An account of the rotula arithmetica invented by Mr. George Brown.

CHAP. VII. Concerning the Rule of 3.

IN the Questions of this Rule, there are allways three Numbers, either ex∣presly given or suposed, to find a Fourth.

Of the given Numbers, there are all∣ways two of the same sort, Species or Notion, & these we shall call Relative Numbers, & the other or third Num∣ber, which falls under a different Notion, we shall call the singular Num∣ber: and the fourth Number demanded (which is allways of the same Species, or Notion with the singular Number) we call the Answer; because, when it is found, it Answers the Question.

Now to save you the trouble of two Rules, one Direct, and another Reverse: I shall lay down such easie Directions, as may render a Man capable to Answer Questions of either, without any such Distinction.

In discerning the Divisor, (which is all the difficulty, and which must all∣ways be one or other of the Relative Num∣bers) you must carefully consider, whe∣ther the Answer ought to be greater, or lesser than the singular Number For,

1st. When the Answer ought to be grea∣ter than the singular Number, then the least of the Relative Numbers must be the Divisor.

2ly. But if the Answer ought to be less than the singular Number, then the greatest of the Relative Numbers must be the Divisor.

Having discovered the Divisor, set it down next your Left-Hand, and the o∣ther two, (which we now call Coeffici∣ents) at a convenient Distance; the one, (it matters not which) before, the other after [∷] which we call the signe of Pro∣portion.

The Numbers thus disposed, Multiply the Coefficients and Divide the Product, by the Divisor and the Quotient, yields you the Answer; Only observe, that

when the Divisor is an Unite, there will be no use for Division; and on the other hand, when one of the Coefficients is an Unite there, will be no Multiplication.

As also, if one of the Coefficients have an Infinite annexed to it, be sure to make that the Multiplicand.

The following Examples, shall Illu∣strate what hath been said.

EXAMPLE, 1st. Concerning the Prices of GOODS.

IF 3 Ells cost 10 L. what will 17¾ Ells cost? In this Example 3 and ••7¾ are the Relative Numbers, because they are both of the same sort, to wit, Ells, and 10 is the singular Number; ••ow because the Price 17¾ Ells, must ••••e greater than the price of three Ells to ••••it, than 10 L. that is the Answer; ••ust be be greater than the singular ••••umber; I thereby understand, that 3 ••••e least of my Relatives must be the Di∣visor,

for which cause I set the Num∣bers as followeth. Taking instead of ¾ it's Decimal 〈 math 〉〈 math 〉.

So that the Answer is 59 L. 3sh. 4d.

You see I first Multiply the Coeffici∣ents, & then Divide the Product by 3 the Divisor, so have I the Answer: You may convert the Question, and so prove your Work, Thus,

2d. EXAMPLE.

If 17¾ Ells cost 59 L. 3sh. 4d. what will 3 Ells cost?

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You see in the Division, that I ought to have 3 Figures in my Quotient, as the Point intimates, of which one ought to be a Decimal, but that being a Cypher I did not think it necessary to set it down.

3d. EXAMPLE.

At 3 sh. 4½ d. per lib. what will 〈 math 〉〈 math 〉 lib. of any thing come to?

The Decimal for 3 sh. 4 pence is 〈 math 〉〈 math 〉 and that for ½ d.

So that 3 sh. 4½ d. is

Then conform to the Rule, the lesson will stand as followeth.

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You see in the lesson, there is no Di∣vision, because the Divisor is an Unite.

4th. EXAMPLE.

If 345½ Ells cost 58 L. 6 sh. o ¾ d. what will one Ell cost?

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In this Example, you see that one of the Coefficients, being an Unite: There is no Multiplication.

5th. EXAMPLE.

If ¾ Ells cost ⅘ lib. what will ⅚ Ells cost?

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Answ. 18 sh. 3 ¼. d. & some little more.

Because the last Dividual is the same the Quotient is Infinite.

In this last Example, the Decimal o•• ⅚ being Infinite; The Product of the dash'••

Figure, is reckoned on the Segment, but the Product of all the rest, is reckoned on the Integer-Circle,

Item, because the Dividend is Infinite, in Prolongation of the Work I have Reite∣rated the last Figure.

6th. EPAMPLE.

At 5½ per Cent, what will 347 lib. 13 sh. 4 d. pay per Annum?

In this lesson, observe, that tho' all the Numbers be of one Denomination or kind of things, yet (the 5½ L. falling under a different Notion, to wit, that of Interest; whereas the other two, to wit, the Cent. or 100, and 347 L. 13sh. 4 d. are Principal Summs) the 5½ is the Singular Number, and so 100 is the Divisor.

