Ans. 5 oz of 15, of 17, and of 18 caracts fine, and 25 oz of 22 caracts fine.* Ex. 2. A vintner has wine at 4s, at 5s, at 5s 6d, and at 6s a gallon; and he would make a mixture of 18 gallons, so that it might be afforded at 5s 4d per gallon ;* how much of each sort must he take? Ans. 3 gal. at 4s, 3 at 5s, 6 at 5s 6d, and 6 at 6s. A great number of questions might be here given relating to the specific gravities of metals, &c. but one of the most curious may here suffice. Hiero, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but suspecting the workman had debased it by mixing it with silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown. Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of silver or copper, and each of the same weight with the former; and by putting each separately into a vessel full of water the quantity of water expelled by them determined their specific gravities; from which, and their given weights, the exact quantities of gold and alloy in the crown may be determined. Suppose the weight of each crown to be 10lb, and that the water expelled by the copper or silver was 921b, by the gold 521b, and by the compound crown 641b; what will be the quantities of gold and alloy in the crown? The rates of the simples are 92 and 52, and of the compound 64; therefore And the sum of these is 12+28 = 40, which should have been but 10; there fore by the Rule, 40:10 12: 3lb of copper) the answer. RULE RULE III. WHEN one of the ingredients is limited to a certain quao. tity; Take the difference between each price, and the mean rate as before; then say, As the difference of that simple, whose quantity is given, is to the rest of the differences severally; so is the quantity given, to the several quantities required. EXAMPLES. 1. How much wine at 5s, at 5s 6d, and 6s the gallon, must be mixed with 3 gallons at 4s per gallon, so that the mixture may be worth 5s 4d per gallon? Ans. 3 gallons at 5s, 6 at 5s 6d, and 6 at 6s. 2. A grocer would mix teas at 12s, 10s, and 6s per lb, with 20lb at 4s per lb. how much of each sort must he take to make the composition worth 8s per lb ? Ans. 20lb at 4s, 10lb at 6s, 10lb at 10s, and 20lb at 12s. 3. How much gold of 15, of 17, and of 22 caracts fine, must be mixed with 5 oz of 18 caracts fine, so that the composition may be 20 caracts fine? Ans. 5 oz of 15 caracts fine, 5 oz of 17, and 25 of 22. In the very same manner questions may be wrought when several of the ingredients are limited to certain quantities, by finding first for one limit, and then for another. The two last Rules can need no demonstration, as they evidently result from the first, the reason of which has been already explained. ROSITION. POSITION. POSITION is a method of performing certain questions, which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of false numbers, to work with the same as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes also called Trial-and-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.-Position is either Single or Double. SINGLE POSITION. SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their result proportional to their suppositions, belong to Single Position such as those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. The rule is as follows: TAKE or assume any number for that which is required, and perform the same operations with it, as are described or performed in the question. Then say, As the result of the said operation, is to the position, or number assumed; so is the result in the question, to a fourth term, which will be the number sought.* The reason of this Rule is evident, because it is supposed that the results are proportional to the suppositions, Thus, na: a: nz: z, EXAMPLES. 1. A person after spending and of his money, has yet remaining 601; what had he at first? 2. What number is that which being multiplied by 7, and the product divided by 6, the quotient may be 21 ? Ans. 18. and 3. What number is that, which being increased by 1, 1, of itself, the sum shall be 75? Ans. 36. 4. A general, after sending out a foraging and of his men, had yet remaining 1000; what number had he in command? Ans. 6000. 5. A gentleman distributed 52 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d, to each woman 4d, and to each child 2d: moreover there were twice as many women as men, and thrice as many children as women. How many were there of each? Ans. 2 men, 4 women, and 12 children. 6. One being asked his age, said, if of the years I have lived, be multiplied by 7, and of them be added to the product, the sum will be 219. What was his age? Ans. 45 years. DOUBLE DOUBLE POSITION, DOUBLE POSITION is the method of resolving certain ques tions by means of two suppositions of false numbers. To the Double Rule of Position belong such questions as have their results not proportional to their positions : such are those, in which the numbers sought, or their parts, or their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought. RULE I. TAKE or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position; and find how much each result is different from the result mentioned in the question, calling these differences the errors, noting also whether the results are too great or too little. Demonstr. The Rule is founded on this supposition, namely, that the first error is to the second, as the difference between the true and first supposed number, is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this Rule.That the Rule is true, according to that supposition, may be thus proved. Let a and b be the two suppositions, and A and B their results, produced by similar operation; also r and ́s their errors, or the differences between the results A and B from the true result N; and let x denote the number sought, answering to the true result N of the question. Then is N-Ar, and N-BS. And, according to the supposition on which the Rule is founded, r: s:: x-a: x-b: hence, by multiplying extremes and means, ra- rb — sx — sa; then, by transposition, rx — sx rb —sa; rb-sa and, by division, ≈ sults are both too little. r-s the number sought, which is the rule when the re If the results be both too great, so that A and B are both greater than N; then NA, and NB- s, or r and s are both negative; hence --:-- $ :: x-a: x- - b, but s::+r+s, therefore rs :: xa: x-b; and the rest will be exactly as in the former case. But if one result a only be too little, and the other в too great, or one error r rb + sa positive, and the other s negative, then the theorem becomes r+s Then |