Descartes' Mathematics
First published Mon Nov 28, 2011
To speak of René Descartes' contributions to the
history of mathematics is to speak of his
La
Géométrie
(1637), a short tract included with the
anonymously published
Discourse on Method
. In
La
Géométrie
, Descartes details a groundbreaking
program for geometrical problem-solving—what he refers to as a
“geometrical calculus” (
calcul
géométrique
)—that rests on a distinctive
approach to the relationship between algebra and geometry.
Specifically, Descartes offers innovative algebraic techniques for
analyzing geometrical problems, a novel way of understanding the
connection between a curve's construction and its algebraic
equation, and an algebraic classification of curves that is based on
the degree of the equations used to represent these curves.
Examining the main questions and issues that shaped Descartes' early mathematical
researches sheds light on how Descartes attained the results presented
in
La Géométrie
and also helps reveal the significance of this work for the debates surrounding early modern mathematics.
The importance of
La Géométrie
for the
history of mathematics is hardly a matter of dispute. The
problem-solving techniques and mathematical results that Descartes
presents in that short tract were both novel and incredibly
influential. However, we can also locate in
La
Géométrie
a philosophical significance: The blending
of algebra and geometry and the peculiar approach to the
“geometrical” status of curves which characterize
Descartes' mathematical program stand as notable contributions to
the on-going philosophical debates that surrounded early modern
mathematical practice. By drawing on the context in which Descartes'
mathematical researches took place, the historical and
philosophical significance of Books One and Two of
La Géométrie
will
be highlighted in what follows.
[
1
]
When Descartes' mathematical researches commenced in the early
seventeenth century, mathematicians were wrestling with questions
concerning the appropriate methods for geometrical proof and, in
particular, the criteria for identifying curves that met the exact and
rigorous standards of geometry and that could thus be used in geometrical
problem-solving. These issues were given an added sense of
urgency for practicing mathematicians when, in 1588, Commandino's Latin translation of Pappus's
Collection
(early fourth century CE) was published. In the
Collection
Pappus appeals to the
ancient practice of geometry as he offers normative claims about how
geometrical problems ought to be solved. Early modern readers
gave special attention to Pappus's proposals concerning (1) how a
mathematician should construct the curves used in geometrical proof,
and (2) how a geometer should apply the methods of analysis and
synthesis in geometrical problem-solving. The construction of
curves will be treated in 1.1 and analysis and synthesis in section 1.2
below.
Pappus's claims regarding the proper methods for constructing
geometrical curves are couched in terms of the ancient classification
of geometrical problems, which he famously offers in Book III of the
Collection
:
The ancients stated that there are three kinds of geometrical
problems, and that some are called plane, others solid, and others
line-like; and those that can be solved by straight lines and the
circumference of a circle are rightly called plane because the lines by
means of which these problems are solved have their origin in the
plane. But such problems that must be solved by assuming one or
more conic sections in the construction, are called solid because for
their construction it is necessary to use the surfaces of solid
figures, namely cones. There remains a third kind that is called
line-like. For in their construction other lines than the ones
just mentioned are assumed, having an inconstant and changeable origin,
such as spirals, and the curves that the Greeks call
tetragonizousas
[“square-making”], and
which we call “quandrantes,” and conchoids, and cissoids,
which have many amazing properties (Pappus 1588, III, §7;
translation from Bos 2001, 38).
We notice in the above remarks that Pappus bases his classification
of geometrical problems on the construction of the curves necessary for
the solution of a problem: If a problem is solved by a curve
constructible by straightedge and compass, it is planar; if a problem
is solved by a curve constructible by conic section, it is solid; and
if a problem is solved by a curve that requires a more complicated
construction—that has an “inconstant and changeable
origin”—, it is line-like. Though a seemingly
straightforward directive of how to classify geometrical problems,
there remained an ambiguity in Pappus's text about whether the
so-called solid and line-like problems—problems that required
the construction of conics and more complicated curves, such as the
spiral—were in fact solvable by genuinely geometrical
methods. That is, there was an ambiguity, and thus, an open
question for early modern mathematicians, about whether problems that
could not be solved by straightedge and compass construction met the
rigorous standards of geometry. (For the special status of
constructions by straightedge and compass in Greek mathematics, see
Heath (1921) and Knorr (1986). For helpful overviews of the historical
development of Greek mathematics, see classics such as Merzbach and
Boyer (2011) and volume 1 of Kline (1972).)
A few examples will help clarify what is at stake here. 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 (Euclid 1956, Volume I, 264–265). Notice here that, to generate the solution, curves are used to construct a point that gives the solution to the problem: by constructing the circles, we identify a point that allows us to bisect the curve. (When dealing with locus problems, such as the Pappus problem, the curves that are constructed are themselves the solution to the problem. See
section 3
below.)
Now, the problem of trisecting an angle was considered a line-like problem, because its solution required the construction of
curves, such as the spiral, which were not constructible by
straightedge and compass. Perhaps most famous among line-like
problems is that of squaring the circle; for those who deemed this
problem solvable, the solution required the construction of a curve
such as the quadratrix, a curve that was proposed by the ancients in
order to solve this very problem (which is how the curve received its
name). Certainly, the generation of such curves could be
described; Archimedes famously describes the generation of the spiral
in Definition 1 of his
Spirals
and Pappus describes the
generation of the quadratrix in Book IV of the
Collection
. However, these descriptions were considered
“more complicated” precisely because they go beyond the
intersection of curves that are generated by straightedge and compass
construction. For instance, according to Archimedes, the spiral
is generated by uniformly moving a line segment around a given point
while tracing the path of a point that itself moves uniformly along the
line segment. And, according to Pappus, the quadratrix is
generated by the uniform motions of two line segments, where one
segment moves around the center of a given circle and the other moves
through a quadrant of the circle. (Cf. Bos 2001, 40–42 for the
details of both these constructions.) In a similar vein, the
construction of conics was considered more complicated: One of the accepted techniques for constructing a conic required cutting a cone in a specified way, which again, went
beyond the consideration of intersecting curves that were constructible
by straightedge and compass.
In the
Collection
, Pappus does not offer a firm verdict on
whether the conics and “more complicated” curves meet the
rigorous standards of geometrical construction and hence, on whether
they are admissible in the domain of geometry. In the case of the
conics, he relies on Apollonius's commentary and reports the
usefulness of these curves for the synthesis (or proofs) of some
problems (Pappus, 116). However, to claim a curve useful is quite
different from claiming it can be constructed by properly geometrical
methods (as we'll see more clearly below). Moreover, in the
case of the quadratrix, Pappus sets out the description of the curve in
Book IV of the
Collection
, and then immediately proceeds to
identify the common objections to the curve's description, e.g.,
that there is a
petitio principii
in the very definition of
the curve, without commenting on whether these objections can be
overcome. Thus, although it was known by the ancients that conics
and other complicated curves could be used to solve outstanding
problems, it was not clear to early modern mathematicians whether the
ancients considered these solutions genuinely geometrical. That
is, it was not clear from Pappus's
Collection
whether
these curves were admissible in geometrical problem-solving and
therefore, whether solid problems (such as identifying the mean
proportionals between given line segments) or line-like problems (such
as trisecting an angle and squaring the circle) had genuine geometrical
solutions.
Consequently, after the publication of Commandino's translation of the
Collection,
early modern mathematicians gave added attention
to question of whether and why these curves should be used in
geometrical problem-solving. The spiral and quadratrix were
prominent in such discussions, because, as noted above, they could be
used to address some of the more famous outstanding geometrical
problems, namely, angle trisection and squaring the circle.
[
2
]
For instance, in his second and expanded (1589) edition of Euclid's
Elements
(which was first published in 1574) as well as in his
Geometria practica
(1604), Christoph Clavius discusses the
status of the quadratrix. Accepting the objections to the
description of the quadratrix detailed by Pappus in the
Collection
, Clavius supplies what he deems a “truly
geometrical” construction of the curve that would legitimize its
use in geometrical problem-solving, and in solving the problem of
squaring the circle in particular. His construction is a
pointwise one: We begin with a quadrant of a circle (as in
Pappus's description) but rather than relying on the intersection
of uniformly moving segments to describe the curve, Clavius proceeds by
first identifying the points of intersection between segments that
bisect the quadrant and segments that bisect the arc of the quadrant.
That is, we identify the several intersecting points of segments which
are constructible by straightedge and compass, and then, to generate the
quadratrix, we connect the (arbitrarily many) intersecting points,
which are evenly spaced along the sought after curve. Therefore,
to construct the quadratrix according to Clavius's method, we
still go beyond basic straightedge and compass constructions
(connecting the points in this case cannot be done by straightedge, as
in the case of bisection), but one need not consider the simultaneous
motions of lines as Pappus's construction requires. (See
Bos 2001, 161–162 for Clavius's construction of the quadratrix
and compare with Pappus's construction on Bos 2001, 40–42.
