and fuppofed; and that the other twe move the Question. As for Inftance in Example 1.

Viz. If 100l. will gain 67. in 12 Months: thefe three Terms are only fuppofed or conditional. Then comes the Queftion; What will 3004. gain in 9 Months? Now, in order to raise the general Theorem, let us suppose, instead of Numbers, thefe Letters.

}{

8=13,5 The Gain.

In the Suppofition
of any proposed
Question.
The three Terms
wherein the Que-
ftion lies.

Then P:G::: GP{The Product of the two Means divided

=

by the first Extream.

Viz. 12 : 18 :: 9: 13,5

Which is the first Part of the Question.

Which is the fecond Part of the Question.

Ergo Tg=Gpt} { That is, the Product of the Extreams is

P

equal to that of the Means,

Confequently, Tg P=Gpt is the Theorem.

This Theorem affords two Rules, by which all Questions in this Double Rule of Three, or rather of five Numbers, may be refolved; due Regard being had to the true placing down of the propofed Terms, which muft be thus:

Always place the three conditional Terms in this Order; let that Number which is the principal Cause of Gain, Lofs, or Action, &c. (viz. P.) be put in the firft Place; that Number which denotes the Space of Time, or Distance of Place, &c. (viz. T.) be put in the fecond Place; and that Number which is the Gain, Lofs, or Action, &c. (viz. G.) be put in the third Place. Now according to thefe Directions, the conditional Terms of the laft Queftion will ftand thus; P.T. G.

That done, place the other two Terms which move the Queftion, underneath thofe of the fame Name,

Then if the Blank or Term fought, fall under the third Place, as in this Question,

It will be

{

Gptg. Which gives this Rule.

TP

Multiply the three laft Terms together for a Dividend, Rule 1. and the two firft together for a Divifor; the Quotient arifing from them will be the fixth Term.

That is, in our proposed Example 1. Thus 6 x 300 x 916200 the Dividend. And 100 x 12 1200 the Divifor. Then 1200) 16200 (13 the Answer, as before,

But if the Blank or Term fought fall under the firft Place,

then

Or if the Blank fall under the fecond Place,

It will be

{

T&P=t. Either of these gives this Rule.
GP

Multiply the firft, fecond, and laft Terms together for a Rule 2. Dividend, and the other two together for a Divifor; the Quotient arifing from them will be the fixth Term.

And because our Example 2. falls under the Confideration both of Direct and Reciprocal Proportion, let it be here proposed again. Viz. If 6 Bushels of Oats will ferve 4 Horfes 8 Days; how many Days will 21 Bufhels ferve 16 Horfes, &c.

If the Terms of this Queftion be placed down as before directed, they will stand

Thus

Horfes. Days. Bufhels.

4. 8. 6 Terms in the Suppofition.

Here the Blank falls under the fecond Place, therefore it must be found by the second Rule.

Thus 4 x 8 x 21 672 the Dividend.

And

16 x 696 the Divifor.

Then 96) 672 (7 the Anfwer, as before.

Q

Quest

Queft. 3. What Principal or Stock will gain 207. in 8 Months at 6 per Cent. per Annum?

Prin. Time. Gain.

In this Queftion the Blank falls under the firft Place, therefore it must be found by the fecond Rule.

Thus 100 x 12 x 20
And

24co the Dividend.

8 x648 the Divifor.

Then 48) 24000 (500l. the Anfwer required.

The Proof of all Queftions in this Double Rule of five Numbers, is best performed by varying the Question; viz. by stating it in another Order, as in the laft Example: Thus,

If 100. gain 6. in 12 Months, what will 500l. gain in 8 Months?

The Answer to this Question must be 20. if the Work of the laft Example be true.

500 × 8 × 6 = 24000. And 100 x 12 = 1200.

Then 1200) 24000 (207. the Answer, &c.

Queft. 4. If two Men can do 12 Rods of Ditching in 6 Days, How many Rods may be done by 8 Men in 24 Days, at the fame Rate of working?

Anfw. 192 Rods.

Queft. 5. If the Carriage of 5 C. 3 qrs. Weight, 150 Miles, colt 31. 7 s. 4 d. What must be paid for the Carriage of 7 C. 2 qrs. 25 lb Weight, 64 Miles, at the fame Rate?

2

Anfw. 11. 18 s. 7 d. Queft. 6. If 8 Men deferve 2 1. Wages for 5 Days Work, How much will 32 Men deferve for 24 Days, at the fame Rate? Anfw. 381. 8 s.

Queft. 7. Suppofe a Hundred Pounds would defray the Expences of five Men for Twenty-two Weeks and fix Days, How long would twelve Men be in fpending of one Hundred and Fifty Pounds, at the fame Rate?

Anf. 14 Weeks and 2 Days.
С Н А Р.

CHAP. VIII.

Of Trading in Company, ufually called the Rule of Fellowfhip; alfo Bartering, and Exchanging of Coins, &c.

TH

HE Rule of Fellowship is that by which the Accompts of feveral' Partners trading in a Company, are so adjusted or made up,' that every Partner may have his juft Part of the Gain, or fuitain his juft Part of the Lofs; according to the Proportion or Share of Money he hath in the Joint-Stock: Now this falls under two Confiderations, called the Single and Double Rules of Fellowship.

Sect. I.

The Single Rule of Fellowship; viz. That without Time.

BY Y the Single Rule of Fellowship is adjusted the Accompts of thofe Partners that put all their feveral and perhaps different Sums of Money, into a common Stock at one and the fame Time; and therefore it is ufually called the Rule of Fellowship without Time: Now all Queftions of this Nature are answered by fo many feveral Operations in the Rule of Three Direct, as there are Partners in the Stock.

For, as the Total Sum of Money in the Stock is in Proportion to the whole Gain, or Lofs: fo is every Man's particular Part of that Stock to his particular Share of that Gain, or Lofs.

Queft. 1. Three Partners, fuppofe A, B, and C, make a JointStock of 961. in this manner.

A puts in 241. B puts in 321. and C puts in 40% with this 967. they trade and gain 121.

Man's true Part of that Gain.

The Operation will ftand, thus

It is required to find each

Proof 31. +41 +5%.

121. the whole Gain.

That is, if the Sum of each Man's particular Gain, amount to the whole Gain, the Work is true; if not, fome Error is committed, which must be found out..

Note, Thefe Operations will be very much abbreviated, if you work them by Theorem 2. page 87. For here 96 is a commo Antecedent, and 12 is the common Confequent in all the three Proportions.

Q 2

There

Therefore 96: 12:: 1: 0,125 a common Multiplicator.

Then 24

31. 32x0,125=341. for B,

B, as before.

40

} 125={

51.

C,

Now this Method is more readily performed than the other, especially when the Partners are many; becaufe one Single Divifion ferves for all the Work.

Queft. 2. Three Merchants, A, B, and C, freight a Ship with 248 Tun of Wine: Thus, A loaded 98 Tun, B 86 Tun, and C64 Tun. By Extremity of Weather the Seamen were forced to caft or throw 93 Tun of it over-board. How much of this Lofs must each Merchant sustain ?

First 248 93: I: 0,375 the common Multiplier.

Now if the Queftion were to find how much of the remaining Wine that was faved, belongs to A, to B, and to C.

That is, ought to have 61 Tun and 63 Gallons, Bought to have 53 Tun and 189 Gallons, And C ought to have 40 Tun of what was left.

Quest. 3. Suppofe fix Men, A, B, C, D, E, and F, make a Joint-Stock of 2558 1.

The whole Stock 2558. 00=2558,00 according to the Quest.