350 
FROM PLATO TO EUCLID 
Tannery thought that we could hardly help attributing it to 
Eudoxus. The suggestion is natural, seeing that Eudoxus 
showed, in his theory of concentric spheres, an extraordinary 
mastery of the geometry of the sphere ; on the other hand, 
as Loria observes, it is, speaking generally, dangerous to 
assume that a work of an unknown author appearing in 
a certain country at a certain time must have been written 
by a particular man of science simply because he is the only 
man of the time of whom we can certainly say that he was 
capable of writing it. 1 The works of Autolycus also serve to 
confirm the pre-Euclidean origin of a number of propositions 
in the Elements. Hultsch 1 2 examined this question in detail 
in a paper of 1886. There are (1) the propositions pre 
supposed in one or other of Autolycus’s theorems. We have 
also to take account of (2) the propositions which would be 
required to establish the propositions in sphaeric assumed by 
Autolycus as known. The best clue to the propositions under 
(2) is the actual course of the proofs of the corresponding 
propositions in the Sphaerica of Theodosius; for Theodosius 
was only a compiler, and we may with great probability 
assume that, where Theodosius uses propositions from Euclid’s 
Elements, propositions corresponding to them were used to 
prove the analogous propositions in the fourth-century 
Sphaeric. The propositions which, following this criterion, 
we may suppose to have been directly used for this purpose 
are, roughly, those represented by Eucl. I. 4, 8, 17, 19, 26, 29, 
47 ; III. 1-3, 7, 10, 16 Cor., 26, 28, 29; IV. 6 ; XI. 3, 4, 10, 11, 
12, 14, 16, 19, and the interpolated 38. It is, naturally, the 
subject-matter of Books I, III, and XI that is drawn upon, 
but, of course, the propositions mentioned by no means 
exhaust the number of pre-Euclidean propositions even in 
those Books. When, however, Hultsch increased the list of 
propositions by adding the whole chain of propositions (in 
cluding Postulate 5) leading up to them in Euclid’s arrange 
ment, he took an unsafe course, because it is clear that many 
of Euclid’s proofs -were on different lines from those used 
by his predecessors. 
1 Loria, Le scienze esatte nelV antica Grecia, 1914, p. 496-7. 
2 Berichte der Kgl. Sachs. Gesellschaft der Wissenschaften zu Leipzig, 
Phil.-hist. Classe, 1886, pp. 128-55.