Theodorus’ proofs of incommensurabilities with Gnomons

Abstract

An ‘infinite decreasing sequence of Gnomons’ is characteristic, according to Proclus, of incommensurability, hence David Fowler's idea to reconstruct Theodorus’ proofs of incommensurabilities, reported in the Theaetetus147d, employing Gnomons, is attractive and solidly based. The ‘preservation of the shape of the Gnomons’ is a form of the Pythagorean principle of the Limited according to Aristotle. In the present paper we propose a reconstruction that employs Gnomons but is free of the drawbacks present in Fowler's reconstruction.

Acknowledgements

We wish to thank the anonymous referee for substantial comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Two line segments are commensurable if they have a common measure, otherwise incommensurable (Definitions, Book X., Elements).

² translation by Harold N Fowler, 1921. The crucial word ‘dunamis’ is rendered as ‘root’.

³ The use of definitions and Propositions in the Elements to reconstruct geometrical Propositions, reported by Plato more than half a century before they were composed, might strike the reader as anachronistic. However Proclus, in In Eucliden 66,8-68,10, arguably provides evidence that much of the Elements were conceived in Plato's Academy, especially by two great mathematicians Theaetetus and Eudoxus.

⁴ The proof of Proposition X.2 in the Elements makes use of Eudoxus’ condition def.4, Book V, of the Elements. For this reason Knorr (1975, Chapter VIII, Section III) believes that it could not be in use by Theodorus. Nevertheless an elementary, not involving pre-Eudoxian, proof of X.2 can be given with a simple argument, similar to the pre-Eudoxian proof, of Proposition VI.1 of the Elements, suggested in Aristotle's Topics 158 b 29–35. The pre-Eudoxian proof of X.2 is as follows: suppose that a,b are commensurable. Then there are natural numbers n,m and line segment c, such that a=mc, b=nc. It is then practically obvious (as Aristotle himself points out) that Anth(a,b)=Anth(m,n); since, by Propositions VII.1 & 2, Anth(m,n) is finite, a contradiction is reached.

⁵ cf. Hardy and Wright 1926; Fowler 1999.

⁶ Fowler 1999, 3.4, 74–83; 10.3, 374–378

⁷ There has been an ongoing controversy as to whether Book II and Propositions 27–29 in Book VI of the Elements can be considered precursors of the modern quadratic equations, with van der Waerden (1976) (who has employed the term geometric algebra for the Propositions of Book II), H Freudenthal (1977), and Andre Weil (1978) the notable exponents of that thesis (with which we firmly side), and Sabetai Unguru (1975), the principal exponent of the opposite thesis.

⁸ cf. Morrow 1970.

⁹ Fowler 1999, 376

¹⁰ Fowler 1999, 76–77

¹¹ This drawback holds as well for all the proposed anthyphairetic reconstructions, employing the theory of ratios, including those by Zeuthen (1910), Becker (1933), von Fritz (1934), van der Waerden (1950), Kahane (1985). Non-anthyphairetic reconstructions of Theodorus’ incommensurabilities include Hardy and Wright (1926) and Knorr (1975), Bashmakova (1986), Conway and Shipman (2013).

¹² 158 b 29–34

¹³ 203 a 1–16

¹⁴ Fowler 1999, 376

¹⁵ Here and elsewhere the bar over a sequence of numbers indicates its periodic recurrence.

¹⁶ Fowler 1999, 83

¹⁷ cf. Knorr 1975, 126, who states ‘it would certainly be neat if Theodorus’ researches had foreshadowed Theaetetus aanthyphairetic theory’, but does not believe that it happened.

¹⁸ cf. concluding remarks after, below.

¹⁹ Meno 84d3–85b7

²⁰ Theaetetus 147 d ‘pos enescheto’. Knorr (1975 Chapter III) reads these words to mean that in fact Theodorus stopped because he encountered difficulty at the case N=17, this being a particularly simple anthyphairesis, in effect provides an argument against an anthyphairetic reconstruction. Accordingly his non-anthyphairetic reconstruction finds some difficulty at N=17 (Knorr 1975, Chapter VI). But there are objections against Knorr's reading, such as: ‘mechri X’ normally means ‘up to and including X’; the anonymous scholiast to the Theaetetus repeatedly replaces ‘enescheto’ with ‘heste’ (simply ‘stopped’) at N=17.