
\chapter{The Transmission of Greek Work on the Construction Problems to Renaissance Europe}
%\maketitle

The Golden Age of Greek mathematics had reached its twilight at the end of the third century BCE.  Over the next two centuries, Greece fell to Roman conquest, with many schools, texts, and intellectual ideas being destroyed and defeated.\footnote{Burton notes that the Romans were more interested in the practical use of mathematics for engineering purposes, disregarding the theory behind it (p. 205).}  Yet, during the third and fourth centuries CE, the mathematical field experienced a brief revival, where many commentators of Greek mathematics expounded on the works of their predecessors.  The majority of these commentators, including scholars such as Hero, Eutocius, and Proclus, contributed very little in the way of original work in their publications.\footnote{As Knorr asserts, ``the prospect of new and interesting discoveries was discouraging,'' (p.340).}  These men, therefore, are not regarded as having advanced the development of geometrical ideas or introducing new concepts on technical solutions.  They are, however, praised for the transmission of the textual works of their precursors, since most of our knowledge of ancient Greek mathematics stems from their commentary.  Additionally, these commentators offered evaluative analysis of mathematical works of both their time and earlier.\footnote{For example, Theon and Eutocius both contributed to the collection of editions of the works of Euclid, Archimedes, and Apollonius, three mathematical giants of Ancient Greece (Knorr, p.341).}  Through the lens of these commentators, modern scholars are able to glimpse the field of geometrical problem solving as it advanced through ancient times.  Nowhere is this more apparent than in the work of Pappus of Alexandria. 

Pappus is the most renowned mathematician of his time, spanning the late third and first half of the fourth century CE.  His most famous work is his \textit{Mathematical Collection}, a series of eight books, in which he originally intended to consolidate much of the geometrical advancements that had taken place in the previous centuries.\footnote{Burton describes that the first and second books of the \textit{Collection} are missing, while Pappus' commentary on Ptolemy's \textit{Almagest} is his only other surviving work (p.221).}  Yet Pappus also offers original proofs, theorems, and concepts, extending and clarifying results of earlier geometers.  One of his most interesting contributions is his classification of problems, where he attempts to assign geometric problems into categories based on the constructions required for their solutions.  In doing so, Pappus has helped ``preserve a shadow of ancient metamathematical thinking,'' where modern scholars can gain a better understanding of the context of the ancients' search for solutions.\footnote{Knorr, p.341.} 

Despite the significance of Pappus as a commentator on ancient geometry, there is little scholarship that coherently traces the \textit{Collection} from its fourth-century CE origin.  Gow notes that Pappus ``is not cited by any of his successors,'' while the works of others, like Ptolemy and Diophantus, were successfully transmitted through a ``continuous history of progress.''\footnote{Gow, p.308.  He goes on to say that ``no Indian or Arab ever studied Pappus or cared in the least for his style or his matter,'' (p.309).}  It is not until Commandino's 1589 Latin translation of the \textit{Collection} that allowed many early modern mathematicians to gain a greater insight of the scope, methods, and results of Greek mathematics.

In two nearly identical passages regarding the problems of the cube duplication and angle trisection, Pappus classifies geometric problems into three distinct categories: planar, solid, and line-like.  Planar problems were those that could be solved by straight lines and circles, and therefore just by the use of the Euclidean tools.  A solid problem, Bos translates, ``employs surfaces of solid figures, namely the conic surfaces.''\footnote{Knorr, p.341.}  Finally, line-like, or linear, problems involve more complex lines and curves than these Euclidean tools or conics provide, such as the quadratrix, spiral, and cissoids.  Several discrepancies in such a classification lead historians to suggest that this was largely Pappus' own categorization rather than that of previous mathematicians.\footnote{For example, Knorr cites that Apollonius is attributed to classifying loci as ``ephectic,'' ``diexodic,'' and ``anastrophic'' based on the dimension required to construct the locus (p.343).}  For example, from the modern point of view, if the planar class consists exclusively of lines having their origins in the plane, than it should also include the spiral, quadratrix, and conchoids.\footnote{The definition and constructions of these and other curves will be provided in Chapter 2.  For now, suffice it to say they are other means the Greeks developed to create curves other than a compass and straightedge.}  Pappus also is clear that the solid class refers only to conic sections, excluding the conic sections of spheres and cylinders.  Such inconsistencies with respect to terminology suggest that even Pappus himself did not appreciate some of these subtleties in the classification he laid out.
	