For Descartes' assessment of Clavius's pointwise
construction see
section 3.3
below.)
According to Clavius's commentary of 1589, this pointwise
construction of the quadratrix was an improvement over that offered by
Pappus, because it was more accurate: Since the pointwise
construction allowed one to identify arbitrarily many points along the
curve, one could trace the quadratrix with greater precision than if
one had to consider the intersection of two moving lines. To
support his case, Clavius relates his pointwise construction of the
quadratrix with the pointwise construction of conics proposed by the
“great geometer” Apollonius and claims that “unless
someone wants to reject as useless and ungeometrical the whole doctrine
of conic sections” proposed by Apollonius, “one is forced
to accept our present description of the [quadratrix] as entirely
geometrical” (cited in Bos 2001, 163). However, in his
later
Geometria practica
(1604), Clavius tempers his
assessment of both the quadratrix and the conics. He maintains
that these more complicated curves could be constructed by pointwise
methods that offered greater precision, but the curves thus generated
were no longer presented as absolutely geometrical. Instead, they
were presented as “more accurate,” “easier,”
and geometrical “in a certain way” (Bos 2001, 164–5).
In his
Supplement of geometry
(1593), François
Viète also addresses the outstanding problems of geometry which
were solvable by curves that could not constructed by straightedge and
compass. He claims that at least some such problems could be solved by
properly geometrically means by adopting as his postulate that the
“neusis problem” could be solved. That is, he assumed
that given two lines, a point
O
, and a segment
a
, it
was possible to draw a straight line through
O
intersecting
the two lines in points
A
and
B
such that
AB
=
a
(Bos 2001, 167–168). In the
Supplement
,
Viète shows that once we accept as a fundamental geometrical
postulate that the neusis problem is solvable, then we can, by
legitimately geometrical means, solve the problems of trisecting a
given angle and of constructing the two mean proportional between two
given line segments. Specifically and importantly, we generate
these solutions without having to rely on the construction of conics or
higher-order curves, such as the spiral or quadratrix (Bos 2001,
168).
The neusis postulate was a powerful tool in Viète's
problem-solving arsenal: By assuming that the neusis problem
could be solved, he expanded the domain of acceptable geometrical
constructions beyond straightedge and compass. However, questions
remained about the acceptability of this assumption as a postulate,
since Viète does not detail the construction of the neusis
problem but simply claims that the “neusis postulate”
should not be difficult for his readers to accept. In making this
assumption, he was taking a significant departure from ancient
geometers, for whom the neusis problem could only be solved by
curves that were not constructible by straightedge and
compass. For instance, Pappus rendered the construction of the
neusis a solid problem and solved it by means of conics in Book IV of
the
Collection
, and Nicomedes rendered the construction of the
neusis a line-like problem and devised the cissoid for its solution.
(See Bos 2001, 53–54 for Pappus's solution and 30–33 for
Nicomedes' solution. See also Pappus 1986, 112–114 for the
classification of the neusis as a sold problem.)
Nonetheless, according to Viète, if a problem could not be
solved by neusis, then questions of legitimacy remained. For
instance, neither the spiral nor the quadratrix—curves used to
square the circle by Archimedes and Pappus, respectively—could
be constructed in the same obvious and “not difficult” way
as the neusis. Viète appears to grant that the pointwise
construction of the quadratrix, such as that presented by Clavius, was
in fact more precise than other constructions of the curve, but,
Viète claims, this greater precision does not legitimize its
status as genuinely geometrical. Indeed, such precise
descriptions relied on instruments and, thus, the mechanical arts and,
as such, were not geometrical. Moreover, Viète claimed, in
general, that curves not constructed by the intersection of curves,
such as the Archimedean spiral, were “not described in the way of
true knowledge” (Bos 2001, 177). Therefore, just as the
quadratrix, these curves were not legitimately geometrical, which left
the problem of squaring the circle an open problem for
Viète.
Viète's program of geometrical problem-solving had an
added significance: By adopting as his postulate that the neusis
problem could be solved, Viète was able to link geometrical
construction with his algebraic analysis of geometrical problems and
show that cubic equations had a genuinely geometrical solution (i.e.,
that the roots of cubic equations could be constructed by consideration
of intersecting geometrical curves). Viète's program
nicely illustrates the merging of algebra with geometrical
problem-solving in early modern mathematics, and moreover, nicely
illustrates an influential way of interpreting Pappus's claims in
the
Collection
regarding how a mathematician should apply the
methods of analysis and synthesis in geometrical problem-solving.
As noted above, Pappus's remarks concerning the two-fold
method of analysis (
resolutio
) and synthesis
(
compositio
) in the
Collection
received a great deal
of attention from early modern readers. And as with his remarks
concerning the construction of geometrical curves, there were
ambiguities in his discussion, which motivated varying interpretations
of the method and its application to geometrical problems. Here
is a portion of what Pappus claims of analysis and synthesis in Book
VII of the
Collection
:
Now analysis is the path from what one is seeking, as if it were
established, by way of its consequences, to something that is
established by synthesis. That is to say, in analysis we assume
what is sought as if it has been achieved, and look for the thing from
which it follows, and again what comes before that, until by regressing
in this way we come upon some one of the things that are already known,
or that occupy the rank of a first principle. We call this kind
of method “analysis,” as if to say
anapalin lysis
(reduction backward). In synthesis, by reversal, we assume what
was obtained last in the analysis to have been achieved already, and,
setting now in natural order, as precedents, what before were
following, and fitting them to each other, we attain the end of the
construction of what was sought. This is what we call
“synthesis” (Pappus, 82–83).
Some of the directives Pappus offers here seem
straightforward. The mathematician begins by assuming what
is sought after as if it has been achieved until, through analysis, she
reaches something that is already known. Then, the mathematician
reverses the steps, and through synthesis, sets out “in natural
order” the deduction leading from what is known to what is
sought after. However, there are ambiguities in Pappus's
discussion. Perhaps most importantly, it is not clear how
reversing the steps of analysis could offer a proof, or synthesis, of a
stated problem, since the deductions of analysis rely on conditionals
(if
x
, then
y
) whereas a reversal would require biconditionals (
x
iff
y
) to achieve synthesis (see Guicciardini 2009, 31–38
for further interpretative problems surrounding Pappus's remarks; for
more on analysis and synthesis in the Renaissance see the classic
Hintikka and Remes 1974, the essays in Otte and Panza 1997, and
Panza 2007). Ambiguities notwithstanding, for Viète
and other early modern mathematicians there was one feature of the
discussion that was incredibly important: Pappus makes clear that the
ancients had a method of analysis at their disposal, and many early
modern mathematicians attempted to align this method from antiquity
with the algebraic methods of geometrical analysis that they were
using.
Prior to the end of the sixteenth century, mathematicians had
already used algebra in the analysis of geometrical problems, but the
program Viète details marks a significant step forward. On
the one hand, in his
Isagoge
[
Introduction to the analytic
art
] of 1591, which was presented as part of a larger project to
restore ancient analysis (entitled
Book of the restored
mathematical analysis or the new algebra
), Viète introduces
a notation that allowed him to treat magnitudes in a general way.
The literal symbols he uses (consonants and vowels depending on whether
the variable in the equation was unknown or indeterminate,
respectively) represent magnitudes generally and do not specify whether
they are arithmetical magnitudes (numbers) or geometrical magnitudes
(such as line segments or angles). He can thus represent arithmetic
operations as applied to magnitudes in general. For instance,
A
+
B
represents the addition of two magnitudes and
does not specify whether
A
and
B
are numbers (in
which case the addition represents a process of counting) or
geometrical objects (in which case the addition represents the
combination of two line segments) (see Viète 1591, 11–27; for
the significance of Viète's “new algebra” for
early modern mathematics see Bos 2001, Chp. 8; Mahoney 1973, Chp. 2;
and Pycior 1997, Chp. 1).
On the other hand, the algebraic, symbolic analysis of geometrical
problems that Viète proposes was offered as the first step in a
three-step process that could render a geometrical solution. The
three stages were: (1)
zetetics
, which involved the algebraic
analysis (or elaboration) of a problem; (2)
poristics
, which
clarified the relations between magnitudes by appeal to the theory of
proportions (see Giusti 1992 on the importance of proportion theory
for Viète's mathematics); and (3)
exegetics
,
which offered the genuine geometrical solution (or proof) of the
problem. To better understand the connection between the stages
of
zetetics
and
exegetics
, which roughly correspond
to the ancient stages of analysis and synthesis, consider the problem
of identifying two mean proportionals. Geometrically, the problem
is as follows:
Given line segments
a
and
b
, find
x
and
y
such that
a
:
x
::
x
:
y
::
y
:
b
, or put differently, such that
In the
zetetic
(analytic) stage of Viète's
analysis, we follow Pappus's directive to treat “what is
sought as if it has been achieved” precisely by naming the
unknowns by variables. Then, by assuming the equivalence between
proportions (as Viète does), we can solve for the variables
x
and
y
and establish that
x
and
y
have the following relationship to
a
and
b
:
- x
2
=
ay
and
- y
2
=
xb
.
