ALLIGATION

, one of the rules in arithmetic, by which are resolved questions which relate to the compounding or mixing together of divers simples or ingredients, being so called from alligare, to tie or connect together, probably from certain vincula, or crooked ligatures, commonly used to connect or bind the numbers together.

It is probable that this rule came to us from the Moorish or Arabic writers, as we find it, with all the other rules of arithmetic, in Lucas de Burgo, and the other early authors in Europe.

Alligation is of two kinds, medial and alternate.

Alligation medial is the method of finding the rate or quality of the composition, from having given the rates and quantities of the simples or ingredients.

The rule of operation is this: multiply each quantity by its rate, and add all the products together; then divide the sum of the products by the sum of the quantities, or whole compound, and the quotient will be the rate sought.

For example, Suppose it were required to mix together 6 gallons of wine, worth 5s. a gallon; 8 gallons, worth 6s. the gallon; and 4 gallons, worth 8s. the gallon; and to find the worth or value, per gallon, of the whole mixture.

	Gal.	s.	products.
Here	6	mult. by 5 gives	30
	8	" by 6 "	48
	4	" by 8 "	32
whole comp.	18		110	sum of prod.

Then 18)	110 (6 2/18 or (6 1/9)s, is the rate sought.
	108
	2

Alligation alternate is the method of finding the quantities of ingredients or simples, necessary to form a compound of a given rate.

The rule of operation is this: 1st, Place the given rates of the simples in a column, under each other; noting which rates are less, and which are greater than the proposed compound. 2d, Connect or link with a crooked line, each rate which is less than the proposed compound rate, with one or any number of those which are greater than the same; and every greater rate with one or any number of the less ones. 3d, Take the difference between the given compound rate and that of each simple rate, and set this difference opposite every rate with which that one is linked. 4th, Then if only one difference stand opposite any rate, it will be the quantity belonging to that rate; but when there are several differences to any one, take their sum for its quantity.

For example, Suppose it were required to mix together gold of various degrees of fineness, viz. of 19, | of 21, and of 23 caracts fine, so that the mixture shall be of 20 caracts fine. Hence,

	Rates	Diffs.	Sum of Diffs.
Comp. rate 20	{	21	1	{	1 of 21 caracts sine,
19	1+3	4 of 19 caracts sine,
23	1	1 of 23 caracts sine.

That is, there must be an equal quantity of 21 and 23 caracts fine, and 4 times as much of 19 caracts fine.

Various limitations, both of the compound and the ingredients, may be conceived; and in such cases, the differences are to be altered proportionally.

Questions of this sort are however commonly best and easiest resolved by common Algebra, of which they form a species of indeterminate problems, as they admit of many, or an indesinite number of answers.

There is recorded a remarkable instance of a discovery made by Archimedes, both by alligation and specific gravity at the same time, namely, concerning the crown of Hiero, king of Syracuse. The king had ordered a crown to be made of pure gold, but when brought to him, a suspicion arose that it was mixed with alloy of either silver or copper, and the king recommended it to Archimedes to discover the cheat without defacing the crown. Archimedes, after long thinking on the matter, without lighting on the means of doing it, being one day in the bath, and observing how his body raised the water higher, conceived the idea that different metals of the same weight would occupy different spaces, and so raise or expel different quantities of water. Upon which he procured two other masses, each of the same weight with the crown, the one of pure gold, and the other of alloy; then immersing them all three, separately, in water, and observing the space each occupied, by the quantity it raised the water, he from thence computed the quantities of gold and alloy contained in the crown.