Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/466

 384 A R I T H Ml E T I C K. The remaining, terms are extremes, which mu ft be claffed. 16 clay! how many bufhels will 20 horfes eat i •i4r into fnnilar pairs, by making each pair conlift of one term claysThe? fuppofition in this queftion is, If 14 horfes eat taken from the fuppofitipn, and another of the fame kind 56 bulhels in 16 days; and the three terms contained in taken from the demand. II. Out of each fimilar pair, joined with the middle it are, 14 horfes, 56 bulhels, and 16 days: The demany bulhels will 20 horles eat in 24 term, form a fimple queftion; and in each Ample que-^ mand is,andHow the two tern^contained in it are 20 horfes, (lion, fo formed, find the divifor; viz. confider from days? 24 days. the nature of the queft on,. whether the anfwer muft be andThe-number fought is bulhels, and the term in the treater or lefs than the middle term; and if the anfwer inu!i be greater,n ' the' rleaft' extreme:Jji-* is the divifor; but if. fuppofition the fame kind 56 is 56bulhels bulhels; according toof Rule I. place in thewherefore, middle. the anfwer muft be lefs than the middle term, the great- The remaining four terms are extremes, which you clafs eft extreme is the divifor. III. Place all the divifors on the left hand, and the into fimilar pairs, by making each pair confift iof one other extremes on the right; then multiply the divifors, term taken from the fuppofition, and another of the fame or extremes on the left, continually, for a divifor, and kind taken from the demand. Thus, 14 horfes, and multiply the extremes bn the right hand and the middle 20 horfes make one pair; again, 16 days, and 24 days term, continually, for a dividend; and, laltly, divide make another pair. the dividend by the divifor; and the quot is the anfwer. Out of the feveral fimilar pairs, joined with the midof the fame name with-the middle term. die term, you form fo many fimple queftions, according The anfwer to queftions in the compound rule o£ to Rule II. viz. by faying, three may alfo be had by working the fimple queftions days,1. Ifhow14many horfesbu/hels eat 56 will bulhels in a certain 20 horfes eat innumber the fameof feparately, or by themfelves, in the following manner. time ? The middle term, with any one pair of fimilar ex- 2. If 16 days eat up, or confume, 5:6, or any other tremes, make the firft fimpie queftion, and the anfwer number ofbulhels, how many bulhels will 24 days conto this queftion muft be made the middle term to the fumeIn ?the firft fimple queftion it is obvious, that the annext fimilar pair of extremes; and the anfwer to this will be greater than the middle term ; for 20 horfes fecond queftion, muft in like manner be made the middle fwer will eat more bulhels than 14 horfes will do in the fame term to the following fimilar pair of extremes, 6c.; time; and fo the leaft extreme, viz. 14, is the divifor; and the anfwer to the laft fimple queftion is the number and becaufe 14 is an extreme found in the fuppofition, fought. is diredt. But the joint operation prefcribed in Rule III. is the theInproportion fimple queftion it is alfo plain, that the fhorter as well as the eafier method; for in working fome anfwerthewillfecond greater than the middle term ; for 24 days of the fimple queftions, there may happen to be a re- will confumebe more bulhels than 16 days; and confemainder, and confequently the middle term of the next quently the leaft extreme, viz. 16, is the divifor; and fimple queftion will have fome.fradional part; which in- becaufe 16 is an extreme found in the fuppofition, the conveniency is avoided by working jointly. is diredl. In every fimple queftion, when the divifor is an ex- proportion Joint operation. According to Rule III. treme found in the fuppofition, 'the proportion is direft; place on the left Horfes. bujhels. horfes. but when the divifor is an extreme found in the demand, hand, theanddivifews the other ex- If 14 : 56 :: 20 the proportion is inverfe. right, and da. 16 24 da. The three rules delivered above are indeed fo calcula- tremesof onthemtheunder one anted, as to make no difference between direft and inverfe, both other, fo that the two upor fo as to render that diftindtion needlefs, the left-hand per ones make a pair, or be extremes being all divifors ; but yet, as queftions confift- of one kind, and the two ing entirely of diredt proportions are the plained and ea- lower another fieft, it will be proper, in the lirft place, to exemplify pair, oronesbe make one kind; the rules by queftiohs of the diredt kind, and afterwards and no matterofwhich of the introduce fuch as are inverfe. be uppermoft: then And as queftions in the rule of five are by far more pairs multiply the divifors, or numerous, and occur much oftener, than quellions in the the extremes on the left rule of feven, nine, or eleven; we (hall, fiiftof all, give for a divifor; and aqueftions in the rule of five, wherein both proportions hand, the extremes are diredt; then thofe wherein one or both proportions ongainthemultiply and the midare inverfe ; and, laftly, give a few examples of the rules dle term,right,continually, for of feven, nine, and eleven. a dividend; and dividing Anf. 120 bulhels. the dividend by the divifor, I. The Rule of Five Direft. the quot or anfwer comes out of the fame name with Quest, i. If 14 horfes eat 56 bulhels of corp in the middle term, viz. 120 bulhels. The