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Descartes' MathematicsFirst published Mon Nov 28, 2011

To speak of René Descartes' contributions to the history of mathematicsis to speak of his La Géométrie (1637), a short tract included with theanonymously published Discourse on Method. In La Géométrie,Descartes details a groundbreaking program for geometrical problem-solving—what he refers to as a “geometrical calculus” (calculgéométrique)—that rests on a distinctive approach to the relationshipbetween algebra and geometry. Specifically, Descartes offers innovativealgebraic techniques for analyzing geometrical problems, a novel way ofunderstanding the connection between a curve's construction and itsalgebraic equation, and an algebraic classification of curves that is basedon the degree of the equations used to represent these curves. Examiningthe main questions and issues that shaped Descartes' early mathematicalresearches sheds light on how Descartes attained the results presented inLa Géométrie and also helps reveal the significance of this work for thedebates surrounding early modern mathematics.

The importance of La Géométrie for the history of mathematics is hardlya matter of dispute. The problem-solving techniques and mathematicalresults that Descartes presents in that short tract were both novel andincredibly influential. However, we can also locate in La Géométrie aphilosophical significance: The blending of algebra and geometry and thepeculiar approach to the “geometrical” status of curves which characterizeDescartes' mathematical program stand as notable contributions to the on-going philosophical debates that surrounded early modern mathematicalpractice. By drawing on the context in which Descartes' mathematicalresearches took place, the historical and philosophical significance ofBooks One and Two of La Géométrie will be highlighted in whatfollows.[1]

1

1. The Background to Descartes' Mathematical Researches1.1 The Construction of Curves and the Solution to GeometricalProblems1.2 Geometrical Analysis and Algebra

2. Descartes' Early Mathematical Researches (ca. 1616–1629)2.1 Texts and sources2.2 Problems and Proposals

3. La Géométrie (1637)3.1 Book One: Descartes' Geometrical Analysis3.2 Book Two: The Classification of Curves and GeometricalSynthesis3.3 The Tensions and Limitations of Descartes' GeometricalCalculus

BibliographyAcademic ToolsOther Internet ResourcesRelated Entries

1. The Background to Descartes' MathematicalResearches

When Descartes' mathematical researches commenced in the earlyseventeenth century, mathematicians were wrestling with questionsconcerning the appropriate methods for geometrical proof and, inparticular, the criteria for identifying curves that met the exact andrigorous standards of geometry and that could thus be used in geometricalproblem-solving. These issues were given an added sense of urgency forpracticing mathematicians when, in 1588, Commandino's Latin translationof Pappus's Collection (early fourth century CE) was published. In theCollection Pappus appeals to the ancient practice of geometry as he offers


2 Stanford Encyclopedia of Philosophy

normative claims about how geometrical problems ought to be solved.Early modern readers gave special attention to Pappus's proposalsconcerning (1) how a mathematician should construct the curves used ingeometrical proof, and (2) how a geometer should apply the methods ofanalysis and synthesis in geometrical problem-solving. The constructionof curves will be treated in 1.1 and analysis and synthesis in section 1.2below.

1.1 The Construction of Curves and the Solution to GeometricalProblems

Pappus's claims regarding the proper methods for constructinggeometrical curves are couched in terms of the ancient classification ofgeometrical problems, which he famously offers in Book III of theCollection:

The ancients stated that there are three kinds of geometricalproblems, and that some are called plane, others solid, and othersline-like; and those that can be solved by straight lines and thecircumference of a circle are rightly called plane because the linesby means of which these problems are solved have their origin inthe plane. But such problems that must be solved by assuming oneor more conic sections in the construction, are called solid becausefor their construction it is necessary to use the surfaces of solidfigures, namely cones. There remains a third kind that is calledline-like. For in their construction other lines than the ones justmentioned are assumed, having an inconstant and changeableorigin, such as spirals, and the curves that the Greeks calltetragonizousas [“square-making”], and which we call“quandrantes,” and conchoids, and cissoids, which have manyamazing properties (Pappus 1588, III, §7; translation from Bos2001, 38).

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We notice in the above remarks that Pappus bases his classification ofgeometrical problems on the construction of the curves necessary for thesolution of a problem: If a problem is solved by a curve constructible bystraightedge and compass, it is planar; if a problem is solved by a curveconstructible by conic section, it is solid; and if a problem is solved by acurve that requires a more complicated construction—that has an“inconstant and changeable origin”—, it is line-like. Though a seeminglystraightforward directive of how to classify geometrical problems, thereremained an ambiguity in Pappus's text about whether the so-called solidand line-like problems—problems that required the construction of conicsand more complicated curves, such as the spiral—were in fact solvable bygenuinely geometrical methods. That is, there was an ambiguity, and thus,an open question for early modern mathematicians, about whetherproblems that could not be solved by straightedge and compassconstruction met the rigorous standards of geometry. (For the specialstatus of constructions by straightedge and compass in Greekmathematics, see Heath (1921) and Knorr (1986). For helpful overviewsof the historical development of Greek mathematics, see classics such asMerzbach and Boyer (2011) and volume 1 of Kline (1972).)

A few examples will help clarify what is at stake here. The problem ofbisecting a given angle is counted among planar problems, because, asdetailed by Euclid in Elements I.9, to construct the line segment thatdivides a given angle into two equal parts, we construct (by compass)three circles of equal radius, and then (by straightedge) join the vertex ofthe angle with the point at which the circles intersect (Euclid 1956,Volume I, 264–265). Notice here that, to generate the solution, curves areused to construct a point that gives the solution to the problem: byconstructing the circles, we identify a point that allows us to bisect thecurve. (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 theconstruction of curves, such as the spiral, which were not constructible bystraightedge and compass. Perhaps most famous among line-like problemsis that of squaring the circle; for those who deemed this problem solvable,the solution required the construction of a curve such as the quadratrix, acurve that was proposed by the ancients in order to solve this veryproblem (which is how the curve received its name). Certainly, thegeneration of such curves could be described; Archimedes famouslydescribes the generation of the spiral in Definition 1 of his Spirals andPappus describes the generation of the quadratrix in Book IV of theCollection. However, these descriptions were considered “morecomplicated” precisely because they go beyond the intersection of curvesthat are generated by straightedge and compass construction. For instance,according to Archimedes, the spiral is generated by uniformly moving aline segment around a given point while tracing the path of a point thatitself 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 theother moves through a quadrant of the circle. (Cf. Bos 2001, 40–42 forthe details of both these constructions.) In a similar vein, the constructionof conics was considered more complicated: One of the acceptedtechniques for constructing a conic required cutting a cone in a specifiedway, which again, went beyond the consideration of intersecting curvesthat were constructible by straightedge and compass.

In the Collection, Pappus does not offer a firm verdict on whether theconics and “more complicated” curves meet the rigorous standards ofgeometrical construction and hence, on whether they are admissible in thedomain of geometry. In the case of the conics, he relies on Apollonius'scommentary and reports the usefulness of these curves for the synthesis(or proofs) of some problems (Pappus, 116). However, to claim a curveuseful is quite different from claiming it can be constructed by properly

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geometrical methods (as we'll see more clearly below). Moreover, in thecase of the quadratrix, Pappus sets out the description of the curve inBook IV of the Collection, and then immediately proceeds to identify thecommon objections to the curve's description, e.g., that there is a petitioprincipii in the very definition of the curve, without commenting onwhether these objections can be overcome. Thus, although it was knownby the ancients that conics and other complicated curves could be used tosolve outstanding problems, it was not clear to early modernmathematicians whether the ancients considered these solutions genuinelygeometrical. That is, it was not clear from Pappus's Collection whetherthese curves were admissible in geometrical problem-solving andtherefore, whether solid problems (such as identifying the meanproportionals between given line segments) or line-like problems (such astrisecting an angle and squaring the circle) had genuine geometricalsolutions.

Consequently, after the publication of Commandino's translation of theCollection, early modern mathematicians gave added attention to questionof 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 morefamous outstanding geometrical problems, namely, angle trisection andsquaring the circle. [2] For instance, in his second and expanded (1589)edition of Euclid's Elements (which was first published in 1574) as wellas in his Geometria practica (1604), Christoph Clavius discusses thestatus of the quadratrix. Accepting the objections to the description of thequadratrix detailed by Pappus in the Collection, Clavius supplies what hedeems a “truly geometrical” construction of the curve that wouldlegitimize its use in geometrical problem-solving, and in solving theproblem of squaring the circle in particular. His construction is apointwise one: We begin with a quadrant of a circle (as in Pappus'sdescription) but rather than relying on the intersection of uniformly



moving segments to describe the curve, Clavius proceeds by firstidentifying the points of intersection between segments that bisect thequadrant and segments that bisect the arc of the quadrant. That is, weidentify the several intersecting points of segments which areconstructible by straightedge and compass, and then, to generate thequadratrix, we connect the (arbitrarily many) intersecting points, whichare evenly spaced along the sought after curve. Therefore, to construct thequadratrix according to Clavius's method, we still go beyond basicstraightedge and compass constructions (connecting the points in this casecannot be done by straightedge, as in the case of bisection), but one neednot consider the simultaneous motions of lines as Pappus's constructionrequires. (See Bos 2001, 161–162 for Clavius's construction of thequadratrix and compare with Pappus's construction on Bos 2001, 40–42.For Descartes' assessment of Clavius's pointwise construction see section3.3 below.)

