# THE ART OF Natural Arithmetick.

## CHAP. I. Of Notation.

1. ARITHMETICK, is the Art of Accounting by Number.

2. There are three kinds of it. 1. Natural. 2. Symbolical. 3. Artificial.

3. Natural Arithmetick, which is the subject of this present discourse, is either Positive or Negative.

4. Positive Arithmetick, is either Single or Comparative.

5. Single is that, which is wrought by numbers considered alone, without relation to one another.

6. The parts of single Arithmetick are Notation and Numeration.

7. Notation is the writing down of any summe to [Page 2] be expressed, or the expressing of any summe set down in writing.

8. The parts of Notation are two; The first doth explain the general value of the Notes, by which numbers are to be expressed.

9. The Notes or Characters by which Numbers are ordinarily expressed, are these; 1. one. 2. two. 3. three. 4. four. 5. five. 6. six. 7. seven. 8. eight. 9. nine. 0. nothing.

10. These Notes are either significant Figures or a Cypher.

11. The significant figures are the first nine, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9. and these have a certain signification when they are put by themselves, but any of these being joyned with it self or any of the rest doth become uncertain in it's signification; for every of these figures alone doth signifie it self once, but if two of them be set together, the first doth signifie it self once, the second doth signifie it self ten times, as in 22 the same figure which in the first place towards the right hand doth signifie only two, doth in the second place signifie ten times two, that is twenty.

12. The Cypher doth indeed signifie nothing of it self, but being set before or after the rest, it doth increase or lessen their value.

13. The Value of these Notes is more fully expressed by Degrees and Periods.

14. The Degrees are three; The first, is the first place of a number towards the right hand, and doth signifie it self once, The second degree, is the second place towards the right hand, and doth signifie it self ten times. The third is the third place towards [Page 3] the right hand, and doth signifie one hundred times it self, so these figures 345 do signifie three hundred fourty five.

15. A Period if it be perfect doth consist of three degrees, if imperfect it doth consist of fewer: and it is either simple or compound. A simple period if it be perfect doth consist of three degrees; a compounded Period, may have all the Periods perfect, or the last towards the left hand may be imperfect. Thus 325 is a simple perfect period, and 325 678 is a perfect compound period, or a number consisting of two perfect periods; 23 is an imperfect simple period, and 23 456 is an imperfect compounded period.

16. The second part of Notation, teacheth how to read a number that is expressed in writing.

17. Every number is either simple or mixt.

18. A simple number is that, whose parts are of one and the same kind, that is, either whole or broken.

19. A whole number is that, which doth consist of integers or entire unities; thus 36 is composed of thirty six integers, or intire unities.

20. Integers of one simple period, are to be read, as hath been shewed in the 14 Rule: and integers which are composed of divers periods, are to be read much after the same manner, the periods being first distinguished from one another, by interposing a short stroke or point between each period, as, 15. 246. 368; whence 15 is an imperfect period, 246 and 368 are perfect periods. This or any other number thus distinguished into periods, may easily be read, by giving to every period its proper denomination.

[Page 4]21. The proper denomination of the first period is Hundreds. 2. Thousands. 3. Millions. 4▪ Thousands of Millions, & so forward increasing by a tenfold proportion as far as you please, but in ordinary practise, we seldom have occasion for the fourth period; the reading whereof will be best learned by young beginners, if they be first exercised in the reading of three or four figures, which being well understood the rest cannot be difficult; as by the Table following, if well considered, it will be manifest.

Thousands of Millions | Millions | Thousands | Hundreds | ||||||||

C | X | I | C | X | I | C | X | I | C | X | I |

9 | |||||||||||

9 | 8 | ||||||||||

9 | 8 | 7 | |||||||||

9 | 8 | 7 | 6 | ||||||||

9 | 8 | 7 | 6 | 5 | |||||||

9 | 8 | 7 | 6 | 5 | 4 | ||||||

9 | 8 | 7 | 6 | 5 | 4 | 3 | |||||

9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | ||||

9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | |||

5 | 6 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 3 | 9 |

22. According to this Table, the fourth number 9. 876 is thus to be expressed, Nine thousand, eight hundred, seventy six, and the last number 569. 876. 543. 239. thus; Five hundred sixty nine Thousands of Millions, eight hundred seventy six Millions, five hundred forty three thousand, two hundred thirty nine, and so of any other.

## CHAP. II. Of Addition.

1. HItherto we have treated of Notation, Numeration follows.

Numeration is that which by certain known numbers propounded, discovereth another number unknown.

2. Numeration hath four species, viz. Addition, Subtraction, Multiplication and Division. In which besides the nine Figures and a Cypher, I shall for the avoiding of many words, make use of these Characters. (+) (−) (=) (×)

3. This Character (+) I use to represent these words more by, and is the signe of addition or affirmation; as 3 + 5 signifies 3 more by 5, or 5 added to 3, or the summe of 3 and 5 that is 8.

4. This Character (-) denotes the words less by, and is the sign of Subtraction or Negation; As 9 − 3 doth signifie 9 less by 3, or 3 Substracted from 9, or the difference between 9 and 3 that is 6.

5. This Character (=) represents the words equal to, and is the note of an [...] ▪ so 6 + 3 = 9 are to be read thus, 6 more by 3 is equal to 9.

6. This Character (×) represents the words multiplied by, as 7 × 6 are thus to be read, 7 multiplied by 6, or the product of 7 and 6.

7. Addition is that, by which divers numbers are [Page 6] added together, to the end that the summe or total may be discovered.

8. Numbers to be added are either simple, as 1. 2. 3. and the rest which may be expressed with one note or figure; or else they are compounded, as 10. 11. 13. 234. which cannot be expressed without two or more figures.

9. The Addition of simple numbers is to be learned by practise rather than precept; as one added to one makes two, two and three make five and so of the rest; this is so well known even to children, as that it needeth no further explication, but yet for Demonstration sake, and to prevent mistakes in the adding of greater summes, I have here exhibited a short Table declaring the sum of all simple Numbers,

1 | 1 | 2 |

1 | 2 | 3 |

1 | 3 | 4 |

1 | 4 | 5 |

1 | 5 | 6 |

1 | 6 | 7 |

1 | 7 | 8 |

1 | 8 | 9 |

1 | 9 | 10 |

2 | 2 | 4 |

2 | 3 | 5 |

2 | 4 | 6 |

2 | 5 | 7 |

2 | 6 | 8 |

2 | [...] | [...] |

2 | 8 | 10 |

2 | 9 | 11 |

3 | 3 | 6 |

3 | 4 | 7 |

3 | 5 | 8 |

3 | 6 | 9 |

3 | 7 | 10 |

3 | 8 | 11 |

3 | 9 | 12 |

4 | 4 | 8 |

4 | 5 | 9 |

4 | 6 | 10 |

4 | 7 | 11 |

4 | 8 | 12 |

4 | 9 | 13 |

5 | 5 | 10 |

5 | 6 | 11 |

5 | 7 | 12 |

5 | 8 | 13 |

5 | 9 | 14 |

6 | 6 | 12 |

6 | 7 | 13 |

6 | 8 | 14 |

6 | 9 | 15 |

7 | 7 | 14 |

7 | 8 | 15 |

7 | 9 | 16 |

8 | 8 | 16 |

8 | 9 | 17 |

9 | 9 | 18 |

[Page 7]This Table doth consist of two columns, in the first are the two simple numbers to be added, and in the second, the summe or total of those numbers, as if 5 and 7 were to be added, right against 5 and 7 in the first columne, I find 12 in the second column for the summe of them.

4. The Addition of composite numbers is something more difficult, and yet it doth depend upon the addition of simple numbers. But that this may be the more commodiously done, the numbers to be added must be so set down, that the unites may stand under the unites, the tens under the tens, and the hundreds under the hundreds, &c. as if the numbers 52364 + 36512 were given to be added, you are to place them as in the Margin. [...]

5. Having thus placed the numbers draw a line under them, and then adde them together, beginning with the unites first, saying 2 + 4 = 6 which I subscribe in the place of unites under the line, and 1 + 6 = 7 which I subscribe in the place of tens, 5 + 3 = 8 which I subscribe in the place of hundreds, and so 6 + 2 = 8 and 3 + 5 = 8 which being subscribed in their respective places, the total summe is 88. 876.

6. But it frequently happens that the several summes of the particular ranks are too great to be expressed with one figure, when this happens write down the first figure of the summe in it's own proper place, and the other figure in the next place above it, and so in the next rank as oft as need shall require, as in the example following.

[Page 8]In which the summe of the figures in the unites place is 25, [...] therefore writing down 5 in the place of unites, and the figure two which may be said to signifie 20 or two decades, I set in the next place above it, that is in the place of tens.

In like manner the summe of the decades is 20, and therefore writing 0 in the rank of tens, the 2 which signifies 20 decades or 200 I set in the place of hundreds, and so the rest in their order; and adding the two last ranks together, the total summe is 2. 278. 825.

7. But this work which for plainness sake is thus set down, may be more briefly performed, if beginning with unites, the last figure of the summe be only set down, and the other kept in mind, and numbered in the next rank, in this manner, because in the rank of unites 8 + 9 = 17, 17 + 6 = 23. [...] 23 + 7 = 30. 30 + 4 = 34, subscribing 4, the other 3 being kept in memory, I presently adde to the next rank thus, 3 + 3 = 6. 6 + 8 14. 14 + 1 = 15. 15 + 6 = 21. 21 + 5 = 26, subscribing 6 I carry 2 to the next rank, saying, 2 + 6 = 8. 8 + 7 = 15. 15 + 4 = 19. 19 + 5 = 24. 24 + 7 = 31 subscribing 1 I carry 3 to the next rank 3 + 2 = 5. 5 + 3 = 8. 8 + 6 = 14. 14 + 9 = 23. and 23 + 4 = 27 subscribing 7 I [Page 9] carry to the next rank, saying, 2 + 7 = 9. 9 + 1 = 10. 10 + 5 = 15. 15 + 8 = 23. 23 + 3 = 26 subscribing 6 I carry 2 to the next rank, saying, 2 + 4 = 6. 6 + 5 = 11. 11 + 8 = 19. 19 + 7 = 26. 26 + 6 = 32 subscribing 2, the other 3 I set in the next rank, because there is nothing more to be added, and so the total summe is, 3267163.

The proof of Addition is thus, draw a line under the first rank of numbers, then adde together all the remaining ranks, as hath been shewed; Lastly, to that total adde the first rank of numbers: if that be the same with the first total found, the Addition is true, otherwise not.

As in the last example, the first rank of numbers is, 472638, and the total of the other four ranks, is, 2794526, to which the first rank being added, the total will be 3267164, as may be before found, therefore the first rank was true. The like may be done in any other.

## CHAP. III. Of Subtraction.

1. SUBTRACTION is that, by which one number is taken out of another which is greater or at least equal to the former, that the residue or remainer may be known.

2. The Subtraction of simple numbers is so easie, that every one is supposed to know that, either naturally or by daily practise; but if this also shall in any case be thought necessary, it may be readily performed by the former table, as if 2 were to be Subtracted out of 9, looking in the former table for 2 in the first column, and 9 in the second, the number between them is 7 which is the remainer sought.

3. But where one composite number is to be Subtracted from another or a simple number from a composite, the two numbers given must be written one under another, the lesser under the greater, as was directed in Addition, that is, the unites under the unites, and the tens under the tens, &c. So the numbers 678945 and 543214 being given to be substracted the one out of the [...] other, they must be placed as in the Margin; then proceeding to the subtraction, I say 5 − 4 = 1 which I place in the same rank under the line; In like manner 4 − 1 = 3 which I likewise set under the line in the next rank, and thus finishing [Page 11] the whole operation, the remainer of 678945 − 543214 is 135731. As in the Example.

4. When any of the figures of the number given to be subtracted is greater than the figure, out of which it is to be subtracted, you must borrow one of the next rank towards the left hand, and then the figure of which they are so borrowed must afterwards be esteemed an unite less, or, which is all one, the figure to be subtracted from it, must be supposed an unite greater, and the one which is borrowed may if you please for memory sake be set between two little lines. [...]

So the numbers 730582 and 51368 being given to be subtracted, the last from the former and placed as before; I am to deduct 8 out of 2 which cannot be, borrowing therefore one decade from the following figure, (which is equal to 10 unites) the 2 is made 12 and 12 − 8 = 4 which I set down under the unites, and the one which I borrowed under the tenths between the lines. And then, saying, 1 + 6 = 7 and 8 − 7 = 1 which I set down in the place of tens; and then 5 − 3 = 2 which I set in the place of hundreds, and then 1 out of 0 I cannot but borrowing one from the next rank 10 − 1 = 9 which I set in the place of thousands, and the 1 which I borrowed under the next rank between the lines, and then, saying, 1 + 5 = 6, but 6 from 3 I cannot, therefore borrowing 1 as before, it is 10 + 3 = 13, & 13 − 6 = 7; lastly 7 − 1 (which was borrowed) = 6 and so the remainer is 679214 = 730582 − 51368.

## CHAP. IV. Of Multiplication.

1. MULTIPLICATION, is that by which we multiply two numbers, by one another, that their product may be discovered.

2. Multiplication hath three parts, the Multiplicand, the Multiplicator and the Product.

3. The Multiplicand is the number given to be multiplied. The Multiplicator is the number by which the Multiplicand is multiplied, and the Product is the number produced by the Multiplication: So if 7 be given to be multipled by 4, the third number which is produced is 28, for 4 times 7 is 28; and here 7 is the Multiplicand, 4 the Multiplicator, and 28 the Product.

4. Multiplication is single or compound.

5. Single Multiplication, is when the Multiplicand and Multiplicator do each consist of single figures, and this is either to be performed by memory, or to be performed by many additions, till use and practise hath made it familiar; but for a help to memory in this particular, the Multiplication of simple numbers may be conceived by this Theoreme.

6. If one of the two numbers given shall be divided into as many parts as you please, the Product of both numbers, shall be equal to all the Products made by one of the numbers and all the parts of the other. As for the small numbers, as [Page 13] how much 2 times 4, 4 times 5, or 3 times 6, even nature it self doth teach unto children. But if thou be not so ready in the greater notes, thou mayest by this means make it familiar. As if the Product of 7 times 8 were required, divide one of them into what parts you please, as 7 into 2. 3. 2. if 8 be multiplied by these parts, the summe of the several products will be 56. And so for 8 times 9 divide 9 into 3. 3. 3▪ and multiply 8 by these parts, the summe of the products shall be 72: and so of any other. [...]

7. The memory▪ may be also helped in the multiplication of simple numbers in this manner. If two numbers be propounded, which together are more than 10, if you multiply both their differences from 10, and also deduct either difference from the other number, the product and remainer set in diverse degrees, shall be the product of the given numbers. As in these Examples.

[Page 14] [...]By either of these ways the product of single figures may be attained, but for the greater ease of young beginers, we have also added this table, in which the product of all the single figures is exprest; the numbers to be multiplied are set in the first column, and the product in the second, as 7 and 8 being given, right against those numbers is 56 which is the product of them.

2.2 | 4 |

2.3 | 6 |

2.4 | 8 |

2.5 | 10 |

2.6 | 12 |

2.7 | 14 |

2.8 | 16 |

2.9 | 18 |

3.3 | 9 |

3.4 | 12 |

3.5 | 15 |

3.6 | 18 |

3.7 | 21 |

3.8 | 24 |

3.9 | 27 |

4.4 | 16 |

4.5 | 20 |

4.6 | 24 |

4.7 | 28 |

4.8 | 32 |

4.9 | 36 |

5.5 | 25 |

5.6 | 30 |

5.7 | 35 |

5.8 | 40 |

5.9 | 45 |

6.6 | 36 |

6.7 | 42 |

6.8 | 48 |

6.9 | 54 |

7.7 | 49 |

7.8 | 56 |

7.9 | 63 |

8.8 | 64 |

8.9 | 72 |

9.9 | 81 |

[Page 15]8. The Multiplication of single figures being thus attained, the Multiplication of compounded numbers cannot be difficult.

9. In Compound Multiplication, set down the given numbers, as in Addition and Subtraction▪ so 3421 being given to be multiplied by 2, place 2 the Multiplicator under 3421 the Multiplicand, and having drawn a line, write [...] the double of every figure in the Multiplicand under the line (because the Multiplicator is 2▪) in the same degree always with that figure of which it is the double. Thus the double of one is 2, the double of 2 is 4, the double of 4 in the place of hundreds is 8, & the double of 3 in the place of thousands is 6 and the product of 3421 multiplied by 2 is 6842.

10. When the product of any of the particular figures exceeds ten, place the excess under the line as before, and the last figure in the next place to it, but a line lower, and so the rest until the multiplication be finished; as if the same number 3421 were to be multiplied by 8. every figure of the multiplicand is to be made 8 times as much as it is, and subscribed in their proper places, and then to be collected into one summe. Thus 8 times 1 is 8, which I place under the 1. and 8 times 2 is 16, I write 6 in the place of tens a line lower, and 1 in the upper line, in the place of hundreds, 8 times 4 is 32, I write 2 in the lower line in the place of hundreds, and 3 in the upper line a place forwarder, that is in the place of thousands, 8 times 3 is 24 thousands, I write 4 therefore in the place of [Page 16] thousands in the lower line, and 2 a place forwarder, in the upper or the lower line it matters not which, because it is the last [...] figure in the product. And these two lines of numbers being added together, the summe is, 27368=3421 × 8.

11. But this may be more compendiously performed, if that every figure to be set in the upper line shall be committed to memory and added to the next rank. So 3421 being given to be multiplied by 8 as before: for 8 times 1 I write 8 under the line in the place of unites: for 8 times 2 being 16 I write 6 under the line in the place of tens, and reserve 1 for the ten it exceeds to be added in the next rank, the place of hundreds. Then I say 8 times 4 is 32, unto which if I adde 1 which I kept in mind the whole is 33, wherefore subscribing 3 in the next rank under the line in the place of hundreds, and carrying 3 in mind for the three [...] tens that it exceeds, I proceed to perform the rest of the work, as you see it in the Example.

12. When the Multiplicand and the Multiplicator are both compounded numbers, for so many figures as are in the Multiplicator, so many several products must be subscribed under the line which at last being added into one summe▪ gives you the total product of all: so 3421 being given to be multiplied by 28, the operation will stand thus.

[Page 17]For 3421 being multiplied [...] by 8 the product is 27368: again 342 I being multiplied by 2 the product is 6842, which several products standing in their due order and added together do make 95788 the product required.

13. When in the Multiplicator there are Cyphers between the significant figures, you must multiply by the significant figures only, neglecting the Cyphers, and set each particular product in its due place, according to what hath been said in the last rule, and as is done in these examples following. [...]

14. When the numbers given to be multiplied do one or both of them end with Cyphers, neglecting the Cyhers, multiply by the significant figures only, and having added together the several products according to the former directions, annex as many Cyphers to the summe of all the products, as were at the end of both the numbers given, so shall you have the product desired as in the examples following. [Page 18] [...]

15. When one of the Numbers given is an unite with nothing but Cyphers annexed, the Multiplication is performed by annexing so many Cyphers to the Multiplicand, as there are in the Multiplicator; so if 4327 were to be multiplied by 1000, the product will be 4327000.

## CHAP. V. Of Division.

1. DIVISION is that, by which we discover how often one Number is conteined in another, and thereby find the Quotient of the greater Number.

2. Division hath three parts, the Dividend, the Divisor and the Quotient. The Dividend is the Number given to be divided; The Divisor is the Number by which the Dividend is divided; and the Quotient is the Number which the Division doth produce: So if 18 were given to be divided by 3, the Number produced would be 6, for 3 is found 6 times in 18, and here 18 is the Dividend, 3 the Divisor, and 6 the Quotient.

3. Division is either single or compound. When the Divisor is a single figure, and the Dividend less than 10 the answer may be given by memory; or if the Dividend be the product of two single figures, the answer may be had from the multiplication table if need be; for finding the Divisor in the first Column and the Dividend in the second, the other figure in the first Column is the Quotient sought; as if 56 were to be divided by 7, in the same line where I find 7 in the first Column, and 56 in the second, I find 8 also, which is the Quotient.

4. When the Divisor is a single figure, and the Dividend a Number of many places, make a [Page 20] crooked line at each end of the Dividend, that on the left hand serving for the place of the divisor, & that on the right, for the Quotient; then proceed gradually, & ask how often the divisor is contained in the Dividend, and place that which answers the question in the Quotient, then multiply the Divisor by that figure in the Quotient, & substract the product from the dividend, setting down the remainer. For example, let 2916 be given to be divided by 4. placing the numbers as hath been said; I ask how often 4 is conteined in 29, the answere is 7 times I therefore put 7 in the [...] Quotient, then I multiply 4 by 7 and it makes 28 which being subtracted from 29 the remainer is 1. Then drawing a stroke between the remainer and the Dividend I draw down the next figure which is 1 also, and so the new Dividend is 11, then I ask how often 4 is conteined in 11, the answer is twice, therefore I put 2 in the Quotient, and 4 multiplied by 2 makes 8 which being deducted from 11 the remainer is 3 which I write under 11 and draw a stroak between them as before, and draw down the next figure which is 6, and so the new divided is [...]. Then I ask how often 4 is conteined in 36 the answer is 9 times, therefore I place 9 in the Quotient, and 4 multiplied by 9 the product is [...], which b [...]ng deducted from [...]6 the remainer is 0▪ and therefore the division is [...] and the Quotient required, is [...]

[Page 21]5. When the Divisor is also a composite Number, though the work be somewhat more troublesome, yet is the manner of working the same with the former; & therefore the former example well considered, will make way for that which follows. Let then 6639642 be a Number given to be divided by 978, and being set down as is before directed, distinguish by a point so many of the foremost places of the Dividend towards the left hand as are either equal to the Divisor, or being greater do yet come nearest to it, which in this Example [...] must be under 9 for 6639 is the least number in this Dividend, from whence 978 can be subtracted. Now therefore I demand how often 978 is conteined in 6639▪ and since to answer this and the like question some trial must be made, write the Divisor in a place by it self, and the product thereof by all the nine figures in this manner.

First, write down the Divisor it self 978 and draw a line on the right hand thereof, and on the right hand of the line right against the Divisor write 1. then underneath the divisor write the double thereof which is 1956, and right against the said double on the other side of the line under 1. write 2. Then adde the Divisor 978 to 1956 the double thereof and their summe 2934, is the triple of the Divisor, which must be placed under the double, and 3 right against on the other side of the line; then adde the Divisor 978 to 2934 their summe 3912 must be put under the triple, and so proceed till you have nine ranks of numbers [Page 22] shewing the product of your divisor by all the 9 figures, and to prove your work, adde the ninth and last number to the first, so will their summe be ten times the Divisor, if the work be true, thus 978 + 8802=9780.

A Table exhibiting the several products of your Divisor by all the 9 digits being thus prepared; an answer to the former question may be easily found, for in that table you may see that the nearest less to 6639 is 5868 and the figure right against it on the other side the line is 6, therefore I place 6 in the quotient and the product of 978 by 6, that is 5868 I write under 6639 and subtract the said product from it, so is the [...]

978 | 1 |

1956 | 2 |

2934 | 3 |

3912 | 4 |

4890 | 5 |

5868 | 6 |

6846 | 7 |

7824 | 8 |

8802 | 9 |

remainer 771. Then deducting 6846=978×7 from 7716, and 7824=978×8 from 8704 and 8802=978×9 from 8802 the Quotient will be 6789, and the remainer nothing.

