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ALLIGATION teaches how to compound or mix together several simples of different qualities, so that the composition may be of some intermediate quality, or rate. It is commonly distinguished into two cases, Alligation Medial, and Alligation Alternate.
ALLIGATION MEDIAL is the method of finding the rate or quality of the composition, from having the quantities and rates or qualities of the several simples given. And it is thus performed :
* MULTIPLY the quantity of each ingredient by its rate or quality; then add all the products together, and add also all
* Demonstration. The Rule is thus proved by Algebra. Let a, b, c be the quantities of the ingredients, and m, n, p their rates, or qualities, or prices; then um, bn, cp are their several values, and am + bn + cp the sum of their values, also a + b + c is the sum of the quantities, and if r denote the rate of the whole composition, then a + b + c X r will be the value of the whole, conseq.
. a + b + cxr=am + bn + cp, and r = am + bn + cp + a + b +c, which is the Rule.
Note, If an ounce or any other quantity of pure gold be reduced into 24 equal parts, these parts are called Caracts; but gold is often mixed with some base metal, which is called the Alloy, and the mixture is said to be of so many caracts fine, according to the proportion of pure gold contained in it; thus, if 22 caracts of pure gold, and 2 of alloy be mixed together, it is said to be 22 caracts 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 wat mixed with wine, and alloy with gold and silver. VOL I. K
the quantities together into another sum ; then divide the former sum by the latter, that is, the sum of the products by the sum of the quantities, and the quotient will be the rate or quality of the composition required.
1. If three sorts of gunpowder be mixed together, víz. 50lb at 12d a pound, 441b at 9d, and 261b at 8d a pound; how much a pound is the composition worth?
Here 50, 44, 26 are the quantities,
9 = 396
120) 1204 ( 10ro = 1036 Ans. The rate or price is 1035d the pound. 2. A composition being made of 5lb of tea at 7s per lbs 9lb at 8s 6d per lb, and 14 lb at 5s 100 per lb; what is a lb of it worth?
Ans. 6s 10 d. 3. Mixed 4 gallons of wine at 4s 10d per galt, with 7 gallons at 5s 3d, per gall, and 9: gallons at 5s 8d per gall; what is a gallon of this composition worth? Ans. 5s 4 d.
4. A mealman would mix 3 bushels of flour at 35 5d per bushel, 4 bushels at 5s 6d per bushel, and 5 bushels at 4s 8d per bushel; what is the worth of a bushel of this mixture ?
Ans. 4s 7d. 5. A farmer mixes 10 bushels of wheat at 5s the bushel, with 18 bushels of rye at 3s the bushel, and 20 bushels of barley at 2s per bushel: how much is a bushel of the mixture worth?
Ans. 35. 6. Having melted together 7 oz of gold of 22 caracts fine, 12oz of 21 caracts fine, and 17 oz of 19 caracts fine : I would know the fineness of the composition ?
Ans. 20%} caracts fine. 7. Of what fineness is that composition, which is made by mixing 31b of silver of 9 oz fine, with 5lb 8 oz of 100% fine, and ilb 10 oz of alloy,
Ans. 765 oz fine.
ALLIGATION ALTERNATE is the method of finding what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate. So that it is the reverse of Alligation Medial, and may be proved by it.
1. Set the rates of the simples in a column under each other.--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 the less.--3. Write the difference between the mixture rate, and that of each of the simples, opposite the rate with which they are 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.
The examples may be proved by the rule for Alligation Medial.
* Demonst. By connecting the less rate to the greater, and placing the difference between them and the rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole is equal, and is exactly the proposed rate : and the same will be true of any other two simples managed according to the Rule.
In like manner, whatever the number of simples may be, and with how many soever every one 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 this 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 dividing each of the quantities found, by 2, or 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 the double or treble, the ļor part, or any other ratio of these quantities, and so on ad infinitum.
These kinds of questions are called by algebraists indeterminate or unlimited problems; and by an analytical process, theorems may be raised that will give all the possible answers.
1. A merchant would mix wines at 16s, at i8s, and at 22s per gallon, so as that the mixture may be worth 20s the gallon : what quantity of each must be taken?
16 2 at 165
2 at 18s
22 4 + 2 = 6 at 22s. Ans. 2 gallons at 16s, 2 gallons at 18s, and 5 at 225. 2. How much wine at 6s per gallon, and at 4s per gallon, must be inixed together, that the composition may be worth 5s per gallon?
Ans. 1 qt, or 1 gall, &c. 3. How much sugar at 4d, at 6d, and at 11d per lb, must. be mixed together, so that the composition formed by them may be worth id
lb ? Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity
of each sort. 4. How much corn at 2s 6d, 3s 8d, 4s, and 4s 8d per bushel, must be mixed together, that the compound may be worth &s 100 per bushel ?
Ans. 2 at 2s 6d, 2 at 3s 8d, 3 at 4s, and 3 at 4s 8d. 5. A goldsmith has gold of 16, of 18, of 23, and of 24 caracts fine : how much must he take of each, to make it 21 caracts fine ? Ans. 3 of 16, 2 of 18, 3 of 23, and 5 of 24.
6. It is required to mix brandy at 12s, wine at 10s, cyder at 1s, and water at O per gallon together, so that the mixture may be worth &s per gallon? Ans. 8 gals of brandy, 7 of wine, 2 of cyder, and 4 of water.
When the whole composition is limited to a certain quantity : Find an answer as before by linking; then say, as the sum of the quantities, or differences thus determined, is to the given quantity; so is 'each ingredient, found by linking, to the required quantity of each.
1. How much gold of 15, 17, 18, and 22 caracts fine, must be mixed together, to form a composition of 40 oz of 20 caracts fine ?
5 and 16 : 40 :: 10 : 25 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 45, 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 52lb, and by the compound crown 64b; 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
12 of copper 52
28 of gold And the sum of these is 12 +28 = 40, which should have been bạt 10; therefore by the Rule,
40 : 10 :: 12 : 31b of copper }the answer.