3. (See Prop. 3, Chap. 8.) 6. —Required the angle included between two lines whose equations are x3z+1 } of the l1st, and x=z+2 } of the 2d. y-2z+6 y= —z+l Ans. V=720 1' 28" (See Prop. 8, Chap. 8.) 23*

Page  270 270 ANALYTICAL GEOMETRY. 7. —Find the angles made by the lines designated in the preceding example, with the co-ordinate axes (See Prop. 7, Chap...) (36~ 42' with RX, 54~44' withY Ans. The 1st line 57~ 41' 20" Y, 2dline 1250 16' Y, 74~ 029' 54" Z, 540 44' Z. 8.-Having given the equation of two straight lines in space, as x=3z+1 of the 1st, and x=z+2 of the 2d y-2z+6 Y= —z+B' f to find the value of f', so that the lines shall actually intersect, and to find the co-ordinates of the point of intersection. Ans. { x2, y7, x-21, Z-+2. (See Prop. 4, Chap. 8.) 9.-Given the equation of a plane 8x-3y+z-4=0, to find the points in which it cuts the three axes, and the perpendicular distance from the origin to the plane. (Prop. 2.) Ans. It cuts the axis of X at the distance of i from the origin; the axis of Y at — 1; and the axis of Z at +4. The origin is.4649+ of unity below the plane. 10. — Find the equations for the intersections of the two planes (Prop. 4.) 3x-4y+ 2z-1=0, 7x-3y-z+ 5=0. Ans. On the plane (xy) 17x —10y+9=0. On the plane (xz) 19x —10z+23=0. 11.-FEind the inclination of these two planes. (Prop. 6.) Ans. 410 27' 41".

Page  271 EQUATION OF A PLANE. 271 12. —The equations of a line in space are x=-2z+l, and y —3z+2. Find the inclination of this line to the plane represented by the equation (Prop. 7.) 8x-3y+z-4=0. Ans. 480 13' 13" 13. —.ind the angles made by the plane whose equation is 8x-3y+z-4=0, with the co-ordinate planes. (Prop. 3.) f 83~ 19' 27" with (xy). Ans. 1100 24' 38" with (xz). 1 21~ 34' 5" with (yz). 14.-The equation of a plane being Ax+By+ Cz+D=O, Required the equation of a parallel plane whose perpendicular distance is (a) from the given plane. Ans. Because the planes are to be parallel, their equations must have the same co-efficients, A, B, and C. In Prop. 2, we learn that the perpendicular distance of the origin from the given plane may be represented by VA2~+B2+ C2 Now, as the planes are to be a distance a asunder, the distance of the origin from the required plane must be D_ +a or D+aV/A2+B2+ C2 V/A2+B2+ 02 VA2+B32+ C' Whence the equation required is Ax+ By+ Cz+ (D+a/A2+AB2+ C2 ) V/A2+_B2+ C2 15. —Find the equation of the plane which will cut the axis of Z at 3, the axis of X at 4, and the axis of Y at 5. Ans. 5x+4!q+60z=20.

Page  272 272 ANALYTICAL GEOMETRY. 16.-Find the equation of the plane which will cut the axis of X at 3, the axis of Z at 5, and which will pass at the perpendicular distance 2 from the origin. At what distance from the origin will this plane cut the axis of Y? Ans. The equation of the plane is 10x+/89y+ 6z — 30=0. 30 The plane cuts the axis of Y at 30. s/89 17. —Find the equations of the intersection of the two planes whose equations are 3x-2y-z —4=0, + 7x+3y+z-2=0. The equation of the projection of the intersection on the plane (xy) is 10x+y-6=0. Ans. On the plane (xz) it is 23x-z-16=0, and that on the plane (yz) is 23y+10z+22=0. 18. —Find the inclination of the planes whose equations are expressed in example 17. Ans. V=60~ 50' 55" or 119~ 9' 5". 19.-A plane intersects the co-ordinate plane (xz) at an inclination of 500, and the co-ordinate plane (yz) at an inclination of 840. At what angle will this plane intersect the plane (xy)? Ans. V=400 38' 6".

Page  273 MISCELLANEOUS PROBLEMS. 273 MISCELLANEOUS PROBLEMS. 1. The greatest diameter or major axis of an ellipse is 40 feet, and a line drawn from the center making an angle of 36~ with the major axis and terminating in the ellipse is 18 feet long; required the minor axis of this ellipse, its area and excentricity. NoTE. —The excentricity of an ellipse is the distance of either focus from the center, when the semi major axis is taken as unity. (The minor axis is 30.8752. Ans?. Area of the ellipse, 969.972 sq. feet. Excentricity.63575. 2. If equilateral triangles be described as the three sides of any plane triangle and the centers of these equilateral triangles be joined, the triangle so formed will be equilateral; required the proof. Let ABC represent any plane A triangle, A, B and C denoting the PI angles, and a, b and c the respective sides, the side a being opposite b the angle A, and so on. B a On A C, or b, suppose an equilat- c eral triangle to be drawn, and let P be its center. Make the same suppositions in regard to the sides c and a, finding P1 and P2. Draw PP1, Pi P2 and PPe; then is PP1 P2 an equilateral triangle, as is to be proved. We shall assume the principle, which may be easily demonstrated, that a line drawn from the center of any equilateral triangle to the vertex of either of the angles, is equal to _ times the side of the triangle. Hence we have AP= —,PC= AP P = —,BP2CP2 V3 v3,/a = Also, the angles PAC=300, PAB=300, PBA=300 s

Page  274 274 ANALYTICAL GEOMETRY. and so on. Now it is obvious that the angle PAP1 is expressed by (A+600), the angle P1BP2 by (B+60~), and PCP2 by (C+60~). We must now show that the analytical expressions for PP, and P, P2 are the same. In analytical trigonometry it was found that the cosine of an angle, A, of a plane triangle would be given by the equation b2+c2 a2 cos. A= 2b~ Whence, a2= b2+c2 —2bc cos. A. That is, The square of one side is equal to the sum of the squares of the other two sides, minus twice the rectangle of the other two sides into the cosine of the opposite angle. Applying this to the triangle PAP1 we have b2 c2be PP1- -+ —- cos. (A+600) (1) - C2 a2 2ac Also, PxP2 + 2=- i cos. (B+600) (2) a b2 2ab And PP.2 —~ +3 ~ -3 cos. (C+600) (3) By trigonometry, cos. (A+60)=cos. A cos. 60-sin. A sin. 60. But cos. 60~0=, and sin. 60= —V3 Whence, cos. (A+60)=- cos. A -3 sin. A This value substituted in eq. (1) that equation becomes 2 b2 c2 be be PP1 3= -3- cos. A+-3- sin. A (4) b2+ c2 -a2 be b2+ c2 -a2 But cos. A — 2be. Whence - cos. A- 6 This value of be cos. A placed in eq. (4), gives 3 2 2b2 2c2 b2 c2 a2 be PP + _-_+-~+ sin. A 6 6 6 6 6 V3 2 a2+ b2+ C2 be 6 3s

Page  275 MISCELLANEOUS PROBLEMS. 275 By a like operation equation (2) becomes 2 6 3(6) But by the original triangle ABC we have sin. A sin. B a a b -, or sin. A=- sin.B Placing this value of sin. A in equation (5) that equation becomes -2 a b22+ ac sin. B. (7) 1 6 3 We now observe that the second members of (6) and (7) are equal; therefore, PP1,-P PP2 And in like manner we can prove PP,=PP2. Therefore the triangle PP1 P2 has been shown to be equilateral. PROBLEM. Given, the excentricity of an Ellipse, to find the difference between the mean and true place of the planet, corresponding to each degree of the mean angle, reckoned from the major axis; the planet describing equal sectors or areas in equal times, about one of the foci, the center of the attractive force. Let AB be the major axis of an Q ellipse, of which CB= CA=A=1 is -x the semi-transverse axis, and also / let C be the common center of the ellipse and of the circle of which A F c n CB is the radius. Then FC=e, and F is the focus of the ellipse. Suppose the planet to be at B, the apogee point of the orbit, (so called in Astronomy). Also, conceive another planet, or material point, to be at B, at the same time. Now, the planet revolves along the ellipse, describing equal areas in equal times, and the hypothetical planet revolves along the circle BPQ, describ

Page  276 276 ANALYTICAL GEOMETRY. ing, in equal times, equal areas and equal angles about the center C. It is obvious that the two bodies will arrive at A in the same time. The other halves of the orbits will also be described in the same time, and the two bodies will be together again at the point B. But at no other points save at A and at B (the apogee and perigee points) will these two bodies be in the same line as seen from F, and the difference of the directions of the two bodies as seen from the focus F is the equation of the center. For instance, suppose the planet to start from B and describe the ellipse as far as p. It has then described the area BFp of the ellipse, about the focus F. In the same time the fictious planet in the circle has moved along the circumference BP to Q, describing the sector B CQ about the center C. Now the areas of these two sectors must be to each other as the area of the ellipse is to the area of the circle. That is, sector BEp: sector B CQ: area Ell.: area Cir. Through p draw PD) at right angles to AB, and represent the are of the circle BP by x. Then CD=cos. x, and PD= sin. x. Draw Cp and CP. But, denoting the semi-conjugate axis by B, we have area DpB: area DPB:: area Ell. area Cir.: B:A:: pD:PD Also we have A CpD: ACPD:: D: PD Hence, area DpB: A CpD:: area DPB: A CPD Therefore, area DpB+ A CpD: area DPB+ ACPD:: B: A or, sector CpB: sector CPB:: B: A: area Ell.: area Cir. Hence it follows that sector FpB: sector CpB:: sector CQB: sector CPB Whence sector FpB-sect. CpB: sect. CQB-sect. CPB:: B: A

Page  277 MISCELLANEOUS PROBLEMS. 277 or, AFpC: sector Q CP:: B: A:: area Ell.: area. Cir. But the area of the ellipse is wrAB and the area of the circle is A2r. But A=1 and B=V1 — c2. The area of the triangle _FCp is le (pD), and the area of the sector is ~y, representing the arc QP by y. Whence E (pD): y: 1-e2: 1. (1) But we have PD: pD:: A: B: 1: Ie 2, and PD=- sin. x. Hence, sin. x: pD:: 1: -e2; pD= sin xl —e2 This value of pD placed in (1) that proportion becomes esin. x1 — e2: y: V1' e2: 1 Or, esin. x:y::1:1. y=esin. x. (2) DEFINITIONS.-lst. The angle x, in astronomy, is called the excentric anomaly. 2d. The angle QCB, or (x+y) is called the mean anomaly. 3d. The angle pFEB is called the true anomaly. 4th. The difference between Q CB or nCB (of the triangle n nC) and nFC (which is the angle n of the triangle CFn) is the equation of the center. The angle QCB, the mean anomaly, is an angle at the center of the ellipse, which is equal to the sum of the angles at n and F; that is, n taken from the angle at the center will give the true angle at the focus, F. We will designate the angle pFB by t. Now, by the polar equation of an ellipse, we have 1 e2 FP=-1 —e cos. t A being 1. Again, by the triangle F.Dp, we find, f P=/EpD2+pD2 But ED2-=(e+cos. x)2=e2+2e cos. x+cos.2x And pD2=sin.2 x (1-e2)=sin.2 x-e2 sin.2 x Therefore, FD2+pD2=e2+ 2e cos. x+1 —e2 sin.2 x But e2 sin.2 x=e2 -e2 cos.2 x. 24

Page  278 278 ANALYTICAL GEOMETRY. Substituting this value of e2 sin.2 x in the preceding expression we have ID2+- pD2-=1+2e cos. x+e2 cos.2 x Whence _Fp-VFD2+pD2=1+e cos. z. Equating these two values of Fp and we obtain 1-e2=(1+e cos. x) (1-e cos. t) e+cos. x Whence cos.t= 1 e (3) 1+e cos. x Here we have a value of t in terms of x and e, but the equation is not adapted to the use of logarithms. By equation (27) Plane Trigonometry, we have 1-cos. t tan.2 t-1+cos. t If the value of cos. t from equation (3) be placed in this we shall have 1 e+cos. x tan.2 _t 1+e cos. x 1+e cos. x-e —cos. x 1+ ecos. x l+e cos. x+e+cos. x I+e cos. x Or, tan.2 t= (1-e)-(1-e) cos. x (1 —e) (1 —cos.x) (1+e)+(l+e) cos. x (1+e) (1+cos.x) That is, tan. t= ( e ) 2 tan.x. (4) 1+e From eq. (2) we obtain Mean Anomaly=x+e sin. x. (5) By assuming x, equation (5) gives the Mean Anomaly. Then equation (4) gives the corresponding True Anomaly. To apply these equations to the apparent solar orbit, the value of e is.0167751 the radius of the circle being unity. But y=e sin. x, and as y is a portion of the circumference to the radius unity, we must express e in some known part of the circumference, one degree, for example, as the unit. Because 180~ is equal to 3.14159265, therefore the value of c, in degrees, is found by the following proportion.

Page  279 MISCELLANEOUS PROBLEMS. 279 3.14159265: 1800::.0167751: d degrees. By log., log. 0167751 -2.2246652 log. 180~ 2.2552725 0.4799377 log. w 0.4971499 Log. e, in degrees, of are, -1,9827878 Add log. 60 1.7781513 Log. e, in min. of arc, 1.7609391 constantlog. Log. 1-e =log. 10.9832249 6751 — 1.992714 cons. log. We are now prepared to make an application of equations (4) and (5) For example, we require the equation of the center for the solar orbit, corresponding to 280 of mean anomaly, reckoning from the apogee. The excentric anomaly is less than the mean- by about half of the value of the equation of the center at any point; and x must be assumed. Thus, suppose x=27~ 32'; then lx=130 46' sin. x-sin. 270 32' 9.664891 Constant, 1.760939 e sin. x= 26' 6518 1.425830 Add x 270 32' Mean Anomaly=270 58' 39"1 Tan. Xx 13~ 46' 9.389178 Const. -1.992714 tan. ~t 130 32' 59" 9.381892 2 True anomaly 27~ 5' 58" Mean Anomaly 27~ 58' 39"1 Equation of center 52' 41"1 corresponding to the mean anomaly of 27~ 58' 39"1, not to 280 as was required.

Page  280 280 ANALYTICAL GEOMET RY. Now let us take x-27~ 40'; then ix —130 50' sin. x 270 40 9.666824 Con. 1.760939 e sin. x 26' 777 1.427763 Add x 270 40' Mean Anomaly, 280 6' 46"6 tan. 2x- 13~ 50' 9.391360 Con. -1.992714 tan. -t 130 36' 43" 9.384074 2 t-270 13' 26" Mean anomaly 28~ 6' 46"6 Eq. center, 53' 20'"6 corresponding to 280 6' 46"6. Now, we can find the equation corresponding to 280 by the following obvious proportion: 28" 6' 46"6 53' 20"6 280 00' 00" 27 58 39 1 52 41 1 27 5 39 1 8' 7"5: 39"5:: 1' 20"9: 4"7 Add 52' 41"1 Equation or value sought, 52' 45"1 In like manner we can find the value of the equation of the center of any and every other degree of the mean anomaly in the orbit of the sun, or any other orbit, when the excentricity is known.

Page  1 LOGARITH3MIC TABLES; ALSO A TABLE OF NATURAL AND LOGARITHMIC SINES, COSINES, AND TANGENTS, TO EVERY MINUTE OF THE QUADRANT.

Page  2 LOGARITHMS OF NUMBERS FROM 1 TO 10000. N. Log. N. Log. N. Log. N. Log. 1 0 000000 26 1 414973 51 1 707570 76 1 880814 2 0 301030 27 1 431364- 52 1 716003 77 1 886491 3 0 477121 28 1 447158 53 1 724276 78 1 892095 4 0 602060 29 1 462398 54 1 732394 79 1 897627 5 0 698970 30 1 477121 55 1 740363 80 1 903090 6 0 778151 31 1 491362 56 1 748188 81 1 908485 7 0 845098 32 1 505150 57 1 755875 82 1 913814 8 0 903090 33 1 518514 58 1 763428 83 1 919078 9 0 954243 34 1 531479 59 1 770852 84 1 924279 10 1 000000 35 1 544068 60 1 778151 85 1 929419 11 1 041393 36 1 556303 61 1 785330 86 1 934498 -12 1 079181 37 1 568202 62 1 792392 87 1 939519 13 1 113943 38 1 579784 63 1 799341 88 1 944483 14 1 146128 39 1 591065 64 1 806180 89 1 949390 15 1 176091 40 1 602060 65 1 812913 90 1 954243 16 1 204120 41 1 612784 66 1 819544 91 1 959041 17 1 230449 42 1 623249 67 1 826075 92 1 963788 18 1 255273 43 1 633468'68 1 832509 93 1 968483 19 1 278754 44 1 643453 69 1 838849 94 1 973128 20 1 301030 45 1 653213 70 1 845098 95 1 977724 21 1 322219 46 1 662578 71 1 851258 96 1 982271 22 1 342423 47 1 672098 72 1 857333 97 1 986772 23 1 361728 48 1 681241 73 1 863323 98 1 991226 24 1 380211 49 1 690196 74 1 869232 99 1 995635 25 1 397940 50 1 698970 75 1 875061 100 ] 2 000000 NOTE. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are now introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the corresponding natural number in the first column stands in the next lower line, and its annexed first two figures of the Logarithms in the second column. I 1

Page  3 LOGARITHMS OF NUMBERS. 3 N. 0 1 2 3 4 5 6 7 8 9 100 000000 0434.0868 1301 1734 2166 2598 3029 3461 3891 101 4321.4750 5181 5609 6038 6466 6894 7321 7748 8174 102 8600 9026 9451 9876.300.724 1147 1570 1993 2415 103 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 104 7033 7451 7868 8284 8700 9116 9532 9947.361.775 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 107 9384 9789.195.600 1004 1408 1812 2216 2619 3021 108 033424 3826 4227 4628 6029 5430 5830 6230 6629 7028 109 7426 7825 8223 8620 9017 9414 9811.207.602.998 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 112 9218 9606 9993.380.766 1153- 1538 1924 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 7666 8046 8426 8805 9185 9563 9942.320 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 117 8186 8557 8928 9298 9668..38.407.776 1145 1514 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 120 9181 9543 9904.266.626.987 1347 1707 2067 2426 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 123 9905.258.611.963 1315 1667 2018 2370 2721 3071 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 1026 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 - 127 3804 4146 4487 4828 5169 6510 5851 6191 6531 6871 128 7210 7549 7888 8227 8565 8903 9241 9579 9916.253 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 0245 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 681 134 7105 7429 7753 8076 8399 8722 9045 9368 9690..12 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 137 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 138 9879.194.508.822 1136 1450 1763 2076 2 289 2702 139 143015 3327 3630 3951 4263 4574 4885 5196 5507 5818 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 141 9219 9527 9835.142.449.756 1063 1370 1676 1982 142 152288 2594 2900 5205 3510 3815 4120 4424 4728 5032 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 144 8362 8664 8965 9266 9567 9868.168.469.b69 1068 145 161368 1667 1967 2266 2564 2863 3161 3460 37 5.8 4055 146 4353 4650 4947 5244 5541 5838 6134 6430 6'i26t, 0t2 147 7317 7613 7908 8203 8497 8792 9086 9380 9(;-,4,99t.b 148 170262 0555 0848 1141 1434 1726 2019 2511. I 5 149 3186 3478 3769 4060 4351 4641 4932 M,.; |t0.

Page  4 4. LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 151 8977 9264 9552 9839.126.413.699.985 1272 1558 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 154 752.1 7803 8084 8366 8647 8928 9209 9490 9771..51 281 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 158 8657 8932 9206 9481 9755..29.303.577.850 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 273 160 4120 4391 4663 4934 5204 5475 5746 6016 6286 6556 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 162 9515 9783..51.319.586.853 1121 1388 1654 1921 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 164 4844 5109 5373 5538 5902 6166 6430 6694 6957 7221 264 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938.193 257 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 172 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 173 8046 8297 8548 8799 9049 9299 9550 9800..50.300 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 9443 9687 9932.176 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 179 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 186 9513 9746 9980.213.446.679.912 1144 1377 1609 16i 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 190 8754 8982 9211 9439 9667 9895.123.351.578.806 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 224 195 290935 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2253 2478 2699 2920 3141 3363 3584 3804 4325 4246 19i 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 199 8853 9071 9289 9507 9725 9943.161.378.695.813

Page  5 OF NUMBERS. 5 N. 0 1 2 3 4 5 6 7 8 9 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 202 5351 5566 5781 5996 6211 6425 6639 6854 7038 7282 203 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417'204 9630 9843..56.268.481.693.906 1118 1330 1542 212 2,05 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 208 8083 8272 8481 8689 8898 9106 9314 9522 9730 9938 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 2219 2426 2633 2839 3046 3252 3458 36 5'3871 4077 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 212 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 213 8380 8583 8787 8991 9194 9398 9601 9805 8.211 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 8656 8855 9054 9253 9451 9650 9849..47.246 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 198 220 2423 2620 2817 3014 3212 3409 3606 13802 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 1 5766' 5962 6157 222 6353 6549 6744 6939 7135 7330 7525 7721) 7915 8110 223 8305 8500 8694 8889 9083 9278 9472 9666 9860..54 224 350248 0442 0836 0829 1023 1216 1410 1603 1796 1989 193 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 |227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 229 9835..25.215.404.593.783.972 1161 1350 1539 190 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 231 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 232 5488 5675 5862 6049 6236 6423 6610 6796 (983 7169 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 234 9216 9401 9587 9772 9958.143.328.513.698.883 185 i 235 371068 1253 1437 1 621 1806 1991 2175 2360 S2544 2728 236 92912 3096 3280 3464 3647 3831 4015 4198 4382 4565 2371 4748 4932 5115 5298 5481 5664 5846 6029 6212 6194 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 1 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 1..30 182 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 242 3815 3995 4174 4353 4533 4712 4891 5070 15249 5428 243 5603 5785 5934 6142 6321 6499 6677 6856 1034 7212 244 7390 7568 7746 7923 8101 8279 8456 81 634 8811 8989 245 9166 9343 9520 969 89 245 9166 9343 9520 9698 9875..51.228.405.582.759 246 390335 1112 1288 1464 1641 1817 1993 2169 2345 2521 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 |: _ |_ z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Page  6 6 LOGARITHMS N. 0 1 2 3 4 | 5 6 7 8 9 250 397940 8114 8287 8461 8634 8808 8981 9154 9328 9501 251 9674 9847.20.192.365.538.711.883 1056 1228 252 401401 1573 1745 1917 2089 2261 2433 2605 2777 2949 253 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 254 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 255 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 257 9933.102.271.440.609.777.946 1114 1283 1451 258 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 259 3300 3467 3635 3803 21970 4137 4305 4472 4639 4806 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 261 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 263 9956.121.286.451.616.781.945 1110 1275 1439 264 421604 1788 1933 2097 2261 2426 2590 2754 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 267 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 269 9752 9914..75.236.398.559.720.881 1042 1203 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 271 2969 3130 3290 3450 3610 3770 3930.4090 4249 4409 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 273 6163 6322 6481 6640 6800 6957 7116 7275 7433 7592 274 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 275 9333 9491 9648 9806 9964.122.279.437.594.752 276 440909 1066 1224 1381 1538 1695 1852 2009 2166 >323 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 278 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 279 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 281 8706 8861 9015 9170 9324 9478 9633 9787 9941..95 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 284 3318 3471 3624 3777 3930 4082 4235 4387 4540 469;,285 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 286 6366 6518 6670 6821 6973 7125 276 7428 7579 731 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 288 9392 9543 96A4 9845 9995.146.296.417.597.748 289 460898 1048 1198 1348 1499 1649 1799 1948 2098 2248 290 2398 2548 2697'2847 2997 3146 3296- 3445 3594 3744 291 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 147 295 9822 9939.116.263.410.557.704.851.998 1145 296 471292 1438 1585 1732 1878 2025 2171 2318 464 2610 291 2756 L903 3049 3195 3341 3487 3633 3779 2925 4071 238 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 239 5jil 5:.1 i 5962 6101 6252 6397 6542 6687 6832 6976

Page  7 OF NUMBERS. 7 N. 0 11 2 3 4 5 6 7 8 9 300 477121 7266 7411 7555 7700 7844 7989 8133 8278 8422 301 8566 8711 8855 8999 9143 9287 9481 9575 9719 9863 302 480007 0151 0294 0438 0582 0725 0869 1012 1156 1299 303 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 304 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157.~ i ~~142 305 4300 4449 4585 4727 4869 5011 5153 5295 5437 5579 306 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 307 7138 7980 7421 7563 7704 7845 7986 8127 8269 8410 308 8551 8692 8833 8974 9114 925o5 9396 9537 9667 9818 309 9959..99.239.380.520.661.801.941 1081 1222310 491362 1502 1642 1782 i922 2062 2201 2341 2481 2621 311 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 312 4155 4294 4433 4572 4711 4850 4989 5128 5267 5406 313 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 315 8311 8448 8686 8724 8862 8999 9137 9275 9412 9550 316 9687 9824 9962..99.236.374.511.648.785.922 317 501059 1196 1333 1470 1607 1744 1880 2017 2154 2291 318 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 319 3791 3927 4063 4109 4335 4471 4607 4743 1878 5014 320 6150 5286 5421 5557 5693 6828 5964 6093 6234 6370 321 6505 6640 6776 6911. 7046 7181 7316 7451 7586 7721 322 7856 7991 8126 8260 8395 8530 8664 8799 8934 9008 323 9203 9337 9471 9606 9740 9874...9.143.277.411 324 510545 0679 0813 0947 1081 1215 1349 1482 1616 1750 134 325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4414 327 4548 4681 4813 4946 6079 5211 5344 5476 5609 5741 328 6874 6006 6139 6271 6403 6635 6668 6800 6932 7064 329 7196 7328 7460 7592 7724 7855 7987 8119 8261 8382 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 331 9828 9959..90.221.353.484.615.745.876 1007 332 521138 1969 1400 1530 1661 1792 1922 2053 2183 2314 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 335 5045 5174 5304 5434 5563 5693 5822 5951 6081 6(10 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 ib;l01 337 7630 7769 7888 8016 8145 8274 8402 8531 8660 8788 338 8917 9045 9174 9302 9430 9559 9687 9815 9943..72 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1L51 340 1479 1607i 1734 1862 1960 2117 2245 2372 2500 2(-27 341 2764 2882 3009 3136 3264 3391 3518 3645 3772 3899 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 343 5294 5421 5547 5674 5800 5927 6053 6180 6305 6432 344 6558 6685 6811 6937 7060 7189 7315 7441 7567 7693 129 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9578 9703 9829 9954..79.204 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1454 348 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 349 28256 2950 3074 3199- 3323 3447 3571 3696 3820 3944 e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,

Page  8 8 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 350 544068 4192 4316 4440 4564 4688 4812 4936 5060 5183 351 5307 5431 5555 5578 5805 5925 6049 6172 6296 6419 352 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 353 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 354 9003 9126 9249 9371 9494 9616 9739 9861 9984.196 122 355 550228 0351 0473 0595 0717 0840 0962 1084 1208 1328 356 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 357 2668 2790 2911 3033 3155 3276 3393 3519 3640 3762 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 359 5094 5215 5346 5457 5578 5699 5820 5940 6061 6182 360 6303 6423 6544 6664 6785 6905 7026 7146.7267 7387 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 363 9907..26.146.265.386.504.624.743.863.982 364 561101 121 1340 1459 1578 1698 1817 1936 2055 2173 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 366 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 369 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 370 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 371 9374 9491 9608 9725 9882 9959..76.193.309.426 372 /570543 0660 0776 0893 1010 1126 1243 1 359 1476 1592 373 1709 1825 1942 2058 2174 2291 2407 2523 1 2C39 2755 374 2872 2988 3104 3220 3336 3452 3568 3z354 i8090 5915 116 375 4031 4147 4263 4379 4494 4610 4726 4141 4957 5072 376 5188 5303 5419 5534 5650 5765 5880 5996 6111 6226 377 63i1 6457 6572 6687 6802 6917 7032 7147 7262 7377 378 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 380 9784 9898..12.126.241.355.469.583.697.811 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 382 2063 2177 2291 2404 2518 2631 2745'858 2972 3085 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 384 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 385 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 387 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9834 1 389 9950..61.173.284.396.507.619.730.842.953 390 591055 1176 1287 1399 1510 1621 1732 1843 1955 26'66 391 2177 2288 2399 2510 2621 2732 2843' 2954 3064 3173 392 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 393 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 394 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487.110 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 3963i 795 17805 7914 8024 8134 8243 8353 8462 8572 8681 S39 8;91 8900 9009 9119 9228 M337 9446 9556 9666 9774 3:98 9863 9992.101.210.319.428.537.646.755.864 390 6003)i3 1082 1191 1299 1408 1517 1625 1734 1843 1951 39_!__5 _0794824138_ 385346187_6_

Page  9 OF NUMBERS. 9 ~N. l 1 2 3 4 5 6 7 8 9 400 602050 2169 2277 2386 2494 2603 2711 2819 2928 3036 401 3144 3253 3361 3469 3573 3686 3794 3902 4010 4118 402 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 108 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 406 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 407 9594 9701 9808 9914..21.128.234.341.447.554 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 415 8048 8153. 8257 8362 8466 8571 8676 8780 8884 8989 416 9293 9198 9302 9406 9511 -9615 9719 9824 9928..32 417 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 419 2214 2318 2421 2525 2628 273! 2835 2939 3042 3146 420 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 6210 422 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 103 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 426 9410 9512 9613 9715 9817 9919..21.123.224.326 427 630428 0530 0631 0733 0835 0936 1038 -1139 1241 1342 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 430'3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 432 5484 5584 5685 5785 5886 5986 6087 6187 6287 6388 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 436 9486 9586 9686 9785 9885 9984..84.183.283.382 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 442 5,122 5521 5619 5717 5815 5913 6011 6110 6208 6306 443 6404 6502 6600 6698 6796 6894 6992 7039 7187 7285 444 7383 7481 7579 7676 7774 7872 7969 8067 8 165 8262 93 445 8360 8458 855 1 8653 8750 8848 8945 9043 A-0140 9237 4 io 9335 9432 9530 9627 9724 9821 9919..16.113.210 4 51 i603508 0405 0502 0599 0696 0793 0890 0987 1084 1181 3i 1278 137.5 1412 i 1569 1666 1762 1859 1953 2053 2150 419!2246 2343 2440 2530 12633 2730 2826 12923 3019 3116

Page  10 10 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 450 653213 3309 3405 3502 3598 3695 t791 3888 3984 4080 451 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 452 5138 5235 5331 5427 5526 5619 5715 5810 5906 6002 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 456 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 457 9916..11.106.201.296.391.486.581.676.771 458 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 459 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 460 2758 2852 2947 3041 3135 3230 3324 3418 3512 3607 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 462 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 463 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 465 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 466 8386 8479 8572 8665 8759 8852 8945 9038 9131 9324 467 9317 9410 9503 9596 9689 9782 9875 9967..60.153 468 670241 0339 0431 0524 0617 0710 0802 0895 0988 1080 469- 1173 1265 1358 41451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 471 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 472 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 473 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 474 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 91 475 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 477 85.18 8609 8700 8791 8882 8972 9064 9155 9246 9337 478 9428 9519 9610 9700 9791 9882 9973..63.154.245 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 481 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 484 4854 4935 5025 5114 5204 5294 5383 5473 5563 5652 485 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 486 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 487 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 489 9309 9398 9486 9575 9664 9753 9841 9930..19.107 490 690196 0285 0373 0362 0550 0639 0728 0816 0905 0993 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 492 1'65 2053 2142 2230 2318 2406 2494 2583 2671 2759 493 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 495 4605 4693 4781 4868 4956 5044 5131 5210 5307 5394 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 497 6356 5444 6531 6618 6706 6793 6880 6968 7055 7142 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883

Page  11 F OF NUMBERS. 11 N. 0 1 2 3 4 5. 6 7 8 9 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 501 9838 9924..11..98.184.271.358.444.531.617'502 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 503 1568 1654 1741 1827 1913 1999 2086 2172 2258'344 504 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 505 3291 3377 3463 3549 3635 3721 3807 3895 3979 4065 506 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 507 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 508 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 509 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 510 7570 7655 7740 7826 7910 7996 8081 8166 8251 8336 511 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 512 9270 9355 9440 9524 9609 9694 9779 9863 9948..33 513 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 514'0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 515 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 616 2650 2734 2818 2902 2986 3070 3154 3238 3326 3407 517 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 518 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 519 5167 5251 5335 5418 5502 5586 5869 5753 5836 5920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 521 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 522 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 523 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 524 9331 9414 9497 9580 9663 9745 9828 9911 9994..77 82 525 720159 0242 0325 0407 0490 0573 0655 0738 0821 0903 526 0986 1068 1151.1233 1316 1398 1481 1563 1646 1728 527 1811 1893.975 2058 2140 2222 2305 2387 2469 2552 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 530 4276 4358 4440 4522 4604 4685 4767 4849 4931 5013 531 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 534 7541 7623 7704 7785 7866 7948 8029 8110 8191- 8273 535 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 536 9165 9246 9327 9403 9489 9570 9651 9732 9813 9893 537 9974..55.136.217.298.378.459.440.621.702 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 539 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 540 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 541 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 542 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 543 4800 4380 4960 5040 5120 5200 5279 5359 5439 5519 544 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 545 6397 6476! 6556 6636 6715 6795 6874 6954 7034 7113 546 7193 7272!7352 7431 7511 7590 7670 7749 7829 7908 547 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 548 8781 8860 i8939 9018 9097 9177 9256 9335 9414 9493 9572 9651 1 9731 9810 9889 9968..47.126.205.284

Page  12 12 LOGARITHMS N. 0 I 1 2 3 4 &' 6 7 8 9 550 740363 0442 0521 0500 0678 0757 0836 0915 0994 1073 551 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860'552 1939 2018 2096 2175 2254 2332 2411 2489 2568 2646 553 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 554 3510 3558 3667 3745 3823 3902 3980 4058 4136 4215 79 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 556 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 557 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 558 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 559 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 561 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 562 9736 9814 9891 9968..45.123.200.277.354.431 563 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 564 1279 1356 1433 1510 1f87 1664 1741 1818 1895 1972 565 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 566. 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 567 3582 3660 3736 3813 3889 3966 4042 4119 4195 4272 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 569 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 570 5875 5951 6027 6103 6180 6256 6332 6408 6484 6560 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 572 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 573 8155 8230 8306 18382 8458 8533 8609 8685 8761 8836 574 8912 8988 9068 9139 9214 9290 9366 9441 9617 9592 74 575 9668 9743 9819 9894 9970..45.121.196.272.347 576 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 577 1176 1251 1326 1402 1477 1552 1627. 1702 1778 1853 578 1928 2003 2078 2150 2228 2303 2378 2453 2529 2604 579'679 2754 2829 2904 2978 3053 3128 2203 3278 3353 580 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 582 4923 4998 5072 6147 5221 5296 5370 5445 5520 5594 583 5669 5743 5818 5892 5966 6041 6115 6190 6264 6338 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 585 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 586 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 587 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 588 9377 9451 9525 9599 9673 9746 9820 9894 9968..42 689 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 590 - 0852 0926 0999 1073 1146 1220 1293 1367 1440 1514 591 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 592 2322 2395 2468 3542 2615 2688 2762 2835 2908 2981 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 1663 4'736 4809 4882 4955 5028 5100 5173 596 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 597 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079,- I I I 0 1

