Introduction to Arithmetic

"It treats, then, on the one hand, that which Archimedes called 'The Cattle Problem,' and on the other hand, 'melite' honey-like and 'phialite' vial-like numbers, the one discussing vials (measures, containers) and the other flocks; and when dealing with other kinds of problems it has regard for the number of sensible bodies and makes its pronouncements as though it were for absolute objects.

"It has for material all numerable objects, and as subdivisions the so-called Greek and Egyptian methods for multiplication and division, as well as the summation and decomposition of fractions, whereby it investigates the secrets lurking in the subject-matter of the problems by means of the procedure that employs triangles and polygons.

"It has for its aim that which is useful in the relations of life and in business, although it seems to pronounce upon sensible objects as if they were absolute."

The philosophical arithmetic of the Greeks, arithmetike arithmetic, of which the arithmetic of Nicomachus is a specimen, corresponds in a measure to our number theory; the subject was designed for mature students as a preparation for the study of philosophy, and was not at all intended for children. Arithmetica is, as the name indicates, the study of that which is implied in number. This branch of arithmetical science developed along two quite distinct lines. On the one hand we have the rigid, mathematical discussion of the properties of numbers, involving the forms of proof and the rigor of the demonstrational geometry, which is the great contribution of Greece to science; on the other hand we have a mystical development, ascribing even magical powers and life-properties to numbers. This pseudo-science which employs the results, but not the demonstrations of the rigid science, is commonly termed arithmology.See Chapter VII, pp. 90 ff., for a discussion of arithmology and of the share of Nicomachus in it. Greek arithmetic must be considered, then, from the point of view of the philosopher and theoretical mathematician, rather than from that of our elementary schools.

Arithmetic was intimately connected by the early Greeks with both geometry and music. The treatise on arithmetic by Euclid, as found in the seventh, eighth, and ninth books of the Elements,Our references to the Elements of Euclid will be to the English edition by Sir T. L. Heath, The Thirteen Books of Euclid's Elements, three volumes, Cambridge, 1908. is wholly from the geometrical standpoint. This point of view is reflected in many ways in later treatises, that of Nicomachus, for instance, which considered arithmetic as an independent science.