Content of type 'historical_note'
Book XII Historical Note
The predominant feature of Book xii. is the use of the method of exhaustion, which is applied in Propositions 2, 3—5, 10, 11, 12, and (in a slightly different form) in Propositions 16—18. We conclude therefore that for the content of this Book Euclid was greatly indebted to Eudoxus, to whom the discovery of the method of exhaustion is attributed. The evidence for this attribution comes mainly from Archimedes. (1) In the preface to On the Sphere and Cylinder I., after stating the main results obtained by himself regarding the surface of a sphere or a segment thereof, and the volume and surface of a right cylinder with height equal to its diameter as compared with those of a sphere with the same diameter, Archimedes adds: “Having now discovered that the properties mentioned are true of these figures, I cannot feel any hesitation in setting them side by side both with my former investigations and with those of the theorems of Eudoxus on solids which are held to be most irrefragably established, namely that any pyramid is one third part of the prism which has the same base with the pyramid and equal height [i.e. Eucl. xii. 7], and that any cone is one third part of the cylinder which has the same base with the cone and equal height [i.e. Eucl. xii. 10]. For, though these properties also were naturally inherent in the figures all along, yet they were in fact unknown to all the many able geometers who lived before Eudoxus and had not been observed by any one.” (2) In the preface to the treatise known as the Quadrature of the Parabola Archimedes states the “lemma” assumed by him and known as the “Axiom of Archimedes” (see note on x. 1 above) and proceeds: “Earlier geometers (οἱ πρότερον γεωμέτραι) have also used this lemma; for it is by the use of this same lemma that they have shown that circles are to one another in the duplicate ratio of their diameters [Eucl. xii. 2], and that spheres are to one another in the triplicate ratio of their diameters [Eucl. xii. 18], and further that every pyramid is one third part of the prism which has the same base with the pyramid and equal height [Eucl. xii. 7]; also, that every cone is one third part of the cylinder which has the same base with the cone and equal height [Eucl. xii. 10] they proved by assuming a certain lemma similar to that aforesaid.” Thus in the first passage two theorems of Eucl. xii. are definitely attributed to Eudoxus; and, when Archimedes says, in the second passage, that “earlier geometers” proved these two theorems by means of the lemma known as the “Axiom of Archimedes” and of a lemma similar to it respectively, we can hardly suppose him to be alluding to any other proof than that given by Eudoxus. As a matter of fact, the lemma used by Euclid to prove both propositions (xii. 3—5 and 7, and xii. 10) is the theorem of Eucl. x. 1. As regards the connexion between the two “lemmas” see note on x. 1.
Book XIII Historical Note
I have already given, in the note to IV. 10, the evidence upon which the construction of the five regular solids is attributed to the Pythagoreans. Some of them, the cube, the tetrahedron (which is nothing but a pyramid), and the octahedron (which is only a double pyramid with a square base), cannot but have been known to the Egyptians. And it appears that dodecahedra have been found, of bronze or other material, which may belong to periods earlier than Pythagoras’ time by some centuries (for references see Cantor’s Geschichte der Mathematik I3, pp. 175—6).