Analytic geometry
A joining of geometry and algebra: a curve is written as an equation and an equation is read as a curve. Viete's symbolic algebra is applied to plane and solid figures.
The Frenchman Rene Descartes printed La Geometrie in 1637 as an appendix to the Discourse on Method. Pierre Fermat arrived at an almost identical system independently in the manuscript Introduction to Plane and Solid Loci (by 1636, printed in 1679); historians regard the priority question as tangled.
Newton and Leibniz studied this geometry in van Schooten's annotated Latin edition (1649, 1659-1661), and the calculus was built in its language.
Why this date
exact for publication: Descartes's Geometrie appeared in 1637 as an appendix to the Discourse on Method; Fermat's manuscript Ad locos planos et solidos isagoge was composed by 1636 (DSB; Britannica) or in 1629 (Science and Its Times), circulated in copies and was printed only in 1679
What it grew from
-
Algebra as a distinct method
prerequisite for · Confidence: high
Analytic geometry is the application of algebra to curves: Descartes and Fermat started from Viete's symbolic algebra and theory of equations, itself descended from al-Khwarizmi's algebra; without an algebraic language a curve cannot be written as an equation.
Dispute: Historians do not disagree here, but the immediate source for Descartes and Fermat is Viete's symbolic algebra (In artem analyticem isagoge, 1591), not al-Khwarizmi's rhetorical algebra of 820.
Link sources
- Knorr W. R. et al. «Mathematics: Analytic geometry», Encyclopaedia Britannica (2024) Verified 20 Sep 2026
After Viète's death the analytic art was applied to the study of curves by his countrymen Fermat and Descartes.
- Mahoney M. S. «Fermat, Pierre de», Dictionary of Scientific Biography (via Encyclopedia.com) (1971) Verified 20 Sep 2026
From Viète he took the new symbolic algebra and theory of equations that served as his basic research tools
- Domski M. «Descartes' Mathematics», Stanford Encyclopedia of Philosophy, §3.1 (2021) Verified 20 Sep 2026
his program for geometry begins where Viète's left off
- O'Connor J. J., Robertson E. F. «René Descartes», MacTutor History of Mathematics (2014) Verified 20 Sep 2026
La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Knorr W. R. et al. «Mathematics: Analytic geometry», Encyclopaedia Britannica (2024) Verified 20 Sep 2026
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The Elements: definition, axiom, proof
influenced · Confidence: medium
Analytic geometry grew on the Greek problem of construction that the Elements codified: Descartes divides problems into the same classes as the ancient geometers, refers to Euclid's constructions, and sets algebraic operations in correspondence with geometric constructions, lifting the old requirement that the product of two line segments must be an area.
Dispute: Historians differ on whether this continues ancient geometry or breaks with it. Descartes himself insisted that the ancients had wrongly restricted geometry to compass and straightedge, and demanded that any curve traced by a continuous motion be admitted; on that reading the Elements were for him a problem he outgrew rather than a foundation. Bos, in his study of Descartes, calls this outright a transformation of the very concept of construction. On the opposite reading the link stands: one can only outgrow what is already there, and Descartes' analysis is conducted in terms set by the Greek tradition.
Link sources
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
geometric problems, previously unsolved since ancient Greek times, to be tackled using algebraic techniques
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
established equivalences between algebraic operations and geometric constructions
- Domski M. «Descartes' Mathematics», Stanford Encyclopedia of Philosophy Verified 20 Sep 2026
The problem of bisecting a given angle is counted among planar problems, because, as detailed by Euclid in Elements I.9, to construct the line segment that divides a given angle into two equal parts, we construct (by compass) three circles of equal radius, and then (by straightedge) join the vertex of the angle with the point at which the circles intersect
- Domski M. Verified 20 Sep 2026
Moreover, in a 13 November 1629 letter written to Mersenne, Descartes refers to the second (1589) edition of Clavius's annotated version of Euclid's Elements
- Mahoney M. S. «Descartes: Mathematics and Physics», Dictionary of Scientific Biography (via Encyclopedia.com) Verified 20 Sep 2026
the Greek mathematicians had established the correspondence between addition and the geometrical operation of laying line lengths end to end
- Descartes R. «Discours de la méthode», 2 (1637) Verified 20 Sep 2026
Ces longues chaînes de raisons, toutes simples et faciles, dont les géomètres ont coutume de se servir pour parvenir à leurs plus difficiles démonstrations, m'avoient donné occasion de m'imaginer que toutes les choses qui peuvent tomber sous la connoissance des hommes s'entresuivent en même façon
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
What grew out of it
-
Differential and integral calculus
prerequisite for · Confidence: high
The calculus of Newton and Leibniz is an algorithm on the equations of curves; Newton taught himself mathematics from van Schooten's Latin edition of Descartes's Geometrie, Leibniz from a reading list by Huygens that included Descartes, and historians call analytic geometry the essential framework of the calculus.
Dispute: Domski (SEP, §3.3) stresses that Descartes excluded infinitesimals and 'mechanical' curves from his program: "Descartes eliminated from his geometrical program the very elements of mathematical and geometrical reasoning that made it possible for Newton and Leibniz to develop the calculus". What Descartes supplied, therefore, was the algebraic language of curves, not the idea of infinitesimals, which came from Cavalieri, Fermat, Wallis and Barrow.
Link sources
- Smith G. «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026, quotation assembled from fragments
By early 1664 he had also begun teaching himself mathematics, taking notes on works by Oughtred, Viète, Wallis, and Descartes ... the latter via van Schooten's Latin translation, with commentary, of the Géométrie.
- «The Development of Analytic Geometry», Science and Its Times (Gale), via Encyclopedia.com (2001) Verified 20 Sep 2026
provided the essential framework for the calculus of Isaac Newton and Gottfried Leibniz
- Cambridge University Library, exhibition «Footprints of the Lion», section «Calculation» Verified 20 Sep 2026
The young Newton owed his greatest intellectual debt to the French mathematician and natural philosopher, René Descartes.
- O'Connor J. J., Robertson E. F. «Gottfried Wilhelm von Leibniz», MacTutor History of Mathematics (1998) Verified 20 Sep 2026
Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint-Vincent, Descartes and Sluze.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.
- Smith G. «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026, quotation assembled from fragments
Node sources
- Domski M. «Descartes' Mathematics», Stanford Encyclopedia of Philosophy (2021) Verified 20 Sep 2026
La Géométrie (1637), a short tract included with the anonymously published Discourse on Method.
- Mahoney M. S. «Fermat, Pierre de», Dictionary of Scientific Biography (via Encyclopedia.com) (1971) Verified 20 Sep 2026, quotation assembled from fragments
a system of analytic geometry almost identical with that developed by Descartes in the Géométre of 1637 ... the two systems stemmed from entirely independent research and the question of priority is both complex and unenlightening
- Knorr W. R. et al. «Mathematics: Analytic geometry», Encyclopaedia Britannica (2024) Verified 20 Sep 2026
Fermat adopted Viète's notation in his paper 'Ad Locos Planos et Solidos Isagoge' (1636; 'Introduction to Plane and Solid Loci').
- Boyer C. B. «History of Analytic Geometry», Dover reprint (cited in Wikipedia «La Géométrie») (2004), 108-109 Verified 20 Sep 2026
Van Schooten published a Latin version of La Géométrie in 1649 and this was followed by three other editions
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.