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ARithmetick is the art of accompting by Numbers; it is either positive or negative.
2. Positive Arithmetick, is that which is wrought by certain and infallible Numbers at first propounded; and this is either single or comparative.
3. Single, which is wrought by Numbers, considered alone, without relation to one a∣nother, and this either in whole Numbers; or in Fractions.
4. The parts of single Arithmetick, are two, Notation and Numeration.
5. Notation hath two parts; the first shew∣eth the value of the Notes, by which all num∣bers are expressed; the second sheweth how to read the Numbers which are expressed by those notes.
6. The Notes or Characters, by which all Numbers are usually expressed are these, 1. one, 2. two, 3. three, 4. four, 5. five, 6. six, 7. seven, 8. eight, 9. nine, 0. nothing.
7. These notes are either significant Fi∣gures, or a Cypher.
8. The significant Figures, are the first nine, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9. The first whereof is more particularly termed an u∣nite or unitie, the rest are said to be compo∣sed of unities; so 2, is composed of two unites 3, of three unites, &c.
9. The Cypher, though it signifie nothing of it self, yet being set before or after any of the rest, increaseth or decreaseth their va∣lue, as shall be farther shewed hereafter.
10. The second part of Notation, is the reading of the Number expressed by these notes; and this is done by distinguising the Number given into Degrees and Periods.
11. The degrees are three, the first is that first place of a number towards the right hand, and is the place of Unity. The second is the second Figure towards the right hand, and this is the place of Tens. The third is the
third Figure towards the right hand, and is the place of Hundreds; so this Character 9, doth signifie Nine; these Notes 27, Twenty seven; and these 235, Two hundred thirty five.
12. A Period, is when a number consisting of more notes than three, hath each three notes thereof (beginning at the right hand) distin∣guished by Points or Commas: The several parts of the Numbers so distinguished, are called Periods; so the Number 38156249, being distinguished into Periods, will stand thus, 38.156.249. of which the first Period is read thus, Two hundred forty nine; the first Figure in the second Period is the place of Thousands, the second Tens of Thousands, and the third Hundreds of Thousands. In the third Period, the Figure is in the place of Mil∣lions, the second Tens of Millions, and so this Number is thus to be read. Thirty Eight Millions, One Hundred Fifty Six Thousand, Two Hundred Forty Nine.
13. Numeration, is that which by certain known Numbers propounded, doth discover another Number unknown.
14. Numeration hath four Species; Additi∣on, Subtraction, Multiplication, and Division.
15. Addition, is that by which divers num∣bers are added together, to the end that the Sum or Total may be discovered. For which purpose, having placed the numbers as in the following Example, begin with those in the
Unity place first, then with these in the place of Tens then of Hundreds, and so forward, according as the Numbers given do consist of places, carrying the Tens, if there be any, to the place of the next greater rank, as here you see.
16. Subtraction is that, by which one num∣ber is taken out of another, so that the Resi∣due or remainder may be known. To per∣form this, you must rank your Numbers, and begin as in Addition; and in case any of the figures of the Number to be subtracted shall be greater than that, from whence the Sub∣traction is to be made, you must borrow one from the next place above it; as in the Ex∣amples following.
17.
18. Multiplication hath three Parts, the Multiplicand, the Multiplicator, and the Pro∣duct.
19. Multiplication, is single or compound.
20. Single Multiplication, is when the Mul∣tiplicand, and Multiplicator, do each of them consist of one only Figure; as if 9 were given to be Multiplied by 6, 9 is the Multiplicand, 6 is the Multiplicator, and 54 is the Product.
21. Compound Multiplication, is when the Multiplicator and Multiplicand do either, or both consist of more Figures than one.
22. When the Product of any of the par∣ticular Figures shall exceed ten, place the Ex∣cess under the Line, and for every ten that it so exceeds, keep in mind one to be added to the next rank: Example; 76147, being to be Multiplied by 5, the Product is 180735, and 39634 being gi∣ven 〈 math 〉〈 math 〉 to be Multiplied by 47, the work will stand as in the Mar∣gin, where the Product by 7 is 277438, and the Product thereof by 4, is 158536, and the Sum of these two Products is 1862798.