Page  13 OF NUMBERS. 13| N. 0 1 2 3 4 5 6 7 8 9 600 778151 8224 8296 8368 8441 8513 8585 8658 8730 8802 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 602 9596 6669 9741 9813 9885 9957..29.101.173.245 603 780317 0389 0461 0533 0305 0677 0749 0821 0893 0965 604 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 606 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 607 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546; 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 613 7460 7531 7602 7673 7744 7815 7885 79o6 8027 8098 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 616 9581 9651 9722 9792 9863 9933...4..74.144.215 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 620 -2392 2462 2532 2602 2672 2742 2812 2882 2952 3022 621 3092 3162 3231 3301 3371 3441 3511 3581 3651. 3721 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 69 625 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 626 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 627 7268 7337 7408 7475 7545 7614 7683 7752 7821 7890 628 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 629 8651 8720 8789 8858 8927 8996 9065 6134 9203 9272 630 9341 9409 9478 9547 9610 9685 9754 9823 9892 9961 631 800026 0098 0167 0236 0305 0373 0442 0511 0580 0648 632 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 633 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 634 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 635 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 636 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 63 7 4139 4208 4276 4354 4412 4480 4548 4616 4685 4753 638 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 639 5501 5669 5637 5705 5773 5841 5908 5976 6044 6112 640 6180 6248 6316 6384 6451 6519 6587 6655 6723 6790 641 6858 6926 6994 7061 7129 7157 7264 7332 7400 7461' 642 7535 7603 7670 7738 7806i 7873 7941 8008 8076 8143 643 8211 8279 8346 8414 8481 8549 8616 8684 8151 88] ]8 644 8886 8.453 9021 9088 9156 9223 9290 9358 9425 9462 645 9560 9627 9694 9762 9829 9896 9964..31..98.165 646 810233 0300 0367 0434 0501 0596 0636 0703 0770 0337 647 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 648 I515 1642 1709 17'76 1843 1910 1977 2044 2111 2178 649 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847

Page  14 14 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 650 812913 2980 3047 3114 3181 3247 3314 3381 3448 3514 651 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 652 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 653 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 654 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 67 655 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 656 6904 6970 7036 7102 7169 7233 7301 7367 7433 7499 657 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 659 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 660 9544 9610 9676 9741 9807 9873 9939...4..70.136 661 820201 0267 0333 0399 0464 0530 0595 0661 0727 0792 662 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 663 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 664 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 666 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 667 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 668 4776 4841 4906 4971 6036 5101 5166 5231 5296 5361 669 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 672 7369 7434 7499 7563 7628 7692 7J757 7821 7886 7951 673 8015 8080 8144 8209 8223 8338 8402 8467 8531 8595 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 65 675 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 676 9947..11..75.139.204.268.332.396.460.525 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 678 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 681 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 683 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 684 5056 5120 5183 5247 5310 6373 5437 5500 5564 5627 685 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 686 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 690 8849 8912 8975 9038 9109 9164 9227 9289 9352 9415 691 9478 9541 9604 9667 9729 9792 9855 9918 9981..43 692 840106 0169 0232 0294 0367 0420 0482 0545 0608 06 1 693 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 694 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 62 695 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 696 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3, 698 3855 3918 3980 4042 4104 4166 4229 4291 4353 4417 699 4477 4539 4601 4664 4726 4788 4850 4912 49,4 )0,

Page  15 OF NUMBERS. 15 N. 0 1 2 3 4 5 6 7 8 9 700 845098 5160 5222 5284 5346 5408 5470 5532 5594 5656 701 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 704 7573 7634 7676 7758 7819 7831 7943 8004 8066 8128 62 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 709 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 713 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 719 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 70 7332 7393 7453 7513 7574 7634 7694 7755 7815 7875 721 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 723 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 724 9739 9799 9859 9918 9978..38.A.98.158.2s18.278 60 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 726 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 727 1534 1594 1654 1714 1773 1833 1893 1952 2012.2072 728 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 7219 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 731 3917 3977 4036 4096 4155 4214 42'74 4333 4392 4452 732 4511 4570 4630 4689 4148 4808 4867 4926 4985 5045 733, 5104 5163 5222 5282 5341 5400'5459 5519 5578 5637 734 5696 5755 5814 5874 5933 5992 6051 6110 6169 622,8 735 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 736 6878 6937 6996 7055 7114, 7173 7232 7291 7350 7409 737 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 738 8056 8115 8174 82,33 8292- 8350 8409 8468 8527 8586 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 741 9818 9877 9935 9994..53.111.170.228.287.345 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 743 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 744 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 59 745 2156 2215 22,73 2331 2389 2448 2506 2564 2622 2681 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 748 3902 13966 4018 4076 4134 4192 4250 4308 4360 4424 749 4482 14540 4598 4656 4714 4772 4830 4888 4945 5003

Page  16 16 LOGARITHMS N. 0 1 2 3 4 6 6 7 8 9 750 8705031 5119 5177 5235 5293 5351 54093 5466 5524 5582 751 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 57 755 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 758 9669 9726 9784 9841 9898 9956..13..70.127.185 759 880242 0299 0356 0413 0471 0528 0580 0642 0699 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 761 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 764 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 765 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 766 4229 4285 4342 4399 4455 4512 4569 4625- 4682 4739 767 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 768 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 769 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6885,6942 6998.771 7054 7111 7167 7233 7280 7336 7392 7449 7505 7561 772 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 773 8179 8236 8292 8348 8404 8460 8516 8573 8629 8655 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 775 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 776 9862 -9918 0974..30..86.141.197.253.309.365 777 890421 0477 0533 0589 0645 0700'0756 0812 0868 0924 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 779 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 782 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 783 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 784 4316 A371 4427 4482 4538 4593 4648 4704 4759 4814 785 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 786 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 789 7077 7132 7187 7242 7297 7352 7407 7462 7517 757'2 790 7627 7683 7737 7792 7847 7902 7957 8012 8067 8122 791 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 794 9821 9875 9930 9985..39..94.149.203.258.312 55 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 796 0913 0968 1022 1077 1131 1186 1240 1295 1249 1404 7 97 1458 1513 1567 1622 1676 1736 1785 1840 14 1948 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036

Page  17 OF NUMBERS. 17 N. 0 1 2 3 4 5 6 7 8 9 800 903090 3144 3199 3253 3307 3361 3416 3470 3524 3578 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 8.92 4174 4229 4283 4337 4391 4445 4499 4553 4607 46t;1 80 3 4716 4'770 4824 4878 4932 4986 5040 5094 5148 52 )2 801 5250 5310 5364 5418 5472 5526 5580 5634 5688 5742 i. 54 P05 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 805 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 807 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 808 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 810 8485 8539 8592 8646 8699 8753 8807 8860 8914 8967 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 812 9556 9610 9663 9716 9770 9823 9877 9930 9984..37 813 910091 0144 0197 0251 0304 0358 0411 0464 0518 0571 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 815 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 816 1690 1743 1797 1850 1903 1956 2009 2063 2115 2169 817 2222 2275. 2323 2381 2435 2488 2541 2594 2645 2700 818 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 819' 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 821 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 822 4872 4925 4977 5030 5083 5136.5189 5241 5594 5347 823 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 825 6454 6507 &559 6612 6664 6717 6770 6822 6875 6927 826 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 827 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 828 8030 8083 8185 8188 8240 8293 8345 8397 8450 8502 829 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 831 9601 9653 9706 9758 9810 9862 9914 9967..19..71 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 833 0645 0697 0749 0801 0853 0903 0958 1010 1082 1114 834 1166 1218 1270 1322 1374 1426 1478 1630 1582 1634 835 1686 1738 1790 1842 1894 1946 1998 2050' 2102 2154 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 838 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 839 3762 3814 3865 3917 3969 4021 4072 4124 4147 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 841 4796 4848 4899 4951 5003 5054 5105 5157 5209 5261 842 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 843 5828 5874 5931 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 52 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 846 7370 7422 7473 7524 7576 7627 7678 7730 7783 7832 817 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 i 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 843 8908 8959 9010 9051 9112 9163 9216 9266 9311 i9368 845[ ________ _. 6_5

Page  18 18 LOGARITHMS N. 0 1 2 3 4 15 6 7 8 9 850 929419 9473 9521 9572 9623 9674 9725 9776 9827 9879 851 9930 9981..32..83.134.185.236.287.338.389 852 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 853 0949 1000 10o51 1102 1153 1204 1254 1305 1356 1407 851 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 859 3993 4044 4094 4145 4195 4246 4269 4347 4397 4448 860 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 861 5003 5054 5104 5154 5205 5255 5306 5356 5406 5457 862 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 864 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 868 8520 8570 8620 8670 8720 8770 8820 8870 8919 8970 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9519 9569 9616 9669 9719 9769 9819 9869 9918 9968871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 874 1611 1561 1611 1660 1710 1760 1809 1859 1909 1958 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 876 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 878 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 881 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 882 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 883 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 884 6452 6501 6551 6600 6649 6698 6747 6796 -6845 6894 885 6943 6992 7041 7090 7140 7189 -7238 7287 7336 7385 886 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8365 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 891 9878 9926 9975..24..73.121.170.219.267.316 892 950366 0414 0462 0511 0560 0508 0657 O0i 0o 54 0803 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 1269 894 1338 1386 1435 1483 1632 1580 162 16 1c2i 17.3 48 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 896 2308 2356 24056 2453 2502 2550 2599 2647 5o93i 2144 897 2792 2841 2889 2938 2986 3034 3083 3131 3180 322c 898 3276 3325 3313 3421 3470 3518 3566 3615 3666 3711 899 3760 3808 3856 3905 3953 4001 4049 40J8 4146 4194

Page  19 O.F NUMBE RS. 19 N. 0 1 2 3 4 5 6 7 8 9 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 901 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 903 5688 5736 5784. 5832 5880 5928 5976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9474 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 912 9995..42..90.138.185.233.280.328.376.423 913 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 915 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 917 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 919 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 922 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 924 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 926 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 927 7080 7127 7173 7220' 7267 7314 7361 7408 7454 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 931 - 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 9882 9928 9975..21..68.114.161.207.254.300 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 937 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 3543 941 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 945 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6533 6579 6925 6671 6717 6763 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 949 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678

Page  20 L20 LOGA RITHMS N. 0 1 2 3 4 5 6 7 8 9 950 977724 7769 7815 7861 7906 7952 7998 8043 8089 8135 951 8181 8226 8272 8317 &363 8409 8454 8500 8546 8591 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 958 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 962 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 963 3626 6671 3716 3762 3807 3852 3897 3942 3987 4032 964 4077 4122 4167 4212 4257 43J2 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 966 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 967 5426 5471 5516 5561 5606 5651 5699 5741 5786 5830 968 5875 5920 5965 6010 6055 6100' 6144 6189 6234 6279 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 762'2 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 973 81'13 8157 8202 8247 8291 8336 8381 8425 8470 8514 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 9005 9049 9093 9138 9183 9227 9272 9316 9361 9405 976 9450 9494 9539 9583 9628.9672 9717 9761 9808 9850 977 9895 9939 9983..28..72.117.161.206.250.294 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 979 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 983 2554 2698 2642 2686 2730 2774 2819 2863 2907 2951 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 988 4757 4801 4845 4886 4933 4977 5021 5065 5108 5152 989 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 990 5635 5679 5723 5767 5811 5854 5898 5942 5986 6030 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779.44 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 99'7 8695 8739 8792 8826 8869 8913 8956 9000 9043 9087 993 9131 9174 9218 9261 9305 9348 9392 9435 9479 9502 EW` 99 9565 960J 9652 j19696 9739 9783 9826 9870 9913 / 95i

Page  21 TABLE IT. Log. Sines and Tangients. (00) Natural Sines. 21 |J Inile'1 10" Cosilne. D.i'/" Tangrt. ID.10" Comami., N.sine. T. COs. 0 0.00000 10.000)000 0.000000 Infiilite. 0000(oo 1000f0 60 1 6. 43 726 000000 6.463726 13.536274 00029 too0003 9 2 764' 4753) 000000 764756 235244 00058 10000I) 58 3 940347 000000 940847 0591.53 00037 100000 57 4 0. O~3756 0)00000 7.055786 12.934214 00116 100000156 5 169I696 000000 162696 837304 00145 10003' 5I 6 ~41877 9.999999 241878 758122 00175 100000 54 7 308824 9 0999999 08825 691175 00204 1000)u1O 3 8 366816 999999 366817 633183 00233 11000)0 52 9 417968 999999 417970 582030 0029 1100030 51 10 463725 999998 463727 536273 00291 10000) 50 11 7.50l118 9.999998 7.505120 12.494880 00320 99999 49 12 542903 999997 542909 457091 00349 999991 48 13 577668 999997 577672 422328 00378 99999 47 14 609853 999996 609857 390143 00407 99999 46 15 639816 999996 639820 360180 00436 99999 45 16 667845 999995 667849 332151 00465 99999 44 17.694173 999995 694179 305821 00495 9999 43 18 718997 999994 719003 280997 00594 99999 42 19 742477 999993 742484 257516 00563 )9998 41 20 764754 999993 764761 235239 00582 (9998 40 21 7.785943 9.999992 7.785951 12.214049 00611 99998 39 22 806146 999991 808155 193845 00340 99998 38 23 8205451 999990 825460 174540 00369 99998 37 24 843934 999989 843944 156056 00698 99998 36 25 8516i63 999988 861674 138326 00727/ 99997 35 26 878695 999988 878708 121292 00756 999971 34 27 895085 999987 895099 104901 00785 999971 33 28 910879 999986 910894 089106 00814 99997132 29 926119 1 999985 926134 073866 00844 99996131 30 940842 999983 940858 059142 00873 999961 3 31 7.955082 9.999982 1 7.955100 12.044900 00902 99996 29 32 968870 2298 999981 0.2 988 2298 031111 00931 9999618 233 227 999980 0.2 227 33 1982233 10. 982253 017747 00960 99995 27 1 34 995198 21061 999979 0.2 995219 2161 004781 00989 99999269 35 8.007787 |209 999977 0. ~ 8.007809 2098 11.992191 01018 999995 N'7 36 020021 2039 999976 02 020045 2039 97995 01047 99995 4 4 3 031919 1'3999975 0'2 1983 95 1983 02 031945 1930 968055 01076 99995 38 04350'1 101 99093 3 054750181 9999731 0*2 043527 1880 956473 i01105 9999492 39 04781 999972 054809 10 945191 01134 9999 1832 0'9 1833 | | 40 | 065776 1 1783 @ 999971 065806 87 934194 01164 99993 2 41 8.076500 1787 9.999969 12 8.076531 17 11.923469 01193 99993119 42 086965 1744 999968 2 086997 1744 9130 01 99993 1 43 097131703 0 1703 913003 01922 99993 18 43 097183 17036 999966 2 097217 6 902783 01251 99992 17 44 107167 1664 999964 2 107202 1664 892797 01280 99992 45 116926 1126 999963 3 116963 883037 0130 99991 1 146 126471 191 999961 03 126510 1591 873490 01338 9999114 47 135810 1557 03 135851 557 8641493 48 144953 1524 999958 0.3 135851 1524 864149 01396 99991 123 49 153907 1833 49 1539071 1462 999956 0.3 153952 1463 846048 1101425 99990 11 50 11626814 999954 0.3 162727 463 837273 101454 99989 10 518 18.71280 1433 9.999952 03 8.171328 14311.828672 01483 99989 9 52 179713 1405 0 3 1406 45 1879 1379 0.3 179763 1406 8202371 01513 99989 8 53 187985 13 999948 188036 1379 81196401542 99988 7 |54 196102 1332| 999946 3 196156 1353 803844 1 1 01571 99988 6 55 204070 13 99994403 204126 2 795874 I01600 9998? 5 1304 0.3 1304 56 211895 999942 03 211953 130 7880471101629 9998? 4 57 219581 128 999940 0.4 219641 1281 780359 01658 99936 3 58 227134 0125 999938 227195 1259 7728050168 999 2 1237 0.4 12384 59- 234557 1 999936 0.4 234621 1238 7653791801716 99985 11 1216 0.4 1.217 i 60 241855 116 99934 241921 1217 7580901745 90985 0 CI osille.; I Sin1e. CI OlIg.'I'.nn. i i N. cn. sine / 89 Degrees.

Page  22 22 Log. Sines and Tangents. (1~) Natural Sines. TABLE II. Sine. D.10" Cosille. D.10"] Tang. ID10t Cotal;..'N.sine. N. cos. | 0 8.241855 1196 9999934 0.4 8.241921 1197 11.758079 01742 99985 60 1 249033 11 999932 9 4 249102 177 750898 01774 99984 59 2 256034 1158 999929 04, 256165 743835 01803 99984 58 3 263042 114 999927 04 263115 736885 01832 99983 57 4 269881 999925'4 269956 1 730044 01862 99983 56 5 276514 1 999922 0. 276691 1105 723309 01891 999821 55 6 283243 8 999920 0'4 283323 10 716677 01920 99982 54 7 289773 999918. 28985 10 710144 01949 99981 53 8 296207 1056 999915 0.4 296292 10 703708 01978 99980 52 9 302546 1041 999913 0'4 302634 1042 697366 02007199980 51 10 308794 1027 999910 0.4 308884 1027 691116 02036 99979 50 11 8.314954 1012 9.999907 0 4 8.315046 10 11.684954 02065 99979 49 12 321027 999905'4 321122 678878 02094 99978 48 13 327016 999902 0'4 327114 985 672886 02123 99977 47 14 332924 971 999899 0'5 333025 972 666975 02152 99977 46 15 338753 959 999897 O s 333856 959 661144 02181 99976 45 16 344504 946 999894 0.5 344610 946 655390 02211 999'76 44 17 350181 999891 0.5 350289 934 649711 02240199975 43 18 355783 922 999888 0.5 355895 644105 02269 99974 42 19 361315 910 999885 O'. 361430 911 638570 02298199974 41 20 366777 899 999882 os 366895 633105i 02327 99973 40 21 8.372171 9.999879 - 8.372292 11.627708 02356 99972 39 22 377499 888 999876 0.5 377622 888 622378 02385 99972 38 23 382762 867 999873 0' 382889 879 617111 02414 99971 37 24 387962 856 999870 0.: 388092 857 611908 02443 99970 36 26 393101 846 999867 6's 393234 847 606766 02472 39969 35 26 398179 837 999864 o 398315 837 601685 02501 99969 34 27 403199 827 999861 o0 403338 82 596662 02530 99968 33 28 408161 818 999858 O.5 408304 818 591696 02560 99967 32 29 413068 809. 999854 O 5 413213 809 586787 02589 99966 31 30 417919 800 999851 0.6 418068 800 581932 02618 99966 30 31 8.422717 791 9.999848 0'6 8.422869 791 11.577131 02647 99965 29 3V2 427462 782 999844 6 427618 783 572382 02676 99964 28 33 432156 999841 0 6 432315 567685 02705 99963 27 34 436800 766 999838 0.6 436962 766 563038 02734 99963 26 35 441394 758 999834 0'6 441560 758 558440 02763 99962 25 36 445941 760 999831 0'6 446110 750 553890 02792 99961 24 37 450440 742 999827 0'6 450613 743 549387 0282.1 99960 23 38 454893 73 999823 0:6 455070 544930 0285099959 22' 39 459301 727 999820 06 459481 728 540519 02879 99959 21 40 463665 i 72 999816.6 463849 720 536151 02908 99958 20 41 8.467985712 9.999812 0.6 8.468172 713 11.531828 02938 99957 19 42 472263 999809 0-6 472454 707 527546 02967 99956 18 43 476498699 999805. 476693 523307 02996 99955 17 44 480693 999801 o 480892 519108 03025 99954 16 45 484848 999797 0.6 485050 514950 03054 99953 15 46 488963 i 999793 0.7 489170 680 510830 10308399952 14 47 493040 67 999790 7 493250 506750 03112 99952 13 48 497078 667 999786 0.7 497293 5027071 03141 99951 12 49 501080 661 999782 0.7 601298 66 498702 031701999.50 11 50 505045 i 999778 505267 494733 03199,99949 10 51 8.508974! 9.99977' 0.7 8.509200 650 11.490800 03228 99948 9 52 512867 64 999769 107 513098 4869021 03257 99947 8 53 616726; 643 999765 0.7 516961 638 483039 03286 99946 7 637 0.7 638 4 20551 632 999761 0.7 520790 633 479210 03316 99945 6 155 524343 62 999757 "7 524586 7 475414 09645199944 5 5666 528102 626 999753 0.7 528349 62 471651 03374199943 4 57 531828 999748 0.7 532080 616 467920 03403199942 3 58 535523 16 999744 0.7 35779 611 464221 03432{99941 2 59 539186 611 999740 0 539447 460553 03461999940 1 60 542819 999735 43084 456916 03490499939 0 Cosiie. Sine. Cia'olag. I N. cos. N.sine. 88 Degrees. I..,,.....

Page  23 TABLE II. Log. Sines and Tangents. (20) Natural Sines. 23 S billle. D.10" Cosine. D. 10" Tang. D. 10" Cotang. N. sine. N. cos. 0 8.542819 609.999735 07 8.543084 602 11.456916 0349099939 60 1 546422 999731 0.7 546691 453309 03519 99938 59 2 549995 5 999726 07 50268 591 4497321 103548 99937 58 3 553539 586 9997/22 0-8 553817 587 446183 1 03577 99936 57 4 557054 581 999717 08 557336 582 442664 03606 99935 56 5 560540 576 999713 0 560828 577 439172 ~ 03635 99934 55 6 563999 999708 0 8 564291 435709.03664 99933 54 7 567431 567 999704 08 567727 568 432273} 03693 99932 53 567 0.8 568 i 8 570836 53 999699 8 71137 4 428863 03723 99931 52 9 574214 999694 0 574520 6 4254840 03752 99930 51 10 577566 999689 08 577877 59 422123 03781 99929 541 0 38 555 1 1 8.580893 554 9.999685 0.8 8.581208 5" 11.418792 03810 99927 49 12 584193 550 999680 84514 ol1 415486 03839 99926 48 13 58746954 999675 O868 587795 7 412205 03868 99925 47 14 590721 542 999670 5.8;91051 543 408949 03897 99924 46 15 693948 538 999665 0.8 594283 539 405717 0392699923 45' 16 5971W 530 999660 8 97492 535 402508 03955 99922 44 17 600332 999655 600677 531 399323 0398499921 43 526 9 038 527 18 60348952 999650.8 603839 27 396161 04013 99919 42 19 606623 999645 606978 23 393022 04042 99918 4 20 609734 515 999640 0.8 610094 619 389906 04071 99917 40 21 8.612823 11 9.999635 0.9 8.613189 611.386811 04100 99916 39 22 615891 58 999629 o.9 616262 508 383738 03129 99915 38 23 618937 04 999324.9 619313 380687 04159913 37 24 621962 50 999619 9 622343 5 377657 04188 99912 36 -25 624965 497 999614 0.9 625352 501 374648 04217 99911 35 26 627948 4 999608 09 628340 498 371660 0424699910 34 27 630911 999603 9 63 495 368692 04275 99909 33 28 633854 490 999603 631308 28 633854 487 9995970.9 634256 491 365744 0430499907 32 29 636776 484 999592.9 637184 488 362816 04333199906 31 30 639680 481 999586 0.9 6400593 48 359907 04362)99905 30 9.999581 0 9 8.642982 482 31 8.642563 477 9.999581.9 8.642982 48 11.357018 0439199904 29 32 645428 999575 645853 475 354147 04420 99902 28 33] 64892744 4648704 4 351296 04449199901 27 34 651102 468 999564 0.9 651537 472 348463 04478 9990026 35 653911 46 999558 654352 469 345648 04507 99898 25 465 999558 1.0 466 36 656702 462 999553 657149 342851 04536 99897 24 37 659475 459. 999547 1.0 659928 463 340072 0456 99896 23 38 662230 456 999541 662689 337311 04594'99894 22 39 664968 999535 10 666433 334567 04623 99893 021 40 667689 41 999529. 668160 45 331840 0465399892 20' 40 667689 51 999529 1 60 453 41 8.670393 9.999524 8.670870 44 11.329130 04682 99890 19 42 673080 448 999518 673563 49 326437 0471] 99889 18 43 675761 999512 1 676239 446 323761 04740 99888 17 44 678405 442~ 999506 1.0 678900 443 321100 04769 99886 16 440 999506 1.0 a 0479899886i442 45 681043 7 999500 681544 438 318456 0479899885 15 46 683665 999493 684172 315828 04827 99883 14 47 686272 1 999487 1.0 686784 35 313216 04856 99882 13 48 688863 432 999481 0 689381 433 310619 04885 99881 12 49 691438 429 691438 999475 1.0 691963 430 308037 04914 99879 11 50 693998 427 999469 694529 428 305471 0494399878 10 51 8.696543 9.999463 1. 8.697081 425 11.302919 0497299876 9 52 699073 999456 699617 423 300383 05001 99875 8 419 1*1 276 0603099873 7 53 701589 419 999450. 1 702139 420 297861 05030 99873 7 54 704090 417 999443 1.1 704246 418 295354 05059 99872 6 64 706577 414 999437 1.1 707140 4115 292860 05088 99870 5 56 706577 9412707140 /1 41 56 709049 999431 709618 413 290382 05117 99869 4 5i7 711507 410 999424 11 702083 411 287917 0514699867 3 8 713952 407 999418 1 71454 408 285465 0517599866 2 59 716383 999411 716972 406 283028 05920599864 1 60 718800 999404 1 7193968 40 280604 05234 9 9863 Cosine. Sine. Cotang. Tang. _N. cos |N.sine. 87 Degrees.!y _

Page  24 24 Log. Sines and Tangents. (.0) Natural Sines. TABLE II. Sine. D. 10'[ Cosine. D. 10''lang. li. 10"I Cotang. I N. sine. N. cos. 0 8.718803 9.999404 1 8.719396 402 11.280604 05234 99863 60 1 721204 999398 721806 278194 05263 99861 59 398 999398 2 723595 396 999391 1.1 724204 97 275796 05292 99860 58 3 725972 394 999384 1.1 726588 3 273412, 05321 99858 57 4 72-8337 39 999378 1.1 728959 3 271041 0535099857 56 5 730388 392 999371 1.1 731317 391 68683 0537999855 55 6 733027 390 999364 1.1 733663 389 266337 05408 99854 54 7 735354 386 999357 1.2 735996 87 264004 05437 99852 53 8 737667 384 999350 1.2 738317 385 261683 05466 99851 52 384 35 0595jC41. 7389 9 739969 38 999343 12 740826 383 259374 05495 99849 51 10 742259 382 999336 1.2 742922 381 257078 05524 99847 50 11 8.744536 380 9.999329 1.2 8.745207 3 11.254793 05553 99846 49 12 746802 999322. 747479 252521 0558299844 48 373 744790557 9984494377 13 749055 376 999315 1.2 749740 375 250260 05611 998421 47 14 751297 372 999308 1.2 75198973 248011 05640 99841 46 15 753528 372 999301 1.2 754227 31 245773 05669 99839 45 16 755747 370 999294 1.2 756453 371 243547 05698 99838 44 17 75755368 1.2 79954 17 757955 366 999286 1.2 758668 36 241332 05727 99836 43 18 760151 36 99927 12 76872 367 239128 05756 99834 42 19 762337 364 999272 1.2 763065 365 236935 05785 99833 41 20 764511 36 999265 76546 364 234754 05814 99831 40 21 8.766675 3 9.999257 1.2 8.767417 36 11.232583 0584499829 39 22 768828 359 999250 1.2 769578 360 230422 05873 99827 38 22 768828 3 99924 38 23 770970 357 999242 1.3 771727 6 228273 05902 99826 37 24 773101 355 999235 1.3 773866 356 226134 05931 99824 36 25 775223 999227 13 775995 224005 05960 99822 35 26 77733352 999220 1.3 778114 3 221886 05989 99821 34 27 779434 350 999212 1.3 780222 351 219778 06018 99819 33 28 781524 348 9992-05 1.3 78232 50 217680 06047 99817 32 29 783605 347 999197 1.3 784408 348 215592 06076 99815 31 30 785675 345 999189 1.3 786486 346 213514 06105 99813 30 31 8.787736 343 9.999181 1.3 8.788554 3 11.211446 06134 99812 29 32 789787 342 999174 1.3 790613 209387 0616399810 28 33 791828 340 999166 1.3 792662 341 207338 06192 99808 27 34 793859 33 999158 1.3 794701 340 205299 06221 99806 26 35 795881 7 999150 1.3 796731 338 203269 0625099804 25 36 797894 5 999142 1.3 798752 201248 06279 99803 24 37 799897 334 999134 1.3 800763 199237 063089980 23 38 801892 332 999126 1.3 802765 197235 0633799799 22 39 803876 331 999118 1.3 804858 332 195242 063669979 21 40 805852 329 999110 1.3 806742 331 193258 0639599795 20 41 8.807819 9.999102 8.808717 329 11.191283 0642499793 19 42 809777 326 999094 1.3 810683 328 189317 0643 99792 18 43 811726 325 999086 1.4 812641 326 187359 06482 99790 1 7 44 813667 323 999077 1.4 814589 325 185411 06511 99788 16 45 815599 322 999069 1.4 816529 323 183471 06540 99786 15 46 817522 320 999061 1.4 818461 322 181539 06569 99784 14 47 819436 319 999053 1.4 820384 320 179616 06598 99782 13 48 821343 318 999044 1.4 822298 31 177702 06627 99780 12 49 823240 316 999036 1.4 824205 318 175795 06656 99778 11 50 825130 315 999027 1 4 826103 316 173897 06685 99776 10 51 8.827011 313 9.999019 1.4 8.827992 315 11.172008 06714 99774 9 52 828884 312 999010 1.4 829874 31 170126 06743 997721. 8 53 830749 311 999002 1.4 831748 312 168252 06773 99770 7 54 832607 309 998993 1.4 833613 311 166387 06802 99768 6 55 834456 308 998984 1.4 835471 310 164529 06831 99766 5 56 836297 307 998976 1.4 837321 308 162679 06860 99764 4 57 838130 306 998967 1.4 839163 307 160837 06889 99762 3 58 839956 304 998958 1 840998 306 159002 05918 99760 2 303 ~82530411.5 59 841774 303 998950 1.5 842825 157175 106947 9975 8 1 302 644 303 60 843585 998941 1.5 8444 3 15536' 0976 99756 0 | Cosine. Sine. Cotanr. - T. (os. N.sine. 86 Degrees.

Page  25 TABLE 11. Log. Sines and Tangents. (40) Nat.ual Sines. 25 Sine. D. 10" Cosine. ID._10" Tang. D. 10" Cotang.!IN. sine. N. cos.1 0 8.843585 300 9.998941 15 8.844644 30 11-.1553561 0697699756 60 1 845387 998932 846455 3 153545 107005 9975459 2 847183 998923 848260 151740!107034199752 58 298 1.5 9 3 848971 998914 850057 8 149943 07063t99750 57 4 850751 295 998905 15 851846 29 148154 0709299748 56 5 852525 294 998896 853628 29 146372i 07121 99746 55 6 854291 998887 855403 1445971107150199744 54 293 1.5 295 853049 292 998878 5 857171 1428291 07179i99742 53 8 857801 998869 858932 1410681 07208199740 52 9 859546 998860 8608692 1393141[07237199738 51 10 861283 290 998851 862433 291 137567][07266199736 50 287 1.5 290 2 9 11 8.863014 9.998841 8.864173 11.13582707129599734 49 12 864738 28 998832 865906 28 1340941 07324 99731 48 13 866455 998823 867632 28 132368 073531 99729 47 285 1.6 287:1 7714 14 868165 284 998813 869351 85 130649i0738299727 46 15 869868 998804 871064 128936 07411:99725 45 283 1.6 284 16 871565 998795 872770 127230:107440 j9723 44 282 1.6 2,83 17 873255 998785 874469 125531 07469 997211 43 281 1.6 282 18 874938 998776 876162 123838i07498 99719142 279 1.6 281 19 876615 998766 877849 122151 107527997161 41 20 878285 99877 879529 120471 107556 99714!40 277 ~ 16 792 279 21 8.879949 27 9.998747 16 8.881202 27 11.118798 1107585 99712 39 22 881607 7 998738 1 882869 27 117131 i107614 997101 38 275 1.6 277 23 883258 998728 884530 115470 J107643 99708 37 274 1.6 276 24 884903 998718 886185 113815 0767299705 36 273 1.6 R~83276 118Si 729753 25 886542 998708 887833274 112167i 07701 99703[35 26 888174 998699 889476 2 110524 0773099701 34 27 889801 27 998689 6 891112 108888 07759 99699 33 270 1.6 W22 28 891421 998679 892742 107 58 0778899696!82 269 1~T6 271 17j 0'B19 9j3 29 893035 998669 894366 105634 107817 99694 31 268 1.7 270 i086962I3 30 894643 998659 1 865984 104016 07846 99692130 W67 1.7 269 31 8.896246 269.998649 8.897596 2 11.102404 0787599689 29 266 1.7 9,68 32 897842 998639 7 899203 100797 07904 S9687 28 265 1.7 267 33 899432 64 9986291 91)0803 2 0991970 07933 99685 27'164 1.7 266 34 901017 263 998619 1 902398 265 097602 0796299683 26 35 902596 2 998609 7 90398' 064 096013 i 07991 99680:25 35 904169 21 998599 9055701 094430 i08020 996781 24 2431 1.7 263 37 905736 260 998589 17 907147 262 092853:08049 996761 23 38 907297 259 998578 7 908719 261 0912811108078 99673122 39 908853 998568 910285 089715 108107 s9671 21 258 1.7 260 40 910404 998558 911846 088154,0813699668 20 257 1.7 259 1~8jg01a366 C 41 8.911949 257 9.998548 17 8.913401 258 11.086599.!08165 996661 19 41 ~~~.91949 9Y8548 8.913401i 42 913488 256 998537 17 914961 257 0(85049 08194 99664 18 43 915022 255 998527 17 916495 26 083505:08223 99661117 44 916550 998516 918034 081966 08?52 99659 16 45 918073 25 998506 1 919568 2 080432i08281 99657 15 4 9103253 9985035 46 919591 252 998495 921096 25 078904110831099654114 472 18 9269254 47 921103 998485 922619 077381110833999652 13 48 922610 25 998474 8 924136 25 075864108368 99649 112 250 i.8 252 49 924112 998464 925649 074351!08397 99647 11 50 925609 249 998453 18 927156 2 072844 08426j99644 10 249 1.8 250; 1Ci14 85~962' 51 8.927100 9.998442 8.928658 2411. 0142 0845599642 9 52 928587 24 998431 930155 49 069845 084684 99639 8 53 93008 24 998421 18 931647 24 068353 08519 i963'g 7 54 9300344 4 998410 933134 066866 08542 99635! 6 246 8 9314$ 55 933015 998399 934616 065384 08571 99662 5 94 4 931 1 244 56 934481 98388 18 936093 24 03907 08600 99630 4 244 1.8 245 57 935942 243 998377 937565 9062435 7v98(19 627 3 243 1.8 9303 24 023 58 937398 243 998366 1 8 939032 244 060968 08658 99625 2 59 938850 24 998355 940494 059506 08687 99622 1. 60 94u296 998344 941962 058048 08716 99619 0 Cosine. Sine. Cotang. I Tang. N. co_.tN.ine. 85 Degrees. 9L~~~~~~~~~~~~, i'

Page  26 26 Log. Sines and Tangents. (50) Natural Sines. TABIE Il. _ Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. IN. sine. N. cos. 08.940296 240 9.998344 19 8.941952 242 11.80481'08716 99619 60 1 941738 239 998333 1 943404 241 056596 08745 99617 59 2 943174 239 99832'2 1.9 944852 240 055148 08774 99614 58 3 944606 238 998311 19 946295 240 053705 08803 99612 57 4 946034 237 998300 9 947734 9 3 052266 08831 99609 56 5 947456 23 998289 19 949168 239 050832 08860 99607 55 6 948874 235 998277 19 950597 237 049403' 08889 99604 54 7 950287 235 998266 19 952021 237 047979 08918 99602 53 8 951696 23 998255. 953441 23 046559 08947 99599 52 9 953100 234 998243 1.9 954856 235 045144 08976 99596 51 10 954499 23 998232 1.9 956267 234 043733 09005 99594 50 11 8.955894 32 9.998220 1 9 8.957674 234 11.042326 09034 99591 49 12 957284 2 998209 19 959075 233 040925 09063 99588 48 13 958670 230 998197 19 960473 233 039527 109092 99586 47 14 960052 229 998186 19 961866 231 038134 09121 99583 46 15 961429 229 998174 19 963255 231 036745 09150 99580 45 16 962801 228 998163 1 9 964639 230 035361 09179 99578 44 17 964170 227 998151 1*9 966019 229 033981 09208 99575 43 18 965534 227 998139 20 967394 229 032606 09237 99572 42 19 966893 226 998128 20 968766 228 031234 09266 99570 41 20 968249 225 998116 20 970133 227 029867 O92999567 40 21 8.969600 224 9.998104 2:0 8.971496 226 11.028504 0932499564 39 22 970947 224 998092 2.0 972855 226 027145 09353 99562 38 23 972289 223 998080 2.0 974209 225 025791 09382199559 37 24 973628 222 998068 o 975560 224 024440 09411199556 36 25 974962 222 998056 2 976906 2 023094 09440199553 35 26 976293 221 998044. 978248 234 021752 09469199551 34 27 977619 220 998032 o 979586 223 020414 09498199548 33 28 978941 220 998020 2 980921 019079 09527199545 32 29 980259 22 998008 20 982251 2 017749 09556995421 31 30 981573 218 997996 2 0 983577 221 016423 095851995401 30 31 8.982883 218 9.997984 2/0 8.984899 220 1 5101 096141995374 29 32 984189 217 997972 2 0 98621783 0964299534 28 33 985491 997959 987632 012468 0967199531 27 34 986789 216 997947.0 988842 011158 09700199528 26 35 988083 216 997935 2 990149 2178 0098511 097291995261 25 36 989374 1 997922 1 991451 2 008549 06758199523 24 37' 99060 214 997910 2.1 992750 216 007250 109787199520 23 38 991943 214 997897 2.1 994045 216 005955 09816199517 22 39 993222 213 997885 2.1 995337 215 004663 098451995141'21 40 994497 212 997872 2.1 996624 214 003376 0987419951120 41 18.995768 211 9.997860 2.1 8.997908 13 11.002092 09903199508 19 42 997036 211 997847 2.1 999188 213 000812 109932 99501 18 43 998299 211 997835 2.1 9.000465 12 10.999535 09961 995031 17 44 999560 210 997822 2.1 001738 22 998262 09990199500 16 45 9.000816 209 997809 2.1 003007 211 9969931 10019j99497 15 46 002069 208 997797 2.1 004272 210 995728 [ 100481994941 14 47 003318 208 997784 2.1 005534 210 994466 1 10077199491 13 48 004563 207 997771 2.1 006792 209 993208 10108199488 12 49 005805 206 997758 2.1 008047 08 991953 10135699485 11 50 007044 2 997745 2.1 009298 208 990702 1016499482 10 951 9.008278 206 9.997732 2.1 9.010546 207 10.989454 10192199479 9 62 009510 205 997719 2.1 011790 207 988210 10221199476 8 53 010737 204 997706 1 013031 206 686969 1 10250:99473 7 54 011962 2 997693 1 014268 985732 1110279199470 6 203 2.2 206 55 013182 203 997680 2.2 015502 20 984498 10308i99467 5 66 014400 202 997667 2.2 016732 204 983268 11033799464 4 57 015613 202 997654 017959 24 983041 I10366199461 3 58 -016824 201 997641 2 2 019183 203 980817 10395199458 2 59 0180311 2 997628 2.2 020403 03 979597 1104249955 1 60 019235f _ 9976142 021620 978380 1045319942 0 Cosine. Sine. Cotang Tang. N. cos.l Nsine. 84 Degrees.,.

