The Birth of Number Theory (Book VII of Euclid's Elements) from the Arithmetization of Pythagorean Music

Abstract

Book VII of Euclid's Elements represents a remarkable mathematical achievement, marking the birth of number theory. Although it falls short of explicitly stating the Fundamental Theorem of Arithmetic (that every natural number is uniquely a product of primes), it contains all the tools necessary for its proof: the Principle of the Least (equivalent to Mathematical Induction) and arithmetical anthyphairesis for finding the greatest common divisor. Our work presents novel arguments that Book VII evolved directly from early Pythagorean arithmetized music. The accounts attributing this arithmetization to Pythagoras' acoustical experiments were shown to be fictitious by Vincenzo Galilei. In contrast, the alternative experiments and Hippasus' 4-chord (bronze cylinders of heights 6, 8, 9, 12) are validated as physically correct by Euler's Law for pipes. This experimental foundation led to the arithmetization of musical intervals and to the discovery of musical (multiplicative) anthyphairesis. Applying Aristotle's Topics 158b24-29 Principle, we reconstruct how musical anthyphairesis was transferred - through an inductive ladder of ratios starting with multiple and epimoric ratios recounted by Theon of Smyrna - to its arithmetical (additive) counterpart. Furthermore, we show that the mathematical peculiarities in the definitions and proofs of Book VII find a convincing explanation only through their musical origin. Ultimately, Hippasus emerges as the pivotal figure who discovered musical anthyphairesis (which evolved into Philolaus' Fragment 6) and transferred arithmetical anthyphairesis to geometry, crucial for Pythagorean incommensurability and the principles of the Infinite and Finite.