According to Clavius's commentary of 1589, this pointwise constructionof the quadratrix was an improvement over that offered by Pappus,because it was more accurate: Since the pointwise construction allowedone to identify arbitrarily many points along the curve, one could trace thequadratrix with greater precision than if one had to consider theintersection of two moving lines. To support his case, Clavius relates hispointwise construction of the quadratrix with the pointwise constructionof conics proposed by the “great geometer” Apollonius and claims that“unless someone wants to reject as useless and ungeometrical the wholedoctrine of conic sections” proposed by Apollonius, “one is forced toaccept 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 theconics. He maintains that these more complicated curves could beconstructed by pointwise methods that offered greater precision, but thecurves thus generated were no longer presented as absolutely geometrical.

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Instead, they were presented as “more accurate,” “easier,” andgeometrical “in a certain way” (Bos 2001, 164–5).

In his Supplement of geometry (1593), François Viète also addresses theoutstanding problems of geometry which were solvable by curves thatcould not constructed by straightedge and compass. He claims that at leastsome such problems could be solved by properly geometrically means byadopting as his postulate that the “neusis problem” could be solved. Thatis, he assumed that given two lines, a point O, and a segment a, it waspossible to draw a straight line through O intersecting the two lines inpoints A and B such that AB = a (Bos 2001, 167–168). In the Supplement,Viète shows that once we accept as a fundamental geometrical postulatethat the neusis problem is solvable, then we can, by legitimatelygeometrical means, solve the problems of trisecting a given angle and ofconstructing the two mean proportional between two given line segments.Specifically and importantly, we generate these solutions without havingto rely on the construction of conics or higher-order curves, such as thespiral or quadratrix (Bos 2001, 168).

The neusis postulate was a powerful tool in Viète's problem-solvingarsenal: By assuming that the neusis problem could be solved, heexpanded the domain of acceptable geometrical constructions beyondstraightedge and compass. However, questions remained about theacceptability of this assumption as a postulate, since Viète does not detailthe construction of the neusis problem but simply claims that the “neusispostulate” should not be difficult for his readers to accept. In making thisassumption, he was taking a significant departure from ancient geometers,for whom the neusis problem could only be solved by curves that werenot constructible by straightedge and compass. For instance, Pappusrendered the construction of the neusis a solid problem and solved it bymeans of conics in Book IV of the Collection, and Nicomedes renderedthe 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 forNicomedes' solution. See also Pappus 1986, 112–114 for the classificationof the neusis as a sold problem.)

Nonetheless, according to Viète, if a problem could not be solved byneusis, then questions of legitimacy remained. For instance, neither thespiral nor the quadratrix—curves used to square the circle by Archimedesand Pappus, respectively—could be constructed in the same obvious and“not difficult” way as the neusis. Viète appears to grant that the pointwiseconstruction of the quadratrix, such as that presented by Clavius, was infact more precise than other constructions of the curve, but, Viète claims,this greater precision does not legitimize its status as genuinelygeometrical. 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 intersectionof curves, such as the Archimedean spiral, were “not described in the wayof true knowledge” (Bos 2001, 177). Therefore, just as the quadratrix,these curves were not legitimately geometrical, which left the problem ofsquaring the circle an open problem for Viète.

1.2 Geometrical Analysis and Algebra

Viète's program of geometrical problem-solving had an addedsignificance: By adopting as his postulate that the neusis problem couldbe solved, Viète was able to link geometrical construction with hisalgebraic analysis of geometrical problems and show that cubic equationshad a genuinely geometrical solution (i.e., that the roots of cubicequations could be constructed by consideration of intersectinggeometrical curves). Viète's program nicely illustrates the merging ofalgebra with geometrical problem-solving in early modern mathematics,and moreover, nicely illustrates an influential way of interpreting Pappus'sclaims in the Collection regarding how a mathematician should apply the

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methods of analysis and synthesis in geometrical problem-solving.

As noted above, Pappus's remarks concerning the two-fold method ofanalysis (resolutio) and synthesis (compositio) in the Collection receiveda great deal of attention from early modern readers. And as with hisremarks concerning the construction of geometrical curves, there wereambiguities in his discussion, which motivated varying interpretations ofthe method and its application to geometrical problems. Here is a portionof what Pappus claims of analysis and synthesis in Book VII of theCollection:

Some of the directives Pappus offers here seem straightforward. Themathematician begins by assuming what is sought after as if it has beenachieved until, through analysis, she reaches something that is alreadyknown. Then, the mathematician reverses the steps, and throughsynthesis, sets out “in natural order” the deduction leading from what isknown to what is sought after. However, there are ambiguities in Pappus's

Now analysis is the path from what one is seeking, as if it wereestablished, by way of its consequences, to something that isestablished by synthesis. That is to say, in analysis we assumewhat is sought as if it has been achieved, and look for the thingfrom which it follows, and again what comes before that, until byregressing in this way we come upon some one of the things thatare already known, or that occupy the rank of a first principle. Wecall this kind of method “analysis,” as if to say anapalin lysis(reduction backward). In synthesis, by reversal, we assume whatwas obtained last in the analysis to have been achieved already,and, setting now in natural order, as precedents, what before werefollowing, and fitting them to each other, we attain the end of theconstruction of what was sought. This is what we call “synthesis”(Pappus, 82–83).



discussion. Perhaps most importantly, it is not clear how reversing thesteps of analysis could offer a proof, or synthesis, of a stated problem,since the deductions of analysis rely on conditionals (if x, then y) whereasa reversal would require biconditionals (x iff y) to achieve synthesis (seeGuicciardini 2009, 31–38 for further interpretative problems surroundingPappus's remarks; for more on analysis and synthesis in the Renaissancesee the classic Hintikka and Remes 1974, the essays in Otte and Panza1997, and Panza 2007). Ambiguities notwithstanding, for Viète and otherearly modern mathematicians there was one feature of the discussion thatwas incredibly important: Pappus makes clear that the ancients had amethod of analysis at their disposal, and many early modernmathematicians attempted to align this method from antiquity with thealgebraic methods of geometrical analysis that they were using.

Prior to the end of the sixteenth century, mathematicians had already usedalgebra in the analysis of geometrical problems, but the program Viètedetails marks a significant step forward. On the one hand, in his Isagoge[Introduction to the analytic art] of 1591, which was presented as part ofa larger project to restore ancient analysis (entitled Book of the restoredmathematical analysis or the new algebra), Viète introduces a notationthat allowed him to treat magnitudes in a general way. The literal symbolshe uses (consonants and vowels depending on whether the variable in theequation was unknown or indeterminate, respectively) representmagnitudes generally and do not specify whether they are arithmeticalmagnitudes (numbers) or geometrical magnitudes (such as line segmentsor angles). He can thus represent arithmetic operations as applied tomagnitudes in general. For instance, A + B represents the addition of twomagnitudes and does not specify whether A and B are numbers (in whichcase the addition represents a process of counting) or geometrical objects(in which case the addition represents the combination of two linesegments) (see Viète 1591, 11–27; for the significance of Viète's “newalgebra” for early modern mathematics see Bos 2001, Chp. 8; Mahoney

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1973, Chp. 2; and Pycior 1997, Chp. 1).

On the other hand, the algebraic, symbolic analysis of geometricalproblems that Viète proposes was offered as the first step in a three-stepprocess that could render a geometrical solution. The three stages were:(1) zetetics, which involved the algebraic analysis (or elaboration) of aproblem; (2) poristics, which clarified the relations between magnitudesby appeal to the theory of proportions (see Giusti 1992 on the importanceof proportion theory for Viète's mathematics); and (3) exegetics, whichoffered the genuine geometrical solution (or proof) of the problem. Tobetter understand the connection between the stages of zetetics andexegetics, which roughly correspond to the ancient stages of analysis andsynthesis, consider the problem of identifying two mean proportionals.Geometrically, the problem is as follows:

In the zetetic (analytic) stage of Viète's analysis, we follow Pappus'sdirective to treat “what is sought as if it has been achieved” precisely bynaming the unknowns by variables. Then, by assuming the equivalencebetween proportions (as Viète does), we can solve for the variables x andy and establish that x and y have the following relationship to a and b:

1. x2 = ay and2. y2 = xb.

Solving (1) for y, we have y = x2/a, and by substitution into (2), we get y2

= (x2/a)2 = x4/a2 = xb, which yields:

Given line segments a and b, find x and y such that a : x :: x : y :: y: b, or put differently, such that

a : x : yx y b



3. x3 = a2b.

Solving (2) for x, we have x = y2/b, and by substitution into (1), we get x2

= (y2/b)2 = y4/b2 = ay, which yields:

4. y3 = ab2.

