Key Takeaway: Diophantus' Arithmetica, a synthesis of Greek, Egyptian, and Mesopotamian mathematics, was influential in the development of Arabic algebra, European number theory, and modern abstract algebra and computer science.

Abstract

Though it was written in Greek in a center of ancient Greek learning, Diophan-tus’s Arithmetica is a curious synthesis of Greek, Egyptian, and Mesopotamian mathematics. It was not only one of the ﬁrst purely number-theoretic and algebraic texts, but the ﬁrst to use the blend of rhetorical and symbolic exposition known as syncopated mathematics. The text was inﬂuential in the development of Arabic algebra and European number theory and notation, and its development of the theory of indeterminate, or Diophantine, equations inspired modern work in both abstract algebra and computer science. We present, in this article, a selection of problems from the Arithmetica , which have been rewritten for ease of reading, and consider Diophantus’s advancements in mathematics and mathematical notation in the context of ancient Greek mathematics. In particular, we examine Diophantus’s use of syncopated mathematics, most notably his use of generic solutions that present an algorithm for solving an entire class of equations through the application of that algorithm to a single representational example, and how these techniques suggest a more extensive use of concrete examples when approaching modern mathematics.