Abstract
An investigation of John Wallis’ mathematical practice and his views about the nature of number and its relationship to magnitude will demonstrate that there were several options available to justify the broadening of the number concept and the breakdown in the separation between number and abstract magnitude. One needs to accept the unity of mathematics under geometry in order to accept Barrow’s foundation for number. But for Wallis, arithmetic and the new algebraic techniques had priority over geometry. Moreover, his practice depended a great deal upon numerical patterns and methods. Although Wallis did not state his opinions of continua concepts in as much detail as Barrow, his point of view will provide us with the ideas of a professional mathematician whose practice was mainly algebraic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Referencias
J.F. Scott, “The Reverend John Wallis, F.R.S. (1616–1703),” Notes and Records of the Royal Society of London 15 (1960) p. 58.
John Wallis, “An Essay of Dr. John Wallis, exhibiting his Hypothesis about the Flux and Reflux of the Sea,” Philosophical Transactions 16 (1666):262–281, is a typical example.
Wallis, Treatise of Algebra, p.399.
Carl Boyer, The History of the Calculus and its Conceptual Development, p.179 and 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 (1985), p. 94.
Whiteside, “Patterns,” pp. 187–188.
Christoph Scriba, ed. “The Autobiography of John Wallis, F.R.S.,” Notes and Records. Royal Society of London 25 (1970): 27.
Grammar schools in England in this period did not provide even a rudimentary mathematical education. Such skills were taught outside by self-styled ‘professors’ of mathematics who were usually almanac-makers, astrologers, surveyors, gunners or gaugers. See Taylor, page 9 – 10.
Scriba, “The Autobiography of John Wallis”, p. 27.
J.F. Scott, The Mathematical Work of John Wallis D.D., F.R.S. (1616–1703) (London: Taylor and Francis, 1938), p. 4.
Hugh Kearney, Scholars and Gentlemen: Universities and Society in Pre-Industrial Britain, 1500–1700 (London: Faber and Faber, 1970), pp. 28 and 63.
Scriba, “The Autobiography of John Wallis, F.R.S.,” p.27.
Scriba, “The Autobiography of John Wallis,” p.27.
Mordechai Feingold, The Mathematicians ‘ Apprenticeship: Science, Universities and Society in England, 1560–1640 (Cambridge: Cambridge University Press, 1984), p. 87.
In A Humble Motion to the Parliament of England Concerning the Advancement of Learning: and a Reformation of the Universities as quoted in Richard Foster Jones’ Ancients and Moderns: A Study of the Background of the Battle of the Books (St. Louis: Washington University Press, 1936), p. 102.
Scriba, “The Autobiography of John Wallis,” p.31.
David Eugine Smith, “John Wallis as a Cryptographer”, Bulletin of the American Mathematical Society, 24 (1917): pp. 83–96
David Kahn, The Codebrakers (New York: Macmillian, 1967), pp. 166–169; See, especially, Peter Pesic, “Secrets, Symbols, and Systems: Parallels between Cryptanalysis and Algebra, 1580–1700” Isis 88 (1997), pp.674–692.
Scriba, “The Autobiography of John Wallis,” p.39.
Christoph Scriba, Studien zur Mathematik des John Wallis (1616 – 1703) (Wiesbaden: Franz Steiner, 1966), p.3. Also see Wallis’ Treatise of Algebra, page 121, where he explains that he afterwards found that his rule was “co-incident with Cardan’s Rules.”
Scriba, Studien, pp. 3–5. 20 Scott, pp.72–81.
Christoph Scriba has explored Wallis’ work with number theory in detail. See his Studien zur Mathematik des John Wallis (1616–1703), especially chapter three “Zaheltheoretische Probleme aus unveröffentlichten Manuskripten.”
Scott, “John Wallis as a Historian of Mathematics,” pp.336–337.
Michael S. Mahoney, “The Beginning of Algebraic Thought in the Seventeenth Century,” in Descartes Philosophy, Mathematics and Physics, ed. Stephen Gaukroger (New Jersey: Barnes and Noble, 1980), p. 148.
Mahoney, “The Beginning of Algebraic Thought,” p. 148.
Compare folilo 57 of Euclides The Elements of Geometrie, trans. by H Billingsley, (London, 1570) with Heath’s The Thirteen Books of The Elements (New York: Dover, 1956), Volume I, p. 382.
Billingsley, Euclides The Elements of Geometrie, fo. 57.
