on of com

Laftly, A common Fraction is turn'd in- 15. to a Decimal, by dividing the Numera- Reducti tor, increas'd with Cyphers (as occafion mon Fra requires) by the Denominator. Thus, &tions inte 15. For 2)10 (5. So 25; for 4) ICO (25. And 75; for 4) 300 (75. Allo1333. For 3) 1000, &c. (333, c. That is, the common Fraction can't be reduc'd into a Decimal exactly equivalent thereto : However the Difference is (or may be by further dividing) render'd fo fmall, as to be inconfiderable. Namely, there is not, wanting to make the Decimal 333, exactly equivalent to the common Fraction. And this is fufficient to our Purpose concerning Reduction.

С НА Р.

1.

С НА Р. XI.

Of Proportion, and more especially of the Rule of Three, or Golden Rule.

A

NY one Number dividing any oProporti-] ther Number, the Quotient thence on, what arifing, fhews the (*) Proportion of one to the other. Thus, because 2)6(3, therefore the greater Number 6 is said to be in a triple Proportion to the leffer 2, and 2 is faid to be in a Sub-triple Proportion to 6.

2.

onal Num

Four Numbers are faid to be proporProporti tional, when the two firft dividing one bers, what the other, give the fame Quotient, as the two laft. Thus, because 2)6(3, and alfo 4)12(3, therefore 2 and 6, 4 and 12, are faid to be proportional one to the other; which their Proportionality is wont to be thus express'd, viz. 2:6:: 4:12, or 6: 2 :: 12: 4.

3.

Of four Proportionals, fome two are A Remark always the Products or Quotients of the Proportion- other two, multiplied or divided by fome

concerning

als.

(*) This is by Mathematicians properly call'd the Ratio. But we ufe the word Proportion in the fame Senfe in common Speech.

one

one and the fame Number. Thus, in the fore going Example, 6 and 12 are the Products of 2 and 4 multiplied by 3; or 2 and 4 are the Quotients of 6 and 12 divided by 3.

The Foun

the Rule

the Ruic; and 3d for prothe ving Num

Hence, if four Numbers be proporti 4. onal, the Product of the two Extreams, ĉation of (i. e. of the ft and 4th Terms or Num- the Golden bers) is always equal to the Product of two Means, (i. e.) of the 2d and Terms. Namely, in both Cafes, bers to le Factors are in effect the fame and con- Proportifequently, the Products must be the fame, onat. For Inftance; 2: (2×3‚i.e.)16 :: 4:(4*31 i. e.) 12. And therefore, 2×12=6×4,

because 2*4*32*3*4

The Gol-..

red.

On the fore-going Property of Pro- 5. portionals, is founded the Rule of Pra den Rule, portion, which from its great Ufe in alio Rule of Arts and Sciences, is commonly call'd the Three diGolden Rule; as alfo the Rule of Three, from the three Proportionals given to find the Fourth. The Rule ftands thus: Of the three Terms or Numbers given, let the Second multiply the Third, and the First divide the Product; the Quotient will be the fourth Proportional fought. For Inftance: Let the three Numbers given be 2, 4, 6; and for the Fourth fought, put Q. Wherefore, 2 : 4 :: 6 : Q. Wherefore, by the fundamental Rule (given, S. 4.) 2*Q=4×6. And therefore, for

afmuch

6.

Of placing

the Terms in the Golden

Rule.

7.

afmuch as

4×6

that 4x6=Q, that is, 2)24(12, the fourth

2

Proportional fought. For 2: 4 :: 6:12. If the Proportionals be of two feveral external Denominations, then they mult be fo plac'd, as that the First and Third may be of one and the fame Denomination, and likewife the Second and Fourth. For Inftance: Suppofing it be demanded, how much will be fpent in 365 Days, at the Rate of spending five Shillings in feven Days; the Terms must be d. s. d. 's.

plac'd thus: 75 :: 365: Q

And if the Proportionals be exprefs'd Further, of by more than two external Denominatithe fame. ons, then they are to be reduc'd to Two. Thus, fuppofing it be demanded, how much Money one fhall spend in a Yer, at the Rate of five Shillings a Week, the four external Denominations here given, (viz. Money, Shillings, Year, and Week,) must be reduc'd to Two, viz. Shillings and Days, as in the Inftance afore.

8.

Direct Proportion, what.

If the Proportion runs fo, that the greater the third Term is, fo much the greater must be the Fourth; or the lefs the third Term is, fo much the less mult be the Fourth; then it is call'd direct Proportion; and the fourth Proporti

onal

109 oħal is found out by the Rule afore laid down, S. 5. Thus, because at whatever Rate one fpends, the more the Time (which conftitutes the third Term) is, the more the Money spent (which conftitutes the fourth Term) will be; hence, in the fore-going Inftance the Proportion is Direct; and confequently, the fourth Proportional is to be found by the Rule laid down, S. 5. Namely,

d. S. d.

1. d.

7: 9 : : 365: (365*5, i, e.) 2604—13. 83, and a

little over.

9.

Indirect.

If the Proportion runs fo, that the greater the third Term is, fo much the Indirect lefs must be the Fourth; or the lefs the Proportion what; and third Term is, fo much the greater mult of the Rule be the fourth Term; then it is call'd of Three indirect (or inverfe, or reciprocal) Proportion ; and the fourth Proportional is found out by a Rule fomewhat different from the former, namely this: The Product of the first and fecond Term being divided by the Third, the Quotient will be the fourth Proportional fought. For Inftance: It has been found by the Rule of direct Proportion, that (rotundè) 13 l. or 260 Shillings will ferve a whole Year (or 365 Days) at the Rate of five Shillings a Week. I would know, how long

the