Solving (1) for
y
, we
have
y
=
x
2
/
a
, and by substitution into (2), we get
y
2
= (
x
2
/
a
)
2
=
x
4
/
a
2
=
xb
, which yields:
- x
3
=
a
2
b
.
Solving (2) for
x
, we have
x
=
y
2
/
b
, and by substitution into (1), we get
x
2
=
(
y
2
/
b
)
2
=
y
4
/
b
2
=
ay
, which
yields:
- y
3
=
ab
2
.
Algebraically, then, the
problem of finding two mean proportional can be elaborated as follows:
Given (magnitudes)
a
and
b
, the problem is to find
(magnitudes)
x
and
y
such that
x
3
=
a
2
b
and
y
3
=
ab
2
.
In this zetetic stage of analysis, the geometrical problem is
transformed into the algebraic problem of solving a standard-form cubic
equation (i.e., a cubic equation that does not include a quadratic
term). However, for Viète, the genuine solution to the
problem must be supplied in the stage of exegetics, which offers the
geometrical construction and thus the synthesis, or
proof.
[
3
]
And it is here that the neusis
postulate supplies the guarantee that such a solution can be found: By
assuming the neusis problem solved, we can construct the curve that
satisfies the two cubic equations above (i.e., we can construct the
roots of the equations) and thereby construct the sought after mean
proportionals. In other words, there was an assumed
equivalence in Viète's program between solving an
algebraic problem that required identifying the roots of specified
cubic equations and solving a geometrical problem that required the
construction of a curve. We also see this in his treatment of
trisecting an angle: To solve the angle-trisection problem is to solve
two standard-form cubic equations, which Viète reveals in his
algebraic elaboration of the geometrical problem (cf. Bos 2001,
173–176). In fact, assuming the neusis postulate, we can solve
any standard-form cubic equation, and since it was already known at the
time that all fourth-degree equations are reducible to standard-form
cubic equations, what Viète supplied with his marriage of
algebra and geometry in his 1594
Supplement
was a program that
solved all line-like problems that could be elaborated in terms of
cubic and quartic equations.
As powerful as Viète's program was, questions remained
for practicing mathematicians. Should we, as Viète urged,
accept the neusis postulate as “not difficult” and thus as
a foundational construction principle for geometry? And should we
follow Viète in claiming that other curves that had significant
problem-solving power in geometry—such as the spiral and
quadratrix—were not legitimately geometrical because they could
not be constructed by neusis? Moreover, there were questions
about the connection Viète forged between algebra and
geometry. For Descartes in particular, there were questions of
whether there was a deeper, more fundamental connection that could be
forged between the solutions of algebraic problems that were expressed
in terms of equations and the solutions of geometrical problems that
required the construction curves. However, these questions did
not come into full relief for Descartes until the early 1630s, after
more than a decade of studying problems in both geometry and
algebra.
Based on the autobiographical narrative included in Part One of the
Discourse on Method
(1637), where Descartes describes what he
learned when he was “at one of the most famous schools in
Europe” (AT VI, 5; CSM I, 113), it is generally agreed that
Descartes' initial study of mathematics commenced when he was a
student at La Fleche. He reports in the
Discourse
that,
when we he was younger, his mathematical studies included some
geometrical analysis and algebra (AT VI, 17; CSM I, 119), and he also
mentions that he “delighted in mathematics, because of the
certainty and self-evidence of its reasonings” (AT VI, 7; CSM I,
114). However, no specific texts or mathematical problems are
mentioned in the 1637 autobiographical sketch. Thus, we rely on remarks
made in correspondence for the more specific details of
Descartes' study of mathematics at La Fleche, and these remarks
strongly suggest that Clavius was a key figure in Descartes'
earliest (perhaps even initial) study of mathematics. For instance, in
a letter of March 1646 written by John Pell to Charles Cavendish, we
have good reason to believe that ca. 1616, while a student at La
Fleche, Descartes read Clavius's
Algebra
(1608).
Reporting on his meeting with Descartes in Amsterdam earlier that same
year, Pell writes in particular that “[Descartes] says he had no
other instructor for Algebra than ye reading of Clavy Algebra above 30
years ago” (cited in Sasaki 2003, 47; cf. AT IV, 729–730 and
Sasaki 2003, 45–47 for other relevant portions of that letter).
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
, in which, as noted above,
Clavius presents his pointwise construction of the quadratrix and uses
the curve to solve the problem of squaring the circle (AT I, 70–71; the
portion of the letter that references Clavius is translated in Sasaki
(2003), 47). And following Sasaki (2003), it is reasonable to
conclude that Descartes was at least aware of Clavius' textbook
Geometria practica
(1604), which was included as part of the
mathematics curriculum of La Fleche. (See Sasaki 2003, Chapter Two on
Clavius' influence on and inclusion in the mathematics curriculum
of Jesuit schools in the early 1600s.)
Although our evidence of the mathematics that Descartes studied at
La Fleche is sketchy, we are quite certain that Descartes'
entrance into the debates of early modern mathematics began in earnest
when he met Isaac Beeckman in Breda, Holland in 1618. Among other
things, Beeckman and Descartes explored the fruitfulness of applying
mathematics to natural philosophy and discussed issues pertaining to
physico-mathematics. It is in this period that Descartes composed his
Compendium musicae
for Beeckman, a text in which he addresses
the application of mathematics to music and also famously discusses the
law of free fall (compare Koyré 1939, 99–128 and Schuster 1977,
72–93 on Descartes' treatment of free fall in this early
text).
Beyond having a common interest in applied mathematics, Beeckman and
Descartes also discussed problems of pure mathematics, both in geometry
and in algebra, and Descartes' interest in such problems extended
to 1628–1629, when he returned to Holland to meet Beeckman after his
travels through Germany, France, and Italy. Our understanding of
what Descartes accomplished in pure mathematics during this eleven year
period relies on the following sources:
-
Five letters written to Beeckman in 1619, which Beeckman
transcribed in his
Journal
. Beeckman's
Journal
was recovered in 1905 and published in 4 volumes by
DeWaard some 35 years later, hereafter Beeckman (1604–1634). The
excerpts of these letters that are relevant to Descartes'
mathematics are included in AT X. (For more details on how these
letters became available to us, see Sasaki 2003, 95–96.)
-
The
Cogitationes privatae
(
Private
Reflections
), which dates from ca. 1619–1620 and which Leibniz
copied in 1676. This text is included in AT X. (For more details
on how this text became available to us, see Bos 2001, 237, Note 17 and
Sasaki 2003, 109.)
-
The
Progymnasmata de solidorum elementis
, a geometry
text which dates from around 1623 and which Leibniz partially copied in
1676. It has been translated into English by Pasquale Joseph
Federico (1982) and into French by Pierre Costabel (1987).
-
A specimen of general algebra, which Descartes gave to Beeckman
after he returned to Holland in 1628. It was transcribed by
Beeckman in his
Journal
under the title
Algebra Des Cartes
specimen quoddam
and can be found in Volume III of Beeckman
(1604–1634).
-
Some texts on algebra that were given to Beeckman in early
1629. These were transcribed by Beeckman in his
Journal
in February 1629 and can be found in Volume IV of Beeckman
(1604–1634).
-
Several letters written to Mersenne in the 1630s in which
Descartes refers to some of the mathematical researches he completed
during the 1618–1629 period.
A look at some of the problems and proposals found in these
mathematical works will help situate Descartes in his early modern
mathematical context and will also help to highlight the results from
this period that have an important connection to what is found in the opening books of the
1637
La Géométrie
. To make these
connections clear, the brief narrative below emphasizes
Descartes' proposals concerning (1) the criteria for geometrical
curves and legitimately geometrical constructions, and (2) the
relationship between algebra and geometry during the 1618–1629
period.