These discrepancies also imply that Pappus' influence would likely have been negligible if his \textit{Collection} had only provided these classifications.  Yet, there are two reasons as to why his work would become so influential.  First, Pappus provides an extensive discussion of geometric solutions by earlier mathematicians.  For many of these geometers, their work survives only in Pappus, and thus he provides essential information explaining the scope and content of Greek mathematics.  Second, he combines his classification with, as Bos states, a ``methodological precept: problems should be constructed with the means appropriate to their class.''\footnote{Bos, p.49.}  This, indeed, is where Pappus' originality is evident.  Previous classifications of problems, whether by Pappus or other geometers, were merely descriptive---problems were differentiated based on the method of construction.  Pappus, however, provides a normative claim to distinguish between problems, insisting that problems be solved with a solution appropriate to their class.

Ironically, there is also a discrepancy among different translations of this precept by historians, leading to significantly different interpretations not only of the passage itself, but also of its impact on the entire mathematical enterprise of problem solving.  Knorr translates the precept as follows:
\begin{quote}
``The following sort of thing somehow appears to be no small error to geometers: whenever a planar problem is found by someone via conic or linear [lines], and on the whole whenever [some problem] is solved from a class other than its own, for instance, the problem on the parabola in the fifth book of Apollonius' \textit{Conics} and the solid \textit{neusis} toward a circle assumed by Archimedes in the book \textit{On the Spiral}; for by using no solid one is able to find the theorem proved by him.''\footnote{Knorr, p.345}.
\end{quote}  

He goes on to call such a precept a ``rather timid pronouncement'' and virtually the only surviving statement that provides a formal preference for planar constructions over others.\footnote{Yet, Knorr describes, this pronouncement is ``almost invariably presented in modern accounts as the principal objective of problem solving throughout the ancient tradition,'' (p.345).}  If this precept were indeed timid, as Knorr suggests, it is unlikely such a restriction would have any prominent influence over the geometers, either in Antiquity or the early modern period.

Commandino, however, provides a much more dramatic translation of the original Greek text.  As Bos translates Commandino's passage:
\begin{quote}
Among geometers it is in a way considered to be a considerable \textit{sin} [emphasis added] when somebody finds a plane problem by conics or line-like curves and when, to put it briefly, the solution of a problem is of an inappropriate kind.\footnote{Bos, p.49. Bos also gives Commandino's Latin translation, in which he uses the word \textit{peccatum}  as a translation of the original Greek $\alpha \mu \alpha \rho \tau \eta \mu \alpha$, which Bos then translates as ``sin.''  Actually, the full semantic range of the Greek word also includes failure, fault, and error (see the Greek-English Lexicon by Liddell and Scott).  That Commandino chooses the term \textit{peccatum} or ``sin'' is perhaps a reflection of sixteenth-century Italy, when Latin was used almost exclusively by scholars and in the Church.} 
\end{quote}  

This translation, therefore, carries a much more compelling tone than Knorr's ``no small error.''  This ``strict directive'' would therefore have a wide influence on the field of problem solving.  Indeed, Bos provides several examples of later geometers who cite Commandino's translation of this precept in their works.\footnote{Ibid. p.50. Bos provides three examples of early modern mathematicians citing Pappus' ``sin'' or ``error'' passage.  Most notably, Descartes states it would be ``an error in geometry...to try in vain to construct some problem by a simpler kind of curves than the nature of the problem allows'' [(Descartes 1637], p. 371).} 

While the influence on later geometers may be apparent, much evidence refutes the assumption that ancient geometers at the height of the Greek enterprise were guided by this rule.  To impose a restriction on problem solving would have served as a detriment to the search for solutions.  In the early stages of geometric development, Knorr states ``the ambition is to find solutions by whatever means one can.''\footnote{Knorr, p.345}  This they did---Hippocrates constructs quadratures of  lunules using a \textit{neusis}, Menaechmus duplicates the cube via conics, and Archimedes trisects the angle with a spiral.  Even Euclid, Knorr argues, should not be considered as having followed Pappus' precept.  Although the postulates in Book I of the \textit{Elements} lead only to planar constructions in Pappus' scheme, Euclid still pursued the study and constructions of solid problems, especially from conic sections, in many of his later works.\footnote{Euclid wrote several other mathematical treatises besides his \textit{Elements}, but many of these works have not survived.  For example, his treatise \textit{Conics} consists of four books that were completed by Apollonius, who added four books himself (Heath, p.438).}  Additionally, when geometers would criticize others for their methods of construction, these criticisms were rooted in issues of practicality and not if they were solved by the means according to their class.  In this light, even if it were a common opinion, the influence of the precept articulated by Pappus seems negligible during the Golden Age of Greek mathematics.  

