Lecture 22 We finished discussing Chapter 3 from Dirk Struik's book (A coincise history of Mathematics), with a view on the upcoming presentations on the contributions by Archimedes and Heron. Please read the chapter!! Section 6 (pp. 47-48) contains a coincise historical description focusing on the Hellenistic period that followed the death of Alexander the Great: a sort of "east meets west" tale! Section 7 (pp. 48-50) talks about the Museum and the Library of Alexandria. Among the most famous scholars associated with Alexandria was Euclid. There is a short description about the XIII books of Euclid's Elements. (Though we know a lot more about each individual book, since we spent about 3 weeks analyzing them in detail!) Section 8 (pp. 50-51) is about the greatest mathematician of the Hellenistic (or even ancient) time: Archimedes. We will learn a great deal about him from the in-class presentation on his work by Austin, Ben and Erin (Lectures 23-24). Section 9 (pp. 51-54) deal with Apollonius, who is one of the three greatest mathematicians of the Hellenistic period along with Euclid and Archimedes. He remained in the annals of history for his magnificent work on Conics (8 books). Section 10 (pp. 54-55) deals with astronomy and its traditions (both the Greek and Babylonian traditions) and its many, many contributors: Eudoxus, Aristarchus of Samos (280 BC) the so called Copernicus of antiquity, and Hipparcus of Nicaea (worked between 141 to 127 BC). Section 11 (pp. 55-57) deals with raise of the Roman Republic (later, Roman Empire). It is succintly explained how it economic structure essentially relied on an agricultural economy based on slavery. Thus, there was no great mathematical development during this period. Original work still continued in Alexandria, but compilation and commentarization became the prominent form of science. Section 12 (pp. 57-58) talks about Ptolomy's Almagest (ca. 150 AD), an astronomical opus which had an influence for many, many centuries. In this period there is also Menelaus, who continued the geometrical tradition in the style of Euclid. We also find Heron, who is the subject of another in-class presentation. Section 13 (pp. 58-59) deals with Diophantus (ca. 250 AD). Only six of his books survive: they show a skillful treatment of equations with a strong influence from the Babylonian tradition. Section 14 (pp. 59-60) talks about Pappus, Proclus, Hypathia (fouth and fifth centuries) and the conquer of Alexandria by the Arabs (630 AD).