“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.

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).

An extraordinarily large proportion of the subject matter of the writings of Archimedes represents entirely new discoveries of his own. Though his range of subjects was almost encyclopaedic, embracing geometry (plane and solid), arithmetic, mechanics, hydrostatics and astronomy, he was no compiler, no writer of textbooks ; and in this respect he differs even from his great successor Apollonius, whose work, like that of Euclid before him, largely consisted of systematising and generalising the methods used, and the results obtained, in the isolated efforts of earlier geometers. There is in Archimedes no mere working-up of existing materials; his objective is always some new thing, some definite addition to the sum of knowledge, and his complete originality cannot fail to strike any one who reads his works intelligently, without any corroborative evidence such as is found in the introductory letters prefixed to most of them. These introductions, however, are eminently characteristic of the man and of his work; their directness and simplicity, the complete absence of egoism and of any effort to magnify his own achievements by comparison with those of others or by emphasising their failures where he himself succeeded: all these things intensify the same impression. Thus his manner is to state simply what particular discoveries made by his predecessors had suggested to him the possibility of extending them in new directions; e.g. he says that, in connexion with the efforts of earlier geometers to square the circle and other figures, it Occurred to him that no one had endeavoured to square a parabola, and he accordingly attempted the problem and finally solved it.

The sources of the text and versions are very fully described by Heiberg in the Prolegomena to Vol. m. of his edition of Archimedes, where the editor supplements and to some extent amends what he had previously written on the same subject in his dissertation entitled Quaestiones Arckimedeae (1879). It will therefore suffice here to state briefly the main points of the discussion.

The MSS. of the best class all had a common origin in a MS. which, so far as is known, is no longer extant. It is described in one of the copies made from it (to be mentioned later and dating from some time between A.D. 1499 and 1531) as ‘most ancient’ (παλαιοτάτου), and all the evidence goes to show that it was written as early as the 9th or 10th century. At one time it was in the possession of George Valla, who taught at Venice between the years 1486 and 1499; and many important inferences with regard to its readings can be drawn from some translations of parts of Archimedes and Eutocius made by Valla himself and published in his book entitled de expetendis et fugiendis rebus (Venice, 1501). It appears to have been carefully copied from an original belonging to some one well versed in mathematics, and it contained figures drawn for the most part with great care and accuracy, but there was considerable confusion between the letters in the figures and those in the text. This MS., after the death of Valla in 1499, became the property of Albertus Pius Carpensis (Alberto Pio, prince of Carpi).