23. Division is that by which we discover how often one Number is contained in ano∣ther, that we may find out the Quotient.
24. Division hath three parts, the Dividend, the Divisor, and the Quotient; thus, if 35 were given to be Divided by 5, 35 is the Dividend, 5 the Divisor, and 7 will be found to be the Quotient.
25. In Division, make a crooked line at each end of your Dividend, that on the left hand serving for your Divisor, and that on the right for the Quotient; then see how oft your Divi∣sor is contained in the first Figure or Figures of your Dividend, and put the answer in the Quotient, then Multiply your Divisor by the Figure in the Quotient, and the Product sub∣tract from your Dividend, then draw down the next Figure of your Dividend, and ask how oft your Divisor may be found in the re∣mainer so increased, & the answer put in the Quotient, and proved as before, till there be no Figures left in your Dividend, and so oft as the Question is repeated, so many places must be in the Quotient, as in manifest by the following Example.〈 math 〉〈 math 〉
Let 1862798, be given to be divided by 47, I ask how often 47 may be had in 186? the Answer is 3, which I place in the Quotient, then I Multiply 47 by 3, the Product is 141, which being Subtracted from 186, the Re∣mainer is 45, to which draw down 2 the next Figure in the Dividend, and then it will be 452, now then I ask how often 47 may be had in 452? the which by the Table made by the continual Addition of 47 unto it self, is 9 times, therefore I place 9 in the Quotient, and the Product of 47 is 423, which being Subtracted from 452, the Remainer is 29, to which I draw 7 the next Figure, and then proceed as before, and so at last I find the Quotient to be 39634.
26. Multiplication and Division, prove one another, for if you Multiply the Quotient by the Divisor, the Product will be equal to the Dividend: so 39634, being Multiplied by 47, the Product is 1862798, and this Pro∣duct being Divided by 47, the Quotient is 39634.
SIngle Arithmetick in a whole Numbers, hath been shewed in the last Chapter; Single Arithmetick in Fractions now followeth.
2. A Fraction is a part of an Integer.
3. Single Arithmetick in Fractions, doth al∣so consist of two Parts, Notation, and Nume∣ration.
4. Notation of Fractions, is that which sheweth how the Fraction part of any Inte∣ger may be expressed in numbers; that is, an Integer on one whole thing being Divi∣ded into any Number of equal parts, Notati∣on sheweth how these parts may be expres∣sed; as if a Yard were Divided into four parts, and it were desired, that I should set down three of these parts; the usual manner is thus, draw a line, & set the Number of parts into which the Integer is supposed to be di∣vided, under the line, and the Number of parts you would express set above the line; thus to express three of four parts, I write 4 under a line, and 3 above it, thus, ¾; and so may you do with any other number pro∣pounded: Where note, that the number a∣bove the line is called the Numerator, and the number under the line the Denominator.
5. A Fraction is either Proper or Improper.
6. A Proper Fraction is that whole Nume∣rator is less than the Denominator, such as are these ¾ 6/12 25/100.
7. A Proper Fraction is either single or com∣pound.
8. A Single Fraction is that which consists of one Numerator and one Denominator, such as are ¾ 6/12 25/100.
9. A Compound Fraction (otherwise called a Fraction of a Fraction) is that which hath more Numerators and more Denominators than one, which kind of Fractions are disco∣verable by this word (of) which is interpo∣sed between their parts; as, ⅔ of ¾ is a Fra∣ction of a Fraction, or a Compound Fracti∣on, and expresseth two thirds of three fourths of an Integer.
10. The things expressed by broken num∣bers or Fractions, are principally the Parts or Fractions of Money, Weight, Measure, Time, and things accounted by the Dozen.
11. The least part or Fraction of Money used in England is a Farthing; and four Far∣things makes a Peny; twelve Pence, a Shil∣ling; and twenty Shillings, one Pound Ster∣ling.