Algebraically, then, the problem of finding two mean proportional can beelaborated as follows:

In this zetetic stage of analysis, the geometrical problem is transformedinto the algebraic problem of solving a standard-form cubic equation (i.e.,a cubic equation that does not include a quadratic term). However, forViète, the genuine solution to the problem must be supplied in the stageof exegetics, which offers the geometrical construction and thus thesynthesis, or proof.[3] And it is here that the neusis postulate supplies theguarantee that such a solution can be found: By assuming the neusisproblem solved, we can construct the curve that satisfies the two cubicequations above (i.e., we can construct the roots of the equations) andthereby construct the sought after mean proportionals. In other words,there was an assumed equivalence in Viète's program between solving analgebraic problem that required identifying the roots of specified cubicequations and solving a geometrical problem that required theconstruction of a curve. We also see this in his treatment of trisecting anangle: To solve the angle-trisection problem is to solve two standard-formcubic equations, which Viète reveals in his algebraic elaboration of thegeometrical problem (cf. Bos 2001, 173–176). In fact, assuming theneusis postulate, we can solve any standard-form cubic equation, andsince it was already known at the time that all fourth-degree equations arereducible to standard-form cubic equations, what Viète supplied with his

Given (magnitudes) a and b, the problem is to find (magnitudes) xand y such that x3 = a2b and y3 = ab2.

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marriage of algebra and geometry in his 1594 Supplement was a programthat solved all line-like problems that could be elaborated in terms ofcubic and quartic equations.

As powerful as Viète's program was, questions remained for practicingmathematicians. Should we, as Viète urged, accept the neusis postulate as“not difficult” and thus as a foundational construction principle forgeometry? And should we follow Viète in claiming that other curves thathad significant problem-solving power in geometry—such as the spiraland quadratrix—were not legitimately geometrical because they could notbe constructed by neusis? Moreover, there were questions about theconnection Viète forged between algebra and geometry. For Descartes inparticular, there were questions of whether there was a deeper, morefundamental connection that could be forged between the solutions ofalgebraic problems that were expressed in terms of equations and thesolutions of geometrical problems that required the construction curves.However, these questions did not come into full relief for Descartes untilthe early 1630s, after more than a decade of studying problems in bothgeometry and algebra.

2. Descartes' Early Mathematical Researches (ca.1616–1629)

2.1 Texts and sources

Based on the autobiographical narrative included in Part One of theDiscourse on Method (1637), where Descartes describes what he learnedwhen 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 ofmathematics commenced when he was a student at La Fleche. He reportsin the Discourse that, when we he was younger, his mathematical studiesincluded some geometrical analysis and algebra (AT VI, 17; CSM I, 119),



and he also mentions that he “delighted in mathematics, because of thecertainty and self-evidence of its reasonings” (AT VI, 7; CSM I, 114).However, no specific texts or mathematical problems are mentioned inthe 1637 autobiographical sketch. Thus, we rely on remarks made incorrespondence for the more specific details of Descartes' study ofmathematics at La Fleche, and these remarks strongly suggest thatClavius was a key figure in Descartes' earliest (perhaps even initial) studyof mathematics. For instance, in a letter of March 1646 written by JohnPell 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 sameyear, Pell writes in particular that “[Descartes] says he had no otherinstructor for Algebra than ye reading of Clavy Algebra above 30 yearsago” (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 November1629 letter written to Mersenne, Descartes refers to the second (1589)edition of Clavius's annotated version of Euclid's Elements, in which, asnoted above, Clavius presents his pointwise construction of the quadratrixand 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 concludethat Descartes was at least aware of Clavius' textbook Geometria practica(1604), which was included as part of the mathematics curriculum of LaFleche. (See Sasaki 2003, Chapter Two on Clavius' influence on andinclusion in the mathematics curriculum of Jesuit schools in the early1600s.)

Although our evidence of the mathematics that Descartes studied at LaFleche is sketchy, we are quite certain that Descartes' entrance into thedebates of early modern mathematics began in earnest when he met IsaacBeeckman in Breda, Holland in 1618. Among other things, Beeckman andDescartes explored the fruitfulness of applying mathematics to natural

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philosophy and discussed issues pertaining to physico-mathematics. It isin this period that Descartes composed his Compendium musicae forBeeckman, a text in which he addresses the application of mathematics tomusic and also famously discusses the law of free fall (compare Koyré1939, 99–128 and Schuster 1977, 72–93 on Descartes' treatment of freefall in this early text).

Beyond having a common interest in applied mathematics, Beeckman andDescartes also discussed problems of pure mathematics, both in geometryand in algebra, and Descartes' interest in such problems extended to1628–1629, when he returned to Holland to meet Beeckman after histravels through Germany, France, and Italy. Our understanding of whatDescartes accomplished in pure mathematics during this eleven yearperiod relies on the following sources:

a. Five letters written to Beeckman in 1619, which Beeckmantranscribed in his Journal. Beeckman's Journal was recovered in1905 and published in 4 volumes by DeWaard some 35 years later,hereafter Beeckman (1604–1634). The excerpts of these letters thatare relevant to Descartes' mathematics are included in AT X. (Formore details on how these letters became available to us, see Sasaki2003, 95–96.)

b. The Cogitationes privatae (Private Reflections), which dates fromca. 1619–1620 and which Leibniz copied in 1676. This text isincluded in AT X. (For more details on how this text becameavailable to us, see Bos 2001, 237, Note 17 and Sasaki 2003, 109.)

c. The Progymnasmata de solidorum elementis, a geometry text whichdates 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).

d. A specimen of general algebra, which Descartes gave to Beeckmanafter he returned to Holland in 1628. It was transcribed by Beeckman



in his Journal under the title Algebra Des Cartes specimen quoddamand can be found in Volume III of Beeckman (1604–1634).

e. Some texts on algebra that were given to Beeckman in early 1629.These were transcribed by Beeckman in his Journal in February1629 and can be found in Volume IV of Beeckman (1604–1634).

f. Several letters written to Mersenne in the 1630s in which Descartesrefers to some of the mathematical researches he completed duringthe 1618–1629 period.

A look at some of the problems and proposals found in thesemathematical works will help situate Descartes in his early modernmathematical context and will also help to highlight the results from thisperiod that have an important connection to what is found in the openingbooks of the 1637 La Géométrie. To make these connections clear, thebrief narrative below emphasizes Descartes' proposals concerning (1) thecriteria for geometrical curves and legitimately geometrical constructions,and (2) the relationship between algebra and geometry during the 1618–1629 period.

2.2 Problems and Proposals

The most famous letter written to Beeckman in 1619 dates from 26 Marchof that year. In this letter Descartes announces his plan to expound an“entirely new science [scientia penitus nova], by which all problems thatcan be posed, concerning any kind of quantity, continuous or discrete, canbe generally solved” (AT X, 156). As he elaborates on how this newscience will proceed, Descartes clarifies that his solutions to the problemsof 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 itsown nature as for example, in arithmetic some questions are

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We notice in Descartes' remarks concerning geometry in particular thatthe “entirely new science” he proposes will provide an exhaustiveclassification for problem-solving, where each of his three classes isdetermined by the curves needed for solution. This suggests an importantoverlap between Descartes' three classes of geometrical problems andPappus's three classes, which, recall, were separated based on the types ofcurves required for solution: Planar problems are solvable by straightedgeand compass, solid problems by conics, and line-like problems by morecomplicated curves that have an “inconstant and changeable origin.”However, there is also a significant difference between theirclassifications insofar as Descartes strongly suggests that those problemsthat require “imaginary” curves for their solution do not have a

resolved by rational numbers, others only by surd [irrational]numbers, and others finally can be imagined but not solved. Soalso I hope to show for continuous quantities that some problemscan be solved by straight lines and circles alone; others only byother curved lines, which, however, result from a single motionand can therefore be drawn with new types of compasses, whichare no less exact and geometrical, I think, than the common onesused to draw circles; and finally others that can be solved bycurved lines generated by diverse motions not subordinated to oneanother, which curves are certainly only imaginary such as therather well-known quadratrix. I cannot imagine anything thatcould not be solved by such lines at least, though I hope to showwhich questions can be solved in this or that way and not anyother, so that almost nothing will remain to be found in geometry.It is, of course, an infinite task, not for one man only. Incrediblyambitious; but I have seen some light through the dark chaos ofthe science, by the help of which I think all the thickest darknesscan be dispelled (AT X, 156–158; CSMK 2–3; translation fromSasaki 2003, 102).



legitimately geometrical solution. Namely, just as some problems ofarithmetic “can be imagined but not be solved,” so too in geometry, thereis a class of problems that require curves that are “certainly onlyimaginary,” i.e., curves generated by “diverse motions,” and thus that arenot geometrical in a proper sense. In this respect, Descartes is movingfrom Pappus's descriptive classification to a normative one that separatesgeometrical curves from non-geometrical curves, and therebydistinguishes problems with a geometrical solution from those that do nothave a legitimate geometrical solution. Just as importantly, we see inDescartes' letter his attempt to expand the scope of legitimate geometricalconstructions beyond straightedge and compass by appealing to themotions needed to construct a curve. Specifically, as we see in thepassage above, Descartes relies on the “single motions” of his “new typesof compasses, which [he says] are no less exact and geometrical…thanthe common ones used to draw circles” in order to mark out a new classof problems that have legitimate geometrical solutions.

