Chapter 30. On the topics.
1. Topica is the science of finding arguments. The division of the topica or the loci from which arguments are derived is three-fold. For some inhere in the very thing that is under discussion; there are others, called affecta (closely connected), which are known to be derived in a certain sense from other things; others, which are taken from outside [the subject]....
18. It is clearly a wonderful thing that whatever the nimbleness and variety of the human mind could discover, searching for ideas in different cases, could have been gathered into unity; that free and spontaneous intelligence is limited. For wherever it turns, whatever thoughts it enters on, the mind must fall upon some of those that have been described.
BOOK III
On the Four Mathematical Sciences
ON ARITHMETIC
INTRODUCTION
In examining Isidore’s De Arithmetica two peculiarities of the development of the subject should be borne in mind. In the first place, the predominant position among the mathematical sciences which Isidore claims for arithmetic was one acquired by it comparatively late. Owing perhaps to the awkwardness of the Greek notation of number[235] geometry had been developed first, and historically arithmetic was an off-shoot from geometry and borrowed its terminology largely from it.[236] It was not given an independent form until the time of Nicomachus (fl. 100 A.D.) whose Introductio Arithmetica was “the first exhaustive work in which arithmetic was treated quite independently of geometry.”[237] Once it become independent, arithmetic, instead of geometry, came to be regarded as the fundamental mathematical science. The old tradition is reflected in Martianus Capella’s order of subjects, in which geometry is placed first and arithmetic second, while the newer tradition is seen in the order of Cassiodorus and Isidore, who both have passages also emphasizing the fundamental character of arithmetic.
The second peculiarity is one which will surprise the modern reader who is familiar with arithmetic as a utilitarian study. The ancient arithmetica had nothing to do with the art of reckoning, which was called logistica.[238] The science and the art of numbers were completely divorced and the latter was excluded from the higher education as we have it in the seven liberal arts. Consequently we can expect nothing practical in Isidore’s De Arithmetica. Nothing is said of methods of calculation, elementary or advanced, and, as a matter of course, nothing is to be found here on such topics as the use of the abacus[239] or the method of computing Easter, though the latter was the greatest mathematical problem of the time.