PYTHAGOREAN ASTRONOMY 
165 
the Pythagoreans 1 ; but he hints at the truer explanation in 
another passage where he says that eclipses of the moon 
were considered to be due sometimes to the interposition 
of the earth, sometimes to the interposition of the counter 
earth (to say nothing, of other bodies of the same sort 
assumed by ‘ some ’ in order to explain why there appear 
to be more lunar eclipses than solar) 2 ; we may therefore 
take it that the counter-earth was invented for the purpose 
of explaining eclipses of the moon and their frequency. 
Recapitulation. 
The astronomical systems of Pythagoras and the Pytha 
goreans illustrate the purely mathematical character of their 
physical speculations ; the heavenly bodies are all spheres, 
the most perfect of solid figures, and they move in circles ; 
there is no question raised of forces causing the respective 
movements ; astronomy is pure mathematics, it is geometry, 
combined with arithmetic and harmony. The capital dis 
covery by Pythagoras of the dependence of musical intervals 
on numerical proportions led, with his successors, to the 
doctrine of the ‘ harmony of the spheres ’. As the ratio 
2 : 1 between the lengths of strings of the same substance 
and at the same tension corresponds to the octave, the 
ratio 3 : 2 to the fifth, and the ratio 4 : 3 to the fourth, it 
was held that bodies moving in space produce sounds, that 
those which move more quickly give a higher note than those 
which move more slowly, while those move most quickly which 
move at the greatest distance ; the sounds therefore pro 
duced by the heavenly bodies, depending on their distances 
(i.e. the size of their orbits), combine to produce a harmony ; 
1 the whole heaven is number and harmony ’. 3 
We have seen too how, with the Pythagoreans, the theory 
of numbers, or 1 arithmetic ’, goes hand in hand with geometry ; 
numbers are represented by dots or lines forming geometrical 
figures ; the species of numbers often take their names from 
their geometrical analogues, while their properties are proved 
by geometry. The Pythagorean mathematics, therefore, is all 
one science, and their science is all mathematics. 
1 Arist. Metaph. A. 5, 986 a 8—12. 
2 Arist. De caelo, ii. 13, 293 b 21-5. 
Arist. Metaph. A. 5, 986 a 2.