=317.= _Ath._ There still remain three studies suitable for freemen. Calculation in arithmetic is one of them; the measurement of length, surface, and depth is the second; and the third has to do with the revolutions of the stars in reference to one another ... there is in them something that is necessary and cannot be set aside, ... if I am not mistaken, [something of] divine necessity; for as to the human necessities of which men often speak when they talk in this manner, nothing can be more ridiculous than such an application of the words. _Cle._ And what necessities of knowledge are there, Stranger, which are divine and not human? _Ath._ I conceive them to be those of which he who has no use nor any knowledge at all cannot be a god, or demi-god, or hero to mankind, or able to take any serious thought or charge of them. --PLATO. _Republic, Bk. 7. Jowett’s Dialogues of Plato (New York, 1897), Vol. 4, p. 334._ =318.= Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. For if they do not actually employ these names, they do not exhibit even the results and the reasons of these, and therefore can be hardly said to make any assertion about them. Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree. And since, at least, these appear to be the causes of many things--now, I mean, for example, order, and that which is a definite thing, it is evident that they would assert, also, the existence of a cause of this description, and its subsistence after the same manner as that which is fair subsists in.--ARISTOTLE. _Metaphysics [MacMahon] Bk. 12, chap. 3._ =319.= Many arts there are which beautify the mind of man; of all other none do more garnish and beautify it than those arts which are called mathematical.--BILLINGSLEY, H. _The Elements of Geometrie of the most ancient Philosopher Euclide of Megara (London, 1570), Note to the Reader._ =320.= As the sun eclipses the stars by his brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them.--BRAHMAGUPTA. _Quoted in Cajori’s History of Mathematics (New York, 1897), p. 92._ =321.= So highly did the ancients esteem the power of figures and numbers, that Democritus ascribed to the figures of atoms the first principles of the variety of things; and Pythagoras asserted that the nature of things consisted of numbers.