This method being practised a while, you may at length shorten the work of the table in this manner. Having doubled the Divisor, as before, and [Page 23] so got the product thereof by 2, double that double, so shall you have the product by 4, and the product of 4 being doubled will give you the product by 8, and then the table 9781195623912478248 will be as here you see it; from whence all the other products may be easily had; for the first and second is 2934 the product by 3, and 1956 + 3912=5868 the product by 6, the summe of the first and third is 4890 the product by 5, the summe of the first second and third is 6846 the product by 7, the first and fourth is 8802 the product by 9, and by using this contraction but a while, you will at last be able to divide any summe without the help of any such table, and use as few figures as in any other method that I yet know to be made publike.

6. When the Divisor hath one or more Cyphers after the significant figures, the work is to be performed by the significant figures only, cutting off so many figures from the Dividend as there are Cyphers at the end of the Divisor, and after the division is finished annex the figures so cut off to the remainer of the Dividend.

7. When the Divisor is an unite with Cyphers, the Division is performed by cutting off with a line so many figures of the Dividend towards the right hand, as there are Cyphers in the Divisor, the rest of the figures in the Dividend towards the left hand is the Quotient sought.

## CHAP. VI. Of the Notation of Fractions.

1. THe same parts of Arithmetick which have been wrought in whole numbers may also be performed in broken numbers, otherwise called parts or fractions, but first it will be necessary to shew what is meant or intended by the word fraction or fractions, and then proceed to their Notation and Numeration.

2. Fractions or broken numbers, are not so properly called numbers as fragments of unitie, f [...]r although an unite is the least number, yet this one may be supposed to be divided, sometimes into more sometimes into fewer equal parts as the thing shall require.

3. And as the equal parts of an unite may be sometimes more, sometimes fewer, so have those parts sometimes one Denomination sometimes another; so that if an unite be supposed to be divided into two equal parts, one of those parts is called half, if into 3, one of those parts is called a third, if into 4, a fourth, if into six a sixth, if into eight, an eighth, if into ten, a tenth, and so of others. And in writing of fractions, this Denomination is expressed, by writing such a figure under a little line as is proper to them. And the number of parts supposed to be taken are expressed by another figure above the line, thus a 2 above the line and a 3 under it do express two thirds. A 3 set above the line and a 4 under it, do express three fourths, and so of any other.

[Page 25]4. A fraction then is to be written with two numbers▪ of which the lowermost is called the Denominator, because it doth denominate the parts, and limits or determines how many parts are to be numbred, and the upper number, is called the Numerator, because this doth number the parts, or shew how many of those parts are to be taken.

5. A number of parts then being propounded to be set down in writing, as the one half of a yard, or three fourths of a foot, they are to be written thus ½ ¾ in which 1. 3. are the numerators, and 2. 4. the denominators.

6. A broken number is either proper or improper.

7. A proper broken number, is that, whose numerator is less than the Denominator: such as are these ¾ 6/ [...]2 5/ [...].

8. A proper broken number is either single or compound.

9. A single broken number is that which consists of one numerator, and one Denominator: such as are ¾ 6/ [...]1 5/1 [...] and the like.

10. A compound broken number (otherwise called a fraction of a fraction) is that which hath more numerators and more Denominators than one, which kind of broken numbers are commonly set down with the particle of between them, as five twelfths of one third is to be written thus 5/12 of 1/ [...], three fourths of seven eighths thus ¾ of 7/ [...] and so of any other.

11. The things to be expressed by broken numbers are chiefly the parts or fractions of money, weight, measure, time, motion, and things accounted by the Dozen. Of the three first of [Page 26] these there are infinite kinds and varieties according to the Laws and Customs of particular Countries; those used in this kingdom are most proper for us to know, and the knowledge of them will be sufficient to direct us in the use of those in forreign nations, if at any time there be an occasion for them.

12. The several pieces of Coine or Denominations of money used in England in reference to account, are pounds, shillings, pence and farthings, whose particular values are as followeth.

- 4. Farthings make 1. Penny.
- 12. Pence make 1. Shilling.
- 20. Shillings make 1. pound Sterling.

And according to these values, a pound Sterling is esteemed an integer, and may be divided into 20 parts called shillings, and therefore one shilling is a broken number of a pound Sterling, & by the former directions is to be written thus 1/ [...] l. that is one twentieth of a pound. Again a shilling may be divided into twelve equal parts called pence, and so one penny is a fraction of a shilling, and is to be written thus [...]/12 s. that is one twelfth of a shilling. Lastly, a penny may be divided into four equal parts called farthings, and so one farthing is a fraction of a penny, and written thus ¼ d. that is one fourth of a penny, or thus ¼ of 1/12 s. that is one fourth of one twelfth of a shilling, or thus ¼ of 1/12 of 1/20 l. that is one fourth of one twelfth of one twentieth of a pound Sterling.

13. Now then though the true and natural [Page 28] way of expressing broken numbers is by their Numerators and Denominators, as hath been shewed, yet the broken numbers or known parts of money, weights, measures and such like, are for more convenient operation, commonly expressed like integers: so that if 12 shillings seven pence half peny farthing were to be expressed in figures, the ordinary & most usual way is thus, 12s−07d, 03 f. but the said twelve shillings seven pence half peny farthing, being distinctly considered as fractions of a pound Sterling, the way to write them properly is thus,

12 Shillings, are twelve twentieths of a pound Sterling, and written thus, 12/2 [...] l.

7 Pence are seven twelfths of one twentieth of a pound Sterling, and written thus 7/12 of 1/20 l.

3 Farthings are three fourths of one twelfth of one twentieth of a pound and written thus ¾ of 1/12 of 1/20 l.

14. The weights used in England are of two sorts, Troy weight and Averdupois.

15. The several pieces or Denominations of Troy weight, are pounds, ounces, peny weights and grains, whose particular values are as followeth.

- 24 Grains make 1 Peny weight.
- 20 Peny weight make 1 Ounce
- 12 Ounces make 1 Pound Troy.

16. The weights used by Apothecaries are derived from a pound Troy, the which is subdivided as in the following Table.

- [Page 24]lb A pound Troy is equal to 12 Ounces.
- ℥ An Ounce is equal to 8 Drams.
- ʒ A Dram is equal to 3 Scruples.
- ℈ A Scruple is equal to 20 Grains.

17. But besides Troy weight, there is another kind of weight used in England call'd Averdupois weight, a pound whereof is equal unto 14 ounces, and twelve-peny weight Troy.

18. This Averdupois weight is either great or small.

19. The great Averdupois weight is, when an hundred consisting of 112 pounds Averdupois is the integer, and subdivided into halves and quarters, each quarter conte [...]ning 28 pounds.

20. The small Averdupois weight is, when a pound is the integer, each pound being subdivided into 16 ounces, each ounce into 16 drams and each dram into 4 quarters, and because many persons have occasion to use both, and are perhaps furnished but with one, I have here exhibited a Table for the speedy converting of the parts of a pound Troy, into the parts of a pound Averdupois, and the Contrary.

Drams Averdupois. | Peny-weights Troy. | Grains. | Decimals of a grain. | Ounces Averdupois. | Ounces Troy. | Peny-weights. | Grains. | ||

1 | 01 | 03 | 375 | 1 | 00 | 18 | 06 | ||

2 | 2 | [...]6 | [...]50 | 2 | 01 | 16 | 12 | ||

3 | 3 | 10 | 125 | 3 | 2 | 14 | 18 | ||

4 | Dram Averdupois | 4 | 13 | 500 | 4 | 3 | 13 | ||

5 | 5 | 16 | 875 | 5 | 4 | 11 | 06 | ||

6 | 6 | 20 | 250 | 6 | 5 | 9 | 12 | ||

7 | 7 | 23 | 625 | 7 | 6 | 7 | 18 | ||

8 | weight is equal to. | 9 | 03 | 000 | 8 | 7 | 06 | ||

9 | 10 | 06 | 375 | 9 | 8 | 04 | 06 | ||

10 | 11 | 19 | 750 | 10 | 9 | 02 | 12 | ||

11 | 12 | 03 | 125 | 11 | 10 | 00 | 18 | ||

12 | 13 | 16 | 500 | 12 | 10 | 19 | 00 | ||

13 | 14 | 19 | 875 | 13 | 11 | 17 | 06 | ||

14 | 15 | 23 | 250 | 14 | 12 | 15 | 12 | ||

15 | 17 | 02 | 625 | 15 | 13 | 13 | 18 | ||

16 | 18 | 06 | 000 | 16 | 14 | 12 | 00 |

21. The measures used in England are of two sorts, Capacity or Length.

22. The measures of Capacity are produced from weight, and are also of two sorts liquid or dry.

23. The liquid measures are those, in which all kind of liquid substances are measured, and are expressed in the table following.

- [Page 30]1 Pound of wheat Troy weight make 1 Pint.
- 2 Pints make 1 Quart.
- 2 Quarts make 1 Pottle
- 2 Pottles make 1 Gallon.
- 8 Gallons make 1 Firkin of Ale, Sope, Herring.
- 9 Gallons make 1 Firkin of Beer.
- 10½ Gallons make 1 Firking of Salmon or Eels.
- 2 Firkins make 1 Kilderkin.
- 2 Kilderkins make 1 Barrel.
- 42 Gallons make 1 Tierce of Wine.
- 63 Gallons make 1 Hogshead.
- 2 Hogsheads make 1 Pipe or But.
- 2 Pipes or Buts make 1 Tau of wine.

24. Dry measures are those in which all kind of dry substances are measured, as grain, Sea-coal, Salt and such like, and are expressed in the table following.

- 1 Pint make 1 Pint.
- 2 Pints make 1 Quart.
- 2 Quarts make 1 Pottle.
- 2 Pottles make 1 Gallon.
- 2 Gallons make 1 Peck.
- 4 Pecks make 1 Bushel land measure.
- 5 Pecks make 1 Bushel water measure.
- 8 Bushels make 1 Quarter.
- 4 Quarters make 1 Chalder.
- 5 Quarters make 1 Wey.

[Page 31]25. Long measures are as followeth.

- 3 Barley Corns in length make 1 Inch.
- 12 Inches make 1 Foot.
- 3 Foot make 1 Yard.
- 3 Foot 9 Inches make 1 Ell.
- 6 Foot make 1 Fathom.
- 5 Yards and ½ make 1 Pole or Perch.
- 40 Poles make 1 Furlong.
- 8 Furlongs make 1 English Mile.

26. In superficial or square measure 40 square poles or perches make 1 Rood or quarter of an Acre, and 4 Roods an Acre.

27. A Table of Time, is this that followeth.

- 1 Minute make 1 Minute.
- 60 Minutes make 1 Hour.
- 24 Hours make 1 Day.
- 7 Days make 1 Week.
- 4 Weeks make 1 Moneth of 28 Dayes.
- 13 Moneths 1 Day and 6 hours make 1 Year, not exactly, but very near.

28. A year is that space of time in which the Sun doth finish course through the circle in the heavens called the Zodiack, which is in 365 days, 5 hours, 4 min.

[Page 32]29. The Zodiack by is Astronomers divided, or supposed to be, into twelve equal parts called signs, whose names and Characters are these, Aries, ♈. Taurus, ♉. Gemini, ♊. Cancer, ♋. Leo, ♌. Virgo, ♍. Libra, ♎. Scorpio, ♏. Sagittarius, ♐. Capricornus, ♑. Aquarius, ♒. Pisces, ♓. And each of these signs into 30 parts called Degrees, so that this and all other Circles are supposed to be divided into 12 times 30 parts or Degrees, that is into 360, each degree into 60 minutes▪ each minute into 60 seconds, each second into 60 thirds, &c.

30. Of things accounted by the Dozen, A gross is the Integer consisting of 12 dozen, and each dozen of twelve particulars,

31. An improper fraction or broken number is that, whose numeration is greater than the Denominator. As 54/12 feet, that is, fifty and four twelfths of a foot; and this may be well called an improper fraction, seeing it will not admit of the definition of a true broken number▪ because it is greater than that whole, whereas a fraction is properly but a part of the whole.

32. A mixt number▪ is that, which besides the integers or intire unities of which it consists, hath also a broken number annexed: As in this improper fraction 54/12 if you divide the numerator 54 by the denominator 12 it will be reduced into the mixt number 4 6/12 of which 4 is the whole part, and 6/12 the broken number or fraction.

33. And this I hope is sufficient to shew what is meant by a fraction, and how all fractions whether proper or improper▪ are to be expressed [Page 33] and read, which is the Notation of them; the next thing propounded concerning fractions is their Numeration, whether such fractions be expressed by their true and natural way, that is by their Numerators and Denominators, or whether they be expressed like integers, as the known parts of money are expressed by pounds, shillings, pence and farthings, and the known parts of Troy weight by pounds, ounces, pennyweights and grains; of these and the like broken numbers which are expressed like integers we speak of first.

## CHAP. VII. Of the Numeration of such broken numbers as are expressed like integers.

1. NUmeration of such as are expressed like integers, is twofold.

2. Accidental and Essential.

3. Accidental Numeration is otherwise called Reduction.

4. Reduction is either descending or ascending.

5. Reduction descending is when a number of a greater Denomination being given, it is required to find how many of a lesser Denomination are equal in value to that given number of the greater: as when it is required to find how many shillings are contained in 34 pounds; or how many [Page 34] pence in 325 shillings, or how many hours in 365 daies.

6. Reduction descending is performed by Multiplication, for if the given number of integers of a greater denomination, be multiplied by the number of integers, contained in the next inferiour denomination, the product shall shew how many of that inferiour denomination, are contained in the integers of the greater denomination given; for example, let 34 pounds be the greater denomination given, and let it be required to shew how many shillings are in 34 pounds, shillings being the next inferiour denomination unto pounds, and that every pound doth contein 20 shillings, as hath been shewed, if you multiply 34 by 20 the product is 680; the number of shillings required. [...] In like manner 680 shillings will be reduced into 8160 pence, if you multiply 680 by 12 the number of pence in a shilling; and 8160 pence will be reduced into farthings 32640, if you multiply 8160 by 4 the number of farthings in a penny.

The like method is to be observed in weights and measures or any thing else that is or may be subdivided, into inferiour denominations; thus, [Page 35] 26 pound Troy will [...] be reduced into 312 ounces, and 312 ounounces into 6240 penny weights, and 6240 penny weights into 149760 grains, as by the operation in the margin it doth appear.

And in this manner may any number of a greater denomination given, be reduced into the least denomination, into which the greater is supposed to be subdivided.

Reduction ascending is, when a number of a lesser denomination being given, it is required to find how many of a greater denomination are equal in value to that given number of the lesser; as when it is required to find how many pence are conteined 32640 farthings, or how many shillings in 8160 pence, or how many daies in 8760 hours.

And this kind of Reduction, called Reduction ascending, is performed by Division; for if the number of integers given be divided by the number of integers in the next superiour or greater denomination, the quotient shall be the number of integers sought; so 32640 farthings being divided by 4, the number of farthings in a penny, [Page 36] the quotient is 8160 the number of pence conteined in 32640 farthings: In like manner if 8160 pence be divided by 12 [...] the number of pence in a shilling, the quotient will be 680, the number of shillings in 8160 pence; and lastly, if 680 shillings be divided by 20 the number of shillings in a pound, the quotient will be 34, the number of pounds in 680 shillings; the like may be done by any other integers of any known denomination given.

## CHAP. VIII. Of the Addition of numbers that are of diverse Denominations.

1. ACcidental Numeration of such broken numbers as are expressed like integers, hath been shewed; that which is essential now followeth.

2. Essential Numeration doth consist of Addition, Subduction, Multiplication and Division.

3. When the numbers propounded to be added are of diverse denominations, you must begin with the least denomination first, and set down their sum under that inferiour denomination, if their sum be fewer than the number of parts in the next greater denomination; but if their sum be more than the number of parts in the next greater denomination, set down the excess; if equal set down a Cypher; and the rest must be added to the next superiour denomination, and for memory sake it may be set down under that denomination to which it is to be added; as in the example following.

[Page 38] lsdf241305307190401608072211 49151In which I begin with the farthings first, and say 3 + 2 = 5 that is one penny and a farthing, wherefore setting 1 down under the denomination of farthings I carry a penny to the denomination of pence, and say 1 + 7 = 8, and 8 + 4 = 12, and 12 + 5 = 17 pence, that is 1 shilling and 5 pence, wherefore I set down 5 in the lowest rank under the denomination of pence and 1 in the line above it under the denomination of shillings, and say 1 + 8 = 9, and 9 + 19 = 28, and 28 + 13 = 41, that is 2 pounds 1 shilling, wherefore I set [...] in the lowest line under the denomination of shillings and 2 in the line above under the denomination of pounds, and say 2 + 16 = 18, and 18 + 7 = 25, and 25 + 24 = 49, which being set down in the lower line under the denomination of pounds, the total of the three sums propounded is 49 pounds 1 shilling 5 pence and 1 farthing.

More examples of this rule are these following.

lib. | oun. | p. w. | Gr. |

23 | 10 | 17 | 19 |

15 | 111 | 08 | 03 |

09 | 07 | 18 | 23 |

04 | 03 | 13 | 12 |

21 | 09 | 18 | 05 |

75 | 07 | 19 | 14 |

51 | 08 | 18 | 19 |

75 | 07 | 16 | 14 |

sig. | deg. | min. | sec. |

11 | 23 | 45 | 16 |

11 | 15 | 55 | 42 |

9 | 03 | 37 | 28 |

10 | 17 | 24 | 36 |

8 | 29 | 59 | 47 |

7 | 12 | 28 | 33 |

11 | 13 | 11 | 22 |

11 | 19 | 26 | 06 |

11 | 13 | 11 | 22 |

lib. | oun. | dr. | f. |

15 | 13 | 12 | 02 |

23 | 11 | 14 | 03 |

10 | 15 | 07 | 01 |

12 | 08 | 15 | 02 |

00 | 14 | 11 | 01 |

64 | 00 | 13 | 01 |

48 | 03 | 00 | 03 |

64 | 00 | 13 | 01 |

sig. | deg. | min. | sec. |

11 | 19 | 21 | 34 |

07 | 18 | 14 | 21 |

09 | 00 | 45 | 36 |

10 | 23 | 53 | 47 |

8 | 28 | 59 | 55 |

7 | 27 | 46 | 32 |

07 | 29 | 01 | 45 |

08 | 09 | 40 | 11 |

07 | 29 | 01 | 45 |

## CHAP. IX. Of the Subtraction of numbers which are of diverse denominations.

1. WHen the numbers propounded to be subtracted are of diverse denominations, you must begin with the least denomination first, and when the number to be subtracted is greater than the number of that denomination from whence it is to be subtracted you must borrow from the next denomination that is greater, and you may if you will set down the number of the parts that it doth contein of the next inferiour denomination, directly over the number from whence subtraction is to be made, setting down the remainer in the lowest line under that denomination, and the one that you borrowed under the next denomination a line higher to be added thereto again; as for example.

20124403051217083111 10382I cannot substract 3 farthings from one farthing, therefore I borrow one from the next greater denomination that is 1 penny out of 8 pence, and because there are four farthings in a penny, I set down the number 4 over the next lesser denomination, that is the [Page 41] denomination of farthings, and then 1 + 4 = 5, and 5 − 3 = 2 which I set in the lower line under the denomination of farthings; and the one that I borrowed, I set in the line above it under the denomination of pence, and say 1 + 8 = 9, and 9 out of 5 cannot be, therefore I borrow 1 shilling from the next rank and set down the number of pence conteined in a shilling, that is 12 over the rank of pence, and say 5 + 12 = 17 and 17 − 9 = 8 which I set down in the lower rank under the denomination of pence, and the 1 shilling that I borrowed in the line above it under the denomination of shillings; then I say 1 + 17 = 18, and 18 out of 3 I cannot take, therefore I borrow 1 pound from the next rank, and set down the number of shillings conteined in one pound, that is 20 over the rank of shillings and say 3 + 20 = 23 and 23 − 18 = 5 which I set down in the lower line under the denomination of shillings, and the I pound that I borrowed in the line above it under the denomination of pounds, and say 1 + 2 = 3, and 4 − 3 = 1, and so 2l. 17 8 3 being deducted from 4l. 03 05 01 the remaner is 0 [...]l. 05 [...]8 2.

2. Another way, when the smaller denominations in the sum to be subtracted, are greater than the like denominations in the sum from whence the Subtraction is to be made, borrow one from the greatest denomination in the question and set down the number of parts less one, that one of the greatest denomination doth contein of the next less over that lesser denomination, and so orderly till come to the least denomination [Page 42] that the greatest can be subdivided in▪ to, and there set down the full number of parts and make your Subtraction as that, so shall you have the remainer sought; so in the former example, because the lesser denominations of the number to be subtracted are greater than the like denominations lib.s.d.f. 19114403051217083105082 in the number from whence the subtraction is to be made I borrow 1 from the greatest denomination in the question, which is the denomination of pounds, and because one pound doth contein 20 shillings, I set 19 over the rank of shillings, 11 over the rank of pence, and 4 over the rank of farthings, and then subtracting the lowest line from the two lines of numbers above it the remainer is 1l. 05 08 2.

More examples of this rule are these following.

lib. | oun. | p.w. | Gr. |

75 | 07 | 16 | 14 |

23 | 10 | 17 | 19 |

51 | 08 | 18 | 19 |

75 | 07 | 16 | 14 |

sig. | deg. | min. | sec. |

11 | 13 | 11 | 22 |

11 | 23 | 45 | 16 |

11 | 19 | 26 | 06 |

11 | 13 | 11 | 22 |

lib. | oun. | dr. | [...]f. |

64 | 00 | 13 | 01 |

15 | 13 | 12 | 02 |

48 | 03 | 00 | 03 |

64 | 00 | 13 | 01 |

sig. | deg. | min. | sec. |

7 | 29 | 01 | 45 |

11 | 19 | 21 | 34 |

8 | 09 | 40 | 11 |

7 | 29 | 01 | 45 |

## CHAP. X. Of the Multiplication of numbers which are of diverse Denominations.

1. WHen a number of diverse Denominations is given to be multiplied by a number of but one denomination, you must begin first with the least denomination, and so by degrees ascend to the greatest, adding still to the greater denomination the integers of the same denomination, which are produced by the multiplication of the lesser, for example, if a man spend 14 pound 13 shillings 8 pence in one day, what will he spend at that rate in seven daies? The answer is seven times as much, and l.s.d141308 744 1021508 therefore the expence of one day must be multiplied by 7 in this manner, 8 × 7 = 56, but because 12 d. = 1 s. 56 d. = 4 s. 48 d. and therefore I write 8 d. in the lowermost line right under the denomination of pence, and the 4 in the line above it under the denomination of shillings. Then 13 × 7 = 91 to which the 4 being added the sum is 95, but because 20 shillings = 1 pound, therefore 95 shillings = 4 pound 15 shillings, and so I [Page 45] write 15 shillings in the lowermost line under the rank of shillings, and 4 in the line above it under the rank of pounds. Lastly 14 × 7 = 98 pounds, to which the 4 pounds being added, the sum is 102. which being under written as before, the expence of the whole week will be found to be 102l. 15 s. 8 d.

2. when a number of diverse denominations is given to be multiplied by a number of one denomination, but greater than can be expressed with one figure, it will be convenient, that the number of diverse denominations, be reduced into a number of the least denomination given, before you begin your multiplication; and after the multiplication is finished, to reduce the product into the former denominations, if need be; for example, if 283 men were to receive 28 pound 13 shillings 9 pence a man, how much money must be paid to the 283 men? Before I begin the multiplication I must reduce the 28 pound 13 shillings 9 pence into pence, which according to the rules of reduction given will be 6885 pence. And then 6885 × 283 = 1948455 as by the work in the margin; and so many pence [...] are required to pay off the 283 men, and 1948455 pence being reduced into the former denominations, they will be found equal to 8118 pound 11 shillings and 3 pence.