Page  27 TABLE II. Log. Sines and Tangents. (6\) Natural Sines. 27 Sine. D. 10" Cosine.'D. 10" Tang. iD. 10"1 Cotang. IN. sine. N. cos. 0 9.019235 200 9.997614 2 9.0216202 10.978380 10453 99452 60 1 020435 199 997601 i 022834 977166 110482 99449 59 2 021632 199 997588 024044 975956 10511 99446 58 3 022825 997574 22 025251 201 974749 1054099443 57 4 024016 198 997561 22 026455 200 973545 10569 99440 56 5 025203 997547! 2 027655 972345 10597 99437 55 6 026386 1 997534 028852 199 971148 11062699434 54 197 0' 763 199 7 027567 196 997520 23 030046 969954 10655 99431 53 8 028744 196 997507 23 031237 198 968763 10684 99428 52 9 029918 997493 032425 967575 10713199424 51 10 031089 195 9974801 23 033609 197 966391 1074299421 50 11 9.032257 94.997466 23 9.034791 196 10.965209 10771 99418 49 12 033421 997452 03969 96 964031 10800 99415 48 13 034582 997439 037144 962856 1082999412 47 14 035741 193 997425 2.3 038316 195 96168411085899409 46 15 036896 192 997411 2.3 039485 194 960515 10887 99406 45 16 038048 19 997397 2.3 0401651 194 959349 10916 99402 44 191 2.3 194 17 039197 997383 041813 958187 10945 99399 43 18 040342 190 997369 23 042973 193 957027 10973 99396 42 19 041486 190 99735 3 044130 192 955870 11002 99393 41 20 042625 189 997341 045284 192 954716 11031 99390 40 21 9.043762 189 9.997327 2 9.046434 191 10.953566 1106099386 39 22 044895 997313 047582 952418 11089 99383 38 23 046026 180 997299 048727 190 951273 11118 99380 37 24 047154 187 997285 24 049869 950131 11147 99377 36 25 048279 997271. 051008 189 948992 11176 99374 35 26 049400 1876 997257 24 052144 189 947856 11205 99370 34 27 050519 186 997242 2.4 053277 188 946723 11234936733 28 051635 997228 054407 945593 11263 99364 32 29 052749 997214 055535 187 944465 11291 99360 31 30 053859 184 997199 056659 943341 1132099357 30 31 9.054966 184.9.997186 9.057781 186 10.942219 11349 99354 29 32 056071 997170 058900 941100 11378 99351 28 33 057172 183 997156 060016 185 939984 11407 99347 27 34 058271 997141. 061130 185 938870 11436 99344 26 35 059367 997127 2.4 062240 937760 11465 99341 25 36 060460 18 997112 2.4 063348 184 936652 11494 99337 24 37 06155 181 997098 2 064453 184 9347 11523 99334 23 38 062639 18 997083 2. 06556 183 934444 1155299331 22 39 063724 180 997068 2.5 066655 183 933345 1158099327 21 40 064806 997053 2 067752 182 932248 1160999324 20 419.06 180 997039 9.06884618 10.931154 11638 99320 19 42 066962 997024 069038 930062 11667 99317 18 43 068036 17 997009 2 071027 181 928973 11696199314 17 441 069107 17 996994 2.5 072113 181 927887 11725 99310 16 45 070176 178 996979 073197 180 926803 11754 99307 15 46 071242 178 996964 074278 180 925722 11783 99303 14 47 072306 177 996949 075356 179 924644 1181299300 13 48 073366 996934 076432 179 923568 11840 99297 12 49 074424 176 996919 2.5 077505 178 922495 11869 99293 11 50 075480 176 996904 2.5 078576 921424 1189899290 10 91 9.0736533 17 9.996889 2:5 9.079644 178 10.920356 11927,99286 9 56 077583 17 996874 2 080710 17 919290 11956 99283 8 53 078631 175 99684 8 2.5 081773 177 918227 11985 99279 7 641 079676 174 996843 5 082833 17 917167 12014 99276 6 655 080719 1 996828 25 08389176 916109 12043 J192;2 5 56 081759 173 996812 2 084947 176 915053 12071 99269 4 5 173 2.6 1075 57 082797 172 99679726 086000 175 914000 12100 99255 3 58 083832 996782 087050 912950 12129199262 2 69 084864 172 996766 2.6 088098 17 91102 1215899258 1 60 085894 996751 2 089144 910856 12187 99255 0 Cosine. Sine. i otTang. ng N.cos. N.siNne.l 83 Degrees. L~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Page  28 28 Log. Sines and Tangents. (70) Natural Sines. TABLE II. Sine. D. I';j Cosine... 1'"'l'anlg. D. so' uotaug.!N.silie... coe. 0 9.085894 171 9.996751 6 9.089144 174 10.910856 12187 99255 60 1 086922 171 996735 2 6 090187 909813 12216 99251 59 2 079471 2.6 173 2 087947 170 996720 091228 908772 1224599248 58 3 0838970 170 996704 099266 3 907734 12274 99244 57 4 08999U 170 996688 76 093302 1 906698 12302 99240 56 5 091008 169 996673 2.6 094336 172 905664 12331 99237 55 6 092024 169 996657 2.6 095367 171 904633 12360 9J233:54 7 093037 168 996641 2,6 096395 171 903605 12389199260 53 8 094047 168 996625 26 097422 1 902578 12418199226 52 9 095056 168 996610 266 098446 170 901554 12447199222 51 10 096062 167 996594 2 - 6 099468 170 900532 12476199219 50 11 9.0970Sa 167 9.996578 27 9.100487 1G9 10.899513 12504 99215 49 12 039806 ~166 998562 2. 101504 169 898496 12533 99211 48 12 098056 62 2.7 101 169 13 099065 166 996546 2.7 102519 897481 12562199208 47 14 1000;2 166 996530 2.7 103532 168 896468 91 99204 46 16 10204865 99514 7 1042 12099200 45 18 104025 164 996466 0,.7 107559 167 892441 1270 99189 42 19 105010 164 996449 27 108560 166 891440 1273 99186 41 20 105992 163 996433. 109559 1 890441 1276499182 40 21 9.106973 163 9.996417 2.7 9.110556 166 10.88944.4 12793 99178 39 22 107951 163 996400 2:7 111551 165 888449 1282299176 38 23 108927 162 996384 27 112543 165 887457 1281 99171 37 24 109901 162 996368 27 113533 165 886467 12880 99167 36 25 110873 162 996351 27 114521 164 8854791 1290899163 35 26 111842 161 996335.7 115507 164 884493 1293799160 34 27 112809 161 996318 2.7 116491 164 883509! 1296 99156 33 28 113774 160 996302 2.8 117472 163 82528 1299599152 32 29 114737 160 996285 118452 163 881548 1302499148 31 30 115698 16 996269.8 119429 162 88057111305399144 30 32 91176656 159 9.996252 2.8 9.120404 162 10.879596 13081 99141 29 32 1176131 159 996235 2.8 121377 162 878623 13110 99137 28 33 118567 159 996219 2:8 122348 161 877652 1313999133 27 34 119519 158 996202. 123317 161 876683 1316899129 26 35 120469 158 996185 28 124284 161 875716 1319799125 25 36 121417 158 996168 2.8 125249 160 874751 1322699122 24 37 122362 157 996151 2.8 126211 160 873789 11325499118 23 38 123306 157 996134 2.8 127172 160 872828 1f3283199114 ~2 39 124248 157 996117 2.8 128130 159 871870 13312699110 21 40 125187 156 99610 28 12087 159 870913 13341 991 20 41 9.12612 56 9.996083 9.13041 159 10.869959 133 99102 19 42 127060 156 996066 9 130994 158 869006 13999098 18 43 127993 1 996049 2.9 13194 158 868056 101342799094 17 44 128925 1 996032 2:9 132893 158 867107 1345699091 16 45 129854 15 996015 133839 17 866161 1348590087 15 ] 154 2.9 157 461 130781 154 995998 2'9 134784 1 7 865216 13514 9-u83 14 47 131706 154 995980 27 9 135436 157 864274 13543 990913 4 1326360 66 157 86 /9 1 153 4 2-9 156 15 995963 35 2.9 7 1 1565 84 491352 199059 12603 49 1335511513 995946 2.9 137605156 862395 1330099111 50 1344'10 3 995928 2 138542 861458 1362999067 10 51 9.135387 1 9 91 911 2 9 91.9476 155 10.860524 1365899063 9 54 158 1522 53 137216 152 995876 29 141340 155 858660| 1371'39055 7 541 1381281 152 9859 2:9 142269 154 8577315 1374[9909 146 55 139037 11 99o841 29 141 154 856804111373 3)9047 5 56 139944 151 9958c3 2.9 144121 154 855879! 13802 19043 4 57 140850 151 99580 2 9 145044 153 8549561 13831 99039 3 58 141-54 150 995188 2.9 145966 153 854034 138609035 2 59 142655 995' 21 1 46885 853115 13889190371 1 60] 3G lV };, 1i3 51 104355 15]995185 14'84 8 75176 1319 7 1 9027 0 0' 1043551 6 Cos0n-. I _______'1 ___ Cotalng. Tang. N. cos.N.seine&' ] 82 Degrees.

Page  29 TABLE II. Log. Sines and Tangents. (8O) Natural Sines. 29 Sine. D. 10' Cosine. D. 1"'lang. 0. In' I Coang. -. siie.l N. cos. 0 9.143555 150 9.995753 0 9.147803 1 10.852197 13917 99027 60 1 144453 149 9935 3.0 148718 8282 1 1394699023 59'2 145349 995717 149632 850368 139i5W9UOl9 68 3 146243 4 995699 3.0 150544 1o5 849456 140114 99()15i 4 147136 148 995681 3.0 151454 848546i 14033 9901156ti 148 3.0 151 145 4 1199006 55 5 148026 148 995664 3.0 152363 lal 847637 1 140o1199003 | 1 6 148985 148 995646 3.0 153269 151 846731 1! 1409099002 54 7 149802 995628 3 0 154174 845826!14119198998 53 8' 150686 ~147 3.0 15 8 150586 14 995610 3:0 155077 [ 844923 i14148 98994 52 9 51569 14 995591 0 155978150 8440221 14177 98990151 10 152451 995573 0 156877 1 843123 14205198986 50 11 9.153330 14 9.995555 3.0 9..157775 150 10.842225 I 142S4 98S82 49 12 154208 99537 158671 841329 14263 989 8 48 146 3.0 1499 13 155083 4 995519 3*'~ 159565 149 840435 | 14292198973 47 14 155957 14 995501 3 1 160457 14 839543 114320 98939 46 15 156830 145 995482 3*1 161347 148 838653 14349 98965 45 16 15770'3 5 995464 162236 148 837764 114378 98961 44 17 158569 4 995446 3.1 163123 148 836877 1 1440, 98957 43 18 159435 144 995427 3.1 164008 14 83992 14439895342 14436 98953 42 19 160301 144 995409 3.1 16482 14 835108 1446498948 41 20 161164 44 995390 3.1 165774 147 834226 1 14493 98944 40 21 9.1620'25 1 9.995372 1 9.166654 14 9.833346 14522 98940 39 22 162885 14 995353 167532 832468 14551 98936 38 3 163743 995334 3 168409 146 831591 14580 98931 37 24 164600 1 995316 3.1 169284 146 830716 14608198927 36 25 165454 142 995297 3 1 170157 145 829843 i 14637 98923 35 26 166307 995278 31 171029 145 828971 14666 98919 34 27 167159 142 995260 3 1 171899 145 828101 14695 98914 33 28 168008 141 995241 3'2 172767 14 827233 14723 98910 32 29 168856 141 995222 3.2 173634 144 826366 14752 98903 31 30 169702 995203 2 174499 825501 14781 98902 30 31 9.170547.99584.17054732 144 10.8246381 14810 98897 29 32 995165 176224 14 3893 33 172230 995146'2 177084 3 823729176 148i78 98893 27 140 3.4 13486798889 27 34 173070 995127 177942 822058111489698884126 3 1798140 9 3.2 143 1 35 173908 139 995108 32 178799 142 821201 1492598880125 36 174744 139 995089 3.2 179655 142 820345 14954 J887(6 24 37 175578 995070 180508 142 99507819492 14982 08871 23 38 176411 139 995051 3 2 181360 142 818640 1501198867 22 39 177242 138 995032 32 182211 1 14 817789 1504098863 21 40 178072 138 995013 3:2 183059 141 816941 1.5069 98858120 41 9.178900 138 994993 2 9.183907 141 10.816093 115097 98854 19 42 179726 994974 184752 815248 1 15126 98849 118 43 180551 994955 185597 140 814403 1515698845 1 17 44- 181374 137 99491 3.2 186439 140 813561 15184 98841 16 45 182196 994916 187280 140 812720 15212i 98836 15 46 183016 994896 188120 140 811880 15241 i8832 1 14 47.183834 136 994877 3 18898 139 811042 1 152709 -827 13 48 184651 136 994857 3 3 189794 139 810206 1 152991,98823 12 136 3.3 139 802061] 1 49 185466 994838 190629 8093711 15327 96618 11 50 186280 13 994818 3 3 191462 139 808538 1 15356 j8814 10 5t 9.187092 135 9.994798 9.192294 138 10.8077061 153858j 8091 9 I187903 135 994779 3 193124 138 806876 j11 15414 98805 8 53 188712 994759 8193953 806047 Ii 1544 9j88001 7 5,1 189519 99473 9 194780 138 805220'11154719 8'796 6 65 190325 134 994719 3.3 195606 137 804394 115500 98791 5 56 191130 1 994700 13 196430 803570 15529 )8877 4 57 191933 134 9946801 197253 137 802747 11155571 )8782 3 58 192734 133 994660 3 198074 137 801926 i15586 98778 2 59 193534 994640 198894 801151115615 98773 1 133 3 3 136 989 C0i-n.- -- Sie3' C-ot- g. 1 3 Tan8. 4 9N. 8cos. N.sine. 81 Degrees.

Page  30 30 Log. Sines and Tangents. (90) Natural Sines. TABLE II.' Sine. D. 10" Cosine. D. 10" Tang. D. 10' Cotang. IN. sine.N. COS.I 0 ).194332 33 9.994620 33 9.199713 16 10.800287 15643 98769 60 1 195129 133 994600 33! 200)529 136 799471 15672 98764 59 2 19a5925 132 994580 3.3 201345 136 798655 15701 98760 58 3 196719 ~132 994560 3 5.4 202159 36 797841 15730198755 57 4 197511 132 994540 3 4 202971 13 797029 15758 98751 56 5 198302 1 32 994519 34 203782 135 796218 15787198746 55 6 199091 131 994499 3.4 204592 35 795408 15816 98741 54 7 199879 131 994479 3 205400 13 794600 15845698737 63 8 200666 13 994459 3. 206207 134 793793 115873 98732 52 9 201451 131 994438 3'4 207013 134 792987 15902 98728 ] 51 10 202234 130 994418 3'4 207817 134 792183 15931198723 50 11 9. 203017 30 9.994397 3 4 9.208619 133 10.791381'15959198718 49 12 203797 130 994377 3 4l 209420 133 790580 15988 98714 48 13 204577 130 994357 3-4 210220 133 789780 16017198709 47 14 205354 129 994336 4 211018 13 788982 16046 98704 46 15 206131 129 994316 3.4 211815 133 788185 16074198700 45 16 206906 129 994295 3.4 212611 132 787389 16103 98695 44 17 207679 129 994274 3.5 213405 132 786595 16132 98690 43 18 208452 128 994254 214198 132 785802 116160198686 42 19 209222 128 994233 214989 132 735011 16189198681 41 20 209992 128 994212 3-5 215780 131 784220 16218198676 40 21 9.210760 128 9.994191 3.5 9.216568 131 10.783432 16246198671 39 22 211526 127 994171 3 217356 131 782644 16275198667 38 23 212291 127 994150 3.5 218142 131 781858 16304 98662 37 24 213055 127 994129. 218926 130 781074 16333 98657 36 25 213818 27 994108 3. 219710 130 780290 16361 98652 35 26 214679 127 994087 3 220492 130 779508 16390[98648 34 27 215338 126 994066 3. 221272 130 778728 16419198643 33 28 216097 126 994045 3;' 222052 130 777948 16447198638 32 29 216854 12 994024 3*5 222830 129 777170 16476 98633 31 126 330 17609 9940129 30 217609 126 994003 5 223606 129 776394 16505198629 30 31 9.218363 12 9.993981 9.224382 129 10.775618 16533198624 29 32 219116 125 993960 225156 129 774844 16562198619 28 33 219868 12 9939391. 225929 129 774071 16591198614 27 34 220618 125 993918 3. 226700 128 773300 16620198609 26 35 221367 12; 993896 3'6 227471 128 772529 1664898604 25 36 222115 126 993875 3:6 228239 128 771761 1667798600 24 37 222861 124 993854 3'6 229007 128 770993 16706198595 23 38 223606 124 993832 3 6 229773 127 770227 116734.98590 22 39 224349 124 993811 3.6 230539 127 769461 16763 98585 21 40 225092 124 993789 3'6 231302 127 768698 116792 98580 20 41 9.225833 123 9.993768 36 9.232065 127 10.767935 16820198575 19 42 226573 123 993746 3.6 232826 27 767174 16849198570 18 43 227311 123 993725 3.6 233586 126 766414 16878 98565 17 44 228048 123 993703 3.6 234345 765655 1"6 f98561 16 45 228784 123 993681 3.6 235103 126 764897 16i35[ 98556 15 46 229518 122 993660 3'6 235859 126 764141 16964198551 14 47 230252 122 993638 3*6 236614 126 763386 1699298546 13 48 23098 4 12 993616 237368 12 762632 17021198541 12 49 231714 1 993594 3. 238120 125 761880 17050198536 11 50 232444 122 993572 37 238872 125 761128 117078198531 10 51 9.233172 121 9.993550 3. 239622 25 10.760378 17107 98526 9 52 233899 121 994528 3.7 240371 125 759629 17136 98521 8 53 234625 121 993506 3.7 241118 124 758882 1 17164{98516 7 54 235349 12o 993484 3 7 241865 124 758135 17193 98511 6 55 23603 120 993462 3'7 242610 124 757390 17222 98506 5 56 236795 120 9)3440 7 243354 124 756646 117250198501 4 57 23751: 120 993418 3' 244097 124 755903 17279198496 3 58 238235 120 993396t3.' 244839 123 755161 1117308 98491 2 59 238953 93374 245579 3 754421 17336198486 1 60 239670 119 993351 246319 12 753681 17365 98481 0 Cosine. Sine. Cotang. I Tang. i N. cos.N.sine. 80 Degrees.

Page  31 TABLE II. Log. Sines and Tangents. (100) Natural Sines. 31,Sine. D. 1 Cosine. [D. 10"' Tang. D. 10"[ Cotang. N.sine. N. cos. V19.2396701 19 9.993351 3-7 9.246319 2 10.753681 17365 98481 60 1 240386 993329 247057 72943 17393 98476 59 241101 119 993307 37 247794 13 752206 1742 98471 68 L 241814 119 993285 3.7 248530 123 751470 1745198466 57 119 993285'3:22 7507361 1 747998461156 4'242526 118 993262 3 249264 750736 17479 98461 56 a 243237 118 993240. 249998 1 750002 1750898455 55 6 243947 118 993217 3.7 250730 192 749270 17537 98450 54 7 244656 1 1 993195 3.8 21461 122 748539 17565 98445 53 8 245363 18 9931721. 252191 122 747809 17594 98440 52 9'246069 118 993149 3.8 252920 121 747080 17623 98435 51 117 3 8 121 17623984351 10) 2406775 117 993127 3 8 253648 121 746352 17651 98430 50 11 9,247478 9.993104 8 9.254374 1 10.745626 1768098425 49 12 248181 117 993081.8 255100 121 744900 17708 98420 48 13 248883 1 99308 38 2558243121 13 248883 117 993059 3.8 2565824 120 744176 17737 98414 47 14 2495831 11 993036 3 8 256547 10 743453 1776698409 46 15 250282 116 993013 3' 8 257269 120 742731 17794 98404 45 16 25980 116 992990 257990 120 742010 17823 98399 44 17 251677 116 9~2967 30 8 2587100 20 741290 117852 98394 43 IS 252373 116 992944 36 8 2 9 1 20 740571 1788098389 42 9 25067 116 992921 - 260146 120 739854 1790998383 41 20 253761 16 992898 260863 19 739137 17937 98378 40 21 9.254453 115 9.992876 3.8 9.261i78 119 10.738422 17966 98373 39 22 255144 115 992852 3:8 262292 119 737708 1799598368 38 23 255834 115 992829 38 263005 119 736995 18023 98362 37 24 25523 115 992806 3.9 263717 119 736283 18052 98357 36 25 257811 114 992783 3'9 264428 118 735572 18081 98352 35 26 257898 114 992759 3'9 265138 118 734862 18109 98347 34 27 258583 14 992736. 265847 118 734153 18138198341 33 28 259268 114 992713 3.9 266555 118 733445 1816698336 32 29 259951 992690' 267267 11 732739 18195 98331 31 30 260633 113 992666 3 9 267967 1 18 732033 18224198325 30 31 9.261314 113 9.992643 3'9 9.268671 117 10.731329 18252198320 29 32 261994 113 992619 3'9 269375 117 730625 1828198315 28 33 262673 113 992596 3 9 2700771 1 729923 18309 98310 27 34 263351 113 992572 3 9 2707791 729221 18338 98304 26 35 264027 113 992549 3'9 271479 117 728521 18367198299 25 272389 116 78 36 264703 112 992525 39 272178 116 727822 18396 98294 24 37 265377 112 992501 3 9 272876 116 727124 184241982E3$ 23 ~112 3.9 116 38 266051 992478 273573 726427 18452 98283 22 39 266723 112 992454 4'0 274269 116 725731 18481 98277 21 40 267395 112 992430 4 0 274964 116 725036 18509 98272 20 41 9.268065 112 9.992406 4.0 9.275658 116 10724342 18538 98267 19 42 268734 111 992382 4.0 276351 115 723649 18567198261 18 43 269402 ill 992359 4.0 277043 115 722957 1118595 98256 I1 44 270069 111 992335 4 277734 15 722266 1862498250 16 45 ill35 117 4.0 115 45 270735 992311 0 278424 115 721576 11865298245 15 46 9271400 1 9992287 4.0 279113 115 720887 18681 98240 14 47 272064 110 992263 4.0 279801 114 7201939 18710 98234 13 48 272726 110 992.239 4.0 280488 114 719512 18738 98229 12 49 273388 110 992214 4.0 281174 718826 18767198223 11 50 274049 11() 992190 4.0 281858 114 718142 18795 98218 10 51 9.27478 110 9.992166 4. 9.2822542 114 10.717458 1882498212 9 52 275367 992142 40 283225 14 71677518852 98207 8 53 276024 1()] 992117 4'1 2839071 716093 111888198201 7 54 276681 10 992093 4-1 284588 13 715412 18910 98196 6'55 277337 992069 4-1 285268 113 714732 18938198190 5 102.1 285947113 7 66 27'7991 1 O 992044 4.1 285947 113 714053 18967 98185 4 109 4.1 113 7 278644 109 992020 4.1 286624 113 713376 18995198179 3 58 27929i 109 991996 4.1 287301 13 712699 1902498174 2 59 279948 108 991971 4.1 287977 112 712023 119052198168 1 60 280599 991947 288652 711348 19081 9816:; 0 Cosine. Si ne. -. _______ _ I 1 1 Cosine. -Sine. Cotag Tang. N. eos. ine. 79 Degrees.

Page  32 32 Log. Sines and Tangents. (110) Natural Sines. TABLE II. S. i. 10 osine. l. 1t''ani. 1). JD u:oniaslg. iN. slne. N. cos. 9.28 9 103 9.991947 9.288652 10.711348' 1908198163 60 1 281248 1 991922 4. 29326 112 710674 19109198157 59 2 281897 1 991897 4.1 289999 1 710011! 19138 98152 58 3 282544 108 991873 4.1 290671 2 709329 1916 798 146 57 4 283190 08 991848 4. 291342 112 708358 19195 98140 56 5 283836 107 991823 4.1 292013 707987 19224 98135 55 6 284480 107 991799 4.1 292682 707318 19252 98t3129 54 7 285124 107 991774 4.6 293350 76650 19281 98124 53 8 285766 107 991749 4.2 294017 705983 19309 98118 52 9 286408 107 991724 4.2 294684 705316 19338 98112 51 10 287048 107 991699 4.2 295349 704651 19366 98107 50 11 9.28'7687 1 9.991674 4. 9.296013 10.703987 19395 98101 49 12 288326 106 991649 4.2 296677 703323 19423198096 48 13 288964 106 991624 4.2 297339 110 702661 19452 98090 47 I 128960 106 4.2 298 16 70 14 289600 O 991599 4.2 298001 70999 19481 98084 46 15 290236 106 991574 4.2 298662 110 701338 19509 98079 45 16 290870 106 991549 4.2 299322 700678 19538 98073 44 17 291594 106 991524 4.2 299980 110 700020 19566 98067 43 18 292137 105 991498 4.2 300638 10 699362 19595 98061 42 19 292768 105 991473 4.2 301295 698705 19623 98056 41 20 293399 10s 991448 4.2 301951 109 698049 19652 98050 40 21 9.294029 105 9.991422 4.2 9.302607 109 10697393 19680 98044 39 22 294658 105 991397 4.2 303261 696739 19709 98039 38 23 295286 l05 991372 4.2 303914 696086 19737198033 37 /'2 413 69 6 4 295913 104 991346 4.3 304567 695433 19766 98027 36 2104 991 4.3 109 9 25 296539 991321 4. 305218 694782 19794 98021 35 26 297164 104 991295 4.3 305869 694131 19823 98016 34 27 297788 104 991270 4.3 306519 693481 19851 98010 33 28 298412 104 991244. 307168 692832 19880 98004 32 29 299034 104 991218 307815 108 692185 19908 97998 31 30 299655 10 991193 308463 691537 19937 97992 30 31 9.300276 1 9.991167 4. 9.309109 107 690891 19965 97987 29 32 300895 103 991141 309754 690246 19994 97981 i28 99103 4.3 107 33 301514 103 991115 4.3 310398 689602 20022 97975 27 34 302132 103 991090 311042 688958 20051 97969 26 35 302748 1 991064 311685 107 688315 20079 97963 25 36 303364 102 991038 4.3 312327 107 687673 20108 97958 24 37 303979 102 991012 4.3 312967 687033 20136 97952 23 38 304593 102 990986 4.3 313608 686392 20165 97946 22 39 305207 102 990960 4.3 314247 685753 2019397940 21 40 305819 2 990934 314885 685115 20222 97934 20 102 4.4 106 41 9.306430 1029 990908 4. 9.315523 10.684477 20250197928 19 42 307041 102 990882 4-4 316159 683841 20279 97922 18 4.2 00 11 90 882 4.1 106 43 3)7650 102 990855 4-4 316795 683205 2030i 97916 17 101 4.4 106 68 44 308259 101 990829 4.4 317430 682570 11 20336197910 16 45 308867 101 990803 4.4 318064 5 681936 20364 97905 15 46 309174 1 990777.4 318697 681303 20397 97899 14 47 310080 1 990750 4.4 319329 105 680671 20421 97893113 48 319O85 101 990724 44 319961 108 680339 20450i 98787 12 49 311289 101 990697 4-4 320592 679408 2047b197881 11 50 311893 100 990671 4.4 321222 678778 200 i851 51 9.129 99964 449.321851 105 10.6781491 2035 97869 9 52 31309 90518 4 322479 104 677521 20ai 9-3 8 100 33986 90 53 313698 100 990591 323106 676894 i20592 9I85;7 7 54 314297 100 990565 4.4 323733 6762i7 120 20(91 l 6 55 314397 1 990538.4 324358 104 675642206499;'840 5 100 4.4 104 6 56 315495 1 990511 324983 675017'2067i9,8 69 4 1 57 316092 990485 325607 674393y 20Oi1J79b1 3 50 3166893 )etS S 990458. 326231 673769 2073419 1b,' 99931 104 673769' 8078492 59 3112981 ) 990431 4 5 326853 673147 1120, 6 6921 1 99 4.5 104 (6 i l i16,9 99J404 32'7475 675255207 91 9J6156 0 -! (!...... — Sine. Cotang. Tang. i N. (os. N.si. / 78 Degrees.

Page  33 TAIBLE II. Log. Sines and Tangents. (120) Natural Sines. 33 Sine. D. 10 Cosine. ID.10" Tanr. D. lu' Ccomg. N. sine. N. cos. 0 9.317879 99 0 9.990404 4 5 9.327474 o 10.6725-261 20791 97815 60 1 318473 98-8 990378. 328035 103 6719051 2082097809 59 2 319036' 87 990351 4.5 328715 103 671285! 20848 97803 58 3 319658 96. 990324 4. 329334 1 3 6703661 2087797797 57 3202498.6 990297 4.5 103 4 30249. 990297 329953 670047 20905 97791 56 98.4 4., 103 5 320840 98'3 990270 4 o 330570 103 669430 i20933 97784 55 6 321430 98'2 990243 4.5 331187 103 6688131 20962 97778 54 7 322019 98 990215 4.5 331803 02 6681971 2099097772 53 8 322607 990188 5 332418 102 66758 21019 97766 52 9 323194 7 990161 4 33303 102 666967! 21047 977560 51 0 323780 976 990134 5 333646 666354 21076 97754 50 11 9.324366 97.5 9.990107 4.6 9.334259 102 10.665741 1 2110497748 49 12 324950 990079 6 3348712 665129 2113297742 48 13 325534 97. 990052 46 335482 664518 1121161 97735 47 14 326117 97.2 990025 46 336093 102 663907 1 2118997729 46 15 326700 989997 4.6 336702 o-) 663298 2 91218 97723 45 16 32 7281 968 989970 4-6 337311 101 662689 2124697717 44 17 327862 96 989942 46 337919 01 662081 121275 97711 43 18 328442 6.6 989915 4*6 338527 101 661473 2130397705 42 19 329021 96.5 989887 4. 339133 101 660867 2133197698 41 20 329599 96.4 989860 4.6 339739 1. 6 660261 21360 97692 40 96.2 4.6 101 21 9.330176 9. 9.989832 4'6 9.340344 101 10.659656 2138897686 39 22 330753 96.1 989804 4. 340948 101 659052 2141797680 38 23 331329 958 989777 4 341552 658448 21445 97673 37 24 331903 989749 7 342155 101 2147497667 36 25 332478 95.6 9897214.7 342757 100 657243 2150297661 35 26 333051 989693 343358 100 656642 2153097655 34'21 333624.3 989665 4!7 343958 100 656042 21559:97648 33 28 334195 989637 344558 0 655442 21587197642 32 29 3347661 989609 345107 654843 21616'97636 31,' 30 335337 989582 347055 654245 1 21644'97630 30 31 9.335903 9.989553 9.346353 10.653647 21672 32 336475 48 989525 346949 310.653051 21701197617 28 94:6 4.7 99.3 337043 94. 989497 4.7 347545 652455 2172997611 27 34 337610 989469 348141 1 651859 21758'7604 26 94.4 I 4.7 99.1:. 94.3 4.7 99.0 n 338176 * 989441 34873 651265 2786 97598 36 33874'2 941 989413 349329 98.8 650671 21814 97592 24; 37 339306 940 989384 349922 987 650078i 21843197585 23 94.0 4.7 98.7 38 339871 989356 350514 649486 21871[97579 22 93.9 4.7 98.6 39 340434 989328' 351106 5 648894 21899 97573 21 40 340996 937 989300 4. 351697 983 648303 21928 97566 20 41 9.341558 936 9.989271 4.7 9.352287 98,2 10.647713 21956197560 19 i1 49 6472421985 97553 18 42 342119 935 989243 47 352876 1 98. 6471241 2189753 18 [l 43 342679 93 989214 3534651 o 646535 If201397547 17 44 343239931 989186 354053 645947 22041 97541 16 93.1 4.7 97.9 45 343797 930 989157 4. 354640 7 64560 22070 97534 15 461 34435592 989128 4. 355227 644773 1 22098197528 14 92.9 4.8 97.6 47? 344912[' 989100 4 355813 644187 212652 48 345469 92.6 989071 4.8 356398 97.5 643602 122155 197521 13 92.6 48 97. 6436022215597515 12 49. 346024 2 5 4 989042 4:8 356982 643018 22183 97508 1 1 92.5 4.8 97.3 - 50[ 346579 92- 989014 4.8 357566 97-1 642434 22212 97502 10 92-4 4.8 97.1 51 9.347134 9-29.988985 48 9.358149 97010.641851 22240197496 9' 347687 921 988956 48 358731 96.9 641269 122268197489 8 92.1 4.8 96.9 5:1 3482401,0 988927 359313 9 8 640687 1!22297 974831 92 989 48 96.8 i 54/ 34897'92 988898 359893 640107 12232597476 6 I 5 3493431 988869 4 1 360474 966 639526 122353197470 5 i 3498,93 * 988840 361053 6389471 223829 463 4 5 3 04391 988811 4 3616321 96 638368 2241097457 3 4.9 96.3 5I,> 3,t2914. 988782 362210 962 6377901122438197450 2 91.4 4.9 96.2 5) 351640191 988753 362787 96W1 637213122467197444 1 60 352088 3 988724 469 363364 63663622495197437 0 Cosine. ||Sine. | Cotang. Tang. N, N 77 Degrees.

