The Sixteenth and Seventeenth Centuries. The Founding of the Infinitesimal Calculus

Bell, John L.

doi:10.1007/978-3-030-18707-1_2

John L. Bell¹⁹

Part of the book series: The Western Ontario Series in Philosophy of Science ((WONS,volume 82))

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Abstract

The early modern period saw the spread of knowledge in Europe of ancient geometry, particularly that of Archimedes, and a loosening of the Aristotelian grip on thinking. In regard to the problem of the continuum, the focus shifted away from metaphysics to technique, from the problem of “what indivisibles were, or whether they composed magnitudes” to “the new marvels one could accomplish with them” through the emerging calculus and mathematical analysis. Indeed, tracing the development of the continuum concept during this period is tantamount to charting the rise of the calculus.

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Notes

1.
Murdoch (1957), p. 325.
2.
A similar, but independent approach was taken by Luca Valerio (1552–1618).
3.
Boyer (1959), pp. 99–100.
4.
Baron (1987), p. 97
5.
Ibid., pp. 108–116; Boyer (1969), pp. 106–110.
6.
Baron (1987), p. 110.
7.
Boyer (1959), p. 107.
8.
Kline (1972) p. 299.
9.
Galilei (1954), p. 55.
10.
Here Galileo is giving expression to the conviction, which he shares with Cantor , that no potential infinity is unaccompanied by an actual one.
11.
Galilei (1954), pp. 33–34
12.
Ibid., p. 48
13.
Ibid., p. 47.
14.
Hermann Weyl makes a similar suggestion in connection with Galileo’s “bending” procedure:

If a curve consists of infinitely many straight “line elements”, then a tangent can simply be conceived as indicating the direction of the individual line segment; it joins two “consecutive” points on the curve. (Weyl 1949, p. 44.)
15.
This conception was to prove fruitful in the later development of the calculus and to achieve fully rigorous formulation in the synthetic differential geometry of the later twentieth century. See Chap. 10 below.
16.
Galilei (1954), p. 37
17.
Ibid., p. 30.
18.
Boyer (1959), p. 116
19.
Galilei (1954), pp. 39–40.
20.
Boyer (1969), p. 117.
21.
Ibid., p. 118.
22.
Ibid., p. 121.
23.
Ibid., p. 122.
24.
Ibid., p. 166.
25.
Ibid., p. 165
26.
Descartes (1927), p. 139.
27.
Ibid., pp. 203–4.
28.
Ibid., p. 209.
29.
Arnauld and Nicole (1996), p. 230.
30.
Ibid., p. 232.
31.
Ibid.
32.
It is interesting to note that this argument fails for geometries in which lines are not indefinitely extensible, for instance in elliptic geometries.
33.
Quoted in Leibniz (2001), p. 361.
34.
In this connection see the discussion at the end of Appendix A,
35.
Ibid., pp. 361–2.
36.
Boyer (1959), p. 137.
37.
Walker (1932), p. 34.
38.
Boyer (1959), pp. 125–6.
39.
Ibid., p. 139.
40.
Baron (1987), p. 205n.
41.
Ibid., pp. 206–7. Baron reports (p. 213) that in Wallis’s A Treatise of Algebra of 1685 he lists the stages in the development of infinitesimal methods as follows: 1. Method of Exhaustion (Archimedes ); 2. Method of Indivisibles (Cavalieri ); 3. Arithmetick of Infinites (Wallis ); 4. Method of Infinite Series (Newton ).
42.
Boyer (1959), p. 155.
43.
Here Fermat seems to recognize that the function f(x) = x(a – x) is locally one-one.
44.
Boyer (1959), p. 156.
45.
Aristotle (1980), VII, 4. See also Heath (1949), pp. 140–142.
46.
Boyer (1959), pp. 162–163.
47.
Child (1916), p. viii. The quotation continues:

Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow’s work, obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow’s book he purchased in 1673.
48.
Article “Infinitesimal Calculus”, Encyclopedia Britannica, 11th edition. But, while Barrow recognized the fact, he failed to put it to systematic use.
49.
Kline (1972), p. 346.
50.
In smooth infinitesimal analysis, the area of a characteristic triangle always reduces to 0. See Chap. 10 below.
51.
Kline (1972), pp. 346–7.
52.
Quoted in Jesseph (1993), p. 63n.
53.
Quoted ibid, p. 63.
54.
Child (1916), pp. 35–7.
55.
Ibid., pp. 38–9.
56.
De analysi, written 1666, published 1711; Methodus fluxionum, written 1671, published 1736; Quadratura, written c. 1676, published 1704.
57.
Quoted in Jesseph (1993), p. 144.
58.
Baron (1987), p. 268.
59.
Newton (1962), p. 29.
60.
Ibid., p. 32.
61.
Ibid., pp. 38–9.
62.
Boyer (1959), p. 200.
63.
Newton (1962), p. 399,
64.
Newton (1952), p. 400.
65.
Quoted from Pyle (1997), p.415.
66.
Ibid.
67.
Actually, as pointed out in Wilson (2015), the metaphor of the labyrinth was a familiar one in seventeenth century writing. In connection with the continuum it had already been employed, for example, by Galileo in Two New Sciences.
68.
Russell (1958), p. 245.
69.
Ibid., p. 246.
70.
Quoted in Weyl (1949) p.41.
71.
Russell (1958), p. 246.
72.
Ibid., p. 245.
73.
Ibid.
74.
Ibid., pp. 245–6.
75.
Leibniz (1951), pp. 107–108.
76.
Quoted in Leibniz (2001), pp. 37, 39.
77.
Russell (1958), p. 242.
78.
Ibid., p. 248.
79.
Leibniz (1951), p. 99.
80.
Russell (1958), p. 247.
81.
Ibid., p. 104.
82.
Strictly speaking, for Leibniz , as Russell points out (p. 106), matter or extended mass is nothing more than “a well-founded phenomenon”, not a substantial unity but a plurality engendered by indifferent monadic aggregation. Monads are not parts of phenomena but rather constitute their foundation. The monads themselves are the sole substantial realities: with the Eleatics Leibniz avers (Russell , p. 242) “What is not truly one being is also not truly a being.”
83.
Russell (1958), p. 105.
84.
Ibid., p. 105.
85.
Ibid.,, p. 247.
86.
Reply to Foucher, 1693, Leibniz (1951), p. 99.
87.
Quoted in Boyer (1959), p. 217.
88.
Letter to Varignon, 1702, Leibniz (1951), pp. 184–5.
89.
Letter to Foucher, 1692, Leibniz (1951), p. 71.
90.
Quoted in Russell (1958), p. 235.
91.
Leibniz’s abilities as a mathematician have been memorably characterized by E. T. Bell:

The union in one mind of the highest ability in the two broad, antithetical domains of mathematical thought, the analytical and the combinatorial, or the continuous and the discrete, was without precedent before Leibniz and without sequent after him. He is the one man in the history of mathematics to have both qualities of thought in a superlative degree. (Bell, E.T. 1965, p. 128).
92.
Indeed, Jakob and Johann Bernoulli , who made many contributions to the Leibnizian calculus and who introduced the term “integral ”, actually defined the operation of integration as the inverse of differentiation.
93.
Quoted in Mancosu (1996), p. 156.
94.
Ibid., p. 156.
95.
Quoted from Bos (1974), p. 14.
96.
Quoted in Boyer (1959), p. 219.
97.
Ibid., p. 239.
98.
Ibid.
99.
Quoted in Mancosu (1996), p. 151.
100.
Quoted ibid., p. 152.
101.
Ibid., p. 159 et seq.
102.
Ibid., p. 159.
103.
Ibid., pp. 153–4.
104.
Ibid., p. 161.
105.
Bell (1998), p. 9. In fact the “nilsquare” property of infinitesimals and L’Hôpital’s Postulate II (for algebraic curves) are equivalent. As remarked above, Nieuwentijdt saw that the first assertion implies the second.
106.
Mancosu (1996), pp. 165 et seq.
107.
Ibid., p. 167.
108.
Boyer (1959), p. 242.
109.
Mancosu (1996), p. 167.
110.
But the other properties have resurfaced in the theories of infinitesimals which have emerged over the past several decades. Appropriately defining the relation, ≈, property 1 holds of the differentials in nonstandard analysis , while properties 1, 2 and 3 hold of the differentials in smooth infinitesimal analysis. See Chaps. 8 and 10 below.
111.
Bayle (1965), pp. 350–94.
112.
Bayle (1965), p. 363.
113.
Op. cit.

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John L. Bell

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Bell, J.L. (2019). The Sixteenth and Seventeenth Centuries. The Founding of the Infinitesimal Calculus. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_2

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