In his 26 March 1619 letter to Beeckman, Descartes does not elaborate onthe “new types of compasses” to which he refers; he simply reports toBeeckman in the early portions of the letter that he has, in a short time,“discovered four conspicuous and entirely new demonstrations with thehelp of my compasses” (AT X, 154). Fortunately, more details aboutthese compasses and Descartes' demonstrations are included inCogitationes privatae, or Private Reflections (ca. 1619–1620), a text inwhich Descartes applies three different “new compasses” (often referredto by commentators as “proportional compasses”) to the problems of (1)dividing a given angle into any number of equal parts, (2) constructingthe roots of three types of cubic equations, and (3) describing a conicsection. In the first two cases, as Descartes treats the angular section andmean proportional problems, the compasses on which he relies are used togenerate a curve that will solve the problem at hand.

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Figure 1

Figure 2

For instance, to solve the angular section problem, Descartes begins bypresenting 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 theinstrument such that they are a distance a from O and are pair-wisehinged 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, wegenerate the curve KLM (figure 2). As Descartes has it, we can constructthe curve KLM on any given angle by appeal to the instrument describedabove, because the angle we are trisecting plays no role in theconstruction of KLM. And once the curve KLM is constructed, the givenangle can be trisected by means of some basic constructions with straightlines and circles. In this respect, the curve KLM is, for Descartes, themeans for solving the angle trisection problem, and moreover, histreatment 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, ormore equal parts. (I borrow my treatment of this construction fromDomski 2009, 121, which is itself indebted to the presentation in Bos2001, 237–239.)

Figure 3: Mesolabe

A similar approach is taken by Descartes when he treats the problem ofconstructing mean proportionals, where in this case, he appeals to hisfamous mesolabe compass, an instrument that is used in Book Three ofthe La Géométrie to solve the same problem. As in 1637, this compass isused to construct curves (the dotted lines in figure 3) that allow us toidentify the mean proportionals between any number of given line

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segments. And as Viète before him, in the Private Reflections Descartesuses this construction of mean proportionals to identify the roots ofstandard-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 toBeeckman: His new compasses generate curves by the single motion of adesignated arm of the compass, and thus, the curves generated in thismanner meet the standard of geometrical intelligibility—the standard bywhich to distinguish geometrical from imaginary curves—that is alludedto in the brief outline of the “entirely new science” that Descartesenvisions. That such motions are completed by instruments does notthreaten the constructed curve's geometrical status. (As we saw above,Viète had leveled this charge against the instrumental, pointwiseconstructions provided by Clavius.) And moreover, we already notice inthe mathematical research of 1619 Descartes' focus on the intelligibility ofmotions 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 relationshipbetween algebra and geometry that will be crucial to the program ofgeometrical analysis presented in Book One of La Géométrie, where atthis early stage of his research, Descartes, like his contemporaries, isexploring the application of geometry to algebraic problems. For instance,as pointed out above, Descartes uses the construction of meanproportionals to solve algebraic equations in the Private Reflections, andin the same text he also shows an interest in the geometricalrepresentation of numbers and of arithmetical operations. The sameinterest appears again in the later Progymnasmata de solidorum elementisexcertpa ex manuscript Cartesii (Preliminary exercises on the elements ofsolids extracted from a manuscript of Descartes, ca. 1623), a text in whichDescartes offers a geometrical representation of numbers and of four of



the five basic arithmetical operations (the four operations he treats areaddition, subtraction, multiplication, and division).

Though there is some dispute among commentators about Descartes' levelof expertise in algebra during this early 1619–1623 period (compare Bos2001, 245 with Sasaki 2003, 126), texts from 1628–1629 show Descartesmaking great advances in algebra in a relatively small amount of time.Two textual sources are of particular interest: (1) The specimen of algebragiven to and transcribed by Beeckman in 1628 upon Descartes' return toHolland, and (2) a text on the construction of roots for cubic and quarticequations given to Beeckman in early 1629.[2] In the Specimen, Descartespresents a rather basic problem-solving program (or schematism) foralgebra that relies on two-dimensional figures (lines and surfaces). Thetexts given to Beeckman several months later, which Descartes composedwhile in Holland, show a great advance over what's found in theSpecimen, since in these texts he appeals to conic sections (or solids) inhis problem-solving regime. For instance, Descartes constructs two meanproportionals by the intersection of circle and parabola (a method he haddiscovered around 1625 according to Bos 2001, 255). More impressively,in a different text from this same period, Descartes offers a method forconstructing all solid problems, i.e., for solving all third- and fourth-degree equations.

While some of the results from this period are connected with theproblem-solving program presented in the 1637 La Géométrie, Rabouin(2010) points out that it is still not clear whether Descartes discovered hismethods for solution using the techniques that are applied in 1637(Rabouin 2010, 456). As such, Rabouin urges us to resist the somewhatstandard reading of Descartes' early mathematical works according towhich there is a linear and teleological progression from the 1619pronouncement of an “entirely new science” to the groundbreakingprogram of La Géométrie (a reading found, for instance, in Sasaki 2003,

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especially 156–176). According to Rabouin, it is not until the early 1630s,when Descartes engages with the Pappus problem—what Bos alsoconsiders “the crucial catalyst” of Descartes' mature mathematicalresearches (Bos 2001, 283)—that he returns to his 1619 project to craft anew science of geometry that is grounded on a new classification ofcurves and problems. Following Rabouin, it is at this point of hismathematical career that Descartes more clearly sees just how crucial theinterplay of algebraic equations and geometry could be for a generalprogram of geometrical problem-solving.

3. La Géométrie (1637)

In late 1631, the Dutch mathematician Golius urged Descartes to considerthe solution to the Pappus problem. Unlike the geometrical problems thatoccupied Descartes' early researches, the Pappus problem is a locusproblem, i.e., a problem whose solution requires constructing a curve—the “Pappus curve” according to Bos's terminology—that includes all thepoints that satisfy the relationship stated in the problem. Generallyspeaking, the Pappus Problem begins with a given number of lines, agiven number of angles, a given ratio, and a given segment, and the taskis identify a curve such that all the points on the curve satisfy a specifiedrelation to the given ratio. For instance, in the most basic two-line PappusProblem (figure 4), we are given two lines (L1, L2), two angles (θ1, θ2),and a ratio β. We designate d1 to be the oblique distance between a pointP and L1 such that P creates θ1 with L1, and we designate d2 to be theoblique distance between a point P in the plane and L2 such that P createsθ2 with L2. The problem is to find all points P such that d1 : d2 = β. Inthis case, all the sought after points P will lie along two straight lines, oneline to the right of L1 and the other to the left of L1. (See figure 5 forBos'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 lineversions of the problem (i.e., the versions of the problem in which webegin with three or four given lines and angles) as well as Apollonius'ssolution to the six-line case, which relies on his theory of conics and thetransformation of areas to construct the locus of points (Pappus, 118–123). However, Pappus does not treat the general (n-line) case, and this isthe advance of the solution Descartes achieves in 1632, a solutionpublished in La Géométrie, where he claims that, unlike the ancients, hehas found a method to successfully “determine, describe, [and] explain thenature of the line required when the question [of the Pappus Problem]involves a greater number of lines” (G, 22). And as Descartes reports toMersenne in 1632, he could not have found his general solution withoutthe help of algebra:

I must admit that I took five or six weeks to find the solution [tothe Pappus Problem]; and if anyone else discovers it, I will notbelieve that he is ignorant of algebra (To Mersenne 5 April 1632;AT I, 244; CSMK, 37).

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According to Bos, consideration of the general Pappus Problem “provided[Descartes], in 1632, with a new ordered vision of the realm of geometry

Figure 5: The General Pappus Problem (from Bos 2001,Fig.19.1, 273)

Given: a Line Li in the plane, n angles θi, a ratio β, a line segmenta. For an point P in plane, let d be the oblique distance between Pand Li such that P creates θi with Li.

Problem: Find the locus of points P such that the following ratiosare equal to the given ratio β:

For 3 lines: (d1)2 : d2d3For 4 lines: d1d2 : d3d4For 5 lines: d1d2d2 : ad4d5For 6 lines: d1d2d3 : d4d5d6

In general,

For an even 2k number of lines: d1…dk : dk+1…d2kFor an uneven 2k+1 number of lines: d1…dk+1 : adk+2…d2k+1



and it shaped his convictions about the structure and the proper methodsof geometry” (Bos 2001, 283). The best evidence we have of the impactthe problem had on Descartes' approach to geometry is La Géométrieitself: in La Géométrie, the Pappus problem is given pride of place asDescartes details his “geometrical calculus” and demonstrates the powerof his novel program for solving geometrical problems. It is treated inBook One, as Descartes explains his geometrical analysis, and then againin Book Two, where Descartes offers the synthesis, i.e., the geometricaldemonstration, of his solution to the Pappus Problem in n-lines, ademonstration which relies on the famous distinction between“geometric” and “mechanical” curves that begins this part of the work.

