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).

“I postulate the following:

1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.

2. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made.

3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken.

4. When equal and similar plane figures coincide if applied to one another, their centres of gravity similarly coincide.

5. In figures which are unequal but similar the centres of gravity will be similarly situated. By points similarly situated in relation to similar figures I mean points such that, if straight lines be drawn from them to the equal angles, they make equal angles with the corresponding sides.

6. If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be in equilibrium at the same distances.

7. In any figure whose perimeter is concave in (one and) the same direction the centre of gravity must be within the figure.”

“Archimedes to Dositheus greeting.

Of most of the theorems which I sent to Conon, and of which you ask me from time to time to send you the proofs, the demonstrations are already before you in the books brought to you by Heracleides; and some more are also contained in that which I now send you. Do not be surprised at my taking a considerable time before publishing these proofs. This has been owing to my desire to communicate them first to persons engaged in mathematical studies and anxious to investigate them. In fact, how many theorems in geometry which have seemed at first impracticable are in time successfully worked out! Now Conon died before he had sufficient time to investigate the theorems referred to; otherwise he would have discovered and made manifest all these things, and would have enriched geometry by many other discoveries besides. For I know well that it was no common ability that he brought to bear on mathematics, and that his industry was extraordinary. But, though many years have elapsed since Conon's death, I do not find that any one of the problems has been stirred by a single person. I wish now to put them in review one by one, particularly as it happens that there are two included among them which are impossible of realisation [and which may serve as a warning] how those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.

It has been often remarked that, though the method of exhaustion exemplified in Euclid xii. 2 really brought the Greek geometers face to face with the infinitely great and the infinitely small, they never allowed themselves to use such conceptions. It is true that Antiphon, a sophist who is said to have often had disputes with Socrates, had stated that, if one inscribed any regular polygon, say a square, in a circle, then inscribed an octagon by constructing isosceles triangles in the four segments, then inscribed isosceles triangles in the remaining eight segments, and so on, “until the whole area of the circle was by this means exhausted, a polygon would thus be inscribed whose sides, in consequence of their smallness, would coincide with the circumference of the circle.” But as against this Simplicius remarks, and quotes Eudemus to the same effect, that the inscribed polygon will never coincide with the circumference of the circle, even though it be possible to carry the division of the area to infinity, and to suppose that it would is to set aside a geometrical principle which lays down that magnitudes are divisible ad infinitum. The time had, in fact, not come for the acceptance of Antiphon's idea, and, perhaps as the result of the dialectic disputes to which the notion of the infinite gave rise, the Greek geometers shrank from the use of such expressions as infinitely great and infinitely small and substituted the idea of things greater or less than any assigned magnitude.