It is also possible, of course, that this precept was not consciously ignored, but completely unknown to the early geometers.  Knorr states that a formal restriction to the use of compass and straightedge constructions would be premature at the time of Hippocrates in the fifth century BCE.\footnote{Knorr goes on to say that the ``enterprise of discovering the solutions to problems could hardly be well served by the imposition of a restriction at this early stage,'' (p.40).}  The body of mathematical work had not yet evolved sufficiently to create such a formal restriction as to narrow the solutions to the use of the Euclidean tools.  Knorr asserts that until the ``geometric corpus had attained a size and diversity meriting'' a restriction, it would be unwise to engage ``in these formal inquiries.''\footnote{Ibid, p.40.}  Such a size, he believes, was reached by the time of Apollonius in the third century BCE.  While previous geometers had not provided a classification of distinct classes of problem-solving methods, Apollonius appears to have provided such a distinction in several of his later works.\footnote{Knorr, p.344. Knorr cites that the terms ``planar locus'' and ``solid locus'' are attributed to Apollonius as evidence that the field of geometry warranted a classification by his time.  In fact, Knorr states, his \textit{On Solid Loci} gives considerable attention to the ``formalization of the results attained within the analytic tradition up to his time,'' (p.348).}  Apollonius had also sparked a contentious environment from his theory of conics, and it seems evident that such an environment merited the formal distinction of appropriate solutions.  For example, Knorr states that Apollonius was ``bound to clash'' with his contemporaries working closely on similar matters and mentions how Apollonius was ``notably ungenerous'' in commenting on the achievements of his predecessors, like Euclid, and his ``grudging acknowledgement of others' merit must surely have been an enormous irritant to many.''\footnote{Ibid, p.329.}

Both Bos and Knorr agree that even Pappus himself did not subscribe to his own rule.  He critiques the works of Archimedes and Apollonius, especially the solid construction of Archimedes' \textit{neusis}.\footnote{Ibid, p.345.}  Pappus goes on, however, to provide an alternative construction using conic sections, which are therefore solid by his distinction.  Elsewhere Pappus gives constructions via the quadratrix and other more complex curves.  He formally provides a construction of a \textit{neusis} from the intersection of conics and uses this construction to trisect the angle.\footnote{Bos adds that Pappus provides this construction after first demonstrating that that a ``\textit{neusis} between perpendicular lines could be performed by means proper to solid problems,'' (p.55).}  Knorr also details that it is not only what Pappus includes in his \textit{Collection} that generated confusion, but also what he omitted.  To classify a problem as solid, a geometer must prove that no planar construction is possible.\footnote{But, a planar construction does indeed establish that the problem is planar (Knorr, p.347).}  Yet neither Pappus, nor any other Greek mathematician, provides such proofs; he merely claims that cube duplication and angle trisection problems are solid in nature.        	
	
Pappus' inconsistencies in claim and practice have led many geometers and historians astray.  It is possible, Knorr speculates, that Pappus had extrapolated the division of loci---which had previously been classified in descriptive accounts---to the entire field of geometric problem solving.\footnote{Knorr adds that the perspective Pappus articulates was ``far from being the ``standard'' ancient view,'' but rather a ``minority opinion held by a few of the late commentators,'' (p.348).}  He further believes that Pappus is responsible for this ``defective conception'' of the classification of problems, calling it a ``misconception of the nature of the 3rd-century [BCE] enterprise of problem-solving.''\footnote{Ibid, p.348.} 

Yet the works of Pappus had a compelling influence on later geometers, particularly after Commandino's Latin translation at the end of sixteenth century.\footnote{Bos provides several quotes by early modern geometers to illustrate this point.  One is by Pierre de Fermat, a seventeenth-century French mathematician, who states ``it has been often declared already, by Pappus and by more recent mathematicians, that it is a considerable error in geometry to solve a problem by means that are not proper to it,'' (p.50).  Additionally, Alexander Anderson, a Scottish mathematician, says ``it was considered no light offence for someone to solve a plane problem by means of conics or line-like curves,'' (p.219).}  His accounts of Greek mathematics sparked a fruitful debate in the early modern period of how to classify geometrical problems and distinguish between legitimate and illegitimate constructions.         	
	