3. When a numbe [...] of diverse denominations is given to be multiplied by another of diverse [Page 46] denominations also, but denominations of the same kind, it will be convenient to reduce both the numbers into numbers of the least denomination given, before you begin your multiplication, and after▪ your multiplication is finished to reduce the product into the former denominations given; for example, let it be required to multiply 2 pound 9 shilling and 6 pence by it self: Reduction being made according to the former directions, the number of pence in each number will be 594, and ▪594 pence being multiplied by 594 pence, the product will be 352836 square pence, which being divided by 57600 the number of square pence in a pound, the quotient will be 6 pound 2 shillings 6 pence, and [...]/ [...] of a farthing, as by the operation it doth appear. [...]

[Page 47]4. Another way in the annexed Diagram, let there be two numbers of three denominations given, and let A F be the square or rect angle made of the greatest denomination in both numbers, E K and B G two rect angles made by multiplying the first denomination by the second, the product being divided by an integer of the greatest denomination reduced into the parts of the second, the quotient shall be of the same denomination with the greatest, and the remainer of the same denomination with the second. F L is the square of the second denomination, which being divided by an integer of the first or greatest reduced into the parts of the second, the quotient shall be of the same denomination with the second; and if there be any remainer it must be multiplied by a number which in the third denomination is equal to an integer in the second, the quotient shall▪ be of the third denomination, and if there be yet a remainer it must be multiplied by a number which in the fourth denomination is equal to an integer in the thid, and divided as before, the quotient shall be of the fourth denomination, and so forward till the remainer cannot be reduced into lesser terms. And thus you have done with the square or rect angle A C LH.

CR and H Q are two rect angles made by the multiplication of the summe of the greatest denomination given, by the summe given which is of the third inferiour denomination, the product shall be of the same denomination with the third, and therefore if that product be greater [Page 48] than an integer of the second denomination, reduced into the parts of the third, it must be divided by a number which in the third denomination is equal to an integer in the second, the quotient shall be of the second denomination, and the remainer shall be of the third. G M and K O are two rect angles made by multiplying the summe of the second denomination by the third,

and the product being divided by one of the integers in the greatest denomination reduced into the parts of the second, the quotient shall be of the same denomination with the third, and the remainer must be multiplied by a number which in the fourth denomination is equal to an integer [Page 49] in the third, the quotient shall be that fourth denomination, and the remainer shall be the numerator of a fraction, whose denominator is that former divisor. Lastly L P is the square of the third denomination which must be divided if it may be by one integer of the greatest denomination, reduced into the parts of the third, the quotient shall be of a fourth inferiour denomination, and the remainer shall be the numerator of a fraction, whose denominator is the same divisor.

From this Diagram thus explained, the Multiplication of pounds, shillings and pence, by pounds, shillings and pence will be plain and easie, for,

1. Pounds multiplied by pounds produce pounds.

2. Pounds multiplied by shillings and the product divided by 20, the quotient will be pounds, and the remainer shillings.

3. Shillings multiplied by shillings the product will be the numerator of a fraction, whose denominator is 400 the square of shillings in a pound, the value of which fraction may be found by multiplying the numerator first by 20, and dividing the product by 400, the quotient shall be shillings, and the remainer the fraction of a shilling, which being multiplied by 12 and the product divided by 400 the quotient shall be pence, and the remainer the fraction of a penny, which being multiplied by 4 and the product divided by 400 the quotient shall be farthings, & the remainer shall be the fraction of a farthing. Example, if 19 shillings were given to be multiplied by 19 shillings the product will be 361 for the numerator of a fraction whose denominator is 400, [Page 50] which fraction being multiplied and divided according to the former directions the value thereof will be 18 shillings 0 pence 2 farthings 4/10. Set the work thus. [...]

Or thus, shillings multiplied by shillings and the product divided by 20, the quotient will be shilings, and the remainer the fraction of a shilling, as the former product 361 being divided by 20 the quotient is 18 shillings, as before, and the remainer is 1/20 which being reduced is 0 d. 2 f. & 4/10.

4. Pounds multiplied by pence the product is the numerator of a fraction, whose denominator is 240 the number of pence in a pound; thus 2 pounds being multiplied by 11 pence the product is 22 for the numerator to 240; now then if you multiply 22 by 20 and divide the product by 240, the quotient will be 1200/ [...]40 shillings and this also being multiplied by 12 and the product divided by 240 the quotient will be 10 pence.

Or thus, pounds multiplied by pence and the product divided by 12 the quotient will be shillings [Page 51] and the remainer pence. Thus re-divided by 12 give 1 shilling 10 pence as before.

5. Shillings multiplied by pence the product will be the numerator of a fraction, whose denominator is 240 the number of pence in a pound, and this numerator being multiplyed by 12 and the product divided by 240, the quotient will be pence, and the remainer the fraction of a penny, which being multiplied by 4 and the product divided by 240, the quotient will be farthings and the remainer the fraction of a farthing. Example, 19 shillings being multiplied by 11 pence the product is 209, which being multiplied by 12 the product is 2508, and this product being divided by 240 the quotient is 10 pence and the remainer 108, which being multiplied by 4 the product is 432 and this product being also divided by 240 the quotient is one farthing and 19 [...]/240 or ⅘

Or thus, shillings multiplied by pence, and the product divided by 20 the quotient will be pence and the remainer the fraction of a penny, thus 209 being divided by 20 the quotient is 10 pence and one farthing and 16/20 or ⅘ as before.

6. Pence multiplied by pence the product will be the numerator of a fraction whose denominator must be 57600 the number of square pence in a pound; which numerator being multiplied by 20, by 12 and by 4 continually, and the last product being divided by 57600, the quotient will be farthings, and the remainer the fraction of a farthing. Example, l [...]t 11 pence be given to be multiplied by 11 pence the product is 121, which [Page 52] being multiplied by 20 the product is 2420, and 2420 being multiplied by 12 the product is 29040, and this also being multiplied by 4 the product is 116160, and this last product being divided by 57600 the quotient is 2 farthings 57 [...]60/6 [...]0, or 1/60. Or thus, 121 being multiplied by 4 the product will be 484, which being divided by 240 the number of pence in a pound, the quotient will be two farthings 4/24 [...] or 1/6 [...] as before. For illustration of these 6 rules which are deduced from the preceeding Diagram, let 2 pounds 19 shillings and 11 pence be given to be multiplied by 2 pound 19 shillings and 11 pence, set the numbers as here you see.

lib. | s. | d. | ||

2 | 19 | 11 | ||

2 | 19 | 11 | ||

4 | 00 | 00 | ||

1 | 18 | 00 | ||

1 | 18 | 00 | 2 | 4 |

0 | 18 | 00 | 0 | 0 |

0 | 01 | 10 | 0 | 0 |

0 | 01 | 10 | 1 | 8 |

00 | 10 | 1 | 8 | |

00 | 10 | 2 | 1667 | |

8 | 19 | 6 | 0 | 1667 |

Where two pounds multiplied by two pounds makes 4 pounds by the first rule, and 2 pounds multiplied by 19 makes 1 l. 185. by the second rule, which must be set down twice, and 19 shillings [Page 53] multiplied by 19 shillings do make 18 0 2 4/10 by the third rule, then 2 pounds multiplied by 11 pence do make 1 shilling 10 pence by the fourth rule, and this must be set down twice, 19 shillings multiplied by 11 pence, do make 10 pence farthing 8/10 by the fifth rule, and this must be set down twice; lastly 11 pence multiplied by 11 pence do make two farthings, 16667 parts of a farthing by the sixth rule, and so the whole product is 8 19 06 0 1667.

### 2 Example in board measure.

Let the length of a board be 14 feet 6 inches and 3 quarters of an inch and let the breadth be 2 foot 9 inches and 3 quarters, and let the content of that board be required. According to the first way you must reduce the length and breadth into the least denomination, so is the the length of this board 699 quarters of an inch, and the breadth 135; and 699 being multiplied by 135 the product is 94365.

And in a [...]oot of board there are 144 inches or 2304 quarters, by which if you divide 94365 the quotient will be 40 feet and 2 [...]05/23 [...] parts of a foot; the numerator of which fraction being multiplied by 12 the product is 2646 [...], and this product being divided by 2304 the quotient is 11 inches and the remainer i [...] 1 [...]16 which multiplied by 4 the product is 4464 which being also divided by 2 [...]01 the quotient is 1 quarter and the remainer is 2 [...]60/ [...] or 1 [...]/1 [...] of a quarter, and so the content [Page 54] of that board is 40 feet 11 inches and ¼ and 15/1 [...] of a quarter.

#### The Second Way.

If you multiply 14 by 2 the product is 28 feet by the first rule. Secondly 14 feet by 9 inches makes 126 inches, and 2 feet by 6 inches makes 12 inches more, in all 138, which being divided by 1 [...], the quotient is 11 feet 6 inches. Thirdly, 6 inches by 9 inches makes 54, which being divided by 12 the quotient is 4 inches and a half. Fourthly, 14 feet by 3 quarters of an inch makes 42, and 2 feet by 3 quarters makes 6 more, in all 48, which being divided by 12 the quotient is 4 quarters or one foot. Fifthly, 6 inches by 3 quarters make 18, and 9 inches by 3 quarters make 27, in all 45, which being divided by 12 the quotient is 3 quarters and 9/1 [...] or ¾ of a quarter or [...]. Sixthly, 3 quarters by 3 quarters makes 9/12, which is the numerator of a fraction whose denominator is 48 the number of quarters in a foot, and [...] is equal to [...]/1 [...] or 1 [...]5/10000 and so the content of this board is 40 feet 11 inches 1 quarter and [...]/ [...] of a quarter, as here you see; for 9/1 [...] more [...]/1 [...] being reduced as shall be shewed in the doctrine of vulgar fractions, are equal to 15/1 [...]. f.w.q.280000110600000402▪010000000003 [...]/11000000 [...]/ [...]

### 3 Example.

Let there be given in Astronomical fractions 23 degrees, 14 minutes, 32 seconds, to be multiplied by 17 degrees, 37 minutes, 25 seconds. The antient Astronomers have divided the Circles in which the Sun and other Planets do move into 12 equal parts and called them signs; each sign into 30 degrees, so that they have divided the whole circle into 360 degrees, each degree into 60 minutes, and each minute into 60 seconds, and each second into 60 thirds, and so forward as far as you please. And hence seconds being multiplied by seconds, produce fourths, seconds by minutes, thirds, and seconds by degrees, minutes; only you remember that if the products shall at any time exceed 60, so many sixties as it doth exceed must be transferred to the product of the next greater denomination, as by the operation it doth appear.

Sig. | Deg. | Min. | Sec. | Thir. | Fourt. |

23 | 14 | 32 | |||

17 | 37 | 25 | |||

115 | 70 | 160 | |||

460 | 280 | 64 | |||

161 | 98 | 224 | |||

690 | 42 [...] | 96 | |||

161 | 98 | 224 | |||

230 | 140 | 320 | |||

391 | 1089 | 1637 | 1534 | 800 | |

18 | 27 | 15 | 13 | ||

4 [...]9 | 1116 | 1652 | 1547 | ||

13 | 19 | 36 | 32 | 47 | 20 |

In which first you have the Indices or exponents of the several denominations, and in the two next [...]e, the Multiplicand and Multiplicator, and beginning with the least denominations first, I multipl [...] ▪ 3 [...] seconds by 25 seconds, and the product I set it under the denomination of fourths, because seconds multiplied by seconds produce fourths, then I multiply 14 minutes by 25 seconds, and the product I set under the denomination of thirds, because minutes multiplied by seconds produ [...] thirds: thirdly I multiply [...] degrees by 25 minutes, and I set their product under the denomination of seconds, because [Page 57] degrees multiplied by seconds, produce seconds; and this must be also done with the second and third numbers in the multiplicator. Then adde together all the products under every particular denomination by themselves, so will the product under the denomination of fourths, be 800, which being divided by 60 the quotient is 13, and the remainer 20, therefore I set down 20 under the denomination of fourths, and carry 13 to the next rank, in which the summe of the product is 1534, to which the 13 being added, it makes 1547 which being divided by 60 the quotient is 15, and the remainer 47, therefore I set down 47 under the denomination of thirds, and carry 15 to the next rank, till I have finished the whole work, and so the product at last is 13 signs, 19 degrees, 36 mintes, 32 seconds, 47 thirds, and 2 [...] fourths.

## CHAP. XI. Of the Division of numbers which are of diverse denominations.

1. WHen a number of diverse Denominations is given to be divided by a number of but one Denomination, you must begin with the greatest Denomination and divide the number given, as hath been shewed in the Division of whole numbers, and when you have pro [...]ded [...]o [...]ar in your Division, that the Dividend is become less than the [Page 58] Divisor, multiply the remaining part of your Dividend by the number of parts in the next inferiour Denomination, and to the product add the parts of that Denomination given, if there be any in the summe propounded, and then proceed as before; so shall you at last produce the whole quotient sought. Example, let it be required to divide 102 pounds 15 shillings 8 pence, into 7 equal parts or among seven men, you must place the numbers as hath been shewed before, then ask how often 7 in 10, the answer is once, therefore I write 1 in the quotient, and deducting 7 out of 10 the remainer is 3, to which I draw down 2, the next figure in the dividend, and it maketh 32, then I ask how often 7 is contained in 32, the answer is 4, which I set in the quotient, then I multiply 7 by 4, the figure last placed in the quotient, and the product 28 being subtracted from 32, the remainder is 4, which being less than 7, I multiply 4 by 20 the number of parts [Page 59] [...] in the next inferiour Denomination, and the product is 80, to which I adde the 15 shillings in the question, and then the summe is 95, which [Page 60] being divided by seven, the quotient is 13, and the remainer 4, which being multiplied by 12, the number of parts in a shilling, the product is 48 pence, to which I adde the 8 pence in the question, and then the sum is 56, which being divided by 7, the quotient is 8, and nothing remains, and so the whole quotient is 14l. 15s. and 8d.

In like manner 8118l. 11s. 3d. being given to be divided by 283, the quotient will be 28l. 13s▪ 9d. as by the following operation doth appear. [...]

[Page 61]3. When a number of diverse Denominations is given to be divided by a number of diverse denominations also, it will be convenient to reduce both the numbers into numbers of the least denomination given, before you begin your Division, and after Division is finished to reduce the quotient into the several denominations in which the number was given.

For example, let it be required to divide 6l. 2s. 6d. 3/1, by 2l. 9s. 6d. To reduce this Dividend into square pence, the ordinary way of Reduction will not serve, for this Dividend is the quotient of a certain number of square pence, divided by 57600 the number of square pence in a pound sterling, and therefore to make this summe given fit for Division, it must be multiplied by 57600▪ and to do this it will be proper to begin with the least denomination first, which in this example is a fraction, and a fraction reduced into its least terms, so that the first work will be to find the greater fraction, to which the fraction given is aequivalent, and this is easily done, for 57600 is the denominator; now then, as 5 to 3, so 57600 to 34560 the numerator sought; and this being divided by 4 the number of farthings in a penny, gives me the last remainer, before this 34560, viz. 8640: now pence being the next greater denomination, I multiply 57600 by 6, the number of pence in the Dividend, the product is 345600, to which adding 8640, the number before found, the total is 354240 which being divided by 12, the number o [...] pence in a shilling, the quotient is 2 [...]520, and [...], the number of shillings in the [Page 62] summe given, is 2, I multiply 57600 by 2, the product is 115200 to which 29520 being added, the total is 144720 which must be divided by 20, the quotient will be 7236, and so we have done with the shillings and pence in the question; lastly I multiply 57600 by 6 the number of pounds propounded, and the product is 345600, to which 7236 being added, the total is 352836, and so at length the number given is fitted and prepared for Division; the Divisor according to the ordinary way of Reduction, will be found to be 594, by which dividing 352836 the quotient will be 594 also, as by the work appeareth. And this [...] quotient being reduced is 2l. [...]s. 6d. the quotient sought.

4. This question and all others of the like nature may be also resolved without reducing both numbers at the first, in this manner. First set down the indices of the several Denominations given, and the Dividend under those indices, and the Divisor under the Dividend, then ask how often the greatest Denomination in the Divisor is conteined [Page 63] in the greatest Denomination in the Dividend, and put the answer in the quotient, then multiply all the Denominations in the Divisor, by the figure in the quotient, and the product being deducted from the Dividend, set down the remainer, and so proceed to the next, and work in like manner, so shall you at last have the quotient desired.

Example, let 6 pound 2 shillings 6 pence ⅗ be given to be divided by 2 pounds 9 shillings 6 pence. The numbers being set as here you see, I ask how often 2 is conteined in 6, and the answer is 3 times, but 2 pound 9 shillings cannot be had but twice in 6 pound 2 shillings, therefore I put 2l. in the quotient, and multiply the Divisor 2l. 9s. 6d. by 2. and the product 4l. 19s. being deducted from 6l. 2s. the remainer is 1l. 3s. 6d and so have I wrought once; then I ask how often 2. is conteined in 1l. 3s. that is in 23s. the answer is 11,

l. | s. | d. | [...]. | ten. | |

6 | 2 | 6 | 00 | 6 | |

2 | 9 | 6 | |||

4 | 19 | 0 | 2 lib▪ | ||

1 | 3 | 6 | 00 | 6 | |

1 | 2 | 3 | 01 | 2 | 9 shil. |

1 | 2 | 3 | 4 | ||

1 | 2 | 3 | 4 | 6 d. |

but I can take but 9s. because of the following figures, therefore I put 9 shillings in the quotient, [Page 64] and multiply 2l. 9s. 6d. by 9. and the product is 1l. 2s. 3d. 1 farthing and 2 tenths, which being deducted from 1l. 3s. 6d. [...] far. and 6 tenths, the remainer is 1 shilling, 2 pence, 3 farthings, and 4 tenths, and so I have wrought twice. Then I ask how often 2. in 1s. 2d. and the answer is 6, therefore I put 6 pence in the quotient, and multiply 2 pounds 9 shillings and 6 pence by 6, and the product is 1 shilling, 2 pence, 3 farthings, and 4 tenths, which being deducted from 1 shilling 2 pence 3 farthings and 4 tenths, the remainer is nothing, and so the quotient is 2l. 9 shillings and 6 pence.

5. In Astronomical fractions, let it be required to divide 13 signs, 19 degrees, 36 minutes, 32 seconds, 47 thirds, and 20 fourths, by 17 degrees, 37 minutes, 25 seconds; first set down the indices or several Denominations of Astronomical fractions given, and under them write the number of the parts belonging to each Denomination, as they are propounded in the Dividend, and set your Divisor either under the Dividend as here you see, or in what other place you please.

S. | D. | M. | S. | T. | F. | |

13 | 19 | 36 | 32 | 47 | 20 | Dividend |

17 | 37 | 25 | Divisor | |||

13 | 15 | 20 | 35 | 23 deg. Quotient | ||

0 | 04 | 15 | 57 | 47 | ||

4 | 06 | 43 | 50 | 14 min. | ||

00 | 09 | 13 | 57 | 20 | ||

00 | 09 | 13 | 57 | 20 | 32 sec. |

Then beginning at the left hand, ask how oft the number of the greatest denomination in the Divisor, can be had in number of the greatest denomination in the Dividend; but if your Divisor be greater than the first number in your Dividend, place your Divisor one step farther, towards your right hand. Thus in our example, because 17 is not conteined in 13, the greatest denomination in the Dividend, therefore I s [...]t it a place forwarder, towards the right hand, that is under 19 deg. and then reducing the 13 signs into deg. by multiplying them by 30, the product is 390, to which the 19 being added, the summe is 409, in which 17 is conteined 23 times, therefore multiplying the Divisor 17 deg. 37 min. 25 sec. by 23, the product is 13 [...]ig. 15 deg. 20 min. 35 sec. which being deducted from the Dividend, the remainer is 04 deg. 15 min. 57 sec. and to this remainer, I draw down the number of parts in the next lesser Denomination, and then I ask how oft 17 is conteined in 4 deg. 14 min. and the answer is 14 times, which I place in the Quotient, [Page 66] and multiplying the Divisor 17 deg. 37 min. 25 sec. by 14, the product is 04 deg. 06 min. 43 sec. 50 thirds, which being deducted from the Dividend, as before, the remainer will be 9 min. 13 sec. 57 thirds, and unto this remainer I draw down the number of parts in the next lesser denomination, and then proceeding as before, I ask how oft 17 in the Divisor is conteined in 9 min. 13 sec. in the Dividend, and the answer is 32, which I set in the Quotient, and thereby also I multiply the Divisor as before, and the product is 9 min. 13 sec. 57 thirds, 20 fourths, there remaineth nothing, and so at last, I find the Quotient to be 23 deg. 14 min. 32 sec.

## CHAP. XII. Of the Reduction of Uulgar Fractions.

1. HItherto hath been shewed the Numeration of such broken numbers or fractions as are many times expressed like integers: Numeration of such broken numbers, or fractions as are expressed by their numerators and denominators now followeth, and these broken numbers, are commonly called natural or vulgar fractions.

2. The numeration of vulgar fractions is either accidental or essential.

3. Accidental numeration, otherwise called Reduction, is that which changeth or reduceth the [Page 67] terms of the parts given, without changing or altering of their value.

4. Reduction or accidental numeration is either of parts to parts, or integers to parts.

5. Reduction of parts to parts, is that which findeth other parts, that are proportional to the parts given, and this three waies.

- 1. By reducing the parts given into their least terms, if they be not in their least terms already.
- 2. By reducing fractions of unequal denominations into fractions of the same value, whose denominator shall be equal.
- 3. By reducing fractions of diverse denominations into fractions of any denomination that shall be desired.

6. Reduction of parts or fractions into their least terms, is performed by finding out a common measure to both parts of the fraction given, that is by finding out the greatest number, which will measure or divide each of the numbers given, without leaving any remainer.

7. The greatest common measure of two numbers given may be found by dividing the greater by the less, and the divisor by the remainer, and so continually untill nothing doth remain, the last divisor is the common measure sought. Example, let it be required to find the greatest common measure unto 63 and 14, set your numbers as hath been directed in Division of whole numbers, thus. [Page 68] [...]

Then I ask how often 14 in 63, the answer is 4 times, therefore I put 4 in the quotiont, and multiply 14 by 4, and the product 56 I set under the Dividend 63, and substracting 56 from 63 the remainer is 7, then I divide 14 by 7 the quotient is 2, and the remainer 0, therefore 7 the last divisor is the common measure sought.

8. The common measure of any two numbers which do constitute a fraction, being thus found, the said fraction may be reduced into its least terms, by dividing the numerator and denominator by the common measure, for the quotients will be the numerator & denominator of the fraction, and in the least terms. So if the fraction 14/ [...] be given to be reduced into the least terms, the common measure being 7, I divide 14 by 7, and the quotient is 2 for a new numerator, and dividing 63 by 7, the quotient is 9 for a new denominator, and so the fraction 14/63 is reduced into its least terms that is into the fraction [...]/ [...]; but if the greatest common measure unto the numerator, and the denominator given shall be 1, such fraction is in its least terms already; so the fraction 13/34 cannot be reduced into lower terms, because the greatest common▪ measure is 1. This is a general rule for the reduction of fractions into their least terms; [Page 69] but yet even this may be sometimes shortened, for when the numerator and denominator are eaven numbers they may be measured or divided by 2. As if 16 were the numerator, and 64 the denominator, draw a line at length between them to separate the numerator from the denominator, and cross the same with a down right stroke near the fraction as here you see in the margin. Then take the half of 16 which [...] is 8 for a new numerator, and the half of 64 which is 32, for a new denominator, again the half of 8 is 4, and the half of 32 is 16, and so proceeding in like manner there will be found ¼ equivalent to 16/64. The like abbreviation will be found, when the numerator and denominator do both end with 5, or one of them ending with 5, and the other with a cypher, for then the fraction may be reduced into its least terms, by dividing both parts of the fraction given, by 5. Thus if it were required to reduce the fraction 125/345 dividing 125 by 5, the quotient will be 5, and dividing 345 by 5, the quotient will be 19, and so the fraction 135/345 is reduced to the fraction 5/19.

