Abstract
Algebra is not solely the product of the evolution of arithmetic. It owes much to geometry. Some algebraic reasoning is present in Greek geometry. Geometric analysis, as well as the theory of proportions, played an important role in the development of algebra in the Renaissance. Until Viète’s algebraic revolution at the end of the 16th century, geometry was a means to prove algebraic rules, and, likewise, algebra was a means to solve some geometrical problems. In this chapter, I discuss some of the relations which, from Euclid to Descartes, bound algebra to geometry.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Notes
Quoted in Unguru (1975, p. 77). Quotation translated from Mahoney (1971, pp. 16–17). See also Mahoney (1972, p. 372) and Mahoney (1980, p. 142).
Descartes (1637/1954, pp. 2, 9).
Euclid (1925/1956, vol. 2, Book III, pp. 71–72).
Ptolemy (1984, pp. 48–56).
This has been challenged in Unguru (1975). A strong reaction to this article has been written by van der Waerden (1976).
Euclid (1925/1956, vol. 1, Book II, p. 372).
Euclid (19251956, vol. 1, Book II, p. 382).
van der Waerden (1961).
Euclid (1925/1956, vol. 1, Book II, pp. 402–403, 409–410).
Euclid (1925/1956, vol. 2, Book V, p. 114).
Idem.
Euclid (1925/1956, vol. 1, Book I, p. 155).
Euclid (1925/1956, vol. 2, pp. 112–113).
Boyer (1956, p. 28–29).
Mahoney (1968, p. 322).
It is an example made famous by Hankel at the end of 19th century. It has been reproduced by Heath in his introduction to The Elements, in Euclid (1925/1956, vol. 1, pp. 141–142). Euclid has only figure 4a. Figures 4b and 4c have been added here to facilitate the understanding of the reasoning. Euclid does not specify the name of the point H (Figure 4c).
Cardano (1968, p. 82). The problem has been translated into a modern form by Witmer.
Cardano (1968, pp. 96–99). This is in chapter XI.
It would have been very interesting to study Fibonacci. Luis Radford pointed out to me that in the Liber Abbaci (1202), the passage from geometry to algebraic reasoning is not as straightforward as in Chuquet.
The rule of “first terms” refers to rules of manipulation of the first (prime) power of the unknowns. See Chuquet (1880, part III) and Flegg, Hay, & Moss (1985, pp. 144–191). La Géométrie was first published in 1979, with an extensive introduction by Hervé l’Huillier in Chuquet (1979). A partial English translation of the whole Chuquet manuscript has been published by Flegg, Hay, & Moss (1985).
I use the translation by Flegg, Hay, & Moss (1985, p. 266).
Regiomontanus (1533/1967). The 1967 book contains a reproduction of the original publication and an English translation.
Ibid. p. 31.
Ibid p. 31
Ibid. p. 35.
This filiation has been revitalized by Mahoney (1973, p. 26; 1980, pp. 147–150). See also Van Egmond (1988, pp. 142–3).
On the difficult task of correctly defining the Poristics, see Viète (1983, pp. 12–13, note 6) and Ferrier (1980, pp. 134–158).
Descartes (1637/1954, p 6). This quotation is immediately followed by the second paragraph of the previous quotation from La Géométrie given at the beginning of my chapter.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Charbonneau, L. (1996). From Euclid to Descartes: Algebra and its Relation to Geometry. In: Bernarz, N., Kieran, C., Lee, L. (eds) Approaches to Algebra. Mathematics Education Library, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1732-3_2
Download citation
DOI: https://doi.org/10.1007/978-94-009-1732-3_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-4168-0
Online ISBN: 978-94-009-1732-3
eBook Packages: Springer Book Archive