The most famous letter written to Beeckman in 1619 dates from 26
March of that year. In this letter Descartes announces his plan
to expound an “entirely new science [
scientia penitus
nova
], by which all problems that can be posed, concerning any
kind of quantity, continuous or discrete, can be generally
solved” (AT X, 156). As he elaborates on how this new
science will proceed, Descartes clarifies that his solutions to the
problems of discrete and continuous quantities—that is, of
arithmetic and geometry, respectively—will vary depending on
the nature of the problem at hand. As he puts it,
[In this new science] each problem will be solved according to its
own nature as for example, in arithmetic some questions are resolved by
rational numbers, others only by surd [irrational] numbers, and others
finally can be imagined but not solved. So also I hope to show
for continuous quantities that some problems can be solved by straight
lines and circles alone; others only by other curved lines, which,
however, result from a single motion and can therefore be drawn with
new types of compasses, which are no less exact and geometrical, I
think, than the common ones used to draw circles; and finally others
that can be solved by curved lines generated by diverse motions not
subordinated to one another, which curves are certainly only imaginary
such as the rather well-known quadratrix. I cannot imagine
anything that could not be solved by such lines at least, though I hope
to show which questions can be solved in this or that way and not any
other, so that almost nothing will remain to be found in
geometry. It is, of course, an infinite task, not for one man
only. Incredibly ambitious; but I have seen some light through the dark
chaos of the science, by the help of which I think all the thickest
darkness can be dispelled (AT X, 156–158; CSMK 2–3; translation from
Sasaki 2003, 102).
We notice in Descartes' remarks concerning geometry in
particular that the “entirely new science” he proposes will
provide an exhaustive classification for problem-solving, where each of
his three classes is determined by the curves needed for
solution. This suggests an important overlap between
Descartes' three classes of geometrical problems and
Pappus's three classes, which, recall, were separated based on
the types of curves required for solution: Planar problems are solvable
by straightedge and compass, solid problems by conics, and line-like
problems by more complicated curves that have an “inconstant and
changeable origin.” However, there is also a significant
difference between their classifications insofar as Descartes strongly
suggests that those problems that require “imaginary”
curves for their solution do not have a legitimately geometrical
solution. Namely, just as some problems of arithmetic “can
be imagined but not be solved,” so too in geometry, there is a
class of problems that require curves that are “certainly only
imaginary,” i.e., curves generated by “diverse
motions,” and thus that are not geometrical in a proper
sense. In this respect, Descartes is moving from Pappus's
descriptive classification to a normative one that separates
geometrical curves from non-geometrical curves, and thereby
distinguishes problems with a geometrical solution from those that do
not have a legitimate geometrical solution. Just as importantly,
we see in Descartes' letter his attempt to expand the scope of
legitimate geometrical constructions beyond straightedge and compass by
appealing to the
motions
needed to construct a curve.
Specifically, as we see in the passage above, Descartes relies on
the “single motions” of his “new types of compasses,
which [he says] are no less exact and geometrical…than the
common ones used to draw circles” in order to mark out a new
class of problems that have legitimate geometrical solutions.
In his 26 March 1619 letter to Beeckman, Descartes does not
elaborate on the “new types of compasses” to which he
refers; he simply reports to Beeckman in the early portions of the
letter that he has, in a short time, “discovered four conspicuous
and entirely new demonstrations with the help of my compasses”
(AT X, 154). Fortunately, more details about these compasses and
Descartes' demonstrations are included in
Cogitationes
privatae
, or
Private Reflections
(ca. 1619–1620), a text
in which Descartes applies three different “new compasses”
(often referred to by commentators as “proportional
compasses”) to the problems of (1) dividing a given angle into
any number of equal parts, (2) constructing the roots of three types of
cubic equations, and (3) describing a conic section. In the first
two cases, as Descartes treats the angular section and mean
proportional problems, the compasses on which he relies are used to
generate a curve that will solve the problem at hand.
Figure 1
Figure 2
For instance, to solve the angular section problem, Descartes begins
by presenting an instrument that includes four rulers (OA, OB, OC, OD),
which are hinged at point O (
figure 1
). We then take four rods
(HJ, FJ, GI, EI), which are of equal length
a
, and attach them
to the arms of the instrument such that they are a distance
a
from O and are pair-wise hinged at points J and I. Leaving OA
stationary, we now move OD so as to vary the measure of angle DOA, and
following the path of point J, we generate the curve KLM (
figure 2
).
As Descartes has it, we can construct the curve KLM on
any
given angle by appeal to the instrument described above, because the angle we are trisecting plays no role in the construction of KLM. And once the curve KLM is constructed, the given angle can be trisected by means of some basic constructions with straight lines and circles. In this respect, the
curve KLM is, for Descartes, the means for solving the angle trisection
problem, and moreover, his treatment suggests that the construction can be generalized further so that, by means of his “new compass,” an angle can also be divided into 4, 5,
or more equal parts. (I borrow my treatment of this construction
from Domski 2009, 121, which is itself indebted to the presentation in
Bos 2001, 237–239.)
Figure 3: Mesolabe
A similar approach is taken by Descartes when he treats the problem
of constructing mean proportionals, where in this case, he appeals to
his famous mesolabe compass, an instrument that is used in Book Three
of the
La Géométrie
to solve the same problem.
As in 1637, this compass is used to construct curves (the dotted lines
in
figure 3
) that allow us to identify the mean proportionals between
any number of given line segments. And as Viète before
him, in the
Private Reflections
Descartes uses this
construction of mean proportionals to identify the roots of
standard-form cubic equations (see Bos 2001, 240–45).
Notice that these constructions illustrate the sort of “single
motion” constructions to which Descartes refers in his 26 March
1619 letter to Beeckman: His new compasses generate curves by the
single motion of a designated arm of the compass, and thus, the curves
generated in this manner meet the standard of geometrical
intelligibility—the standard by which to distinguish
geometrical from imaginary curves—that is alluded to in the
brief outline of the “entirely new science” that Descartes
envisions. That such motions are completed by instruments does
not threaten the constructed curve's geometrical status. (As
we saw above, Viète had leveled this charge against the
instrumental, pointwise constructions provided by Clavius.) And
moreover, we already notice in the mathematical research of 1619
Descartes' focus on the intelligibility of motions as a standard
for identifying legitimately geometrical curves. This theme will
reemerge in Book Two of
La Géométrie
.
In addition, we find in Descartes' early work an interest in
the relationship between algebra and geometry that will be crucial to
the program of geometrical analysis presented in Book One of
La
Géométrie
, where at this early stage of his
research, Descartes, like his contemporaries, is exploring the
application of geometry to algebraic problems. For instance, as
pointed out above, Descartes uses the construction of mean
proportionals to solve algebraic equations in the
Private
Reflections
, and in the same text he also shows an interest in the
geometrical representation of numbers and of arithmetical
operations. The same interest appears again in the later
Progymnasmata de solidorum elementis excertpa ex manuscript
Cartesii
(Preliminary exercises on the elements of solids
extracted from a manuscript of Descartes, ca. 1623), a text in which
Descartes offers a geometrical representation of numbers and of four of the five
basic arithmetical operations (the four operations he treats are addition, subtraction,
multiplication, and division).
Though there is some dispute among commentators about
Descartes' level of expertise in algebra during this early
1619–1623 period (compare Bos 2001, 245 with Sasaki 2003, 126), texts
from 1628–1629 show Descartes making great advances in algebra in a
relatively small amount of time. Two textual sources are of
particular interest: (1) The specimen of algebra given to and
transcribed by Beeckman in 1628 upon Descartes' return to
Holland, and (2) a text on the construction of roots for cubic and
quartic equations given to Beeckman in early
1629.
[
2
]
In the
Specimen
, Descartes
presents a rather basic problem-solving program (or schematism) for
algebra that relies on two-dimensional figures (lines and
surfaces). The texts given to Beeckman several months
later, which Descartes composed while in Holland, show a great advance
over what's found in the
Specimen
, since in these texts he
appeals to conic sections (or solids) in his problem-solving
regime. For instance, Descartes constructs two mean proportionals
by the intersection of circle and parabola (a method he had discovered
around 1625 according to Bos 2001, 255). More impressively, in a
different text from this same period, Descartes offers a method for
constructing all solid problems, i.e., for solving all third- and
fourth-degree equations.
While some of the results from this period are connected
with the problem-solving program presented in the 1637
La
Géométrie
, Rabouin (2010) points out that it is still not
clear whether Descartes discovered his methods for solution using the
techniques that are applied in 1637 (Rabouin 2010, 456).
As such, Rabouin urges us to resist the
somewhat standard reading of Descartes' early mathematical works
according to which there is a linear and teleological progression from
the 1619 pronouncement of an “entirely new science” to the
groundbreaking program of
La Géométrie
(a
reading found, for instance, in Sasaki 2003, especially 156–176).
According to Rabouin, it is not until the early 1630s, when Descartes
engages with the Pappus problem—what Bos also considers
“the crucial catalyst” of Descartes' mature
mathematical researches (Bos 2001, 283)—that he returns to his
1619 project to craft a new science of geometry that is grounded on a
new classification of curves and problems. Following Rabouin, it
is at this point of his mathematical career that Descartes more clearly
sees just how crucial the interplay of algebraic equations and geometry
could be for a general program of geometrical problem-solving.
