The English academy, or, A brief introduction to the seven liberal arts grammar, arithmetick, geometrie, musick, astronomie, rhetorick & logic : to which is added the necessary arts and mysteries of navigation, dyaling, surveying, mensuration, gauging & fortification, practically laid down in all their material points and particulars, highly approved to be known by the ingenious, and as such are desirous to profit, or render themselves accomplished : chiefly intended for the instruction of young scholars, who are acquainted with no other than their native language, but may also be very useful to other persons that have made some progress in the studies of the said arts / by John Newton.

CHAP. V. Of the Compound Rule of Three.

THe Compound Rule of Three, is when more than three Terms are propoun∣ded.

2. Under the Compound Rule of Three is comprehended the Double Rule of Three, and divers Rules of plural proportion.

3. The Double Rule of Three, is when five Terms are propounded, and a sixth in pro∣portion to them is required.

4. In this Rule the five Terms given do consist of two parts; first a Supposition, and then a Demand; the Supposition is expres∣sed by three of the Terms propounded, and the demand by the other two.

5. And here the greatest difficulty is in placing of the Terms; for which observe amongst the Terms of Supposition, which of them hath the same Denomination with the Term required, reserve that for the se∣cond place, and write the other two Terms in the Supposition one above another in the first place; and lastly, the Terms of De∣mand one above another, likewise in the third place, in such sort, that the upper∣most may have the same denomination with the uppermost of those in the first place.

Example.

If 6 Clerks can write 45 sheets of Paper in 5 Dayes; How many Clerks can write 300 sheets in 72 Dayes? Here the Questi∣on is concerning the number of Clerks , the 6 Clerks must therefore possess the second place, and the Dayes and Paper in the Sup∣position must be set in the first, one over the other, of which, if Paper be the uppermost in the other Terms, the Paper must be set over the Dayes in the third place, and then the Number in the Question will stand thus,

45	6	300
5		13

6. The Terms propounded being thus placed, the Question may be resolved by two Single Rules of Three in this manner.

1. As the uppermost Term of the first place is to the middle, so is the uppermost Term in the last place to a fourth Number.

2. As the lower Term of the first place is to that fourth Number, so is the lower Term of the last place to the Term requi∣red.

But in both these Proportions, conside∣ration must be had to the Term required, namely, whether it must be more or less than the middle Term given.

In our present Question, the fourth term in the first proportion must be greater than the second; for it is plain, that more work will require more men; therefore I say,

as 45 . 6 ∷ 300 . 40 Clerks.

But in the second proportion, it is like∣wise plain, that the more time is given, the fewer persons are required; and therefore in this proportion, 5.40.13. I multiply the middle term by the first, and the pro∣duct 200 I divide by 13, the last, and the Quotient is 15 10/13.

2. Example: If 100 l. gain 6 l. in 12 months, what shall 276 l. gain in 18 months? In this Question the terms must be thus pla∣ced.

100	6	276
12		18

• 1. 100 . 6 ∷ 276 . 16 . 56.
• 2. 12 . 16 . 56 ∷ 18 . 24 . 84.