“There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognising that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe. Now you are aware that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth.

So far as the language of Archimedes is that of Greek geometry in general, it must necessarily have much in common with that of Euclid and Apollonius, and it is therefore inevitable that the present chapter should repeat many of the explanations of terms of general application which I have already given in the corresponding chapter of my edition of Apollonius' Conies. But I think it will be best to make this chapter so far as possible complete and selfcontained, even at the cost of some slight repetition, which will however be relieved (1) by the fact that all the particular phrases quoted by way of illustration will be taken from the text of Archimedes instead of Apollonius, and (2) by the addition of a large amount of entirely different matter corresponding to the great variety of subjects dealt with by Archimedes as compared with the limitation of the work of Apollonius to the one subject of conies.

One element of difficulty in the present case arises out of the circumstance that, whereas Archimedes wrote in the Doric dialect, the original language has been in some books completely, and in others partially, transformed into the ordinary dialect of Greek. Uniformity of dialect cannot therefore be preserved in the quotations about to be made; but I have thought it best, when explaining single words, to use the ordinary form, and, when illustrating their use by quoting phrases or sentences, to give the latter as they appear in Heiberg's text, whether in Doric or Attic in the particular case.

THE Republic of Plato touches on so many problems of human life and thought, and appeals to so many diverse types of mind and character, that an editor cannot pretend to have exhausted its significance by means of a commentary. In one sense of the term, indeed, there can never be a definitive or final interpretation of the Republic: for the Republic is one of those few works of genius which have a perennial interest and value for the human race; and in every successive generation those in whom man's inborn passion for ideals is not quenched, will claim the right to interpret the fountain-head of idealism for themselves, in the light of their own experience and needs. But in another sense of the word, every commentator on the Republic believes in the possibility of a final and assured interpretation, and it is this belief which is at once the justification and the solace of his labours. Without desiring in any way to supersede that personal apprehension of Platonism through which alone it has power to cleanse and reanimate the individual soul, we cannot too strongly insist that certain particular images and conceptions, to the exclusion of others, were present in the mind of Plato as he wrote.

It has often been explained how the Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others because the conception of a negative quantity was unknown to them. The quadratic equation was regarded as a simple equation connecting areas, and its geometrical expression was facilitated by the methods which they possessed of transforming any rectilineal areas whatever into parallelograms, rectangles, and ultimately squares, of equal area; its solution then depended on the principle of application of areas, the discovery of which is attributed to the Pythagoreans. Thus any plane problem which could be reduced to the geometrical equivalent of a quadratic equation with a positive root was at once solved. A particular form of the equation was the pure quadratic, which meant for the Greeks the problem of finding a square equal to a given rectilineal area. This area could be transformed into a rectangle, and the general form of the equation thus became x2 = ab, so that it was only necessary to find a mean proportional between a and b. In the particular case where the area was given as the sum of two or more squares, or as the difference of two squares, an alternative method depended on the Pythagorean theorem of Eucl. 1. 47 (applied, if necessary, any number of times successively).