29 Stobaeus, Eclogues, ii. 31. Anthologiurn recensvervnt Curtis Wachsmuth. Berolini, 1958.
30 Commentary on Aristotle's Soul (Greek), Joannes Philoponus, Berlin 1897, p 117.
31 Science Awakening, Van der Waerden, Kluwer Academic Publishers, p 155.
32 Plutarch. The Age of Alexander. Ian Scott-Kilvert. Penguin Classics. 1973. p 259.
33 See Renaissance and Renascences ..., Erwin Panofsky, Icon, 1972
34 Nicomachus of Gerasa, Palestine, circa 100 AD
35 Iamblichus, thought to have been born in Syria in mid-third century.
36 Proclus lived from 410 to 485 AD
37 Diogenes Laertius, third century AD
38 sydonic month: period of time between conjunctions of Sun and Moon.
39 Commentary on Elements, Book I. Proclus.
40 Greek Mathematics, Vol. 1, The Loeb Classical Library, edited by G.P. Goold, pp. 145-149.
41 Thaetatus was contemporary with Plato and is believed to have had a major influence on Plato's understanding of mathematics and hence Platonic thought.
42 Euclid, Heath, Vol. HI, p 261-62.
43 Dictionary of Scientific Biography.
44
45 The Pythagorean Life, Iamblichus, Clark, Liverpool Univ. p 6.
49 Science Awakening, p 95-96.
50
54 "In right-angled triangles, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."
55 In order to draw a right-angled triangle, draw a semicircle first, for: "In a circle the angle in the semicircle is right, etc..." Euclid, Preposition 31, Book III.
56 These rectangles are known as the dynamic rectangles. If the side of the square is set at one unit, their diagonals, including that of the square, are expressions of the sides of squares with the following areas: 2,3,4,5 etc....
57 In Plato's dialogue Theaetetus, the protagonist is given to say: "Here our Theodorus drew something about sides of squares and showed that those of three or five feet are not commensurable in length with those of one foot, and in this manner he took up one after the other up to the one of 17 feet; here something stopped him." Theodorus is dated about 430 BC, sixty years after Pythagoras. Each spoke gives the value of each successive square root. He probably stopped at 17, because it seems to be a limit. But the spiral is endless.
58 The Evolution of the Euclidean Elements. Wilbur Richard Knorr. D. Reidei Pub. Co. Boston. p34.
59 "Insofar as mathematics was developed in Greek times, the structure consisting mainly of Euclidean geometry, proved to be stable. One fault did show up, namely, that certain line segments, such as the diagonal of an isosceles triangle whose arms are I unit long -would have to have a length of root 2 units. Since the only numbers the Greeks recognized were the ordinary whole numbers, they would not accept such entities as root 2. The resolved the dilemma by ostracizing these irrational "numbers," and abandoning the idea of assigning numerical lengths to line segments, areas and volumes. Hence they built no additions to arithmetic and algebra beyond the whole numbers and what could be incorporated in to the structure of geometry. It is true that some Alexandrian Greeks, notably Archimedes, did operate with irrational numbers, but these were not incorporated into the logical structure of mathematics." Mathematics, The Loss of Certainty, Morris Kline, Oxford, 1980, p 308.
60 Also known as the Divine Proportion, The Section, the Golden Cut, Mean and Extreme Ratio.
61 "The Pythagorean community from which the poem derives, consisted paradoxically of moral rationalists with a predilection for Orphic doctines and an interest in Empedocles. If as 1 suggested, the Golden Verses originated between 350 and 300 BC, it provides us with valuable evidence about the Pythagorean tradition in a period about which we otherwise know very little." The Pythagorean Golden Verses. Johan C. Thorn. Leiden, New York & Koln. 1995. p 92.
63 In Defense of a Slip of the Tongue in Greetings, Lucian, Kilburn, p 177.
64 Pappus, Commentary... on Book X of Euclid's Elements, ed. G Junge and W. Thompson, pp. 63, 64. Also a Greek Scholium to BookX: of Euclid, Elementa, ed J. L. Heiberg, V, p417.
65 "However this may be, it is a fact that the Pythagorean Society, and consequently the dominant influence of the Pythagoreans on political life in Southern Italy, survived till the middle of the fifth century. The catastrophe mentioned in Polybius II39, the destruction by fire of the synedrion and the political disturbances throughout Southern Italy can be assigned with certainty to that period. Lysis, who is named as one of the two survivors, lived at this time..." Pythagoras and Early Pythagoreanism. C.J. Vogel. Van Gorcum & Co. p 24.
66 The Evolution of Euclidean Elements. Wilbur Richard Knorr, D. Reidel Publishing Co. p49.
68 The Physics and De generatione et corruptione.
69 The Greek Commentators' Treatment of Aristotle's Theory of the Continuous, David J. Furley. From Infinity and Continuity in Ancient and Medieval Thought, edited by Norman Kretzmann. Cornell University Press, p 32.
70 "Aristotle commits himself to denying that there can ever be an infinitely large spatial magnitude and to denying that an infinitely numerous set of parts can ever be actually produced by dividing a magnitude..." Ibid., p 34.
71 "The infinite or unlimited only exists potentially, not in actuality. The infinite is so in virtue of its endless changing into something else, like a day or the Olympic Games...The infinite is manifested in different forms in time, in Man, and in the division of magnitudes. For, in general, the infinite consists is something new being continually taken, that something being itself finite but always different. Therefore the infinite must not be regarded as a particular thing....but as being always in course of becoming or decay, and, though finite at any moment, always different from moment to moment...Thus in no sense does the infinite exist, but only in the sense just mentioned, that is, potentially and by way of diminution." Euclid, Heath, Vol. I, p 233.
72 "Thus, excepting the topological materials contained in Book I, III, VI and XI, virtually the whole of the Elements can be understood as the product of Euclid's effort to present the entire formal theory of irrationals within a self-sufficient compilation of treatises." The Evolution of the Euclidean Elements. W.R. Knorr. p 288.
73 Vol. I, II & III. Sir Thomas Heath, Dover Reprint, 1956. Euclid.
74 Phare: light-house, headlight. French.
75 Each book could be comprised often to twenty rolls.
76 This story is distrusted by Gibbon.
77 syllogism: a logical mathematical argument.