9. Reduction of fractions to the same denomination is performed by multiplying the numerator of each fraction into all the denominators continually except its own, for so the several products which arise by such continual multiplication, shall be the new numerators; and the product found by multiplying all the denominators continually, shall be the common denominator to all those new numerators. As in the [Page 70] following examples ¾ and 5/6 I multiply 3 by 6, and they make 18, and 5 by 4, and they make 20, for the two numerators, then I multiply 4 by 6 and the product is 24 for the common denominator, and so the fractions ¾ and ⅚ are reduced into the fractions 18/24 and 20/24, or being reduced into their least terms 9/12 and 10/12, as by the work appears.

Thus if the fractions ⅖ 3/7 and 4/9 [...] were propounded, here I multiply 2 by 7 and 9 continually, and they make 126, also I multiply 3 by 5 and 9, and the product is 135, in like manner I mulmultiply 4 by 5 and 7 and the product is 140, and these three products are three new numerators, lastly I multiply 5 7 and 9 continually, and the product 315 is the common denominator to all the numerators. [...]

10. But if the fractions propounded, be compounded fractions, or fractions of fractions, they must be reduced into single fractions, by multiplying all the numerators continually for a new numerator, and all the denomniators continually for a new denominator. Thus if 2/3 of ¾ of 4/5 were given, the product of the numerators 234 is 24; and the product of the denominators 34 and 5 is 60, and so the fractions ⅔ of ¾ of ⅘ are reduced to [...]4/ [...] or [...]/ [...]

[Page 71]11. The Reduction of integers to parts is either of whole or mixt numbers.

12. A whole number is reduced into an improper fraction by placing the whole number given as a numerator, and one as a denominator, thus twelve integers will be reduced into the improper fraction 12/1

13. A mixt number may be reduced into an improper fraction, by multiplying the whole part of the mixt number, by the denominator of the fraction given, and adding to the product the numerator of [...] s [...]id mixt number thus 47/12 is reduced into the improper fraction 55/12, for 4 being multiplied by 12 the product is 48, to which if you add 7▪ the summe is 55 for a new numerator, which being placed over the denominator 12, gives the improper fraction 55/12 equal to the mixt number 47/12

14. Fractions of diverse denominations may be reduced into fractions of any denomination desired by multiplying the numerator given by the denominator required, and dividing the product by the denominator given, for so the quotient shall be the numerator required.

Example, let the fraction given be ⅞ of a pound sterling, and let it be desired to know what that is in the 15/12 of a pound, multiply 7 by 15, the product is 105, which being divided by 8 the Quotient is 1331/1 [...] that is 13 shillings and 13 groats, that is 17 shillings and 4 pence and 1/ [...] of a pound that is 16 pence, which being also divided by 8 the quotient is 2 pence, which being added to 17 s. 4 d. the whole is 17 s. 6 d. and so the fraction▪ [Page 72] is reduced to the equivalent fraction 105/15 or which is all one to the improper fraction 13 1/15 as was propounded. And by this rule may any other fraction given, be reduced to another equivalent fraction, that shall have any denomination assigned, and is of excellent use for the converting of vulgar fractions into decimals, and the contrary, as shall be farther shewed in its proper place.

## CHAP. XIII. Of the Addition and Subtraction of vulgar fractions.

1. REduction of fractions I call accidental numeration, yet is the knowledge thereof very essential, for that being well understood, the essential numeration of vulgar fractions cannot be difficult; whose parts as in whole numbers are these four, addition, subtraction, multiplication and division.

2. Vulgar fractions whether proper or improper, being reduced to fractions of the same denomination, if need be, may be added like whole numbers, still keeping the same denominator, so 3/9 and 2/9 being given to be added, their summe is 5/9.

3. Vulgar fractions whether proper or improper being reduced into fractions of the same denomination, if need be, may be subtracted one from another like whole numbers, keeping still [Page 73] the same denominator. Thus if the fraction 2/9 were to be subtracted from the fraction 5/9 the difference will be 3/9.

## CHAP. XIV. Of the Multiplication and Division of vulgar fractions.

1. WHen Vulgar fractions whether proper or improper are given to be multiplied, (being reduced to single fractions, if need require) multiply the numerator of the one, by the numerator of the other, so is the product a new numerator, also multiply the denominator of the one, by the denominator of the other, and so shall the product be a new denominator, and this new fraction is the product sought. Example.

Let the fractions 5/7 and ¾ be given to be multiplied, if you multiply the numerator 5 by the numerator 3, the product is 15 for a new numerator, also if you multiply the denominator 7 by the denominator 4, the product is 28, for a new denominator, and so the fraction 15/2 [...] is the product of the fractions given.

2. When vulgar fractions whether proper or improper be given to be divided, they must be reduced into single fractions if need require, then multiply the denominator of that fraction w^{ch} is the divisor by the numerator of that fraction which is [Page 74] the dividend, so is the product a new numerator, also multiply the numerator of that fraction which is the divisor, by the denominator of that fraction which is the dividend, so is the product a new denominator, and this new fraction is the quotient sought. Example.

Let the fraction 2/ [...] be given to be divided by the fraction ¾, the quotient will be 8/9, for multiplying 4 by 2, the product is 8 for a new numerator, also multiplying 3 by 3 the product is 9 for a new denominator, and the fraction 8/9 the quotient sought; and the work standeth thus [...]/ [...]) 2/ [...] (8/9.

2. But if the fraction ¾ of ⅗ of ½ be given to be divided by the fraction ⅔ of 4/7 these compounded fractions must be reduced into single fractions by the tenth rule of the twelfth Chapter, and then divided as before; thus the fraction ¾ of ⅗ of ½ will be reduced to 9/40 and the compounded fraction ⅔ of 4/ [...] will be reduced to 8/ [...] and the fraction 9/ [...] being divided by the fraction 8/21 the quotient will be 189/320 as by the work appeareth▪ 8/ [...]1) 9/40 (189/320

## CHAP. XV. Of the Notation of Decimal fractions.

1. THe Notation of fractions in the general hath been shewed in the sixth Chapter, and these which for their excellent use are esteemed, as it were another part of Arithmetick, and from their denominator called Decimals, are in reality no other than vulgar fractions, and expressed like them, as 5/10 25/100 125/1000 but have also this property belonging to them, which vulgar fractions in the general have not; namely, that they may be as well expressed without their denominators as with them▪ by setting only a point before them; thus [...]5 doth signifie five tenths, as fully, and as plainly, as the writing of it thus 5/10.

2. When the Numerator hath not so many places as the Denominator hath Cyphers, supply those places by setting so many cyphers as there are places wanting, before the numerator given, thus 25/1000 may be as well expressed thus .025.

3. Decimal fractions are so like unto integers, that the parts of single Arithmetick in whole numbers being well understood, the same parts of Arithmetick in Decimal fractions, may easily be conceived; the greatest difference is in the reading of the value of their notes and periods; which is directly contrary unto that of whole numbers, for in the one, to wit, in whole numbers, we reckon [Page 76] unites, tens, hundreds, thousands and so forward from the right hand to the left, but in Decimal fractions we reckon tenths of unity, hundreds of unity, thousands from unity, and so forward from the left hand to the right, as doth very clearly appear by the following Table.

9 | 6 | 3 | 4 | 7 | 6 | 5 | 8 | 3 |

Fifth place | Fourth place | Third place | Second place | First place | First place | Second place | Third place | Fourth place |

In this Table you may observe that the places of Decimal parts are separated from the place of unity in whole numbers by a line drawn between them, so that the number on the left hand of that line expresseth 9 6 3 4 7 integers or unities; and the number on the right hand of the line 6 5 8 3 expresseth the parts of an unite, or an integer divided into 10000 equal parts, in like manner this number 7 6 doth signifie 7 integers, and eight tenth parts of an integer, and this number 3 4 7 6 5 doth signifie 347 integers or unities, and 65 decimal parts of an integer.

## CHAP. XVI. Of the Reduction of vulgar fractions into decimal fractions.

1. HAving shewed the notation of Decimal fractions, I come now unto their numeration, the which as in vulgar fractions, is accidental or essential.

2. Accidental numeration, is the Reduction of vulgar fractions into Decimals, and this may be performed by the fourteenth rule of the twelfth Chapter, where is shewed how fractions of diverse denominations, may be reduced into fractions of any denomination desired; namely by multiplying the numerator given, by the denominator required, and dividing the product by the denominator given; now then to apply this rule to decimal fractions, let it be required to reduce this vulgar fraction 5/ [...] of a pound into a fraction, whose denominator shall be 1000; multiply 5 the numerator given by 1000 the denominator desired, and the product is 5000, which being divided by 8 the denominator given, the quotient is 625 the numerator answering to 1000 the denominator desired, and so the vulgar fraction [...]/ [...] is reduced into the decimal fraction 625/1000.

3. Upon this ground the known or accustomary parts of money, weight, measure, time, motion, &c. may be reduced unto decimals, and first for money, if you desire to know what decimal fraction is equal to one shilling▪ or 1/20 of a [Page 78] pound, assigning the denominator of that decimal to be 100, the numerator thereto by the last rule will be found to be 5 so that 5/100 or .05 is a decimal fraction equal in value, to one shilling or 1/20 part of a pound sterling.

In like manner if the Decimal answering to 9/ [...] of 1/ [...] of a pound were required, that is the decimal of 9 pence, the compounded fraction 9/ [...] of 1/ [...]0 being reduced to a single fraction is 9/240▪ now then if you assign the denominator of that decimal to be 1000, the numerator thereto by the last rule, will be found to be 375, so that 375/10000 or .0375 is a decimal fraction equal in value to 9 pence or 9/240 of a pound sterling.

Or if the decimal answering to ¾ of 1/12 of 1/20 of a pound were required, that is the decimal of 3 farthings, the compounded fraction ¾ of 1/12 of 1/20 being reduced into a single fraction is 3/960; now then if you assign the denominator of that decimal to be 10.000.000 the numerator thereto by the last rule will be found to be 31250, so that 31250/1000000 or .0031250 is a decimal fraction equal in value to [...]/960 of a pound sterling. And hence it appears, that to find the decimal answering to any number of shillings, you must multiply the shillings propounded by the denominator 10000 &c. and divide the product by 20, the number of shillings in a pound sterling.

And to find the decimal answering to any number of pence, you must multiply the pence propounded by the denominator 10000 &c. and divide the product by 240 the number of pence in a pound sterling.

[Page 79]And to find the decimal answering to any number of farthings, you must multiply the number of farthings propounded by the denominator 100000, and divide the product by 960, the number of farthings in a pound sterling.

The decimal answering to any known parts of money, weight, measure, &c. may be otherwise found in this manner; suppose a pound sterling to be divided into 1.0000000 equal parts, then doth .1.0000000 represent one integer, or 20 shillings, and the half thereof 0.50000000 is the decimal of 10 shillings & 0.500000 is the decimal of one shilling, and the half thereof 0.0250000 is the decimal of six pence, and the half thereof is the decimal of 3 pence, to wit, 0.01250000, and a third of 0.01250000 to wit 0.00416667 is the decimal of a penny, the half of the decimal of a penny viz. 0.00208333 is the decimal of half one penny, and the half of that, to wit, 0.00104166 is the decimal of a farthing, and by either of these waies may a Table be made to shew the decimal of any number, of shillings, pence and farthings.

5. But if any dislike this, because the decimals of some parts of a pound cannot be exactly found (though it may be thus found so infinitely near that all questions may be resolved without considerable, though not without sensible errour) yet to satisfie the curiosity of such as shall make this objection, let them make the third part of a pound to be the integer, that is six shillings and eight pence, so may they have the decimal of any number of shillings, pence and farthings exactly, as by the following operation doth appear.

- Let 6s. 8d. be divided into
- 1.000000
- Then shall the decimal of 8d. be
- 0.100000
- The decimal of 4d. shall be
- 0.050000
- The decimal of 2d.
- 0.025000
- The decimal of 1 penny
- 0.012500
- The decimal of 2 farthings
- 0.006250
- The decimal of 1 farthing
- 0.003125

6. The like may be done for weights and measures, but as for weights there will be no great advantage by turning them into decimals, because all questions concerning them may be more conveniently resolved without them; and as for measures, especially long measures, there needs no reduction of them, for the foot, yard or chain for measuring of land, being divided first into ten equal parts, and then each of those into ten other equal parts, they will be more useful, than as they are now usually divided.

7. And in Astronomical fractions, the Reverend, Learned, and the famous Oughtred in the 1 Chap. of his key to the Mathematicks, doth tell us, that this Decimal Logistica or accounting is much easier and nearer than that Sexagenary commonly used▪ which he plainly perceived, saith he, whatsoever he was, that first reduced the Canon of Sines, from a Semidiameter of 60 parts, unto 1 with circles or cyphers annexed, and wisheth that the same were also done in all other Astronomical Canons; how such fractions therefore may be converted into decimals, the following examples will sufficiently declare.

[...] as much as 60 minutes make a degree, 60 [Page 81] seconds a minute, 60 thirds a second, and so forward as far as you please; if you divide the minutes given with cyphers, annexed by 60, the quotient shall be the decimal sought; thus if the decimal of 45 minutes were required, I divide 450000 &c. by 60, and the quotient is 75 for the decimal of 45 minutes, if the decimal of seconds be required, your divisor must be 3600, if the decimal of thirds be required, your divisor must be 216000, and 12960000 for fourths.

But if there be diverse Sexagesimal Species joined to integers, suppose 127 deg. 32 min. 00 sec. 09 thirds, 45 fourths, you may use this compendium; under the integers 127 set the Sexagesimal species in an oblique descent, as here you see in the margin, then beginning at the lowest, divide every of them continually 127 d. 32 m. 00 s. 09 t. 45 f. by 6, and setting the quotients over their heads join them to the row next above until you come to the integers, so shall the last quotient be the Decimal sought, Oughtred's Clavis, Chap. 6.

Thus 45 fourths being divided by 6 the quotient is 7, and the next dividend 09.75 which being also divided by 6 the quotient is 1625, and the next dividend 00.1625 [...] which being also divided by 6, the quotient is 002708333 and the next dividend 32.002708333 fourths which being divided by 6 the quotient 5333784722 is the decimal sought.

[Page 82]8. When a decimal fraction is given, and the value thereof in the known parts of the Integer is required, multiply the decimal propounded, by the number of known parts of the next inferior denomination, which are equal to the Integers, and from that product cut off as many figures towards the right, as the Decimal given doth consist of places, the figures remaining are the value of that fraction in the next inferior denomination; and the value of the remaining part of the decimal given in the next inferior denomination may be found by the same rule, and so forward till you come to the last known parts.

Example, let the value of this decimal .98563, be required in the known parts of a pound sterling, this decimal 98563 being multiplyed by 20, that product is 19) 71260, from which I cut off five figures, the rest are 19 shillings, and the figures cut off namely 71,260, being multiplied by 12 the product is 8.55120, that is 8 [...] pence and 55120 parts of a shilling, which being multiplyed by 4, the product is 2,220480 that is 2 farthings, and the remainer 20480 are the decimal parts of a farthing, and so the value of the decimal fraction given is 19 shillings 8 pence ½ penny.

But the value of a decimal fraction in the known parts of a pound, may be known by inspection, for the first figure doubled, gives the number of shillings the two next figures are the farthings, thus in the decimal given 98563 the figure 9 being doubled is [Page 83] 18 shillings, and 85 farthings that is 21 pence and a farthing; but more exactly thus, from 85, deduct 75 which is the decimal of 18 pence the remainer is 10 farthings or 2 pence half penny; that is in all 19−8d, 2 farthings as before.

If the decimal fraction given whose value is inquired be the decimal of the degree for a circle, you must divide the decimal given by six continually, and write the products underneath in every row cutting off one place towards the right hand, so shall you have the value desired, let the decimal fraction given be 127.5333784723 and let the value thereof in minutes and seconds be desired, this fraction being multiplyed by 6 the product is 32.002708332 from [...] which cutting off one place of the fraction given, the figures remaining give 32 minutes, and so multiplying of the remainer by 6 continually, the products give 00 seconds, 09 thirds, and 45 fourths, almost as by the operation in the margent, doth appear.

T. W. | Penc. | 0 | 1 | 2 | 3 |

46 | 23 | 095833 | 096875 | 097916 | 098958 |

44 | 22 | 091666 | 092708 | 093750 | 094791 |

42 | 21 | 087500 | 088541 | 089583 | 526090 |

40 | 20 | 083333 | 084375 | 085416 | 086458 |

38 | 19 | 079166 | 0 [...]0208 | 081250 | 082291 |

36 | 18 | 075000 | 076014 | 077083 | 078125 |

34 | 17 | 070833 | 071874 | 072916 | 073958 |

32 | 16 | 066666 | 067708 | 068750 | 069792 |

30 | 15 | 062500 | 063542 | 064583 | 065625 |

28 | 14 | 058333 | 059375 | 060416 | 061458 |

26 | 13 | 054166 | 055208 | 065250 | 057292 |

24 | 12 | 050000 | 05104 [...] | 052083 | 053125 |

22 | 11 | 045833 | 046875 | 047016 | 048958 |

20 | 10 | 041666 | 042708 | 043750 | 044791 |

18 | 9 | 037500 | 038541 | 039583 | 040625 |

16 | 8 | 033333 | 034375 | 035416 | 036458 |

14 | 7 | 029166 | 030208 | 031250 | 032291 |

12 | 6 | 025000 | 026014 | 027083 | 028125 |

10 | 5 | 020833 | 021874 | 022916 | 023958 |

8 | [...] | 016666 | 017708 | 018750 | 019791 |

6 | 3 | 012500 | 013541 | 014583 | 015925 |

4 | 2 | [...]08333 | 009375 | 010416 | 011458 |

2 | 1 | 004166 | 005208 | 006250 | 007291 |

0 | 0 | 0000 [...]0 | 001 [...]42 | 002083 | 003125 |

### The use of the Table.

THis Table doth consist of six Columns, in the first, you have each other Grain in two penny weight Troy, in the second every penny in two shillings, in the other four Columns you have the decimals, answering to the grains in Troy weight, and to the pence and farthings not exceeding two shillings in English mony. Example, let the decimal 1 penny weight, and 16 grains, be required, or the decimal of 40 grains, I look for 40 grains in the first Column, and in the third, I find the decimal thereof to be 083333. Or let the decimal of 1 penny weight, and 11 grains be required, that is the decimal of 35 grains; 34 grains I find in the Table; and therefore in the same line in the first Column, I find 072916, for the decimal thereof▪ if the decimal of 17 penny weight and 13 grains were required, the decimal of 16 penny weight is the half thereof; viz. 8 and that decimal of one penny weight 13 grains is 077083 and so the decimal required, is 8.077083, the like may be done for any other; and so for mony you have the pence in the second Column and the farthings in the top of that page, and in the angle of meeting the decimal answering thereunto, thus the decimal of 19 pence 3 farthings is 08 [...]291 and so for any other.

## CHAP. XVI. Of the Addition and Subtraction of Decimals.

1. THe essential numeration of decimal fractions, in all the four speceies, to wit of addition, Subtraction, Multiplication and Division is the same as in whole numbers, only if the numbers given to be added, subtracted, multiplyed or divided, be any of them mixt numbers, that is one part of a whole number, and the other a decimal fraction, or that in the several operations, integers shall arise in the working, the greatest difficulty is to distinguish the Integers from the Decimal parts, which yet in the three first Species of numeration is very easie, and in the fourth, not very hard to effect, as by example, will appear.

Let the Decimals of these several sums of mony be given to be added together.

[...] ▪ | 1 [...] | 08— | 03— | 10. | 8864583 |

8. | 12— | 09— | 02— | 8. | 6395833 |

[...]. | [...]9— | 11— | 01— | 6. | 4968750 |

1 [...]. | 19— | 10— | 02— | 13. | 9937590 |

40. | [...]0▪ | [...]4— | 00 | 40. | 0166666 |

Draw a downright line between the whole numbers and the Decimal fractions, and then add [Page 87] them as in whole numbers, so will the summ be 40l. 00s. 04 as by the operation doth appear.

2. The Subtraction of Decimals is as easie, only, as before, you are to distinguish the whole number from the fraction by a stroke or point, and then proceed as in whole numbers hath been directed.

Example. | Decimals. | |||

From 29. 11— | 03— | 01— | 29.5635416 | |

Deduct 14. 19— | 11— | 02— | 14.9979166 | |

Remain. 14. 11— | 03— | 03— | 14.5656250 |

## CHAP. XVII. Of the Multiplication of Decimals.

1. THough the Multiplication of Decimals, doth not at all differ from the Multiplication of whole numbers, as to the manner of the work, and so I might be as brief in the explanation thereof, as in Addition and Subtraction, yet that the excellent use of decimal fractions may appear, I shall illustrate [...] Rule with more examples, than the former

2. And to distinguish the Integers, from the Decimal parts, when the Multiplication is finished, consider how many decimal parts there are in both numbers given, and cut off so many figures, or places from the product towards the right hand the rest towards the left are, Integers, [Page 88] and if the whole product have not so many places, as there are parts in both the numbers given, you must prefix as many Cyphers before the product towards the left hand as there are places wanting, to make it even with the number of the places in the parts of both the numbers given.

Example, let 2 pound 9 shillings and 6 pence be given to be multiplyed, by 2 pound 9 shilling 6 pence, the Decimal of 9 shillings & 6 pence, is, .475, and therefore the Multiplicand is 2.475 And the Multiplicator also is 2.475 [...] the product is—6.125625 that is, 6 pound 2 shillings 6 pence, and 625 parts.

### 2 Example.

Let a board in length be 14 feet 6 inches 3 quarters, and in breadth, 2 foot 9 inches 3 quarters, and let the content of that board be required: the Decimal answering to 6 inches 3 quarters, is 5625, and therefore the Multiplicand is .14 .5625 and the Decimal of 9 inches 3 quarters is, 8125, and therefore the Multiplicator is 2.8125, which being multiplyed together, as whole numbers the [Page 89] [...] Product is—40.95603125 that is, 40 feet 11 inches and 1 quarter of an inch, as it was before found in Chap. 10.

### 3. Example.

Let the Decimal .3625 be given to be multiplied by .2387 [...]The product is—8652875 which having but seven places of figures, whereas there are eight places of parts in both the numbers given, I set a Cypher before the product towards the left hand to make them equal, and then the product is .08652875.

3. By these few examples the easie resolution of some questions by decimal fractions doth appear, which are any other way troublesome enough; and as to the manner of working, more [Page 90] needs not be said neither, but yet there are some contractions in Multiplication which are in astronomical questions very useful, and also in making of Tables of interest and such like, which are set down by Mr. Oughtred in his Clavis, Chap. 4 It is to find any part of the product towards the left hand without finding the remainer thereof towards the right hand, as if 583614 were given to be multiplied by 472963 and that the six first figures of the product were required towards the left hand, without any consideration had to the use of the figures of the product towards the right hand. The way is thus, set the figures of your Multiplicator under the figures of your Multiplicand, but the quite contrary way, as here you see.