12. The least Fraction of weight used in England, is a Grain; that is, the weight of a Grain of Wheat, well dryed and gather'd out of the middle of the Ear, whereof 32 make a peny weight, and twenty peny weight an ounce Troy, and twelve ounces a Pound; but
a peny weight being thus ascertained, it is now subdivided into twenty four Grains.
13. The weights used by Apothecaries are de∣rived from a Pound Troy, which is subdivi∣ded in this manner.
| lb | A Pound Troy, is | 12 Ounces. |
| ℥ | An Ounce, is | 8 Drams. |
| ʒ | A Dram, is | 3 Scruples. |
| ℈ | A Scruple, is | 20 Grains. |
14. Besides Troy weight, there is another kind of weight used in England, called Aver∣dupois weight, a Pound whereof is equal to fourteen Ounces, twelve peny weight Troy, the which is subdivided into 16 Ounces, each Ounce into 16 Drams, and each Dram into 4 Quarters. Of this weight 112 makes a Hundred.
15. The Measures used in England are of Capacity or Length.
16. The Measures of Capacity are Liquid or dry; Liquid Measures are according to this Table.
| One pound of Wheat Troy weight— | One Pint. |
| Two Pints | One Quart. |
| Two Quarts | One Pottle. |
| Two Pottles | One Gallon. |
| Eight Gallons | One Firkin Of Ale. |
| Nine Gallons | One Firkin Of Beer. |
| Two Firkins | One Kilderkin. |
| Two Kilderkins | One Barrel. |
| Forty two Gallons | One Tearce of wine |
| Sixty three Gallons | One Hogshead. |
| Two Hogsheads | One Pipe or But |
| Two Pipes | One Tun. |
17. Dry Measures are those in which all kind of dry substances are Meted; as Grain, Sea-coal, Salt, and the like; their Table is this that followeth.
| One Pint | Makes | One Pint. |
| Two Pints | Makes | One Quart. |
| Two Quarts | Makes | One Pottle. |
| Two Pottles | Makes | One Gallon. |
| Two Gallons | Makes | One Peck. |
| Four Pecks | Makes | 1 Bushel land measure. |
| Five Pecks | Makes | 1 Bushel water measure. |
| Eight Bushels | Makes | One Quarter. |
| Four Quarters | Makes | One Chaldron. |
| Five Quarters | Makes | One Wey. |
18. Long Measures are expressed in the Table following.
| Three Barley-corns in length | Make | One Inch. |
| Twelve Inches | Make | One Foot. |
| Three Foot | Make | One Yard. |
| Three Foot 9 Inches | Make | One Ell. |
| Six Foot | Make | One Fathom. |
| Five yards & a half | Make | One pole or pearch. |
| Forty Poles | Make | One Furlong. |
| Eight Furlongs | Make | One English Mile. |
Note that a Yard, as also an Ell is usually subdivided into four quarters, and each quarter into four Nails.
19. A Table of Time is this that followeth.
| Sixty Minutes | Make | One Hour. |
| Twenty four hours | Make | One Day natural. |
| Seven Days | Make | One Week. |
| Four Weeks | Make | 1 month of 28 days |
Fifty two Weeks, one Day, and six hours make one Year.
And these Fractions of Money, Weight, Measure, &c. are usually written under their several Denominations, instead of having their Denominators written under them thus;
| lib. | shill. | pence. | farth. |
| 23. | 19. | 08. | 3. |
And as their Notation is two-fold, so is their Numeration also, First, then I will shew you the Numeration of parts when written, as Integers, and then as vulgar fra∣ctions.
20. Numeration of parts when written, as Integers, is Accidental or Essential.
21. Accidental Numeration, otherwise cal∣led Reduction, is either descending or ascen∣ding.
22. Reduction Descending, is when a num∣ber of greater Denomination being given, it is required, to find how many of a lesser de∣nomination, are equal in value to that gi∣ven Number of the greater. And this is performed by Multiplication; as if it were required to Reduce 329 Shillings into Pence, if you Multiply 329 by 20, the number of shillings in a pound, the Product will be 6580 shillings, and 6580 shillings being mul∣tiplied by 12, the number of pence in a shil∣ling, the Product will be 78960 pence.
