§ 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: