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 A R I T H M E T I C K. 38i 2. AVOIRDUPOIS WEIGHT. PROOF. In 727 guineas 13 (hillings, how many pounds? Quejl, i. In C. 47 : i : 20 how many ounces? Guineas. Jhill. C. lb. 727 13 47 i 20 21 4 730 189 I45S 28 2jo)l528|o 1512 380 764 pounds. 5312 lb. 16^ Chap. VII. The Rule of Three. 3x872 The Rule of Three, called alfo, on account of its 5312 * excellence, the Golden Rule, from certain numbers given finds another ; and is divided into fimple and com84992 oz. pound, or into Angle and double. PROOF. Sect. I. The Simple or Single Rule of Three. In 84992 ounces how many lb. Q^and C. The fimple rule of three, from three numbers given, 28) 4) C. lb. finds a fourth, to which the third bears the fame propor*6(84992 (5312 (189 (47 i 20 tion as the firft does to the fecond. So’-’ 23•• 16 The nature and properties of proportional numbers may be underftood fufficiently for our purpofe from the 49 251 29 following obfervations. 48 224 28 In comparing any two numbers, with refpeft: to the proportion which the one bears to the other, the firtb 19 272 (o) number, or that which bears proportion, is called the 16 252 antecedent; and the other, to which it bears proportion,' is called the confquent; and the quantity of the propor-* 32 (20) lb. tion or ratio is eftimated from the quot arifing from divi32 ding the antecedent by the confequent. Thus the ratio or proportion betwixt 6 and 3 is the quot arifing from (°) dividing the antecedent 6 by the confequent 3 ; namely,Mixt Reduflian. 2 ; and the ratio or proportion betwixt 1 and 2 is the quot arifing from the divifionr onof the antecedent 1 by the In working mixt reduction obferve the following confequent 2 ; namely ° e half. Rule. By redudtion defcending bring the given are faid to be proportional when the name to fome fuch third name as is an aliquot part both Fourof numbers the firft to the fecond is the fame as that of the' of the name given and of the name fought, and then by ratio third to the fourth proportional numbers are redudtion afcending bring the third name to the name ufually didinguilhed ;fromandonetheanother as in the following fought. examples. Mixt redudtion, as well as redudtion defcending and 4 : 2 :: 16 : 8 6 : 9 :: 12 : 18. afcending, extends to money, as follows. ^uejl. In 7641. how many guineas ? Proportional numbers, or numbers in proportion, are Here the given name is ufually denominated terms; of which the firrt and laft ■ 764 pounds. pounds, the name fought is are called extremes, and the intermediate ones get the 20 guineas, and the third name, name of means, or middle terms. to which the pounds are reIf four numbers are proportional, they will alfo be* guineas. duced, is (hillings ; for a 21)15280(727 inverfely proportional ; that is, the fir(t conlequent will; 147 •• (hilling is an aliquot part be to its own antecedent as the fecond confequent is both of a pound and of a to its antecedent; or the fourth term will be to the guinea. third as the fecond is to the firft. Thus, if 6 : 3 :: 10 : 5, then by inverfion, 3 : 6 :: 5 : 10, or 5 : 10 :: 3 : 6. 160 Euclid v. 4. cor. By either of thefe kinds of inveifioa may any- queftion in the rule of three be proved. 147 If four numbers are proportional, they will alfo be al(13) (hillings. ternately proportional; that is, the firft antecedent will Vol. I. No. 16. 5D be 3