23. Reduction Ascending, is when a num∣ber of a lesser Denomination being given, it is required, to find how many of a greater Denomination, are equal to that given num∣ber of the lesser: And this is done by Divi∣sion; as if it were required to find how ma∣ny Pounds there were in 78960 pence; if 78960 pence be divided by 12, the number of pence in a shilling, the Quotient will be 6580 Shillings, and if 6580 shillings be divi∣ded
by 20, the number of shillings in a pound, the Quotient will be 329 Pounds, and so for any other.
24. Essential Numeration, doth consist of four Species, Addition, Subtraction, Multipli∣cation, and Division.
25. In Addition of Numbers of several Denominations, you must begin with the least first, and when the sum of any of the Denominations amounts to an Integer, add it to the next Denomination that is greater.
〈 math 〉〈 math 〉
26. In Subtraction of Numbers of several Denominations, when any of the parts of the greater Number are less than the parts of the lesser Number subscribed, Deduct the parts of the lesser Number from the parts of the greater, increased with an Integer, of the next superiour Denomination, and keeping one in mind, add to the next place of the number given to be Subtracted.
〈 math 〉〈 math 〉
27. In Multiplication of numbers of several Denominations, you must first reduce the numbers given to their least Denominations and then Multiply them as hath been shewed in whole numbers, the Product divided by the square of the parts of an Integer, reduced to the least Denomination, shall in the Quo∣tient give the Product required.
Let the Product of 17 l. 19 s. 6 d. by 5 l. 13 s. 6 d. be required. 17 l. 19 s. 6 d. being reduced to make 4314 Pence. And 5 l. 13 s. 6 d. reduced do make 1362 Pence.
The Multiplicand.〈 math 〉〈 math 〉
The Multiplicator.〈 math 〉〈 math 〉
The Product.〈 math 〉〈 math 〉
The number of pence in a pound are 240, and the square thereof is 57600, by which dividing 5875668 the Quotient; 102 lib. 00 shill. 01 peny. 3 farthings, and 4608/5760 is the Quotient sought.
28. In Division of numbers of several De∣nominations, first reduce your Divisor to its number of parts in the least Denomination, then Multiply your Dividend, by the square of the parts in an Integer reduced to the least Denomination; & if there be any parts annexed, to the Integers of the Dividend, they must be reduced to the highest Fracti∣on, that the square of the parts in an Inte∣ger reduced to its least denomination will bear, and added to the former Product, the whole being divided by your divisor redu∣ced, will give you the Quotient sought.
Let 102:00:01:3 4608/1760 be given to be divided by 5:13:6. First I reduce the divi∣sor given to its number of parts in the least denomination, and it makes 1362 pence, then I Multiply 102 the Integral part of my dividend, by 57600, the square of pence in a pound, the Product is 58752, and the Fra∣ction of my dividend 00:01:3 4608/5760 being reduced, is 468/57600, which being added to the former Product 58752, the sum is 5875668, for the dividend; which being divided by
1362, the Quotient is 4314 pence, that is 17 lib. 19 shill. 6 pence.
29. Numeration of Fractions, when written with their Numerators and Denominators, is also Accidental and Essential.
30. Accidental Numeration, otherwise called Reduction, is three-fold.
1. To Reduce one Fraction which is not already in its least terms, to a lesser denomi∣nation.
To do this, divide the numerator and de∣nominator by their greatest common mea∣sure, the two Quotients shall be one of them, a new numerator, and the other a new de∣nominator of a Fraction equal to the Fra∣ction given, and in its least terms.
Example 91/117 being given to be Reduced, the greatest common measure is 13, by which dividing 91, the Quotient is 7, for a new numerator, and dividing 117 by 13, the Quotient is 9 for a new denominator, and so 91/117 is reduced to 7/9.
