ARISTOTLE 
341 
by being put together fill up space in a plane are the equi 
lateral triangle, the square and the regular hexagon. 
(5) Curves and solids known to Aristotle. 
There is little beyond elementary plane geometry in Aris 
totle. He has the distinction between straight and 4 curved ’ 
lines (Ka/xTrvXcu ypa/ifiai), but the only curve mentioned 
specifically, besides circles, seems to be the spiral 1 ; this 
term may have no more than the vague sense which it has 
in the expression ‘ the spirals of the heaven ’ 2 ; if it really 
means the cylindrical helix, Aristotle does not seem to have 
realized its property, for he includes it among things which 
are not such that ‘any part will coincide with any other 
part’, whereas Apollonius later proved that the cylindrical 
helix has precisely this property. 
In solid geometry he distinguishes clearly the three dimen 
sions belonging to ‘ body *, and, in addition to parallelepipedal 
solids, such as cubes, he is familiar with spheres, cones and 
cylinders. A sphere he defines as the figure which has all its 
radii (‘ lines from the centre ’) equal, 3 from which we may infer 
that Euclid’s definition of it as the solid generated by the revo 
lution of a semicircle about its diameter is his own (Eucl. XI, 
Def. -14). Referring to a cone, he says 4 ‘the straight lines 
thrown out from K in the form of a cone make GK as a sort 
of axis (dxrTrep d£ova) ’, showing that the use of the word 
‘ axis ’ was not yet quite technical; of conic sections he does 
not seem to have had, any knowledge, although he must have 
been contemporary with Menaechmus. When he alludes to 
‘ two cubes being a cube ’ he is not speaking, as one might 
suppose, of the duplication of the cube, for he is saying that 
no science is concerned to prove anything outside its own 
subject-matter; thus geometry is not required to prove ‘that 
two cubes are a cube’ 5 ; hence the sense of this expression 
must be not geometrical but arithmetical, meaning that the 
product of two cube numbers is also a cube number. In the 
Aristotelian Problems there is a question which, although not 
mathematical in intention, is perhaps the first suggestion of 
1 Phys. v. 4. 228 b 24. 2 Metaph. B. 2. 998 a 5. 
3 Phys. ii. 4. 287 a 19. 4 Meteorologica, iii. 5. 375 b 21. 
5 Anal. Post. i. 7. 75 b 12.