Abstract

Summary In terms of control of composition, the fabrication of money was arguably the most demanding of all pre-Industrial Revolution metallurgical practices. The calculations involved in such control needed arithmetical computations involving repeated multiplications and divisions, not only of integers but also of mixed numbers. Such computations were possible using Roman numerals, but with some difficulties. The advantages gained by employing arithmetic using Indo-arabic numerals for alloying calculations would have been the same as for other types of commercial calculations. A method of calculating the proportions of two coinage alloys of different compositions that were needed to produce a third alloy of specified intermediate composition, eventually known as the rule of alternate alligation, was however a specific requirement in the minting of money. This requirement made a particular demand on medieval mathematics, as a solution was not possible using arithmetic alone. Furthermore, the problem became indeterminate when more than two components were available for mixing. All the necessary methods of calculation were described by Fibonacci in his Liber abbaci. These methods are to be found in a sequence of practical arithmetics that stretch through the trattati d'abbaco of the Italian city states during the fourteenth and fifteenth centuries, to the first printed books of Continental authors such as Pacioli. Of the earliest sixteenth-century printed English books on practical arithmetic, The Grounde of Artes by Robert Recorde contained the first advance in the methodology of alternate alligation since Fibonacci. In the seventeenth century, formal proof of the rule of alternate alligation was given by Kersey in his practical arithmetics and in his treatise on algebra. The topic was also addressed by several authors of academic algebras. It was in this century that the problem of indeterminacy in the solution of alternate alligation involving the mixture of more than two components, was finally resolved. English mint accounts provide little direct information on the alloying calculation carried out, and nothing on the methods used. However, the continued description of such calculations in other mint documents, merchants notebooks, and in goldsmiths' handbooks spanning four centuries imply that the calculations described in practical arithmetics were used in practice. Monetary matters had high economic and political profiles during the whole of the time span covered in this study. The influence of such considerations pervades the history of the technology of minting and the mathematics on which it called, and is discussed in context