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We now return to pure mathematics, and consider more closely the apparatus of ideas out of which the science is built. Our first concern is with the symbolism of the science, and we start with the simplest and universally known symbols, namely those of arithmetic.

Let us assume for the present that we have sufficiently clear ideas about the integral numbers, represented in the Arabic notation by ${\displaystyle 0}$,${\displaystyle 1}$,${\displaystyle 2}$,…, ${\displaystyle 9}$, ${\displaystyle 10}$, ${\displaystyle 11}$,… ${\displaystyle 100}$, ${\displaystyle 101}$,…and so on. This notation was introduced into Europe through the Arabs, but they apparently obtained it from Hindoo sources. The first known work^[1] in which it is systematic ally explained is a work by an Indian mathematician, Bhaskara (born 1114 A.D.). But the actual numerals can be traced back to the seventh century of our era, and perhaps were originally invented in Tibet. For our present

1. ↑ For the detailed historical facts relating to pure mathematics, I am chiefly indebted to A Short History of Mathematics, by W. W. R. Ball.