GREEK ARITHMETIC.
Greek mathematicians were in the habit of discriminating between the science of numbers and the art of calculation. The former they called arithmetica, the latter logistica. The drawing of this distinction between the two was very natural and proper. The difference between them is as marked as that between theory and practice. Among the Sophists the art of calculation was a favourite study. Plato, on the other hand, gave considerable attention to philosophical arithmetic, but pronounced calculation a vulgar and childish art.
In sketching the history of Greek calculation, we shall first give a brief account of the Greek mode of counting and of writing numbers. Like the Egyptians and Eastern nations, the earliest Greeks counted on their fingers or with pebbles. In case of large numbers, the pebbles were probably arranged in parallel vertical lines. Pebbles on the first line represented units, those on the second tens, those on the third hundreds, and so on. Later, frames came into use, in which strings or wires took the place of lines. According to tradition, Pythagoras, who travelled in Egypt and, perhaps, in India, first introduced this valuable instrument into Greece. The abacus, as it is called, existed among different peoples and at different times, in various stages of perfection. An abacus is still employed by the Chinese under the name of Swan-pan. We possess no specific information as to how the Greek abacus looked or how it was used. Boethius says that the Pythagoreans used with the abacus certain nine signs called apices, which resembled in form the nine "Arabic numerals." But the correctness of this assertion is subject to grave doubts.
The oldest Grecian numerical symbols were the so-called Herodianic signs (after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them). These signs occur fre-
