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tfi AP. XI, | Ai, is G.47 föW. 3 that Rule whereby we resolve question: goto the mixing of several Simples, Groo minudities into one Compound Quantity. - Aligation is either Medial or Alternate. Aiiigation #edia!, is when having the several on". and rates of divers simples proposed, we discover the rate to a mixture competinded of these sittiples, - - - Rule, Find according to the given rates, the value of each given quantity, then taking the suo (5f these quantities; ind the sum of their values, say, if that sufi of quan" ties give that sum of values what will the quantity (proposed) give - o * - - - Application, i. Suppose 15 bushels of wheat, at 5s, the bushel, and 13 bushëls of rye, at 3s. 6d. #' bushel, were mixed logo ther, what is the mean rate or price it may be sold at" bushel, without loss or gain. - - ? s, d. 1. s. 15 Busheds, at 5 () come to 3 15 12 ditto, -- 3 6 — 2 2 - - - 3) tonsequently 2% bush. their sum come to (–5 it what is that for 1 bushed; 27 & 1--— 9) I Ig * * * * * * - - * ... Answ. 4s. 4d. ‘the reason of this operation is manifest in itself. To prote the truth of the Work. Find the value of all the flixture at the meaii ta'e Likewise the value of each particular quantity proposed to je mixed at its give" rates, and collect these particular

values into one sum, if the said sum is equal to the value of All ilic filixture before foulid, the work is fight, as

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8. A refiner has 10th. of gold of 20 caracts fine, and

melts it with 16th. of 18 caracts fine. The question is, how much alloy must be put to it to make it 22 caracts fine? - A a Answ.

* An ounce of pure gold being reduced into 24 equal pars, these ..

parts are called caracts: but gold is often mixed with some baser metal, which in the mixture is called the ailoy ; and according to the proportion of pure gold which is in every ounce, so the mixture is said to be so many caracts fine: Thus, if only 22 caracts of pure gold, and two of alloy, it is 22 caracts fine; if 20 caracts of pure gold and 4 of alloy it is 20 caracts fine; if there is no alloy, it is 24 caracts fine, or pure gold.

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Answ. 'Tis not fine enough by 3% caracts, so no alloy must be put to it but more gold. .

SECT II.

ALLIGATION ALTERNATE

LLIGATION Alternate is, when we have the several ingredients to be mixed, and the mean rate of the mixture given, to find such quantities of the simples or ingredients, as being mixed together, shall bear that common

rate. Rule.

The rates being all of, or reduced to one denomination, 2. Set the rates of the simples in a column under one another, and the mean rate on the left-hand of these. 3. Connect or link together the several simple rates, so that every one less than the mean be linked to one or more greater than it, and every one greater with one or more less. 4. Take the difference between the mean rate and that of the several simples, and write it over against all the simples with which that one (whose difference it is) is linked, then the sums of these differences standing against every simple rate, are such quantities of the several simples against which they stand as answer the question.

I. When the simple rates do not exceed three, there can be but one way of linking them; because the mixture or mean rate must be between the highest and lowest rate of the simples; else, ’tis plain the mixture would not bear that rate, but would be of a greater or lesser rate, as the simples were either all of a greater or lesser: So then of two simples one will be greater and one less than the mean rate, and of three one greater and two less, or two greater and one less, which cases can admit but one way of linking.

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The Proof.

Questions in this Rule are proved by Alligation Medial, as follows. "

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4. How many oz, of silver, of 11 oz. fine, and 8 oz., fine, must be melted together to make the mass or mixture 9 oz. fine Answ. 1 oz. of 11oz. fine, and 2 oz. of 8 oz. fine.

II. When one rate is joined to two others, the sum of the differences of the said two and the mean rate, will be the quantity sought, at that rate to which the two are linked.

5. A merchant hath sugar of 5d. 10d. and 12d. #2, #3. How much of each sort must he take that he may sell a pound for 8d.

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7. A grocer would mix three sorts of sugar together, viz. one sort at 10d. 4p' ib. another at 7d. and another at 6d. How much of each sort must he take, that the whole mixture may be sold for 8d.: #2 ib. ib. d

$3 at 10 £' fh. Answ. K. 2 at 7 Q2 at 6 8. A vintner has brandy, at 3, 5, and 6s. the gallon, and has a mind to mix a quantity of them together, so that it may stand him in 5s. 6d. the gallon. I demand how many gallons he must take of each sort Answ. 4 at 3s. 4 at 5s and 3 at 6s. III. If the number of simple rates exceed 3 there may be several ways of linking them, and every way brings different answers; but all giving such numbers as will answer the thing required. Examples. 0. A merchant would mix wines, at 14, 15, 19 and 228. d?’ dozen, so as the mixture may be worth 18s, What quantity of each may be taken? This sum may be linked 7 different ways as follow, viz.

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* Besides the different answers produced by this different manner of linking the terms or simple prices, questions in Alligation Alternate (being of that kind Algebraists term unlimited problems) have an infinite variety of other answers: for any other numbers in proportion to those found by this Rule (as above) will answer the questions as well as those. - - 10. A

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