One of these later mathematicians was Christopher Clavius, a German Jesuit geometer and astronomer of the late sixteenth and early seventeenth centuries.  He was an admired publisher and composer of textbooks, teaching for several decades at the esteemed Roman College in Italy.  In his treatise on the quadratrix curve first published in 1589, Clavius claims he had successfully provided the legitimately geometrical solution for the quadrature of the circle from his distinct construction of the quadratrix.\footnote{Bos describes the constructions Clavius provides were essentially the same as Pappus', he even goes so far to explain the objections to it (p.161).}  Although other geometers had used the quadratrix curve centuries before, Clavius provided a unique, pointwise construction that differed from the uniform-motion procedure of Hippias.  Clavius' construction involves plotting an arbitrary number of points evenly distributed along the curve, with a close approximation for the endpoint.\footnote{Bos translates a claim by Clavius that this approximation is ``without…an error which can be detected by the senses,'' (p.162).}  Additionally, Clavius justified his claim by asserting that his construction was more accurate than that of Pappus and equally as valid as the pointwise constructions of Apollonius, Menaechmus, and Nicomedes.\footnote{Ibid, p.163-4.}  
	
From Clavius' quadtratrix construction and claim of a legitimate solution, one can speculate that he did not seem to think it a ``considerable sin'' to provide non-planar solutions to the construction problems.  He does, however, change his stance on the validity of his construction.  While in 1589 Clavius authoritatively asserted that he solved the squaring of the circle, he later relaxed his claim and adopted a more cautious perspective.  The reason for his change of mind is unknown, yet it can be speculated that perhaps the impracticality of the construction led other mathematicians to critique his work.  Yet, undoubtedly, the extent of Clavius' influence was great, ultimately providing the foundation for Descartes' work on algebraic analysis. 
	
While Clavius was teaching and writing in Rome, the French mathematician Francois Vi\`ete was developing his own contributions to the field of legitimate geometric constructions.  In 1593, just four years after Clavius' edition of Euclid's \textit{Elements}, Vi\`ete published his \textit{Supplement of geometry} in which he suggested that \textit{neusis} constructions be admitted as a postulate.  Indeed, the opening sentence of the work addresses this issue.  As Bos translates
\begin{quote}
[To supply the defect of geometry, let it be conceded] To draw a straight line from any point to any two given lines, the intercept between these being any possible predefined distance.\footnote{Ibid. p.168.}
\end{quote}  
This claim would therefore make a \textit{neusis} construction as legitimately geometrical as the use of the Euclidean tools.  The very title of his work reflects this necessity---geometry, to this point, was deficient and thus needed a supplement, one capable of allowing for solutions beyond just straightedge and compass.  By admitting the \textit{neusis} as a postulate, Vi\`ete was effectively avoiding the issue of its construction, asserting it was as obvious as drawing straight lines and circles.  As a result of this new postulate, Vi\`ete proved that any problem producing third- or fourth-degree equations could be reduced to finding a mean proportion or by trisecting an angle.\footnote{Ibid, p.169.}  Hence, all non-planar problems, ``legitimated by a new geometrical postulate,'' could now be constructed by a \textit{neusis}.\footnote{Ibid. p.176. Bos also states that with the acceptance of the \textit{neusis} postulate, ``all problems leading to equations of degree less than five were duly brought with the power of legitimate geometry,'' (p.169).}  In this way, Vi\`ete demonstrated how algebraic analysis could extend the field of Euclidean geometry.

Vi\`ete, however, not only asserted his new postulate as geometrical, but also commented on other methods of construction that were resurfacing at the turn of the seventeenth century.  In regards to shifting rulers, for example, Vi\`ete described that they are mechanical and not geometrical, despite their accuracy.\footnote{For example, Vi\`ete describes the construction of a regular heptagon by Francois Foix de Candale as ``accurate, but not geometrical'' for its use of a shifting ruler (Bos, p.176).}  Vi\`ete also gives a pointwise construction of the quadratrix curve yet does not, unlike Clavius, claim it as a geometrical solution.  In his 1600 publication \textit{Apollonius from Gaul}, Vi\`ete comments on non-planar constructions from several ancient geometers.  Bos translates that Dinostratus was able to square the circle via the quadratrix curve and Archimedes via the spiral, ``but is the circle thereby geometrically squared?  No geometer would make that proposition.''\footnote{Ibid, p.178.}  Here, Vi\`ete is not only classifying problems as planar and non-planar, but definitively stating that non-planar solutions are not geometrical.  Vi\`ete, therefore, not only classified problems based on their construction, but also demarcated between legitimate and illegitimate geometrical constructions as a consequence of his supplemental postulate.
	