In late 1631, the Dutch mathematician Golius urged Descartes to
consider the solution to the Pappus problem. Unlike the
geometrical problems that occupied Descartes' early researches,
the Pappus problem is a locus problem, i.e., a problem whose solution
requires constructing a curve—the “Pappus curve”
according to Bos's terminology—that includes all the
points that satisfy the relationship stated in the problem.
Generally speaking, the Pappus Problem begins with a given number of
lines, a given number of angles, a given ratio, and a given segment,
and the task is identify a curve such that all the points on the curve
satisfy a specified relation to the given ratio. For instance, in
the most basic two-line Pappus Problem (
figure 4
), we are given two
lines (
L
1
,
L
2
), two angles (
θ
1
,
θ
2
), and a ratio
β
. We designate
d
1
to be the oblique distance between a point
P
and
L
1
such that
P
creates
θ
1
with
L
1
, and we designate
d
2
to be the oblique distance between a
point
P
in the plane and
L
2
such that
P
creates
θ
2
with
L
2
. The problem is to find
all points
P
such that
d
1
:
d
2
=
β
. In
this case, all the sought after points
P
will lie along two straight
lines, one line to the right of
L
1
and the other to the left
of
L
1
. (See
figure 5
for Bos's presentation of the
general problem.)
Figure 4: A Two line Pappus problem
In the
Collection
, Pappus presents a solution to the three
and four line versions of the problem (i.e., the versions of the problem in
which we begin with three or four given lines and angles) as well as
Apollonius's solution to the six-line case, which relies on his
theory of conics and the transformation of areas to construct the locus
of points (Pappus, 118–123). However, Pappus does not treat the
general (
n
-line) case, and this is the advance of the solution
Descartes achieves in 1632, a solution published in
La
Géométrie
, where he claims that, unlike the
ancients, he has found a method to successfully “determine,
describe, [and] explain the nature of the line required when the
question [of the Pappus Problem] involves a greater number of
lines” (G, 22). And as Descartes reports to Mersenne in
1632, he could not have found his general solution without the help of
algebra:
I must admit that I took five or six weeks to find the
solution [to the Pappus Problem]; and if anyone else discovers it, I
will not believe that he is ignorant of algebra (To Mersenne 5
April 1632; AT I, 244; CSMK, 37).
Figure 5: The General Pappus Problem (from Bos 2001,Fig. 19.1, 273)
Given: a Line
L
i
in the plane,
n
angles
θ
i
, a ratio
β
, a line
segment
a
. For an point
P
in plane, let
d
be the
oblique distance between
P
and
L
i
such that
P
creates
θ
i
with
L
i
.
Problem: Find the locus of points
P
such that the following ratios are equal to the given ratio
β
:
For 3 lines:
| (
d
1
)
2
| :
d
2
d
3
|
For 4 lines:
| d
1
d
2
| :
d
3
d
4
|
For 5 lines:
| d
1
d
2
d
2
| :
a
d
4
d
5
|
For 6 lines:
| d
1
d
2
d
3
| :
d
4
d
5
d
6
|
In general,
For an even 2
k
number of lines:
|
d
1
…
d
k
| :
d
k
+1
…
d
2
k
|
For an uneven 2
k
+1 number of lines:
|
d
1
…
d
k
+1
| :
a
d
k
+2
…
d
2
k
+1
|
According to Bos, consideration of the general Pappus Problem
“provided [Descartes], in 1632, with a new ordered vision of the
realm of geometry and it shaped his convictions about the structure and
the proper methods of geometry” (Bos 2001, 283). The best
evidence we have of the impact the problem had on Descartes'
approach to geometry is
La Géométrie
itself: in
La Géométrie
, the Pappus problem is given pride
of place as Descartes details his “geometrical calculus”
and demonstrates the power of his novel program for solving geometrical
problems. It is treated in Book One, as Descartes explains his
geometrical analysis, and then again in Book Two, where Descartes
offers the synthesis, i.e., the geometrical demonstration, of his
solution to the Pappus Problem in
n
-lines, a demonstration which relies
on the famous distinction between “geometric” and
“mechanical” curves that begins this part of the work.
Book One of
La Géométrie
is entitled
“Problems the construction of which requires only straight lines
and circles,” and it is in this opening book that Descartes
details his geometrical analysis, that is, how geometrical problems are
to be explicated algebraically. In this respect, what we find in
Book One is similar to the algebraic elaboration of geometrical
problems presented by Viète in his 1594
Supplement of
geometry
as he explains the stage of exegetics. That said,
Descartes' approach to analysis rests on innovations in notation
and formalism as well as in the merging of geometry and arithmetic
which move him beyond Viète's analysis, lending some
credence to Descartes' remark to Mersenne that, in
La
Géométrie
, his program for geometry begins where
Viète's left off (To Mersenne, December 1637, AT I, 479;
CSMK 77–79).
Book One commences with the geometrical interpretation of algebraic
operations, which, we saw above, Descartes had already explored in the
early period of his mathematical research. However, what we are
presented in 1637 is, as Guicciardini aptly describes, a
“gigantic innovation” both over Descartes' previous
work and the work of his contemporaries (Guicciardini 2009, 38).
On the one hand, Descartes offers a geometrical interpretation of root
extraction and thus treats five arithmetical operations (as opposed to
the four operations of addition, subtraction, multiplication, and
division that were treated in his early work). On the other hand, and more
significantly, his treatment relies on an interpretation of
arithmetical operations according to which these operations are taken
to be closed operations on line segments. Traditionally, for
instance, the product of two segments
a
*
b
was interpreted as a
rectangle, but for Descartes, the product is interpreted as a
segment. This allows Descartes to translate geometrical problems
into equations (that include products such as
a
*
b
) and treat each
term of the equation as similar in kind. Finally, Descartes uses
a new exponential notation as he sets forth equations of multiple terms
in Book One, and this notation, which replaces the traditional cossic
notation of early modern algebra, allows Descartes to tighten the
connection between algebra and geometry, and more specifically, between
the algebraic representation of curves through equations with the
geometrical classification and geometrical solution of stated problems
(as we will see more clearly below in
section 3.2
).
With his new geometrical interpretation of the five basic
arithmetical operations at his disposal, Descartes proceeds to
describes how, in the stage of geometrical analysis, one is to give an
algebraic interpretation of a geometrical problem:
If, then, we wish to solve any problem, we first suppose the
solution already effected, and give names to all the lines that seem
needful for its construction,—to those that are unknown as well
as to those that are known. Then, making no distinction between
unknown and unknown lines, we must unravel the difficulty in any way
that shows most naturally the relations between these lines, until we
find it possible to express a single quantity in two ways. This
will constitute an equation, since the terms of one of these two
expressions are together equal to the terms of the other (G, 6–9).
We notice that the key to Descartes' analysis is to make no
distinction between the known and unknown quantities in the problem:
Both kinds of quantities are granted a variable (generally,
a
,
b
,
c
… for known quantities and
x
,
y
,
z
… for unknown quantities), and thus, we
treat the unknowns as if their values were already found. Or, as
Descartes puts it, we “suppose the solution already
effected.” The task then is to reduce the problem to an
equation (in contemporary terms, to a polynomial equation in two
unknowns) that expresses the unknown quantity, or quantities, in terms
of the known quantities. For instance, take the
following
problem:
[
3
]
Given a line segment AB
containing point C (see
figure 6
), the problem is to produce AB to D
such that the product AD*DB is equal to the square of CD. Let AC
=
a
, CB =
b
, and BD =
x
, which yields AD =
a
+
b
+
x
and CD =
b
+
x
. Thus, the problem to find BD such that AD*DB =
(CD)
2
is algebraically equivalent to finding
x
such
that: (
a
+
b
+
x
)*(
x
) = (
b
+
x
)
2
. Or, solving for
x
, the
problem is to find
x
such that, given
a
and
b
,
x
=
b
2
/ (
a
—
b
).
Figure 6:
In this example, we are dealing with a determinate problem, i.e., a
problem to which there are a finite number of solutions, and we can therefore
reduce the problem to a single equation that expresses the unknown
quantity in terms of the known quantities. However, as Descartes points
out, there are also indeterminate problems that involve an infinite
number of solutions. (Locus problems, such as the Pappus Problem,
are of this sort, because the solution includes the infinitely many
points that lie along a curve.) When dealing with an
indeterminate problem, Descartes instructs us that “we may
arbitrarily choose lines of known length for each unknown line to which
there corresponds no equation” (G, 9), i.e., we are to set the
unknown lines as oblique coordinates that have a stated value. We
then generate several equations that express the unknown quantities in
terms of one or more known quantities, and solve the equations
simultaneously. This is precisely the approach that Descartes
takes as he treats the Pappus Problem in Book One.
