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PYTHAGOREAN GEOMETRY 
It is this identification of mathematics (and of geometry 
in particular) with science in general, and their pursuit of it 
for its own sake, which led to the extraordinary advance of 
the subject in the Pythagorean school. It was the great merit 
of Pythagoras himself (apart from any particular geometrical 
or arithmetical theorems which he discovered) that he was the 
first to take this view of mathematics; it is characteristic of 
him that, as we are told, ‘ geometry was called by Pythagoras 
inquiry or science ’ (e/caAefro Se rj yecoyerpia 7rpos Ilvdayopov 
'lottopia)} Not only did he make geometry a liberal educa 
tion ; he was the first to attempt to explore it down to its 
first principles ; as part of the scientific basis which he sought 
to lay down he ‘ used definitions A point was, according to 
the Pythagoreans, a ‘ unit having position ’ 2 ; and, if their 
method of regarding a line, a surface, a solid, and an angle 
does not amount to a definition, it at least shows that they 
had reached a clear idea of the differentiae, as when they said 
that 1 was a point, 2 a line, 3 a triangle, and 4 a pyramid. 
A surface they called xpoid, 1 colour ’; this was their way of 
describing the superficial appearance, the idea being, as 
Aristotle says, that the colour is either in the limiting surface 
(7repay) or is the nepas, 3 so that the meaning intended to be 
conveyed is precisely that intended by Euclid’s definition 
(XI. Def. 2) that ‘ the limit of a solid is a surface ’. An angle 
they called yXcoffs, a ‘ point ’ (as of an arrow) made by a line 
broken or bent back at one point. 4 
The positive achievements of the Pythagorean school in 
geometry, and the immense advance made by them, will be 
seen from the following summary. 
1. They were acquainted with the properties of parallel 
lines, which they used for the purpose of establishing by 
a general proof the proposition that the sum of the three 
angles of any triangle is equal to two right angles. This 
latter proposition they again used to establish the well-known 
theorems about the sums of the exterior and interior angles, 
respectively, of any polygon. 
2. They originated the subject of equivalent areas, the 
transformation of an area of one form into another of different 
1 Iambi. Vit. Fyth. 89. 
3 Arist. De sensu, 3. 439 a 31. 
2 Proclus on End. I, p. 95. 21. 
4 Hevon, Def. 15.