Johannes Molther, another seventeenth century mathematician, offered his own interpretation of geometrical exactness in his 1619 work, \textit{The Delian Problem.}  Molther begins by criticizing the work of other scholars---he condemns pointwise constructions of the quadratrix curve and he regards many constructions as approximate at best, therefore impractical and not geometrical, for geometrical constructions to Molther were rooted in precision and accuracy.   In attempting to construct two mean proportions, Molther proposes the acceptance of the \textit{neusis} postulate as the necessary solution, precisely echoing the work of Vi\`ete.  Yet Vi\`ete provided no argument as to why the \textit{neusis} postulate should be accepted, he only claims it as necessary.  Molther, however, justifies his claim, believing it to be just as acceptable as the use of the Euclidean tools.  Indeed, Molther's construction of the \textit{neusis} uses both a straightedge and compass, using both motion and the judgement of the senses.   This, he argues, is exactly what the Euclidean postulates assume, for in constructing both a line and a circle the geometer must move the required tools and succumb to the judgement of his senses.  While Molther's point is true, he does not address the issue of demarcating between valid and invalid geometric constructions, for surely not all geometric constructions of the motion and judgement of the senses were to be accepted.   

Perhaps no early modern geometer had a more rigid interpretation of geometrical exactness than Johannes Kepler, the famed mathematician and astronomer.  The underlying principle of Kepler's philosophical view of constructions was harmony---harmonious ratios were those that could be constructed with the Euclidean tools.\footnote{Bos calls such ratios ``knowable.''  Ratios that could not be constructed with the Euclidean tools, such as the regular heptagon, were unharmonious and therefore not knowable (p.183).}  These ratios, Bos asserts, occurred ``in nature as signs of God's deliberate choices,'' and hence to extend the legitimacy of construction beyond straightedge and compass was a philosophical, almost religious wrongdoing.\footnote{Ibid, p.184}  In this sense, Kepler imposes a strict interpretation of Commandino's ``considerable sin,'' insinuating that any inappropriate solution is both geometrically illegitimate and morally wrong.  Yet Kepler does not provide a detailed argument as to why his strong demarcation rests solely on the Euclidean tools.  Bos states that Kepler sought to ``appeal to authority and tradition,'' namely the works of Euclid and Proclus.\footnote{As Bos describes, Euclid provided the proper geometrical constructions through the restriction to straightedge and compass, and Proclus detailed the arguments to this restriction in his commentary of the \textit{Elements} (p.194).}  The entirety of Euclid's \textit{Elements} stems from the restriction of constructions of straight lines and circles, and therefore to extend this restriction would derail not only Euclid's work, but the greater field of geometry.  
	
Kepler, like other mathematical scholars, went on to criticize his contemporaries for their studies of non-planar constructions.  He believed Pappus' method of hyperbolic constructions, Apollonius' procedure of forming conics, and the pointwise construction of the quadratrix curve were both impractical and imprecise, and therefore rejected these methods of construction as geometrically illegitimate.\footnote{The flaw with the conic constructions of Apollonius rests in finding the hyperbola from the intersection of the cone with the plane (p.188).}  Additionally, Kepler opposed the incorporation of algebra into geometric problems.  Although reducing these problems to equations of varying degrees was a powerful and useful tool, this would, at best, provide an approximation, where the search for precise solutions would be compromised.  
	
