the golden ratio....excerpt

THE GOLDEN RATIO: The Story of Phi, The World's Most Astonishing Number, by Mario Livio, (Broadway Books, New York: 2002), p.87-88 “The last great Greek geometer who contributed theorems related to the Golden Ratio was Pappus of Alexandria. In his “Collection (Synagogue;” ca. A.D. 340), Pappus gives a new method for the construction of the dodecahedron and the icosahedron as well as comparisons of the volumes of Platonic solids, all of which involve the Golden Ratio. Pappus' commentary on Euclid's theory of irrational numbers traces beautifully the historical development of irrationals and is extant in Arabic translations. However, his heroic efforts to arrest the general decay of mathematics and geometry in particular were unsuccessful, and after his death, with all the overall withering of intellectual curiosity in the West, interest in the Golden Ratio entered a long period of hibernation. The great Alexandrian library was destroyed by a series of attacks, first by the Romans and then by Christians and Muslims. Even Plato's Academy came to an end in 529 A.D., when the Byzantine emperor Justinian ordered the closing of all the Greek schools. During the depressing Dark Ages that followed, the French historian and bishop Gregory of Tours (538-594) lamented that the 'study of letters is dead in our midst.' In fact, the whole enterprise of science was essentially transferred in its entirety to India and the Arab world. A significant event of this period was the introduction of the so-called Hindu-Arabic numerals and of decimal notation. The most important Hindu mathematician of the sixth century was Aryabhara (476-ca. 550). In his best-known book, entitled Aryabhatiya, we find the phrase 'from place to place eac is ten times the preceding,' which indicates an application of a place value system. An Indian plate from 595 already contains writing (of a date) in Hindu numerals using decimal place notation, implying that such numerals has been in use for a long time. The first sign (albeit with no real influence) of Hindu numeral moving west can be found in the writings of the Nestorian bishop Severus Sebokht, who lived in Keneshra on the Euphrates River. He wrote in 662: 'I will omit all the discussion of the science of the Indians... and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs.' “With the ascendancy of Islam, the Muslim world became an important center for mathematical study. Had it not been for the intellectual surge in Islam during the eighth century, most of the ancient mathematics would have been lost. In particular, Caliph al-Mamun (786-833) established in Baghdad the Beit al-hikma (House of wisdom), which operated in a similar fashion to the famous Alexandrian university or 'Museum.' Indeed, the Abbasid empire subsumed any Alexandrian learning that had survived. According to tradition, after having a dream in which Aristotle appeared, the caliph decided to have all the ancient Greek works translated.”