PREFACE.

as I have said, I indicated the genuine writing of Archimedes in the notes, wherever it was either certain or could be restored by probable conjecture, I also enclosed many additions in the Greek text in brackets [ ], which I omitted in the translation; conversely, in the translation, those things are enclosed in brackets [ ] which I myself added to illustrate the words of Archimedes and the method of the demonstration. Regarding those additions, cf. what I wrote in Quaest. Arch. pp. 69—78 and Neue Jahrbücher Suppl. XI pp. 384—398. In those places, which I later understood to be interpolated, I have always briefly indicated the reason in the notes; regarding the rest, it should be sufficient to have cited those two discussions once here. One place, however, in which a longer discussion is needed, I wish to treat more fully here.

On the Sphere and Cylinder I, 41 (in Torelli I, 47) p. 172, 8: καὶ ὡς ἄρα τὸ πολύγωνον πρὸς τὸ πολύγωνον, ὁ Μ κύκλος πρὸς τὸν Ν κύκλον and as, therefore, the polygon is to the polygon, so is circle M to circle N] let there be rectangular spaces contained by the sides of the polygons (P, p) and the lines joining the angles, S and s; which are equal to the radii (R, r) squared of circles M and N. And equal to circles N and M are the surfaces of the circumscribed and inscribed figures (O, o). Now Archimedes, from the fact that

wishes to conclude that O : o = EK² : AΛ². If those words were genuine, he would reason thus: S : s = EK² : AΛ², but S : s = R² : r² = M : N, and EK² : AΛ² = P : p; wherefore P : p = M : N; but M : N = O : o and P : p = EK² : AΛ²; wherefore O : o = EK² : AΛ². How flawed this is, no one fails to see; for the mention of the polygons has been introduced quite perversely, since it should have been concluded thus:

but S : s = EK² : AΛ²; wherefore O : o = EK² : AΛ².