3.1 Book One: Descartes' Geometrical Analysis

Book One of La Géométrie is entitled “Problems the construction ofwhich requires only straight lines and circles,” and it is in this openingbook that Descartes details his geometrical analysis, that is, howgeometrical problems are to be explicated algebraically. In this respect,what we find in Book One is similar to the algebraic elaboration ofgeometrical problems presented by Viète in his 1594 Supplement ofgeometry as he explains the stage of exegetics. That said, Descartes'approach to analysis rests on innovations in notation and formalism aswell as in the merging of geometry and arithmetic which move himbeyond Viète's analysis, lending some credence to Descartes' remark toMersenne that, in La Géométrie, his program for geometry begins whereViète's left off (To Mersenne, December 1637, AT I, 479; CSMK 77–79).

Book One commences with the geometrical interpretation of algebraicoperations, which, we saw above, Descartes had already explored in theearly period of his mathematical research. However, what we arepresented in 1637 is, as Guicciardini aptly describes, a “giganticinnovation” both over Descartes' previous work and the work of his

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contemporaries (Guicciardini 2009, 38). On the one hand, Descartesoffers a geometrical interpretation of root extraction and thus treats fivearithmetical operations (as opposed to the four operations of addition,subtraction, multiplication, and division that were treated in his earlywork). On the other hand, and more significantly, his treatment relies onan interpretation of arithmetical operations according to which theseoperations are taken to be closed operations on line segments.Traditionally, for instance, the product of two segments a * b wasinterpreted as a rectangle, but for Descartes, the product is interpreted as asegment. This allows Descartes to translate geometrical problems intoequations (that include products such as a * b) and treat each term of theequation as similar in kind. Finally, Descartes uses a new exponentialnotation as he sets forth equations of multiple terms in Book One, and thisnotation, which replaces the traditional cossic notation of early modernalgebra, allows Descartes to tighten the connection between algebra andgeometry, and more specifically, between the algebraic representation ofcurves through equations with the geometrical classification andgeometrical solution of stated problems (as we will see more clearlybelow in section 3.2).

With his new geometrical interpretation of the five basic arithmeticaloperations at his disposal, Descartes proceeds to describes how, in thestage of geometrical analysis, one is to give an algebraic interpretation ofa geometrical problem:

If, then, we wish to solve any problem, we first suppose thesolution already effected, and give names to all the lines that seemneedful for its construction,—to those that are unknown as well asto those that are known. Then, making no distinction betweenunknown and unknown lines, we must unravel the difficulty in anyway that shows most naturally the relations between these lines,until we find it possible to express a single quantity in two ways.



We notice that the key to Descartes' analysis is to make no distinctionbetween the known and unknown quantities in the problem: Both kinds ofquantities are granted a variable (generally, a, b, c… for known quantitiesand x, y, z… for unknown quantities), and thus, we treat the unknowns asif their values were already found. Or, as Descartes puts it, we “supposethe solution already effected.” The task then is to reduce the problem toan equation (in contemporary terms, to a polynomial equation in twounknowns) that expresses the unknown quantity, or quantities, in terms ofthe known quantities. For instance, take the following problem:[3]

Figure 6:

In this example, we are dealing with a determinate problem, i.e., aproblem to which there are a finite number of solutions, and we cantherefore reduce the problem to a single equation that expresses theunknown quantity in terms of the known quantities. However, asDescartes points out, there are also indeterminate problems that involve aninfinite number of solutions. (Locus problems, such as the Pappus

This will constitute an equation, since the terms of one of thesetwo expressions are together equal to the terms of the other (G, 6–9).

Given a line segment AB containing point C (see figure 6), theproblem is to produce AB to D such that the product AD*DB isequal 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 tofind BD such that AD*DB = (CD)2 is algebraically equivalent tofinding 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 = b2 / (a—b).

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Problem, are of this sort, because the solution includes the infinitely manypoints that lie along a curve.) When dealing with an indeterminateproblem, Descartes instructs us that “we may arbitrarily choose lines ofknown length for each unknown line to which there corresponds noequation” (G, 9), i.e., we are to set the unknown lines as obliquecoordinates that have a stated value. We then generate several equationsthat express the unknown quantities in terms of one or more knownquantities, and solve the equations simultaneously. This is precisely theapproach that Descartes takes as he treats the Pappus Problem in BookOne.

Figure 7: The Four-Line Pappus Problem in Book One (G, 27)

Descartes begins with consideration of the problem when we are giventhree or four lines, which, borrowing from Guicciardini (2009), can bestated as follows (see figure 7):

Having three or four lines given in position, it is required to findthe locus of points C from which drawing three or four lines to the



In Book One, Descartes applies his geometrical analysis to the four-linecase of the Pappus problem. He begins by designating two given linesegments (of unknown length) AB and BC as oblique coordinates x and y,respectively, such that all other lines needed to solve the problem will beexpressed in terms of x and y.[4] Then, by considering the angles given inthe problem and the properties of similar triangles, he generates analgebraic expression of the sought after points C in terms of the twounknowns x and y and the known quantity z (where z designates the ratiogiven in the problem) (G, 29–30).

Importantly, the analytic method that Descartes uses in the four-line caseis generalized to apply to the general, n-line version of the PappusProblem. That is, Descartes' claim is that no matter how many lines andangles are given in the problem, it is possible, by means of his analyticmethod, to express the sought after points C in terms of two unknownquantities (in contemporary terms, to reduce the problem to a polynomialequation in two unknowns) (G, 33). As a result, for any n-line version ofthe Pappus Problem, we can generate values for C and construct thesought after Pappus curve by assigning different values to x and y, andthereby describe the curve in a pointwise manner. As Descartes puts it,

three or four lines given in position and making given angles witheach one of the given lines the following condition holds: therectangle [or product] of two of the three lines so drawn shall beara given ratio to the square of the third (if there be only three), or tothe rectangle [or product] of the other two (if there be four)(Guicciardini 2009, 54; based on G, 22).

Furthermore, to determine the point C, but one condition isneeded, namely, that the product of a certain number of lines shallbe equal to, or (what is quite as simple), shall bear a given ratio tothe product of certain other lines. Since this condition can be

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The result of Descartes' analysis, as indicated by the remarks above, isthat the curve that includes the sought after points C can be pointwiseconstructed by using ruler and compass to solve for the roots of a second-degree equation in two unknowns. He then generalizes this result andclaims that the solution points for any problem that can be reduced to asecond-degree equation can be constructed by ruler and compass. Ifinstead a problem is reduced to an equation of third or fourth degree, thepoints are constructed by conics, and if a problem is reduced to anequation of fifth or sixth degree, the points are constructed by a curve thatis “just one degree higher than the conic sections” (G, 37). In otherwords, Descartes' claim is that if a problem can be reduced to a singleequation of degree not higher than six, in which the unknown quantity orquantities are expressed in terms of a known quantity, then the roots ofthe equation be constructed by straightedge and circle, or by conic, or by amore complicated curve that does not have degree higher than four. Basedon this result, Descartes suggests a way to generalize further and solve then-line Pappus Problem, for no matter how many given lines and angleswith which a Pappus Problem begins, it will be possible to reduce the

expressed by a single equation in two unknown quantities, we maygive any value we please to either x or y and find the value of theother from this equation. It is obvious that when not more thanfive lines are given, the quantity x, which is not used to expressthe first of the lines can never be of degree higher than the second.

Assigning a value to y, we have x2 = ± ax ± b2, and therefore xcan be found with ruler and compasses, by a method [forconstructing roots] already explained. If then we should takesuccessively an infinite number of different values for the line y,we should obtain an infinite number of values for the line x, andtherefore an infinity of different points, such as C, by means ofwhich the required curve can be drawn (G, 34).



problem to an equation and then pointwise construct the roots of theequation, i.e., the sought after points C of the problem (G, 37).

What Descartes achieves here by means of his geometrical analysis is nodoubt significant. He has outlined a way of solving the Pappus Problemfor any number of given lines. However, questions about proving thePappus Problem solved still linger come the end of Book One. As inViète's analysis, Descartes has shown that a solution to the generalproblem exists but the algebraic elaboration of the problem does not untoitself give a clue to how we are to geometrically construct the curve thatsolves 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 curvesthat include the roots are constructed pointwise. But this leaves us thequestion: Are the Pappus curves of Book One legitimately geometrical?That is, can the curves that solve the n-line version of the Pappus Problemthemselves be constructed by legitimately geometrical methods? This isan issue broached in Book Two, the main focus of which is how to enacta synthesis, or construction, of a geometrical problem.

3.2 Book Two: The Classification of Curves and GeometricalSynthesis

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 theprogram of La Géométrie as well as the attention this distinction hasdrawn from commentators, it is worth examining the proposals made inthe opening pages of Book Two with some care.

