bscure, that made possible the development of modern technique. Of this
discovery, or rediscovery from the Hindoos, together with the zero
symbol, Cajori (_Hist. of Math._, p. 11) has said "of all mathematical
discoveries, no one has contributed more to the general progress of
intelligence than this." The notation no longer merely records results,
but now assists in performing operations.

The origins of geometry are even more obscure than those of arithmetic.
Not only is geometry as highly developed as arithmetic when it first
appears in occidental civilization, but, in addition, the problems of
primitive peoples seem to have been such that they have developed no
geometrical formulae striking enough to be recorded by investigators, so
far as I have been able to discover. But just as the commercial life of
the Phoenicians early forced them self-consciously to develop
arithmetical calculation, so environmental conditions seem to have
forced upon the Egyptians a need for geometrical considerations.

It is almost platitudinous to quote Herodotus' remark that the invention
of geometry was necessary because of the floods of the Nile, which
washed away the boundaries and changed the contours of the fields. And
as Proclus Diadochus adds (_Procli Diadochi, in primum Euclidis
elementorum librum commentarii_--quoted Cantor, I, p. 125): "It is not
surprising that the discovery of this as well as other sciences has
sprung from need, because everything in the process of beginning
proceeds from the incomplete to the complete. There takes place a
suitable transition from sensible perception to thoughtful consideration
and rational knowledge. Just as with the Phoenicians, for the sake of
business and commerce, an exact knowledge of numbers had its beginning,
so with the Egyptians, for the above-mentioned reasons, was geometry
contrived."

The earliest Egyptian mathematical writing that we know is that of Ahmes
(2000 B. C.), but long before this the mural decorations of the temple
wall involved many figures, the construction of which involved a certain
amount of working knowledge of such operations as may be performed with
the aid of a ruler and compass. The fact that these operations did not
earlier lead to geometry, as ruler and compass work seems to have
done in Japan in the nineteenth century (Smith and Mikami, index,
"Geometry"), is probably due to the stage at which the development of
Egyptian intelligence had arrived, feebly ad