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In this you have 6 Decimals in the Answer to wit, four for those of the Coe∣fficients, and two for the Cyphers of the Divisor.

7th. EXAMPLE.

At 4 sh. 10½ per Crown, how many Pounds Sterl. must one have for 758½ Crowns.

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Answer. 184 lib. 17sh. 8¼d.

8th. EXAMPLE.

If 40 Men are able to finish a piece of Work in 8 Dayes, how many may do the same in 5 Dayes? here, because the Answer ought to be greater than the sin∣gular Number 5, the last of the Rela∣tives

is the Divisor, and so you are free from the cumbersome Reflections, on a Reverse Rule.

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9. EXAMPLE.

If the penny Loaff, ought to VVeigh 18 Ounces, when the VVheat sells at 10 sh. Sterling. per Boll, what ought the same to VVeigh, when the VVheat sells at 15 sh. per Boll;

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In Questions of 5 Numbers, you have, for the most part, two paires of Relatives, and but one singnlar Number, wherefore you may take any one of the paires of 〈◊〉〈◊〉 with the singular Number, to

find the first Answer, and that Answer will serve for a singular Number, for the other pair of Relatives in finding the last Answer.

9th. EXAMPLE.

At 5 per Cent per Annum, what will 197 L. 15 sh. 8 d. come to in 7 Years? In this the Relatives are 100 L. and 197 L. 15 sh. 8 d. for the first pair and, 1, Year and 7 Years for the 2d. pair, and 5 is the Singular.

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So that the Answer is 76 L. 12 sh. 9½ d. and very little more.

10th. EXAMPLE.

At 2 Dollars per L. Flemish, Ex∣change

at 35⅚ sh, Flemish per L. St. how many L. Ster. will 666⅔ Dollars come to?

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In this Division, finding my 3d. Di∣vidual, the same with the 2d. and, (be∣cause of the Reiterated Figures of the Dividend) understanding that it will allways be the same in Infinitum, I there∣fore Reiterate the 2d. of my Figures in the Quotient; to wit, 6, till I have five Figures in all, as the Dividual-Point Intimates: And seeing I have 3 Deci∣mal in my Dividend, and but one in my Divisor, I understand that the last two, of my five Figures in the Quo∣tient, must be Decimals, and the da∣shed Figure is added, because, (as hath been allready said) Infinites cannot be

limited under Decimal thirds; so that my 666⅔ Dollars, makes just 266 L. 13 sh. 4 d. Flem. for my first Answer, and this must be one of the Coefficients for the 2d. operation, because in this we have now two Parcells of Flemish Money, and but one L. Sterl.

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In this last Operation the third Di∣vidual. viz.. 0000 being less than the Divisor, I put 0 in my Quotient, and so the Answer is 150 Lib. Ster.

11th. EXAMPLE.

At 5 L. 10 sh. per Cent per An what will the Interest of 756 L. 13 sh. 4d. a∣mount to in 7 Yeares, and 7 Months?

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Vide. my Com∣pend. system a∣nent Multiplicati-L. In. = on, with an Infinite in both Coefficients.

The Answer is 317 L. 11 sh. 10 ½d. very near

12th. EXAMPLE.

Interest at 5 ½ per Cent. What will 456 L. 13 sh. 4 d. (payable 3 Years hence) be worth in present Money?

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Add this Interest to its principal, and the 2 d. Operation will stand thus.

〈 math 〉〈 math 〉 Ans. 378 L 11 sh. 8 d. & less than ¼ d. more.

I shall conclude with an Example of un∣equall Division, which may be very useful in fellowship.

13th. EXAMPLE

There is to be Divided amongst 14 Men 458 L. 5 sh. 11 d. with Proviso, that 9 of the Number (whose stocks or hazards were equull) have equall shares, but the other 5 are to have, one of them ½ share another ⅓ another ¼ another ⅙ another ⅚ of an equall share: The equall share, and conse∣quently the severall Fractions of the equall share, is demanded?

In this you must add the Decimals of the severall Fractions in the Questi∣on to 9, and so you will find your Di∣visor to be 〈 math 〉〈 math 〉 and not 14 thus 〈 math 〉〈 math 〉

The Sum of of all which is

Hence the Question must be thus stated.

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If any Difficulty occur in these les∣sons, it may be easily overcome, and at a very Reasonable Rate, by a little converse with the AUTHOR.