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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 some∣times four, sometimes five, sometimes more, and the terms required sometimes more than one.
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 pro∣pounded) 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 adven∣ture, 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 que∣stion.
Example, Three Farmers hired a Shepherd to keep their sheep, for 8 pound 15 shillings per An∣num.
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,
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 eve∣ry 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 par∣ticular product is the third, the whole gain or loss
is the second in both Rules, and therefore the pro∣portion to be observed in the solution of such que∣stions as fall under this Rule is
As the sum of the products found by multiply∣ing 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 mul∣tiply 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.
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 de∣mand what each mans part of the gains comes to.
In Questions of this nature, two things are principally to be observed.
| 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 ad∣ding the terms required, whose sum ought to be e∣qual 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 Chap∣ter; 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. |