The fifty years between Commandino's 1588 \textit{Collection} and Descartes' 1637 \textit{G\'eom\'etry} generated much mathematical activity as geometrical problem solving flourished among early modern mathematicians.  Yet, on the verge of the breakthroughs Descartes provided, Bos notes that the field of geometrical problem solving still contained three issues regarding the interpretations of legitimacy and exactness.  First, the works of the early modern geometers sparked a growth in the number of constructions beyond the use of the Euclidean tools.  Still, there lacked a precedence among these order, with hardly any available criteria to distinguish genuinely geometrical constructions.\footnote{Bos notes Willebrord Snellius, the seventeenth-century Dutch mathematician and astronomer, as an exception.  Snellius often cites ruler and compass constructions as legitimately geometrical.  In regards to the squaring of the circle, he states, a solution had not been found ``authoratively and by ruler and compass according to the rules of the art,'' despite the many solutions that existed (p.220).}  Thus, Bos asserts, ``the objective of problem solving had become opaque and the practice lacked a clear direction.''\footnote{Bos, p.221.}  Second, while many geometers recognized the use of algebraic methods in relating problems, equations, and solutions, the nature of these relations was not well articulated or understood before Descartes.  Third, the field required a more ``definite and refined interpretation of the exactness of constructional procedures'' to help both clarify the differing perspectives and progress the field of problem solving.\footnote{Ibid, p.221.}  The necessity to clarify these issues and advance the geometrical field was the impetus for the mathematical breakthroughs of Descartes. 
	
Ren\'e Descartes is considered by many to be the catalyst behind modern mathematics.  Born in 1596 in La Haye, France, Descartes received an elite education at the prestigious Jesuit College in La Fl\`eche, where he was properly trained in the Classics and humanities.\footnote{Hollingdale notes that Descartes attended the Jesuit College in France, where his best philosophical and mathematical insights came not from his classes, but when he was lying in bed in the morning (p.126).}  He spent much of his life as a solider and a philosopher, embracing a Catholic faith despite believing in what were then considered radical and heretical claims.  He applied his philosophical inquiries to his mathematics, where he was primarily concerned with issues of methodology and exactness.\footnote{Bos notes that the famed \textit{La G\'eom\'etrie} was in fact an appendix to his philosophical treatise \textit{Discourse on Method}, where he tackled the method of ``rightly conducting one's reason and seeking the truth in the sciences,'' (p.228).}  Consequently, his results in determining geometrical exactness were much more thorough than his predecessors'.  His 1637 publication of \textit{La G\'eom\'etrie} as an appendix to his \textit{Discourse on Method} was necessitated, as he saw it, for ``a more precise and reasoned definition of exactness in geometry.''\footnote{Ibid, p.225.}  Written in the vernacular of French rather than Latin, Descartes addressed his work to the general public rather than a small portion of the educated elite.  Indeed, the influence of his work lead many to assert his career as the ``the turning point between medieval and modern mathematics.''\footnote{Burton also claims that Descartes ``laid the foundations for the growth of mathematics in modern times;'' indeed, shortly after his death, Newton and other mathematicians advanced Descartes' work to lead to the development of calculus (p.337).}
	
\textit{La G\'eom\'etrie}, divided into three books, served to unify the disciplines of algebra and geometry, bridging the gap between the mathematics of the ancient Greeks and the more recent works of Descartes' contemporaries.\footnote{Indeed, Hollingdale provides the context of Descartes' motivation with the opening sentence of \textit{La G\'eom\'etrie}: ``Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction,'' (p.131).}   The first book introduces algebraic terms and notations that are still used today.  Additionally, Descartes provided an algebraic solution to a problem that concerned the works of Pappus, where the fourth-century geometer claimed without proof that the required locus was designed from conic sections.\footnote{Burton, p.342.}  The second book, titled \textit{On the Nature of Curved Lines}, provides a distinction between two types of curves.  Geometrically acceptable curves were those that were constructed by the intersection of two lines, ``each moving parallel to the one coordinate axis with ``commensurable'' velocities.''\footnote{Burton, p. 343. Perhaps Hollingdale provides a simpler definition: ``Geometric curves are those that can be expressed by an algebraic equation (of finite degree) in x and y,'' (p.134).}  Legitimate geometrical curves were those that produced algebraic equations of finite degree.  Other curves, such as the quadratrix and spiral, were deemed mechanical and therefore illegitimate, for they required ``two simultaneous motions whose relation does not admit of precise determination.''\footnote{Such pointwise methods only provide accurate constructions of a finite number of points on the curve (Burton, p.343).}   Descartes then provides instructions for constructing the roots of equations of degrees less than six.  The third book of \textit{La G\'eom\'etrie} introduces the solutions of equations, with several consciously omitted proofs to ``give others the pleasure of discovering for themselves.''\footnote{Ibid, p.348.}

Thus the mathematical commentary of both the fourth and fifth centuries and early modern period stimulated a revival of the search for solutions for the construction problems of antiquity.  Not only did these problems demonstrate the intellectual fortitude of the Greeks, but they also exemplified the distinction between legitimate and illegitimate geometric constructions that would last for the next two millennia. 

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