7
Quoted in Ignatin and Smith, 1976, p. 80. The quotation is from Book I, Chapter X, of
8
Keynes, 1936, p. 159.
9
10
This entire paragraph is from Bolen, 1976.
11
Excellent background on all this may be found in David, 1962, pp. 2–21.
12
See David, 1962, p. 34.
13
Hayano, 1982.
14
Johnson, 1995.
15
See David, p. 2.
16
Sambursky, 1956, p. 36.
17
18
19
Rabinovitch, 1969.
20
Frankfort, 1956; quoted in Heilbroner, 1995, p. 23. See also David, 1962, pp. 21–26.
21
Peter Kinder has pointed out to me a great historical irony in all this. The Vikings and other Norsemen who laid waste to Roman civilization and destroyed the repositories of learning in the ninth century reappear in history as the Normans who brought back to the West the achievements of Arabic learning in the twelfth century.
22
See Eves, 1983, p. 136.
23
Most of the background and biographical material on Fibonacci comes from the
24
One of chose odd quirks that numbers can produce reveals that you can derive 0.618 if you take the square root of 5, which is 2.24, subtract 1, and then divide by 2; this result is the algebraic proof of Fibonacci’s sequence of numbers.
In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole.
The Arabic term survives even in Russian, where it appears as
25
In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole.
The Arabic term survives even in Russian, where it appears as
26
Two stimulating commentaries on the Fibonacci numbers are Garland, 1987, and Hoffer, 1975. The examples here are drawn from those two sources.
27
The background material presented here comes primarily from Hogben, 1968, Chapter I.
28
See Hogben, 1968, p. 35; also Eves, 1983, Chapter I.
29
See Hogben, 1968, p. 36 and pp. 246–250.
30
The background material on Diophantus is from Turnbull, 1951, p. 113.
31
32
33
See Hogben, 1968, pp. 244–246.
34
The Arabic term survives even in Russian, where it appears as
35
From Newman, 1988a, p. 433.
36
The background material on al-Khowârizmî is primarily from Muir, 1961, and Hogben, 1968.
37
Hogben, 1968, p. 243.
38
See Hogben, 1968, Chapter VI, for an extended and stimulating discussion of the development of algebra and the uses of zero.
39
The background material on Omar Khayyam is from Fitzgerald.
40
Hogben, 1968, p. 245.
41
The background material on Paccioli comes primarily from David, 1962, pp. 36–39, and Kemp, 1981, pp. 146–148.