Figure 7: The Four-Line Pappus Problem in Book One (G, 27)
Descartes begins with consideration of the problem when we are given
three or four lines, which, borrowing from Guicciardini (2009), can be
stated as follows (see
figure 7
):
Having three or four lines given in position, it is required to find
the locus of points C from which drawing three or four lines to the
three or four lines given in position and making given angles with each
one of the given lines the following condition holds: the rectangle [or
product] of two of the three lines so drawn shall bear a given ratio to
the square of the third (if there be only three), or to the rectangle
[or product] of the other two (if there be four) (Guicciardini 2009,
54; based on G, 22).
In Book One, Descartes applies his geometrical analysis to the
four-line case of the Pappus problem. He begins by
designating two given line segments (of unknown length) AB and BC as
oblique coordinates
x
and
y
, respectively, such that
all other lines needed to solve the problem will be expressed in terms
of
x
and
y
.
[
4
]
Then, by considering the angles given in the
problem and the properties of similar triangles, he generates an
algebraic expression of the sought after points C in terms of the two
unknowns
x
and
y
and the known quantity
z
(where
z
designates the ratio given in the problem) (G,
29–30).
Importantly, the analytic method that Descartes uses in the
four-line case is generalized to apply to the general,
n
-line version
of the Pappus Problem. That is, Descartes' claim is that no
matter how many lines and angles are given in the problem, it is
possible, by means of his analytic method, to express the sought after
points C in terms of two unknown quantities (in contemporary terms, to
reduce the problem to a polynomial equation in two unknowns) (G,
33). As a result, for any
n
-line version of the Pappus Problem,
we can generate values for C and construct the sought after Pappus
curve by assigning different values to
x
and
y
, and
thereby describe the curve in a pointwise manner. As Descartes
puts it,
Furthermore, to determine the point C, but one condition is needed,
namely, that the product of a certain number of lines shall be equal
to, or (what is quite as simple), shall bear a given ratio to the
product of certain other lines. Since this condition can be
expressed by a single equation in two unknown quantities, we may give
any value we please to either
x
or
y
and find the
value of the other from this equation. It is obvious that when
not more than five lines are given, the quantity
x
, which is
not used to express the first of the lines can never be of degree
higher than the second.
Assigning a value to
y
, we have
x
2
=
±
ax
±
b
2
, and therefore
x
can be found with ruler and compasses, by a method [for
constructing roots] already explained. If then we should take
successively an infinite number of different values for the line
y
, we should obtain an infinite number of values for the line
x
, and therefore an infinity of different points, such as C,
by means of which the required curve can be drawn (G, 34).
The result of Descartes' analysis, as indicated by the remarks
above, is that the curve that includes the sought after points C can be
pointwise constructed by using ruler and compass to solve for the roots
of a second-degree equation in two unknowns. He then generalizes
this result and claims that the solution points for any problem that
can be reduced to a second-degree equation can be constructed by ruler
and compass. If instead a problem is reduced to an equation of
third or fourth degree, the points are constructed by conics, and if a
problem is reduced to an equation of fifth or sixth degree, the points
are constructed by a curve that is “just one degree higher than
the conic sections” (G, 37). In other words,
Descartes' claim is that if a problem can be reduced to a single
equation of degree not higher than six, in which the unknown quantity
or quantities are expressed in terms of a known quantity, then the
roots of the equation be constructed by straightedge and circle, or by
conic, or by a more complicated curve that does not have degree higher
than four. Based on this result, Descartes suggests a way to
generalize further and solve the
n
-line Pappus Problem, for no matter
how many given lines and angles with which a Pappus Problem begins, it
will be possible to reduce the problem to an equation and then
pointwise construct the roots of the equation, i.e., the sought after
points C of the problem (G, 37).
What Descartes achieves here by means of his geometrical analysis is
no doubt significant. He has outlined a way of solving the Pappus
Problem for any number of given lines. However, questions about
proving the Pappus Problem solved still linger come the end of Book
One. As in Viète's analysis, Descartes has shown
that a solution to the general problem exists but the algebraic
elaboration of the problem does not unto itself give a clue to how we
are to geometrically construct the curve that solves the problem.
Notice in particular that in Book One the roots (i.e., the points along
the sought after curves) are constructed by straight edge, compass,
conics, and higher order curves, such that the Pappus curves that
include the roots are constructed pointwise. But this leaves us
the question: Are the Pappus curves of Book One legitimately
geometrical? That is, can the curves that solve the
n
-line
version of the Pappus Problem themselves be constructed by legitimately
geometrical methods? This is an issue broached in Book Two, the
main focus of which is how to enact a synthesis, or construction, of a
geometrical problem.
Book Two of
La Géométrie
is entitled
“On the Nature of Curved Lines” and commences with
Descartes' famous distinction between “geometric” and
“mechanical” curves. Given its importance for
understanding the program of
La Géométrie
as
well as the attention this distinction has drawn from commentators, it
is worth examining the proposals made in the opening pages of Book Two
with some care.
Descartes begins with reference to the ancient classification of
problems and offers his interpretation of how ancient mathematicians
distinguished curves that could be used in the solution to geometrical
problems from those that could not:
The ancients were familiar with the fact that the problems of
geometry may be divided into three classes, namely, plane, solid, and
linear problems. This is equivalent to saying that some problems
require only circles and straight lines for their construction, while
others require a conic section and still others more complex
curves. I am surprised, however, that they did not go further,
and distinguish between different degrees of those more complex curves,
nor do I see why they called the latter mechanical, rather than
geometrical. If we say that they are called mechanical because
some sort of instrument has to be used to describe them, then we must,
to be consistent, reject circles and straight lines, since these cannot
be described on paper without the use of compasses and a ruler, which
may also be termed instruments. It is not because the other
instruments, being more complicated than the ruler and compass, are
therefore less accurate, for if this were so they would have to be
excluded from mechanics, in which accuracy of construction is even more
important than in geometry. In the latter, exactness of reasoning
alone is sought, and this can surely be as thorough with reference to
such lines as to simpler ones (G, 40–44).
Descartes implies that the terms “mechanical” and
“non-geometrical” were synonymous in ancient mathematics;
however, it is not at all clear that this was the intended meaning of
the term “mechanical.” Namely, it is not clear that
the classification of curves into “geometrical” and
“mechanical” was intended to serve as a normative claim
concerning the legitimacy of a curve's use in geometrical
problem-solving or simply as a descriptive moniker that captures the
different ways in which curves were constructed (see Molland 1976 on
this issue; see
section 2.2
above for Descartes' blending of the
descriptive and the normative in his 1619 proposal for a “new
science” of geometry).
Descartes' reading of the ancients aside, important for
understanding his own peculiar interpretation of geometrical curves is
the distinction he draws between the “accuracy of
construction” of a curve, which he renders an issue for
mechanics, and the “exactness of reasoning,” which he deems
as the sole requirement for accepting a curve as legitimately
geometrical. In making this claim, Descartes is carving out a
unique place for his notion of geometrical curves: He abandons the
“accuracy of construction” criterion that Clavius adopted
in his early works to render a curve acceptable in geometrical
problem-solving and also the claim forwarded by Viète that
instrumentally-constructed curves were not to be considered geometrical
(see
section 1.1
above). As Descartes' presentation
implies, both these sorts of criteria confuse issues of mechanics with
the “exactness of reasoning” that is the sole concern of
geometry. Thus, as Book Two continues, Descartes reiterates that
to determine the geometrical status of a curve we must lay our focus on
issues of exact and clear reasoning and, specifically, on the question
of whether a curve can be constructed by exact and clear
motions
. After presenting the postulate that “two
or more lines can be moved, one upon the other, determining by their
intersection other curves,” Descartes explains,
It is true that the conic sections were never freely received into
ancient geometry, and I do not care to undertake to change names
confirmed by usage; nevertheless, it seems very clear to me that if we
make the usual assumption that geometry is precise and exact, while
mechanics is not; and if we think of geometry as the science which
furnishes a general knowledge of the measurement of all bodies, then we
have no more right to exclude the more complex curves than the simpler
ones, provided they can be conceived of as described by a continuous
motion or by several successive motions, each motion being completely
determined by those which precede; for in this way an exact knowledge
of the magnitude of each is always obtainable (G, 43).
