(AZ/BZ) · (BX/CX) · (CY/AY) = 1.
The most popular writer on astronomy among the Greeks was Ptolemy (Claudius Ptolemaeus, 87-165 A.D.), who lived at Alexandria. He wrote a work entitled "Megale Syntaxis" (The Great Collection), which his followers designated as Megistos (greatest), on which account the Arab translators gave it the name "Almagest" (al meaning "the"). He advanced the science of trigonometry, but did not contribute to geometry.
At the close of the third century Pappus of Alexandria (295 A.D.) wrote on geometry, and one of his theorems, a generalized form of the Pythagorean proposition, is mentioned in [Chapter XVI] of this work. Only two other Greek writers on geometry need be mentioned. Theon of Alexandria (370 A.D.), the father of the Hypatia who is the heroine of Charles Kingsley's well-known novel, wrote a commentary on Euclid to which we are indebted for some historical information. Proclus (410-485 A.D.) also wrote a commentary on Euclid, and much of our information concerning the first Book of Euclid is due to him.
The East did little for geometry, although contributing considerably to algebra. The first great Hindu writer was Aryabhatta, who was born in 476 A.D. He gave the very close approximation for π, expressed in modern notation as 3.1416. He also gave rules for finding the volume of the pyramid and sphere, but they were incorrect, showing that the Greek mathematics had not yet reached the Ganges. Another Hindu writer, Brahmagupta (born in 598 A.D.), wrote an encyclopedia of mathematics. He gave a rule for finding Pythagorean numbers, expressed in modern symbols as follows:
He also generalized Heron's formula by asserting that the area of an inscribed quadrilateral of sides a, b, c, d, and semiperimeter s, is
The Arabs, about the time of the "Arabian Nights Tales" (800 A.D.), did much for mathematics, translating the Greek authors into their language and also bringing learning from India. Indeed, it is to them that modern Europe owed its first knowledge of Euclid. They contributed nothing of importance to elementary geometry, however.
The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi (820 A.D.). He lived at Bagdad and Damascus. Although chiefly interested in astronomy, he wrote the first book bearing the name "algebra" ("Al-jabr wa'l-muqābalah," Restoration and Equation), composed an arithmetic using the Hindu numerals,[20] and paid much attention to geometry and trigonometry.
Euclid was translated from the Arabic into Latin in the twelfth century, Greek manuscripts not being then at hand, or being neglected because of ignorance of the language. The leading translators were Athelhard of Bath (1120), an English monk; Gherard of Cremona (1160), an Italian monk; and Johannes Campanus (1250), chaplain to Pope Urban IV.