All the results of this Pythagorean research lived through into the days of Plato and Aristotle and then, as we know from Euclid and Theon, into the learning of Alexandria. The importance recognised by them in the numbers ten and twelve was shown by the general adoption of a decimal system of notation, and of the division of time on a duodecimal system. You will ask me what symbols the Greeks had which could enable them to treat arithmetical figures of any complexity, and on this I could give you now a very definite reply, but the details would lead us away from our subject, seeing that this notation was lost in the Dark Ages and was ultimately replaced by the Arabic numerals. But we now know that they had a very practical system of decimal notation based on the use of the letters of the alphabet; and the fact that several letters obsolete in the alphabet of the fifth century B.C. appear as symbols, proves that it was current as early as Pythagorean days. The sign for 6 is the _digamma_, that for 90 is the _koph_ of the Phœnician alphabet, which is still found in Locrian inscriptions; the Phœnician letter known as _sampi_ is used for 900. We know the practical management of this easy notation perfectly from the mass of accounts both private and public found on Egyptian papyri. It can express large numbers far more compendiously than the Roman system, often more compendiously even than ours. Suppose you desire to express any large number, say 20,050, here it is β/ΜΝ; say 47,678, it is δ/ΜΖΧΟΗ, and if there be small gain in simplicity here, I will give you 800,000 = 10,000 × 80 = π/Μ. But these are practical matters, though without an easy notation even the most scientific thinkers could not make large progress.[34] The next great step was to pass from arithmetic to geometry as the science of space and to show how far the same laws governed both. If we are not well informed upon the beginnings of arithmetic, we are more fortunate in the case of geometry, and here, if anywhere, the old Greeks have been the acknowledged teachers of modern Europe. For we have in the so-called _Elements_ of Euclid, composed most probably at Alexandria about 300 B.C., a summary of all that had been discovered up to his day, doubtless with many new things of his own. He had distinctly built upon his predecessors; he has before him all through his book a problem discussed in Plato, that of the possible number of regular polyhedra, and its solution forms the climax of his work. But he begins from the very beginning and builds up his whole doctrine with such accuracy that a flaw in the demonstration is hard to be found.