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 ALL ALL ( I2i ) ■Covered AhhtY, that over which the branches of trees In this fenfe, alliances may be diftinguiihed into meeting, form a (hade. fuch as are offenfive, whereby the contra&ing parties Alley of compartment, that which divides the fquares oblige themfelves jointly to attack fome other power; of a parterre. See Parterre. and into defenlive ones, whereby they bind themfelves Alley, among builders, denotes a narrow pafiage lead- to hand by and defend each other, in cafe they are*ating from one place to another tacked by others. Alley, in perfpe&ive, that which, in order to have a Alliance, in a figurative fenfe, is applied to any kind greater appearance of length, is made wider at the en- of union or conneftion: thus we fay, there is an allitrance than at the termination ance between the church and hate. ALLIANCE, in the civil and canon law, the relation ALLIGATI, in Roman antiquity, the bafefl: kind of contradted between two perfons or two families by (laves, who were ufually kept fettered. See Slave. marriage. ALLIER, a river of France, which, arifing in LangueAlliance is alfo ufed for a treaty entered into by fove- doc, waters part of Auvergne and Bourbonnois, and reign princes and hates, for their mutual fafety and falls into theJLoire, a little below Nevers. defence. ALLIGATION, A LLIGATION, the name of a method of folving all queftions that relate to the mixture of one ingredient with another. Though writers on arithmetic generally make alligation a branch of that fcience; yet, as it is plainly nothing more than an application of the common properties of numbers, in order to folve a few queftions that occur in particular branches of bufinefs, we chufe rather to keep it diftind from the fcience of arithmetic. Alligation is generally divided into medial or alternate. I. Alligation Medial. Alligation medial, from the rates and quantities of the ftmples given, difcovers the rate of the mixture. Rule. As the total quantity of the fimples. To their price or value ; So any quantity of the mixture. To the rate. Examp. A grocer mixeth 30 lb. of currants, at 4d. per lb. with 10 lb. of other currants, at 6A. per lb : What is the value of 1 lb. of the mixture. Anf. 4Td. lb. d. d. 30, at 4 amounts to 120 ro, at 6 —— 60 4° 180 lb. . d. lb. d. If 40 : 180 :: 1 : 4J Note 1. When the quantity of each (imple is the fame, the rate of the mixture is readily found by adding the rates of the (imples, and dividing their fum by the number of fimples, thus. Suppofe a grocer mixes feveral forts of fugar, and of Vol. I. No. 6, 3

each an equal quantity, viz. at 50 s. at yq s. and at 60 s. per Cwt. the rate of the mixture will be 54 s. 8 d. /(?rCwt.; for, /. s. d. 5°+J4+6o—164, and 3)164(54 8 . Note 2. If it be required to increafe or diminifti the quantity of the mixture, fay. As the fum of the given quantities of the fimples, to the feveral quantities given; fo the quantity of the mixture propofed, to the quantities of the fimples fought. Note 3. If it be required to know how much of each (imple is in an alligned portion of the mixture, fay. As the quantity of the mixture, td the feveral quantities of the fimples given ; fo the quantity of the alligned portion, to the quantities of .the (imples fought. Thus, Suppofe a grocer mixes 10 lb. of raifms, with 30 lb. of almonds, and 46 lb. of currants, and it be demanded, how many Ounces of each fort are found in every pound, or in every fixteen ounces of the mixture, fay. 80 : 10 : : 16 : 2 raifins. 80 : 30 : : 16 : 6 almonds. 80 : 40 : : 16 : 8 currants. Proof 16 Note 4. If the rates of two fimples, with the total value and total quantity of the mixture be given, the quantity of each fimple may be found as follows, viz. multiply the lefler rate into the total quantity, fubtradt the produft from the total value, and the remainder will be equal to the produdt of the excefs of the higher rate above the lower, multiplied into the quantity of the higher-priced fimple; and confequently the faid remainder, divided by the difference of the rates, will quote the faid quantity. Thus, Suppofe a grocer has a mixture of 400 lb weight, that coft him 7 1. 10 s. confifting of raifins, at 4 d. per lb, Hh and