Then beginning with the first figure [...] towards the right hand I set down the product thereof as was directed in the multiplication of whole numbers, which done I multiply by the second figure, but begin with that figure of the Multiplicand which stands directly over it, and set the product under the first figure of the first product, and so the rest of the figures in the Multiplicator, having still regard to that increase, which would arise out of the figure preceeding that with which you begin your Multiplication, so shall there be effected what was propounded, as by the operation doth appear; and for the clearer manifestation [...] I have added the wh [...]le product also. [Page 91] [...]

## CHAP. XVIII. Of the Division of Decimals.

1. THis is the last Species of essential numeration, and the most difficult, but not much more difficult in decimal fractions than in whole numbers; in the Arithmetick work there is no difference between them at all, that difference that is, between them and all the difficulty too, is in discovering the nature of the quotient, as whether it be only a fraction, or a mixt number, or which be integers and which fractions, or if fractions only whether the first place be tenths or hundreds, or of some other denomination.

2. For the resolving of this question in all the cases or varieties that can possibly happen, take this general rule.

The first figure in your quotient will be always [Page 92] of the same degree or place with that figure or Cypher in your Dividend, which standeth over the place of unites in your Divisor.

To illustrate this rule, I will adde an example or two; let it be required to divide a mixt number by a mixt number, let the product of the first example in the last Chapter 6.125625 which is a mixt number, be given to be divided by the Multiplicator there given, which is a mixt number also, to wit, 2.475, division being made according to the rules given for the dividing of whole numbers, the figures in the quotient will be 2475. Now it remaineth to separate the integers in this quotient from the Decimal parts, to perform which I subscribe the Divisor 2.475 underneath the first dividual 6.125 and I find that the figure 2 which stands in the unites place in the Divisor, doth also stand under the place of unites in the Dividend, and therefore I conclude according to the rule given, that the first figure in the quotient must also stand in the first place of Integers, and consequently that the next place on the right hand must be the beginning of the fraction, and the note of separation must be put between 2 and 4, as here you see. [...]

2. Example, let the mixt number 612.5625 be given to be divided by .2475 the quotient will be 2475 as before, but placing the Divisor [Page 93] under the Dividend, I find that the place of unites in the Divisor which is now supplied by a Cipher, doth stand [...] under the fourth place of Integers in the Dividend, and therefore the first figure in the quotient, is the fourth place of Integers also, and consequently in this example there are no decimal parts in the quotient.

3. Example, Suppose the Decimal .3675 were given to be divided by this Decimal .025 the quotient will be, 147, and to discover the nature thereof I set the Divisor under the Dividend thus [...] from whence it appeareth that the unites place in that Divisor, is supposed to stand under the place of tens in the Dividend, and therefore there are two places of Integers in the quotient. And by these few examples, the use of the rule in any case is▪ I hope, sufficiently declared.

The benefit of Division by Decimals may partly appear by the following propositions.

1 Proposition, Let 8118 pounds 11 shillings 3 pence be given to be divided amongst 283 men, 11 shillings 3 pence being turned into a Decimal, the Dividend is 8118.5625 which being [Page 94] divided by 283 the [...] quotient is 28.6882 which being reduced is 28 pound 13 shillings 9 pence for each man,

2. Proposition, In Astronomical fractions let 13 signs 19 degrees 36 minutes 32 seconds 47 thirds and 20 fourths be given, to be divided, by 17 degrees, 37 minutes, 25 seconds, these numbers being converted into Decimals stand thus. [Page 95] [...]

[Page 96]And the quotient is and being [...] reduced is 23 deg. 14 min. 32 sec. as by the operation doth appear, in which you may see not only the excellent use of decimal fractions, but an excellent contraction of common division▪ for as in Multiplication I shewed you how to shorten every particular product, so may you by considering the manner of this Division, see how to shorten your Divisor, and yet produce the same quotient, as would arise by using the Divisor at the whole length given.

3. Proposition, If the content of a board be 40 feet 11 inches 1 quarter of an inch and 15 sixteenths of a quarter, and the breadth 2 foot 9 inches 39, and let the length of that board be required.

The Decimal of 11 inches 1 quarter 15/1 [...] by the second example of the last Chapter is .95703125. So that 40.59703125 is the Dividend, and 2.8125 is the Divisor, and the quotient is (14.5625 [Page 97] [...]

Compare these examples with those in the 11 Chapter, and use that way which liketh you best.

## CHAP. XIX. Of the relation of numbers in Quantity.

1. HItherto we have spoken of single Arithmetick, Comparative followeth; in which numbers are to be considered as they have relation to one another, in quantity or in quality.

2. Relation in quantity is of equal or unequal numbers▪ and hence it is called Relation of equal or unequal reason, in both which the numbers propounded are always two, the first of which is called the Antecedent, and the other the Consequent.

3. Equal reason, is the Relation that equal numbers have to one another, as 5 to 5, 6 to 6, 12 to 12.

4. Unequal reason is the relation, that unequal numbers have to one another, and this is either of the greater to the less, or of the less unto the greater.

5. Unequal reason of the greater to the less, is when the greater number is the Antecedent, and the lesser, the Consequent, that is, when the greater number is first set down, and the lesser after it, or when the greater number is set over the less, with a line between them in manner of a vulgar fraction, as of 6 to 3, 5 to 2, where 6 and 5 are Antecedents, 3 and 2 are Consequents, and may be otherwise written thus 6/ [...] 5/ [...].

[Page 99]6. Unequal reason of the lesser to the greater, is when the lesser number is the antecedent, and the greater the consequent; as 5 to 6, 2 to 5.

7. The relation that these several kinds of numbers have to one another in quantity, is either in respect of the difference between the antecedent and the consequent, or in respect of the reason that is between them.

8. The difference between the antecedent and the consequent is found by subtracting the consequent from the antecedent; thus the difference between equal numbers as of the relation of 5 to 5, 6 to 6, or 12 to 12, is 0. but the difference of the relation of 6 to 3 is 3, of 5 to 2 is 3, for 3, deducted from 6, the remainer is 3, and 2 deducted from 5 is 3 also, but the difference of the relation of 3 to 6 and 2 to 5 is 3 less than nothing, that is the antecedent is less by 3 than the consequent.

9. The rate or reason between 2 numbers i [...] found by dividing the antecedent by the consequent.

10. The rate or reason between two equal numbers is always an unite, for if it be demanded how often 5 is conteined in 5, the answer is 1.

11. The rate or reason between 2 unequal numbers is greater than unity, if the antecedent be the greater, and less than unity if the antecedent be the lesser number; thus the rate or reason of 8 to 5 is 1⅕ but the rate, or reason of 5 to 8 is [...]/ [...].

12. Both these kinds of unequal reason are either simple or compound.

13. The simple kinds of unequal reason are [Page 100] of three varieties. 1. Manifold, 2. Superparticular, 3. Superpartient.

14. Manifold simple reason, is when the greater number doth contain the less a certain number of times, without leaving any remainer, as 4 to 2, 9 to 3, 16 to 4 are numbers which have manifold reason, as double, triple, and fourfold reason, for the greater numbers being divided by the less, the quotients are 2. 3. 4.

15. Superparticular, is a simple rate or reason in which the greater number doth contein the lesser once and one part more, as 3 to 2, 6 to 5, 8 to 7, in which the antecedent doth contein the consequent once, and one part more of the consequent, as by dividing the antecedents by the consequents, the quotients clearly shew; to wit, 1 [...]/ [...], 1 [...]/5 11/ [...]. There is the same rate or reason in these 12 to 8, and 15 to 10, for though the quotients are 14/ [...], 15/1 [...] yet being reduced, they are 1½.

16. Superpartient, is a simple rate or reason in which the greater number doth contein the lesser once, and several parts of the lesser; as 5 to 3, 7 to 4, 13 to 8, in which the antecedents being divided by the consequents, the quotients are 12/ [...], 1¾, 15/ [...].

17. The compounded kinds of unequal reason are either Manifold Superparticulars or Manifold Superpartient. That these are reasons compounded of the preceding doth plainly appear by their very names, the one being the second joyned unto the first, and the other the third joyned unto the first.

18. Manifold Superparticular reason, is a compounded [Page 101] reason, in which the greater number doth contein the lesser divers times, and one part more of the lesser; as in these unequal reasons, 5 to 2, 10 to 3, 13 to 4, the greater numbers being divided by the lesser the quotients are 2½, 31/ [...], 3¼.

19. Manifold Superpartient reason, is a compounded reason, in which the greater number doth contein the lesser several times, and several parts more of the lesser, as in these unequal reasons 11 to 4, 12 to 5, and 22 to 6, the greater numbers being divided by the lesser, the quotients are, 2¾, 2⅖, 32/ [...].

## CHAP. XX. Of the relation of numbers in Quality.

1. RElation of numbers in Quality, (otherwise called Proportion,) is either Arithmetical or Geometrical.

2. Arithmetical proportion, is a Relation that numbers have unto the equality of their differences; as in 8. 7. 6. and in 12. 9. 6. 3 there are two proportions which have equal differences; in that first rank of numbers, the equal or common difference is 1, and in the second 3.

3. Arithmetical proportion, is either continued or interrupted.

4. Arithmetical proportion continued, is when the same difference is continued from the [Page 102] first numbers to the last, as in this rank of numbers. 5. 7. 9. 11. the common difference is 2.

5. Arithmetical proportion interrupted, is when the difference between the first and second is the same with the third and fourth, but not between the second and third, or when the difference is discontinued in any part of the rank, as in these numbers, 10. 8. 4. 2. the difference between 10 and 8 is 2, and the difference between 2 and 4 is 2, but the difference between 4 and 8 is 4.

6. In Arithmetical progression, two things may be inquired, either the terms that constitute the progression, or the sum that the terms in a certain progression made.

7. Sometimes all the terms in an Arithmetical progression at a certain given rate may be required to be expressed in their natural order; or the progression being interrupted, some particular term in that progression may be required.

8. If it be required to express the terms in any rate propounded, add the rate to any given term continually and you have what is required; as if it were required to make a rank of numbers, in Arithmetical proportion, answerable to the rate or difference that is between 1 and 5, add 4 which is the difference between the numbers given unto 5, and so continually to their sum and you shall constitute this rank of numbers 1. 5. 9. 13. 17. 21. which is the rank of numbers in Arithmetical proportion required.

9. If some particular term in any progression be required without the rest, deduct, from the term required, the remainer being multiplied by [Page 103] the common difference in that progression, and the product added to the first shall give the term desired: as in this progression 4. 7. 10, let the 12 term from 4 the first term given be required, deducting one from 12 the remainer is 11, which being multiplied by 3 the common difference the product is 33, to which adding 4 the first term, the sum is 37 for the 12 term required: and in like manner the 30 term wil be found to be 91: the like may be done for any other.

The Terms. | 4 | 2 | 10 | 37 | 91 |

The number. | 1 | 2 | 3 | 12 | 30 |

10. If the sum of the terms in any rank of numbers, in Arithmetical proportion continued, multiply the sum of the first and last terms by the number of the terms, half the product shall be the sum required, as in this example, 1. 5. 9. 13. 17. 21, the sum of the first and last terms is 22, which being multiplied by 6 the number of the terms the product is 132, the half whereof is 66 the sum of the terms required.

11. Three numbers being given which differ by Arithmetical proportion continued, the mean number being doubled, shall be equal to the sum of the extreams, so 5. 9. 13. being given, the double of 9 is 18 and the sum of 5 and 13 is 18 also.

12. Four numbers being given that differ by Arithmetical proportion continued or interrupted, the sum of the two means shall be equal to [Page 104] the sum of the two extreams, so 5. 9. 13 17 being given, the sum of 9 and 13 the two mean numbers, is equal to the sum of 5 and 17 the 2 extreams.

13. Geometrical proportion is a relation that numbers have to the equality of their rate or reason: as in 2. 6. 4. 12, where the rate between 2 and 6 is the same with that which is between 4 and 12; for as 6 is three times as much as 2, so 12 is three times as much as 4.

14. Geometrical proportion is either continued or interrupted.

15. Geometrical proportion continued, is when the same rate or reason is still kept in the whole rank of numbers given, as 2. 4. 8. 16. 32, in which the progression is continued by double reason, for as 4 is twice 2, so 8 is twice 4, 16 is twice 8, and 32 is twice 16.

16. In numbers that increase by Geometrical proportion continued, if you multiply the last term by the rate or reason by which the rank of numbers is created, and from the product subtract the first, the remainer being divided, by the rate less one shall give you in the quotient, the total sum of all the terms. Example, let the rank of numbers propounded be 3. 9. 27. 81. 243, the rate or reason by which this rank of numbers is created is 3, by which the last term 243 being multiplied the product is 729, out of which deducting 3 the first term, the remainer is 726, which being divided by the triple less one that is by 2, the quotient gives me 363, for the sum of the terms propounded.

[Page 105]17. Three numbers being given which do increase by Geometrical proportion, the square of the mean shall be equal to the product of the extreams; as 9. 27. 81. being propounded, the square of 27 is 729, and the product of 81 by 9 the two extreams is 729 also.

18. Geometrical proportion interrupted, is when the rate or reason which is between the first and second and between the third and fourth, is not to be found between the second and third; as 6. 3. 16. 8. in which 3 is half 6, and 8 half of 16, but the same rate is not found between 3 and 16.

19. Four numbers being given which do increase by Geometrical progression continued or interrupted, the product of the two mean numbers shall be equal to the product of the extreams: let the four numbers given be 6. 3. 16. 8. the product of 16 by 3 is 48, and the product of 8 by 6 the extream numbers is 48 also.

## CHAP. XXI. Of the Rule of Three.

1. From the last rule of the preceeding chapter doth arise this precious Jewel in Arithmetick, which for its excellent use is called the Golden Rule, otherwise and more commonly the Rule of Three; because it teacheth from three numbers given, how to find a fourth that is in Geometrical proportion to them.

2. This Rule of three is either simple or compound.

3. The single Rule of three is, when three numbers are propounded, and a fourth in proportion to them is required.

4. This Rule of three is either direct or inverse.

[Page 107]5. The rule of three Direct is when the third term hath such proportion unto the fourth, as the first hath to the second.

6. The Rule of three Inverse is, when the first term hath such proportion to the fourth, as the third hath to the second.

7. The difficulty of this Rule lieth in two things, first in stating the question or placing the numbers given, and then secondly in discovering, whether the first or third term must be the Divisor.

8. The question may be easily stated by considering, by which of the three numbers given the question is moved, for that must be always the third term, and that which is of the same denomination with the third, must be always set first, and the number remaining is to be set between them, and must be always of the same denomination with the fourth that is required.

9. The Question being stated, to discover whether the first or third number given must be the divisor, consider again whether the fourth term required, must be more or less than the second term given, for if it must be more, the lesser extream must be the divisor, but if less, the greater extream must be the divisor.

Example, If 12 men spend 36 pound, how much shall 18 men spend; here the question is concerning the 18 men, this is therefore the third number, and 12 men is the number of the same denomination, and must therefore possess the first place, and the middle number is 36 pounds: so that the numbers in the question must stand thus. [Page 108] And now to discoverMenlib.Men12.36.18. whether the first or last of these numbers must be the Divisor, I consider again whether the fourth term required must be more or less than 36. the middle term or second number, and my reason tells me that 18 men must needs spend more than 12, therefore the fourth number must be greater than the second, and so 12 the lesser extream must be the Divisor; now then forasmuch as the product of the two mean numbers is equal to the product of the two extreams by the last rule of the preceding chapter, it followeth that if I multiply 36 by 18 and divide the product by 12, the quotient will be the fourth term required, and so the answer will be found to be 54 pounds.

Men | lib. | Men | lib. |

12 | 36 | 18 | 54 |

2. Example, If 25 shillings in Wine will serve 36 men when Wine is at 12 pounds the Tun, how many men will the same sum serve, when Wine is at 18 pounds the Tun; Here the question is concerning the 18 pounds for a Tun of Wine; this is therefore the third number, & 12 pounds the Tun is of the same denomination, and must therefore possess the first place, and the middle number is the 36 men, and so the numbers in the question must stand thus.

lib. | Men | lib. |

12 | 36 | 18. |

And now to discover whether the first or third of these numbers must be the Divisor, I consider if the higher the price of Wine is, the fewer persons will 25 shillings serve, and therefore 18 the greater extream must be the Divisor, and then by the Rule of three inverse it is as 18. 36. 12. 14.

These two examples well considered will be sufficient to shew the nature of this Rule, and the manner of working: but because there is such variety of questions to be resolved thereby, in which sometimes one part of single Arithmetick is exercised, sometimes another, and of Comparative Arithmetick also, by way of preparation for the better stating of the question, and more plain and easy solution, it will not be amiss to add an example in each variety, or in so many of them as are of daily practice.

3. My first instance shall be in such questions, in which the addition of whole numbers is required in order to the better stating and resolving of them by this Rule.

A Merchant buyes 120 Tun of Wine for 2400l. and the fraight thereof from Spain to London cost him 1200l. for loading and unloading 30l. for custom 50l. and for expences in the voyage 64l. and he designs to get 800l. by the bargain. The question is at what rate this wine must be sold per Tun.

In such examples, the price of the fraight with [Page 110] all the expences and gain must by addition be brought into one sum, which in this particular is 4544; then I say if 120 Tun cost 4544 pound, what shall 1 Tun cost, and the answer is 37104/120 or 52/60 or [...]6/3 [...] or 13/15 of a pound.

4. Sometimes a question belonging to the Rule of three is prepared for solution by subtraction, as in this following. If 24 Gallons of water do in one hours time run from one pipe into a Cistern conteining 250 Gallons, and run out 16 Gallons by another in the same time, in how many hours will that Cistern be filled.

Here one of the terms propounded, must be explained by subtraction, for it is plain that the enquiry is, by how much the Cistern doth fill more than it empties; if therefore you subtract 16 Gallons from 24 the remainer is 8 Gallons, which the Cistern recieves in an hours time; hence the question must be thus stated.

If a Cistern recieve 8 Gallons in one hours time, in how many hours will it receive 250 Gallons: facit, 32 ¼.

5. Sometimes a question belonging to this rule must be prepared by addition and subtraction both, as in this following. A Merchant buyes [...] bags of Pepper.

lib. | |||

The first marked A containing | 86 | tare | 06 |

The second marked B containing | 76 | tare | 05 |

The third C containing | 98 | tare | 12 |

The fourth D containing | 75 | tare | 05 |

335 | 028 |

And he is to pay 35 shillings per. Cent. neat. The question is how much the 4 bags cost.

In the Trade of Merchandize, there are certain allowances and abatements used, which are known by the names of Tare and Tret, by Tare they understand the weight of the chest, bag, but, &c. And by Tret the overweight which is allowed the buyer, in this question we have to do with the former only; where first you must get the gross weight of all the parcels by addition and they are 335 lib. and also the sum of the allowances for the Tare which is 28 lib. then deducting the 28 lib. Tare from 335 gross weight, the remainer is 307 lib. neat; hence the question must be stated thus.

If 100 pound cost 35 shillings what shall 307 lib. cost: facit 107 4 [...]/ [...].

6. Sometimes a question belonging to this Rule must be prepared for solution by Multiplication, as in this following. How many Spanish Pistolets at 14 shillings sterling the piece, ought to be received for 128 Escus d'or, at 7 shillings sterling the piece. Here the 128 Escus d'or must be reduced into English by multiplying them by 7 shillings, the value of one Escus d'or, whose product [Page 112] is 896 shillings, and then the question must be thus stated.

If 14 shillings be worth 1 Pistolet, how many Pistolets are 896 worth. Facit 64 Pistolets.

7. Sometimes a question belonging to this rule must be prepared for solution by multiplication and addition, sometimes by multiplication and subtraction; as in these examples following, the first of which requireth multiplication and addition.

A Butcher sends his man with 216 pound sterling to buy Cattel, Oxen at 11 pound apiece, Cows at 2 pound a piece, Colts at one pound 5 shillings, Hogs at 1 pound 15 shillings a peice, and of each a like number, the question is how many of each he might buy for that mony. To resolve this question I must first reduce the pounds sterling into shillings by multiplication, and 216 pound being multiplied by 20 the product is 4320 shillings, and the several prizes of the Cattel must be reduced into shillings also, and thus 11 pounds for an Ox is 220 shillings, the price of a Cow is 40 shillings, the price of a Colt 25 shillings, and the price of a Hog is 35 shillings, the sum of these several prizes is 320 shillings, this done the question may be thus stated.

If 320 shillings buy 1 of each sort, how many shall 4160 buy, facit 13.

2. Example, In the which the question is to be prepared by multiplication and subtraction.

If 128 Gallons of water run into a Cistern in an hours time from one Cock, and 174 Gallons of water run into another Cistern, in an hours time [Page 113] also, but not begin to run till 4 hours after the first, in what time will the quantity of water run out of both be equal? To resolve this question, one of the terms given must be prepared by Multiplication, and another by Subtraction; first there must be computed how much water was run out of the first Cock before the second began, and that is 4 times 128 Gallons that is 512: secondly it must be considered, how many Gallons more doth run out of the last Cock than doth out of the first in an hours time, which by subtracting 128 from 174 I find to be 46, and hence this question is to be thus stated.

If I gain 46 Gallons in one hour, in how many hours shall I gain 512 Gallons. Facit 11.

8. Sometimes a question belonging to this rule, must be prepared for solution, by Arithmetical or Geometrical proportion, as in these questions following.

A man going a journey spends one shilling the first day, 3 the second, 5 the third, 7 the fourth, still encreasing his expences by this proportion till he hath spent 1500 shillings, I demand in how many days that sum is spent.

To resolve this question the sum of such an Arithmetical progression for some certain time must be first computed; as suppose for thirty days, which will be found to be 900 shillings; and hence it may be thus concluded.

If 900 shillings last me 30 days, how long shall 1500 shillings last me. Facit 50.

Another question may be this, If a piece of Cloth conteining 40 yards did cost me 30 pounds. [Page 114] at what rate must that Cloth be sold to gain 12 pound in the hundred?

Before this question can be resolved, I must find the price of one Yard, by a proportion Geometrical by way of preparation thus, If 4 [...] Yards cost 50 pound, what shall one Yard cost, facit 1¼. Then I say.

If 100 pounds be increased to 112, to what shall 1¼ be increased? Answer 1⅖.

## CHAP. XXII. The Rule of Three in fractions.

1. FRactions as I have already shewed, are of 3▪ sorts, viz. such as are many times expressed like integers, such as are expressed by their numerators and denominators, and such as have a unite with Cyphers annexed for their denominator, and now known by the name of Decimal fractions: in each of these I will exemplify this Rule by some few examples; and first in such fractions as are expressed like Integers.

2. Your question being stated bring your numbers propounded into the least name mentioned, or as low as you desire the question to be answered in, then multiply the second and third and divide the product by the first, if your question belong to the Rule of Three direct; or multiply the first and second and divide by the third, [Page 115] if it belong to the Rule of Three inverse, the quotient being reduced shall be the answer required.

### Example.

If 7 lib. 8 ounces of Currants, cost 2s. 7d. what shall 100½ 13 lib. cost. In this question the first and third terms must be reduced into ounces, and the second term into pence, and then the question will stand thus.

If 112 ounces Cost 31 pence, what shall 1267 ounces cost. Facit 359d. 1 64/112 farthings; that is 1l. 9s. 11d. 1f. ⅘ of a farthing.

