§ 39. We pass to the teaching of elementary science. Geometry was still an advanced study, and, though in high esteem among the Greeks as one of the most elegant and perfect, seems not to have been taught in schools. Arithmetic was regarded either as the abstract science of numbers (ἀριθμητική), and as such one of the most difficult of sciences, or as the art of reckoning (λογιστική) to be employed in the ordinary affairs of life. Mercantile Greeks, like the Athenians and Ionians generally, among whom banking was well developed, must have early found this a necessity; but even in Greek art, architectural perfection was attained by a very subtle and evidently conscious application of arithmetical proportions. This was first shown in the accurate measurements of the Parthenon by Penrose, and was, no doubt, expounded in the treatise written on this building by its architect, Ictinus. In the great temple of Zeus at Olympia, the use of multiples of 7 and 5 has been shown so curiously applied by an American scholar that he suspects the application of Pythagorean symbolism by the architect Libon. But of course this was ἀριθμητική in the strict sense, and is only here mentioned to show how the Greeks must have been led to appreciate the value of the science of numbers. Ordinary schoolboys were taught to add, subtract, multiply, and divide, as they now are, but without the advantage of our admirable system of notation.
Starting from the natural suggestion of the fingers—a suggestion preserved all through later history by such words as πεμπάζεσθαι (literally, to count by fives, but used of counting generally)—the Greeks represented numbers by straight strokes, but soon replaced ||||| either by a rude picture of a hand, V (as we find in Roman numbers), and made two such symbols joined together to represent 10 (X), or else the higher numbers were marked by the first letter of their name—viz., M and C, in Latin mille and centum. So in Greek, Χ (χίλιοι), Μ (μυρίοι), etc. The smaller numbers were represented in ordinary counting by the fingers of the hand, not merely as digits (a suggestive word, in itself a survival of the process), but, according as they were bent or placed, fingers represented multiples of 5, and so were sufficient for ordinary sums. Aristophanes even contrasts[39] this sort of reckoning, as clearer and more intelligible, with reckoning on the abacus, or arithmetical board, which has still survived in our ball-frames. We are told that the fingers were sufficient to express all figures up to thousands, which is indeed strange to us; but both the finger signs and the abacus failed in the great invention we have gained from Arabic numerals, the supplying of the symbol ○. The abacus used in Greek schools appears to have had several straight furrows in which pebbles or plugs were set, and at the left side there was a special division where each unit meant 5. Thus, 648 (DCXXXXVIII) was represented in the following way:
| M | ||||||||
| o | o | C | ||||||
| o | o | o | o | X | ||||
| o | o | o | o | |||||
This abacus was ascribed to Pythagoras, but was in all probability older, and derived from Egypt, where elementary science was well and widely taught from very early times. When initial letters were used for numbers, as Π for πέντε, and Δ for δέκα, combinations such as
meant 50. Last of all, we find in our MSS. a system of using the letters of the alphabet for numbers, preserving ϛ (ἐπίσημον) for 6, and thus reaching 10 with ι, proceeding by tens through κ (20), λ (30), etc., to ρ (100), σ (200), and for 900 using Ϡ. This notation must not be confused with the marking of the twenty-four books of the Homeric epics by the simple letters of the alphabet.
Further details as to the technical terms for arithmetical operations, and the amount to be attributed to a nation using so clumsy a notation, must be sought in professed hand-books of antiquity.[40]
As regards geometry, all we can say is that in the days of Plato and Aristotle both these philosophers recognize not only its extraordinary value as a mental training, but also the fact that it can be taught to young boys as yet unfit for political and metaphysical studies.
§ 40. Having thus disposed of the severer side of school education, we will turn to the artistic side, one very important to the Greeks, and suggestive to us of many instructive problems.