Descartes begins with reference to the ancient classification of problemsand offers his interpretation of how ancient mathematicians distinguishedcurves that could be used in the solution to geometrical problems from

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those that could not:

Descartes implies that the terms “mechanical” and “non-geometrical”were synonymous in ancient mathematics; however, it is not at all clearthat this was the intended meaning of the term “mechanical.” Namely, itis not clear that the classification of curves into “geometrical” and“mechanical” was intended to serve as a normative claim concerning thelegitimacy of a curve's use in geometrical problem-solving or simply as adescriptive moniker that captures the different ways in which curves wereconstructed (see Molland 1976 on this issue; see section 2.2 above forDescartes' blending of the descriptive and the normative in his 1619

The ancients were familiar with the fact that the problems ofgeometry may be divided into three classes, namely, plane, solid,and linear problems. This is equivalent to saying that someproblems require only circles and straight lines for theirconstruction, while others require a conic section and still othersmore complex curves. I am surprised, however, that they did notgo further, and distinguish between different degrees of thosemore complex curves, nor do I see why they called the lattermechanical, rather than geometrical. If we say that they are calledmechanical because some sort of instrument has to be used todescribe them, then we must, to be consistent, reject circles andstraight lines, since these cannot be described on paper without theuse of compasses and a ruler, which may also be termedinstruments. It is not because the other instruments, being morecomplicated than the ruler and compass, are therefore lessaccurate, for if this were so they would have to be excluded frommechanics, in which accuracy of construction is even moreimportant than in geometry. In the latter, exactness of reasoningalone is sought, and this can surely be as thorough with referenceto such lines as to simpler ones (G, 40–44).



proposal for a “new science” of geometry).

Descartes' reading of the ancients aside, important for understanding hisown peculiar interpretation of geometrical curves is the distinction hedraws between the “accuracy of construction” of a curve, which herenders an issue for mechanics, and the “exactness of reasoning,” whichhe deems as the sole requirement for accepting a curve as legitimatelygeometrical. In making this claim, Descartes is carving out a unique placefor his notion of geometrical curves: He abandons the “accuracy ofconstruction” criterion that Clavius adopted in his early works to render acurve acceptable in geometrical problem-solving and also the claimforwarded by Viète that instrumentally-constructed curves were not to beconsidered geometrical (see section 1.1 above). As Descartes' presentationimplies, both these sorts of criteria confuse issues of mechanics with the“exactness of reasoning” that is the sole concern of geometry. Thus, asBook Two continues, Descartes reiterates that to determine thegeometrical status of a curve we must lay our focus on issues of exact andclear reasoning and, specifically, on the question of whether a curve canbe constructed by exact and clear motions. After presenting the postulatethat “two or more lines can be moved, one upon the other, determining bytheir intersection other curves,” Descartes explains,

It is true that the conic sections were never freely received intoancient geometry, and I do not care to undertake to change namesconfirmed by usage; nevertheless, it seems very clear to me that ifwe make the usual assumption that geometry is precise and exact,while mechanics is not; and if we think of geometry as the sciencewhich furnishes a general knowledge of the measurement of allbodies, then we have no more right to exclude the more complexcurves than the simpler ones, provided they can be conceived of asdescribed by a continuous motion or by several successivemotions, each motion being completely determined by those which

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We see in these remarks that the precision and exactness of geometry isintimately tied with the geometer's consideration of motions that can beprecisely and exactly traced. Namely, the geometer is justified in usingsimple curves as well as more complex curves, so long as the constructionof these curves proceeds by “precise and exact” motions. Descartesclarifies how a complex curve “can be conceived of as described by acontinuous motion or by several successive motions, each motion beingcompletely determined by those which precede” by presenting themesolabe compass that he first developed in 1619:

precede; for in this way an exact knowledge of the magnitude ofeach is always obtainable (G, 43).

Consider the lines AB, AD, AF, and so forth, which we maysuppose to be described by means of the instrument YZ [Figure8]. This instrument consists of several rulers hinged together insuch a way that YZ being placed along the line AN the angle XYZcan be increased or decreased in size, and when its sides aretogether, the points B, C, D, E, F, G, H, all coincide with A; but asthe size of the angle is increased, the ruler BC, fastened at rightangles to XY at the point B, pushed toward Z the ruler CD whichslides along YZ always at right angles. In a like manner, CDpushes DE which slides along YX always parallel to BC; DEpushes EF; EF pushes FG; FG pushes GH, and so on. Thus wemay imagine an infinity of rulers, each pushing another, half ofthem making equal angles with YX and the rest with YZ.

Now as the angle XYZ is increased, the point B describes thecurve AB, which is a circle; while the intersections of the otherrulers, namely, the points D, F, H describe the other curves, AD,AF, AH, of which the latter are more complex than the first andthis more complex than the circle. Nevertheless I see no reason



Figure 8: Mesolabe

A couple points are worth emphasizing. First, Descartes presents the morecomplex curves generated by his compass as described by motions thatcan be as “conceived as clearly and distinctly” as the motions required toconstruct the more simple circle. And because of the clear and distinctmotions needed for their construction, these curves are legitimatelygeometrical. That is, consistent with Descartes' general criterion forconstructing geometrical curves, these complex curves can be used in thesolution of geometric problems. Second, we see that although Descartestakes care to distinguish the concerns of geometry from those ofmechanics, he does not steer away from the construction of curves bymeans of instruments. Although instrumental constructions aremechanical constructions, they can nonetheless give rise to geometricalcurves precisely because the motions of the instruments are “clearly and

why the description of the first cannot be conceived as clearly anddistinctly as that of the circle, or at least as that of the conicsections; or why that of the second, third, or any other that can bethus described, cannot be as clearly conceived of as the first: andtherefore I see no reason why they should not be used in the sameway in the solution of geometric problems (G, 44–47).[5]

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distinctly” conceived. That the motions are generated by instruments doesnot render the resultant curve non-geometrical. (For more on the use ofinstruments in La Géométrie, see Bos 1981.)

In a similar vein, curves that are non-geometrical by Descartes' standardare curves that require more complicated, less clear and distinct motionsfor their construction. He explains:

Descartes explicitly names the spiral and quadratrix as those curveswhose construction “must be conceived of as described by two separatemovements whose relation does not admit of exact determination.” Laterin Book Two he clarifies why such descriptions fail to be clearly anddistinctly conceived:

Given these remarks, the fundamental problem with the spiral, thequadratrix, and “lines that are like strings” is that their constructionrequires consideration of the ratio, or relation, between a circle and

Probably the real explanation of the refusal of ancient geometersto accept curves more complex than the conic sections lies in thefact that the first curves to which their attention was attractedhappened to be the spiral, the quadratrix, and similar curves,which really do belong only to mechanics, and are not among thecurves that I think should be included here, since they must beconceived of as described by two separate movements whoserelation does not admit of exact determination (G, 44).

geometry should not include lines that are like strings, in that theyare sometimes straight and sometimes curved, since the ratiosbetween straight and curved lines are not known, and I believecannot be discovered by human minds, and therefore noconclusion based upon such ratios can be accepted as rigorous andexact (G, 91).



straight line. Consider the spiral. As we saw above, its constructioninvolves two uniform motions: the uniform rectilinear motion of a pointalong a segment and the uniform circular motion of the segment around apoint. These two motions must simultaneously be considered in order forthe moving point's path to describe the spiral, and this, for Descartes, iswhat is ultimately problematic. The human mind can think aboutsimultaneous rectilinear and circular motions, but it cannot do so with theclarity and distinctness required to meet the exact and rigorous standardsof geometry. (This claim is not without its problems, which will bediscussed in section 3.3 below.)

After presenting his construction criterion for geometrical curves,Descartes develops his novel connection between geometricalconstruction and the algebraic representation of these curves. Whereas inBook One Descartes details how to use algebra to establish that a solutiona geometrical problem exists, here, in Book Two, Descartes proposes astronger connection between algebra and geometry and famously claimsthat any legitimately geometrical curve can be represented by an equation:

He then proceeds to classify these “geometric” curves according to thedegree of their corresponding equations, claiming:

I could give here several other ways of tracing and conceiving aseries of curved lines, each curve more complex than anypreceding one, but I think the best way to group together all suchcurves and then classify them in order, is by recognizing the factthat all the points of those curves which we may call “geometric,”that is, those which admit of precise and exact measurement, mustbear a definite relation to all points of a straight line, and that thisrelation must be expressed means of a single equation (G, 48).

If [a curve's] equation contains no term of higher degree than therectangle [product] of two unknown quantities, or the square of

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The same point is made later in Book Two, where Descartes emphasizesthat “no matter how we conceive a curve to be described, provided it beone of those which I have called geometric,” it will always be possible tofind an equation determining all of the curve's points (G, 56). Hereiterates that geometric curves can be classified according to theirequations but also points out that within a specific class, a curves'simplicity should be ranked according to the motions required forconstruction. For instance, although the circle belongs to the same class asthe ellipse, hyperbola, and parabola, these latter curves are “equallycomplex” whereas the circle “is evidently a simpler curve” and will thusbe more useful in the construction of problems (G, 56).