A second example may be this; A man doth borrow of his friend 246 lib. 7 shillings and 6 pence for 27 weeks, for how long time must he lend his friend 329 lib. 13 shillings 4d. to require his kindness. Here it is plain that the fourth term required, must be less than the second term given, and therefore the greater extream must be the Divisor, and the numbers being reduced and placed as hath been directed they will stand thus 59130—187—79120▪ in which the greater extream being in the third place doth plainly shew that this question doth belong to the Rule of Three inverse, and therefore as [Page 116] [...]

3 Example in Sexagenary numbers, If 48 firsts, 28 seconds, give 60 minutes, what shall 25 minutes, 12 seconds give. According to the former Rules given in Multiplication 25 firsts, 12 seconds being multiplied by 60 minutes, the product is 25 degrees, 12 firsts, 00 seconds, which being divided by 48 firsts, 28 seconds the quotient will be 31 firsts, 08 seconds.

4. Example, in vulgar fractions. If ⅔ of a Yard of Velvet be sold for ⅘ of a pound sterling, what shall ⅞ of a Yard cost?

If respect be had to the Rules of the Multiplication and Division of fractions already delivered in the 14th Chapter, the working of this Rule in fractions will be the same as in whole numbers; thus ⅘ and ⅞ being multiplied together the product will be 28/40 which being divided by ⅔ 7 ⅔ the quotient is 84/ [...]0 or 1 lib. 4/ [...]0 or 1/20; and so the answer is 1 pound and one shilling.

Otherwise thus, Multiply the Denominator of the first number by the numerators of the second and third continually, so shall the last product be a new numerator; then multiply the numerator of the first number by the denominators of the second and third numbers continually, so shall the last product be a new denominator, and this fraction shall be the quotient desired. Thus the denominator 3 being multiplied first [Page 117] by 4 and then by 7 the last product is 84; and the numerator 2 being multiplied first by 5 and then by 8 the last product will be 80 for a new denominator, and so the new fraction will be 84/ [...]0 as before.

5 Example, If 4¼ of a Yard of Cloth in length, being 1 Yard and half broad, will make a Cloak, how much Plush that is ¾ of a Yard in breadth will serve to line that Cloak. Here the fourth term required must be greater than the second term given, and therefore the lesser extream must be the Divisor, and the less extream being in the third place, plainly sheweth, that this question doth belong to the Rule of Three inverse, and the numbers must stand as here you see.

breadth | length | breadth |

1/1 [...] | 4¼ | ¾ |

or 3/ [...] | 17/4 | ¾ |

And ¾ being the Divisor, 4 being multiplied first by 17 and then by 3 the last product is 204, and 3 being multiplied by 4 and by 2, the last product is 24, and so the new fraction or quotient sought is 204/24, which being reduced, is 8½ yards.

6. Example, in Decimal fractions. If 7 ounces [Page 118] 3 penny weight and 12 grains be worth 21l. 11s. 6d. what is the value of 1½ ounce?

- The Decimal of 3 penny weight is
- 0.15
- And the Decimal of 12 ounces is
- 0.025
- And the sum of these 2 Decimals is
- 0.515
- Wherefore the first number in the Rule of Three is
- 7.175
- Again the Decimal of 10s. 6d. is
- 0525
- Wherefore the second number is
- 21.515
- And the third number is
- 1.5

Now to resolve this question, the 3 given numbers will stand thus. 7.175 − 21.525 − 1.5.

Lastly, multiplying the second by the third, the product is 32.2875 which being divided by 7.175 the quotient is 4.5 that is 4 pounds and ten shillings, which is the price of an ounce and half of that Gold.

## CHAP. XXIII. Of the Rules of Practice.

1. WHen the Rule of Three direct hath 1, or an Integer for the first term, it is commonly called a Rule of practice, not only for the speedy, but the practical resolution of such questions: for whereas other questions belonging to this Rule, do for the most part require Multiplication and Division, these in the ordinary way are resolved by Multiplication; and whereas Multiplication is generally much easier than Division, these questions are in an accustomary or practical way, for the most part performed by Division, and yet done with more ease than they can be by Multiplication: nay where Multiplication and Division are both used, the resolution is yet easier, than it is or can be by Multiplication only.

2. The questions to be resolved by these abbreviations of the Rule of Three, or practical operations, do for the most part come under one or other of these five cases. The price of 1 or an Integer doth consist either

- 1. Of a certain number of shillings under twenty.
- 2. Of a certain number of pounds and shillings.
- 3. Of a certain number of pence under twelve.
- [Page 120]4. Of a certain number of shillings and pence.
- 5. Of a certain number of pounds, shillings and pence with the parts of a penny.

3. Now for the resolution of such questions which may or do fall under any of these cases; it is necessary that the Aliquot parts of a pound and shilling be first known, that is, how often the shillings of pence, which are the given price of one Integer are conteined in a pound or shilling, without leaving any remainer; thus 4 shillings is an aliquot part of a pound or 20 shillings, because 4 may be taken 5 times in 20, without either excess or defect; and 3 is an aliquot part of a shilling or 12 pence, for 4 times 3 doth make 12; but 7 is not the aliquot part of 20, for two times 7 is less than 20; and 3 times 7 is more. In like manner 5 is not the aliquot part of 12, for 2 times 5 is but 10, and 3 times 5 is 15, the one wants of 12, and the other exceeds it; but yet any number of shillings under 20 may be reduced to the aliquot parts of a pound, and any number of pence under 12 may be reduced to the aliquot parts of a shilling.

Thus 7 shillings is one fift more, 3 twenties of a pound: and 5 pence, is one third more, one twelveth of a shilling, or one fourth, more one sixth. Such aliquot parts of a pound or shilling as are most useful in the resolution of questions of this nature, are here expressed in two Tables; with which I have joyned a third, shewing the product of 12 by any number not exceeding it [Page 121] self, which in many questions will be found as useful as either of the other.

Aliquot parts of a pound. | Aliquot parts of a Shilling. | The product of 12 by any number less. | |||

Pence. | |||||

10 | 0 | ½ | 11 | ⅓+⅓+¼ | 2 × 12. 24 |

6 | 8 | ⅓ | 10 | ½+⅓ | 3 × 12. 36 |

5 | 0 | ¼ | 9 | ½+¼ | 4 × 12. 48 |

4 | 0 | ⅕ | 8 | ⅓+ [...]/ [...] | 5 × 12. 60 |

3 | 4 | ⅙ | 7 | ¼+⅓ | 6 × 12. 72 |

2 | 6 | ⅛ | 6 | ½ | 7 × 12. 84 |

2 | 0 | 1/10 | 5 | ¼+1/ [...] | 8 × 12. 96 |

1 | 8 | 1/12 | 4 | ⅓ | 9 × 12. 108 |

1 | 4 | 1/15 | 3 | ¼ | 10 × 12. 120 |

1 | 3 | 1/ [...] | 2 | 1/ [...] | 11 × 12. 132 |

1 | 0 | 1/ [...] | 1½ | [...]/ [...] | 12 × 12. 144 |

Shil. | Pence. | 1 | 1/20 |

These things premised, I will now shew how any question coming under the aforesaid five cases may be resolved.

### 1. Case, where the price of an Integer is shillings only.

4. Where the price of 1, or an Integer is two shillings, the price of as many Integers as you will may be discovered by bare inspection, for two shillings being the tenth of a pound, the double of the first figure (towards the right hand) [Page 122] is the number of shillings required, and the rest of the figures are so many pounds. Example, 567 yards at 2 shillings the Yard will cost 56 pounds 14 shillings; for the double of 7 is 14 which I write down by it self as shillings, then taking the rest of the figures towards the left hand for pounds, the answer is 56l. 14s.

5. When the given price of 1 or an Integer is an even number of shillings greater than two, multiply the number of Integers whose price is required, by half the number of shillings given; the double of the first figure towards the right hand in the product being set down for the shillings apart, all the other figures towards the left hand shall be the pounds required.

Example, let the price of 365 Yards be required, at 14 shillings per 1 Yard: if you multiply 365 by 7 (which is the half of 14 the number of shillings given) the product will be 2562, now the double of 2, the first figure y.s.y.1.14365 7 256. 2 towards the right hand is 4, the other figures are 256 pounds, and so the answer is 256l—4s.

Here note that 4 shillings being the 5th part of a pound, if that be the price of an Integer, it will be all one to multiply by 2 or divide by 5, if the double of the first figure in the product towards the right hand be taken for the shillings according to this rule.

6. When the given price of 1 or an Integer is an odd number of shillings, for the odd shilling [Page 123] take the ½. of the price propounded, and add it to the product of the price given by half the number of shillings remainer, taking the double of the last figure of the products for shillings according to the former directions, their sum shall be the answer required. Example, let the price of 367 Yards be required at 9 shillings the Yard. The twentieth part of 367 y.s.y.1.9.367 420▪367—1871468—14616 16503 is 18 pound 7 shillings, and the product of 367 by 4 (half the number of the shillings remaining) is 1468 that is 146 pound 16 shillings, which being added to 18l. 7 shillings, the answer is 165 pound, 3 shillings.

Note, When 5 shillings is the given price of an Integer the shortest way will be to divide the number whose price is required by 4, because 4 is the first part of a pound; thus if the worth or price of 367 Yards at 5 shillings the Yard were required, the answer would be 91¾ that is 91l. 15s.

### 2. Case, Where the price of an Integer is pounds and shillings.

7. When the price of 1 or an Integer doth consist of pounds and shillings; first multiply the number of Integers whose price is required, by the number of pounds in the price given, and subscribe the product as pounds; then proceed with the shillings in the price propounded, according [Page 124] to the 5 or 6 rules of this Chapter, and subscribe the sum or sums so found under the number of pounds, the total of these sums shall be the answer required. Example, let the price of 1238 hundred weight at 3l. 12s. per C. be required, 1238 being Cls 1.2.13.1238 3714 742.16 619.18 5067.14. multiplied by 3 the product is 3714, and the product thereof by 6 is 7428 that is 742l. 16s. and the twentieth part of 1238 is 619l. 18s. the which several sums being added together their total 5067l. 14s. is the answer sought.

### 3. Case, Where the price of an Integer is a number of pence under 12.

8. When the given price of 1 or an Integer is such a number of pence, as is an Aliquot part of a shilling; divide the number of integers, whose value is required, by such aliquot part; so will the quotient be the number of shillings, which answers the question, and may be reduced into pounds (if need require) by dividing the same by 20.

Example, Let the value of 3947 pounds weight, be required at 3 pence the pound, because [Page 125] 3 is the 4 part of 12 pence, lib.dlib.1.3.3947 20.986¾ 49. 6. 9 I divide 3947 by 4 and the quotient is 986¾ that is 986 shillings 9 pence, or the shillings being divided by 20 the answer is 49l. 6s. 9d.

9. When the given price of 1 or an Integer is composed of aliquot parts of a shilling, divide the number of Integers whose price is required, by the several parts of which the given price of one Integer is composed; the several quotients being added together, shall be the value or sum inquired.

Example, let the price of 1638 Yards at 7 pence the Yard be inquired; for as much 7 is composed of one third, and one fourth of a shilling, if I divide 1638, by 4 which is one third of a shilling, the quotient will be 4022/4 shillings, again dividing 1638 by 3 which is the fourth of a shilling, the quotient will be 546 shillings, and the sum of these two quotients 9482/4 is the answer to the question, or 94l. 16s. 6d. [...]

### 4. Case, Where the price of an Integer is a number of shillings and pence.

10. When the given price of an Integer consists of shillings and pence, multiply the number of integers, whose value is required, by the number of shillings given, and set down the product as shillings; then divide the said number of Integers, by the Aliquot parts of a shilling of which the pence are composed, and set their quotients under the former product; the sum of the product and quotients shall be the answer sought.

Example, let the value of 836 Yards at 17 shillings 9 pence the yard be required, the product of 836 by 17 is 14212, that ysdy1179836174836 58522)836 83604)836 418 20920) 14839 741l. 19s. is the product by 7 is 5852 and the product by 10, 8360, then dividing 836 by 2 and 4 severally the quotients are 418, and 209, and the sum of all is 14839 that is 741l. 19s.

11. When the given price of an Integer consists of shillings and pence, and that such shillings and pence considered together do make the aliquot part of a pound; the question may sometimes be more expeditiously answered, if we divide the number of Integers whose value is required, by such aliquot part of a pound, than by the former rule. Example, let the price of one integer be 6 shillings 8 pence, which is the third part [Page 127] of a pound, and let the value of 983 Yards at that rate be required: the product of 983 by 6 is 5898, and 983 being divided ysd 168983 6983 5898 3)983 327 [...]83)983 327 820) 6553 4 327134 by 3 twice, because 4 pence is the third of a shilling, the quotient is 327⅔ that is 327 shillings 8 pence, which being set down twice, the sum of the product and quotients 6553 shillings 4 pence, and this being divided by 20, the value of 983 Yards will be 327l. 13s. 4d. But now if you divide 983 by 3, the quotient is 327⅔ that is 327 lib. 13 shillings 4 pence, as before.

### 5. Case, Where the price of an Integer doth consist of pounds, shillings and pence, and the parts of a penny.

12. Where the given price of an Integer doth consist of pounds, shillings and pence; and the price of a certain number of Integers less than 10 be required; multiply the given price of one Integer, by the number of Integers whose value is desired, as hath been shewed in the Multiplication of pounds, shillings and pence, so shall the product be the answer required.

Example, let the price of 8 hundred weight be required, one hundred being 4 lib. [...]7 shillings 9 pence, first I multiply 9 pence by 8 and they make 72 pence or 6 shillings; therefore I set down 0 pence [Page 128] and carry 6 to the next denomination: then I multiply 17 shillings by 8 and the product is 136, to which I add the 6 I carried and then the product is 142 that is 7 pound 2 shillings, therefore I set down 2 shillings and carry 7 to the pounds: then I multiply 4 lib. by 8 and they make 32, and the 7 I carried makes 39, and so the whole product is 39 lib. 2 shillings, lib.shil.pence4—17—939—2—0 0 pence, as by the work appeareth: and so much is the value of 8 C. Weight; as was required.

13. When the given price of an Integer doth consist of pounds, shillings and pence; and the price of a certain number of integers more than 10 is required; reduce the pounds into shillings and add the shillings in the given price to them; then proceed according to the tenth rule of this Chapter.

Example, let the price of one hundred weight be 5▪ lib. 18 shilling and 4 pence, and let the price of 543 be required; the pounds and shillings being reduced to one denomination are 118 shillings; now therefore I multiply 543 by 118, & the product is 64074 [...] and then because 4 is the third part of a shilling [...] divide 543 by 3 and the quotient is 181 shillings▪ which being added to 64074 the whole number of shillings is 60255, which being divided by 20 the quotient is 3212 lib. 15 shillings, 0 pence, which is the price required. [Page 129] [...]

14. When the given price of an integer doth consist of pounds, shillings and pence, with the parts of a penny, reduce the pounds into shillings, as in the last example; and for the pence and parts of a penny, work by the aliquot parts of a shilling, contained in the pence, except 1 penny, and the parts annexed in the question, if the parts of a penny be 2 farthings or more; but if it be but one farthing proceed with the whole numbers of pence as in the former examples; and for the farthing take one fourth of the number whose value is inquired, the quotient divided by 12 will be the shillings and pence desired; if 2 farthings be annexed to the pence they are with the penny not computed the eight part of a shilling, if 3 farthings you must for the odd farthing take the sixth part of the last quotient; all these quotients added together shall be the price required.

Example, let the price of one hundred weight [Page 130] be, 3 lib. 15 shillings 7¼, and let the price of 95 C. be required. The pounds and shillings being reduced unto one denomination do make 75 shillings: by which if you multiply 95 the product will be 71.25, and 95 divided by 4 and 3 severally, [...] the quotients are 43 shillings, 9 pence, and 31 shillings, 8 pence, and 95 divided by four for one fourth of a penny, the quotient is 43 pence 3 farthings, which being divided by 12 the quotient is 3 shillings and 7 pence 3 farthings, which several quotients being added to the former product, the total is 7204 shillings 0 pence 3 farthings, and this again divided by 20 gives the answer to the question, 360 lib. 4 shillings 0 pence 3 farthings.

### 2. Example.

[...]

15. When the given price of an Integer is given, and the price of many Integers of the same name, together with one fourth, one half, or 3 fourths of an Integer is required, the value of those Integers may be first found by some of the precedent rules, and then for the half of an Integer take the half of the given price, for one fourth of an Integer take one fourth of the given price, and for 3 fourths of an Integer, take the sum of one half and one fourth of the given price. Example, let it be required to know what 127 hundred weight, 3 quarters, 24 pound will cost [...]t [Page 132] 4l. 17s. 7d. ¾ per Cent. The answer by the following operation will be 625 lib. 5 shillings 4 pence 2 farthings, and somewhat more.

[...] Where you may observe the price of 127 C. to be found after the manner of former examples, and the 3 quarters is the sum of one half and one fourth of the given price; then for the 24 lib. I take first the half 1 quarter of C. for 14 thereof; and the half of that for 7 pound more thereof, and for the 3 pound remaining I take 3/7 of the price of [Page 133] 7 pound, thus: the price of 7 lib. is 6 shillings and a penny and something more; or 73 pence, which being multiplied by 3, the product is 219, which being divided by 7, the quotient is 31 2/7 that is 2 shillings 7 pence and 2/7 of a penny. And hence it is apparent, that in resolving of questions after this practical way, some error will be committed, yet the loss for the most part will be less than a farthing, which is not considerable.

16. When the price of one pound weight is known, and the price of one hundred is required (that is 112) there is a rule differing from all the former, and somewhat more expeditious by which the value thereof may be discovered, and it is this: Twice so many shillings as there are farthings and once so many Groats in the price of one pound is the value of 1C. weight. Example, let the price of one pound be 3 pence or 12 farthings,

Twice 12 shillings is | 24s | ||

Once 12 groats is | 4 [...]. | ||

lib. | |||

The Total is | 1 | 8 | 0 |

The proof is plain. 1 lib. 12 far. 112 1344 in which there are 336 pence, that is 28 shillings.

17. By all the preceding rules, the price of one Integer being given, I have shewed how the price or value of any certain number of Integers may be found at that rate; but for as much as the converse of this is no less necessary to be known, I will now shew how from the known price of many Integers, to find the price or value of one.

[Page 134]If 123 Ells cost 61 lib. 17 shillings, 6 pence, what is the price of 1 Ell: Reduce the pounds and shillings into shillings & divide by 123, the quotient shall be the answer to the question; thus 61 lib. 17 shillings being reduced do make 1237 shillings, which being divided by 123 the quotient is 10 shillings, and 7 remains, which being reduced into pence is 84, which with the 6 pence in the question do make 90 just, which being less than the divisor I reduce into farthings and they make 360, which being divided by 123 the quotient is 2 farthings 9268 parts of a farthing, as by the work appeareth.

[Page 135] [...] [Page 136] But if the first term be 100, the question will be answered by Reduction only, as if 100 Ells cost 74. 13. 9d. what is the price of 1 Ell? [...]

18. In the Rule of Three direct or inverse, when the Divisor with either of the other two given numbers may be severally divided, by some common measure, without leaving any remainer, the quotients may be taken for new terms, and proceeding in like manner as often as is possible, the operation will be much contracted.

Example, If 25 Yards, cost 75 pound, what shall 85 Yards cost?

y. | |||

25 | 75 | 83 | |

5 | 15 | 83 | |

1 | 3 | 83 | (249 |

Another Example, if 27 men will finish a work in 25 days, in how many days will 15 men finish the same work? Answer in 45 days.

men | days | ||

27 | 25 | 15 | |

9 | 25 | 5 | |

9 | 5 | 1 | (45 |

In the first rank you may observe, that the Divisor 15 (for the rule is inverse) and the first term 27 being severally divided by their common measure 3, the three new terms (in the second rank) will be 9. 25. 5. Again in the second rank, the Divisor 5 and the second term 25 being divided by their common measure 5, the three new terms in the third rank, will be 9. 5. 1. Lastly working with these as the Rule of Three inverse requires, the answer to the question will be 45.

19. In the Rule of Three direct or inverse, when the Divisor and either of the other two terms are fractions, having a common denominator, the said▪ denominators may be rejected, and their numerators retained as new terms: Example, if ⅝ of an Ell cost 65 pence, what shall ⅞ cost, the answer will be found 91 pence, and the work will stand as here you see.

⅝ | 65 | ⅞ | |

5 | 65 | 7 | |

1 | 13 | 7 | (91 |

20. In the rule of three direct or inverse, when one of the given terms is a fraction, if it be not the Divisor, the Divisor may be turned into a fraction having the same denominator, or if the Divisor only be a fraction, one of the other terms may be turned into a fraction having the same denominator; and in either case the common denominators may be cancelled.

An Example of the first case may be this▪ if 12 Yards in length will make a Cloak, the stuff or Cloth being ¾ of a Yard broad; how much Stuff of one Yard broad will make a Cloak of the same length and compass? Answer, 9 Yards.

¾ | 12 | 1 | |

¾ | 12 | [...]/4 | |

3 | 12 | 4 | 36/4 or 9 Yards |

An Example of the second case. If ⅞ of a yard cost 16 shillings, what shall one Yard cost? Answer

Yard | |||

⅞ | 16 | 1 | |

⅞ | 16 | [...]/ [...] | |

7 | 16 | 8 | (18 [...]/7 |

## CHAP. XXIV. Of the double Golden Rule.

1. IN the two last Chapters I have shewed the nature of the single Golden Rule, with the ordinary and the practical manner of working the same: I come now to speak of the compounded Golden Rule.

2. The compounded Golden Rule or rule of Proportion is, when more than three terms are propounded.

3. Under the compounded Golden Rule, is comprehended the double Golden Rule, and divers Rules of plural Proportion.

4. The double Golden Rule is, when five terms being propounded, a sixt in proportion to them is demanded: the greatest difficulty whereof is in placing of the terms; for which observe, as in the former Rule of Three,

- 1. That the first and third numbers must be both of one kind.
- 2. That the two first terms in the question do consist of a supposition, and the third of a demand.

Example, if 10 [...]l. gain 6 pound in 12 months, how much will 65 pounds gain in 8 months?

Here you see the supposition is, If 100l. gain 6; the demand is, how much will 65 pounds gain: & therefore these terms must stand thus, [...] and the other two terms must be set under [Page 140] the numbers to which they have relation, the 12 months under 100, and the 8 months under 65.

5. The terms of the question being thus placed, a resolution may be made, either by two single Rules of Three, or by one Rule of Three compounded of the five numbers given.

6. If you make the resolution by two single Rules of Three, you must consider, whether the terms in both Rules be in a direct proportion or not, by observing whether the Demand be more or less, as hath been taught in the single rule of Three; in this example they are both direct, and the proportions are,

- 1. As the uppermost term of the first place, is to the middle term; so is the uppermost term of 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 required.

See in the example before set down; using the lower term of the first place as a common number in the first proportion say, If 100 pounds gain 6 pounds in 12 months, how much will 65 pounds gain, in the same time: which being a direct proportion, the fourth term proceeding from the said 3 given numbers 100. 6. 65 is 3. 90.

Again, to find the term required, using the uppermost term of the third p [...]e, as a common number in this proportion, I say, 2. If in 12 months 65 pounds gain 3. 90, how much would 65 pounds gain in 8 months, which being also a direct proportion, the fourth proportional proceeding from the said 3 numbers given 12. 3. 90. 8. is 2. 60. So I [Page 141] conclude that if 100 pounds gain 6 pounds in 12 months, 65 pounds will gain 2. 60 pounds in 8 months, as you may observe by the work. [...]

