ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led on by such easy stages that the difficulty of the original problem, as presented at the outset, is scarcely appreciated.
As Plutarch says, “It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations”. But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that any one could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Although each step depends upon the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark of Wallis to the effect that he seems “as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results”.
A partial exception is now furnished by the Method; for here we have (as it were) a lifting of the veil and a glimpse of the interior of Archimedes’s workshop. He tells us how he discovered certain theorems in quadrature and cubature, and he is at the same time careful to insist on the difference between (1) the means which may serve to suggest the truth of theorems, although not furnishing scientific proofs of them, and (2) the rigorous demonstrations of them by approved geometrical methods which must follow before they can be finally accepted as established.
Writing to Eratosthenes he says: “Seeing in you, as I say, an earnest student, a man of considerable eminence in philosophy and an admirer of mathematical inquiry when it comes your way, I have thought fit to write out for you and explain in detail in the same book the peculiarity of