PYTHAGOREAN ARITHMETIC 
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Numbers and the universe. 
We know that Thales (about 624-547 B.c.) and Anaximander 
(born probably in 611/10 b.c.) occupied themselves with 
astronomical phenomena, and, even before their time, the 
principal constellations had been distinguished. Pythagoras 
(about 572-497 b.c. or a little later) seems to have been 
the first Greek to discover that the planets have an inde 
pendent movement of their own from west to east, i.e. in 
a direction contrary to the daily rotation of the fixed stars ; 
or he may have learnt what he knew of the planets from the 
Babylonians. Now any one who was in the habit of intently 
studying the heavens would naturally observe that each 
constellation has two characteristics, the number of the stars 
which compose it and the geometrical figure which they 
form. Here, as a recent writer has remarked, 1 we find, if not 
the origin, a striking illustration of the Pythagorean doctrine. 
And, just as the constellations have a number characteristic 
of them respectively, so all known objects have a number ; 
as the formula of Philolaus states, ‘all things which can 
be known have number; for it is not possible that without 
number anything can either be conceived or known ’. 2 
This formula, however, does not yet express all the content 
of the Pythagorean doctrine. Not only do all things possess 
numbers ; but, in addition, all things are numbers ; ‘ these 
thinkers ’, says Aristotle, ‘ seem to consider that number is 
the principle both as matter for things and as constituting 
their attributes and permanent states ’. 3 True, Aristotle 
seems to regard the theory as originally based on the analogy 
between the properties of things and of numbers. 
‘ They thought they found in numbers, more than in fire, 
earth, or water, many resemblances to things which are and 
become ; thus such and such an attribute of numbers is jus 
tice, another is soul and mind, another is opportunity, and so 
on ; and again they saw in numbers the attributes and ratios 
of the musical scales. Since, then, all other things seemed 
in their whole nature to be assimilated to numbers, while 
numbers seemed to be the first things in the whole of nature, 
1 L. Brunschvicg, Les étapes de la philosophie mathématique, 1912, p. 38. 
2 Stob. Ecl. i. 21, 7b ( Vors. i 3 , p. 310. 8-10). 
3 Aristotle, Metaph. A. 5, 986 a 16. 
* F 2