We see in these remarks that the precision and exactness of geometry
is intimately tied with the geometer's consideration of motions
that can be precisely and exactly traced. Namely, the geometer is
justified in using simple curves as well as more complex curves, so
long as the construction of these curves proceeds by “precise and
exact” motions. Descartes clarifies how a complex curve
“can be conceived of as described by a continuous motion or by
several successive motions, each motion being completely determined by
those which precede” by presenting the mesolabe compass that he
first developed in 1619:
Consider the lines AB, AD, AF, and so forth, which we may suppose to
be described by means of the instrument YZ [
Figure 8
]. This
instrument consists of several rulers hinged together in such a way
that YZ being placed along the line AN the angle XYZ can be increased
or decreased in size, and when its sides are together, the points B, C,
D, E, F, G, H, all coincide with A; but as the size of the angle is
increased, the ruler BC, fastened at right angles to XY at the point B,
pushed toward Z the ruler CD which slides along YZ always at right
angles. In a like manner, CD pushes DE which slides along YX
always parallel to BC; DE pushes EF; EF pushes FG; FG pushes GH, and so
on. Thus we may imagine an infinity of rulers, each pushing
another, half of them making equal angles with YX and the rest with
YZ.
Now as the angle XYZ is increased, the point B describes the curve
AB, which is a circle; while the intersections of the other rulers,
namely, the points D, F, H describe the other curves, AD, AF, AH, of
which the latter are more complex than the first and this more complex
than the circle. Nevertheless I see no reason why the description
of the first cannot be conceived as clearly and distinctly as that of
the circle, or at least as that of the conic sections; or why that of
the second, third, or any other that can be thus described, cannot be
as clearly conceived of as the first: and therefore I see no reason why
they should not be used in the same way in the solution of geometric
problems (G,
44–47).
[
5
]
Figure 8: Mesolabe
A couple points are worth emphasizing. First, Descartes
presents the more complex curves generated by his compass as described
by motions that can be as “conceived as clearly and
distinctly” as the motions required to construct the more simple
circle. And because of the clear and distinct motions needed for
their construction, these curves are legitimately geometrical.
That is, consistent with Descartes' general criterion for
constructing geometrical curves, these complex curves can be used in
the solution of geometric problems. Second, we see that although
Descartes takes care to distinguish the concerns of geometry from those
of mechanics, he does not steer away from the construction of curves by
means of instruments. Although instrumental constructions are
mechanical constructions, they can nonetheless give rise to geometrical
curves precisely because the
motions
of the instruments are
“clearly and distinctly” conceived. That the motions
are generated by instruments does not render the resultant curve
non-geometrical. (For more on the use of instruments in
La
Géométrie
, see Bos 1981.)
In a similar vein, curves that are non-geometrical by
Descartes' standard are curves that require more complicated,
less clear and distinct motions for their construction. He
explains:
Probably the real explanation of the refusal of ancient geometers to
accept curves more complex than the conic sections lies in the fact
that the first curves to which their attention was attracted happened
to be the spiral, the quadratrix, and similar curves, which really do
belong only to mechanics, and are not among the curves that I think
should be included here, since they must be conceived of as described
by two separate movements whose relation does not admit of exact
determination (G, 44).
Descartes explicitly names the spiral and quadratrix as those curves
whose construction “must be conceived of as described by two
separate movements whose relation does not admit of exact
determination.” Later in Book Two he clarifies why such
descriptions fail to be clearly and distinctly conceived:
geometry should not include lines that are like strings, in that
they are sometimes straight and sometimes curved, since the ratios
between straight and curved lines are not known, and I believe cannot
be discovered by human minds, and therefore no conclusion based upon
such ratios can be accepted as rigorous and exact (G, 91).
Given these remarks, the fundamental problem with the spiral, the
quadratrix, and “lines that are like strings” is that their
construction requires consideration of the ratio, or relation, between
a circle and straight line. Consider the spiral. As we saw
above, its construction involves two uniform motions: the uniform
rectilinear motion of a point along a segment and the uniform circular
motion of the segment around a point. These two motions must
simultaneously be considered in order for the moving point's path
to describe the spiral, and this, for Descartes, is what is ultimately
problematic. The human mind can think about simultaneous
rectilinear and circular motions, but it cannot do so with the clarity
and distinctness required to meet the exact and rigorous standards of
geometry. (This claim is not without its problems, which will be
discussed in
section 3.3
below.)
After presenting his construction criterion for geometrical curves,
Descartes develops his novel connection between geometrical
construction and the algebraic representation of these curves.
Whereas in Book One Descartes details how to use algebra to establish
that a solution a geometrical problem exists, here, in Book Two,
Descartes proposes a stronger connection between algebra and geometry
and famously claims that any legitimately geometrical curve can be
represented by an equation:
I could give here several other ways of tracing and conceiving a
series of curved lines, each curve more complex than any preceding one,
but I think the best way to group together all such curves and then
classify them in order, is by recognizing the fact that all the points
of those curves which we may call “geometric,” that is,
those which admit of precise and exact measurement, must bear a
definite relation to all points of a straight line, and that this
relation must be expressed means of a single equation (G,
48).
He then proceeds to classify these “geometric” curves
according to the degree of their corresponding equations, claiming:
If [a curve's] equation contains no term of higher degree than
the rectangle [product] of two unknown quantities, or the square of
one, the curve belongs to the first and simplest class, which contains
only the circle, the parabola, the hyperbola, and the ellipse; but when
the equation contains one or more terms of the third or fourth degree,
in one or both of the two unknown quantities (for it requires two
unknown quantities to express the relation between two points) the
curve belongs to the second class; and if the equation contains a term
of the fifth or sixth degree in either or both of the unknown
quantities the curve belongs to the third class, and so on indefinitely
(G, 48).
The same point is made later in Book Two, where Descartes emphasizes
that “no matter how we conceive a curve to be described, provided
it be one of those which I have called geometric,” it will always
be possible to find an equation determining all of the curve's
points (G, 56). He reiterates that geometric curves can be
classified according to their equations but also points out that within
a specific class, a curves' simplicity should be ranked according
to the motions required for construction. For instance, although
the circle belongs to the same class as the ellipse, hyperbola, and
parabola, these latter curves are “equally complex” whereas
the circle “is evidently a simpler curve” and will thus be
more useful in the construction of problems (G, 56).
As in Book One, Descartes uses the Pappus Problem to illustrate the
power of his geometrical calculus, where in Book Two, his aim is to
show how his algebraic classification of curves makes it easy “to
demonstrate the solution which [he has] already given of the problem of
Pappus” (G, 59). The specific goal here is to establish
that the curves which solve the general Pappus Problem are legitimately
geometrical curves, i.e., to show that the Pappus curves meet the exact
and rigorous standards of geometrical construction that he has just
laid out. Descartes' discussion of the Pappus Problem in
Book Two begins as follows:
Having now made a general classification of curves, it is easy for
me to demonstrate the solution which I have already given of the
problem of Pappus. For, first, I have shown [in Book One] that
when there are only three or four lines the equation which serves to
determine the required points is of the second degree. It follows
that the curve containing these points [i.e., the Pappus curve] must
belong to the first class, since such an equation expresses the
relation between all points of curves of Class I and all points of a
fixed straight line. When there are not more than eight given
lines the equation is at most a biquadratic, and therefore the
resulting [Pappus] curve belongs to Class II or Class I. When
there are not more than twelve given lines, the equation is of the
sixth degree or lower, and therefore the required curve belongs to
Class III or a lower class, and so on for other cases (G, 59).
As indicated in the passage above, Descartes establishes in Book Two
that Pappus curves fall into the specified classes of geometric curves
he has designated, where the class into which a Pappus curve falls
depends on the number of lines given in the problem and thus, on the
degree of the equation to which the problem is reduced. For
instance, when Descartes treats the four-line Pappus Problem in Book
Two, he shows that, by varying the coefficients of the second degree
equation to which the problem has been reduced (through the analysis of
Book One), we can construct either a circle, parabola, hyperbola, or
ellipse (G, 59–80). That is, he shows that the Pappus curve that
solves the four-line problem is either a circle or one of the conic
sections, the very “geometric” curves that he has grouped
into Class I.
In two stages, then, Descartes has demonstrated the solution to the
general Pappus Problem. In Book One he offers his algebraic
analysis of the problem, and in Book Two he claims to provide the
synthesis (or demonstration) that the curves that solve the general
problem are legitimately geometrical curves which meet his stated
standard for geometrical exactness and precision. And with these
two stages completed, Descartes claims to Mersenne six months after
La Géométrie
is published that his treatment of
the general Pappus Problem is
proof
that his new method for
geometrical-problem solving is an improvement over the methods of his
predecessors:
I do not like to have to speak well of myself, but because there are
few people who are able to understand my
Geometry
, and since
you will want me to tell you what my own view of it is, I think it
appropriate that I should tell you that it is such that I could not
wish to improve it. In the
Optics
and the
Meteorology
I merely tried to show that my method is better
than the usual one; in my
Geometry
, however, I claim to have
demonstrated this. Right at the beginning I solve a problem which
according to the testimony of Pappus none of the ancients managed to
solve; and it can be said that none of the moderns has been able to
solve it either, since none of them has written about it, even though
the cleverest of them have tried to solve the other problems which
Pappus mentions in the same place as having been tackled by the
ancients (To Mersenne, end of December 1637; AT 1, 478; CSMK,
77–78).
