vanced on the road to higher
abstract thinking. It is everywhere characteristic of Egyptian genius
that little purely intellectual curiosity is shown. Even astronomical
knowledge was limited to those determinations which had religious or
magically practical significance, and its arithmetic and geometry never
escaped these bounds as with the more imaginative Pythagoreans, where
mystical interpretation seems to have been a consequence of rather than
a stimulus to investigation. An old Egyptian treatise reads (Cantor, p.
63): "I hold the wooden pin (Nebi) and the handle of the mallet (semes),
I hold the line in concurrence with the Goddess S[a.]fech. My glance
follows the course of the stars. When my eye comes to the constellation
of the great bear and the time of the number of the hour determined by
me is fulfilled, I place the corner of the temple." This incantation
method could hardly advance intelligence; but the methods of practical
measuring were more effective. Here the rather happy device of using
knotted cords, carried about by the Harpedonapts, or cord stretchers,
was of some moment. Especially, the fact that the lengths 3, 4, and 5,
brought into triangular form, served for an interesting connection
between arithmetic and the right triangle, was not a little gain, later
making possible the discovery of the Pythagorean theorem, although in
Egypt the theoretical properties of the triangle were never developed.
The triangle obviously must have been practically considered by the
decorators of the temple and its builders, but the cord stretchers
rendered clear its arithmetical significance. However, Ahmes' "Rules for
attaining the knowledge of all dark things ... all secrets that are
contained in objects" (Cantor, _loc. cit._, p. 22) contains merely a
mixture of all sorts of mathematical information of a practical
nature,--"rules for making a round fruit house," "rules for measuring
fields," "rules for making an ornament," etc., but hardly a word of
arithmetical and geometrical processes in themselves, unless it be
certain devices for writing fractions and the like.

II

THE PROGRESS OF SELF-CONSCIOUS THEORY

A characteristic of Greek social life is responsible both for the next
phase of the development of mathematical thought and for the
misapprehension of its nature by so many moderns. "When Archytas and
Menaechmus employed mechanical instruments for solving certain
geometrical problems, 'Plato,' says Plutarch,