PYTHAGOREAN ASTRONOMY
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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.