Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/154

 122 A L L I G and almonds at 6 d. how many pounds of almonds were in the mixture ? lb. Rates, 400 6 d. X. s. d. 44 d. 7 10 = 1800 1600 i6ood. 2d. L. s. 2)200(100 lh. of almonds at 6d. is, 2 10 And 300 Itr. of raifins, at 4 d. is, 5 o Total 400 Proof 7 10 II, Alligation Alternate. Alligation alternate, .being the con^erfe of alligation medial, from the rates of the fimples, and rate of the mixture given, finds the quantities of the fimples. RULES. I. Place the rate of the mixture on the left fide of a brace, as the root; and on the right fide of the brace fet the rates of the feveral fimples, under one another, as the branches. II. Link or alligate the branches, fo as one greater, and another lefs than the root may be linked or yoked together. III. Set the difference betwixt the root and the feveral branches, right againfi: their refpediye yoke-fellows. Thefe alternate differences are the quantities required. JVate 1. If any branch happen to have two or more yoke-fellows, the difference ketwixt the root and thefe yoke-fellows mult be placed right againft the faid branch, one after another, and added into one fum. Arote 2. In fome queftions, the branches maybe alligated more ways than one; and a queftion will always admit of fo many anfwers, as there are different ways of linking the branches. Alligation alternate admits of three varieties, v/z. 1. The queftion may be unlimited, with refped both to the quantity of the fimples, and that of the mixture. 2. The queftion may be limited to a certain quantity of one-or more of the fimples. 3. The queftion may be limited to a certain quantity of the mixture. Variety I. When the queftion is unlimited, with refped both to the quantity of the fimples, and that of the mixture, this is called Alligation Simple. Examp. A grocer would mix fugars, at 5 d. yd. and 10 d. per lb. fo as to fell the mixture or compound at 8*d. per lb: What quantity of each muft he take ? lb.

A T I O N. fide of the brace, as the root; and on the right fide of the fame brace are fet the rates of the feveral fimples, vizaccording7>to 10, Ruleunder I. One another, as the branches; The branch 10 being greater than the root, is alligated or linked with 7 and 5, both thefe b^eing lefs than the root; as direded in Rule II. The difference, between the root 8 and the branch 4, viz. 3, is fet right againft this branch’s yoke-feilow 10. The difference between' 8 and 7 is likewife fet right againft the yoke-fellow 10. And the. difference betwixt 8 and 10, viz. 2, is fet right againft: the two yoke-fellows 7 and 5 ; as preferibed by Rule III. As the branch 10 has two differences on the right, viz. 3 and 1, they are added ; and the anfwer to-the queftion is, that 2 lb. at 5 d. 2 lb. at 7 d. and 41b. at 10 d. will make the mixture required. The truth and reafon of the rules will appear by confidering, that whatever is loft upon any one branch is gained upon its yoke-fellow. Thus, in the above example, by felling 4 lb. of 10 d. fugar at 8 d, per lb. there is 8 d. loft: but the like fum is gained upon its two yoke-fellows; for by felling 2 lb. of 5 d. fugar at 8 d. per lb. there is 6 d. gained; and by felling 2 lb. of 7 d. fugar at 8 d. there is 2 d. gained; and 6 d. and 2 d. make 8 d. Hence it follows, that the rate of the mixture muft always be mean or middle with refpedl to the rates of the fimples; that is, it muft be lefs than the greateft, and greater than the leaft; otherwife a folution would be impdllible. And the price of the total quantity mixed, computed at the .rate of the mixture, will always be equal to the fum of the prices of the feveral quantities call up at the refpedive rates of the fimples. Variety II. When the queftion is limited to a certain quantity of one or more of the fimples, this is called Alligation Partial. If the quantity of one of the fimples only be limited, a’ligate the branches,, and take their differences, as if there had been no fiich limitation; and then work by the following proportion. K the difference right againft the rate of the fimple whofe quantity is given. To the other differences refpe&ively; So the quantity given. To the feveral quantities fought. Examp. A diftiller would, with 40 gallons of brandy at 12 s. per gallon," mix rum at 7 s. per gallon, and gin at 4 s. per gallon: How much of the rum and gia muft he take, to fell the mixture at 8 s. per gallon ? Gal. C1i, 4 I 5 ! 40 of brandy.^ 8 < ^ J4 4 32 of rum. C Anf. C 4—''4 I 4 I 32 of gin. 3 Here the rate of the mixture 8 is placed on the left The operation gives for anfwer, 5 gallons of brandy, 4 of