 or more probably the Pythagorean s added seven more proportions. Here are a few, given in modern notation:
Note: each of these is expressible in the notation of proportion, as above.
Allowing A , G, and H denote the arithmetic, geometric and harmonic means, the Pythagorean s called the proportion
the perfect proportion. The proportion
was called the musical proportion.
The Discovery of Incommensurables
This discovery is usually given to Hippasus of Metapontum ( cent B.C.). One account gives that the Pythagorean s were at sea at the time and when Hippasus produced an element which denied virtually all of Pythagorean doctrine, he was thrown overboard.
Theorem. is incommensurable with 1.
Proof. Suppose that , with no common factors. Then
Thus , and hence . So, a=2c and it follows that
whence by the same reasoning yields that . This is a contradiction. width .1in height .1in depth 0pt
But is this the actual proof known to the Pythagorean s? Note: Unlike the Babylonnians or Egyptians, the Pythagorean s recognized that this class of numbers was wholly different from the rationals.
``Properly speaking, we may date the very beginnings of ``theoretical" mathematics to the first proof of irrationality, for in ``practical" (or applied) mathematics there can exist no irrational numbers." Here a problem arose that is analogous to the one whose solution initiated theoretical natural science: it was necessary to ascertain something that it was absolutely impossible to observe (in this case, the incommensurability of a square's diagonal with its side).
The discovery of incommensurability was attended by the introduction of indirect proof and, apparently in this connection, by the development of the definitional system of mathematics. In general, the proof of irrationality promoted a stricter approach to geometry, for it showed that the evident and the trustworthy do not necessarily coincide.
Other Pythagorean Geometry
Quadrature of certain lunes was performed by Hippocrates of Chios. He is also credited with the arrangement of theorems in an order so that one may be proved from a previous one (as we see in Euclid).
The large lune ABD is similar to the smaller lune (with base on one leg of the right isosceles triangle . Proposition:The area of the large lune ABD is the area of the semi-circle less the area of the triangle .