The greatest common measure between two numbers is found thus; divide the grea∣ter number by the less, and your divisor by the Remainer, if there be any, your last di∣visor is the common measure sought, as in the following Example.
〈 math 〉〈 math 〉
2. To Reduce many Fractions of divers Denominations into one Denomination.
To do this, Multiply each Numerator by all the Denominators except its own, the Products shall be the new Numerators, then Multiply all the Denominators together, and the Product shall be the common Deno∣minator sought.
Example. ⅔ ⅘ 6/7 will be reduced to 70/105 84/105 90/105.
3 To Reduce any Fraction from one De∣nomination, to any other Denomination desired. And to do this Multiply the Nu∣merator given, by the Denominator requi∣red, & divide the Product by the Denomina∣tor given, the Quotient shall be the Numera∣tor desired.
Example, let it be desired to Reduce 17/20 to a Fraction, whose Denominator shall be 100, first Multiply 17 by 100, the Product is 1700 which being divided by 20, the Quotient is
85, for the new Numerator desired.
31. Essential Numeration of Fractions hath four Species, Addition, Subduction, Multiplication and Division.
32. In Addition of Fractions, the Fracti∣ons given must be first Reduced to one De∣nomination, and then add the Numerators together, so have you the Sum of the Fra∣ction, so 2/9 and 5/9 make 7/9.
33. Subtraction of Fractions is thus, if of one Denomination, deduct the less from the greater, their difference is the remainer, so 2/9 taken from 7/9 rest 5/9.
34. Multiplication of Fractions, is thus, Multiply all the Numerators together, so is their Product a new Numerator, then Mul∣tiply all the Denominators together, and their Product is a new Denominator.
Thus if 7/12 and ⅝ were to be Multiplied, the Product will be 35/96.
35. Division of Fractions is thus, Multi∣ply the Numerator of the dividend by the denominator of the divisor, the Product shall be a new Numerator; also Multiply the nu∣merator of the divisor, by the denominator of the dividend, so shall the Product be a new denominator, and this new Fraction is the quotient sought; so if 4/9 were to be di∣vided by 3/5, the Product will be 20/27.
36. When the denominator of a Fracti∣on is an Unite with Cyphers, the Fraction is more particularly called a Decimal; and
such Fractions may be expressed without their denominators as well as with them, thus, 5/10 may be written thus, 5.
37. When the Numerator doth not con∣sist of so many places, as the denominator hath Cyphers, fill up the void places of the Numerator with Cyphers, so, 5/100, 25/1000, are written thus, .05, and .025.
38. Numeration of Decimal-Fractions, is likewise two-fold, Accidental and Essential.
39. Accidental Numeration, otherwise called Reduction, is performed, by the third way of Reduction; shewed in the Twenty seventh Rule of this chapter.
40. Essential Numeration, hath in it the four usual Species, Addition, Subtraction, Multiplication, and Division.
41. Addition of Decimals is the same with Addition of whole Numbers, if a point or line be set between the Integers and the Parts, as in the following Examples.
42. Subduction of Decimals doth differ from Subduction in whole Numbers, but by a point to distinguish the whole number from the broken; as in the Example fol∣lowing.
43. Multiplication of Decimal Fractions, is the same with Multiplication in whole numbers, but when the work is finished, to distinguish the Integers from the Decimals, do thus; so many places of parts as are in both the numbers given, being separated by a point, the rest of the figures towards the left hand are Integers, and those towards the right are Decimal parts; as in these Exam∣ples. 〈 math 〉〈 math 〉
44. Division of Decimal Fractions is the same with Division in whole numbers, but when the Work is finished, to distinguish the Fractional part from the Integers, ob∣serve this general Rule.
The first figure in your quotient will be always of the same degree or place with that Figure or Cypher in your dividend, which standeth over the Unites place in your divisor.
