The Book of Mechanics

that the two definitions of the center of gravity provided are either diminished or are repugnant to those things that have just been said by us concerning the center of gravity; since we showed that the center of gravity is sometimes either on the perimeter of the figure or outside the figure, whereas the definitions provided always assume it to be placed within them. The difficulty is confirmed, since we must not think that this kind of center established outside the figure was entirely unknown to Archimedes; as can be gathered from the ninth postulate of this book, when he says: Every figure whose perimeter is concave toward the same side must have its center of gravity within itself. As if it were not repugnant for a figure whose perimeter is not concave toward the same side to have its center of gravity outside it.

This objection can be met in such a way if we say that, although in figure C, for example, it was said that the center of gravity D exists outside the figure, it can also be affirmed that it is within the figure, since the perimeter of figure C contains the center D within itself, such that in respect to the whole, it is within. The same must be said of the other figure A. This, however, is most evident in figure E. And this is the meaning of the definitions of the center of gravity. These things having been known first, the intention of Archimedes in these books must be considered, which usually becomes clear for the most part from the inscriptions of the books.

ON THE PURPOSE OF THESE BOOKS

If the purpose of Archimedes in these books is to be investigated from the very inscription of the work, as is usually done for the most part in the volumes of other authors, it will seem in part indeed to be conspicuous, but in part so unknown that Archimedes declares he will have a discourse on almost nothing at all. For what (I pray) could be meant by those words which are at the beginning of the first book: Ἀρχιμήδους ἐπιπέδων ἰσορροπιῶν ἢ κέντρων βάρους ἐπιπέδων; that is, "Archimedes on Plane Equilibriums, or the Centers of Gravity of Planes," since Archimedes seems to propose for himself to contemplate a matter altogether useless, indeed repugnant to nature. For while he promises