The Book of Mechanics

and judged that the demonstrations conclude only concerning planes or only concerning solids, and not concerning whichever ones [in general], but only concerning rectilinear or homogeneous ones, or those which are of the same species among themselves, he wanders far from the scope and mind of Archimedes. For in these he always speaks either of heavy things simply, as in the first three theorems, or of magnitudes, as in the remaining five. Which name is indeed common to any planes and solids whatsoever, as even those who are little versed in mathematics know well enough. Just as Euclid also, while he treated the propositions of the fifth book, comprehended continuous quantity under the name of magnitude. That the name "heavier" is common, however, is already sufficiently clear in itself. It is therefore evident that these first eight theorems are common to both planes and solids, and not only to solids of the same species and homogeneous, but even to solids of diverse species and heterogeneous, as will become manifest in its own place. And this foundation having been laid, which Archimedes demonstrated in two propositions, namely the sixth and the seventh, in the eighth he collects it as a corollary. Next, he treats particularly of the center of gravity of planes, and no longer names planes by the name of magnitude, but by their own proper names, such as parallelogram, triangle, and others of this kind. And in this part he descends to particulars, since he teaches us the center of gravity of any particular plane, if not perhaps in act, yet in virtue. For in the first book it seemed to him sufficient to have shown the centers of gravity of triangles and parallelograms, from which it will not be very difficult to investigate the centers of gravity of other figures, such as the pentagon, hexagon, and other similar ones, since planes of this kind are divided into triangles, as we shall touch upon at the end of the first book. In the second book, however, he raises himself higher, and in his own manner concerns himself with the most subtle theorems, namely concerning the center of gravity of a conic section which is called a parabola. And he premises some theorems which are as it were preliminary preparations for investigating the demonstration of the center of gravity in the parabola. Thus it is evident that Archimedes properly hands down mechanical elements,