For Example: 78925, being given to be divided by 32, the quotient will be 2466, & because the place of Unites in the divisor, doth stand under the place of seconds in the dividend, therefore the first figure in the quotient, will be in the place of seconds, and the first must be supplied with a Cypher, and then the quotient will be 0.02466.
THus much hath been said concerning Single Arithmetick, Comparative fol∣lows, which is wrought by Numbers, as they are considered to have relation to one ano∣ther.
2. This Relation consists either in quan∣tity or in quality.
3. Relation in quantity is the reference that the Numbers themselves have one to another; as when the comparison is made be∣tween 8 and 2, or 2 and 8; 7 and 3, or 3 and 7.
And here the Numbers propounded are always two, whereof the first is called the Antecedent, the other the Consequent.
4. Relation in quantity, consists either in the difference, or in the rate or reason found between the Numbers propounded; the one is found by Subtracting the less from the greater; so 6 is the difference between 8 and 2; but the other, to wit, the rate or reason, is found by dividing the greater by the less, and thus the rate between 8 and 2 is four-fold, because 2 is found four times in 8; Or the rate may be also found by di∣viding the less by the greater, or setting the Numbers given in manner of a Fraction, and thus the rate between 2 and 8 is 4 also, or 2/8 that is ¼.
5. This rate or reason of Numbers is ei∣ther equal or unequal; equal reason, is the relation that equal Numbers have one to another, as 5 to 5, 6 to 6. Unequal Rea∣son is the relation that Unequal Numbers have one to another, and this is either of the greater to the less, or of the less unto the greater.
In the one the greater Number is the An∣tecedent, and the less the Consequent; and in the other the lesser Number is the Ante∣cedent, and the greater is the Consequent.
6. Relation in quality, (otherwise called Proportion) is the reference or respect that the reasons of Numbers have one to ano∣ther,
and therefore the Numbers must be more than two, or else three cannot be the comparing of reasons in the Plural Number.
7. Proportion is two-fold, Arithmetical and Geometrical.
8. Arithmetical Proportion, is when num∣bers differ according to equal reason; that is, have equal differences; as 2, 4, 6, 8, 10, or 3, 6, 9, 12, in the first rank the common dif∣ference is 2, and in the second 3.
9. Arithmetical Proportion, is either con∣tinued, or interrupted.
10. Arithmetical Proportion continued, is when divers numbers are linked together by a continued Progression of equal differ∣ence; and in such a Progression, the sum of the first and last Terms being Multiplied by half the number of the Terms, the Product will be the sum of all the Terms; as in this Progression, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, the sum of the first and last is 13, which be∣ing Multiplied by 6, half the number of the Terms the Product is 78, the sum of all the terms in that Progression.
11. Three Numbers being given in A∣rithmetical Proportion, the mean number be∣ing doubled is equal to the sum of the Ex∣treams; so 3, 6, 9, being given, the double of 6, the mean number is equal to the sum of 3 and 9, the two Extreams.
12. Arithmetical Proportion Interrupted, is when the Progression is discontinued,
as in these numbers, 2, 4, 8.10.
13. In Arithmetical Proportion continued, or discontinued, the sum of the Means is e∣qual to the sum of the Extreams, as in 3, 6, 9, 12, being given, the sum of 6 and 9 is e∣qual to the sum of 3 and 12; or 3, 6, 12, 15, being given, the sum of 6 and 12, is e∣qual to the sum of 3 and 15.
14. Geometrical Proportion is, when di∣vers numbers differ by the like reason; as, 1, 2, 4, 8, 16, which differ one from ano∣ther by double reason; for as 1 is the half of 2, so 2 is the half of 4, 4 of 8, 8 of 16.
15. Geometrical Proportion is either con∣tinued or interrupted, Geometrical Proportion continued, is when divers numbers are lin∣ked together, by a continued Progression of the like reason; as 1, 2, 4, 8, 16, or 3, 6, 12, 24, 48.
