318 
FROM PLATO TO EUCLID 
Theory of numbers (Speusippus, Xenocrates). 
To begin with arithmetic or the theory of numbers. Speu 
sippus, nephew of Plato, who succeeded him as head of the 
school, is said to have made a particular study of Pythagorean 
doctrines, especially of the works of Philolaus, and to have 
written a small treatise On the Pythagorean Numbers of 
which a fragment, mentioned above (pp. 72, 75, 76) is pre 
served in the Theologumena Arithmetices} To judge by the 
fragment, the work was not one of importance. The arith 
metic in it was evidently of the geometrical type (polygonal 
numbers, for example, being represented by dots making up 
the particular figures). The portion of the book dealing with 
‘ the five figures (the regular solids) which are assigned to the 
cosmic elements, their particularity and their community 
with one another ’, can hardly have gone beyond the putting- 
together of the figures by faces, as we find it in the Timaeus, 
To Plato’s distinction of the fundamental triangles, the equi 
lateral, the isosceles right-angled, and the half of an equilateral 
triangle cut off by a perpendicular from a vertex on the 
opposite side, he adds a distinction (‘passablement futile’, 
as is the whole fragment in Tannery’s opinion) of four 
pyramids (1) the regular pyramid, with an equilateral triangle 
for base and all the edges equal, (2) the pyramid on a square 
base, and (evidently) having its four edges terminating at the 
corners of the base equal, (3) the pyramid which is the half of 
the preceding one obtained by drawing a plane through the 
vertex so as to cut the base perpendicularly in a diagonal 
of the base, (4) a pyramid constructed on the half of an 
equilateral triangle as base; the object was, by calling these 
pyramids a monad, a dyad, a triad and a tetrad respectively, 
to make up the number 10, the special properties and virtues 
of which as set forth by the Pythagoreans were the subject of 
the second half of the work. Proclus quotes a few opinions 
of Speusippus; e. g., in the matter of theorems and problems, 
he differed from Menaechmus, since he regarded both alike 
as being more properly theorems, while Menaechmus would 
call both alike problems. 1 2 
1 Theol. Ar., Ast, p. 61. 
2 Proclus on Eucl. I, pp. 77. 16; 78. 14.