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 382 A R I T H M E T I C K. be to the fecond antecedent as the iirft confequent is to the feeond confequent; or the firfl: term will be to I. The Simple Rale of Three Dircfi. the third term as the fecond term is to the fourth. Thus, if 8 : 4 :: 24 : 12, then, by alternation, 8 : 24 :: 4 : 12. Quest, i. If 4 yards coft 12 (hillings, what will 6 yards coft at that rate? Euclid v. 16. But the celebrated property of four proportional num- The fuppofition and demand of this queftion have albers is, that the produdt of the extremes is equal to the ready been diftinguiflied, and the two terms in the former produft of the means. Thus, if 2 : 3 :: 6 : 9, then are 4 yards 12 (hillings, and th* only term in the latter is 6 yards. 2X9 = 3X6=18. Euclid vi. 16. Hence we have an eafy method of finding a fourth pro- The number fought is the price of fix yards, and the term in the fuppofition of the fame kind is the portional to three numbers giten, viz.. Multiply the middle number by the laft, and divide the price of 4 yards, viz. 12 (hillings, which place in the produft by the firft, the quot gives the fourth propor- middle, as djreded in Rule I. and the two remaining terms are extremes, and of the fame kind, viz. both tional. Examp. Given 6, 5, and 36, to find a fourth pro- lengths. portional; put x equal to the fourth proportional, then It is eafy to perceive Yds. . s. yds. 6 : 5 :: 36 : x, and 5 X 36 = 180 = 6 X x; where- that the anfwer muft be If 4 : 12 6 6 fore, dividing the produdi 180 by the fadtor 6, the quot greater than the middle gives the other fadtor x, namely 30, the fourth propor- term; for 6 yards will coft more than 4 yards; there4)72(18 (hillings Anf. tional fought. leaft extreme, Every queftion in the rule of three may be divided fore 4theyards, is the diviinto two parts, viz. a fuppofition and a demand; and of viz. the three given numbers, two are always found in the fup- for, according to Rule II. 33 ppfition, and only one in the demand. 32 Examp. If 4 yards coft 12 ftiillings, what will 6 yards coft at that rate? (°) In this queftion the fuppofition is, If 4 yards coft x 2 place the divifor 4 yards on the left hand, {hillings; and the two terms contained in it are 4 yards andWherefore the other extreme 6 yards on the right; and multiand 12 (hillings: The demand lies in thefe words, What plying fecond and third terms, divide their produd will 6 yards coft? and the only term found in it is 6 by the the firft term, and the quot 18 is the anfwer, and of yards. with the middle term, viz. (hillings, acThe fuppafition and demand being thus diftinguilhed, the fametoname Rule III. proceed to ftate the queftion, or to put the terms in due cording And becaufe the divifor is the extreme found in the order for operation, as the following rules diretft. the proportion is dired. Rule I. Place that term of the fuppofition, which fuppofition, Quest. 2. is of the fame kind with the number fought, in the mid- will 5 C. coft at Ifthat7C.rateof? pepper coft 211. how much dle. The two remaining terms are extremes, and always The fuppofition in this queftion is, that 7 C. of pepof the fame kind. cofts 211. and the two terms in it are 7 C. and 21 1.; II. Confider, from the nature of the queftion, whether per demand is, How much will 5 C- coft? and the term the anfwer muft be greater or lefs than the middle term; the in it Mid if the anlwer muft be greater, the leaft extreme is Theis 5C. number is the price of $ C. and the term the divifor; but if the anfwer muft be lefs than the mid- in the fuppofitionfought of the fame kind is the price of 7C. dle term, the greateft extreme is the divifor. viz. 211. which place in The two remainIII. Place the divifor on the left hand, and the other terms are extremes, andtheofmiddle. the fame kind, viz. quanextreme on the right; then multiply the fecond and third ing tities of pepper. terms, and divide their produft by the firft; and the quot It is obvious, that the anfwer C, L. C. gives the anfwer; which is always of the fame name with muft be lefs than the middle If 7 : 21 :: 5 the middle term. for 5 C. will coft lefs than 5 When the divifor happens to be the extreme found in term; 7 C.; and therefore the greateft the fuppofition, the proportion is called dirc£i; but when extreme, 7)105(15 /. Anf u/z. 7 C. h the divifor. the divifor happens to be the extreme in the demand, the 7 proportion is inverfe. The three rules delivered above are indeed fo framed, 35 as to preclude the diftitsftion of direft and inverfe, or 35 tender it needkfsj. the left-hand term being always the divifor; but yet the direil queftions being plainer in their own nature, and more eafily comprehended by a learn- Accordingly place the divifor 7C. on (”) the left hand, er, (hall, in of the first place, kind,exemplify rules by a and the other extreme 5C. on the right; and fet ofwequeftions the direft and (hallthe afterwards having multiplied the fecond and third terms, divide their product adduce an example or two of fuch as are inverfe. by