16. In numbers Geometrically proporti∣onal, If you Multiply the last Term by the common rate by which they differ, and from the Product deduct the first Term, and di∣vide the Remainer by the former rate less by an Unite, the Quotient shall be the sum of all the Progressions; So 2, 6, 18, 54, 162, 486, 1458, being propounded the last term 1460, being multiplied by 3 the rate, the Product is 4374 out of which deducting 2 the first Term, the Remainer is 4372, which being divided by 2 the rate less one, the quotient 2186 is the sum of that Progression.
17. Three Proportionals being given, the square of the Mean is equal to the Pro∣duct of the Extreams; so 4, 8, 16, being given, the square of 8 is equal to four times 16.
18. Geometrical Proportion interrupted, is when the Progression of like reason is dis∣continued; as, 2, 4, 16, 32, where the Term between 4 and 16 is wanting, and therefore the rate between 4 and 16 is not the same that is between 2 and 4, or 16 and 32.
19. Four Proportional Numbers what∣soever being given, the Product of the two Means is equal to the Product of the two Extreams; so 2, 4, 16, 32, being propound∣ed, 4 times 16 is equal to 2 times 32, which is 64.
FRom the last Rule of the former Chap∣ter ariseth that precious Gem in Arith∣metick, the Rule of three, which for its ex∣cellency, deserves the name that is given to it, The Golden Rule.
2. The Golden Rule, is that by which cer∣tain numbers being given, another number Geometrically proportional to them may be found out.
3. The Golden Rule is either single or compound.
4. The single Rule, is when three terms or numbers are propounded, and a fourth in proportion to them is desired.
5. The Terms of the Rule of Three con∣sist of two Denominations; two of the Terms propounded have one Denominati∣on, the third propounded and fourth requi∣red, have another.
6. Of those two numbers given which are of one Denomination, that which moves the Question must possess the third place, the other number of the same Denominati∣on, must be put in the first place, and con∣sequently, the other known Term, which is of the same Denomination with the fourth required, must possess the second place.
7. The three Terms propounded being thus placed, consider whether your third doth require more or less; if it requires more, Multiply the middle number by the greater of the two Extreams, and divide the Product by the lesser, the Quotient is the fourth Number or Term desired.
But if the third Term in the Question re∣quire less, Multiply the middle Term by the lesser of the two Extreams, and the Pro∣duct
Divide by the greater, the Quotient shall be the fourth Term desired; An Exam∣ple in each Case will sufficiently explain the Rule.
If 7 Pound of Sugar cost 2 s. 7 d. What shall 28 Pound of Sugar cost? The Terms must stand thus,
| lb sugar. | s. | d. | lb sugar. |
| 7 | 2 | 7 | 28 |
Where it is plain, that 28 pound of Su∣gar must needs cost more than 7, therefore I Multiply 2 s. 7 d. or 31 pence, by 28, the Product 868 being Divided by 7, and the Quotient is 124 d. or 10 s. 4 d.
2. Example: If 7 Men will digg a Gar∣den in 31 Dayes, In how many Dayes will 28 Men digg the same Garden? Here the Terms must stand thus,
| Men. | Dayes. | Men. |
| 7 | 31 | 28. |
And by the state of the Question it plain∣ly appears, that the third Term requireth less: therefore I Multiply 31, the middle Term, by 7, the lesser Extream, and the Product 217 being Divided by 28, the Quo∣tient 7 21/28 is the fourth Term desired.
THe Compound Rule of Three, is when more than three Terms are propoun∣ded.
2. Under the Compound Rule of Three is comprehended the Double Rule of Three, and divers Rules of plural proportion.
3. The Double Rule of Three, is when five Terms are propounded, and a sixth in pro∣portion to them is required.
4. In this Rule the five Terms given do consist of two parts; first a Supposition, and then a Demand; the Supposition is expres∣sed by three of the Terms propounded, and the demand by the other two.
5. And here the greatest difficulty is in placing of the Terms; for which observe amongst the Terms of Supposition, which of them hath the same Denomination with the Term required, reserve that for the se∣cond place, and write the other two Terms in the Supposition one above another in the first place; and lastly, the Terms of De∣mand one above another, likewise in the third place, in such sort, that the upper∣most may have the same denomination with the uppermost of those in the first place.
