214 THE ELEMENTS DOWN TO PLATO’S TIME 
the geometric, and the harmonic (a name substituted by 
Archytas and Hippasus for the older name ‘ sub-contrary’). 
From his mention of sphaerlc in connexion with his state 
ment that ‘ the mathematicians have given us clear knowledge 
about the speed of the heavenly bodies and their risings and 
settings’ we gather that in Archytas’s time astronomy was 
already treated mathematically, the properties of the sphere 
being studied so far as necessary to explain the movements 
in the celestial sphere. He discussed too the question whether 
the universe is unlimited in extent, using the following 
argument. 
‘ If I were at the outside, say at the heaven of the fixed 
stars, could I stretch my hand or my stick outwards or not ? 
To suppose that I could not is absurd ; and if I can stretch 
it out, that which is outside must be either body or space (it 
makes no difference which it is, as we shall see). We may 
then in the same way get to the outside of that again, and 
so on, asking on arrival at each new limit the same question ; 
and if there is always a new place to which the stick may be 
held out, this clearly involves extension without limit. If 
now what so extends is body, the proposition is proved ; but 
even if it is space, then, since space is that in which body 
is or can be, and in the case of eternal things we must treat 
that which potentially is as being, it follows equally that there 
must be body and space (extending) without limit.’ 1 
In geometry, while Archytas doubtless increased the number 
of theorems (as Proclus says), only one fragment of his has 
survived, namely the solution of the problem of finding two 
mean proportionals (equivalent to the duplication of the cube) 
by a remarkable theoretical construction in three dimensions. 
As this, however, belongs to higher geometry and not to the 
Elements, the description of it will come more appropriately 
in another place (pp. 246-9). 
In music he gave the numerical ratios representing the 
intervals of the tetrachord on three scales, the anharmonic, 
the chromatic, and the diatonic. 2 He held that sound is due 
to impact, and that higher tones correspond to quicker motion 
communicated to the air, and lower tones to slower motion, 3 
1 Simplicius in Phys., p. 467. 26. 2 Ptol. harm. i. 18, p. 81 Wall. 
3 Porph. in Ptol. harm., p. 286 (Vors. i s , p. 232-3); Theon of Smyrna, 
p. 61. 11-17.