Page  34 34 Log. Sines and Tangents. (130) Natural Sines. TABLE II. Sine. D. 10" Cosine. D. 10" Tang. D. 10"1 Cotang. iN.sine N. cos. 0 9.352088 91 1 9.988724 9 9.363364 96.0 10.636636 22495 97437 60 1 352635 910 988695 4 9 363940 636060 22523 97430 59 2 353181 90*9 988666 4'9 364515 95.9 635485 22552 97424 58 3 353726 90.8 988636 4.9 365090 95.8 634910 22580 97417 57 4 354271 0. 988607 365664 95.7 634336 122608 97411 56 5 354815 90.7 988578 9 366237 95.5 633763 122637 97404 55 6 355358 904 988548 4.9 366810 95.4 633190 2266597398 54 7 355901 903 988519 9 367382 95.3 632618 22693 97391 53 8 356443 90.2 988489 A' 367953 95.2 632047 22722 97384 62 9 3569840. 988460 368524 95.1 631476 22750 97378 51 10 357524 90.1 988430 369094 95. 630906 22778197371 50 11 9.358064 8.988401 9 9 663 94. 10 630337 22807 97365 4 12 358603 8971 988371 4.9 370232 94,8 629768 22835 97358 48 13 359141 896 988342 49 370799 94.6 629201 22863 97351 47 14 359678 89.5 988312 60 371367 94.5 628633 22892197345 46 15 360215 893 988282 0 371933 94.4 628067 22920 9738 45 16 360752 89. 988252 0 372499 94.3 627501 22948 97331 44 17 361287 988223 0 373064 94.2 626936 22977 97325 43 18 361822 988193 373629 94.1 626371 23005 97318 42 19 362356 889 9988163 0:0 374193 94.0 625807 23033 97311 41 20 3ti2889 88 988133 *0 374756 93. 625244 23062 97304 40 21 9.363422 7 9.988103'0 9.375319 93. 10.624681 23090 97298 39 22 363954 885 988073' 375881 93.7 624119 2311897291 38 23 364485 88.4 988043' 0 376442 93.5 6235581 23146 97U84 37 24 365016 883 988013 5'0 377003 93.4 622997 23175 97278 36 25 365546 88 2 987983' 0 377563 93. 622437 i 23203 97271 35 26 866075 881 987953,' 378122 93.2 621878 2323197264 34 27 366604 88'0 987922,, 378681 93.1 621319 23260 9727 33 28 367131 879 987892'~ 379239 93.0 620761 2328897251 32 29 367659 877 98782' 379797 92.8 620203 23316 97244 31 30 368185 1 987832 1 380354 92.8 619646 23345 97237 30 31 9.368711 875 9.987801 5 1 9.380910 2. 10.619090 2337397230 29 32 36923 987771 1 381466 618634 123401 97223 1 28 33 369761 987740 5' 38202092 617980 2342997217 27 34 370285 87 77 92.4 6 34 370285 87 987710 1 382575 92 4 617425 23458 97210 1 8712 2348 97203 35 370808 87.1 987679' 1 383129 616871 2348697203 25 36 371330 70 987649 383682 92.2 616318 23514 97 96 24 37 371852 87 987618' 1 384234 93 1 615766 23642 97189 23 38 372373 8679 987588'; 384786 92.0 615214 23571 97182 22 39 372894 866 987557 1 385337 91'9 * A14663 1 23599 97176 21 40 373414 86' 987526 385888 91.8 614112 23627 97169 20 41 9.373933 865 9.987496 5 1 9.386438 91 7 10.613562 23656 97162 19 42 374452 86'3 987465' 386987 91' 613013 2368497155 18 43 374970 86.2 987434 1 387536 91.4 612464 23712 97148 17 44 375487 86 987403'2 1 388084 91 3 611916 23740 97141 16 45 376003 861 9873472 52 388631 91.2 611369 23769 97134 15 46 376519 586 987341 5.2 389178 91.' 610822 23797 97127 14 47 377035 987310 52 389724 91.' 0 610276 23825 97120 13 48 377549 987279 -2 390270 90.9 609730 23853 97113 12 9 378 857 15 90.8 49 378063 856 987248'2 390815 6 8 609185 23882 97103 11 60 3785771 987217 5.2 391360 90.7 608640 23910 97100 10 51 9.379089 9.987186 5-2 9.391903 90.6 10.608097 23938 37093 9 52 379601 8532 987155 5.2 392447 90.5 607553 23966 970S6 8 53 380113 987124 52 392989 90.4 607011 23995 97079 7 54 380624 851 987092 5 2 393531 90.3 606469 24023 970i2 6 55 381134 987061 52 394073 90.2 605927 24051 97065 5 56 381643 987030 52 394614 90.1 605386 24079 )305 1 4 57 382152 8 986998 52 395154 9 0 604846 24108 7 051 3 68 382661 847 986967 &.2 395694 89.9 604306 24136 97044 2 59 383168 986936 5.2 396233 89. 603767 24164 97037 1 60 383675 986904 396771 897 603229 2419297030 0 Cosine. Sine. Cotang. Tang. _ N. cos.jN.sine. 76 Degrees.

Page  35 TABLE II. Log. Sines and Tangents. (140) Natural Sines. 35 Sine. D. 1l" Cosine. D. 1("I Tang. D. 10' Cotang. N. sine. N. cos0 9.383675 84 4 9.986904 2 9.396771 89 6 10.603229 124192 97030 60 1 384182 84.3 986873. 397309 89.6 602691 24220 97023 69 2 384687 84 2 986841 397846 896 602154 24249 97015 68 3 385192 84- 1 986809 5 398383 89 4 601617 24277 97008 57 4 385697 * 81 986778 3 398919 83 601081 24305 97001 56 5 386201 83.9 986746 5 399455 89 2 600545 24333 96994 55 6 386704 83.8 986714 3 399990 1 600010 2436296987 64 7 387207 83 7 986683 4v40524 0 599476 2439096980 53 8 387709 83.6 986651. 4010581 88' 598942 24418 96973 52 9 38810 83. 986t19. 4401591 88i8 598409 24446 96966 51 10 388711 83-4 986587 583 402124 887 597876 24474!96959 50 11.389211 833 9.986555 9.40265 10.597344 24503 96952 49 12 389 711 3. 986523 5.3 40318&7 88.6 596813 24531196945 48 13 390210 83.2 9864951.3 403718 88 5696282 24559 96937 47 831 986491 14 390708 83. 986459 404249 88.3 6595751 24587 96930 46 15 391'205 83.8 986f427. 404778 8. 595222 24615 96923 45 i16 391703 28 98639 5 405308 88. 94692 24644 96916 44 8267 563 88.1 1 7 392199 82.6 986633 5. 406836 888. 594164 24672 96909 43 IS 1 392695 82.5 986331 406364 878 593636 2470096902 42 19 393191 82.4 986299 4 406892 8 78 693108 24728196894141 393685 9862ii 407419 599581 24756i96887140 21 9394179 2. 9.986234 5 9.407945 87. 10.592055 24784196880! 39 2'2 394673 82.1 986226 4 408471 87. 591529 24813 96873 38 23 395166 80 986169 514 408997 87.4 591003 24841 96866 37 24 395658 82. 0986137 5 409521 87.4 590479 24869 96858 36 25 396150 81 986104 tl 4 410045 87.- 589955 2489796851 35 26B 396641 81.7 986(072 *4 410569 87.3 589431 2492596844 34 27 397132 81.7 98639 5 411092 87.2 688908 2495496837 133 817 986039 28 397621 81.7 9860071 64 411615 87.1 588385 2498296829 32 29 398111 81-5 985974 412137 87.0 587863 25010 9682231 30) 398600 81. 985942 412658 86.9 587342 2503819681 30 31 9.399088 819 985909 54 9.413179 86.8 10.586821 12506696807 29 32 399575 81.2 985876 413699 86. 586301 2509496800 28 33 400062 81.2 9858343 414219 86.6 585781 2512296793 27 134 400549 811 985811 414738 86 585262 25151196786 26 401035 80 985778 415257 86.4 584743 25179 96778 25 36t 401520 808 985745 * 415775 86.4 584225 25207196771 24 37 402005 80. 985712 956 416293 86.3 683707 25235 96764 23 38 402489 80.6 985679 416810 86.1 583190 2526396756 22 39 402972 806 985646 5.6 417326 86.1 682674 25291 96749 21 40 403455 80.4 985613.5 1 417842 86.0 682158 2532096742 20 41 9.403938 9.985580 9.418358 85. 10.6581642 25348 96734 19 12 404420 8023 985547 418873 85.8 581127 25376196727 18 43 4(04901 802 985514 5.6 419387 85.7 580613 25404196719 17 80.1 985 81.6 44 405382 so.: 985480 419901 85. 580099 2543296712 16 45 405862 0.0 5447 420415 85.5 579585 25460 96705 15 46 406341 7899 985414 - 420927 85.6 579073 248896697 14 i47 405820 798 985380 56 5421440 84 B78560 2551696690 13 48 40299 79.7 985347 5.b 421952 85.3 678048 925545 96682 12 49 407777 7 6 985314.6 422463 8'5.2 577537 25573|96676 11 50 403254 79-4 985280 5 422974 85.1 577026 25601 96667 10 1 9.408731 9.985247 6 9.423484 8.09 10.576516 2562996660 9 e 526 4092079.' 986213.6 423993 84.9 576007 25657 96653 8 53 409682 79.3 985180.6 424503 84.8 575497 25685 96645 7 54 410157 7 9.2 985146 5.6 426011 84.8 574989' 2571396638 6 55 410632 79.1 595113.6 425519 84.7 574481 25741 96630 5 66 41110 98950790 42602784.6 573973'2576696623 4 78.9 5.6 0 84 ||57 411579 78.8 985045 66 426534 84.4 573466 2579896615 3 68 412052 78 7 985011 427041 84. 672959 V 25826 96608 2 41259 9851 1 6 4 7843 59 412524 76 984978 427547 672453 2585496600 1 78 6 54 6 84.3 5 ||60 412996 |*|984944 * 428052 4 571948 l25882 96593 0 Cosine. Sine. | |CotL Tang. Tag N. cgos.| N.ine. 75 Degrees. 20

Page  36 36 Log. Sines and Tangents. (150) Natural Sines. TABLE II. Sine. D. 1_"1 Cosine. D. 10"1 Tang. D. 10" totafn N. sine N. co.= 0 9.41299 78.5 9.984944 5.7 9.428052 84 2 10.571348 25882 96 5 93 60 1 413467 784 984910 428557 84.1 51443 2591 96583 59 2 413938 78. 984876 5.7 429062 84 0 5709138 25931 9578 58 3 414408 783 984842 429566 570434 2596o965i0 57 4 414878 c 984808 5.7 430070 83.9 569930' 20994 96o62 56i 5 415347] 984774 430573 88 519427 2602 90 -3 5 78,1~ 5, 55947 832295585 6 4158150 984740 431075. 5,892.5 2605096547 54 7 416283 984706 431577 568423'26079 96540 53 77.9 ~5.7 8. 8 416751 984672 432079 567921 2610i 96532 i52 77.8 98472 7 43258 9 417217 777 984637 7 432580 83.45 567420i 26135 96524 51 3 5 43007 5 69"0 2616396517 50 10 417684.6 984603 5.7 433080 83.3 56690 26163 96517 5 11 9.418150 9.984569 9.433580 10.56420 2619196509 49 12 418615 984535 430 8321.6 77.5 ~5.7 8. 712 418615 774 984535.7 434080 83.2 5659201 26219 96502 48 13 419079 73 984500.7 434579 83. 55421 26247 96494 47 5.7 81 56542 649444 14 419544 7 984466. 435078 564922 126275 96486 46 15 420007 72 984432 83 15 420007 77.3 984432 5.7 435576 832. 564424 i 26303 96479 45 16 420470 77.1 984397 58 436073 828 563927i 26331 96471 44 17 420933 770 984363 58 436570 82.8 563430 26359 96463 43 18 421395. 984328 5.8 437067 82.7 562933 26387 96456 42 19 421857 76. 984294 5.8 437563 82. 562437 2641596448 41 76.8 - 5.8 8. 20 422318 767 984259 5 8 438059 82.5 561941 26443 96440 40 21 9.422778 767 9.984224 5.8 9.43864 824 10.561446 26471 96433 39 22 423238 766 984190 5.8 439048 8 3 560952 26500 96425 38 *23 423697 765. 984155 5.8 439543 823 560457 26528 96417 37 24 424156 76.4 984120 5.8 440036 82.2 559964 26556 96410 36 25 424615 76.3 984085.8 440529 821 559471 26584 96402 35 26 425073 76.2 984050 5.8 441022 2. 558978 26612 96394 34 27 425530 76 1 984015 8 441514 8 558486 26640 96386 33 28 425987 76.0 983981 5.8 442006 819 557994 26668 96379 32 29 426443 760 983946 5.8 442497 818 557503 26696 96371 31 30 426899 7. 983911 5.8 442988 817 557012 2672496363 30 31 9.427354 75. 9 983875 5.8 9.443479 81. 10.556521 26752 96355 29 32 427809 75.7 983840 5. 443968 81.6 556032 26780 96347 28 33 428263. 983805 5.9 444458 81.5 555542 26808 96340 27 34 428717r 75.6 983770.9 444947 81.4 555053 26836 96332 26 35 429170 983735 445435 81. 554565 26864 96324 25 36 429623 75.3 983700 5.9 445923 81.2 554077 26892 96316 24 37 430075 75. 983664 5.9 446411 812. 553589 26920 96308 23 38 430527 75.2 983629 5.9 446898 81.1 553102 26948 96301 22 39 430978 75.2 983594 5.9 447384 81.0 552616 26976 96293.40 431429 75.1 983558 5.9 447870 80.9 552130 27004 96285 20 41 9.431879 75.0 9.983523 5.9 9.448356 8. 10.551644 27032 96277 19 42432329 809810.55 42 432329 74.9 983487 5.9 448841 80.8 551159 27060 96269 18 43 432778 748 983452.9 449326 807 550674 27088 96261 17 44 433226 74.8 983416 5. 449810 806 550190 27116 9253 16 45 433675 74. 983381 5.9 450294 806 549706 2714496246 15 46 4341522 7456 983345 5.9 450777 805 549223 27172 96238 14 47 434569. 983309 451260 48740 127200 96230 13 48 435016744 983273 5.9 451743 804 548257 2722896222 1 4 435462 98328 45225 547775 2725696214 11 49 435462 9 873 3 6.0 4 80.2 50 435908 983202 6.0 452706 547294 27284 9620 10 1 74.2 6080.2 51 9.436353 74.1 983166 6.0 9.453187 801 1.546813 2731296198 9 52436798 60 810.54 0 ~ 52 436798 -0983130 6 453668 5463,2 27340 96190 8 53 437242 4.0 983094 0 454148 545852 2736896182 54 437686 74.0 983058 6.0 454628 59 45372 27396 961 4 6 1 55 438129 3- 983022 6.0 455107 79.9 544893 27424 96166 5 563438526.0 679.8 56~ 438572 73.8 982986 6.0 455586 79.7 5 444145 2745283 6158 4 5 7 439014 73 982950 6.0 456064 r96 543936 27480 96150 3 58 439456 73.6 982914 6.0 456542 79.6 43458 27608 96142 2 439897 73 5 982878 6.0 457019 79.6 5 542981 27536 6134 1 601 440338 982842 445796 542504i 27564 J6126 0 | | Cosine. J Sine. Cotang. |Tan0. I N. cos.iN sine. I 174 Degrees. 57 391 6.0

Page  37 TABLE II. Log. Sines and Tangents. (160) Natural Sines. 37 Sine. D. 10" Cosine. D. 1QD" Tang. D. 10" Cotang. iN. sine. N. cos. 0.410338 9.982842 9.457496 10.542504! 27564 96126 60 440778 982805 60 457973 79.4 5420271 27592 96118 59 9 441218 982769 ( 458449 641551 1 22760 96110 58 3 441658 982733 1 458925 79 541075127648 96102 57 40 97 6.1 79 2 I 4 44209 3.1 982696 61 459400 1 5406001 27676 96094 56 5 442535 730 982660 6. 459875 9. 540125 1127704 96086 55 6 442973 72. 982624 6.1 460349 979 0 539651 27731 96078 54 7 443410 728 982587 6 1 460823 9 539177 27759 96070 653 8 443847 72 7 982551 61 461297 7 538703 27787 96062 52 9 444284 982514 6. 461770 78 9 538230 27815 96054 51 1 0 44472i) 72.6 9824'77 1 462242 787 537758 27843 96046 50 11 9.445155 9.982441 9.462714 10.537286 27871 96037 4 19i 445590 10. 5372.5 6.1 78.6 12 445590 72.54 982404 6.1 463186 8. 536814 2789996029 48 13 446025 2.3 982367 61 463658 78.5 536342 27927 96021 47 14 446459 72.3 982331 6:1 464129 78. 535871 27955 96013 46 15 446893 72.2 982294 6.1 464599 78.3 535401 27983 96005 45 16 447326 72.1 98227 6.1 465069 8 34931 28011 95997 44 17 44'/759 982220 6. 465539 78 634461 28039 95989 43 18 448191 72.0 982183 6.2 466008 78.2 533992 2806Q 95981 42 19 448623 72.0982146 6.2 466476 78 53324 2809595972 41 20 449054 71 982109 2 466945 8. 533055 28123 95964 40 21 9.449485 71 9.982072 6.2 9.467413 78. 10.532587 28150 95956 39 22 449915 71.7 982035 6.9 467880 77. 532120 28178 95948 38 23 450345 71.6 981998 6.2 468347 77.8 531653 28206 95940 37 24 450775 71.6 981961 6., 468814 77.8 531186 28234 95931 36 25 451204 71.5 981924 6.2 469280 77.7 530720 28262 95923 365 26 451632 71.4 981886 6.2 4697466 530254 28290 95915 34 27 452030 71.3 981849 6.2 470211 529789 28318 95907 33 28 452488 71.3 981812 6.2 470676 77.5 529324 28346 95898 32 29 452915 71. 1 981774 6.2 471141 77.4 528859 28374 95890 31 30 453342 71.1 981737 6. 471605 77.3 528395 28402195882 30 1 4 71.0 6.2 77.3 31 9.453768 9.981699 6.3 9 472068 77 10.527932 28429 95874 29 32 454194 981662 6. 472532 77. 527468 28457195865 28 3 454619 708 981625 6.3 472995 77.1 527005 28485 95857 27 34 4 70.8 981587 6. 434577.1 5265431 28513|95849 26 3 455469 07 981549 6.3 473919 77.0 526081 28541195841 25 36 455893 0 981312 6.3 474381 76.9 5256191 2856995832 24 i37 456316 70. 981474 6.3 474842 76.9 525158 28597195824 23 389 456739 70.4 981436 6.3 475303 76.8 524697 28625 95816'2 39 457162 981399 6.3 47563 76 524237 286524t5807!21 40 457584 7 981361 6.3 476223 76.7 523777 2868095799 20 41 9.458006 70. 9.981323 6 3 9 476683 10 523317 28708 95191 19 42 4584uS327 981285 6. 477142 6 522858 28736 9578&2 18 43 458848 01 81247 6.3 477601 76.5 522399 28764195774117 44 459268 01 981209 6.3 4780 51941 287927657(4116 7009811 4705 76.4 521941 2879295766116 45 45988 981171 6.3 478517 76.3 521483 128820 91557 1 15 46 460108 981133 6. 478975 76 521025 128847195749 114 47 460527.981095 4 479432 762 520568 128875195740 13 48 4609469.7 981057 4 479889 76.1 520111 28903195732 12 49 461364 981019 6.4 480345 76. 519655 28931 95724 11 69.6 6.4 76.0 50 461782.6 98098 480801801 76 519199 28959195715 10 51 9).462199 69.5 9.980942 6. 9 481257 79 10.518743 28987195707 9 52 462616 69.5 980904 6.4 481712 175.9 618288 29015 95698 8 53 463032 9 980866 6.4 482167175.8 517833 290421 95690 7 1654 463448 693 930827 6.4 4826211757; 517379 29070 95681 6 55 463864 69.3 980789 6.4 483075 7 516925 12909895673 5 55 463i649. 98i5o 6.4 483529 75 6 516471 ib91~!95664 4 6901 6.4 75.5 7 464694 69 980712807 483982;) 5160181129154955656 3 58 465108 98073 6 - 484435 7 54 515565 129182195647 2 59 465522 980635 484887 515113 1i2920995639 1 60 465935 980596 6 4 485339 75:3 514661 129247 95630 0 Cosine. Sine Cotang. Tang. i N. cos. Nine. T 73 Degrees.

Page  38 [ - - _ I 38 Log. Sines and Tangents. (170) Natural Sines. TABLE II. Sine. D. 10" Cosine. D. 10'1 Tang. D. 10" Cotang. 1N. sine.IN. cos. 0 9.465935 9.980596 9.485339 10. 514661 i 2923 956-30 (i 1 466348 980558 64 485791 75 614209!i 2'92i51o22 59 2 466761 980519 486242 75 5137581 2291956l3 58 3 467173 980480 65 486693 751 51330 7 i 2%L21 95605 7 4 467585 68.6 980442 65 487143 750 512857j 29348 95596 56 4679968.5 98040 5 48793 512407 I 29376 95588 55 6 468407 68.5 980364 6B5 488043 511957 129404 95579 54 7 468817 68.4 980325 65 488492 511508 29432 95571 53 8 469227 980286 488941 511059 29460 95562 52 68.3 65 1 29460 74.7 9 469637 6.3 980247 6.5 489390 7 510610 29487 95554 51 10 470046 980208 5 489838 5101621 29515 95545 5o 11 9.470455 68.1 9 980169 6 9.490286 10. 509714 129543 95536 49 12 470863 68.0 980130 6.5 490733 74.6 509267 29571 95528 48 13 471271 68.0 980091 6.5 491180 745 508820 29599195519 47 14 471679 67.9 980052 6.5 491627 74* 508373 29626 95511 46 15 472086 67.8 980012 6.5 492073 7 507927 2965495502 45 16 472492 67.8 979973 6.5 492519 7.3 507481 29682 95493 44 17 472898 67. 979934 6 492965 507035 2971095485 43 18 473304 67.6 979895 6.6 493410 74.2 506590 29737 95476 42 19 473710 67.6 979855 6.6 493854 74.0 506146 29765 95467 41 20 474115 67.5 979816 6.6 494299 740 505701 29793 95459 40 21 9.474519 67.4 979776 6.6 9 494743 740 10. 505257 29821 95450 39 22 474923 674 979737 495186 73.9 504814 2984 95441 38 23 475327 67.3 979697 6 495630 3' 504370 2987695433 37 24 475730 67.2 979658 66 496073 73.7 503927 2990495424 36 25 476133 67.2 979618 6 496515 77' 503485 29932 9541 35 26 476536 979579 66 496957 76 50343 29960 9407 34 27 476938 67.0979539 6 497399 6 502601 29987 95398 33 28 477340 66.9 979499 497841 73.5 502159 3001595389 32 29 477741 66.9 979459 6 468282 501718 3004395380 31 30 478142 66.8 979420 6. 498722 73.4 501278 30071 95372 30 31 9.478542 66.7 500837 3009895.4363 29 66.79380 9.49916 1 0.500837 30098 32 478942 66.7 979340 66 499603 7 533 500397 3012695354 28 33 479342 66.6 979300 67 500042 7332 499958 3015495345 27 34 479741 66.5 979260 500481 73.1 499519 30182 95337 26 35 480140 6 979220 6. 500920 73.1 499080 30209 95328 36 480539 979180.7 501359 73. 498641 30237 95319 24 37 480937 979140 501797 0 498203 30265 9531 23 38 481334 66.3 979100 7 02235 7. 497765 30292 95301 22 39 481731 662.97909 67 502672 72.9 497328 30320 95293 21 39 481731.9790569 502672 328 40 482128 661 979019 67 503109 72.8 496891 30348 95284 120 41 9.482525.978979 9.503546 7 10.496454 30376 95275 1 42 482921 978939 503982 72.7 496018 3040395266 18 43 483316 659 978898 6 504418 73 495582 3043195257 65.9 6.7 72.6 495146 44 483712 978858 504854 49514 30459 95248 16 45 484107 978817 505289 72.5 494711 3048695240 15 46 484501 978777 6.7 505724 494276 30514 95231 14 47 484895 978736 506159 72.4 493.41 3054295222 13 48 485289 65.6 978696 6.7 506593 72.4 493407 3051095213 12 49 485682 65.5 978655 6.8 507027 72.3 492973 305(9 95204 11 50 486075 65.5 9786151 6.8 507460 722 492540 30625.95 51 19.486467 65.978574, 8.507893 7.10.492107i1 30653 5110 62. 486860 66.3 978533 6 508326 72.1 491674 30O80951 7 | 3 1487251 65.3 8493 6.8 508 72.1 4912411 30i08 95166 63 487261 6 S4Y50 9893 6 408769 7513. 9515. 54 487643 65.2 978452 6.8 509191 72.0 490809 |301369159 55 488034 65.1 978411 6.8 509622 71. 490378 30)76' 95160 56 488424 978370 510054 71 489946 30791 95142 57 488814 65'0 9783291 68 510485 71.8 489515 3081995133 3 58 489204 978288 6.8 510916 71.7 489084 308469524 59 489593 978247 8 511346 71 488654 30874 951 1 i 60 489982 64.8 978203 6.8 511776 71.6 488224 3090d1J Cosine. Sine. Cotang. Tang. N. co. 79 Degrees.

Page  39 TABLE II. Log. Sines and Tangents. (180) Natural Sines. 39 I Sine. D. 10"' Cosine. D. 10"' Tang. D. 10" Cotang. N. sine.N. eos.[ 0 9.489982 9.978206 9.511776 71.6 10.488224 30902 95106 60 1 490371 64.8 978165 6.8 512206 71'6 4877941 3092995097 59 2 490759 7 978124 8 51263 71.5 487365 3095795088 58 3 491147. 978083 513064 486936 30985 95079 57 4 491535 64.6 978042 6.9 513493 71.4 486507 31012 95070 56 5 491922 64.6 978001 6.9 13921 71.4 486079 31040 95061 55 6 492308 64. 977959 9 514349 71 485651 31068 95052 54 514349 71.7 492695 4 977918 6 514777 71.3 485223 31095 95043 53 8 493081 64.4 977877 69 15204 712 484796 31123 95033 52 9 493466 643 977835 6 1.215631 484369 31151 95024 51 10 493851 642 977794 69 6160571.1 483943 31178 95015 50 10 49385164'2 516057 11 9.494236 64. 9.977752 6.9 9.516484 710 0.483516 31206 95006 49 12 494621 641 977711 6.9 516910 71 483090 31233 94997 48 13 495005 6 977669 6-9 517335 709 482665 31261 94988 47 14 495388 6.0 977628 06.9 517761 70.9 482239 31289 94979 46 15 495772 977586 518185 15 49538 639 77.9 6 18185.8 481815 3131694970 45 6 496154 6 9775644 6. 18610 481390 31344 94961 44 63.8 7.0 70.7 480966 31372 2 43 17 496537 637 977503 7.0 519034 18 496919 i 977461 i 519458 8 496919 3 977461 519458 76 480542 31399 94943 42 19 497301 63.7 977419 70 519882 70.6 480118 31427 94933 41 20 497682 63.6 977377 5.0 620305 70.5 479695 31454 94924 40 21 9.4980o4 63. 9.977335 7.0 9. 20728 705 10.479272 31482 94915 39 22 498444 63. 977293 7.0 521151 70.4 478849 3151094906 38 23 498825 63.4 977251 7 0 6215737 24 499204 63.4 977209 70 21995 70.3 478005 31565 94888 36 2 499584 63.3 977167 70 22417 70.3 477583 31593 94878 35; 26 499963 977125 7.0 522838 70.2 477162 3162094869 34 27 00342 632 7.0 52359 70.2 476741 31648 94860 33 631 60977083 523259 28 500721 63. 977041 7.0 23680 70.1 476320 31675 94851 32 29 501099 631 976999 7 0 524100 70.1 475900 31703 94842 31 30 501476 63.0 97697.0 52450 70.0 475480 31730194832 30t 31 950184.976907 524520 31 9.501854 62 9.976914 7. 9.524939 699 10.475061 31758!94823 29 321 502231 62. 9768727' 525359 699 474641 31786194814 28 33 607 976830 525778 69.8 474222 31813,94805 27 34 0984 2.8 976787 7.1 2197 69.8 473803 31841i94795 26 35 03360 62.7 976745 7.1 526615 69.7 473385 31868194786 25 36 50378 626 976702 7.1 527033 69.7 472967 131896194777 24 62:6 976702 37 504110 976660 7.1 527451 69.6 472549 31923!94768 23 38 504485 62. 976617 7.1 527868 9' 6 4721331951 94758 22 39 504860 62.6 976574 7.1 528285 69.5 471715 31979:94749 21 40 505234 62.4 9765.1 628702 69.5 471298 32006!94740 20 41 9.505608 623 9.976489 7.1 9.529119 69 10.470881 32034!94730 19 46293 9 7.1 69.3 0 47 13014 19 42 505981 976446 529535 470465 3'206194721 18 43 506354 62.2 976404 7.1 529950 69.3 470050 32089194712 17 441 06727 62.21 976361 7.1 630366 69.3 469634 32116194702 16 45 507099 6. 976318 7 1 530781 69.2 469219 32144194693 15 46 507471 62.0 976275 7.1 531196 69.1 468804 3217194684 14 47 07843 62.0 976232 7.1 531611 69.1 468389.32199194674 13 48 508214 61 976189 7.2 532025 69.0 3222794665 12 49 508585 61 976146 7.2 532439 69.0 467561 32250!94656 11 61 8 7,2 68,9 467147 3228294646 10 50 508956 61. 976103 7. 632853 68.9 467147 3228294646 61.8 7 8.9 10.466734 3230994637 9 51 9.609326 61. 9.976060.2 9.633266 688 10.4667341 32309 94637 9 52 609696 61.6 976017.72 533679 68.8 466321 32337 94627 8 63 510065 61:6 9769 7.2 534092 68.8 465908 3236494618 7 5451043461 69 75974.2 534092 5451 0434 61l5 975930 7.2 534604 68.7 465496 32392 94609 6 5 510803 61 975887 7 34916 68 465084 32419 94599 5 66 611172 61 9.2 535328 68.6 464672 32447194590 4 57 511540 613 975800 7.2 3573968 464261 3247494580 3 58 511907 61 957 7.2 36150 68.5 463850 3250294571 2 59 512275 61 975714.2 36561 68. 463439 3252994561 1 60 261.2 9 75714 7.2 5369672 60 5, 512642' 975670. 536972 463028 3255794552 0 |_ Cosine. | Sine. = Cotang. Tang. N. cos.Nine.ie ~L~~~~~ ~~71 Degrees.

Page  40 40 Log. Sines and Tangents. (190) Natural Sines. TABLE II. M oe. 10/ Cosine. U. 10' Tang. 1). 1o"'] Cotano. iN. sine. A. cos.! 0 9.512842 9.975670 9.536972 8 10.463028 i 32551 94552 60 1 51330 51.2 975627 537382 68.3 462618 1 32584 94542 59 2 513315 975583 537792'3 4622081 32612 94533 58 i 561.1 7.3 68.3 3 513741.1 975a39 538202 461798 3263i 94523 57 61.0 7.3 68.3 4 514101 61.0 975490 7.3 538611 68.2 461389 11326617 94514 56 5 514472 975452 539020 68. 460980 32694 94o04 55 6 514837 60. 975408 539429 68.1 460571 32722 94495 54 7 515202 60.8 975365 539837 68.0 4601631 32749 94485 53 8 5155660. 975321 3 540245 68.0 459755 32777 94476 5 9 515930 975277 540353 68.0 459347 32804 94466 51 10 516294 60.7 975233 541061 67.9 458939 3283294457 50 11 9516 60.6 9.975189 7.3 541468 67. 10.4585321 3285994447 49 12 517020 60 975145 541875 67.8 4581251 3288' 94438 48 60.5 7.3 67.8 13 517382 975101 542281 67.7 457719 132914 94428 47 14 5177456 97507 542688 677 4573121 32942 94418 46 15 518107 60.3 975013 73 543094 676 456906 32969 94409 45 16 518468 60.3 97499 4 543499676 456501 3299794399 44 17 518829 62 974925 543905 67. 456095 33024 94390 43 18 519190 60.2 974880 7.4 544310 675 4556901 33051 94380 42 19 519551 60.1 974836 544715 67.4 455285 330i9 94370 41 20 519911 60.1 974792 545119 674 454881 33106 94361 40 21 9.520271. 9 974748 9.545524 673 10.454476 3313494351 39 22 520631 60.0 974703 545928 67.3 454072 33161 94342 38 23 520990 974659 546331 67-2 453669 33189 94332137 24 521349 974614 546735 672 453265 33216 94322 36 25 521707 59. 974570 7 54 547138 67.1 452862 33244 94313 35 26 522066 59.7 974525 547540 671 452460 33271 94303 34 27 522424 974481 7 547943 670 452057 33298 94293 33 28 522781 596 974436 548345 67.0 451655 33326 94284 32 29 523138 974391 548747 451253 33353 94274 31 30 523495.5 974347 549149 66.9 450851 33381 94264 30 595 75 6.9 430 31 9.523852 59. 974302 9,549550 66.8 10.450450 33408 94'254 29 32 524208 4 974257 549951.8 450049 33436 94245 28 59.4 705 668 4 33 524564 974212 7. 550352 667 449648 33463 94235 27 34 524920 59 974167 550752 66.7 449248 133490 94225 26 35 525275 974122 7.5 551152 66.7 448848 i 33518 94215 25 ~6 025050 59.2 7.5 66.6 36 525630 592 74077 7.5 551552 666 448448 133545 94203 24 37 525984 59.1 974032 7 551952 66*5 448048 133673 94196 23 38 526339 973987 552351 66 447649 33600 94186 22 39 1 526693 1 59.0 973942 7.5 552750 j666 447250 1o3627 94176 21 40 52i046 59 973897 553149 66 446851 i 133655o94167 20 41 9.52740908. 9.973852 9.553548 166.4 10. 446452 i33682194157 19 42 5277533 58.9 9738(07 7. 553946 66.4 446054 1 33710 94147 18 58.8 7.5 66.3 3 43 52810 5 973761 5 54344 66.3 445656 13373794137 17 44 528458 973716 554741 445259 33764!94127 16 58.7 7.6 66.2 45 528810 7 97361 555139 662 44861 i 33792194118 15 46i 529161 58 973625 7.6 555553 66 444464 1 33819934108 14 47 529513 8.6 973580 7.6 555933 6 444087 3384o 94098 13 48 529884 973535 556329 66 443671 33844 9408 12 -58.66 7.6 6 49 530215 973489 6 556725 66.0 443275' 33901 1408M 11 50 5305i6 58. 973444. 557121 66. 442879 i3392994068 10 51 9.530915 58.4 9.973398 7.6 9.557517 65.9 10.442483 13395,'94058 9 52 5312 58 973352 557913 65 442087 33983 94049 6 58.3 9 7.6 65.9 53 531614 91 730. 558308 65.9 441692i 34011194039 7 54 531963 8I 95 1 6 5.5802 65 8 441298 134038 94019 6 55 532312| 973213 7.6 559097 65.8 440903Q' 34065 94019 5 58.1 7.6 65.7 66 53261 l 93169 7.6 559491 65.7 44009 34093194009 4 57 5330VJ 51 9i3124 559885 440115 13412093999 3 58 5381)5 1 o8,0,90 7.6 502i965 6 4397'21'34141i93989 2 591 5:38704 58 0 97393 7.6 560873.1 439327!34175 939'79 1 ) 53455- 9 97298C5 561086 65. 438934 13420.:)391i` 0 \,Coboain e. Sine. Cotan. I a'iang. tN. cos.l.sine. 70 Degrees.