The example is, “Take any number as 20, divide into 2 equal partes 10 and 10 and then into two unequal parts 13 and 7..,” the example, although numerical, is given in words — not equations.
Niccoló Tartaglia’s, Euclides Solo introdttore della scientie mathematice: diligentemente reassettato, et alla ridotto per il degno prefessure di tal scientie, (Vinegina, 1543) fo. xxxii.
Wallis’ discussion of numbers will be examined in the section of the chapter describing his continua concepts.
At this point in his career, Wallis still respected Descartes. He was especially enthusiastic about Descartes’ exponential notation. See John Wallis, Mathesis Universalis, p. 71 in Opera Mathematica, Vol. 1. His treatment of Book II, however, utilised Oughtred’s symbolism.
Indeed, in his A Defense of the Treatise of the Angle of Contact, p. 99, he says, “we find Euclide’s Geometry (good part of it) employed about Numbers.”
Wallis, Treatise of Algebra, p.90.
Euclid, Elements, p. 382.
Wallis, Mathesis Universalis, p. 124.
Wallis, Mathesis Universalis, p. 124. If a straight line (Z) be cut into equal (S,S) and unequal (A, E, or S + V, S-V) segments, the rectangle contained by the unequal segments (AE) of the whole together with the square of the straight line (Vq) between the points of the sections is equal to the square on the half (Sq).
Wallis, Mathesis Universalis, p. 124.
Wallis, Mathesis Universalis, p. 124..
Wallis, Treatise, p.114.
Wallis, Treatise, p.113.
Wallis, Mathesis Universalis, p.69.
See Chapter IV pp. 174–178.
The date on the title page was 1655, but the work was issued as part two of operum mathematicorum pars altera (Oxford, 1656).
The modern equation for the ellipse, for example, refers to the vertex as origin, i.e. y2 = kx (a ‒ x). Putting ka = p, one obtains y2 = px -(p/a)x2, p is the latus rectum.
Whiteside, “Patterns,” p.295. Wallis, however, did utilise Descartes’ notation, together with some of the symbols of Oughtred, but his explanations are still mainly verbal. Yet Scott, Mathematical Works, p. 24, quotes Hobbes as saying that this work “is so covered with the scab of symbols, that I had not patience to examine whether it were well or ill demonstrated.”
Whiteside, “Patterns,” p. 236.
Whiteside, “Patterns,” p. 236.
Wallis, Opera Mathematica, p. 365.
Wallis, Opera Mathematica, p.366.
Wallis, Opera Mathematica, p.373.
Wallis, Opera Mathematica, p.382.
Wallis, Opera Mathematica, p.389.
Wallis, Opera Mathematica, p.390.
53Wallis, Opera Mathematica, p.390, as translated in Scott, page 38.
Wallis, Opera Mathematica, p.392.
Scott, Mathematical Work, pp. 42–46. Scott provides a detailed examination of Wallis’ difficulties.
Wallis, Treatise, p.68.
Scott, Mathematical Work, pp. 66–67.
Wallis, Opera Mathematica, p. 20.
Klein, Greek Mathematical Thought, p.211.
Wallis, Opera Mathematica, pp. 24–27.
Klein, Greek Mathematical Thought, pp. 214–215 for a detailed discussion of this point.
Klein, Greek Mathematical Thought, p. 220.
Whiteside, “Patterns,” pp. 187–188.
Mahoney, “Barrow’s Mathematics,” p. 189.
Wallis, Treatise, introduction.
Barrow, p.47. It is interesting that Barrow brings up Wallis’ ideas on this issue.
Mahoney, “The Beginning of Algebraic Thought,” p.190.
Wallis, Treatise, p.265.
Wallis, Treatise, p.317.
Wallis, Treatise, p.92.
Wallis, Treatise, p.92.
C. Jones, p.47.
Wallis, Treatise, p.285.
Wallis, Treatise, p.286.
Wallis, Treatise, p.305.
Klein, Greek Mathematics, p.220.
Klein, Greek Mathematics, p.223.
Wallis, Mathesis universalis, p.57.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Neal, K. (2002). John Wallis. In: From Discrete to Continuous. Australasian Studies in History and Philosophy of Science, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0077-1_7
Download citation
DOI: https://doi.org/10.1007/978-94-017-0077-1_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5993-2
Online ISBN: 978-94-017-0077-1
eBook Packages: Springer Book Archive