Neither ancient nor modern: wallis and barrow on the composition of Continua. Part one: Mathematical styles and the composition of Continua

Notes and Records of the Royal Society of London

Volumn 50, Issue 2, 1996, Pages 165-178

  Hill, Katherine ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 14144252276     PISSN: 00359149     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsnr.1996.0019     Document Type: Article

References (44)

1. Helena Pycior, 'Mathematics and philosophy: Wallis, Hobbes, Barrow and Berkeley', Journal of the History of Ideas 48, 266 (1987). The main focus of her article is the seventeenth-century roots of Berkeley's mathematical views. Thus she does not explore the details of how the dispute between ancients and moderns influenced the transition to an algebraic mode of thought.
2. Chikara Sasaki, 'The acceptance of the theory of proportions in the sixteenth and seventeenth centuries: Barrow's reaction to the analytic mathematics', Historia Scientiarum 29, 83 (1985).
3. For the Greeks a number was a multitude of units. Numbers were discrete. 'One' was not a number, but the origin which was used to generate numbers. 'One' was a unit, and, as such, it was not thought to be divisible. Aristotle, Physics, 12 2209 27.
4. Consequently, the Greeks could in no way develop a real number system. Many of the mathematical details of the transition from numbers being considered as collections of discrete units to numbers being treated as similar to continuous magnitude have been thoroughly explored in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (New York, 1992).
5. Wallis's and Barrow's mathematics are both discussed in D.T. Whiteside, 'Patterns of mathematical thought in the later seventeenth century', Arch. Hist. Exact Sci. 1, 178-388 (1960).
6. See also: Christoph Scriba, Studien zur Mathematik des John Wallis (1617-1703) (Wiesbaden, 1966);
7. and Michael Mahoney, 'Barrow's mathematics: between ancients and moderns' in Before Newton: The Life and Tunes of Isaac Barrow, ed. Mordechai Feingold, pp. 179-249 (Cambridge, 1990).
8. J.F. Scott, The Mathematical Work Of John Wallis D.D., F.R.S. (1616-1703) (London, 1938).
9. D.T. Whiteside, 'Patterns of mathematical thought in the later seventeenth century', Arch. Hist. Exact Sci. 1, 263.
10. John Wallis, Opera Mathematica, vol. 1, 365 (Oxford, 1695).
11. Ibid., p. 366.
12. John Wallis, A Treatise of Algebra, Both Historical and Practical, p. 68 (London, 1685).
13. Isaac Barrow, Geometrical Lectures, (tr. Edward Stone), p. 100 (London, 1734).
14. 2 or the square on DC. Also the justifications given for each step were not supplied by Barrow, he merely provided the equations.
15. Michael Mahoney, 'Barrow's mathematics: between ancients and modems', in Before Newton: The Life and Times of Isaac Barrow, (ed. Mordechai Feingold), p. 266 (Cambridge, 1990).
16. Quoted in op. cit. (11), p. 189.
17. Wallis, op. cit. (9).
18. Ibid.
19. This was also pointed out by Barrow in The Usefulness of Mathematical Learning Explained and Demonstrated: Being Mathematical Lectures Read in the Publick Schools at the University of Cambridge (tr. John Kirby, London), p. 47 (London, 1734).
20. Quoted in Mahoney, op. cit. (12), p. 190.
21. Barrow, op. cit. (16), p. 20.
22. Ibid., p. 27.
23. Wallis, op. cit. (9), p. 3.
24. Barrow, op. cit. (16), p. 28.
25. Ibid., p. 243.
26. F. Vieta, 'Analytic art', in J. Klein, Greek Mathematical Thought and the Origin of Algebra (tr. Eva Brann), p. 318 (New York, 1992).
27. Michael S. Mahoney, 'The beginning of algebraic thought in the seventeenth century', in Descartes' Philosophy, Mathematics and Physics, (ed. Stephen Gaukroger), p. 148 (New Jersey, 1980).
28. Wallis, op. cit. (9), p. 265.
29. Ibid., p. 92.
30. Ibid.
31. Barrow, op. cit. (16), p. 37.
32. Ibid., p. 41.
33. Ibid., p. 44.
34. Wallis, op. cit. (9), p. 92.
35. b 15-20.
36. Wallis, op. cit. (9), p. 285. The method of exhaustion was a rigorous geometrical technique that was perfected by Archimedes.
37. Ibid., p. 286.
38. Barrow, op. cit. (16), p. 152.
39. Ibid., p. 186.
40. Ibid., p. 198.
41. Barrow, op. cit. (10), p. 176.
42. Ibid., p. 256.
43. Quoted in Mahoney, op. cit. (12), p. 205.
44. Ibid.