specifically to Pythagoras and only discussed the work of the Pythagoreans as a group. Almost half of the material in Euclid's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BC) and Theodorus (fl. 450 BC). The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BC), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics. Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BC), Theaetetus (c. 417–369 BC), and Eudoxus (c. 408–355 BC). Greek mathematics also drew the attention of philosophers during the Classical period. Plato (c. 428–348 BC), the founder of the Platonic Academy, mentions mathematics in several of his dialogues. While not considered a mathematician, Plato seems to have been influenced by Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids. He also believed that geometrical proportions bound the cosmos together rather than physical or mechanical forces. Aristotle (c. 384–322 BC), the founder of the Peripatetic school, often used mathematics to illustrate many of his theories, as when he used geometry in his theory of the rainbow and the theory of proportions in his analysis of motion. Much of the knowledge about ancient Greek mathematics in this period is thanks to records referenced by Aristotle in his own works. Hellenistic and Roman periods The Hellenistic era began in the late 4th century BC, following Alexander the Great's conquest of the Eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to a Hellenistic mathematics. Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BC), Archimedes (c. 287–212 BC), Apollonius (c. 240–190 BC), Hipparchus (c. 190–120 BC), and Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small circle. There is also evidence of combining mathematical knowledge with technical or practical applications, as found for instance in the work of