7. But to resolve this question by one Rule of Three compounded of the five numbers given, they being placed as hath been already directed, you must take the product of the two numbers in the first place for the first term; and the product of the two numbers in the last place for the third term, then working as in the single Rule of Three the answer will be 2. 60. as before, as by the operation doth appear. [Page 142] [...]

8. A second question or example may be this.

If 8 Clerks write 154 sheets in 6 days, how many Clerks will write 462 sheets in 12 days?

Here the supposition is, if 154 sheets be writ in 6 days; the demand is, in what time 462 sheets may be writ; the terms of the question must therefore stand thus.

Sheets | Clerks | Sheets. | |

154 | 8 | 462 | (24 |

6 | 12 |

And then to resolve this question by two single Rules of Three, according to the directions of the fifth Rule of this Chapter, I say: by the Rule of 3 direct,

If 154 sheets canbe writ by 8 Clerks in 6 days, How many Clerks will write 462 sheets in that time, and the fourth term proceeding from the said 3 numbers given 154. 8. 462 is 24.

[Page 143]2. I say, if 24 Clerks write 462 sheets in 6 days, how many Clerks will write the same number of sheets in 12 days? Now here more time being allowed a lesser number of Clerks will serve to do the work; and therefore by the Rule of Three inverse, the fourth term proceeding from the said 3 numbers given 6 24 12 is 7 12; so I conclude, if 8 Clerks write 154 sheets in 6 days, that 7 Clerks will write▪ 462 sheets in 12 days, as by the work appeareth. [...]

9. But to resolve this question by one rule of Three compounded of the five numbers given, they being placed as before, one of the single Rules being inverse, you must multiply the lower numbers in the first and last terms, by the upper numbers cross wise, that is the upper number of the first term, by the lower number of the last, and the uppermost of the last terms by the lower of [Page 144] the first, and write each product under the lower term by which it is produced; and then if the inverse proportion be found in the uppermost line using those products as single terms, proceed to find the term required by the single Rule of Three direct: but in case you find the inverse proportion in the lower line, perform the work by the single Rule of Three inverse. So in this example the terms standing as before, I multiply 154 by 12 and the product 1848 I set under 12: again I multiply 462 by 6, and the product 2772 I set under 6, then because the inverse proportion is in the lower line, I proceed to find the fourth term required, by the Rule of Three inverse, and find the fourth proportion all, from the 3 given numbers 2772. 8. 1848 to be 12, as by the following work appears. [...]

But the terms of this question being so ordered as that the uppermost numbers in the first and last places be set in the place of the lower, and the [Page 145] lower in the place of the first, the inverse proportion will be found in the upper line, and then working by 2 single rules of three, the fourth proceeding by an inverted proportion from the said three numbers given 6. 8. 12 will be 4.

And the fourth term proceeding from the 3 given numbers 154. 4. 462 in a direct proportion, will be 12, as by the work appeareth. [...]

Again the resolution of this question by one rule of three compounded of the five numbers thus placed will be by a direct proportion, because the inverted proportion is in the first line, as by the work appeareth. [Page 146] [...]

## CHAP. XXV. Of the Rule of Fellowship.

1. THe Rules of plural proportion are those, by which we resolve questions, that are discoverable by more Golden Rules than one, and yet cannot be performed by the double Golden Rule mentioned in the last Chapter.

2. Of these Rules there are divers kinds and varieties according to the nature of the question propounded: for here the terms given are sometimes four, sometimes five, sometimes more, and the terms required sometimes more than one.

[Page 147]3. The particular Rules of plural proportion which I shall here treat of are these, the Rule of Fellowship, the Rule of Company, the Rule of Barter and Exchange, and the Rule of Alligation.

4. The Rule of Fellowship is that by which in accounts amongst diverse men (their several stocks together with the whole gain or loss being propounded) the gain or loss of each particular man may be discovered.

5. This Rule is either single or double.

6. The single Rule of Fellowship is, when the stocks propounded do all continue in the adventure, for equal times, that is, the one as long as the other.

7. In the single Rule of Fellowship, the total of all the stocks must be the first number in the Rule of Three, the whole gain or loss the second; and each particular mans stock the third; and therefore the proportion is this.

As the whole stock is to the whole gain or loss, so is every particular mans stock to his particular gain or loss: and by this proportion you must work as often as there are particular stocks in the question.

Example, Three Farmers hired a Shepherd to keep their sheep, for 8 pound 15 shillings per Annum.

A committed to his care 357 sheep, B 465 and C 543, I demand how much each man must pay of the 8 lib. 15s.

Here 357, 465 and 543 are the several stocks propounded, whose total 1365 is the first term: 8 lib. 15s. the shepherds wages is the second, [Page 148] and 357 the first mans stock, is the third term, in the first proprtion, and therefore I say as 1365 to 8 lib. 15 shillings, or to 175s; so is 357 to

357 | 357—45.77 | ||

465 | 1365 | 175 | 465—59.61 |

543 | 543—69.62 | ||

17500 |

2. Example, Four Merchants adventured to Sea a stock of 4858 pounds: A put in 2315 pounds; B put in 946 l. C put in 834 l. D put in 763 l. but the Mariners meeting with a storm at Sea, were constrained to cast overboard as much goods as did amount to 879 l. The Question is, what each man's loss is? The which is thus resolved.

A. 2315 | 2315—418.872 | ||

B. 946 | 946—171.167 | ||

C. 834 | 4858 | 879 | 834—150.902 |

D. 763 | 763—138.066 | ||

878.997 |

8. The double Rule of Fellowship is, when every mans stock hath a relation to a particular; that is when every particular stock is multiplied by the time, for which it doth continue in Fellowship, and hath no other difference from the single Rule of Fellowship but only this, that the first number is the sum of these several products, and each particular product is the third, the whole gain or loss [Page 149] is the second in both Rules, and therefore the proportion to be observed in the solution of such questions as fall under this Rule is

As the sum of the products found by multiplying each stock by its own time, is to the whole gain or loss: so is each particular product, to its particular gain or loss.

1 Example, A B and C hold a piece of ground in common, for which they are to pay 36l. 10s. 6d. Into this pasture A put in 23 Oxen 27 days, B put in 21 Oxen 35 days, and C put in 16 Oxen 23 days, the question is what each man is to pay of the said rent of 36 lib. 10s. 6 pence. First multiply each stock by its own time, and the several products will be as followeth.

Products | Total | lib. | s. | d |

A. 23 × 27 = 521 | ||||

B. 21 × 35 = 420 | 1309 | 36. | 10. | 6 |

C. 16 × 23 = 368 |

And hence the particular proportions are

1309 | 36.525 | 521 | 14.53745 |

1309 | 36.525 | 420 | 11.71925 |

1309 | 36.525 | 368 | 10.26829 |

36.52499 |

2. Example may be this, Three Merchants company for 18 months; A put in 500l. and at 5 months end took out 200l. and at 10 months end put in 300l. and at 14 months, cook out 130 pound. [Page 150] B put in 400l. and at 3 months put in more 270l. and at 7 months took out 140l. and at 12 months put in more 100l. and at 15 months took out 99l. C put in 900l. and at 6 months took out 200l. and at 11 months put in 500l. and at 13 months took out 600 pounds, and they gained 200l: I demand what each mans part of the gains comes to.

In Questions of this nature, two things are principally to be observed.

- 1. The whole time for Partnership.
- 2. The respective time belonging to each mans stock. So here it is evident that the whole time is 18 months, and the particular stocks and times belonging to each Merchant are as followeth.

A | 500 lib. 5 | Months. | 2500 | 8280 |

300 lib. 5 | 1500 | |||

600 lib. 4 | 2400 | |||

470 lib. 4 | 1880 | |||

B | 400 lib. 3 | Months. | 1200 | 10293 |

670 lib. 6 | 4020 | |||

530 lib. 3 | 1590 | |||

630 lib. 3 | 1890 | |||

531 lib. 3 | 1593 | |||

C | 900 lib. 6 | Months. | 5400 | 14300 |

700 lib. 5 | 3500 | |||

1200 lib. 2 | 2400 | |||

600 lib. 5 | 3000 |

A. 8280 | 8280—50.376 | ||

B. 10293 | 32873 | 200 | 10293—62.623 |

C. 14300 | 14300—87.001 | ||

200.000 |

9. The Rule of Fellowship is proved by adding the terms required, whose sum ought to be equal to the second term in the Question, or else the whole work is erroneous.

3 Example, There is 20 shillings to be divided amongst 4 men, of which A. is to have ⅓, B. ¼, C. ⅕ and D. ⅙. The question is what every mans share is.

To answer this question, and those of the like nature, these fractions ⅓ ¼ ⅕ and ⅙ must be reduced to fractions of one and the same denomination, as hath been shewed in the 9th rule of the 12th Chapter; so will ⅓ be equal to 120/300, ¼ will be equal to 90/ [...], ⅕ will be equal to 72/300 and ⅙ will be equal to 60/300. Now then neglecting the denominators the sum of the numerators will be, and then the question may be resolved as here you see.

342 | 120 | 84. | 72. | 342. | |

90 | 63. | 54. | |||

72 | 240 | 50. | 180. | ||

60 | 42. | 36. |

which being added make 240 pence.

## CHAP. XXVI. Of the Rule for the exchange of Coyns, Weights and Measures.

1. THe rate and proportion between Coyns Weights and Measures of different kinds being known, either from some good Author, or rather by experience, it will be easie for such as understand the Rule of Three, to convert one Species into another; when the question concerns no more than two or three sorts: but when the comparison is made between more than three, one single Rule of three without some other help will not resolve the question.

2. Now because there is nothing more usual with Merchants and Tradesmen, than to exchange the Coyn of one Country for the Coyn of the like value in another, and in like manner to Barter for commodities of different weights and measures, I will here shew how such exchange or Barter is to be computed, and then especially when more is required to resolve the question, than one single Rule of Three only.

3. The questions of this kind being all resolvable either by one single Rule of Three, or by the Rule of Three often repeated; there needs no other directions, than what hath been already given in the Rule of Three, to wit, that care be had to make the first and third numbers of one kind, that is, if the first be sterling mony, the third must [Page 153] be so too; if the first be Flemish, the third must be Flemish. Example, How much Flemish mony must be received for 340 lib. Sterling; every 20 shillings sterling being valued at 34 shillings 7 pence Flemish; here the proportion is.

s. | Flemish | ster. | Flemish | |

As | 20 | 34s. 7d. | 340 lib. |

Or reducing the 34s. 7d. Flemish into a decimal.

s. | Flemish | ster. | Flemish |

1 | 1.729166 | 340 | 587.916440 |

That is 587 lib. 18 shillings 4 pence, that is 45 lib. 12s. sterling.

Spanish Pistolet, | ster. | Spanish Pistolet | ||

2. | As 73/ [...] | 1 | 4560/ [...] | 6234/ [...] |

But if the terms be placed according to the following directions, this and all other questions of the like nature may be resolved by one single Rule of Three.

1 Spanish Pistolet = | 73/ [...] shillings sterling. |

38/ [...] shillings sterling = | 1 Crown Genoa. |

120 Crowns Genoa = | Spanish Pistolets. |

Which order of placing the said given numbers being observed,

2. Example, If 1 Spanish Pistolet be equal in value to 14⅗ shillings sterlings and 7⅗ shillings sterling [Page 155] equal to 1 Crown Genoa; how many Spanish Pistolets are equal to 120 Crowns Genoa? For the more easy resolution of this question and all others of the like nature take these directions; set down the numbers which make the supposition in a Column one under another; and the numbers which answer the supposition right against them in another Column one under another also; then consider whether it be required to find, how many pieces of the first Coin, are equal in value to a given number of pieces of the last Coin. Or whether it be required to find, how many pieces of the last Coin are equal in value to a given number of pieces of the first.

In the first case place the number that makes the Question under the first Column, under the several suppositions, and in the last case place the said number that makes the question, in the second column, under the several numbers that answer to the suppositions; then proceed to answer the question by the Rule of three in this manner.

When you are to find, how many pieces of the first Com are equal in value to a given number of pieces of the last, as in this question you are, The first term in the Rule of Three must still be one of the numbers in the second Column; and look how many several numbers there are there, so often must the Rule of Three be repeated, which in the present question are two: the second and third terms in the first Rule must be the first and second numbers in the first Column, and the fourth term in proportion to these three numbers given must be the one of the middle terms in the second Rule, [Page 154] and so continually, as oft as the Rule of Three shall be repeated: In this question, the terms of the first proportion stand thus.

sterling | Pistolet | sterling | Pistolets |

73/5. | 1. | 38/5. | 38/7 [...]. |

And then | Crown | Pistolet |

1 Crown Genoa [...]8/73 Pistolets that is 62 34/73 Pistolets. | 120 | 45 60/76 |

But the terms being placed into two Columns according to the former directions, this question, and all other of the like nature, how many numbers soever there be in each Column, may be more briefly resolved thus.

Multiply all the given terms in the first Column according to the Rule of continual Multiplication, and reserve the last product for a Dividend: again, Multiply continually all the terms in the second Column, so shall the last product be a Divisor, and the Quotient arising from this Dividend and Divisor, shall be the answer desired.

In this present question the numbers in the first Column are, 1 38/5. 120. and the product of them is 45 60/5 which being divided by 73/ [...] the product of the numbers in the second Column, the Quotient is 62 [...]4/ [...], as before.

But if the question had been, how many pieces of the last Coin were equal in value to a given number of the first, that is, how many Crowns Genoa were equal to 120 Spanish Pistolets, the last number 120 Pistolets, must have been placed in the second Column, as here you see.

[Page 157]1. Column 2. Column

1 Spanish Pistolet = [...]3/ [...] shillings sterling.

3 [...]/ [...] shillings sterling = 1 Crown Genoa.

Crown Genoa = 1 [...]0 Spanish Pistolets.

And then to resolve this question by two Rules of Three in the ordinary way; the first in the second Column must be the first term, in the first Rule, and the first and second numbers in the first Column, must be the second and third terms as before; and the fourth proportional arising from those three numbers given, must be one of the first terms in the second, to which the second number in the second Column must be one of the middle terms in the last proportion.

Ster. | Pistolet. | Ster. | Pistolet. |

[...]/ [...] | 1 | 38/5 | 38/75 |

Pistolet | Crown Genoa | Pistolet | Crown Genoa. | |

2. | 78/ [...]3 | 1 | 120/ [...] | 23. [...]0/38 |

Note, That when the same numbers happen to be multiplicators in the Dividend and also in the Divisor, such multiplicators may be neglected in both, and much labour in questions of this nature may be many times saved; and then the two proportions will be as here you see them; and how oft soever the Rule of Three must be repeated, you must proceed as in the former directions till you come to the last, and then make the [...]m fourth proportional, [Page 156] to be the first term in the last Rule of three, and by that dividing the product of the second and third terms, the quotient shall be the answer required; as by the operation it doth appear.

Or thus: Multiply 73/5. 1. 120 the numbers in the second Column continually, so will the product be 876 [...]/5 for a dividend, and the numbers in the first Column 1. 38/5 being almost multiplied together the product is 38/5 for a Divisor, and the quotient arising thence is 42 [...]/ [...] or 23 20/3 [...].

3. Example: If 27 Ducatons Florence, be equal in value to 5 lib. 19 shillings, 3 pence sterling; and 4 lib. 5 shil. 4 pence sterling, equal to 16 Crowns at Lyons; how many Ducatons Florence are equal to 43 Crowns at Lyons? The numbers being placed according to the former directions will stand thus.

27 Ducatons = | 5 lib. 19 shillings, 3 pence sterl. |

4l. 5s. 4d. sterling = | 16 Crowns Lyons. |

43 Crowns Lyons = | Ducatons Florence. |

The which question by two Rules of Three is thus resolved.

Pence | ster. | Ducatons | d. sterling | Ducatons |

1431 | 27 | 1024 | 19.32075 |

Crowns | Lyons | Ducatons | Cr [...]tyons | Ducatons |

16 | 19.32075 | 43 | [...]1.9 [...]45 |

Or thus: 1431 multiplied by 16, the product is [Page 159] 22896 and 27. 1024. 43 being multiplied continually the product is 1188864, which being divided by the former Product the quotient is 51.9245 Ducatons as before.

But if the question had been, how many Crowns Lyons are equal to 27 Ducatons; the second proportion would have been 19. 32075. 16. 43. 35. 60937: Or the product of [...]431. 16. 43 is 984528, and the product of 1024. 27 is 27648, by which dividing the former product the quotient is 35. 60937.

4. Example: If ½ Pistolet of Spain be valued at 3l. 13s. 6d. Tournois, 6l. Tournois at 141 Flemish, 28l. 14s. 7d. Flemish at 24l. 12s. 6d. sterling, how many Pistolets ought I to receive for 27l. 6s. 9d. sterling? Answer 98 41/ [...]1 Pistolets. For

Tournois | Pistolet | Tournois | Pistolets | |

1. | 147s. | 1 | 120 [...]. | 120/147 or 40/ [...] |

Flem. | Pistolets | Flem. | Pistolets | |

2. | 168/ [...]2 | 40/49 | 689 [...]/12 | 49728/ [...] |

that is 33 74/147 Pistolets. Lastly

Flem. | Pistolets | Flem. | Pistolets | |

3. | 1970/4 | 4925/147 | 5787/ [...] | 48462/ [...] |

Or 98 [...]2/ [...] Pistolets.

Or placing the terms according to the former directions they will stand thus.

1. Pistolet = | 1764d. Tournois. |

1440d. Tourn. = | 168d. Flemish. |

6895d. Flem. = | 5910d. sterling. |

17361d. ster. = | Pistolets. |

The product of 1764. 168. 5910 being multiplied continually is 1751440320. And the numbers 17361.6895.1440 being multiplied continually, the product is 172373896800; which being divided by the former product, the Quotient is 98 73274544/ [...]5144 [...] or 98 41/9 [...] as before.

5. Example shall be of several measures compared with one another.

Suppositions, 45 Ells of Norimberg = | 36 at Vienna▪ |

Suppositions, 17 Ells of Vienna = | 21 at Venice |

Suppositions, 280 Ells of Venice = | 208 at Colen. |

Question, 95 Ells of Colen = | Norimberg. |

The numbers in the first Column 4517. 180.95 being multiplied continually, the product is 130 81500. And the numbers 36. [...]1.208 in the second Column being also multiplied continually the product is 157248, by which dividing the former product, the quotient 83. 1902 is the number of Ells at Norimberg, equal to 95 Ells at Colen.

### 6. Example.

15 lib. Averdup. London. = | 13½ lib. Amsterdam. |

60 lib. Amsterdam = | 65½ at Bruges. |

122½ at Bruges = | 145 at Dentzick. |

Dentzick = | 112 at London. |

[...]

7. Example, May 1. 1669. A Merchant stocked his Factor with 1000 lib. sterling and 64 pieces of Cloth, Worcester Whites 38 Ells English in every piece, and 16 shillings sterling by the Ell, all which the Factor in way of Traffique disposeth of in manner following.

942 lib. sterling = | 2355 Ells of Taffity London. |

1547 Ells Taffity = | 045 Pipes of Wine |

015 Pipes of Wine = | 0173 pieces of Dowlace |

0138 pieces of Dowlace = | 55 pieces of Holland |

64 pieces of Cloth = | 275 Spansh Pistolets. |

198 Spanish Pistolets = | 5821 Florins. |

275 Florins = | 114158 lib. Flemish. |

114. 158 lib. Flem. = |

November 17. 1669. The Merchant and his Factor are to come to Ballance. The question is twofold.

- 1. What the Factor hath remaining in his hands?
- 2. What it is worth in pounds sterling?

- To the first the Factor is Debtor
- Ells
- Imprimis for London Ells of Taffity
- 808
- Item pieces of Dowlace
- 35
- Item pounds Sterling
- 58
- Item pieces of Holland
- 55
- Item Spanish Pistolets
- 77
- Item Florins
- 5546
- Item pounds Flemish
- 114. 358

To find the value of 808 Ells of Taffity.

Ells Taffity | ster. | Taffity | lib. | |

2355 | 942 | 808 | 323 | 2 sterling |

that is 323 lib. 4 shillings.

Then to find the value of the 35 pieces of Dowlace in pounds sterling. The terms of the questions will stand thus.

942 lib. sterling = | 2355 Ells Taffity |

1547 Ells Taffity = | 15 Pipes of Wine |

15 Pipes = | 173 pieces of Dowlace |

35 pieces Dowlace = | pounds sterling. |

Here according to the Method prescribed; the four numbers in the first Column 942. 1547. 15. 35 being multiplied continually the product is 765053850, for a dividend, and the numbers in the second Column 2555. 15. 173, being also multiplied continually the product is 6111225 for a Divisor, and the quotient arising from these two products is 2 [...]. 1882 lib. sterling for the value of the 30 pieces of Dowlace.

But because that one of the numbers in the first [Page 162] Column is the same with one of the numbers in the second, according to the note at the end of the second example, that number may be neglected in both, and then the numbers in the first Column are 942. 1547. 35, which being multiplied continually the product is 51003590 for a Dividend, and the numbers in the second Column to be multiplied are 2355. 173, and their product is 407415 for a Divisor, and the quotient arising from these two numbers is 125. 1882 pounds sterling as before.

3. To find the value of the 55 pieces of Holland. The terms of the question will stand thus.

942 lib. sterling = | 2355 Ells of Taffity |

1547 Ells of Taffity = | 15 Pipes of Wine |

15 Pipes of Wine = | 173 pieces of Dowlace |

138 pieces of Dowl. = | 55 pieces of Holland |

55 pieces of Holland = | pounds sterling. |

Here the 15 Pipes of Wine, and the 55 pieces of Holland may be neglected in both Columns, and then the numbers to be multiplied in the first Column are 942. 1547. 138, and the product of them is 201103872, and the numbers in the second Column are 2355. 173, and the product 407415 as before, and the quotient arising from these numbers is 493. 6092 pounds sterling, which is the value of the 55 pieces of Holland.

[Page 163]The Factor then hath in his hands.

- Imprimis ready mony of the 1000l.
- 58.0000
- Item 808 Ells of Taffity worth
- 323▪2000
- Item 35 pieces of Dowlace worth
- 125.1882
- Item 55 pieces of Holland worth
- 493.6092
- That is 999 lib. 19 shil. and 11½ pence
- 999.9974

The account for the 64 pieces of Cloth which cost 30.4 pounds sterling per piece, that is in all 1945.6 lib. ster. is thus: first then for the worth of the 77 Pistolets.

Pistolet | ster. | Pist. | ster. | |

275. | 1945. | 6 | 77 | 544.768 |

Secondly for the worth of the 5546 Florins. The terms of the question will stand thus.

1945.6 lib. sterling = | 275 Spanish Pistolets |

198 Span. Pisto. = | 5821 Florins |

5546 Florins = | lib. sterling. |

Here the numbers in the first Column to be multiplied together are 1945. 6. 198. 5546, and the product is 21364789248 for the Dividend, and the numbers in the second Column to be multiplied are 275.5821, whose product is 1600775 for the Divisor, and the quotient arising from these two products is 1334.6528 which is the value of 5546 Florins in pounds sterling.

[Page 164]Lastly for the worth of 114. 158 Flemish, the terms of the question will stand thus.

1945. 6 lib. sterling = | 274 Spanish Pistolets |

198 Spanish Pisto. = | 5821 Florins |

275 Florins. = | 114. 158 lib. Flemish |

114. 158 lib. Flem = | pounds sterling. |

Where neglecting the 275 Pistolets in one Column and 275 Florins in the other and 114. 158 lib. Flemish, in both, the numbers in the first Column to be multiplied are 1945. 6 and 198, whose product is 3852288 for the Dividend, and 5821 in the other Column is the Divisor, and the quotient arising from two numbers is 66. 1791 lib. sterling for the value of 114. 158 lib. Flem.