Page  41 TABLE II. Log. Sines and Tangents. (200) Natural Sines. 41 | Sine. D. s10" Cosine. D. 10"'ang. ). 10 Costti- N. sine. N. cos. 0 19.534052 57.8 l 972986 7.7 9.561066 65. 10.4 934 3420293969 60 1 534399 57.7 972940 7'7 561459.5 438541 3422) 93959 59 2 534745 972894 561851,. 438149 3425 93949 58 3 535092 972848 562244 5.4 437756 34284 93939 57 57 -7 7.7 65 3 4 535438 97280-2 562636 43 364 34311 93929 56 57.6 7.7 3543 5 535783 95 2755 563028 ^5 4369262 343.9 93919 55 57.6 7.7 65.3 6 536129 9'7270 563419' 436581 34366 93909 54 57.5 7.7 65.2 7 536474 92663 563811 65.2 436189 1 343953 93899 63 8 536818 a 972617 564202 435798 i34421 93889 92 57.4 7.7 65.1 9 5327163 573 972570 56492.1 435408 1134448 93879 I1 10 5370i.3 972524 564983 435017 3 1 34475 93869 50 11 9.537851 9.972478 9.665373 10.434627 - 1 345473 93863 49'.2 9 9 7.7 65.0 12 538194 57 972431 8 565763 64 9 434'237'3450 93849 48 57 2 748 6439 13 385538 71 972385 7 8 566153 649 433847 3455, 93839 47 14 538880 57. 972338 7:8 566542 64.9 433458 34584 93829 46 57.1 7.8 64.9 39 15 539223 5 912291 7 8 566932 4 8 433068 34612 93819 45 57.0 7.8 64 8 16 539565 972245 567320 432680 34639 83809 44 17 539907 57.0 972198 7.8 567709 64. 432291 34666 9399 43 18 540249 569 972191 7.8 568098 64. 431902 34694 937 89 42 19 540590 56.9 972105 7'8 568486 64- 431514 34721 93779 41 20 540931 56.8 972058 7. 568873 646 431127 34748193769 40 1 9.541272 5.. 972011 78 9569261 646 10.430739 13475 93759 39 22 541613 56.7 971964 78 69649 64.5 430362 134803193;48 38 23 541953 56.6 971917 7.8 570035 64.5 429965'3488093738 37 24 54'"293 56.6 971870 7'8 570422 64.4 429578 3485i 93728 36 566 78 57 2 64 25 54t263-' 56. 971823 78 570809 64.4 429191 34884 93718 35 26 642971.. 5 971776 7.8 571195 64.3 428805 34912193708 34 27 543310 5. 971729 571581 64.3 428419 34939193698 33 28 543649 97168 571967 64.2 428033 34966 93688 32 29 543987 971635 7 9 572352 64-2 427648 34993193677 31 30 544325 56.3 971588 7.9 572738 64-2 427262 35021 93667 30 31 9.544663 562 9.971540 7 9 9.573123 64 1 10.426877 35048193657 29 32 545000 971493 573507 426493 1 3505193647 28 33 545338 5 971446 7.9 63892 64.1 426108 35102593637 28l 34 545674 56-1 971398 7- 574276 64 0 425724 1135130193626 26 35 546011 56. 971351 9 574660. 425340 35157193616 21 36 546347 560 971303 7-. 575044 63. 424956 135184193605 24 37 546683156.0 971256 79 575427 63,9 424573 35211193596 23 38 647019 5'79 971208 7/9 575810 638 424190 35239}919t85 22 39 4736455 971161. 9 576193 6.8 423807 35266313575:21 40 54768955.81 971113 7.9 576576 63.8 423424 3529393565 20 41 9.548024 65 7 9.971066 7, 9.576958 63.7 10.423041 135320193555 19 42 548359 5-"1 971018 8 5677341 63.7 422659 35341 93544 18 43 548693 55. 970970 8'0 577723 63.6 422277 35375193534 17 44 5.49027 556 970922 8S0 578104 63'6 4218961 35401 936524 16 45 549360 5. 970874 80 678486 63.6 421514 135429193514 15 46 549693 970827 80 578867 635 4211331 354566193503 14 55.5 8.0 63.5 1 47 550026 970779 8o 579248 63.4 4207521 35484193493 13 43 5503569 554 970731 8-0 579629 634 420371 35511193483 12 1 49'550692 970683 580009 6 4 419991 35538193472 11 50 551024 53 970635 8,0 580389 63.4 419611 3556593462 10 51 9.5513561 5'2 9.970586 8*0 9.580769.3 10.419231 35592193452 9 52 551687 652 970538 8:0 | 81149 63.3 418851 356193441 8 55.2 8.0 1 63. 53 552018 55 970490 8 581528 63.2 418472'!35647193431 7 54 552349 55-1 970442 810 581907 63.2 418093 135674193420 55 552680 5 970394 0 582286 63.1 4177141 35701193410 5 56 553010 5 1 970345 8 1 582665 63. 1 417335 35728193400 4 57 553341 970297 81 583043 63. 416957 3575593889 3 58 563670 - 970249 583422 416578 35782493379 2 54 9 8 1 63.0 59 554000, 5 9 970200 8. 583800 6. 416200 35810193368 1 9 54.9 81 62.93839338 (6O 554329 970152 584177 415823235393858 0 C1 osine. 1 |Sine. Cotang. Tang. I N. cos.lsine.' 69 Degrees.

Page  42 42 Log. Sines and Tangents. (210) Natural Sines. TABLE II. Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. I N.sine. N. cos. 0 9.554329 54 8 9.970152 8.1 9.584177 62 9 10.415823 i 35837 93368 60 1 554658 970103 8 5845552 415445 35864 93348 59 2 b54987 54.7 970055 8.1 584932 62.8 415081 3589.1 93337 58 3 555315 54.7 970006 8.1 585309 62.8 414691 135918 93327 5 i 4 555643 54. 969957 81 585686 62.7 414314 35945[93316 55 5 5565971 4.6 969909 8.1 586062 627 413938 3597393306 55 6 55629) 544.5 969860 8.1 586439 62.7 413561 36000 93295 54 7 556626 969811 8'1 586815 626 413185 36027 9325 3 8 556953 54.4 969762 8.1 5871902.6 412810 36054 93274 9 557280 54.4 969714 88 587566 62.6 412434 36081193264 51 10 557606 54.3 969665 8:1 587941 62.5 412059 36108 93253 50 11 9.557932 54.3 9.969616 8 2 9.588316 62.5 10.411684 36135193243 49 12 558258 54.3 969567 8.2 588691 62.5 411309 36162 93232 48 13 558583 54.2 969518 8.2 589066 62.4 410934 36190193222 47 14 558909 54'2 969469 8.2 589440 62.4 410560 36217193211 46 15 559234 54.1 969420 8.2 589814 62.3 410186 36244193201 45 16 559558 54-1 969370 82 590188 6.3 409812 36271 93190 44 17 559883 54-0 969321 8-2 590562 62.3 409438 36298 93180 43 18 560207 4 969272 590935 62.2 409065 36325] 3169 42 19 560531 53. 969223 8.12 591308 62.2 408692 36352 93159 41 20 560855 53.9 969173 8.2 591681 62.2 408319 36379 93148 40 21 9.561178 53.8 9.969124 8.2 9.592054 62.1 10.407946 36406193137 39 22 561501 53.8 969075 8.2 592426 62.1 407574 36434 93127 38 63.8 8.2 62.0 407574 36434193127 38 23 561824 537 969025 8. 592798 62.0 407202 36461 93116 3 24 562146 57 968976 82 593170 61.9 406829 3648893106 ti 25 562468 3. 968926 8.3 593542 61.9 406458 36515 93095'5 26 562790 53.6 968877 63 593914 61.9 406086 36542193084 34 bk46 8 3 U1 8 27' 563112 53.6 968827 8.3 594286 61.8 40715 36569 93074 33 28 563433 53.5 968777 8.3 594656 61.8 405344 36596 693063 3'. 29 563755 53.5 968728 8.3 595027 61.8 404973 36623 930532 1 30 564075 968678 595398 404602 36650 93042 30 31 9.564396 53.4 9.968628 8 3 9.5957681 7 10. 404232 136677 93031 2I 53.4 8.3 61.7 32 564716]53-3 968578 596138 403862 1136704 93020 28 33 665036 533 968528 8.3 596508 61.6 403492 36731193010 2 7 34 665356 533 968479 8. 596878 61.6 403122 36758 92999 26 35 565676 53. 968429 8.3 697247 61.6 402753 36785 92988 25 /63,2 8.3 36 565995 53.1' 968379 8.3 697616 61. 402384 36812 92978 24 37 566314 53.1 968329 8.3 597985 61.5 402015 3683992967 23 38 666632 53'1 968278 8.3 598354 61.5 401646 36867 92956 22 39 566951 53.1 968228. 598722 61.4 401278 36894 92945 21 40 5672691 530 968178 8.4 699091 61.4 400909 1136921 92935 20 41 967587 529.968128 4 999459 613 10 40041 36948 92926 19 42 567904 12-9 968078 8.4 599827 61.3 400173 1 36975 92913 18 43 568222 - 968027 8 600194 61 399806 37002 92902 1 1 44 568539 52.8 967977 8.4 60052 61.2 399438 3702992892 16 52.8 8.4 61.2 45 56885612'81 967927 8.4 600929 6 399071 -370O56 2881 15 46 569172 528 967876 84 601296 61 398704 i 37083 2870 14 5237 3 o28.4 61.1 47 569488 527 967826 8.4 601662 61.1 3983338 37110 92859 13 48 569804 526 967775 84 602029 61. 397971 37137 92849 12 49 670120 1526 967725 8.4 602395 61.0 3976051 371649283811 50 570435 52 967674 4 602761. 397239 1137191 92827 10i 5256 8.4 61.0 51 9.57075152 9.967624 84 9.603127 601.9 10.3968731 37218 92816 9 2 571066 652 967573 84 603493 60.9 3965071 37245 92805 8 52571066 524 95 53 571380 52*4 967522 845 603858 60.9 3961421 37272 92794 7 54 571695 1 967471 85 604223 60.8 3957771137299 92784 6 55 572009 52'3 96.421. 604588 60 395412 1137326 92773 5 56 5 72323 52-3 967370 8. 604953 60:7 395047 37353 9276-2 4 57 672636 52. 967319' 605317 607 3946831 37380 927,1 3 581 672950 2 2 9672681 85 605682 60.7 394318 113740712740 2 59 573263' 521 9672171.06046 60.6 393954tl3743492729 1 601 573575 l 96166 606410 393590113746192718 0 Cosine. Sine. | Cotang. | Ta ng. N. n. Dco. 68 Degrees.

Page  43 TABLE II. Log. Sines and Tangents. (220) Natural Sines. 43 Sine. D. 1" Cosine. D. 10" Tang. D. 10"1 Cotang. -N. sine. N. cos. 0 9.573575 52 1 9.967166 8. 9.606410 60 6 10.3935901 3746192718 60 1 573888 520 967115 85 606773 60 393227!1 3748892707 59 2 574200 2.0 967064 607137.6 3928631 37515 92697 68 3 574512 52.0 967013 8.~ 607500 60. 392500 t137542 92686 57 4 574824 51.9 966961 8.5 607863 60.5 392137 13756992675 56 5 575136 51.9 966910 8.5 608225 60.4 391775:137595 92664 55 6 575447 51 966859 8 5 608588 604 391412 37622 92653 54 51.8 ~ 8.5 60.4 7 675758 *1.8 966808 8. 608950 60.3 391050 9: 37i49'92642 53 8 576069 51'7 966756 8.6 609312 60-3 390688 1137676 92631 52 9 576379 51.7 966705 8.6 609674 60. 3903261 37703 92620 51 10 576689 51.6 966653 8.6 610036 611 389964 1 37730 92609 50 11 9.576999 51.6 9.966602 8.6 9.610397 60.2 10.389603 37757 92598 49 12 577309 51.6 966550 8.6 610759 60~ 2 389241 37784 92587 48 13 577618 966499 6 611120 0.2 388880 37811 92576 47 14 577927 51.5 966447 8. 611480 60.1 3885201 37838 92565146 15 578236 5.4 966395 8.6 611841 60.1 3881591 37865 92554 45 16 678545 1.4 966344 8.6 6122 01 60. 387799 1 37892 92543 44 17 578853 51.3 966292 8.6 612561 60.O 3874391 37919 92532 43 18 679162 513 966240. 612921 60: 387079 37946 92521 42 19 579470 516. 966188 8.6 613281 60. 386719 37973 92510 41 20 579777 51.3 966,368.6 613641 59 3863 3799992499 40 21 9.580085 51.2 9 966085 8.6 9.614000 59. 10.386000 38026 92488 39 22 580392 1.2 966033 8.7 614359 59.8 385641 38053 92477 38 23 580699 11 965981 8.7 614718 59.8 385282 3808092466 37 24 581005 611 965928 8 615077 59.7 384923 38107 92455 36 25 581312 965876.7 615435 59 384565 38134192444 35 26 581618 s:o 965824 8.7 615793 3 84207 38161 92432 34 27 581924.09. 965772 8. 616151 59 383849 3818892421 33 28 582229 50.9 965720 8 7 616509 59.6 383491 38215 92410 32 29 582535 50.9 965668 8.7 616867.6 383133 38241 92399 31 30 582840 50.8 965615 8.7 617224 59.6 382776 38268 92388 30 31 9.583145 50.8 9.965563 8. 9.617582.6 10.382418 38295 92377 29 32 583449 i,'7 965511 8.7 617939 382061 38322 92366 28 33 683754 0.7 965458 8'7 618295 9,4 381705 38349 92355 27 34 584058 50.7 965406 8.7 618652 5.. 381348 38376 92343 26 35 584361 50.6 965353 8. 619008 694 380992 38403 92332 25 50{6 6 8 5354 36 584665 965301 8.8 619364 B 380636 38430 92321 24 37 584968'0.6 965.248.8 619721 9 380279 38456192310 23 38 585272 50'5 965195 8.8 620076 59"3 379924 38483192299 22 39 685574 5,. 965143 88 620432 59.3 379568 3851092287 21 40 86877 0.4 965090.8 620787 59.2 379213 3853792276 20 41 9.586179 50.4 9.966037 8.8 9.621142 59.2 10-378858 3856492265 19 42 586482 B0. 964984 8.8 621497 59. 2 378503 3859192254 18 43 586783 50.3 964931 8.8 62185'2 59.1 378148 38617'92243 17 44 5687085 50 964879 8.8 622207 59.1 377793 38644'92231 16 44 58 70858 50 09 7 45 587386.2 964826 8.8 622561 59.0 377439l 38671192220 15 46 587688 502 964773 8.8 622915 69.0 377085 38698 92209 14 47 587989 ~501 964719 8.8 623269 59.0 376731 13872592198 13 48 58-8289 60-1 964666 8.8 623623 58.9 376377 3875292186 12 49 588590,.5 964613 8.9 623976 58.9 376024 3877892175 11 50 588890 50 8 964560.9 624330 5.9 375670 38805 92164 10 51 9 589190 50. 9 964507 8.9 9624683 58.8 10.375317 38832192152 9 652 689489,49'9 964454.9 625036 58.8 374964 3885992141 8 53 589789 49.9 964400 8.9 625388 58.8 374612 3888692130 7 54 690088 49.9 964347 8.9 625741 58.7 374259 38912 92119 6 55 590387 49.8 964294 8.9 626093 58.7 373907 38939 92107 66 590686 49 964240.9 626445 58.7 373555 38966 92096 4 67 690984 49.7 964187 8.9 626797 58.6 373203 38993 92085 3 68 691282 49.7 964133 89 627149 8.6 372851 39020i92073 2 9149 77 8.9 58.66 4 59 691680 964080 627501 37249939046 92062 1 49~6 89 58.5 60 691878 964026 627862 5 72148390392050 0 Cosine, I Sine. Cotang. Tang. N. cos. Nsine,' 67 Degrees.

Page  44 44 Log. Sines and Tangents. (23~) Natural Sines. TABLE II. Sine. D. lV' Coslne. D. 10i"'TIan 1).li' Cotang. I N. sine. N. cos. 0 9.591878 9.964026 9.627852 10.37-2148,-39073 92050 60 49. 6 8.9 58.9 4 I 592176 49. 963972 8.9 628203 5 5 371797V 39100 92039 59 2 59247349 963919 628554 585 371446 39127 92028 58 3 592770.5 9638665 9 0 628905 58'4 37109511391539'2()16 57 4a 59303749'4 963811 90 62925 58-4 370745 i 39180192005 156 5 593363 49Q*4 963757 9. 0 629606 58*3 370394 1 39207 91994 55 6 593659 49 963704 9'0 629956 58.3 370044 139234 91982 54 7 593955.3 963650 90 630306 583 369694 139260 91971 53 8 594251 4 963596 9 630656 583 369344 39287 91959 52 9 594547 49.2 963542 9.0 631005 58-2 368995 139314 91948 51 1 0 594S42 49.2 963488 9.0 63135 58.2 3686451 39341191936 50 11 9.595137 491 9.963434 9 0 9.631704 58-2 10.368296 3936791925 49 12 595432 49.1 963379 9 632053 58:1 367947 139394 91914 48 13 595727 49.1 963325 9. 632401 58 1 367599 139421 91902 47 14 696021 4910 963271 9I 632750 581 367250 39448 91891 46 15 596515 49.0 963217 9.0 633098 58o0 366902 39474 91879 45 16 596609 48.9 963163 9.0 633447 5810 366553 39501 91868 44 17 596903 48.9 963108 9.1 633795 58-0 366205 139528 91856 43 18 597196 489 963054 9.1 634143 57.9 365857 139555 91845 42 19 59;4904 962999 9'1 634490 3655102139581 91833 41 20 59778348.8 962945 9.1 634838 57 9 365162 39608 91822 40 21 9.598075 48 * 9.962890 91 9.635185 5 8 10.364815 39635191810 39 22 598368 48.7 962836 91 6355327 578 364468 39661191799 38 23 59866 48.7 962781.1 635879 578 364121 139688 91787 37 24 598952 48.7 962727 9'1 636226 57*8 363774 139715 91775 36 25 599244 48.6 962672 9'1 636572 57'7 363428 139741 91764 35 26 599536 486 9362617 9.1 636919 57-7 363081 39768 91752 34 27 599827 48.5 962562 9-1 637265 57-7 362735 39795 91741 33 28 600118.5 962508 9.1 637611 5776 362389 139822 91729 32 29 609049 48.5 962453 9.1 637956 57.6 362044 I39848 91718 31 30 6003700 48.4 962398 9'2 638302 57-6 361698 39875 91706 30 31 9.600990 48.4 9,962343 9 2 9.638647 57.6 10.361353 39902 91694 29 32 601280 48. 962288 9.2 638992 361008 39928191683 28 33 601570 48.3 962233 9:2 639337 57:5 360663 39955 91671 27 34 601860 48.3 962178 9 2 639682 57-4 360318 39982 91660 26 35 602150 48.2 962123 9*2 640027 57-4 359973 40008 91648 25 36 602439 48.2 962067 9-2 640371 57-4 359629 40035 91636 24 37 602728 48. 962012 9'2 640716 57'3 359284 40062 91625 23 38 603017 48.1 961957 9.2 641060 57.3 358940 40088 91613 22 39 603305 48.1 961902 9,2 641404 57. 358596 40115191601 21 40 603594 48.1 961846 9'2 641747 57'2 358253 40141191590 20 41 19.603882 48.0 9.961791 92 9.642091 57-2 10.357909 401681915'i8 19 42 604170 47.9 961735 9-2 642434 572 357566 4019591566 18 43 604457 47.9 961680 9-2 642777 57.2 357223 40221 915556 17 44 604745 47.9 961624 9"3 643120 571 356880 40248 91543 16 45 605032 47 8 961569 1 643463 571 356537 40275191531 15 46 605319 47.8 961513 9'3, 643806 57.1 356194 40301[91519 14 47 605606 47 8 961458 9'3 644148 57.0 355852 40328 91508 13 48 605892 47.7 961402 9'3 644490 57.0 355510 40355 91496 12 49 603179 47.7 961346 9'3 644832 57.0 355168 4038191484 11 50 60O654 47.5 961290 9'3 645174 66'9 354826 40408191472 10 51 19.606751 47.6 9.961235 9'3 9.645516 56.9 10.354484 40434191461 9 52 607036 47.6 961179 9.3 645857 56.9 354143 40461191449 8 53 60,322 47- 961123 9 646199 569 353801 40488191437 7 54 60M607 475 961037 /93 646540 56.8 353460 1140514191425 6 55 607892 47 961011. 646881 56'8 353119 140541191414 5 4754 6 0;7 9609 56 608177 47-4960955 9.3 647222 168 352778 40567191402 4 571 608461 47.4 960899 93 647562 5667 352438 40594|91190 3 58'{ 608745 47 3 960843 9'4 647903 56-7 352097 40621191378 2 59 609029 47 3960786 6482431567 351757 40647191366 1 60 609313 930730 648583 3514171 403741491355 0 Cosine. Sine. Cotan,, T*ang. r 1N. (cOr. N.sir.e. 66 Degrees.

Page  45 TABLE II. Log. Sines and Tangents. (240) Natural Sines. 45 I iSin:u. ID. 10' Cosine. D. 10' Tang. D. 10' Cotang. iN. sine.JN. cos.1 o09.603313 47.3 3.930730 9.4 9.648583 6 10.351417 40674131355 60 1 603597 472 950574 648923 6 351077 4070091343 59 2 609880 472 960618 4 649263 5.6 350737 40727 91331 58 3 610164 472 96051 649602 56.6 350398 40753 91319 57 4 610447 471 960505 9.4 649942 56 6 350058 40780 91307 56 5 610729 471 960448 9 4 650281 56.5 349719 40806 91295 55 6 611012 470 960392 9 650320 56 349380 40833 91283 54 7 611294 470960335 9 4 650959.5 349041 40860 91212 53 8 611576 4 96079 9 4 651297 564 348703 I]40886 91260 52 47.0 9.4 56.4 9 611858 960222 651636 64 348364 40913191248 51 10 612140 469 960165' 4 651974 56.4 348026 40939 91236 50 46.9 9.4 62.3 11 9.612421 46.9 9.960109 9.652312 56.3 10.347688' 40966 91224 49 12 612702 46'8 960052' 6 652650 563 347350 40992 91212 48 13 612983 468 959995 9 652988 56.3 347012 41019 91200 47 14 613264 7 959938 9' 653326 56 2 346674 41045 91188 46 15 613545 959882. 653663 56.2 346337 410i2 91176 45 16 613825 467 95982 9 654000 56 2 346000 410981.1164 44 17 614105 466 959768 654337 561 345663 41125 91152 43 18 614385 46' 959711 9 5 654174 56.1 345326" 41151 91140 42 19 614665 46 6 959654 9 655011 56.1 344989, 1 4117S891128 41 20 614944 465 959596. 655348 56.1 344652 41204191116 40 21 9.615223 46 9.9539 9. 655684 560 10.344316 41231 91104 839 22 615502 46 5 959482 9 656020 56.0 343980 41257 91092 38 23 615781 959425. 656356 56. 343644 41284191080 37 24 616030 464 959368 656692 343308 41310191068 36 25 616338 46 4 959310 9. 657028'9 342972 41337 910o6 5 26 616616 463 959253 9.6 657364 55 9 342636 4136391044 34 27 616894 46 3 959195 9.6 657699 55 9 342301 41390191032 33 28 617172 462 959138 96 658034 5 341966 41416]91020 32 29 617450 462 959081 9 -6 658369 o5-8 341631 41443191008 31 ~ 30 617727 462 959023 9'6 658704 55*8 341296 41469190996 30 31 9.6184jf04 4621 9.958965 9-6 9.659039 55.8 10.340961 41496{90984 t9 32] 618281 461 958908 9.6 659373 55.: 340627 41522 90972 28 33 618558 1 958850 659708 " 340292 41549190960 27 34 618834 46:0 958792 9 6 660042 55.7 339958 415'i5909484 26 35 619110 46. 0 958734 9-6 660376 55.7 339624 416t02190936 25 36 619386 46'0 958677 9-6 660710 55.6 339290 41628 90924 24 37 619662' 958619 9 6 661043!5'6 338957 41655190911:3:3 38 619938 9 958561 9.6 661377 55.6 338623 4168119089c' i2 39 620213 958503 6 661710 55.6 338290 4107 )990887|1?1 4539 9.7 55.5 40 620488 458 958445 662043 337957 41734!908i5 0 41 9. 620763 4568 9.958387 9.7 9.662376 65. 10.33762411417609086 I 19 i 42 621038 45 958329,7 662709 55 337291 14178i 90651 118 1 43 621313 45-7 958271 9 7 663042 6!.4 336968 141813190;396 17 44 621587 i7 958213 9*7 663375 55'4 336625 141840908(bit 16 45 6221861 45O 958154 9 663707 i554' 336293 41866!90614 15 45.6 9.7 565.4 46 622135 958096 6640391 335961 41892 90802 1.4 1 47 622409' 958038 664371 335629 41919;90)190 13 48 622682 45 957979. 664703 | 335297 41945190(77 12 49 622956 455 957921 7 665035 553 334965 4197290i66 11 50 623229 45 957863 7 6653661 552 3346341 41998190i65 10 51 ).623512 4 4 9.957804 9 9.665697 55*2 10.334303] 4202490 i41 9 52 (623774 454 957746 9 666029 2 3339 9 41 40lf51 92 8 45.4 ~ 957804 ~ 666029 55. 39 4205 1 190429 53 624047 95768 666360552 333620 142057 1 191i 7 4 45.4 9.8 6663601 333620]142077 90'1i 7! 54 624319 45 957628 98 666691 5.1 333309 11421041Y9074 6 55 624591 3 957570 9.8 667021 51 3329191 42130190692 5 56 624863 3 957511 667352 332648 42l15 90i 80 4 4 667.3 918 5,9i4 51 625135.2 957452 98 667682 55.0 332318 4218390668 3 58 62 2 9573 9.8 668013 331987 422099065 2 59 625677 9 J57335 668343 ~ 331657 42235 90i4L 1 6) - 625948 5.2 957276 8 668672'60 331328 4226'2190,., 1 0 (iCosielh. I I -Sine. Cotanr. Tan — Ig. I 7l —N i 65 Degrees.

Page  46 146 Log. Sines and Tangents. (250) Natural Sines. TABLE II. - Sine. {D. 10" Cosine. _D. 10"i Tang. D. 10" Cotang. IN.sine. N. cos. 0 9.625948 9.957276 9.668673 10.331327 4226290631 60 1 626219 4 1 957217 98 669002 5 330998 4228890613 59 2 626490 45.1 957158 98 669332 4.9 330668 42315 90606 68 3 6267,0 4. o 957099 9.8 669661 54.9 330339 42341 90594 57 4 627030 45 957040 9 8 6699914 8 330009| 42367 9582 56 62 4300-'0 42394/85568 5 627300 45.0 956981 9.8 670320 54'8 329680 4239490569 55 6 627570 96921 670649 548 329351 4242090557 54 7 627840 956862 670977 548 329023 42446 90545 3 8 628109, 953803 671306 328694 42473 90532 5 9 628378' 955744 671634 32';r[ 9 62837 44.8 955744 9. 671634 47 328366 42499 9052) 51 10 628647 4.8 956684 9 671963 328037 42626 90 5 11 9.628916 44. 9.956625 9.9 9.672291 54:7 10.327709 42552 90495 49 12 629185 44. 956566. 672619 54 6 327381 42578 904t3 48 13 629453 447 9565 99 672947 546 327053 42604190470 47 14 629721 446 956447 9.9 673274 546 326726 4263190458 46 15 629989 44.6 956387 69 3602 54 6 326398 42657 90446 45 16 630257 446 956327 673929 326071 42683190433 44 17 630524 44. 955268 674257 325743 42709190421 43 18 630792 44. 9562 10.0 674584 325416 4273690408 42 19 631059 9564 1ioo 674910 4 325090 42762 90396 41 20 631326 956089 io 0 675237 - 324763 423788 90383 40 219.631593 44. 956029 10.0 9.675564 410.3 24436 4281590371 39 22 631859 44 955896 100 675890 324110 4284190358 38 23 632125 4 9o59 10 0 676216 323784 42867 90346 37 24 632392 4 955849 10.0 676543 323457 4289490334 36 25 632658 3 955789 1o 6768 594.3 323131 42920190321 35 26 63-2923 95729 1 677194 322806 42946 90309 34 27 63318944.2 955669 0oo 677520 54.2 322480 42972190296 33 28 633454'4.2 955609 10.0 677846 54.2 322154 4299990284 32 29 633719 44.2 955548 10-0 678171 542, 321829 43025190271 31 30 633984 4431 955488 10' 678496 564.2 321504 4305190259 130 31 9.634249 441 9.955428 1'1 9.678821 5431 10.321179 43077[90246 29 32 6345144.0 95536810.1 679146 541 320854 43104190233 28 33 634778 440 955307 1.1 679471 54.1 320529 4313090221127 34 635042 4.0 955247 10'1 679795 64.1 320205 4315690208 26 35 635301 43'9 955186 10 1 680120 54'6 319880 43182 90196 25 36 635570 955126 0o1 680444 0 319556 143209190183 24 37 635834 4319 955065 10'1 680768 5401 319232 43235190171 23 38 636097 438 955005 10o1 681092 540 318908 43261/90158 122 39 636360 43'8 954944 10 1 681416 53 318584 43287190146 21 40 636623 43.8 954883 10.1 681740 53.9 318260 43313/90133 201 41 9.636886.954823 9.682063 10.317937 4334090120 19 42 637148 437 994762 10'1 682387 53 317613 43366190108 16] 43 637411 43 954701 10 1 682710 538 317290 4339290095 1i 44 637673 43 954640 101 683033 531 316967 43418190082 11 45 63793543 954579 10.1. | 683356 538 316644 4344590070 15 46 638197 43.6 954518 10.2 683679 3'.8 316321 43471 90057 14 47 638458. 954457 10;2 68400153 315999 4349790045 13 48 638720 954396 102 684324 37 3156761143523190032 12 49 638981 954335 102 684646 315354 4354990019 11 50 639242 954274 102 684968 315032 43575|90007 10 51 9.639503 434 9.954213 1029.685290 536 10.314710 l43602[89994 9 52 63976443 954152 102 685612 536 314388 143628189981 8 53 640024 954090 10.2 685934 536 314066 43654 89968 7 54 64028443 954029 102 686255 536 313745 4368089956 6 55 640544 95396810 686577 313423i4370689943 5 56 64080443 953906 102 686898 313102437389930 4 57 641064 432 953845 10'2 687219' 312781 4375989918 3 58 641324 432 953783 102 687540 312460 4371589905 2 59 641584 432 953722 103 6878613 312139 481189892 1 60 641842 953660 688182 311818 43837 89879 0 Co-sine. Sine. dCotang. - Tang. N. cog. Neir.' _I~~~~~~ ~ ~64 Degrees. I