As in Book One, Descartes uses the Pappus Problem to illustrate thepower of his geometrical calculus, where in Book Two, his aim is to showhow his algebraic classification of curves makes it easy “to demonstratethe 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 thegeneral Pappus Problem are legitimately geometrical curves, i.e., to showthat the Pappus curves meet the exact and rigorous standards ofgeometrical construction that he has just laid out. Descartes' discussion ofthe Pappus Problem in Book Two begins as follows:

one, the curve belongs to the first and simplest class, whichcontains only the circle, the parabola, the hyperbola, and theellipse; but when the equation contains one or more terms of thethird or fourth degree, in one or both of the two unknownquantities (for it requires two unknown quantities to express therelation between two points) the curve belongs to the second class;and if the equation contains a term of the fifth or sixth degree ineither or both of the unknown quantities the curve belongs to thethird class, and so on indefinitely (G, 48).



As indicated in the passage above, Descartes establishes in Book Twothat Pappus curves fall into the specified classes of geometric curves hehas designated, where the class into which a Pappus curve falls dependson the number of lines given in the problem and thus, on the degree of theequation to which the problem is reduced. For instance, when Descartestreats the four-line Pappus Problem in Book Two, he shows that, byvarying the coefficients of the second degree equation to which theproblem has been reduced (through the analysis of Book One), we canconstruct either a circle, parabola, hyperbola, or ellipse (G, 59–80). Thatis, he shows that the Pappus curve that solves the four-line problem iseither a circle or one of the conic sections, the very “geometric” curvesthat he has grouped into Class I.

3.3 The Tensions and Limitations of Descartes' GeometricalCalculus

In two stages, then, Descartes has demonstrated the solution to the general

Having now made a general classification of curves, it is easy forme to demonstrate the solution which I have already given of theproblem of Pappus. For, first, I have shown [in Book One] thatwhen there are only three or four lines the equation which servesto determine the required points is of the second degree. It followsthat the curve containing these points [i.e., the Pappus curve] mustbelong to the first class, since such an equation expresses therelation between all points of curves of Class I and all points of afixed straight line. When there are not more than eight given linesthe equation is at most a biquadratic, and therefore the resulting[Pappus] curve belongs to Class II or Class I. When there are notmore than twelve given lines, the equation is of the sixth degree orlower, and therefore the required curve belongs to Class III or alower class, and so on for other cases (G, 59).

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Pappus Problem. In Book One he offers his algebraic analysis of theproblem, and in Book Two he claims to provide the synthesis (ordemonstration) that the curves that solve the general problem arelegitimately geometrical curves which meet his stated standard forgeometrical exactness and precision. And with these two stagescompleted, Descartes claims to Mersenne six months after La Géométrieis published that his treatment of the general Pappus Problem is proof thathis new method for geometrical-problem solving is an improvement overthe methods of his predecessors:

As great as Descartes' confidence in his solution to the Pappus Problem,there are questions that surround his synthesis of the general problem inBook Two.

As indicated above, Descartes attempts to establish via his synthesis thatthe curves that solve the Pappus Problem are “geometric” by his ownstated standard, that is, that the Pappus curves are constructible by the

I do not like to have to speak well of myself, but because there arefew people who are able to understand my Geometry, and sinceyou will want me to tell you what my own view of it is, I think itappropriate that I should tell you that it is such that I could notwish to improve it. In the Optics and the Meteorology I merelytried to show that my method is better than the usual one; in myGeometry, however, I claim to have demonstrated this. Right atthe beginning I solve a problem which according to the testimonyof Pappus none of the ancients managed to solve; and it can besaid that none of the moderns has been able to solve it either,since none of them has written about it, even though the cleverestof them have tried to solve the other problems which Pappusmentions in the same place as having been tackled by the ancients(To Mersenne, end of December 1637; AT 1, 478; CSMK, 77–78).



“precise and exact” motions needed to construct genuinely geometriccurves. 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 distinctas he constructs the Pappus curves that solve the problem (in this case,the circle, parabola, hyperbola, and ellipse). Rather, he relies onApollonius's theory of conics, which requires that a cone be cut at adesignated point in the plane, and as Bos remarks, this Apolloniantechnique for constructing conics “is not a method of construction thatimmediately presents itself to the mind as clear and distinct” (Bos 2001,325). Specifically, since it was not evident to mathematicians at the timewhether constructions that required locating a cone in the plane met theexact and rigorous standards of geometrical reasoning, Descartes'treatment of the Pappus curves in this four-line case does notconvincingly demonstrate their “geometric” status. Later in Book Two,when he treats the five-line Pappus Problem, matters get morecomplicated.

Recall that in addition to his emphasis on the “precise and exact” motionsthat can be used to describe legitimately geometrical curves, Descartesalso claims that these curves “can be conceived of as described by acontinuous motion or by several successive motions.” As such, we wouldreasonably expect that the geometrical construction of these curves shouldnot proceed pointwise in the manner of Book One, where Descartesconstructed the Pappus curves by solving the equations to which theproblem had been reduced. However, when Descartes treats the five-linePappus Problem in Book Two, he in fact offers a pointwise constructionof the Pappus curve. He then remarks that the pointwise construction ofthis “geometric” Pappus curve is importantly different from the pointwiseconstruction of non-geometrical, “mechanical” curves:

It is worthy of note that there is a great difference between this

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The suggestion from Descartes is that when we pointwise construct ageometric curve, we can identify any possible point on the curve, andimmediately after the above remarks, he proceeds to equate curvesconstructed in this manner with curves that could possibly be constructedby continuous motions: “this method of tracing a curve by determining anumber of its points taken at random applies only to curves that can begenerated 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 programof La Géométrie: (1) Descartes can establish that the Pappus curves hehas pointwise constructed are in fact “geometrical” and thereby completehis synthesis (or demonstration) of the general Pappus Problem, and (2)he can maintain a boundary between intelligible “geometric” curves andunintelligible “mechanical” curves. Without a clear indication of why thepointwise construction of a Pappus curve is “geometric,” Descartes wouldhave to allow “mechanical” curves such as the spiral and quadratrix intothe domain of geometrical curves, since these curves can also bepointwise constructed. Recall for instance Clavius's pointwiseconstruction of the quadratrix. According to Clavius's description, webegin with a quadrant of a circle and then identify the points ofintersection between segments that bisect the quadrant and segments thatbisect the arc of the quadrant (see figure 9). That is, we identify the

method in which the [Pappus] curve is traced by finding severalpoints upon it, and that used for the spiral and similar curves. Inthe latter, not any point of the required curve may be found atpleasure, but only such points as can be determined by a processsimpler than that required for the composition of the curve…Onthe other hand, there is no point on these [“geometric”] curveswhich supplies a solution for the proposed problem that cannot bedetermined by the method I have given (G, 88–91).



several intersecting points of segments which are constructible bystraightedge and compass, and then, to generate the quadratrix, weconnect the intersecting points, which are evenly spaced along the soughtafter curve. Why is such a pointwise construction not “geometric”?Because, according to Descartes, if we proceed as Clavius does, “not anypoint of the required curve may be found at pleasure.” Specifically, giventhe restrictions of Euclidean construction, we are only able to divide thegiven arc into 2n parts. As such, what Descartes suggests is that it is notpossible to divide the arc any way we please, and we cannot thereforelocate any arbitrary point along the curve by use of pointwiseconstruction. In the case of the “geometric” curves, however, we can findany arbitrary point on the curve by appeal to the equations correspondingto the problem; or borrowing Bos's terminology, Descartes is claimingthat “geometric” curves, and the Pappus curves in particular, can begenerated by “generic” pointwise constructions.

Figure 9

While consideration of Clavius's construction of the quadratrix offerssome reason to accept Descartes' distinction between the different sorts ofpointwise constructions, there remains the controversial claim that curvesdescribed by “generic” pointwise constructions are curves that can beconstructed by continuous motion. This identification allows Descartes to

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establish Pappus curves as “geometric” curves, but he offers no proof ofthe identity, and thus, there is question of whether Descartes has in factdemonstrated that the Pappus curves are “geometric” by his ownstandards. (See Grosholz 1991 and Domski 2009 for alternative ways ofaddressing this tension.)

There is a further question surrounding Descartes' criterion for“geometric” curves. As we have seen above, Descartes' explicit concernin Book Two is to offer a standard for geometrical curves that is boundwith intelligible, clear and distinct motions needed for their construction.However, Mancosu (2007) has recently offered a compelling case thatbehind Descartes' explicit remarks in La Géométrie lies a morefundamental concern: To ensure that those curves mathematicians hadused to square the circle, such as the spiral and quadratrix which areexplicitly mentioned in Book Two, are rendered non-geometrical.Mancosu supports his case with evidence from Descartes' correspondencethat shows, for Descartes, it is in fact possible in some instances to clearlyand distinctly conceive the relation between a straight line and a circle, arelation he had considered inexact in La Géométrie. Namely, in a 1638letter to Mersenne, Descartes writes,

In La Géométrie, the relation between straight and curves lines wasconsidered inexact because, as Descartes put it, “the ratios betweenstraight and curved lines are not known, and I believe cannot be

You ask me if I think that a sphere which rotates on a planedescribes a line equal to its circumference, to which I simply replyyes, according to one of the maxims I have written down, that isthat whatever we conceive clearly and distinctly is true. For Iconceive quite well that the same line can be sometimes straightand sometimes curved, like a string (To Mersenne, 27 May 1638;AT 2, 140–141; translation from Mancosu 2007, 118).



discovered by human minds” (G, 91). That Descartes later admits clearlyand distinctly conceiving such a relation suggests, according to Mancosu,that the stated criteria for geometrical curves presented in La Géométriereveals only part of Descartes' mathematical agenda. A more completeportrait 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 inMancosu 2007, 120; see also Mancosu and Arana 2010 for furtherevidence in support of the position of Mancosu 2007).

