In the 3rd century BCE, the premier center of mathematical education and research was the Musaeum of Alexandria.^[36] It was there that Euclid (c. 300 BCE) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.^[35]

Considered the "father of geometry", Euclid is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.^[37] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,^[37] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.^[38]

The Euclidean algorithm is one of the oldest algorithms in common use.^[93] It appears in Euclid's Elements (c. 300 BCE), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. Centuries later, Euclid's algorithm was discovered independently both in India and in China,^[94] primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.