Page  47 T AI\I,1 11. Log. Sines and Tangents. (260) Natural Sines. 47 Sine.. D. 10" Cosine. D. 10" Tang. D. 10o Cotang. iN. sine. N. os. 0 9.641842 43.1 9.953660 10.3 9.688182 10.311818 143837 89879 60 1 642101 43.1 953599 688502 53.4 311498 4386389867 59 2 642360 9543 3537 88823 311177 4388989854 58 3 642618 43 953475 689143 4 310857 43916 89841 67 4 642877 43.0 953413 689463 310537 4394289828 56 5 643135 43. 953352 689783 310217 43968 89816 55 6 643393 43.0 953290 690103 309897 43994 89803 54 7 643650 43.0 953228 103 6904;23 309577 4402089790 53 8 643908 429 953166 690742 53 309258 44046 89777 52 9 644165 42' 953104 691062 308938 44072 89764 51 10 644423 42.9 953042 10.3 691381 53.2 308619 44098 89752 50 11 3.644680 42 9.952980 103 9.691700 3.210.3083001 44124 89739 49 12 (i44936 42.8 952918 692019 53.1 307981 44151 89726 48 13 645193 427 952855 692338 307662 44177 89713 47 14 645450 427 962793 104 692656.1 307344 44203 89700 46 15 645706 42 952731 692975 3.1 307025 44229 89687 45 1 (645962 4276 952669 10.4 693293 3. 306707 44255 89674 44 1 7 646218 4 952606 10.4 693612 53 306388 44281 89662 43 18 646474 4 952544 693930 3060701 44307 89649 42 42.6 10.4 53.0 19 646729 952481 694248 305752 44333 89636 41 20 64698 42.5 952419 694566 0 305434 4435989623 40 21 9.647240 42.5 9.952356 10.4 9.694883 52.9 10.305117 44385 89610 39 42.5 9.952356 10.14 352.9 22 647494 952294 104 695201 9 304i99 44411 89597 38 23 6477493 952231 10'4 695518 529 304482 44437 89584 37 24 648004 4 952168 10. 69836 52 304164 44464189571 36 25 648258 42.4 952106 10. 696153 303847 4449089558 35 26 (48512 42.4 952043 696470 52 8 303550 4451689545 34 27 648766 423 951980 6 696787 303213 44542 89532 33 8 649020 42*3 951917 10. 697103 52.8 302897 4456889519 32 29 649274 94 1854 697420 302580 44594 89506 31 30) 649527 951791 697736 302264 44620 89493 30 31 9.649781 2. 9.951728 10. 9.698053 2.7 10.301947 44646 89480 29 32 650034 42.2 951665 10.5 698369 52.7 301631 44672189467 28 33 650287 42 951602 10. 698685.7 301315 44698 89454 27 34 650539 42.1 951539 10.5 699001 52.6 300999 44724 89441 26 35 660792 42.1 951476 10. 699316 62.6 300684 44750!8942825 36 651044 421 951412 0.5 699632 52.6 300368 44776 89415 24 37 661297 42 951349 106 699947 526 300053 44802 89402 23 38 651549 42 951286 700263 52.6 299737 44828 89389122 39 651800 4. 951222 10.6 700578. 299422 4485489376 21 40 652052 49 951159 106 700893 52.5 299107 44880 89363 20 41 9.652304 9.951096 10.6 9.701208 2.5 10.298792 44906189350 19 42 652555 41.9 951032 10.6 701523 298477 44932 89337 18 43 652806 41.8 950968 10.6 701837 298163 44958189324 17 44 653057 418 950905 10.6 702152 52.4 297848 44984|89311 16 45 653308 41.8 950841 106 702466 5 297534 45010 89298 15 46; 6~330841.8 10.6 5214 46 653558 41 950778 10 702780 29220 4503689285 14 47 653808 417 950714 106 703095 523 296905 45032189272 13 648 54059 41:7 950650 10.6 703409 523 296591 45088189259 12 049 54309 41 950586 10.6 703723 523 296277 45114189245 11 0 654558 4146 950522 107 704036 52 295964 4514089232 10 51 9.654808 41. 9.950458 10.704350 522 10.295650 45166189219 9 5-2 655058 41 950394 107 704663 5-2 295337 45192 89206 8 53 65530 41 950330 704977 -52 295023 45218 89193 7 54 655556 41 950366 10 705290 2522 294710 45243189180 6 55 65-5805 41.5 950202 10.1 705603 522 294397 45269 89167 6 55i 65054 41.5 950138 10.7 705916 521 294084 45295|89153 4 5i 656302 41 950074 1 06228 521 293772 45321189140 3 58 656551 41 950010 10:7 706541 293459 45347189127 2 59 4156799 949945 06854 293146 4537389114 1 60 65i7047 949881 107 07166 52.1 292834 45399 89101 0 Cosinl.,ine. Cotang. Tang. N. cos. N.sine. 63 Degrees. l i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Page  48 48 Log. Sines and Tangents. (27~0) Natural Sines. TABLE II. Sine. D. 11' Cosine. L). lW'ailag. 1D. lo Cotang. N. sine.jN. cos.I 0 9.657047 9.949881 9.707166 10.292834 145399 89101 60 1 657295 41 3 94981 10.7 707478 52' 2.92522 4542589087 59 2 657542 41. 949752 107 707790 52.0 292210 45451 89074 58 3 657790 412 949688 108 708102 520 291898 45477 89061 57 4 658)37 412 949623 8 708414 52.0 291586 45503 89048 56 5 658284 41 2 949558 108 708726 51. 291274 45529 89035 55 6 658531 949494. 709037 51.9 290963 45554 89021 54 7 658778 411 949429 108 709349 1 5 9 290651 45580 89008 53 8 659025 1 949364 8 709660 51.9 290340 45606 88995 52 9 659271 410 949300. 709971 51' 290029 45632 88981 51 10 659517 410 949235 108 710282 51.8 289718 45658 88968 50 11 9.659763 41. 9.949170 108 9.710593 51.8 10.289407 45684 88955 49 12 660009 40. 949105 1 710904 51.8 289096 45710 88942 48 13 660255 409 949040 10:8 711215 51.8 288785 45736 88928 47 14 660501 9 948975 8 711525 51.7 288475 4576288915 46 15 660746 409 948910 711836 51.7 288164 45787 88902 45 16 660 408 9488410.8 712146 51.7 287854 45813 88888 44 17 661236 408 948780 109 712456 51.7 287544 45839 88875 43 18 661481 40.8 948715 109 712766 51.7 287234 45865 88862 42 19 661726 40. 948650 109 713076 61.6 286924 45891 88848 41 20 661970 407 948584 109 713386 51.6 286614 45917 88835 40 21 9.662214 47 9.948519. 9.713696. 10.286304 45942 88822 39 22 662459 407 948454 109 714005 51.6 285995 45968 88808 38 23 662703 40. 948388 109 714314 51.6 285686 45994 88795 37 24 662946 40:6 948323 10.9 714624 51.6 285376 46020188782 36 25 663190 40 6 948257 9 714933 51.5 285067 46046 88768 35 26 663433 40. 948192 10. 715242 51.5 284758 46072 88755 34 27. 663677 405 948126 10.9 715551 51.5 284449 46097 88741 33 28 663920 40 5 948060 109 715860 51.4 284140 46123 88728 32 29 664163 405 94799 1 716168 51.4 283832 4614988715 31 30 664406 40.4 9479291 *10 716477 51.4 283523 46175 88701 30 31 9.664648 40 9.947863. 9.716785 51.4 10.283215 46201 88688 29 32 664891 404 947797 11.0 717093 51.4 282907 46226 88674 28 33 665133 40. 947731 i.0 717401 51.3 282599 46252 88661 27 34 665375 40.3 947666 1 5. 717709 51.3 282291 46278 88647 26 35 665617 40.3 947600 11.0 718017 51.3 281983 46304 88634 25 36 665859 4. 94733 718325 51.3 281675 46330 88620 24 37 666100 402 947467 1 718633 51.3 281367 46355 88607 23 38 666342 40. 947401 11 718940 51.2 281060 46381 88593 22 39 666583 947335 719248 512 280752 46407188580 21 40 666824 40 947269 0 719555 51 2 280445 46433 88566 20 41'9.667065 401 9.947203 11. 9.719862 51 2 10.280138 46458 88553 19 42 667305 401 947 \36 1 720169 51 2 279831, 46484188639i 18 43 667546 401 947070 1 720476 51.1 279524' 46510 88526 17 44 667786 40 947004 1 720783 51.1 279217 46536 88512 16 45 668027 400 946937.1 721089 51.1 278911 46561188499 15 46 668267 1 946871 11. 721396 51.1 278604 46587188485 14 47 668506. 946804 11.1 721702 }1.1 278298 46613 88472 13 48 668746 399 946738 722009 51.0 277991 46639188458 12 49 668986 39 946671 1 722315 51.0 277685 46664188445 11 50 669225 399 946604 11 722621 51.0 277379 46690188431 10 61 9.669464 398 9.946638 9.722927 51.0 10.277073 46716188417 9 52 669703 39.8 946471 1 723232 509 276768 46742188404 8 53 669942 39 946404 1 723538 50'9 276462 46767188390 7 54 670181 397 946337 1 723844 509 276156 46793188377 6 55 670419 946270 112 724149 275851 46819 88363 5 56 670558 946203 12 724454 1 09 275546 46844188349 4 57 670896 946136 11 724759 275241 4687088336 3 58 671134 39 7 946069 112 725065 274935 46896188322 2 59) 671372 39.6 946002 11.2 725369 50 8 274631 46921188308 1 39.6 11.2 50.8 ) 671609 945935 725674 274326 1 46947188295 0 _| % f_,inie. _ Sine. Cotang. M r Tang. N. cos.I.isll. 62 Degrees.

Page  49 TABLE II. Log. Sines and Tangents. (280) Natural Sines. 49 _ Sine. D. 10',Cosine. D. 10" Tang. D. 1i0" Cotang. N. sine.IN. cos. 0 9.67169 99 6.945935 11 9.725674.8 10.274326i 4694788295 60 1 671847 945868 725979 274021i146973188281 59 2 672084 39.5 945800 11.2 726284 50.8 2737161 46999188267 58 3 672321 945733 726588 507 27341241 7024i88254 57 4 672558 39.5 945666 726892 50.7 273108 1147050 88240 56 5 672795 945598 727-197 2728031147076 88226 55 6 673032 39 4 945531 727501 5 272499 147101 88213 54 7 673268 945464. 72780 27219514712788199 53 39'4 11.3 50.6 8 673505 39'4 945396 11.3 728109 50.6 271891 147153 88185 52 9 673741 39.3 945328 11.3 728412 50.6 271588 4717888172 51 1 0 673977 39..3 945261 728716 50.6 271284 472048815850 11 9.674213 9.3.945193 11.3 9.72902' 10.270980 47229188144 49 12 674448 339.2 945125 11.3 729323 50.5 270677 47255188130 48 13 674684'399-2' 945058 11.33 729626 50.5 270374 147281 88117 47 14 674919 39 944990 11.3 729929 5 270071 47306 88103 46 15 675155 39.2 944922 11.3 730233 50.5 269767 47332 88089 45 16 675390 39.1 944854 11.3 730535 50.5 269465 4735888075 44 17 675624 391 944786 11.3 730838 50.5 269162 47383 88062 43 18 675859 39.1 944718 11 3 731141 50.4 268859 1147409 88048 42 19 676094 39.1 944650 113 731444 0.4 268556 14743488034 41 20 676328 39.0 944582 11.4 731746 50.4 268254 1 47460 88020 40 21 9.676562 39.0 9.944514 11.4 9.732048 50.4 10.2679521i4748688006 39 22 67679 39.0 944446 11.4 732351 0. 2676491 47511 87993 38 23 677030 ~39 944377 11.4 732653 50.3 267347 1 47537 87979 37 24 677264 9 944309 4 73295 50 267045 1 4756287965 36 25 677498 389 944241 114 733257 503 266743 147588187951 35 26 677731 38 9 944172 11.4 733558 266442!47614 87937 34 27 677964 38:8 944104 11.4 733860 50.3 266140 147639 87923 33 28 678197 38.8 944036 734162 50.2 265838 4765 87909 32 29 678430 38.8 943967 11.4 734463 50.2 265537 147690 87896 31 30 678663 38.8 943899 11.4 734764 50.2 65236! 47716 87882 30 31 9.678895 7 9.94380 9.735066 50.2 10.264934 47741 87868 29 32 679128 38 7 943761 1 735367 5. 264633 47767 87854 28 33 679360 38'7 943693 5 735668 50 264332 47793187840 27 34 679592 3857 943624 1 735969 1 264031 47818 87826 26 35 679824 386 943555. 736269 50.1 263731 47844187812 25 36 680056 386 943486 736570 50.1 263430 4786987798 24 37 680288 38.6 943417 1 736871 2631291!47895187784 23 38 680519 38.6 943348 737171 50.1 262829 4792087770 22 39 680750 386 943279 737471 262529 4794687756 21 3856 7137.0 262229114797187743 20 40 680982 385 943210 737771 50 2 47971 87743 20 41 9.681213 38. 9.943141 5 9.738071 50.0 10.261929 47997i87729 19 42 681443 38.4 943072 1 738371 50.0 261629 148022 87715 18 43 681674 38 943003 738671 261329 I 48048187701 17 44 681905 38.4 942934 738971 49.9 261029 148073187687 16 45 682135 384 942864 11.5 739271 9 260729 148099187673 15 46 682365 38.4 942795 11.5 73970 49.9 2604301 48124 87659 14 47 682595 38 942726 11.6 739870 9.9 260130 48150187645 13 48 682825 8.3 9426 11 740169 49.9 259831 1148175187631 12 49 683055 38.3 942587 116 740468 49. 259532 148201 87617 11 50 683284 38 942517 740767 259233 1!48226187603 10 51 9.683514 382 9.942448 119.741066 10. 2589341 48252187589 9 52 683743 382 942378 11 741365 49.8 258635! 48277187576 8 53 683972 38.2 942308 11.6 741664 49.8 258336 4830387561 7 54 684201 38 1 942239 11.6 741962 258038 148328 87546 6 55 6844308 381 942169 11.6 742261 4 257739 48354 87532 5 -56 684658 38.1 942099 742559 49.7 257441 48379 87518 4 57 684887 380 942029 742858 25 142 48405 87504 3 685116 38 0 941959 11.6 743156 497 256844 4843087490 2 59 685343 38 941889 743454 2565461 48456 87476 1 60 686571 3 941819 743752 256248' 2 i4848187462 0 - Cosine. Sie. | Cotang. = Tanz. N. ico.IN.. ine. 61 Degrees.

Page  50 50 Log. Sines and Tangents. (290) Natural Sines. TABLE II. Sine. D. 10" Cosine. D. 1Y' Tang. D. 10" Cotang. N. sine.N. cos.0 9.685571 38 0 9.941819 117 9.743752 496 10.256248 4848187462 60 1 685799. 941749 117 744050 46 255950 4850687448 59 2 686027 9 941679 11 74434896 255652 4853287434 58 3 686254. 941609 11 74464 496 255355 4855787420 57 4 686482 9 941539 11 74494396 255057 4858387406 56 5 686709 378 941469 745240 496 254760 4860887391 55 6 686936 37.8 941398 117 745538 254462 4863487377 54 7 687163 378 941328 117 745835 254165 4865987363 53 8 687389 3 941258 746132 253868 48684'87349 52 37.8 11.7 49.5 9 687616 37941187 117 746429949 253571 4871087335 51 10 687843 37:7 941117 117 746726 253274 4873587321 50 9.688059 37 79.941046 11.8 9.747023 4410.252977 4876187306 49 II.6808 1:,'J 99.9414 12 688295. 940975 118 747319494 252681 4878687292 48 13. 688521 376 940905 118 747616 252384 4881187278 47 37.6~~~~~~228 4881,872787 14 688747 376 940834 118 747913 252087 4883787264 46 15 688972 3 940763 748209 251791 4886287250 45 37.6 11.8 6 49,4 16 689198 376 940693 f 748505 251495 4888887235 44 17 689423 37'5 940622 118 748801 49 251199 4891387221 43 748801 26119948913872214~3 18 689648 940551 118 749097 250903 4893887207 42 19 689873 940480 749393 250607 4896487193 41 37.5 11.8 49.3 20 690098 940409 749689 250311 4898987178 40 21 9.690323 9.940338. 749985 493 10.250015 4901487164 39 22 690548' 940267 11.8 7502814 249719 4904087150 38 37.4 11.8 49.2 23 690772 4 940196 11.8 750576 42 249424 4906587136 37 24 690996 940125 11. 750872 492 249128 4909087121 36 37.~~~~~~4911~s 49008l8 25 691220 3 940054 11.9 751167 492 248833 4911687107 35 26 6913444A7 939982 11.9 751462 49.2 248538 4914187093 34 27 p91668 33 939911 11.9 751757 492 248243 4916687079 33 28 691892 3 939840 11.9 752052 491 247948 4919287064 32 29691159 I,' 913 19768] i 29 69211 5 7. 939768 19 752347 491 247653 4921787050 31 30 692339 3. 939697 752642 247358 4924287036 30 31 9.692562 37.2 9.939625 11.9 9 752937 4110.247063 4926887021 29 37. 19.69252~! 99.9365 32 692785 37.293 11.9 753231 491 246769 492937007 28 37.1 11.~~~~2679 49298002 33 693008 37.1 939482 11.9 753526 491 246474 4931886993 27 37.1 11.9 49.1 34 693231. 939410 753820 246180 4934486978 26 35 693453 37.1 939339 11.9 75411 490 245885 49369 869649 25 37.1 11.9 ~ ~ ~ 9488 49368962 36 693676 37.1 939267 11.9 754409 490 245591 4939486949 24 37.0 12.0 49.0 37 693898 37.0 939195 12.0 754703 49.0 245297 4941986936 23 38. 69120'~. 38 694120 37.0 939123 12.0 754997 490 245003 4944586921 22 39 694342 37.0 939052 12.0 755291 490 244709 494708696 21 40 694564 1 ~I 40 694564 37.0 938980 12.0 75585 489 244415 4949586892 20 41 9.694786 36.9 938908 12.09755878 489 10.244122 4952186878 19 36~~~~~~~I.244120 489 51671 42 695007 36.9 938836 12.0 756172 489 243828 495486863 18 36,9 12.0 ~~~~438 28 9 5 4883 43 695229 36.9 938763 12.0 75645 489 243535 4957186849 17 44 695450 36.9 938691 12.0 756759 48.9 243241 4959686834 16 44. 6 9540 ~'i';9389 45 695671 36.8 938619 12.0 757052489 242948 4962286820 15 46 695892 36.8 938547 12.0 757345 242655 4964786805 14 47 696113 36.8 938475 12.0 77638 488 242362 4967286791 13 48 6 633 ~', 9384 0~ 48 48 696334 36.8 938402 12.0 757931 48 242069 4969786777 12 49 696554 36.7 938330 12.1 758224 48.8 241776 4972386762 11 50 696775 36.7 938258 12.1 758517 48.8 241483 4974886748 10 51 19.696995 36.7 9938185 12.1 9.758810 48.8 10.241190 49773!86733 9 52 697215 36. 938113 12.1 79102 487 240898 4979886719 8 53 697435 36.6 938040 12.1 5939 240605 4982486704 7 36.6 12.1 ~~~~~~498~4870 4 54 67654 36.6 93796 2.1 759687 47 240313 4984986690 6 55 697874 36.6 937895.1 9979 240021 4987486675 5 36.61 12 1 4. 56 698094 937822 121 760272. 239728 4989986661 4 57 698313 36. 93774 11 760564 487 239436 4992486646 3 ~~~~~~~~239,5 144 4 9 90863 58 698532 365 937676 121 760856 486 239144 4995086632 2 59 698751 36.-5 937604 121 761148 46 238852 4997586617 1 60 698970 3. 937531 761439 238561 5000086603 0 Cosine.' -Sine. Cotang. Tang. N. cos.. N.ine. 60 Degree&s.

Page  51 TABLE II. Log. Sines and Tangents. (300) Natural Sines. 51 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. i N. sine. N. cos. 0 9.698970 9.937531 1 9.761439 48 6 10.238561 15000086603 60 1 699189 364 937458 2 761731 4 238269 1 5002586588 59 2 699407 36.4 937385 12.2 762023 48.6 237977i 500508 6573 58 3 699626 364 937312 122 762314 486 237686si 50076 86559 57 4 699844 36 937238 122 762606 485 237394 150101 86544 56 5 700062 363 937165 12. 762897 4856 237103j150126 86530 55 6 700280 36.3 937092 12.2 763188. 236812 150151 86515 54 7 700498 36.3 937019 12,2 763479 48.5 236521 )50176 86501 53 8 700716 363 936946 122 763770 4 236230 50201 86486 52 9 700933136.2 936872 122 764061 *85 235939 50227 86471 51 10 701151 36..2 936799 12.2 764352 48.4 235648 50252 86457 50 11 9.701368 36 2 9.936726 12 2 9.764643 4 10.235357 50277 86442 49 12.701585 36.2 936652 12.3 764933 48.4 235067 50302 86427 48 13 701802 36.1 936578 123 765224 4 24 234776 50327 86413 47 14 702019 36.1 936505 12.3 765514 48. 234486 50352 86398 46 15 702236 36.1 936431 12.3 765805 48.4 234195 50377 86384 45 16 702452 36.1 936357 12.3 766095 48.4 233905 50403 86369 44 17 702669 360 936284 12.3 766385 43 233615 50428 86354 43 18 702885 360 936210 123 766675 483 233325 50453 86340 42 19 703101 360 936136 123 766965 483 233035 50478 86325 41 36.0 12.3 48.31040 20 703317 360 936062 123 767255 483 232745 5050386310 40 21 9.703533 3 9.936988 123 9.767545 10.232455 5052886295 39 22 703749 35.9 935914 123 767834. 232166 5055386281 38 23 703964 9358412 768124 48 231876 5057886266 37 24 704179 35.9 935766 12.4 768413 48-2 231587 5060386251 36 25 704395 35 935692 124 768703 48. 231297 50628 86237 35 26 704610 35.9 935618 124 768992 482 231008 5065486222 34 27 704825 35.8 935543 12 4 769281 48.2 230719 50679 86207 33 28 705040 35.8 93469 124 769570 48.2 230430 50704 86192 32 29 705254 358 935395 769860 48 230140 5072986178 31 34 705469 35.7 935320 12.4 770148 48.1 229852 50754 86163 30 31 9.705683 357 935246 124 770437 481 10.229563 50779 86148 29 31 3 124 229274508048613328 3N2 705898 357 935171 124 770726 48.1 229274 5 04 86133 28 33 706112 35.7 935097 12.4 771015 48. 228985 50829 86119 27 34 706326 35.6 7935022 12.4 771303 48.1 228697 5085486104 26 35 706539 35.6 934948 12.4 771592 48.1 228408 5087986089 25 36 706753 35.6 934873 12.4 771880 48.1 228120 50904 86074 24 37 705967 35.6 934798 12.4 772168 48.0 227832 5092986059 23 38 707180 35.6 934723 12. 772457 4.0 227543 5095486045 22 39 707393 35.5 934649 12.5 772745 48.0 227255 50979 86030 21 40 707606 35.6 934574 12.5 773033 48.0 226967 5100486015 20 41 9.707819 35. 9934499 12.5 9.773321 48.0 10226679 5102986000 19 42 70803235 934424 12.5 773608 48.0 226392 5105485985 18 43 708245 354 934349 2. 773896 47.9 226104 5107985970 17 44 708458 35.4 934274 12.6 774184 47.9 225816 51104!85956 16 45 708670 35-4 934199 12.5 774471 47.9 225529 51129185941 15 46 708882 35 934123 12. 7747 47.9 225241 51154 85926 14 47 709094 353 934048 12. 77046 224954 51179 85911 13 48 709306 353 93973 12. 775333 47.9 -224667 5120485896 12 49 709518 353 933898 12.5 775621 47.8 224379 5122985881 11 60 709730 353 933822 12.6 775908 47.8 224092 5125485866 10 1 9 709941 35 1.933747 126 9.776195 47.8 10.223805 5127985851 9 62 710163 35.2 933671 12.6 776482 47.8 223518 5130485836 8 53 710364 35.2 933596 12.6 776769 47.8 223231 5132985821 7 54 710576 35.2 933520 12.6 777055 47.8 222945 5135485806 6 65 710786 35.2 933445 12.6 777342 47 8 222658 5137985792 5 50 35.1 12.6 5 4748 5 777 4 56 710367 35.1 933369 12.6 777628 47 2223721 5140485777 4 7 711208 351 312.6 777915 222085 51429 85762 3 58 711419 35.1 9330q17 12.6 778201 247-7 221799 51454]85747 2 59 711629 -351 933141 12.6 778487 221512 l51479835732 1 60 711839 35.0 933066 12.6 778774 47*7 221226 1 F150485717 0 -= Cosine. Sine. Cotang. Tang. N. cos.iN.eine. 59 Degrees. 21

Page  52 52 Log. Sines and Tangents. (310) Natural Sines. TABLE II.' Sine ID. 10" Cos;ne. D. 10"o Tang. D. 10"O Cotang. N.sine.lN. cos.! 0.711839 35.0 9.933036 12.6 9.778774 477 10.21226 5104185717 60 1 712050 35 932990 12.7 779050 4 220940 51529185702 5S 2 712260 350 932914 127 779346 476 220654 5155485687 5b 3 712469 932838 127 779632 476 220368 51679185672 57 4 712679 932762 127 779918 476 220082 51604185657 56 5 712889 932685 127 780203 476 219797 51628185642 55 6 713098 34.9 932609 127 780489 476 219511 51653 85627 54 7 713308 349 932533 127 780775 476 219225 51678 85612 63 8 713517 932457 781060 476 218940 51703 85697 52 34.8 12.7 47.6 9 713726 34. 932380 127 781346 2186541 5172885582 51 10 713936 3 932304 781631 218369 15176385567 50 34.8 12.7 47.5.. 11 9.714144 4:8 9.932228 1 9.781916 10.218084 651778 85551 49 12 714352 34. 932151'782201 4 *5 217799 ( 51803 85536 48 13 714561 34. 932075 12.7 782486 475. 217514 151828185521 47 14 714769 34 7 931998 12.8 782771 47*6 217229 115185285506 46 15 714978 34.7 931921 12.8 783056 47.5 216944 51877185491 45 16 715186 4 7 931845 12.8 783341 47.5 216659 51902185476 44 17 716394 34. 931768 12.8 783626 474. 216374 51927185461 43 18 715602 34.6 931691 12.8 783910 47.4 216090 1 51952185446 42 19 715809 34.6 931614 12.8 784195 47-4 215805 161977185431 41 20 716017 346 93137 12.8 784479 47 2155211 152002185416 40 21 9.716224 3. 9.931460 12.8 9 784764 47.4 10.215236 52026185401 39 22 716432 34 931383 12 785048 47,4 214952 52051185385 38 23 716639 34- 931306 12.8 786332 47 214668 52076185370 37 24 716846. 931229 12.8 78o616 47 214384 52101185365 36 fL5 346 5 12.9 7859 5 717053 34.5 931152 129 786900 47.3 214100 152126185340 36 26' 717259 34 931075 129 7 86184 47.3 213816 62151186325 34 27 717466 4.4 930998 129 786468 47.3 213532 1!62175185310 33 28 717673 3 930921 12.9 786752 47'3 213248 5522001852V4 32 29 717879 34- 930843 12-9 787036 47-3 212964 1521225S52i9 31 30 718085 34 930766 129 787319 472 212681 52250185264 30 31 9.718291 9.930688 129 9.787603 472 10.212397 ]52275686249 29 32 718497 343 930611 129 787886 472 212114 52299185234 28 33 718703 3 3 930533 12 9 788170 47 2 211830 52324185218'27 34 718909 134 930456,12'9 788453 47.2 211547 152349t8520):3 26 35 719114 342 93037812.9 788736 472 211264 2374 85188 2 36 719320 34.2 930300 13290 789019 47.2 210981 52399185173 2 4 37 719525 342 930223 130 789302 471 210598 52423185167 23 381 719730 342 930145 130 789585 471 210415 52448185142 22 39 719935 3421 930067 13.0 789868 471 210132 5247385127 21 40 720140 341 929989 130 790151 471 209849 52498185112'20 41 9.720345 341 9.929911 130 9.790433 471 10.209567 l5625'2185096 19 42 720549 34'1 929833 130 790716 471 209284 5254ii!85081 18 14.1 13.0 47.1 43 720754! 929755'130 790999 471 209001 5257i1856006 17 44 72095834'0 929677 130 791281 47.1 208719 6259i 85061 16 45 721162 929599 13 791563 47 208437 52621185035 15 46 721366 134 9'2921 13'0 791846 47.0 208154 52964685020 14 )34.0 13.0 47.0 47 721570340 929442 13 792128 07872 52671 8506 13 48 721774 9 929364 1 792410 0 207'590 52696184989 12 )33.9 13.1 47.0 49 721978 929286 1 792692 207308 52720[84974 11 50 722181133 9 929207 131 792974 470 207026 52745 84909 10 51 9.722385 33 9.929129 9.793266 4710.206744 52770!84943 9;33-9 13.1 47.0 62 722588 929050 11 793538' 206462 527941849%8 8 53 722791 33 928972 131 793819 469 206181 52819[84913 7 91'33-8 13.1 46.9 54 722994 338 928893 131 794101 469 205899 115244184897 6 65 723197 928815 3 794383 469 205617 15289184882 5 56 723400 338 928736 131 794664 469 205336 152893184866 4 67 723603 33 7 928657 131 794945 469 20551152918 848651 3 58 7238033 7 928578 i3.1 796227 204773 1152943184836 2 59- 724007 37 928499 131 795508 46 8 20449211 52967 84820 1 6010 724210 928420 795789 2042111152992!848056 0 Cosine. Sine. ] Cotang. Tang. N. i.s'e. Cosne. Sine. 58 Degrees.

Page  53 TABLE II. Log. Sines and Tangents. (320) Natural Sines. 53 I Sine. D. 10" Cosine. D. 10"1 Tang. D. 10"! Cotang. *iN. sine.iN. cos. 0 9.724210 9.928420 1 9.795789 6 8 10.204211 5299284805 60 1 724412 33. 928342 132 796070 4.8 203930' 53017 84789 69 2 724814 33. 7 928263 3 2 796351 468 203649 153041 84774 58 3 724816 33.6 928183 13-2 796632 468 203368! 53056 84759 o7 4 725017 33.6 928104 132 796913 46.8 203087 53091 84743 56 5 725219 33.6 928025 13.2 797194 468 202806 53115 84728 55 6 725420 33. 5 927946 13. 2 797475 4 8 202525 153140184712 54 7 725522 33.5 927867 13.2 797755 46.8 202245 53164 84697 53 8 725823 33.5 927787 3.2 798036 46 7 201964 53189 84681 52 9 726024 33.5 927703 13.2 793316 467 201684 53214 84666 51 10 72622 33. 927629.2 798596 46 7 201404 53238 84650 50 11 9.726426 33. 927549 9.798877 467 10.201123 5323 84635 49 12 726626 33.4 927470 13 799157 467 200843 53288 84619 48 13 726827 33.4 927390 13.3 799437 4.7 200563 53312 84604 47 14 727027 33.4 927310.3 799717 467 200283 153337 84588 46 16 727228 33.4 927231 3 46 799997 46 6 200003 53361 84573 45 16 727428 33.4 927151 13 3 800277 466 199723 53386 84557 44 17 727628 33.3 927071 13.3 800557 46 6 199443 53411 84542 43 18 727828 33.3 926991 3.3 800836 46 6 199164 53435184526 42 19 728027 33. 926911 13: 801116 46'6 198884 53460[84511 41 20 728227 926831 801396 198604 5348484495 40 21 9.72842 7 33.3 9.96751 13.3 9.801675 4.6 10.198325 53509 84480 39 22 728626 33.2 13.3 801955 46.6 198045 53534 84464 38 24 729024 33.2 926591 133 8012234 23 728825 33.2 926591 13.3 802234 46.6 197766 53558 84448 37 24 729024 332 926511 13 46.5 197487 53583[84433 36 25 729223 33. 92431 13 802792 465 197208 53607 84417 35 27 729601 33.1 926351 13:4 26 729422 33.1 926351 13.4 803072 465 196928 53632 84402 34 27 729621 33.1 926270 13.4 803351 46.5 196649 5365684386 33 28 729820 33.1 926190 803630 465 196370 53681 8430 32 29 730018 33.0 926110 13.4 803908 46:5 196092 53705 84355 31 30 730216 33.0 926029 13.4 804187 46.r 195813 53730184339 30 31 9.730415 33.0 9.925949 13.4 9.804466 4.4 10.195534 5375484324 29 32 730613 33.0 925868 13.4 804745 46.4 195255 5377984308 28 33 730811 33.0 925788 134 805023 46.4 194977 53804 84292 27 34 731009 32.9 925707 805302 194698 53828184277 26 805130~'. 35 731206 32.9 925626 13.4 805580 46.4 194420 53853]84261 25 36 731404 32.9 925545 l, 805859 46.4 194141 53877 84245 24 37 731602 32. 925465 13 061.37 464 193863 53902184230 23 38 731799 32. 925384 3. 806415 3 193585 53926 84214 22 39 731996 32.8 925303 806693 46.3 193307 53951184198'21 40 732193 32.8 925222 5 *806971 463 1.93029 5397584182 20 41 9.732390 32.8.9251411 9.807249 10.192751 54000 84167 19 42 732587 32.8 925060 13 807527 4.3 192473 54024184151 18 43 732784 924979 807805 46 192195 5404984135 17 32.8 13.5 46.3! i. 44 732980 32.7 924897 135 808083 43 191917 54073184120 16 44 733177 327 1248 5 808361. 4 733177 32.7 924816 13.5 808361 46. 191639 54097184104 15 46 733373 32.7 924735 808638 46.3 191362 5412284088 14 47 733569 32.7 924654 136 808916 45.2 191084 541468402 13 48 733765 32.7 924572 1 3 809193 4.2 190807 54171 84057 12 49 733961 32.7 924491 13.6 809471 46. 190529 54195[84041 11 50 734157 32.6 924409 13.6 809748 46.2 190252 54220 84025 10 1 9. 734353 32. 9.924328 13.6 9.810025 46.2 10.189975 5424484009 9 52 734549 32.6 924246 13.6 810302 46.2 189698 1 54269 83994 8 53 734744 32. 924164 13.6 810580 46.2 189420 l 54293 83978 7 54 734939 3 924083 13.6 810857 46.2 189143 5431783962 6 5 | 735135 32.5 924001 13.6 811134.2 188866 5434283946 6 56 735330 32.5 923919 13.6 811410 46.1 188590 54366|83930 4 57 735525 32.5 923837 13.6 811687 46.1 188313 54191183915 3 68 735719 32. 923755 13.6 811964 46.1 188036 54415183899 2 59 735914 32.4 923673 13.7 812241 4 187759 54440183883 1 60 736109 32.4 923591 13.7 812517 6.1 187483 5446483867 0 CosinSine. Si Cotang. Tang.!N. cos.iN.sine. 57 Degrees.

Page  54 54 Log. Sines and Tangents. (330) Natural Sines. TABLE II. / Sine. D. 10' Cosine. D. 10" Tang. D. 10" Cotang. iN. sine. N. cos. 0 9.736109 9.923591 9.812517 10.187482 54464 83867 60 1 736303 324 923509 137 812794 461 187206 5448883851 59 2 736498 324 923427 137 813070 461 186930 54513 83835 58 3 736692 32. 923345 13. 813347 0 186653 5453783819 57 4 736886 323 923263 137 813623 460 186377 54561 83804 56 5 737080 323 l23181 137 813899 460 186101 54586183788 55 6 737274 323 923098 137 814175 185825 5461083772 54 7 737467 323 923016 137 814452 46.0 185548 54635 83756 53 8 737661 322 922933 13.7 814728 46.0 185272 5465983740 52 9 737855 32.2 922851 815004 46.0 184996 54683 83724 51 10 738048 32. 922768 1 815279 0 184721 54708 83708 50 11 9.738241 32 2 9.922686 13.8 9 815555 46. 10.184445 54732i83692 49 12 738434 322 922603 815831.9 184169 54756 83676 48 13, 738627 922520 816107 183893 54781[83660 47 14 738820 32.1 922438 13.8 816382 183618 54805183645 46 15 739013 32.1 922355 13.8 816658 B 183342 5482983629 45 16 739206 32.1 922272 13.8 816933 9 183067 5485483613 44 17 39398 31 922189 13.8 817209 4 182791 5487883597 43 18 739590 32. 922106 13.8 817484 45*9 182516 5490283581 42 19 739783 32.0 922023 13.8 817759 182241 54927 83565 41 20 739975 921940 13.8 818036 181965 54951 83549 40 32.0 39.8 45.8 21 9.740167 32. 9921857 9818310 45.8 10.181690 54975 83533 39 22 740359 32:.0 921774 13.9 818585 45.8 181415 54999 83517 38 23 740550 31 921691 13.9 818860 45.8 181140 5502483501 37 24 740742 31' 921607 13.9 819135 48 180865 55048 83485 36 25 740934 319 921524 13.9 819410 45.8 180590 55072 83469 35 26 741125 319 921441 13.9 819684 45.8 180316 55097 83453 34 27 741316 31.9 921357 13.9 819959 458 180041 55121 83437 33 28 741508 31. 921274 13.9 820234 458 179766 55145 83421 32 29 741699 31.8 921190 13.9 82008 179492 55169 83405 31 30 741889 31 92107 13. 820783 179217 5519483389 30 31 9.742080 31.8 9.921023 13.9 9.821057 457 0.178943 65218 83373 29 318 13.9 45 10. 178943 55218337329 32 742271 31.8 920939 821332 7 178668 24283356 28 33 742462 31. 920856 14.0 821606 45.7 178394 55266 83340 27 34 742652 31.7 920772 14.0 821880 45. 178120 5529183324 26 3147 8120 55291 8332426 35 742842 31.7 920688 14.0 822154 7 177846 5531583308 25 31777846 55315 83308 25 36 743033 31.7 920604 14.0 822429 457 177571 533983292 24 31677571 5533983292 24 37 743223 31.7 920520 14.0 822703 45.7 177297 5536383976 23 3167 1 1770523 5538883260 22 38 743413 920436 14. 822977 45.6 17023 39 743602 31.6 920352 14.0 823250 176750 5541283244 21 31.6 45.6 17647655436 83228 20 40 743792 31.6 920268 14.0 823524 45.6 41 9.743982 3 6 9.920184 14.0 9.823798 10.176202 55460[83212 19 42 744171 920099 14.0 175928 42 744171 31.6 920099 14.0 824072 45.6 175928 5548483195 18 43 744361 920015 824345 45. 175655 5550983179 17 31.5 93114.0 45555823417 91 44 744550 31.5 919931 14.1 824619 45.6 175381 55533 83163 16 45 744739 31 919846 14.1 824893 175107 46 744928 919762.1 825166 46 744928 31. 919762 14.1 825166 45.6 174834 5581183131 14 47 745117 5 91967714.1 825439 5. 174561 5560583115 13 48 745306 314 919593 14.1 825713 4.5 174287 55630 83098 12 49 745494 919508 14.1 825986 174014 55648382 11 31.4 14. 1 45.5 5565 4 83082 11 50 745683 314 919424 14.1 826259 173741 L5678183066 10 51 9.745871 314 9.919339 14.1 9.826532 10.173468 55702!83050 9 52 746059 31.4 919254 14.1 826805 45.5 173191 572683034 3154 173195 [ 5572683034 8 53 746248 31. 919169 14.1 827078 45. 172922 557083017 3173 172922 55750!83017 7 54 746436 313 14.1 827351 45. 172649. 55775 83001 6 55 746624 313 919000 14.1 827624 45.5 172376 55799 8285 5 56 74812 313 14.1 827897 172103 55823 82969 4 57 746999 918830 14. 828170 171830 5584782953 3 58 747187 918745 14.2 828442 45.4 171558 55871 82936i 2 59 747374 918659 828715 171285 55b95182920 1 60 747562 31 918574 14.2 828987 171013 55919k82904 0 Cosine. Sine.' Cotang. Tang.'N. cos Nsine. 56 Degrees. _=...