So that for the 64 pieces of Cloth the Factor hath in his hands.

- Sterling
- Imprimis 77 Pistolets worth
- 544. 7680
- Item 5546 Florins worth
- 1334. 6528
- Item 114. 158 lib. Flemish, worth
- 66. 1791
- And the ballance is
- 1945. 5999

## CHAP. XXVII. The Rule of Aligation.

1. ALligation is an Art, by which we resolve questions that concern the mixing of divers simples together.

2. Alligation is twofold Medial, and Alternate.

3. Alligation Medial is, when the several quantities and rates of divers simples being, we find out a mean rate for which a mixture made of these simples may be afforded: to effect this, the sum of the quantities being given with the value of all the simples, the proportion is.

As the sum of the quantities

Is to the value of all the simples,

So is any part of the mixture propounded

To the mean rate that is required.

Example: let 12 Gallons of Canary at 4 shil. the Gallon, 36 Gallons of Sherry at 3 shillings the Gallon, and 52 Gallons of White-Wine, at 2 shillings 8 pence the Gallon be mixt together; and let the price of one Gallon of that mixture be required. The sum of 12, 36 and 52 Gallons is 100 Gallons: and the value of 12 Gallons of Canary at 4 shillings or 48 pence by the Gallon is 576d. the value of 36 Gallons of Sherry at 3 shil. or 36 pence is 1296d. and the value of 52 Gallons of White-Wine at 2 shillings 8 pence the Gallon, or 32 pence is 1664 pence; and all these values added [Page 166] together do make in the whole, 3536d. Now then by the Rule of Three.

100. 3536 1 35 36/104 that is 35 pence, and 36 hundreds of a penny.

For proof of the work, compare the total value of the several simples, with the value of the whole mixture; if their sums agree the work is true as in this example.

l | s | d | |

The value of 12 Gallons of Canary at 4 shillings per Gallon is | 2 | 08 | 00 |

The value of 36 Gallons of Sherry at 3 shillings the Gallon is | 5 | 08 | 00 |

And the value of 52 Gallons of White Wine at 2 shil. 8 pence the Gallon is | 6 | 18 | 08 |

All which amount to | 14 | 14 | 08 |

And the value of 100 Gallons at 35d. and 36 hundreds of a penny by the Gallon is 14 14 08 as before.

5. Alligation Alternate is, when several rates of divers simples being given, such a mixture is required, as may be sold at a certain mean rate propounded.

6. Alligation Alternate is either partial or total.

7. Alternation partial is, when the several rates of divers simples, with the quantity of one of them is given, and the several quantities of the rest are required, in such sort that a mixture being made [Page 167] according to the given quantities, and the quantities so found, that mixture may bear a certain rate propounded.

8. Alternation total is, when the total quantity of all the simples with their several rates being given, we find out their several quantities, in such sort, as that a mixture of them being made according to the quantities so found, that mixture may bear a certain mean rate propounded.

9. In both these kinds of alternation, there is some preparation to be made, before the Rules by which such questions may be resolved can be well delivered.

10. The preparation to be made doth partly consist in placing the given rates of the things propounded and taking the differences between those rates, and the mean price you would have them bear.

11. The given rates of the things propounded must be placed one under another so as that they may orderly increase or decrease, and the mean price that you would have them bear, may stand on the right hand of the rank about the middle: For example, let the given rates be 12. 24. 36. 48. and the mean price 28d. which being placed one under another descending I [...] draw a line of connexion, and on the left hand thereof I set 28 by the mean price by it self, as here you see. The terms being thus ranked in their due order, link them together by certain arches, in such sort, as that one that is greater than the mean price, may be still coupled with another that is [Page 168] less, so in the premised example 12 may be linked with 36 or 48, and 24 may be linked with 48 or 36, and the work will stand thus. [...] or thus [...]

The terms being thus ranked and linked together, proceed to find the differences, between the given rates and the mean price, and write that difference just against his respective yoke-fellow: thus the difference between 12 and 28 being 16, I place 16 against 48 its respective yoke-fellow; and the difference between 24 and 28 being 4, I place 4 against 36: the difference between 28 and 36 being 8, I place 8 against 24: lastly 20 being the difference between 28 and 48, I place 20 against 12, and the work will stand according to the first way of linking the rates propounded, as here you see them.

But the branches being linked after the other manner the differences must be otherwise placed; for here 24 hath 48 for his Yoke-fellow, and 12 hath 36 for his. [...]

[Page 169]12. And many times it so falls out, that one of the given rates, may be linked with two or more of the other rates given, in which case the differences ought to be as often set down, as it is or may be diversly linked: thus in the premised rates given, if you change the mean price to 16, the rates must be linked and the differences set as here you see them: or being 40 thus. [...] [...]

13. The given rates being thus ranked and linked and their differences taken, all questions in Alternation partial may be resolved by this proportion.

As the difference against the first rate

Is to the several differences that are under it,

So is the quantity propounded

To the several quantities that are required.

Example, Suppose a man were to mix 10 bushels of Wheat at 48 pence the bushel, with Rye at 36d. the bushel, Barley at 24 pence the bushel, and Oats at 12d. the bushel, so that the whole mixture may be sold at 28d. the bushel, the question [Page 071] is, how much Rye, how much Barly, and how much Oats ought to be added to the 10 bushels of Wheat. It is evident by the differences before taken, that for every 16 bushels of Wheat, I ought to take 4 bushels of Rye, 8 bushels of Barley, and 20 bushels of Oats, and therefore I say

- 1. As 16. 4. 10. 28/ [...]
- 2. As 16. 8. 10. 5
- 3. As 16. 20. 10. 128/ [...]

And from hence it appears that 2 bushels and a half of Rye, 5 bushels of Barley, and 12 bushels and a half of Oats being put to 10 bushels of Wheat, when these several sorts of grain do bear the prices as in the 11 rule, may be sold one with another for 28 pence, or 2 shillings and 4 pence the bushel.

Or the differences being taken according to the second way of linking them expressing the same rule, I say

- 1. As 4. 16. 10. 40 bushels of Rye.
- 2. As 4. 20. 10. 50 bushels of Barley.
- 3. As 4. 8. 10. 10. 20 of Oats, and a mixture being made according to these proportions, the whole may be sold at 2 shilling 4 pence the bushel.

14. In Alternation partial, the proof is▪ likewise by comparing the total value of the several simples, with the value of the whole mixture: so in the last example of the former rule, the total value of 10 bushels of Wheat 40 bushels of Rye, 50 bushels of Barley and 20 bushels of Oats amounts [Page 171] to 14 pound, which is also the value of the whole mixture at 2 shillings and 4 pence the bushel.

15. In Alternation total, the several rates and mean price, being so placed, and their differences taken as hath been shewed, all questions belonging thereto may be answered by this proportion.

As the sum of all the differences

Is to the total quantity of all the simples;

So is the correspondent difference of each rate

To the respective quantity of the same rate.

Example, A Goldsmith having diverse sorts of gold, viz. some of 24 Caracts fine, some of 22 Caracts, some of 18 Caracts and some of 16 Caracts fine, is desirous to melt of all these sorts, so much together, as may make a Mass containing 60 Ounces of 21 Caracts fine. The numbers being placed and differenced as hath been shewed, and is here expressed, I say [...]

- [Page 172]1. As 12. 60. 5. 25 Ounces.
- 2. As 12. 60. 3. 15 Ounces.
- 3. As 12. 60. 1. 5 Ounces.
- 4. As 12. 60. 3. 15 Ounces.

Whereupon I do conclude that 25 ounces of 24 Carects fine, 15 ounces of 22 Carects, 5 ounces 'of 18 Carects, and 15 ounces of 16 Carects fine, being all melted together, will produce a Mass of Gold containing 60 ounces of 21 Carects fine, as was required in the question propounded.

16. Here the work is true, when the sum of the quantities found are equal to the total quantity propounded: thus in the preceeding example, the quantities found were 25. 15. 5. 15. which being added together do make 60, and the quantity propounded was 60 also.

17. When a question in Alligation Alternate doth consist of three ingredients, it is capable of an infinite Series of Affirmative and Negative answers▪ and when the question doth consist of more than three ingredients, it is capable of as many infinite Series of affirmative and negative solutions, as you please to impose upon it.

18. In Alligation Alternate consisting of three ingredients, (their values or prices being orderly placed, and the differences between them being taken as hath been already shewed) if you take the difference of every two values, and place them against the third, you shall constitute two continual addends, and one continual subducend, that is, the two continual addends are outmost, and the [Page 173] continual subducend is inmost; or if you please, you may let the two outmost be continual subducends, and then the inmost will be a continual addend: or any three numbers in the same proportion will effect the same thing: but it is best to take the three least numbers, that be in the same ratio: for example, let the value or prices of the three ingredients be 12. 24. 36, and let the mean price be 16, these prices being orderly placed, and the differences between them and the mean price will be as here you see them. [...]

Now then deducting 12 from 24 the difference is 12, which I place against 36, towards the left hand, and the difference between 24 and 36 is 12 also, which I place against 12, lastly the difference between 12 and 36 is 24, which I place against 24, and because these differences are large and so not so convenient for this work, I reduce them to the least numbers in the same proportion, and then they are 1. 2. 1. the two outmost to wit 1. and 1. shall first be continual addends, and the inmost or middle number, to wit 2, let be a continual subducend, and then some of the many answers that might be given to this question will be these following.

[Page 174] [...] Of which the first is affirmative and the two last negatives, but maketh the two outmost to wit 1 and 1 to be continual subducends, and the inmost to wit 2 a continual addend; the answers to this question will be these following.

1 | 16 | 12 | 28 | 27 | 26 | 25 | 24 | 23 |

+ 2 | 24 | 4 | 6 | 8 | 10 | 12 | 14 | |

1 | 36 | 4 | 3 | 2 | 1 | 0 | 1 | |

36 | 36 | 36 | 36 | 36 | 36 |

And thus you may proceed in Infinitum: but the negative answers, being improper and of no use in practice, it will be sufficient to go as far as they are affirmative and proper, taking still something of each ingredient in the question.

19. In Alligation Alternate consisting of more than three ingredients, having set down the several prices, and taken the difference between them, and the mean price, you may constitute your runners from any three that are together, and keep the other proportions for standing numbers; thus the prices of 4 Ingredients being 48. 36. 24. 12, and the mean price 16, the prices and differences will be.

[Page 175] [...] And the runners for the three lower prices will be 1. 2. 1. as before and therefore, the answers these following.

+ 1 | 16 | 48 | 4 | 4 | 4 |

2 | 36 | 4 | 5 | 6 | |

+ 1 | 24 | 4 | 2 | 0 | |

12 | 60 | 61 | 62 | ||

72 | 72 | 72 |

1 | 16 | 48 | 4 | 4 | 4 | 4 |

+ 2 | 36 | 4 | 3 | 2 | 1 | |

1 | 24 | 4 | 6 | 8 | 10 | |

12 | 60 | 59 | 58 | 57 | ||

72 | 72 | 72 |

20. In Alligation Alternate the prices of several ingredients being given, with the total quantity and value thereof, the several quantity of each ingredient may be otherwise found in this manner: let A. B. C. represent the prices of three ingredients, and let S. be equal to the proposed quantity of those ingredients, and F. the value of that quantity: moreover let H. represent the difference between [Page 176] A and B, let K represent the difference between B and C, then multiply the proposed quantity represented by S by the price of the middle ingredient represented by B, and to the product adde the difference represented by K continually, until the difference represented by H being continually added to the value of the quantity propounded represented by F, shall make those numbers to be equal: which being done, the number must be reserved for a common Dividend, I call it D. The unknown quantities of A B and C, I call X. Y. Z, now then to find X, deduct F from D and divide the remainer by H.

And to find Z deduct B S from D, and divide the remainer by K, the quotients shall be X and Z, and Y must be the complement of them to S.

Example, suppose the prizes of three ingredients were A 5. B 7. C 10, the proposed quantity S 50, and the value of that quantity F 307, S 50 being multiplied by B 7 the product is B S 350, the difference between B7 and C 10 is K 3, which must be added continually unto B S 350; the difference between A 5 and B 7 is H 2 which must be added continually unto F 307, which being done the common Dividend will be D 353, as by the work appeareth.

[Page 177]In this continual addition, you must [...] keep the least side going first, till both be equal. But if both sides will not be equal by the continual addends, which will fall out in these two cases

1. When both addends are equal, 2. When one side runs on with continually odd numbers and the other with continual even, you must make them even by taking a part of one of the continual addends: here the common Dividend is D 353; from whence deducting F 307 the remainer is 46, which being divided by H 2 the quotient is X 23.

Again from D 353 deducting B S 350, the remainer is 3, which being divided by K 3 the quotient is Z 1. Now the complement of these two quotients unto S 50 is Y 26: the work and proof will stand thus. [...]

And to constitute the runners, the difference between A 5 and B 7 is 2, which I place against C 10, the difference between B 7 and C 10 is 3, which I place against A 5, and the difference between A 5 and C 10 is 5, which I place against B 7; and by [Page 178] adding and subducting these runners to and from X. Y. Z. some of the many answers are these following.

+ 3 | A 5 | X 23 | 26 | 29 | 32 | 35 | 38 |

− 5 | B 7 | Y 26 | 21 | 16 | 11 | 6 | 1 |

+ 2 | C 10 | Z 01 | 03 | 05 | 07 | 9 | 11 |

S 50 | 50 | 50 | 50 | 50 | 50 |

21. In Alligation Alternate when the prizes of more than three ingredients are given, you may set by any three at pleasure, and upon the prizes remaining you may impose any part of the whole quantity propounded, less than the sum of the other unknown quantities, and upon the value any sum less than the value of the other unknown quantities, and let the sum of the three quantities set by, be the complement of the sum of all the unknown quantities propounded; and the value of those three quantities be the complement of the value of the whole quantity propounded; this done you may proceed with the three prices set by as was before directed.

Example, let the given prices of the several ingredients be, A 2, B 3, C 5, D 7, E 10, and let the sum of all their quantities be S=80, the value of all that sum F 380; now then setting by C 5, D 7, E 10, impose upon A 2 for its quantity, T 17, and upon B 3, V 13, both these quantities being deducted from S8 othe remainer will be X+Y+Z=50

Then multiply T 17 by A 2, the product will be A T = 34, and V 13 by B 3, the product will be B V 39, and these two products being deducted [Page 179] from F 380, the remainer is the value of C X + D Y + E Z = 307: but the particular quantities of X Y Z and their particular values were before found to be as are here expressed.

X | 23 | C X = 115 |

Y | 26 | D Y = 182 |

Z | 01 | E Z = 10 |

S | 50 | F 307 |

The which with their runners will stand thus,

2 | T = 17 | 17 | 17 | 17 | 17 | 17 | |

3 | V = 13 | 13 | 13 | 13 | 13 | 13 | |

+ 3 | 5 | X = 23 | 26 | 29 | 32 | 35 | 38 |

− 5 | 7 | Y = 26 | 21 | 16 | 11 | 06 | 01 |

+ 2 | 10 | Z = 01 | 03 | 05 | 07 | 09 | 11 |

80 | 80 | 80 | 80 | 80 | 80 |

Or placing two sets of runners thus.

− 2 | 2 | T = 17 | 15 | 13 | 11 | 09 | 07 |

+ 3 | 3 | V = 13 | 16 | 19 | 22 | 25 | 28 |

+ 3 | 5 | X = 23 | 25 | 27 | 29 | 31 | 33 |

− 5 | 7 | Y = 26 | 21 | 16 | 11 | 06 | 10 |

+ 2 | 10 | Z = 01 | 03 | 05 | 07 | 09 | 11 |

80 | 80 | 80 | 80 | 80 |

If you would have the unknown quantities to [Page 180] be all affirmatives, it is absolutely necessary, that the sum of all the quantities being divided by the sum of the unknown quantities, the quotient be less than the greatest, and greater than the least of the given prices; which quotient is the mean price, for which the mixture may be afforded.

22. By this last way of working such questions as do belong to Alligation Alternate, may such sporting questions be also answered, which some refer to the Rule called Ceres and Virginum; such as are these two questions following.

1. Quest. A Maid being sent to Market to buy fowl did with 20 pence buy 20 Birds, Larks for farthings a piece, Pigeons for halfpence, and Chickins for groats; The question is how many she bought of a sort?

To answer this question I place the names of the birds with their several prices in this manner.

A | 1 | L. | 2 | 2 | 80 |

B | 2 | P. | 15 | 30 | |

C | 16 | C. | 3 | 48 | |

80 |

Then I multiply the number of Birds, to wit 20 by 2 the middle price, and the product is 40, and the 20 pence reduced to farthings make 80; then H=B−A viz. 1 I place under 80 the whole value o [...] expence, and K=C−B viz. 14 I place under

40 | 80 |

14 | 1 |

54 | 81 |

28 | 1 |

82 | 82 |

40 the product of B S as here you see: then deducted 54 from 81 the difference is 27, and twice 14 is 28, which being added to 54 the total is 82; so then twice H being added to F 80, makes that also to be 82, now then I say that the Maid bought 2 Larks and 3 Chickens which are 5, and by consequent 15 Pigeons, the truth whereof doth thus appear; 2 Larks make 2 farthings, 15 Pigeons 30 farthings, and 3 Chickens 48 farthings which being added together do make 80 the sum expended.

2. Question, A Gardener paid 20 persons 20 shillings, to every man 20 pence, to every woman 15 pence, to every boy 8 pence: The question is how many men, how many women and how many Boys?

To answer this question, I place the persons with their several rates or wages thus.

A | 8 | M | 10 | 80 | 240 |

B | 15 | W | 8 | 120 | |

C | 20 | M | 2 | 40 |

Then I multiply the number of persons, viz. 20 by 15 the middle price, and the product is B S 300, and the 20 shillings being reduced to pence do make F 240, H=B−A viz. 7 I place under F 240, and K=C−B viz. 5 I place under B S 300; he difference between these two numbers is 60, now 10 times 7 makes 70 which being added to F 240 the sum is 310, and twice 5 being added to B S 300 [Page 182] the sum is 310 also, now then D 310-F 240=70 which being divided by H=7 quotes 10 for the number of the Boys, and D-310 B S 300=10 being divided by K 5 quotes 2 for the number of men, both which make I 2, and therefore the number of women must be 8 as by the work appears.

And here you may observe that so oft as you add H to F so much or so many you must take of the lowest prices, and so oft as you adde K to B S, so much or so many you must take of the greatest price, and what these two do want of the whole number, must be the number or quantity of the third; if there be more ingredients propounded you must proceed as was before directed.

## CHAP. XXVIII. Of the Rule of False.

1. HItherto we have spoken or treated of positive Arithmetick; Negative now follows, otherwise called the Rule of false position, in which by false terms supposed, we have a means afforded by which to discover the true terms required.

2. The Rule of false Position, is either single or double.

3. The Rule of single position is, when by one false Position, we may discover the true resolution of the question propounded.

Example, suppose 400l. were to be distributed [Page 183] among three persons, of which the second man is to have three times as much as the first, and the third four times as much as the second, the question is, how much of the 400l. must be given to each person? Here you may suppose any sum at pleasure, and fit the proportions to it according to the state of the question: by this sum so stated, a means will be afforded to answer the question propounded, though the supposition doth not do it. Say then that the first man had 3 pound, the second man must then have 9 pound, and the third 36 pound, and these three sums in the total are but 48 pound, and yet by this false supposition, we may by the Rule of Three find out the true answer to the question propounded.

For as 48 | 3 | so 400 | 25 |

9 | 75 | ||

36 | 300 | ||

48 | 400 |

The first man then had 25l. the second 75l. and the third 300l; the which sums do together make 400l. as was propounded.

4. The Rule of double position is, when two false positions are propounded, by which to discover the true resolution of the question.

5. In this Rule one supposition must be made at pleasure, and examined according to the state of the queston, whether you have guest right or not.

If this supposition do not answer the question, observe whether your error be in the excess, or in the defect. [Page 184] Then make another supposition and examine this second position as you did that first, and if this be also false, observe again whether the error of this supposition be in that excess or in the defect, that is, whether you did suppose too much or too little; for these errors will either be both too much, or both too little, or one too much and the other too little, and according to this diversity of the errors, a diverse manner of working must be used for the resolution of the question propounded.

6. If the errors of both suppositions, be both too little or both too much, deduct the less from the greater, and note the remainer for your Divisor; then multiply the first position by the error of the second, and the second position by the error of the first, and deduct the lesser product from the greater, the remainer being divided by the difference of the errors, shall in the quotient give you the true answer of the question propounded. Example, if from the half of a certain sum of mony I take away another half, and also a fourth, leaving 13 pound, how much was the whole sum? To resolve this question, first I suppose the whole sum to be 40 pound, from the half whereof to wit 20 pound, I deduct another half that is ten pound and there remaineth 10, and also a fourth of 20 pound, and then there remaineth 5 pound, whereas the remainer should have been 13 pound, therefore this error is in the defect 8.

Secondly I suppose the whole sum to be 64 pound the half whereof is 32, and the half of that is 16 and the fourth 8, and the remainer also 8, and therefore the second error is also in the defect, and [Page 185] the defect is 5. the which being placed as here you [...] see cross-wise with the errours under each position. I multiply the first position 40 by the second error 5, and the product is 200: then I multiply the second position 64 by the first error 8, and the product is 512; and the lesser product being deducted from the greater the remainer is 312, which being divided by 3 the quotient is 104 the number sought, whose half is 52, and the half of that 26, and the fourth of 52, 13, and the remainer 13 also as was required.

7. If the errors in your suppositions, be one too much and the other too little, adde both the errors together, and the whole shall be your Divisor; then multiply your first position by your second error, and your second position by your first error as before; the sum of these products being divided by the sum of the errors shall in the quotient give you the answer to the question.

2. Example, the same question being again propounded, for my first supposition I take 56 pounds, the half whereof is 28, and the half of 28 is 14, and the fourth 7, and so the first error is 6 in the defect; And for my second position I take 128 pound, the half whereof is 64, and half of that is 32, and the fourth 16, and so the second error is [Page 186] 3 too much, the sum of [...] these errors is 9: now the first position 56 being multiplied by 3 the second error, the product is 168, and the second position 128 being multiplied by the first error 6 the product is 768, the sum of these two products is 936, which being divided by 9 the sum of the errors, the quotient is 104, the answer to the question as was desired.

And thus we have passed through all the chief parts of Natural Arithmetick, (the extraction of the square and cube roots only excepted, and that purposely omitted, because it may be not only more briefly but also more plainly shewed, in Specious or Symbolical Arithmetick) and as for artificial or Logarithmetical Arithmetick, so much as may serve to resolve all ordinary questions, hath been already shewed in my scale of interest, only the construction of the numbers themselves is there omitted, which yet I have shewed by the continual extraction of the square Root in my Mathematical institution, and by multiplication in my Trigonometria Britanica; since which, there are some excellent ways of making those numbers lately found out and published in Latine by Nicholas M [...]rc [...]tor: so much whereof as may give thee satisfaction in that particular. I purpose God willing, to adde to a small introduction to Specious Arithmetick, [Page 187] if the Author himself find not encouragement or opportunity to do it in english himself; and that I be not suddenly prevented in the other, by some skilful hand, whose zeal for propagating the knowledge of the Arts in our own tongue, is at least equal to, if not exceeding mine.

Soli Deo Gloria.