Whether Descartes had the hidden agenda that Mancosu suggests, theexplicit claims used to define the program of problem-solving presentedin La Géométrie point to the limitations of Descartes' mathematics.Namely, as we have seen above, Descartes' primary focus is on a standardfor geometry's “exactness of reasoning” that is bound to clear and distinctmotions for construction and to the finite equations to represent curves soconstructed. There is thus no room in the program of La Géométrie to useinfinitesimals in the construction of curves or to treat curves representedby infinite equations, and as such, Descartes had eliminated the veryelements of mathematical and geometrical reasoning that made it possiblefor Newton and Leibniz to develop the calculus come the late seventeenthcentury. Nonetheless, given how quickly Descartes honed hismathematical skills and how quickly he developed his innovative programfor geometry, it seems safe to follow Descartes' self-assessment andmaintain some confidence that the calculus would have been in his reachhad he considered the infinitesimal and the infinite:

having determined as I did [in La Géométrie] all that could beachieved in each type of problem and shown the way to do it, Iclaim that people should not only believe that I have accomplishedmore than my predecessors but should also be convinced thatposterity will never discover anything in this subject which I could

Mary Domski


Bibliography

Beeckman, Issac, 1604–1634, Journal tenu par Isaac Beeckman de 1604a 1634, 4 volumes, introduction and notes by C. de Waard. La Haye:M. Nijhoff, 1939–1953.

Bos, Henk J.M., 1981, “On the representation of curves in Descartes'Géométrie,” Archive for History of Exact Sciences 24: 295–338.

–––, 2001, Redefining Geometrical Exactness: Descartes' Transformationof the Early Modern Concept of Construction. New York, Berlin,Heidelberg: Springer-Verlag.

Descartes, René, 1637, The Geometry of Rene Descartes with a facsimilieof the first edition, translated by David E. Smith and Marcia L.Latham. New York: Dover Publications, Inc., 1954. [cited as Gfollowed by page number]

–––, 1985, The philosophical writings of Descartes. Two volumes,translated by John Cottingham, Robert Stoothoff, and DugaldMurdoch. Cambridge: Cambridge University Press. [cited as CSMfollowed by volume and page number]

–––, 1987, Exercices pour les éléments des solides: Essai en complémentd'Euclide: Progymnasmata de solidorum elementis. Edition critiqueavec introduction, traduction, notes et commentaries par PierreCoatabel. Paris: PUF.

–––, 1991, The philosophical writings of Descartes: The correspondence.Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch,and Anthony Kenny. Cambridge: Cambridge University Press. [citedas CSMK followed by volume and page number]

–––, Ouevres de Descartes. Eleven Volumes, edited by Charles Adam andPaul Tannery. Paris: J. Vrin, 1996. [cited as AT followed by volumeand page number]

not have discovered just as well if I had bothered to look for it (ToMersenne, end of December 1637; AT 1, 478; CSMK, 78–79).



Domski, Mary, 2009, “The intelligibility of motion and construction:Descartes' early mathematics and metaphysics, 1619–1637,” Studiesin History and Philosophy of Science 40: 119–130.

Euclid, The Thirteen Books of the Elements, translated with introductionand commentary by Sir Thomas L. Heath. New York: DoverPublications, Inc., 1956, second unabridged edition.

Federico, Pasquale Joseph, 1982, Descartes on Polyhedra: A Study on theDe Solidorum Elementis. New York, Heidelgerg, Berlin: Springer.

Grosholz, Emily, 1991, Cartesian method and the problem of reduction.Oxford: The Clarendon Press.

Guicciardini, Niccolò, 2009, Isaac Newton on Mathematical Certaintyand Method. Cambridge, MA, London: The MIT Press.

Giusti, Enrico, 1992, “Algebra and geometry in Bombelli and Viète,”Bollettino di storia delle scienze matematiche 12: 303–328.

Hintikka, Jakko and U. Remes, 1974, The method of analysis, itsgeometrical origin and its general significance. Dordrecht: Reidel.

Kline, Morris, 1972, Mathematical Thought from Ancient to ModernTimes, three volumes. New York, Oxford: Oxford University Press.

Koyré, Alexandre, 1939, Études galiléennes. Paris: Hermann.Mahoney, Michael S., 1973, The Mathematical Career of Pierre de

Fermat (1601–1665). Princeton: Princeton University Press.Mancosu, Paulo, 2007, “Descartes and Mathematics,” in A Companion to

Descartes, Janet Broughton and John Carriero (eds.), Malden, MA:Blackwell, pp. 103–123.

Mancosu, Paulo and Andrew Arana, 2010, “Descartes and the cylindricalhelix,” Historia Mathematica 37: 403–427.

Merzbach, Uta C. and Carl C. Boyer, 2011, A History of Mathematics.Hoboken, NJ: John Wiley and Sons, Inc, 3rd edition.

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Mathematics. Boston Studies in the Philosophy of Science, Volume196. New York: Springer.

Panza, Marco, 2007, “What is new and what is old in Viète's analysisresituta and algebra nova, and where do they come from? Somereflections on the relations between algebra and analysis beforeViète,” Revue d'histoire des mathèmatiques 13: 365–414.

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Notes to Descartes' Mathematics

1. Given the primary goal to situate La Géométrie in the philosophicaldebates surrounding early modern mathematical practice, there will be nodiscussion of Book Three, the section of Descartes' work that stands as aprimary impetus in the development of modern algebra. For anilluminating treatment of Book Three, see Bos (2001), Chapter 27.

2. This is not to say that these more complicated curves were required forsolutions to these problems. See, for instance, Panza (2011), 59-61 forsolutions to the problem of angle trisection that did not rely on the spiral.

3. Note here the difficulty of mapping Viète's stages of exegetics andzetetics onto Pappus's stages of analysis and synthesis. Pappus claims that

How to cite this entry.Preview the PDF version of this entry at the Friends of the SEPSociety.Look up this entry topic at the Indiana Philosophy OntologyProject (InPhO).Enhanced bibliography for this entry at PhilPapers, with linksto its database.

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synthesis involves the reversal of analysis, but for Viète, there is no suchreversal. Rather, the analysis that is applied in the stage of exegeticsinvolves an elaboration of a problem in terms of equations, whereas thesynthesis in the stage of zetetic involves constructions in the plane, andimportantly, the analysis does not indicate how to construct the curverequired to complete the synthesis. A similar issue arises in Descartes' LaGéométrie (see section 3.2). As such, what we find in Viète (and later inDescartes) is that an essential ingredient of their early modern algebraicanalysis is treating what is sought after as known, which is accomplishedby the use of variables to represent both known and unknown quantities.

4. When Descartes presents Beeckman the specimen of algebra in 1628,he promises to supply his more complete Parisian Algebra at a later time.However, it is not clear whether Descartes actually gave Beeckman thecomplete Algebra, since Descartes reports to Mersenne in a letter from 25January 1638 that no one has a copy of his Algebra (AT I, 501). What iscertain is that he gave Beeckman at least some parts of that project inearly 1629, because they were transcribed by Beeckman in his Journal.

5. The problem I present here is one that van Schooten uses to clarifyDescartes' analytic procedure in his 1683 annotated Latin version of LaGéométrie, and it is glossed over by Smith and Latham in G, p. 9, Note12. It is worth noting that Smith and Latham's treatment of the problem issomewhat misleading. They claim that the problem illustrates Descartes'directive that we are to reduce a determinate geometrical problem to a setof equations, which we must solve simultaneously (see G, p. 9, Note 11).However, this directive from Descartes only applies when we are dealingwith multiple unknowns, in which case we establish an equation for eachunknown in the problem. When there is only one unknown, as in theexample above, there is only equation to which the problem must bereduced. Smith and Latham claim, in contrast, that we reduce the aboveproblem to x = b2 / (a – b) and also to (a + b + x)*(x) = (b + x)2.



However, these two equations are equivalent and thus, do not offer a setof equations that can be solved simultaneously.

6. In this case and throughout La Géométrie, Descartes uses obliquecoordinates that are intrinsic to the problem. That is, the coordinatesdesignate distances that are given naturally by the figures in the problem.In contemporary analytic geometry, we use typically use orthogonal axes(with x as the horizontal axis and y as the vertical axis) for our coordinatesystem, and these axes are, as it were, extrinsic to the problem.

7. The mesolabe compass is presented again at the opening of BookThree, where Descartes uses the compass to solve the problem ofconstructing mean proportionals, the same problem for which thecompass was used in 1619 (see section 2.2 and G, 152–157).

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