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DESCARTES, RENÉ DU PERRON (b. La Haye,
Touraine, France, 31 March 1596; d. Stockholm,
Sweden, 11 February 1650), natural philosophy, scientific
method, mathematics, optics, mechanics, physiology.
7. Both the term and the concept it denotes are certainly anachronistic.
Descartes speaks of the indeterminate equation that
links x and y as the “relation [rapport] that all
the points of
a curve have to all those of a straight line”
(Géométrie, p. 341).
Strangely, Descartes makes no special mention of one of the
most novel aspects of his method, to wit, the establishment of
a correspondence between geometrical loci and indeterminate
algebraic equations in two unknowns. He does discuss the
correspondence further in bk. II, 334-335, but again in a way
that belies its novelty. The correspondence between determinate
equations and point constructions (i.e., section problems)
had been standard for some time.
8. For problems of lower degree, Descartes maintains the classification
of Pappus. Plane problems are those that can be
constructed with circle and straightedge, and solid problems
those that require the aid of the three conic sections. Where,
however, Pappus grouped all remaining curves into a class he
termed linear, Descartes divides these into distinct classes of
order. To do so, he employs in bk. I a construction device
that generates the conic sections from a referent triangle and
then a new family of higher order from the conic sections, and
so on.
9. Two aspects of the symbolism employed here require comment.
First, Descartes deals for the most part with specific examples
of polynomials, which he always writes in the form
xn + a1xn - 1 + ... +
an = 0; the symbolism P(x) was unknown
to him. Second, instead of the equal sign, = , he used
the symbol ?, most probably the inverted ligature of the first
two letters of the verb aequare (“to equal”).
10. One important by-product of this structural analysis of equations
is a new and more refined concept of number. See Jakob
Klein, Greek Mathematical Thought and the Origins of Algebra
(Cambridge, Mass., 1968).
11. Here again a totally anachronistic term is employed in the
interest of brevity.
12. Ironically, Descartes's method of determining the normal to
a curve (bk. II, 342 ff.) made implicit use of precisely the same
reasoning as Fermat's. This may have become clear to Descartes
toward the end of a bitter controversy between the two
men over their methods in the spring of 1638.
13. Cf. Vuillemin, pp. 35-55.
14. Ibid., pp. 11-25; Joseph E. Hofmann, Geschichte der
Mathematik,
II (Berlin, 1957), 13.
15. The anaclastic is a refracting surface that directs parallel rays
to a single focus; Descartes had generalized the problem to
include surfaces that refract rays emanating from a single point
and direct them to another point. Cf. Milhaud, pp. 117-118.
16. The full title of the work Descartes suppressed in 1636 as a
result of the condemnation of Galileo was Le monde, ou Traité
de la lumière. It contained the basic elements of Descartes's
cosmology, later published in the Principia philosophiae (1644).
For a detailed analysis of Descartes's work in optics, see A. I.
Sabra, Theories of Light From Descartes to Newton (London,
1967), chs. 1-4.
17. “One must note only that the power, whatever it may
be, that causes the motion of this ball to continue is
different from that which determines it to move more
toward one direction than toward another,” Dioptrique
(Leiden, 1637), p. 94.
18. Cf. Descartes to Mydorge (1 Mar. 1638), “determination
cannot be without some speed, although the same speed can
have different determinations, and the same determination can
be combined with various speeds” (quoted by Sabra, p. 120).
A result of this qualification is that Descartes in his proofs treats
speed operationally as a vector.
19. See the summary of this issue in Sabra, pp. 100 ff.
20. Cf. Carl B. Boyer, The Rainbow: From Myth to Mathematics
(New York, 1959).
21. For a survey of Descartes's work on mechanics, which includes
the passages pertinent to the subjects discussed below, see René
Dugas, La mécanique au XVIIe siècle
(Neuchâtel, 1954),
ch. 7.
22. Presented in full in the Principia philosophiae, pt. II, pars.
24-54.
23. Cf. Milhaud, pp. 34-36.
BIBLIOGRAPHY
I. ORIGINAL WORKS.
All of Descartes's scientific writings
can be found in their original French or Latin in the critical
edition of the Oeuvres de Descartes, Charles Adam and Paul
Tannery, eds., 12 vols. (Paris, 1897-1913). The
Géométrie,
originally written in French, was trans. into Latin and
published with appendices by Franz van Schooten (Leiden,
1649); this Latin version underwent a total of four eds. The
work also exists in an English trans. by Marcia Latham and
David Eugene Smith (Chicago, 1925; repr., New York,
1954), and in other languages. For references to eds. of
the philosophical treatises containing scientific material, see
the bibliography for sec. I.
II. SECONDARY LITERATURE.
In addition to the works
cited in the notes, see also J. F. Scott, The Scientific Work
of René Descartes (London, 1952); Carl B. Boyer, A History
of Analytic Geometry (New York, 1956); Alexandre Koyré,
Études galiléennes (Paris, 1939); E. J. Dijksterhuis,
The
Mechanization of the World Picture (Oxford, 1961). See also
the various histories of seventeenth-century science or
mathematics for additional discussions of Descartes's work.
MICHAEL S. MAHONEY
DESCARTES: Physiology.
Descartes's physiology grew and developed as an
integral part of his philosophy. Although grounded
at fundamental points in transmitted anatomical
knowledge and actually performed dissection procedures,
it sprang up largely independently of prior
physiological developments and depended instead on
the articulation of the Cartesian dualist ontology, was
entangled with the vagaries of metaphysical theory,
and deliberately put into practice Descartes's precepts
on scientific method. Chronologically, too, his physiology
grew with his philosophy. Important ideas on
animal function occur briefly in the Regulae (1628),
form a significant part of the argument in the Discours
de le méthode (1637), and lie behind certain parts of
the Principia philosophiae (1644) and all of the Passions
de l'âme (1649). Throughout his active philosophical
life, physiology formed one of Descartes's
most central and, sometimes, most plaguing concerns.
Descartes hinted at the most fundamental conceptions
of his physiology relatively early in his philosophical
development. Already in the twelfth regula,
he suggested (without, however, elaborating either
more rigorously or more fully) that all animal and
subrational human movements are controlled solely
by unconscious mechanisms. Just as the quill of a pen
moves in a physically necessary pattern determined
by the motion of the tip, so too do “all the motions
