The Early Study of Incommensurability: Theodorus

Abstract

Via the unit-mode of representing numbers, the fifth-century Pythagoreans succeeded in studying many properties of numbers: oddness and evenness; the interrelated aspects of the figured numbers: triangular, square and oblong; the divisibility of square numbers; and the description of the number-triples. But much of this same arithmetic material was collected and presented via a different representational mode. In Elements II one finds an alternative conception of the ‘gnomon’, as the geometric difference of parallelograms, used to yield results already standard in the Pythagorean arithmetic. For instance, 11,4 establishes geometrically that for given lines A, B, (A+B)² = A ²+2AB+B ². If we set B as the unit line and A as some integral multiple, we obtain the Pythagorean assertion that the square numbers are formed from the consecutive odd integers as gnomons.