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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.