time they find they have gained 5261. what is each just share? man's 1 2 5 49 58 1 2 5 4 Ans. A 1921. 19s. od. 672 B 3331. os. 11 d. 4. A, with a capital of 1000l. began trade January 1, 1776, and meeting with success in business he took in B as a partner, with a capital of 150ol. on the first of March following. Three months after that they admit C as a third partner, who brought into stock 2800l. and after trading together till the first of the next year, they find, there has been gained since A's commencing business, 17761. 10s. How must this be divided among the partners? Ans. A 4571. 95. 4 d. B 5711. 16s. 8d. ALLIGATION. ALLIGATION teaches how to mix several simples of dif ferent qualities, so that the composition may be of a middle quality; and is commonly distinguished into two principal cases, called Alligation medial and Alligation alternate." ALLIGATION MEDIAL. Alligation medial is the method of finding the rate of the 'compound, from having the rates and quantities of the sev eral simples given. RULE.* Multiply each quantity by its rate; then divide the sum of the products by the sum of the quantities, or the whole composition, *The truth of this rule is too evident to need a demonstration. NOTE. If an ounce or any other quantity of pure gold be reduced into 24 equal parts, these parts are called carats; but gold is composition, and the quotient will be the rate of the compound required. EXAMPLES. 1. Suppose 15 bushels of wheat at 5s. per bushel, and 12 bushels of rye at 3s. 6d. per bushel were mixed together: how must the compound be sold per bushel without loss 2. A composition being made of 5lb. of tea at 75. per pound, 9lb. at 8s. 6d. per pound, and 14 lb. at 5s. 10d. per pound, what is a pound of it worth? Ans. 6s. 10d. 3. Mixed 4 gallons of wine at 4s. 10d. per gallon, with 7 gallons at 5s. 3d. per gallon, and 94 gallons at 5s. 8d. per gallon; what is a gallon of this composition worth? Ans. 5s. 44d. 4. A goldsmith melts 8lb. 5oz. of gold bullion of 14 carats fine, with 12lb. 8 oz. of 18 carats fine how many carats fine is this mixture? : Ans. 1620 508 carats. 5. A is often mixed with some baser metal, which is called the alloy, and the mixture is said to be of, so many carats fine, according to the proportion of pure gold contained in it: thus, if 22 carats of pure gold and 2 of alloy be mixed together, it is said to be 22 carats fine. If any one of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing, as water mixed with wine, and alloy with gold or silver. 5. A refiner melts 1clb. of gold of 20 carats fine with 16ib. of 18 carats fine; how much alloy must he put to it to make it 22 carats fine? Ans. It is not fine enough by 3 carats, so that no alloy must be put to it, but more gold. ALLIGATION ALTERNATE. Alligation alternate is the method of finding what quan tity of any number of simples, whose rates are given, will compose a mixture of a given rate; so that it is the re-verse of alligation medial, and may be proved by it. 1 RULE I. 1. Write the rates of the simples in a column under cach other. * 2. Connect DEMONSTRATION. By connecting the less rate to the greater, and placing the differences between them and the mean rate alternately, the quantities resulting are such, that there is précisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole are equal, and ́are exactly the proposed rate and the same will be true of any other two simples, managed according to the rule. In like manner, let the number of simples be what it may, and with how many soever each is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole, Q. E. D. It is obvious from the rule, that questions of this sort admit of a great variety of answers; for, having found one answer, we may find as many more as we please, by only multiplying or di viding each of the quantities found by 2, 3, or 4, &c. the reason of which is evident; for, if two quantities of two simples make a balance of loss and gain, with respect to the mean price, so must also 2. Connect or link with a continued line the rate of each simple, which is less than that of the compound, with one or any number of those, that are greater than the compound; and each greater rate with one or any number of 3. Write the difference between the mixture rate and that of each of the simples opposite the rates, with which they are respectively linked. 4. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity. EXAMPLES. 1. A merchant would mix wines at 14s. 195. 15s. and 22s. per gallon, so that the mixture may be worth 18s. the gallon what quantity of each must be taken? 2. How much wine at 6s. per gallon and at 4s. per gallon, must be mixed together, that the composition may be worth 5s. per gallon? Ans. 12 gallons, or equal quantities of each. 3. How also the double or treble, the or part, or any other ratio of these quantities, and so on, ad infinitum. Questions of this kind are called by algebraists indeterminate or unlimited problems, and, by an analytical process, theorems may be raised, that will give all the possible answers. 3. How much corn at 2s. 6d. 35. 8d. 4s. and 4s. Sd. per bushel, must be mixed together, that the compound may be worth 3s. 10d. per bushel? Ans. 12 at 2s. 6d. 12 at 3s. 8d. 18 at 4s. and 18 at 45. 8d. 4. A goldsmith has gold of 17, 18, 22 and 24 carats fine how much must he take of each to make it 21 carats fine ? Ans. 3 of 17, 1 of 18, 3 of 22 and 4 of 24. 5. It is required to mix brandy at 8s. wine at 7s. cider at Is. and water at o per gallon together, so that the mixture may be worth 5s. per gallon? Ans. 9 of brandy, 9 of wine, 5 of cider, and 5 of water.. RULE 2.* When the whole composition is limited to a certain quantity, find an answer as before by linking; then say, as the sum of * A great number of questions might be here given relating to the specific gravity of metals, &c. but one of the most curious, with the operation at large, may serve as a sufficient specimen HIERO, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but suspecting the workmen 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, one of pure gold, the other of silver or copper, and each of the same weight with the former; and each being put separately into a vessel full of water, the quantity of water expelled by them determined their specific bulks: 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 rolb. and that the water expelled by the copper or silver was 92lb. by the gold 52lb. and by the compound crown 64lb. what will be the quantities of gold and alloy in the crown? The |