If 6 Clerks can write 45 sheets of Paper in 5 Dayes; How many Clerks can write 300 sheets in 72 Dayes? Here the Questi∣on is concerning the number of Clerks , the 6 Clerks must therefore possess the second place, and the Dayes and Paper in the Sup∣position must be set in the first, one over the other, of which, if Paper be the uppermost in the other Terms, the Paper must be set over the Dayes in the third place, and then the Number in the Question will stand thus,
| 45 | 6 | 300 |
| 5 | 13 |
6. The Terms propounded being thus placed, the Question may be resolved by two Single Rules of Three in this manner.
1. As the uppermost Term of the first place is to the middle, so is the uppermost Term in the last place to a fourth Number.
2. As the lower Term of the first place is to that fourth Number, so is the lower Term of the last place to the Term requi∣red.
But in both these Proportions, conside∣ration must be had to the Term required, namely, whether it must be more or less than the middle Term given.
In our present Question, the fourth term in the first proportion must be greater than the second; for it is plain, that more work will require more men; therefore I say,
as 45 . 6 ∷ 300 . 40 Clerks.
But in the second proportion, it is like∣wise plain, that the more time is given, the fewer persons are required; and therefore in this proportion, 5.40.13. I multiply the middle term by the first, and the pro∣duct 200 I divide by 13, the last, and the Quotient is 15 10/13.
2. Example: If 100 l. gain 6 l. in 12 months, what shall 276 l. gain in 18 months? In this Question the terms must be thus pla∣ced.
| 100 | 6 | 276 |
| 12 | 18 |
THe Rules of plural proportion are those, by which we Resolved Questions that are discoverable by more Rules of Three than one, and cannot be performed by the double Rule of Three mentioned in the last chapter.
Of these Rules there are divers kinds and varieties, according to the nature of the Question propounded; I will only mention one, and refer the rest to my larger treatise of this Subject.
2. The Rule of plural proportion that I mean to mention, is the Rule of Fellowship.
3. And the Rule of Fellowship is that by which in Accompts amongst divers Men, (their several stocks together) the whole Loss or Gain being propounded, the Loss or Gain of each particular man may be disco∣vered.
4. The Rule of Fellowship is either single or double.
5. The Single Rule of Fellowship is, when the stocks propounded are single numbers; As in this Example: A and B were Partners in an Adventure to Sea, A put in 25. l. B 56, and upon return of the Ship, they sold the
Fraight for 50 l. profit; the question is, What part of this 50 l. is due to A, and what to B? to resolve this and the like Que∣stions, the sum of the stocks must be the first term in the Rule of Three, the whole gain the second, and each particular stock the third; this done repeating the Rule of Three, as often as there are particular stocks in the Question, the fourth term pro∣duced by these several operations are the respective Gains or Losses of those particu∣lar stocks propounded; so in the present question, the Resolution will be as here you see. 〈 math 〉〈 math 〉
6. The Double Rule of Fellowship is, when the stocks propounded are double numbers, that is, when each stock hath relation to a particular time. A, B, and C, hire a piece of Ground for 45 l. per Annum, in which A had 24 Oxen 32 days, B 12, for 48 days, C 16, for 24 days; now the question to be resolved is, What part of the Rent each person must pay?
For this purpose you must first Multiply each particular stock by its respective time, and take the total of their Products for the first term, the Gain or Loss for the second, and every man's particular stock and time
for the third; this done repeating the Rule of Three so often as there are Products of the double Numbers; the fourth terms pro∣duced upon those several operations are the numbers sought. So then in the question propounded, the Product of 24 and 32 is 768; the Product of 12 and 48 is 576, and the Product of 16 and 24 is 384, the sum of these Products is 1728, which is the first term, 45 l. the Rent is the second, and each particular Product the third; 〈 math 〉〈 math 〉
By which three Operations the question is Resolved.