Page  55 TABLE II. Log. Sines and Tangents. (340) Natural Sines. 55 _ Sine. D. 10"_ Cosine. D. 10" Tang. D. 10"' Cotang. I N.sine N. cos. 0 9.747562 31.2 9.918574 9.828987 45 4 10.171013 55919 82904 60 1 747749 918489 14. 82920 54 170740 5594382887 59 2 747936 31.2 918404 14.2 82932 170468 55968 82871 58 31.2 14.2 829532 3 748123. 918318 29805 170195 55992 82855 57 4 748310 3.1 918233 14.2 830077.4 169923 56016 82839 56 5 748497 31.1 918147 14.2 830349 45. 169651 56040 82822 55 6 748683 31.1 918052 14.2 830621 45.3 169379 56064 82806 54 7 748870 31 1 917976 43 830893 169107 5608882790 53 8 749056 311 917891 143 831165 168835 56112 82773 52 9 749243 310 917805 143 831437 168563 56136 82757 51 10 749426 31.0 917719 831709 168291 56160 82741 50 11 9.749615 31.0 9.917634 14 9.831981 453 10.168019 56184 82724 49 12 749801 31 0 917548 4.3 832253 3 167747 6620882708 48 13 749987 31.0 917462 14.3 832625 45.3 167475'56232 82692 47 14 750172 309 917376 143 832796 3 167204 56256 82675 46 15 750358 30.9 917290 143 833068 45- 166932 5628082659 45 16 750543 309 917204 143 833339 45 2 166661 56305 82643 44 17 -750729 30.9 917118 4.34 833611 45.2 166389 56329 82626 43 18 750914 3 917032 4 833882 45 166118 56353 82610 42 19 751099 30.8 916946 14'4 834154 45.2 165846 56377 82593 41 20 751284 30 8 916859 144 834425 452 165575 56401 82577 40 21 9.751469 30.8 9.916773 144 9.834696 45 10.165304 56425 82561 39 22 751654 3 916687 834967 452 165033 5644982544 38 23 751839 30.8 916600 4 83523845 164762 5647382528 37 751839308 9 0 144 8 53383 24 7520231 3.8 916514 144 835509 *.2 164491 56497 82511 36 25 752208 3 7 916427 144 835780 452 164220 56521 82495 36 26 752392 30-7 916341 14.4 36051 45.1 163949 5654582478 34 27 752576 3-7 916254 836322 163678 65656982462 33 30.7 14.4 45.3 28 752760 30'7 916167 14. 836093 451 163407 56593 82446 32 29 752944 306 916081 836864 451 163136 56617 82429 31 30 753128] 36915994 14. 837134 1 162866 5664182413 30 31 9.763312 306.915907 14.5 837405 45.1 10.162595 56665 82396 29 32 753495!' 91582014 32 753495 30*6 915820 14 837675 4.1 162325 56689 82380 28 33 7536 9, 916733 837946 162054 56713 82363 27 34 753862 3056 915646 14.5 838216 45.1 161784 56736 82347 26 35 754046 130' 915559'5 838487 41 161513 56760 82330 25 36 754229 30.5 915472 145 83857 0 161243 56784 82314 24 37 754412! 915385 14.5 839027 45.0 160973 56808 82297 23 1 555305 14.5 6 0.0 38 7545W953 915297 839297 160703 5685282281 22 39 7547781 304 916210 14.5 839568 45.0 160432 56856 82264 21 40 754960.4 91123 146 839838 0 160162 5688082248 20 41 9.755143i0 9.915035 146 9.840108 10.159892 56904 82231 19 42 7553261304 914948 6 840378 0 159622 569288224 18 430 4 14.6 45.0 43 75550 914860 14. 840647 159353 56952 82198 17 44 75563903 4 914773 6 840917 45.0 159083 56976182181 16 B 75 wB2 304 Y1' 145G81 4459 7002165715 45 756872 30 914685 841187 158813 570008265 15 46 75605430 914598 84147 158543 57024 82148114 47 756236 30 3 914510 14.6 841726 44 158274 5704782132 13 48 756418 30 914422 14. 841996 44 158004 67071 82115 12 49 756600 30 3 914334 14. 842266 44. 157734 57095182098 11 50 7567820 303 914246 14.6 842535 157465 57119 82082 10 50 9.756783 302 9914158 14.7 9842805 4 10.157195 57143 82065 9 52 757144 302 914070 147 843074' 16926716782048 S 53 757326 913982 14 843343 156657 5719182032 7 54 7571507 3 913894 7843612. 156388 57215182015 6 55 757688 30 913806 1 43882 156118 57238 81999 5 56 75789 3011 913718 1. 844151 44| 155849 57262 81982 4 57 758050 1 913630 14.7 844420 44 155580 728681965 3 1 3531407'18 58 758230 1 913541 4 844689 44 155311 57310 81949 2 59 758411 13453 844958 155042 57334 81932 1 60 753591.1 913365 1 845227 44.8 154773 57358819151 0 Cooine. ine Cotang. ang. S1i.,N Sne.o 55 Degrees.

Page  56 56 Log. Sines and Tangents. (350) Natural Sines. TABLE II. t Sine. D.. 1U' Cosine. 0D. l"i; Tang. D. 10" Cotang. [N. sine. N. cos. 0 9.758591 30 1 9.913365 14 7 9.845227 48 110.154773 57358 81915 60 1 758772 3010 913276 84549o 164504 57381 81899 59 2 758952 300 913187 14.7 84576 4. 154236 16740 81882 58 3 759132 0*0 913099 14.8 846033 44. 163967 1 57429 81865 57 4 759312 30 913010 44.8 846 13698 5745381848 56 5 759492 30. v 912922 14.8 846570 44.8 153430 57477 81832 5 6 769672 9 912833 846839 7 153161 57501 81815 54 7 769852 299 912744 14.8 847107 4 152893 57524 81798 53 8 760031 299 912655 14.8 847376 44. 152624 67548 817i82 52 9 760211 29.9 912566 4.8 847644 44.7 152356 57572181765 51 10 760390 29:9 912477 14.8 847913 44.7 15'Z087 57596181748 50 11 9.760569 29 8 9.912388 148 9.848181 44.7 10.151819 57619 81731 49 12 760748 298 912299 848449 44. 151551 57643 81714 48 13 760927 298 912210 4.9 848717 44. 151283 57667 81698 47 14 761106 29-8 912121 14.9 848986 44.7 151014 57691{81681 46 15 761285 298 912031 14.9 849254 44.7 150746 57715 81664 45 16 761464 29 8 911942 849522 4 150478 5773881647 44 17 761642 297 911853 14.9 849790 44. 150210 5776281631 43 18 761821 2997 911763 14.9 850058 44.6 149942 57786 81614 42 19 761999 29 7 911674 14.9 850325 149675 57810i81597 41 20 762177 297 911584 14.9 850593 44 149407 57833 81580 40 21 9.762356 9911495 9 850861 44.6 10.149139 57857 81563 39 22 762534 29:7 911405 149 81129 446 148871 57881 81546 38 29.6 14.9 44.6 23 762712 911315 851396 148604 57904[81530 37 ~4 7628 9 ~. { 9116 35. 1 44.6 14 24 762889 129.6 911226 15.0 851664 148336 57928181513 36 25 763067 29.6 911136. 1931 44. 148069 57952181496 35 26 763245 29.66 911046 15. 852199 44.6 147801 5797681479 34 27 763422 9 910956 852466 44. 147534 15799981462 33 28 763600 29.6 910866 15.0 852733 44.6 147267 58023181445 32 29 763777 29.5 910776 15.0 853001 146999 5804781428 31 30 763954 29.5 910686 15.0 853268 44.5 146732 158070 81412 30 3 1 19.76413 1 99.5 991096 11. 09.853536 10146465 5809481395 29 3-2 764308 29.5 910506 15.0 8538)2 44.5 146198 5811881378 28 33 764485 29.5 910415 15.0 854069 145931 5814181361 27 34 764662 29.4 910325 15.0 854336 44 145664 i 58165 81344 26 35 764838 29.4 910235 151 854603 145397 58189181327 25 36 765015 29.4 910144 15.1 854870 44. 145130'58212|81310 24 37 765191 29.4 910054 15.1 855137 44. 144863 5823681293 23 38 765367 29.4 909963 15.1 855404 44.t 144596 5826081276 22 39 765544 29.4 909873 1651 855671 4 44329 582831812.9 21 40 1765720 129.3 909782 15-1 855938 444 144062 583078124 20 41 9.765896 29.3 9 909691 15.1 9.856204 10 143796 58330181225 19 42 766072 29.3 909601 15.1 856471 44.4 143529' 58354181208 18 43 766247 29.3 909510 |151 856737 44 143263 58378181191 17 44 766423 29.3 909419 15-1 857004 44.4 142996 58401 81174 16 45 766598 29.3 909328 5.1 857270 44.4 14230 58428117 15 46 766774 29.2 909237 152 857537 44.4 142463 58449 81140 14 47 766949 292 909146 152 857803 4 142197| 58472 81123 13 48 767124 29.2 909055 1592 858069 444 141931 5849 1 10 12 49 767300 292 908964 15.2 858336 4 141664. 5851981089 l11 50 767475 29. 908873 15.2 858602 444 1413981 5843 81 I 0,i 10 51 9.767649 29.1 9.908781 15.29.858868 10.141132 15856( 81055 9 62 767824 29.1 908690 1 859134 44'3 1408661 58590810o8 S 53 767999 291 9059 15.2 8940 443 140500 586141810-21 7 54 768173 291 908507 1 859666 44 140334 [ 58034i61004 6 55 768348 2. 908416 15 859932 4 1400o8'158661{li098 5 56 768522 290 908324 153 860198 139802 58684180j0 4 57 768697 29 908233 860464 44 139536 15870S1 10953 3 58 768871 29.0 908141 153 86730 44 139270 1 58Iii1 1[036 2 59 769045 29 908049 86995 4 139005 58'5ojog913i 1 60 769219 2 907958 15.3 861261 44. 13839 58i3i:09i 0 Cosine. Sine.1 I Cotan-. n-..I-' S;e. 54 Degrees.

Page  57 TABLE II. Log. Sines and Tangents. (360) Natural Sines. 57 I Sine. D. 10" Cosine. D. 10"1 Tang.. D. 10" Cotang. N. sine. N. cos. 0 9.769219 29 0 9.907958 15 9.861261 10.138739 1 58779 80902 60 1 769393 28.9 907866 153 861527 4 1384'73 158802,80885 59 2 769a566 28 9 907774 15.3 861792 138208 4 5882680867 58 3 769740 28.9 907682 15*3 862058 44 2 137942 i 58849'80850 57 4 769913 28*9 907590 15.3 862323 44.2 137677 l58873 80833 56 5 77008 7 907498 862589.2 137411 158896 80816 55 28.9 15.3 44.2 6 770260 28.8 907406 15.3 862854 442 137146 1 58920 80799 54 7 770433 907314 863119 442 136881 58943'80782 53 28.8 15.4 44.2 8 7706()0628:8 907222 15:4 863385 136615 158967 80765 52 9 770779 288 907129 15 4 863650 44.2 136350 58990 80748 51 10 770952 288 907037 154 863915 44. 136085 59014 80730 50 11 9.77112 288 9.906945 154 9.864180 44.2 10.135820 59037180713 49 12 771298 28.7 906852 15.4 864445 442 135555 59061180696 48 13 771470 28*7 906760 154 864710 442 135290 59084180679 47 14 771643 28.7 906667 15*4 864975 441 135025 59108180662 46 15 771815 287 906575 154 865240 4 1 134760 1 59131180644 45 16 771987 287 906482 154 865505 441 134495[ 59154 80627 44 28.7 15.4 44.1 17 772159 28*7 906389 15*5 865770 4. 134230 59178 80610 43 18 772331 28.6 906296 15.5 866035 441 133965 59201 80593 42 19 772503 28:6 906204 5.5 866300 44.1 133700 59225 80576 41 20 772675 28 6 906111 15 5 866564 44 1 133436 59248 80558 40 21 9.772847 2836 9.906018 15. 9.866829 44.110.133171 592780541 39 22 773018 28-6 905925 5.5 867094 441 132906 59295 80524 38 23 773190 286 905832 155 867358 441 132642 59318 80507 37 24 773361 285 905739 15 867623 441 132377 59342 80489 36 25 773533 285 905645 155 867887 44.1 132113 59365 80472 35 26 773704 28.5 905552 15:5 868152 131848 59389 80455 34 27 773875 28*5 905459 15*6 868416 4.0 131584 59412180438 33 28 774046 28.5 905366 156 868680 440 131320 59436]80422 32 29 774217 28*5 905272 15*6 868945 440 131055 59459 80403 31 30 774388 28.4 905179 15.6 869209 440 130791 5948280386 30 31 9.774558 28*4 9.905085 15*6 9.869473 44.0 10.130527 59506 80368 29 32 774729 28:4 904992 15:6 869737 44.0 130263 59529 80351 28 33 77489928 4 904898 156 87000 44 0 129999 59552 80334 27 34 775070 284 904804 5.6 870265 440 129735 5967680316 26 35 775240 28-4 904711 15.6 870529 440 129471 59599180299 25 36 775410 28.3 904617 15.6 870793 129207 59622180282 24 37 775580 28.3 904523 156 871057 44.0 128943 59646 80264 23 38 775750 28.3 9044'29 15.7 871321 44.0 128679 59669180247 22 39 775920 28.3 904335 15.7 871685 4.0 128415 59693180230 21 40 776090 28.3 904241 15.7 871849 4. 128151 59716180212 20 41 9.776259 283 9.904147 15.7 9.872112 10.127888 59739180195 19 42 776429 282 904053 157 872376 127624 59763 801rD8 18 43 776598 903959 872640 127360 59786 80160 17 44 776768 157 43. 1 44 l 776768 28-2 903864 15.7 872903 127097 59809180143 16 45 776937 28:2 903770 873167 126833 59832980125 15 28 2 15 7 43.9 46 777106 28.92 903676 15.7 873430 439 126570 1 59856 80108 14 47 777275 28-1 903581 15.7 873694 3.9 126306 59879 80091 13 48 777444 28.1 903487 15*7 873957 43.9 126043! 59902180073 12 49 777613 281 903392 158 874220 125780 59926 80056 11 50 777781 81 903298 18 87448 43.9 125516 59949180038 10 51 ~1.777950 28-1 9.903202 15*8 9.874747 10.125253 59972180021 9 52 778119 281 903108 158 875010 43.9 124990 59995180003 8 53 778287 28.0 903014 15-8 875273 124727 60019179986 7 541 778455 28 902919 158 875536 43.8 124464 60042179968 6 55 778624 28.0 902824 15.8 875800 438 124200.60065 79951 5 56 778i92 28.0 9027629 15.8 87606343.8 123937 60089]79934 4 57 778960280 902634 15 876326 43.8 123674 60112 79916 3 58 779128 28:0 902539 159 876589 43.8 123411 6013579899 2 59 77192955 902444 1 876861 123149 60158 79881 1 0 27.9 15.9 43.8 60 779463 2 9 902349 877114 122886 l60182 79864 0 Cosine. Sine. I Cotang. Tang. NcosN. N-sine. 53 Degrees.

Page  58 58 Log. Sines and Tangents. (370) Natural Sines. TABLE II.' Sine. D. 10}" Cosine. D. 10" Tang. ID. 10"[ Cotang. lN.sine. N. cos. 0 9.779463 29. 9.9023499 9.877114 438 10.122886 160182179864 60 1 779631 279 902253 159 877377 43-8 122623 160205 79846 59 2 779798 27.9 90'2158 15.9 877640 438 12236060228 79829 58 3 779966 27.9 902063 159 87790 43 122097 160251 79811 57 4 780133 279 901967 159 878165 438 121835 60274 79793 56 5 780300 278 901872 159 878428 438 121572 60298 79776 55 6 780467 278 901776 159 878691 438 121309 60321 79758 54 7 780334 8 901681 159 878953 121047 60344 79741 53 8 780801 27.8 901585.9 879216 437 120784 160367 79723 52 9 78098 278 901490 9 8794781 437 120522!i60390179706 51 10 781134 27.8 901394 159 879741 43'7 12029 1 60414179688 50 11 9.781301 27.8 991298160 9.880003 43.7 10.119997 6043779671 49 12 7814'58 27.7 901202 160 88026 43.7 119735 60460 79658 48 13 781634 27.7 901106 16 880528 43 7 119472 60483 79635 47 14 781800 27.7 901010 16.0 880790 4'7 119210 60506 79618 46 15 781966 27.7 900914 16.0 88105 437 118948 60529 79600 45 16 782132 27.7 900818 16.0 881314 43.7 118686 60553 79583 44 17 782298 27.7 900722 16.0 881576 43.7 118424 60576 79565 43 18 782464 27.6 90626 16 881839 437 118161 6059979547 42 19 782630 27.6 900529 16 882101 117899 60622179530 41 20 782796 27.6 900433 16.0 882363 437 117637 60645 79512 40 21 9.782961 27.6 900337 16.1 9.882625 43.6 10.117375 60668179494 39 22 783127 27.6 900242 16l1 882887 43.6 117113 60691179477 38 23 783292 7.6 900144 16.1 883148 43.6 116852 60714 79459 37 24 783458 27.5 900047 16.1 883410 43.6 116590 60738 79441 36 25 783623 27.5 899951 16.1 883672 43.6 116328 60761179424 35 26 783788 27.5 899854 16.1 883934 43.'6 116066 60784 79406 34 27 783953 27.5 899757 16.1 884196 43.6 115804 60807 79388 33 28 784118 27.5 899660 16.1 884457 43.6 115543 60830179371 32 20 784282 27.5 899564 16.1 884719 43.6 115281, 60853179353 31 39 784447 27.4 899467 16.1 884980 43.6 115020 60876 79335 30 31 9.784612 27.4 9.899370 16.2 9.885242 43.6 10.114758 60899 79318 29 32 784776 27.4 899273 16.2 885503 43.6 114497 60922 79300 28 33 784941 27.4 899176 16.2 885765 43.6 114235 60945 79282 27 34 785105 27.4 899078 16.2 886026 43.6 113974 60968179264 26 35 785269 27.4 898981 16.2 886288 43.6 113712 60991 79247 25 36 785433 27.3 898884 16.2 886549 43.6 113451 61015 79229 24 37 785597 27.3 898787 162 886810 435 113190 61038 79211 23 38 785761 27.3 898689 16.2 887072 43.5 112928 6106179193 22 39 785925 27.3 898592 162 887333 435 112667 61084 19176 21 40 786089 27.3 898494 16 88759 43 5 112406 61107 79158 20 41 9.786252 27.3 9.898397 16.3 1988785 43.510 112145 6113079140 19 42 786416 27.2 898299 16.3 888116 43.5 111884 61153i79122 18 43 786579 27.2 898202 16.3 888377 43.5 111623 61176 79105 17 44 7867427.2 898104 16.3 88863943 5 11136t- 61199 79087 16 45 786906 27.2 898006 16.3 888900 43.5 111100 61222 79069 15 46 787069 27.2 897908 16.3 889160.5 110840 1 61245 79051 14 47 787232 27. 897810 16.3 889421. 110579 161268 79033 13 48 787395 27.1 897712 16.3 88968 43.5 110318 161291 79016 12 49 78755 27.1 897614 16.3 889943.5 110057 6131478998 11 50 787720 27.1 897516 16.3 89020 43. 109796; 61337 78980 10 51 9.787883 279.897418 16. 9890465 10.109535116136078962 9 52 7880451 1 897320 890725 43.4 109275!61383 78944 8 53 7882U0827 1 89722 16.4 890986 4 1090178926 7 4 7883.0 897 891247 43.4 108753 161429!78908 6 55 788531 3270 897123 164 - [ 108493 1 61451 78891 5 56 788694 27.0 89692 i 89176 8 108232 i 61474 78873 4 57 788856 27.0 896828 164 89202 43.4 107972i 61497 78855 3 58 789018 27.0 896729 164 892289 43.4 107711[ 61520 78837 2 59 789180 27.0 896631 164 892549 43.4 107451 61543[78819 1 60 789342 896532 892810 107190 1161566 78801 0 Cosine. Sine. Cot-n-_. Tang. N. cos.IN.sine. 52 Degrees.

Page  59 TABLE II. Log. Sines and Tangents. (380) Natural Sines. 59 I Sine. D. 10" Cosine. ID. 10" Tang. D. 10'' Cotang. N. sine.lN. cos.' 0 9.789342 6 9 9.896532 16 4 9.892810 3 10.107190 6166 78801 60 1 789504 26. 896433. 893070 4.4 106930 6158978783 59 2 789665 896335 6.5 893331 43.4 106669 61612 78765 58 3 789827 269 896236 16.5 893591 4 4 106409 61635 78747 57 4 789988, 9 896137 89381 434 106149 61658 78729 56 5 790149 65 89896038.5 894111. 105889 61681 78711 55 6 79010 26.9 895939 16.5 894371 43.4 105629 61704 78694 54 7 790471 26.8 895840 16.5 894632 433 105368 61726 78676 53 8 790632 268 895741 165 894892 43.3 105108 61749 78658 52 9 7 69793. 895641 16.5 895152 4.3 104848 61772 78640 51 10 I 790954 26.8 895542 895412 43.3 104588 61795178622 50 11 9.791115 26. 9.895443 16 9.895672 3.3 10.104328 61818 78604 49 1'2 79127 2 5343 16.6 95932 104068 61841 78586 48 13 791436 26:77 895244 166 96192433 103808 6186478568 47 14 791596 267 895145 16.6 896452 43.3 103548 61887 78550 46 15l 791757 2.7 895045 16.6 896712 43.3 3288 6190978532 45 16 791917 26.7 894945 16.6 896971.3 103029[1 61932 78514 44 11! 792077 67 894846 897231 43.3 102769 61965578496 43 18 792237 266 894746 6 897491 4.3 102509 61978 78478 42 19 792397 26 894646 16.6 897751 3.3 102249 62001 78460 41 20 792557 26.6 894546 16.6 898010 43.3 101990 62024 78442 40 21979271626 6 894546.6 8927 21 9.79271 266 9. 894446 16.7.898270 3 10.101730 62046 78424 39 22 792876 266 894346 16.7 898530 43 101470 62069178405 38 896.6 16,7 43.3 23 793035 126.6 894246 17 898789 43*3 101211 62092 78387 37 24 793195 i 26 89904 9 43.2 100951 62115 78369 36 251 793354! 226 894046 899308 100692 62138 78351 36 25 79335142615 16 7 26 793514 26 * 893946 16-7 899568 43.2 100432 62160 78333 34 27 793673 265 893846 167 899827 43. 100173 62183 78315 33 28 793832 26 893745 17 900086 43.2 099914 62206 78297 32 291 7939 893645 6.7 900346 4.2 099654 62229 78279 31 301 794150 1 893544 16. 900605 43.2 099395 62251 78261 30 31 9.794308.893444 167.900864 2 10.099136 62274 78243 29 * 26-4 16.8 4390 321 79446726-4 t 893343 901124 3.2 098876 62297 78225 28 33 794626!b4 893243 * 901383 43 098617 62320 78206 27 34 7947841*64 893142 901642 098358 62342 78188 26 35 79494264 893041 18 901901 43.2 098099 62365 78170 25 361 795101 264 892940 902160 43.2 097840 62388178152 24 37 7959265923 892839 168 902 432 097581 62411 78134 23 38 795417 892i39 168 902679 43 097321 62433 78116 22 39 795575 2 892638 902938 43 97062 6245678098 21 o263?? 40, 795733 26 3 892536 1 903197 43.2' 096803 62479 78079 20 41 19.79891 263 9.892435 1 903455 43 10.096545 162502 78061 19 42! 796049 3 892334 * 903714 43.1 096286 62524 78043 18 43 796206 W892233 903973 43 09602762547 78025 17 44 796364 892132 19 904232 431 095768 6257078007 16 45 796521 262 892030 169 904491 4.1 095509 62592177988 15 46 796679 262 891929 1 904750 43.1 0952501626159 7970 14 47 796836 262 891827 19 905008 43'1 094992116263877952113 48 796993 262 891 726 ( 905267 43 1 094733 62660177934112 49 797160 2 891624 905526 431 094474 62683 77916 11 50 797307 61 891523 170 905784 4 1 0942161 62706 77897 10 264 1707.0 43.1 51 t9.797464 6. 1421.9089143 1 -110.093957 62728 77879 9 521 797621 6 1 91319 1 906302 431 093698 16275177861 8 1536 797777 21 891217 17 0 906560 43 1 093440 162774 77843 7 54 9934 61 891115 170 906819 431 093181 162796 77824 6 i 55 798091 201 89101 170 907077 43.1 092923 62819 77 80 5 56 798247 26.1 890911 907336 43'1 092664 162842177788 4 67 798403 ( 17.0 9075943 092406 162864177769 31 68 79856260 826. 8 7 * 907852143*1 092148 62887 77751 2 59 798716 26 0 890605 17.0 908111 43.1 09188911 62909 77733 1 60 798872 6.0 80 17.0 8369 43.0 01631 6293277715 890 38oz, 0oa908369 091631 62932 77715 0 Cosine. Cotang. Tang. I co. cN.sine...... 51 Degrees.

Page  60 60. Log. Sines and Tangents. (390) Natural Sines. TABLE II.' Sine.. D. 16" Cosine. D. 10'n. t uorang Sine.. o s. 0 9.798772 26 0 9.890503 17 0 9.903369 43 10.091631 62932 77715 60 -1 799028 260 890400 17. 903628 430 01372 629.5 77696 59 2 799184 260 890298 171 908886 43 091114 62977 i7678 58 3 799339 2569 890195 1 90J144 43. 090855s 63009 77660 57 4 799495 29 890093 17.1 9040 430 090598 63022 77641 56 5 799651 25.9 889990 17-1 909660 430 0903401 63045 77623 55 6 799806 215.9 889888 7.1 909918 43.0 0909j2 6308 77605 154 7 799962 2.9 88978, 171 910177 430 089823 63090 77586 53 8 800117 25.9 889682 91035 430 0895OJ5 63113 77568 52 9 800272 2.8 889579 171 910693 43.0 089307 63135177550 51 10 800427 25.8 889477 17.1 910951 430 039049 63158 77531 50 1 9.800582 2 9.8 889374 172 9.911209 43.0 10.088791 93180 77613 49 12 800737 25.8 889271 172 911467 430 088533 63203 77494 48 13 800892 25.8 889168 17.2 911724 430 088276 63225 77476 47 14 801(47 25.8 889064 17'2 911982 430 088018 163248 77458 46 14 801047 120 91198 15 801201 25.8 888961 17'2 912240 430 087760 63271 77439 45 16 801356 25.8 888858 172 912498 430 087502 6329377421 44 17 801511 25.7 888755 172 912756 430 087244 63316 77402 43 18 801665 25.7 888651 17.2 913014 42, 086986 63338 77384 42 19 801819 25.7 888548 17.2 913271 429 086729 68361 77366 41 20 801973 25.7 888444 173 913529 06471 63383 77347 40 21 9.802128 25.7 9.888341 173 9.913787 10.036213 63406 7739 39 22 802282 25.7 888237 17:3 914044 42.9 085956 63428 77310 38 23 802436 25.6 888134 17 3 914302429 08698 63451 77292 37 24 802589 25.6 888030 173 914560 42. 085440 63473 77273 36 25 802743 25.6 887926 173 914817 429 085183 63496 77255 35 26 802897 25.6 887822 173 915075 429 084925 63518 77236 34 27 803050 25.6 887718 173 915332 429 084668 6354077218 33 28 803204 25.6 887614 173 915590 429 084410 63563 77199 32 29 803357 25.6 887510 173 915847 429 084153 I 63585 77181 31 30 803511 25.5 887406 174 916104 429 083896 63606 77162130 31 9.803664 25.5 9.887302 174 9.916362 429 10.083638 63630 77144 29 32 803817 25.5 887198 174 916619 429 083381 63653 77125 28 33 803970 2.5 887093 17.4 916877 429 083123 6367577107 27 34 804123 25. 886989 14 917134 429 082866,63698 770O8 26 35 804276 25.5 886885 174 917391 429 082609 63720177070 25 36 804428 25.4 886780 174 917648 429 082352 6374277051 24 37 804581 25.4 886676 174 917905 429 082095 63765 770633 23 38 804734 25.4 886571 174 918163 42,8 081837 63787 77014 22 39 804886 25.4 886466 17 918420 41 081580 63810 76996 21 40 805039 25.4 886362 175 918677 4; 081323 638321769i7 20 41 9.805191 25.4 9 886257 175 9.918934 428 10.081066 6385476959 | 42 805343 25.3 886152 17 91919142 0809116387, 76940 18 43 805495 25.3 886047 7 919448 428. 0805521 63899 76921 17 44 805647 25.3 885942 1.5 919705 42.8 080295 63922760903 116 45 805799 25.3 885837 17.5 919962 428 080038. 639447;j884 15 46 805951 25.3 885732 17.5 920219l4) 8 079781 639661jI7ob6i 14 47 806103 25.3 885627 17.5 920476 42.8 079524 6398.1j76847 13 48 806254 25.3 885522 17.5 920733 42.8 079267 64011 7t6828 12 49 803406 25.3 885416 17.5 920990 42.8 079010116403316810 11 50 806557 25.2 885311 17.6 921247 42.8 078753 i 64056 76791 101 51 9.806709 25. 9.885205 17.6 9.921503 42.8 10.078497 611 4078176772 9 652 806860 25.2 885100 1 921760 42.8 078240 t6410076754 8 53 807011 2 864994 922017 0742 983. i 64128 76735 7 54 807163 2.2 884889 17.6'9222i4 42.8 077726 64145 76 1lI 6 55 807314 884783 6 922530 07747'.i64186 7~60 5 56 807465 25.2 17.6 42.8 7 64160 769 5 56 807465 251 884677 17 922787 42.8 077213 64190 766,9 4 57 807615 25.1 884572 17.6 923044 42.8 07696 116421S 76166 3 58 807766 2 884466 1 923300 42 076700 64234 i6642 2 59 807917 2 1 884360 176 92357 42 076443 64256 76623 1 60 808067 884254 1 923813 0761871164279 i6604 0 Cosine. Sine. Cotang. Tang. N. cs. 50 Degrees.

Page  61 TABLE II. Log. Sines and Tangents. (400) Natural Sines. 61 I Sine. _D. 10.' Cosine._ ID. 9[ Tang. D. 10" Cotanw. I N.sing. N. cos. 0 9.808037 25.1 9.884254 17.7 9.923813 42.7 10.076187 64279 76604 60 2I' 17 7' 42. 70068 429~646 1 808218 251 884148 177 92407Q 42.7 075930 64301176586 59 2 80836812. 884042 177 924327 42.7 0756,73 64323 76667 58 3 808519 25.1 883936 177 924583 427 075417 64346176548 57 4 803669 25.0 883829 177 924840 42.7 075160 643687'i6530 56 5 808819 25.0 883723. 92509642.7 074904 64390176511 55 6 8 6 177 925096 6[ 80969 25.0 883617 17.7 925352 427 074648 64412 76492 54 z5 0 17 7 42 7 7 803119 25.0 883510 177 925609 427 074391 64435176473 53. ~o o 17 7 4~'l 8 809269 88340417 925865 2 074135 64457 76455 52 9 809419 25.0 883297 17. 926122 427 073878 64479176436 51 10 809569 24.9 883191 178 926378 427 073622 64501176417 50 11 9.809718 24.9 9 883084 17,8 9.926634 42:7 10.073366 64524176398 49 12 809868 24.9 8829771 92689042.7 073110 64546/76380 48 24.9 17,8 92689 13 810017 24/9 882871 178 927147 427 072853 64568176361 47 24,9 ~17,8,~Z7 14 810167 249 882764 178 927403 427 072597 64590176342 46 18136 24, ss 1sv,8g'~/ 15 810316 24 882657 178 92765941 072341 6461217632345 16j 810465 2, 784. 17 810461 24:8 882550 178 927915 427 072085 64635 76304 44 1 81071 24.8 882443 178 928171 427 071829 64657176286 43 24 8 17 8 422 1 8107631 248 882336 17 928427 427 071573 64679176267 42 l4 813 2, 17 9 42. 19 819128 882229 179 928683 427 071317 64701176248 41 20 81~~~~17ti 2, 19 42 7 20 8110i61 248 882121 179 928940 427 071060 64723176229 40 19.811210 248 9.882014 179 9.929196 427 10.070804 64746]76210 39 2~2 8i.1js 24.817 9' 42 7 22 81358 24 881907 17 929452 427 070548 64768176192 38 23 811507 247 881799 179 99708 427 070292 6479076173 37 2 179 9293708'070292 i6479017i61' 173 24 811655 24.7 17.9 42.7 245 811655 247 8816924 179 929964 42.6 070036 64812176154136'5818424 7 17 9 42 6] 254818 7 881584179 930220 426 069780 64834176135 36 26 811952 247 881477 1 930475 069525 64856[76116 34 21 812~~~~8103 6 19 18854 2 812100 247 881369 179 930731 426 069269 64878176097 33 [ 248 7 17.9 42'7 28 812248 247 881261 180 930987 46 069013 64901 76078 32 2y 81259 4, 8 41~' 29 812396 246 881153 18:0 931243 426 068757 64923 76059 31 30 812644 246 881046 1. 931499 426 0685011,64945176041 30 I19~ 812~92 l:, 9 804. 18. 93149 31 9812692 2469.880938 18 30 61755 10.0682451 64967176022 29 3 812840 246 880830 180 932010 426 067990 6498976003 28 33 8~~~~~128 2, 80 42 6 33 812988 246 880722 ~s 932266 426 067734 65011175984 27 34 8138135 246 880613 180 932522 426 06747816503375965 26'861 932522'067478 1 6 35 8132831 24.6 880505 18.0 932778 42.6 067222 65055759462 136 813430 245 80397 i 933033 426 066967 65077 75927124 37 813578 245 880289 181 933289 426 066711 6510075908 23' 880289'933289'061 38 8137251?4 5 1651 [ 23 245 880180 181 33545 6 066455 65122175889 2 349 81401 24~5 880072 181 933800 426 0662001 65144175870 21 18796 930655 596 4 81387 245 879963 181 934056 426 065944165166 75851 20 41 9.814166 24' 9.879855 1 9.934311 4 10.0656891 65188175832 19 42 814313 245 879746 181 934567 426 065433 65210175813 18 879637 ~383 657 143 814460 244 879637 181 934823 426 065177 65232 594 17 44 814607 244 879529 181 935078 426 064922 65254 7575 16 45 t311753 24.418 1 42 6 4 814753 244 879420 18 935333 426 054667 1165276 5,56 15 45 814900 24.4 18l 42.6I 244 879311 181 935589 426 064411 652985'38 14 41 81504612 4 ~4. 18 1 42. t~i41 5Yl 5i8 4 4/ 8151)4i 244 879202 935844 064156 65320 /5119 13 24,4 ~18 2 49 6 48 8015193 244 879093 180 936100 42 063900 65342 5700 12 4) 815839 244 878984 182 936355 426 063645 65364 5680 24.4 18 2 063645 1 65364 0 560 11 50 81548 878875 182 936610 063390 658675661 10 51 9.815631 2 9.878766.936866 1,63134 540 42 9 9.936866,,,, 0. 0631334 165408 5042~ 6~ 1i824.3 18.2 42.6 0;8915;u ~i3 t 52 815/78 243 878656 9371214 062879 6543Ui5623 8 53 1924 )24.3 878547 18 2 425 062624 6545 504 7 53 ib 1 8i213 243878328 937881 7. 062113 65-i9u,. 5juU a 24 31 1 3 938142 425 0 8 655102 354 4 o1650, 24"2 648109 1 938398 425' 061029 U55-*I55.8 3 58 81652 2218785478 93718653 42:5 061347 65454i 5509 24 3 18 2 42 5 59 816096 24~ 2 8178904 938908 42 06109-2'558-ii5490 1 60 816943 1 18.3 42.5 60 8t,,=16943 ___.877180 166 939163 060637 t560,,i54 0'4~, 3i,~ 878521. 35'8'/. -i' It 34.8 1~'2 0 611 6~9 5 8~ ol,iatiul:.'_, l S~819-~'981421 -- 061858,sSalo],00-t, c 0160, 4 3 t83,42 0' obine. - b01 Sine. Cotan38398g. Tang. N. cos. 0.iu, 49 Degrees. 94;2 18 3 42 5 Ii8 t~`'is~Y938fj618~is~I ( 06134 b55~6~[,5509 59[ 8116,98,42', 8i790.'. 38iYO908,'"' 06109'2 ual~ 5 458~154 24 21 1 83 42 5 60 1 810943't87T'180 98_.13' 000u83'/UtOtb, — 7 7osfi~-. - [ ine. —I l-Cotangg. I i' Tang.~ N. Pcos4NK.siue.l 49 Degrees.

