There are some, king Gelon, who think that the numberof the sand is infinite in multitude; and I mean by the sandnot only that which exists about Syracuse and the rest of Sicilybut also that which is found in every region whether inhabitedor uninhabited. Again there are some who, without regardingit as infinite, yet think that no number has been named whichis great enough to exceed its multitude.… Read the rest of this passage →

" Archimedes to Dositheus greeting. On a former occasion I sent you the investigations whichI had up to that time completed, including the proofs, showingthat any segment bounded by a straight line and a section of aright-angled cone [a parabola] is four-thirds of the trianglewhich has the same base with the segment and equal height.Since then certain theorems not hitherto demonstrated… Read the rest of this passage →

I POSTULATE the following: 1. Equal weights at equal distances are in equilibrium,and equal weights at unequal distances are not in equilibriumbut 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 inequilibrium but incline towards that weight to which theaddition was made.… Read the rest of this passage →

In this book I have set forth and send you the proofs of theremaining theorems not included in what I sent you before, andalso of some others discovered later which, though I had oftentried to investigate them previously, I hekd failed to arrive atbecause I found their discovery attended with some difficulty.And this is why even the propositions themselves were notpublished with the rest. But… Read the rest of this passage →

The area of any circle is equal to a right-angled triangle inwhich one of the sides about the right angle is equal to the radius,and the other to the circumference, of the circle. Let ABCD be the given circle, K the triangle described. i^ ^ -^ II \. \ C ) B ^«$=^.<=^ o Then, if the circle is not equal to K, it must be eithergreater or less. I. If possible, let the circle be greater than K.… Read the rest of this passage →

Of most of the theorems which I sent to Conon, and ofwhich you ask me from time to time to send you the proofs, thedemonstrations are already before you in the books brought toyou by Heracleides; and some more are also contained in thatwhich I now send you. Do not be surprised at my taking aconsiderable time before publishing these proofs. This hasbeen owing to my desire to communicate them first… Read the rest of this passage →

ERATOSTHENES "Archimedes to Eratosthenes greeting. I sent you on a former occasion some of the theoremsdiscovered by me, merely writing out the enunciations andinviting you to discover the proofs, which at the momentI did not give. The enunciations of the theorems which Isent were as follows. 1. If in a right prism with a parallelogrammic base acylinder be inscribed which has its bases in the…