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On the Role of Equality and Inequality in the History of Mathematics

Published online by Cambridge University Press:  05 January 2009

R. C. H. Tanner
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Extract

The following essay is adapted from one with the same title read to the British Society for the History of Science on 20 October 1958—the anniversary, by a striking coincidence, of the birth of W. H. Young (1863–1942). To his memory I dedicated the talk, and now rededicate its publication, not only because I am his daughter and of all that means, but because he invented a method, the method of monotone sequences, which shows the powerfulness of inequalities as a mathematical tool supremely.

Type
Research Article
Information
The British Journal for the History of Science , Volume 1 , Issue 2 , December 1962 , pp. 159 - 169
Copyright
Copyright © British Society for the History of Science 1962

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References

1 Cp. Tanner, R. C. H., ‘W. H. Young et la méthode des suites monotones’, Actes Soc. helv. sc. nat., 1960.Google Scholar

2 Thomas Harriot, b. Oxford, 1560, d. 1621.

3 Robert Recorde, b. Tenby, 1510(?), d. 1558.

4 A language in which comparison is possible only through an extremely cumbersome form of circumlocution might be expected to provide correspondingly less ready access to basic mathematical notions. A remarkable example is cited by French, A., Math. Gazette, 1962, xlvi, 177CrossRefGoogle Scholar, of a seemingly supremely anthropomorphic African language, in which two compared entities have always to be assimilated to two human rivals. Even such extreme metaphorical phrase-forms of comparative, however, still embody the required hierarchical structure. The development of mathematical notions in the Bantu context would appear to offer a fruitful field for research.

5 Le Bourgeois gentilhomme, 1st ed., 1671Google Scholar, Acte II, Scene iv, Monsieur Jourdain: ‘Il y a plus de quarante ans que je dis de la prose sans que j'en susse rien.’

6 Hobson, E. W., b. Derby, 1856, d. Cambridge, 1933.Google Scholar

7 The word ‘medieval’ is inapplicable in its usual sense as we want to include the inception of the infinitesimal calculus, with Newton and Leibniz. The ‘Middle Ages’ thus have to be extended by the characteristically transitional periods called Renaissance (c. 1400–1540) and Early Baroque (1550–1650) in Hofmann, J. E.'s pocket Geschichte der Mathematik (Göschen Bd. 226, 1953)Google Scholar, precisely the periods that interest us most here.

8 Eudoxus of Cnidus (?408–?355), whose works are known only by report, and who had a period in Egypt after a short time with Plato, later returning to Athens with his own School.

9 The main evidence appears to be in two texts, one in Paris (Seleucid period?), the other in Berlin (Ancient?). Cp. Thureau-Dangin, F., Textes mathématiques babyloniens, Leiden, 1938Google Scholar, Introduction p. xvii, and pp. 78, 130.

10 Groningen, , 1954, pp. 184–240.Google Scholar

11 Teubner, , 1880.Google Scholar

12 Stifel, Michael (1487?–1567)Google Scholar, German theologian, follower of Luther, and notable mathematician.

13 Cp. Tanner, R. C. H., ‘La controverse sur l'origine commerciale des signes + et — en algèbre’, Actes Soc. helv. Sc. nat., 1959, pp. 191–2.Google Scholar

14 Cauchy, Augustin-Louis (1789–1857)Google Scholar, son of a poet and of considerable literary talent himself, one of France's greatest pure mathematicians.

15 Such cannot have been the intended meaning of Cunninghame-Greene, R., ‘Industrial mathematics’, Math. Gazette, 1962, xlvi, 107.Google Scholar

16 Passages expanding these points earlier in the spoken lecture are omitted here as already printed in the Mathematical Gazette, 1961, xlvGoogle Scholar, under the title ‘Mathematics begins with Inequality’. The original of this was read to the International Mathematical Congress at Edinburgh, 1958.

17 It appears from Cajori, , History of Mathematical Notation, Open Court, La Salle, Illinois, 1928, vol. i, p. 298, p. 111Google Scholar and figure 53, p. 129, that the two parallel dashes as symbol of equality appear in a manuscript in the Library of Bologna, conjectured to be lessons of Pompeo Bolognetti, who died in 1568. The discovery was by Prof. E. Bortolotti.