Page  62 62 Log. Sines and Tangents. (410) Natural Sines. TABLE II., Sine. D. 10" Cosine. D. 10" Tang. ID. 10" Cotang. tiN. sine_. N. cos. 0 9.816943 9.877780 3 9.939163 425 10.050837 i 65io0;i io471 60 1 817088 24.2 877670 939418 425 060582 656"8 [5452 59 2- 817233 242 877560 1 939673 42'5 060327 65,;o5 1o433 58 3 817379 4. 877450 18.3 939928 42.5 060072, tizi2 i5414 57 24.2 18.3 42.5 4 817524 877340 940183 059817 9' 65o94 75395 56 6 817668 24.1 877230 8.3 940438 42' 059562 i, 6o716 75376 55 6 817813 877120 18.4 94094 425 059306 165738175356t 54 7 817958 24.1 877010 18.4 94049 42.5 059051 1 65759175337 53 8 818103 876899 941204 42.5 058796 1165781 75318 52 24.1 18.4 42.5 9 81824724.1 876789 18.4 941458 42.5 058542 iI 5803 71299 51 10 818392 24.1 876678.4 941714 42.5 058286i 65825 75280 50 11 9.818536 9.876568 18.4 941968 42.5 10.0803265847 75261 4 24.0 18.4 42.5 12 818681 24.0 876457 18.4 942223 42.5 057777! H65869 75241 48 13 818825 24.0 876347 18.4 942478 42.5 057522 I 65891 75222 47 14 818969 24.0 876236 18.4 942733 425 057267 65913 15203 46 24.0 876125425018.7 15 819113 24.0 18.5 942988 42.5 057012 65935 75184 45 16 819257 876014 943243 425 056757 65956 75165 44 24.0 18.5 42.5 17 819401. 875904 943498 42 056502 i 65978 75146 43 24.0 18.5 42.6 18 819545 24.0 875793 18.5 943752 42.5 056248 166000 75126 42.19 819689 23.9 875682 18.5 944007 42.5 055993 66022 75107 41 20 819832 23.9 875571 18.5 944262 42.5 055738!1 i 66044 75088 40 21 9.819976 23.9 9 875459 18.5 9944517 425 10.055483 166066 75069 39 22 820120 23.9 875348 944771 42.4 055229 166088175050 38 23 820263 23.9 875237 18. 945026 42.4 0549741 66109 75030 37 24 8'20400 23.9 875126 18.5 945281 42.4 054719 166131 75011 36 25 820550 3. 875014 18.6 9453542.4 054465 66153 74992 35 26 82 3 23. 874903 18.6 94790 42.4 05410 16617574973 34 27 820836 28 874791 18.6 94604 42.4 053955 166197 74953 33 23.8 18.6 42.4 28 820919 23.8 8740 946299 42 053701 1 66218 74934 32 18.6; 29 821122 3. 874568 946554 424 053446;66240 74915 31 30 821265 23.8 874456 18.6 946808 424 053192 66262 74896 30 31 9.82140i 23.8 9874344 9.947 42 100529376 29 23.8 18.6; 42:4 10.052937 66284 7487 32 821550 23.8 874232 18.7 947318 44 052682 663065 74857 28 33 821693 23. 874121 18.7 94757 052428 166327 74838 12i 23.7 18.7 42' 3-1 8218;35 23.7 87409 18.7 947826 42.4 02174 6349 74818 2 35 821973 873896 187 948081 01919 i 66371 74799 25 23.7 18.7 42.4 "O19 - 36 822120 873784 1 948336 42.4 0o1664 1i 66393 74i80 124 37 82262 23.7 873672 18.7 948 01410 1166414 74760 | 23 38 822404 23.7 83660 18.7 948844 42.4 01156 664364741 22 2 23.7 418.7 42.4 39 8225 23.7 873448 18.7 949099 424 050901!66458 i422!1 1 40 822688 23.6 873335 18.7 94933 24 0504766480 403 41 9.8228 23 9.873223.9496 3 66501 4683 1 19 23.6 18.7 42.4' 42 8:229i2 2 44873110 1 949862' 050138 66523 74563 1li 23.6 82 18.8 42.4 43 82)3114 872998 9501 1 4.4 049884!i 66545 /4t644 1 7 44 823255.6 87288.8 95037 42.4 049630! 66b66b 462t 5 16 23.6 18.8 42.4 45 82339 23. 872772 188 95062 42.4 049375 66588 74603 15 46 823539 23.6 872659 18.8 950879 42.4 049121 116661t0 456t 0 14 47 823680 23.6 872547 18.8 951133 424 048867 1! 66632 7456 13 48 823821 872434 951388 42.4 48612 ii666i53 i4s548 12 49 823963 23.5 872321 18.8 951642 42'4 048358!i 6667 - i 45 11 50 824104'23.5 872208 18.8 951896 42.4 048104!i 6669 746 10 51 9.82424 23. 9.872095 9.952150 10 047850 667181,4483 9 52 82438623." 8i1981 952405 424 041595 bi6740 744i0 8 53 824527 23.5 871868 18.9 952659 424 047341 66i6,.445o 1 7 7 54 824668 5 81155 1.9 952913 0408 i 6678i i4431 6 1 23.4 18.9 42.4 55 824808 4 8i1641 953167 046833 6680l5 i44121 5 56 824949.4 81156 1 953421. 4'3 4Ji9 6681, 74392 4 234 81414 18 9 3 42.93 i 57 825090 871414 92344 0453426i 43 t60846 i4;3 3 23 4 18.9 422. 1 5 58 825230 234 871o01 18.9 953929 423 i 1 6 ti 04!; 3 2 59 82531 2 8711 1 954183 045817,668t91,43; 1 23 4 18.9 60 825511 8710i3 954437 045563 6ti691o14, i1 0 Cosine. Sine. Cotang.'lang. 48 Degrees.

Page  63 TABLE II. Log. Sines and Tangents. (42~) Natural Sines. 63 Sine. ID' Coine. D. 10"1 Tang. ID. 10" Cotang. N. sine.lN. cos.: 0 9.825511 23.4 9.871073 1 954437. 10.046563 66913 74314 60 1 825651 233 870960 954691 423 045309 1 66935 74295 59 2- 825791 2 3 870846 19. 954945 4. 3 045055 1166956174276 58 3 825Ju3 23.3 19.87 0 429 3 0448001166978174256 57 4 826071 23.3 870618 19.0 955 42.3. 044546 1166999i74237 56 5 826'211 23.3 870504 19.0 955707 42.3 0442931! 67021174217 55 6 826351 23.3 870390 19.0 955961 42.3 044039 67043 74198 54 7 826491 23.3 870276 19.0 956215 42.3 043785 167064 74178 63 8 826631 2 3 870161 19.0 956469 043531 167086i74159 52 9 8261 70 3 870047 19.0 956723 42.3 043277 67107 74139 51 10 826910 23.2 89933. 19.1 956977 42 3 0430231 i67129 74120 50 1.827049 23.2 9'869818. 9.957231 42.3 10.042769 67151 74100 49 12 82718923.2 869704 19.1 748542.3 042515 67172 74080 48 13 8273823.2 869589 19.1 95773942.3 04'2261 67194174061 47 14 827467 23.2 869474 1911 957993 42.3 042007 67215[74041 46 15 827606 23.2 869360 19.1 958246 42.3 041754 67237174022 45. 16 827745 23, 869245 19.1 958500 42.3 041500 167258 74002 44 17 827884 23.2 869130 19.1 958754 42.3 041246167280 73983 43 18 828023 23.1 869015 191 959008 42.3 040992 67301 73963 42 19 828162 23.1 868900 19.2 959262 42.3 040738 67323173944 41 I0/ 828301 23.1 868785 19.2 959516 42.3 040484 67344173924 40 2 3.1 19:2 42.3 [2129 828439 1 9 868670' 9 9597694-*-10 67040231]6,t366]73904 39 22 828578 23.1 868555 19.2 960023 42.3 039977 67387 73885 38 23 828716 23.1 868440 19.2 960277 42.3 039723 67409'73865 37 24 828855 23.1 868324 19.2 960531 42.3 039469 67430173846 36 25 828993 23.0 868209 19.2 960784 42.3 039216 67452 73826 35 26 829131 23.0 868093 19.2 961038 42.3 038962 67473 73806 34 27 8292693 0 8697 192 96291 42.3 038709 67495173787 33 28 829407 23.0 867862 19.'3 96154 42.3 038455 67516 73767 32 291 829545 23.0 867747 19.3 961799 42.3 038201 67538173747 31 30 8296831 230 867631 19.3 962052 42.3 037948 67559 73728 30 31 99.829821 23-0 9.867515 19.3 9.962306 42.3 10.037694 67580173708 29 3229 8 19.3 4298 32 829959 22.9 867399 19.3 962560 42.3 037440 67602 73688 28 33 83097 122.9 867283 19.3 962813 42.3 037187 67623 3669 27 34 830234 i229 867167 19.3 963067 42.3 036933 67645 73649 26 35 830372 29 86701 19.3 963320 42.3 036680 67666 73629 25 36 83009 229 866935 19.3 963574 42.3 036426 1167688 73610 24 37 830646 2-9 66819 1.4 963827 42.3 036173 67709 73590 23 22. 19.4 42.3 38 830784 229 866703 19. 964081 42.3 03591911 67730 73570 22 39 8309211228 866586 19.4 964335 2.3 035665 67752173561 2'1 40 831058 228 866470 19.4 964588 42.2 0354121 6777373531 20 41 1 9.8653'9 964 42 41 9.8311951 228 9.866353 14 9.9648 42.2 10 035158 6779573511 13 42 831332' 22-8 66237 19.4 965095 42.2 034905 67816173491 18 43 8314691 22.8 866120 19.4 965349 2.2 034651 6783773472 17 44 831606228 866)004 19. 965602 42.2 034398 67859173452 16 45 8317420 2 865887 965855 42. 034145 16788073432 1 6 46 831879 -8 865770 9195 66109 033891 67901173413 14 47 832015 122.7 865653 19.5 966362 42.2 033638 167923173393 13 48 8321621 22.7 865436 19 5 966616 42.2 033384 1679441'73373 112 48 8321522 7 8 65o36' 1956 42.2 49 832288'2 865419 966869 0331311 67966173531 11 22.7 19.5 42.2 50 8324 227 865302 967123.2 032877 1 67987173333 10 1 9 61i.865185 9.967376 42. 10.032624 168008173314 9 1521 1832697,.,'7 18 9 9.967376 52 8326971 2 27 865068 19.5 967629 42.2 0323711 68029173294 8 122;.7 19. 42.2 63 832833 864950 967883 032117 68051}73274 7 5 ~*7 19.b 42.2 54 83296922 7 864833 96813642.2 031864 168072173254 6 22 6 19.6 429 2 551 833105 -2'6 8647161 19.6 968389 42.2 031611 68093/73234 5' 22.6 19.6 42.2,56 8324 2 864598 968643 031357 68115 3215 4 57 833377,22.6 864481 19.6 968896 42.2 03110411 68136173195 3 226 86448 19.6 9696 42.2 58 83,3512 226 8643631- 19 969149 030851 68157|73175 2 22.6' 19.6 42:2 59 833648, 864245 196 969403 42.2 030597 68179173155 1 60 833783 864127 1 969656 42 03344 168220073135 0 Cosine. Sine. - Cotang. Tang. IlN. cos.|Nsine. 47 Degrees.......................... _. s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Page  64 64 Log. Sines and Tangents. (430) Natural Sines. TABLE II.' Sine. D. 10" Cosine. D. 10"l Taig. Dr..lu"_ Cotang. INe.sine.c. cos. 0 9.833783 2 9.864127 1 9 9.969356 1 0.030344 1 682-0073135 60 1 83.3919;.6 864010 19.6 969909 42'2 030091 68221173116 o59 2 834)154 2' 863892 19.6 970162 42 029838 1 68242 73096 58 3 834189 25 863774 19.7 970416 422 0295841 68264 i3076 57 4 83 132- 22 5 863656 19.7 970569 422 0293311 68285113056 56 5 834460 863538 19 7 970922 422 029078 68306 i3036 55 6 834.59 5o 22 55 863419 19.7 971175 42-2 028825 683'27 i3016 54 7 834730 22.5 863301 19.7 971429 42 028571 68349 72996 53 8 834865 22 863183 197 971682 42 028318 68370 72976 52 9 834999 224 863064. 197 971935 42 2 028055 68391 72957 51 10 835134 22.4 862946 19.8 972188 422 027812 6841272937 50 11 9.835269 22.4 9.862827 19 9.972441 422 10.027559 68434 72917 49 12 835403 862709 972694 027306 68455172897 48 13 835538 22.4 862590 198 972948 422 027052 68476172877 47 22.4 19.8 42.2 14 83567 2 4 862471 8 973201 026799 684971720857 46 16 83587 22.4 862353 198 734 422 026546 68518 72837 45 161 835941 22.4 862234 19. 973707 422 026293 68539 72317 44 171 836075 22.4 862115 198 973960 422 026040 68561 72797 43 18 836209 22.3 86196 198 974213 42-2 025787 68582 72777 42 191 836343 22.3 861877 198 974466 422 025534 68603172757 41 20 836477 22.3 861758 19.9 974719 422 025281 68624172737 40 21 9.836611 22. 9.861638 199 9.974973 422 10.025027 68645172717 39 22 836745 831519 191 9226 422 024774 68666172697 38 23 8368i8 22.3 861400 19 97 479 42.2 024521 168688172677 37 24 837012 22.3 861280 199 75732 2 024268 68709172657 36 22.2 19.9 42.2 25 837146 2 861161 19.9 975985 422 024015 68730 72637 35 26 837279 22.2 861041 19 9 76238 422 023762 68751 72617 34 27 837412 22.2 860922 19 976491422 023509, 68772 72597 33 1.28 837546 22 860802 976744 0232561 6879372577 32 I29 837679 222 860682 20 976997422 023003i 68814172557 31 30 8378122 860562 200 97720 022'50 68835 72537 30 31 9.83794 22. 9.860442 200 9.977503 422 10.024971 688 72517 29 32 83808 221 860322 200 97756 422 022244 68878172497 28 33 838411 1 860202 97800942 021991 168899172477 27 34 838344 860 978262 2 0217381 68920172457 26 22.1 ~ 20.0 42.2 35 83847i 22. 8599i2 20 0 978515 422 0214851 68941172437 25 35 838610 221 859842 978768 021232i 68962172417 24 37 838742 221 859'21 20: 979021 422 0209791 68983172397 23 38 8388i5 22.1 8;t591 20-1 979274 942.2 020726 69004172377 22 39 83901 5948 201 9792527 42.2 0204731 69025172357 21 40 839140 2. I 859360 201 979780 42.2 020220 69046!72337 120 41 9.8392i2 22. 9 859-239 20 1 9.980033 42.2 10.019967 69057172317 19 42 839404 22.0 859119 201 980286 42 019'7141i69088172297 18 43 839536 22. 85899 21 98038 42 019462i 691091722 77 17 44 839368 2. 58877 1 93091 4 019209 69130'72257 16 45 839800 22. 858i56122 981044 018956 169151172236 15 22.0 20.2 42.1 0182031 691721722 46 839932 58635 2 98129714 47 84093)4 858514 20 981550 1 018450 1i69193;72196 13 48 84019 21.9 858393, 981803 018197 1 69214!72176 12 49 840328 858222 9820 426 0179441 69235172156 11 50 840459 858151 2 9823091 4 017691 i 69256172136 10 51 9.840591 219 858029 8262 10.017438 69277 72116 9 21.9 20.2 8042.1 41 69277172116 9 52 840722 21.9 8579 20:2 982814 42 1 017186 i69298172095 8 53 840854i 21.9 8577ti 202 983067 421 016933 1169319172075 7 54 840985 857665 983320.1 016680 1169340172055 6 55 841116 21 85754 23 983573 4.1 0164271 69361172035 5 56 84124721 8542220 983826 42.1 0161741 69382172015 4 5I7 841378 j21.8 85730 20.3 9834079 42.1 0159211 69403171995 3 58 841509 21 8 857178 3984331 1 015669! 69424j71974 2 59 841640 1 857056 20.3 98484 42.1 015416 69445 71954 1 60 841771 21.8 853934 9203 98483 1 42.1 6151631 -69466 71934 0 s - ine. Sine. Cotang.' Tran. 1; N.c-s. N.sine.' 46 Degrees.

Page  65 TABLE II.L Log. Sines and Tangents. (440) Natural Sines. 65 Sine. D. 10" Cosine. D. 10' Tang. D. 10" Cotang. f"N. sine. N. cos. 0 9.841771 21 8 9.856934 20-3 9.984837 421 10.015163 i69466 71934 60 1 841902 21.8 856812 203 985090 421 014910 1'6948. 194 59 2 842033 21. 856690 O24 985343 42. 014657 16950o 11894 58 3 842163 21.8 856568 20.4 985596 4. 1 014404 6952971873 7 4 842294 21.7 856446 290.4 985848 42. 1 014152l 69549 71853 56 5 842424 21.7 856323 20.4 986101 42.1 013899 69350 71833 55 6 842555 2 856201 4 986354 421 013646 69591 171813 54 7 842385 21.7 856078 20.4 986607 42'1 013393 6961271792 53 8 842815 21.7 855956 20:4 9863860 421 013140 69633171772 52 9 842946 21.7 855833 2041 987112 42.1 012888 69654 71752 51 10 843076 21.7 855711 1 987365 42.1 012635 69675 71732 50 11 9.843205 21.79 855588 20.5 9.987618 42.1 -0.012382 69696171711 49 12 843336 21.6 855465 20.5 987871 42.1 0121291 69717 71691 48 13 843466 21.6 855342 20 988123 42.1 011877 169737171671 47 14 843595 21.6 855219 2 988376 42.1 011624 69758 71650146 15 843725 21.6 855096 0.0 988629 42.1 011371 169779 71630 45 16 843855 21.6 8549 i 3 20.5 988882 42.1' 011118 69800 71610 44 17 s843984 21.6 854850 20.5 989134 42.1 010866 169821 71590 43 18 844114 21.6 854727 20.6 989387 42.1 010i13 1169842 71569 42 19 844243 851603 989640' 010360 1698G2 i14Y9 41 20 844372 21.5 854480 20.6 989893 42.1 010101 69s883 71529 40 21 19.844502 21.59 9.854356 20.69. 99014 42.1 10.009855 169904 71508 39 22 844631 21.5 854233 20.6 990398 42.1 096021 6992511488 38 23 844760 21.5 854109 20.6 990651 42.1 009349 169946i 1468 37 24 844889 21.5 853986 20.6 990903 42.'1 009097 116996 71447 36 26 845018 21. 853862 20.6 991156 42.1 008844 69987 11427 35 26 845147 21.5 853738 20.6 991409 42.1 00859 1170008 71407 34 27 21.5 8 20.6 99162 1 008338117029 71386 33 2 8452176 853614 991662 28 84405 21.41 853490 20.7 991914 42.1 008086 70049 11366 32 29' 845533 21.4 853366 20.7 992167 42.1 007833 70070111345 O31 30 845662 21.4 853242 20.7 992420 4.1 007580 70091 11325 30 31!9.845790 21.4 19.853118 20.7 9.992672 42.1 10.00'7328 70112 i1305 129 32 845919 21.4 852994 20.7 992925 42.1 00705 1170132 71284 128 331 846047 21.4 82869/ 20.7 993178 42.1 006822 70153 712641 27 34 846175 21.4 852745 20.7 993430 42.1 00570 70174 71243 26 351 846304 21.4 8526 20.7 993683 42.1 006317 70195 71223 125 36 846432 21.4 852496 20.7 993936 42.1 006064 70215 71203 24 37 846560 21.3 852371 20.8 994189 42.1 0058111 70236 71182 123 38 846688 21.3 852247 20.8 994441 42.1 0055591 70257 71162 22 39 846816 21.3 852122 20.8 994694 42.1 005305 70277 71141 21 40 846944 21.3 851997 20.8 994947 42.1 005053 1 70298 11121 20 4119.847071 21.3 9.851872 20.8 9.995199 42.1 10.004801 70319 71100 19.42 847199 21.3 851747 20.8 995452 42.1 0045481 70339 11080118 21 3 20.8 42.1 ~ 43[ 847327 21.3 851622 20.8 995700 42.1 004295 1170360(7109 117 44 847454 21.31 851497 20.8 995957 42.1 004043 70781 710,91 16 45 847682 21.2 851372 20.9 996210 42.1 003i90!70401 71019 15 46 847709 21.2 851246 20.9 996463 42.1 003537 170422 70998 114 47 847836 21.2 851i21 20.9 996715 42.1 003285 170443 i0978 113 48 847964 21.2 850996 20.9 996968 42.1 003032 70463 70957 12 49 848091 21.2 850870 20.9 997221 42. 1 0027791 70484170937 11 60 848218 21.2 850745 20.9 997 42.1 0025271 70505170916 10 51 9.848345 21.2 19 850619 20.9 9.997726 42.1 10.002274 70525 70896 9 621 848472 21.2 850493 20.9 997979142.1 002021 70546 708751 8 53 848599 21.1 850368 21.0 998231| 42.1 001769 7056 7i0855! 7 54 848726 21.1 850242 21.0 998484 42.1 001516 70587 70834 6 55 848852 21.1 850116 21.0 998737 42.1 001263 7050810813 5 56 848919 21.1 849990 21.0 998989 42.1 001011 7062810i93 4 71 849106 21.1 849864 21.0 99924214,'.1 007581 70649 10772 3 58 849232 21.1 849738 21.0 99949 2.1 000505! 70670 i0752 2 91 8493569 2.1 849611 21 -0 9997481421 00253 11 70690170731 1 60 84948 21211 8494851 10.000000 42:1 0000001 70711 0;11 O Cosine. Sine. Cotang. T a Tn N. os,. 45 Degrees.![ —

Page  66 66 LOGARITHMS TABLE III. LOGARITHMS OF NUMBERS. FROM I TO 200, INCLUDING TWELVE DECIMAL PLACES. N. Log. i N. Log. N. Log. 1 -)000300 000000 f1 41 612783 856720 81 908485 018879 2 301029 995664 I 42 623249 290398 82 913813 852384 3 477121 254720 43 633468 455580 83 919078 092376 4 602059 991328 44 643452 676486 84 924279 285062 5 698970 004336 46 653212 513775 85 929418 925714 6 778151 250384 46 662757 831682 86 934498 451244 7 845098 040014 47 672097 857926 87 939519 252619 8 903089 986992 48 681241 237376 88 944482 672150 9 954242 509439 49 690193 080028 89 949390 006645 10 Same as to 1. 50 Same as to 5. 90 Same as to 9. 11 041392 685158 51 707570 176098 91 959041 392321 12 079181 246048 52 716003 343635 92 963787 827346 13 113943 352307 63 724275 869601 93 968482 948554 14 146128 035678 54 732393 759823 94 973127 853600 15 176091 259056 55 740362 689494 95 977723 605889 16 204119 982656 56 748188 027005 96 982271 233040 17 230448 921378 57 755874 855672 97 986771 734266 18 255272 503103 68 763427 993563 98 991226 075692 19 278753 600953 59 770852 011642 99 995635 194598 20 Same as to 2. 60 Same as to 6 100 Sanme as to 10, 21 322219 2947 61 785329 835011 101 004321 373783 22 342422 680822 62 792391 699498 1()2 008600 1717'62 23 361727 836018 63 799340 549453 103 012837 224705 24 380211 241712 64 806179 973984 104 017033 339299 25 391940 008672 65 812913 356643 105 021189 299070 26 414973 347971 66 819543 935542 10; 025305 865265 27 431363 764159 67 826074 802701 10 ()W19383 777685 28 447158 031342 68 832508 912706 10-. 03)3423 755487 29 462397 997899 69 838849 090737 109 037426 497941 30 Stime as to 3. 70 Same as to 7. 110 Sanie as to 11. 31 491361 693834 71 851258 348719 111 045322 978787 32 505149 978320 72 857332 496431 112 049218 022670 33 518513 939878 73 863322 860120 113 053078 443483 34 531478 917042 74 869231 719731 114 056904 851336 35 44068 044350 75 875061 263392 115 060397 840354 36 556302 500767 76 880813 592281 116 064457 989227 37 568201 724()67 77 886490 725172 1 117 068185 861746 38 579783 596617 78 892094 602690 118 071882 0073(6 39 591064 607026 79 897627 091290 119 075546 961393 40 Same as to 4. 80 Same as to 8. 120 Same as to 12. ~~~~~~~~~~~~~ I~~~~~~~~~~~~~~~~~~~~~~~

Page  67 OF NUMBERS. 67 N. Log. N. Log. N. Log 121 082785 370316 148 170261 715395 175 243038 048686 122 086359 830675 149 173186 268412 176 245512 667814 123 08990E5 111439 150 176091 59056 i 177 247973 266362 124 093421 685162 151 178976 947293 178 250420 002309 125 096910 013008 152 181843 587945 179 ~5~F853 030980 126 100370 545118 153 184t(91 430818 180 2-5:72 505103 127 103803 720956 I 154 187520 7c0836 181 257t'i 574469 128 107209 969648 1 155 190331 698170 182 2t(0),71 387';85 129 110589 710299 156ti 193124 588354 183 262451 089730 130 Same as to 13. 157 195899 652409 184 2C4n 17 8Q3010 131 117271 295656 158 198657 086954 185 f267171 728403 132 120573 931206 159 201397 124320 186 269512 944218 133 123851 640967 160 204119 982656 187 271841 606536 134 127104 798365 161 206825 876032 188'274157 849264 135 130333 768495 162 209515 014543 189 276461 804173 136 133538 908370 163 212187 604404 1 190 278753 600953 l37 136720 567156 164 214843 848048 191 281033 367248 138 139879 086401 165 217483 944214 192 283301 228704 139 143014 800254 166 220108 088040 193 285557 309008 140 146128 035678 167 222716 471148 194. 287801 729930 141 149219 112655 168 225309 281726 195 2900%4 611362 142 152288 344383 169 227886 704614 196 292256 071356 143 155336 037465 170 230448 921378 197 294466 226162 144 158362 492095 171 232996 110392 198 296665 190262 145 161368 002235 172 235528 446908 199 298853 0i6410 146 164352 855784 173 238046 103129 147 167317 334748 174 240549 248283 LOGARITHMS OF THE PRIME NU MBERS FROM 200 TO 1543, INCLUDING TWELVE DECIMAL PLACES. N. Log. N. Lg. N. Log. 201 303196 057420 2-77-1 442479 769064 379 578639 209968 203 307496 037913 281 1 448705 319905 383 583198 773968 207 315970 345457 283 451786 435524 389 589949 601326 209 320146 286111 293 466867 620354 397 598790 506763 211 324282 455298 307 487138 375477 401 603144 372620 223 348304 863048 311 492760 389027 409 611723 308007 227 356025 857193 313 495544 337546 419 622214 022966 229 359835 482340 317 501059 262218 421 624282 095836 233 367355 921026 331 519827 993776 431 6344i7 270161 239 378397 900948 337 527629 900871 433 636487 896353 241 3820'7 042575 347 540329 474791 439 642424 520242 251 399673 721481 349 542825 426959 443 646403 126223 257 409933 123331 353 F47774 705388 i 449 652246 341003 263 419955 748490 359 555094 448578 i 457 659916 200070 269 429752 280002 367 564666 064252' 461 663;'00 925390 271 492969 290874 373 571708 831809 463 665580 991018 _- -;.

Page  68 68 LOGARITHMS N Lo N. Lo. N. Log. 467 6,,.1L;',o 6 821 914345 15 1 19 11-1 L _95072 479 ti68033- 13414 823 91599 835 12 1181 02'249 807613 487 ti8i;28 96121o5 8W -2i 9155i) 60-) 5553 11i7 0744-'0 718955 491 69108, 192123 829 918554 530550 11)3 076640 443670 499 6d.old)j A4oo23 839 923761 9608&9 1201 0.9543 00735 503 7015t;7 95056 853 930949 031168 1213 0838;0 800845 509 706 17 7i2337 857 932980 821923 121 085290 578 10 621 716-.37. 3303 859 933993 163331 1223 087426 458017 623 18501 688867 863 936010 795715 1229 089551 882866 541 33h)''oo)107 877 942999 693356 1231 090258 052912 547 73i9'387 326333 881 944975 908412 1237 092369 699609 557 745855 195174 883 945960 703578 1249 096562 438356 563 75)5.)8 39451 887 947923 619832 1259 10125 729204 569 755112 1 26L.39 ) 907 957607 287030 1277 103190 896808 571 756636 168243 911 959518 376973 1279 1068 v0 542450 577 761175 813156 919 963315 511386 1283 108226 656362 587 7t8638 101248 929 958015 713994 1289 110252 917337 593 773054 693364 937 971739 590888 18291 110926 242517 599 77'7426 8'2389 9411 973589 623427 1297 112939 986066 601 778874 472092 947 6976349 979003 1301 114277 296540 607 783138 6910:5 953 9790932 900638 1303 114944 415712 613 78i460 474518 967 985426 474083 1307 116275 587564 617 790285 1640:3 9 1 987219 229908 1319 120244 795568 619 791690 649020 977 989894 563719 1321 120902 817604 631 8030'29 359244 983 992553 517832 1327 122870 922849 641 806858 029519 991 996073 654485.1361 133858 125188 643 808210 972924 997 998695 158312 1367 135768 514554 647 810904 280569 1009 003891 166237 1373 137670 537223 653 814913 181275 1013 O05309 445360 1381 140193 678544 659 818885 41.4594 1019 008174 1840)6 1399 145817 714122 661 810201 459486 1021 009025 742087 1409 148910 994096 673 828015 064224 1031 013258 665284 1423 153204 896557 677 830588 668685 1033 014100 321520 1427 154424 012366 683 834420 703682 1039 016615 547557 1429 155032 228774 691 839478 047374 1049 020775 488194 1433 156246 402184 701 845718 017967 1051 021602 716028 1439 158060 793919 709 850646 235183 1061 025715 383901 1447 160468 531109 719 853728 890383 1063 026533 264523 1451 161667 412427 727 8i1534 410859 1069 028977 705209 1453 162265 614286 733 835103 974'42 1087 036229 544086 1459 164055 291883 739 858644 488395 1091 037824 750588 1471 167612 672629 743 8 0988 8137T1 1093 038620 161950 1481 170555 058512 751 855ii39 3'7)004 10137 00206ti 627575 1483 171141 151014 757 8;-90:1)5 679500 1103 042595 512440 1487 172310 968489 761 881384 656771 | 1109 044931 546149 1489 172894 731332 769 885926 339801 1117 018053 173116 1493 174059 807708 773 888179 493918 1123 050379 756261 1499 175801 632866 787 895974.32359 1129 052693 941925 1511 179264 464329 797 901458 321396 1151 031075 323630 1523 182699 903324 809 907948 521612 1153 061829 307295 1531 184975 190807 811 909020 854211 1163 035579 714728 1543 188365 926053

Page  69 OF NUMBERS. 69. AUXILIARY LO GARITHMS, N. Log. N. l N. Log. 1.009 003891166237 1.O009 000390689248 1.008 003460532110 01.008 000347296684 1.007 003029470554 1.0007 000303899784 1.003 002598080685 1.0006 003260498547 1.005 002166031756 A 1.0005 003217092970 B 1.004 001733712775 1.0004 000173683057 1.003 001300933020 1.0003 000130268804 1.002 000867721529 1.0002 000086850211 1.001 000434077479 1.0001 000043427277 C N. Log. N. -Log. 1.00009 000039083266 1.000009 000033908628 1.00008 000034740691 1.000008 000003474338 1.00007 000030398072 1.000007 000003040047 1.00006 000026055410 1.000006 000002605756 1.00005 000021712704 1.000005 000002171464 1.00004 000017371430 1.000004 000001737173 1.00003 000013028638 1.000003 000031302880 1.00002 000008685802 1.000002 030000868587 1.00001 000004342923 1.000001 000000434294 N. Log. 1-. 00001 000000043429 (n) 1.00000001 000000004343 (o) 1.000000001 000000000434 (p) 1.0000000001 00000000043 (q) m=0.4342944819 log. -1.637784298. By the preceding tables -and the auxiliaries A, B, and C, we can find the logarithm of any number, true to at least ten decimal places. But some may prefer to use the following direct formula, which may be found in any of the standard works on algebra: Log. (z+l)=log.z+(0.8685889638(- + ) The result will be true to twelve decimal places, if z be over 2000. The log. of composite numbers can be determined by the combination of logarithms, already in the table, and the prime numbers from the formula. Thus, the number 3083 is a prime number, find its logarithm.'We first find the log. of the number 3082. By factoring, we discover that this is the product of 46 into 67.

Page  70 70 NUMBERS. Log. 46, 1.6627578316 Log. 67, 1.8260748027 Log. 30)82 3.4888326343 Log. 3383=3.4888326343+0.8685889638 6165 NUMBERS AND THEIR LOGARITHMS, OFTEN USED IN COMPUTATIONS. Circumference of a circle to dia. 1' Log. Surface of a sphere to diameter 1 -3.14159265 0.4971499 Area of a circle to radius I Area of a circle to diameter I -.7853982 -1.8950899 Capacity of a sphere to diameter 1.5235988 -1.7189986 Capacity of a sphere to radius 1 =4.1887902 0.6220886 Are of any circle equal to the radius -57~29578 1.7581226 Arc equal to radius expressed in sec. =206264"8 5.3144251 Length of a degree, (radius unity)=.01745329 -2.2418773 12 hours expressed in seconds, - 43200 4.6354837 Complement of the same, =0.00002315 -5.3645163 360 degrees expressed in seconds, = 1296000 6.1126050 A gallon of distilled water, when the temperature is 62~ Fahrenheit, and Barometer 30 inches, is 277. g:74- cubic inches. 4/277.274=16.651542 nearly. /277.274_ =18.78925284 231 - 15.198684..775398 J 282 =16.792855. 282. -_=18.948708..785398 The French Metre=3.2808992, English feet linear measure, =39.3707904 inches, the length of a pendulum vibrating seconds.

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