Alligation Calculator

Alligation alternate is a shortcut for solving a single conservation-of-solute balance. If a volume Vh of strength Ch is combined with a volume Vl of strength Cl to reach a target Cd, then Ch times Vh plus Cl times Vl equals Cd times the total, and rearranging gives Vh over Vl equal to Cd minus Cl divided by Ch minus Cd. The cross diagram simply draws those two subtractions. The parts are relative, so any total can be split by the fractions: with 90 and 50 percent stock aiming at 70 percent, the parts are 20 and 20, or 200 mL of each in a 400 mL batch. The method is only valid when the target lies between the two source strengths, and it degenerates when both sources are identical. Where the weaker source is pure solvent at 0 percent, the relation collapses to the familiar C1V1 equals C2V2 dilution equation. Units must be consistent but need not be percentages: milligrams per millilitre, molarity and percent weight in volume all work provided both sources and the target share a basis. The main practical caveat is that volumes are treated as additive, which is only approximately true. Ethanol and water contract on mixing, so combining 50 mL of ethanol with 50 mL of water yields roughly 96 mL rather than 100, and alcohol dilutions in pharmacy are therefore calculated by weight or corrected using specific gravity. The companion procedure, alligation medial, runs the arithmetic in the other direction, computing the strength that results from mixing known quantities.

Alligation belongs to the body of commercial arithmetic that European merchants, assayers and mint officials developed in the late medieval and Renaissance periods. The name derives from the Latin alligare, to bind or tie together, describing the lines drawn between the paired figures in the cross diagram. The original problems were metallurgical and monetary rather than chemical: how much silver of one fineness must be blended with silver of another to strike coin at the standard, or how to bring a batch of gold to a required carat without wasting bullion. Errors here had legal consequences, since mint output was assayed against a fixed standard, and so reliable mixing arithmetic was a valued professional skill. By the sixteenth and seventeenth centuries alligation had become a standard chapter in English arithmetic textbooks alongside the Rule of Three, appearing in widely used works such as Robert Recorde's The Ground of Artes, first published in 1543, and Edward Cocker's Arithmetick of 1678, with the two forms distinguished as alligation medial and alligation alternate. As school mathematics was reorganized around algebra during the nineteenth and twentieth centuries the topic disappeared from general curricula, because the same problems are handled by a linear equation. It survived, however, in pharmacy practice and pharmaceutical calculation textbooks, where compounding to a prescribed strength from available stock concentrations remains a routine bench task and alligation is still taught and examined.