Chapter XX: 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.

[235] “The cumulative evidence is surely very strong that the
alphabetic numerals were first employed in Alexandria early
in the third century B.C.” J. Gow, _A Short History of Greek
Mathematics_ (Cambridge, 1884), p. 48.

[236] We have in Isidore, for example, the terms _numerus
trigonus_, _numerus quadratus_, _numerus quinquangulus_, and
_linealis_, _superficialis_, and _circularis numerus_.

[237] Cajori, _Hist. of Math._, p. 72.

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.

[238] Gow, speaking of the Greek ἀριθμητική, says: “Its aim was
entirely different from that of the ordinary calculator, and it
was natural that the philosopher who sought in numbers to find
the plan on which the creator worked, should begin to regard with
contempt the merchant who wanted only to know how many sardines
at ten for an obol he could buy for a talent.” Gow, _op. cit._,
p. 72.

[239] Cantor believes that the use of the abacus had been
forgotten before Isidore’s time, _cf._ “calculator a calculis,
id est a lapillis minutis quos antiqui in manu tenentes numeros
componebant.” _Etym._, 10, 43. See Cantor, _Vorlesungen über
Geschichte der Mathematik_ (Leipzig, 1894–1900), vol. i, p. 774.

Isidore’s source in the _De Arithmetica_ was Cassiodorus,[240] whom he copies with little change; while Cassiodorus’ work was apparently a bare abstract of Boethius’ translation of Nicomachus. Isidore’s account is of great brevity and contains a number of unexplained technical terms.

[240] Isidore adds to the account as found in Cassiodorus a few
remarks about numbers in the Scriptures, some derivations of
numbers, and the sections on the means and on infinity.

EXTRACTS

PREFACE. Mathematics is called in Latin _doctrinalis scientia_. It considers abstract quantity. For that is abstract quantity which we treat by reason alone, separating it by the intellect from the material or from other non-essentials, as for example, equal, unequal, or the like. And there are four sorts of mathematics, namely, arithmetic, geometry, music and astronomy. Arithmetic is the science of numerical quantity in itself. Geometry is the science of magnitude and forms.[241] Music is the science that treats of numbers that are found in sounds. Astronomy is the science that contemplates the courses of the heavenly bodies and their figures, and all the phenomena of the stars. These sciences we shall next describe at a little greater length in order that their significance may be fully shown.

[241] Du Breul has _magnitudinis et formarum_; Arevalo,
_magnitudinis formarum_.

Chapter 1. On the name of the science of arithmetic.

1. Arithmetic is the science of numbers. For the Greeks call number ἀριθμός. The writers of secular literature have decided that it is first among the mathematical sciences since it needs no other science for its own existence.

2. But music and geometry and astronomy, which follow, need its aid in order to be and exist.

Chapter 2. On the writers.

1. They say that Pythagoras was the first among the Greeks to write of the science of number, and that it was later described more fully by Nicomachus, whose work Apuleius first, and then Boethius, translated into Latin.