As great as Descartes' confidence in his solution to the
Pappus Problem, there are questions that surround his synthesis of the
general problem in Book Two.
As indicated above, Descartes attempts to establish via his
synthesis that the curves that solve the Pappus Problem are
“geometric” by his own stated standard, that is, that the
Pappus curves are constructible by the “precise and exact”
motions needed to construct genuinely geometric curves. However,
it is not at all clear that Descartes has proven this point. Even
when addressing the basic four-line Pappus Problem in Book Two,
Descartes does not appeal to motions that are evidently clear and
distinct as he constructs the Pappus curves that solve the problem (in
this case, the circle, parabola, hyperbola, and ellipse). Rather,
he relies on Apollonius's theory of conics, which requires that a
cone be cut at a designated point in the plane, and as Bos remarks,
this Apollonian technique for constructing conics “is not a
method of construction that immediately presents itself to the mind as
clear and distinct” (Bos 2001, 325). Specifically, since it
was not evident to mathematicians at the time whether constructions
that required locating a cone in the plane met the exact and rigorous
standards of geometrical reasoning, Descartes' treatment of the
Pappus curves in this four-line case does not convincingly demonstrate
their “geometric” status. Later in Book Two, when he treats
the five-line Pappus Problem, matters get more complicated.
Recall that in addition to his emphasis on the “precise and
exact” motions that can be used to describe legitimately
geometrical curves, Descartes also claims that these curves “can
be conceived of as described by a continuous motion or by several
successive motions.” As such, we would reasonably expect
that the geometrical construction of these curves should not proceed
pointwise in the manner of Book One, where Descartes constructed the
Pappus curves by solving the equations to which the problem had been
reduced. However, when Descartes treats the five-line Pappus
Problem in Book Two, he in fact offers a pointwise construction of the
Pappus curve. He then remarks that the pointwise construction of this
“geometric” Pappus curve is importantly different from the
pointwise construction of non-geometrical, “mechanical”
curves:
It is worthy of note that there is a great difference between this
method in which the [Pappus] curve is traced by finding several points
upon it, and that used for the spiral and similar curves. In the
latter, not any point of the required curve may be found at pleasure,
but only such points as can be determined by a process simpler than
that required for the composition of the curve…On the other
hand, there is no point on these [“geometric”] curves which
supplies a solution for the proposed problem that cannot be determined
by the method I have given (G, 88–91).
The suggestion from Descartes is that when we pointwise construct a
geometric curve, we can identify any possible point on the curve, and
immediately after the above remarks, he proceeds to equate curves
constructed in this manner with curves that could possibly be
constructed by continuous motions: “this method of tracing a
curve by determining a number of its points taken at random applies
only to curves that can be generated by a regular and continuous
motion” (G, 91).
This distinction between the pointwise construction of
“geometric” and “mechanical” curves serves two
rather important purposes in the program of
La
Géométrie
: (1) Descartes can establish that the
Pappus curves he has pointwise constructed are in fact
“geometrical” and thereby complete his synthesis (or
demonstration) of the general Pappus Problem, and (2) he can maintain a
boundary between intelligible “geometric” curves and
unintelligible “mechanical” curves. Without a clear
indication of why the pointwise construction of a Pappus curve is
“geometric,” Descartes would have to allow
“mechanical” curves such as the spiral and quadratrix into
the domain of geometrical curves, since these curves can also be
pointwise constructed. Recall for instance Clavius's
pointwise construction of the quadratrix. According to Clavius's
description, we begin with a quadrant of a circle and then identify the
points of intersection between segments that bisect the quadrant and
segments that bisect the arc of the quadrant (see
figure 9
). That is,
we identify the several intersecting points of segments which are
constructible by straightedge and compass, and then, to generate the
quadratrix, we connect the intersecting points, which are evenly spaced
along the sought after curve. Why is such a pointwise
construction not “geometric”? Because, according to
Descartes, if we proceed as Clavius does, “not any point of the
required curve may be found at pleasure.” Specifically, given the
restrictions of Euclidean construction, we are only able to divide the
given arc into 2
n
parts. As such, what Descartes
suggests is that it is not possible to divide the arc any way we please, and
we cannot therefore locate any arbitrary point along the curve by use
of pointwise construction. In the case of the
“geometric” curves, however, we can find any arbitrary
point on the curve by appeal to the equations corresponding to the
problem; or borrowing Bos's terminology, Descartes is claiming
that “geometric” curves, and the Pappus curves in
particular, can be generated by “generic” pointwise
constructions.
Figure 9
While consideration of Clavius's construction of the
quadratrix offers some reason to accept Descartes' distinction
between the different sorts of pointwise constructions, there remains
the controversial claim that curves described by “generic”
pointwise constructions are curves that can be constructed by
continuous motion. This identification allows Descartes to
establish Pappus curves as “geometric” curves, but he
offers no proof of the identity, and thus, there is question of whether
Descartes has in fact demonstrated that the Pappus curves are
“geometric” by his own standards. (See Grosholz 1991 and
Domski 2009 for alternative ways of addressing this tension.)
There is a further question surrounding Descartes' criterion
for “geometric” curves. As we have seen above,
Descartes' explicit concern in Book Two is to offer a standard
for geometrical curves that is bound with intelligible, clear and
distinct motions needed for their construction.
However, Mancosu (2007) has recently offered a compelling
case that behind Descartes' explicit remarks in
La
Géométrie
lies a more fundamental concern: To
ensure that those curves mathematicians had used to square the circle,
such as the spiral and quadratrix which are explicitly mentioned in
Book Two, are rendered non-geometrical. Mancosu supports his case
with evidence from Descartes' correspondence that shows, for
Descartes, it is in fact possible in some instances to clearly and
distinctly conceive the relation between a straight line and a circle,
a relation he had considered inexact in
La
Géométrie
. Namely, in a 1638 letter to
Mersenne, Descartes writes,
You ask me if I think that a sphere which rotates on a plane
describes a line equal to its circumference, to which I simply reply
yes, according to one of the maxims I have written down, that is that
whatever we conceive clearly and distinctly is true. For I
conceive quite well that the same line can be sometimes straight and
sometimes curved, like a string (To Mersenne, 27 May 1638; AT 2,
140–141; translation from Mancosu 2007, 118).
In
La Géométrie
, the relation between
straight and curves lines was considered inexact because, as Descartes
put it, “the ratios between straight and curved lines are not
known, and I believe cannot be discovered by human minds” (G,
91). That Descartes later admits clearly and distinctly
conceiving such a relation suggests, according to Mancosu, that the
stated criteria for geometrical curves presented in
La
Géométrie
reveals only part of Descartes'
mathematical agenda. A more complete portrait must, as Mancosu
argues, take into account Descartes' commitment to the
impossibility of squaring the circle (see Descartes' letter to
Mersenne, 13 November 1629, AT 1, 70–71; translated in Mancosu 2007,
120; see also Mancosu and Arana 2010 for further evidence in support of
the position of Mancosu 2007).
Whether Descartes had the hidden agenda that Mancosu suggests, the
explicit claims used to define the program of problem-solving presented
in
La Géométrie
point to the limitations of
Descartes' mathematics. Namely, as we have seen above,
Descartes' primary focus is on a standard for geometry's
“exactness of reasoning” that is bound to clear and
distinct motions for construction and to the finite equations to
represent curves so constructed. There is thus no room in the
program of
La Géométrie
to use infinitesimals in
the construction of curves or to treat curves represented by infinite
equations, and as such, Descartes had eliminated the very elements of
mathematical and geometrical reasoning that made it possible for Newton
and Leibniz to develop the calculus come the late seventeenth
century. Nonetheless, given how quickly Descartes honed his
mathematical skills and how quickly he developed his innovative program
for geometry, it seems safe to follow Descartes' self-assessment
and maintain some confidence that the calculus would have been in his
reach had he considered the infinitesimal and the infinite:
having determined as I did [in
La Géométrie
]
all that could be achieved in each type of problem and shown the way to
do it, I claim that people should not only believe that I have
accomplished more than my predecessors but should also be convinced
that posterity will never discover anything in this subject which I
could not have discovered just as well if I had bothered to look for it
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[Please contact the author with suggestions.]
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