2. Ancients and Moderns revisited

“The spirit of modern Geometry is to always elevate truths, whether old or new, to the greatest possible generality” (Chasles Reference Chasles1837, 269). ^{Footnote 7} Chasles significantly chose this quotation from Bernard Le Bovier de Fontenelle as a finale for his own 1837 prize-winning essay, the Aperçu historique. The sentence is excerpted from the tract “Sur les spirales à l’infini,” presented at the Académie Royale des sciences in 1704. One of the main protagonists of the French quarrel between the Ancients and the Moderns, Fontenelle took advantage of his friend Pierre Varignon’s memoir on spirals, published that same year, to draw an eloquent parallel between ancient and modern geometry and compare their respective merits. Varignon had indeed generalized Fermat’s parabolic spirals into spirals of all kinds, obtained from any generating curve. ^{Footnote 8} From the so-called “general equation of infinitely many spirals” ^{Footnote 9} (Varignon Reference Varignon1707, 72), he could then retrieve the equation in polar coordinates of any specific spiral, by merely substituting for the ordinate a certain algebraic expression yielded by the equation of the generating curve, ^{Footnote 10} and thus, in each case, compute the subtangents and spiral areas algebraically. As Fontenelle pointed out,

Chasles took up similar metaphors, but diverted them from their original meaning by making them serve another purpose: instead of promoting analytic Geometry as Fontenelle intended, he offered pure Geometry, enhanced by the new methods he had created. Duality and homography would henceforth endow modern Geometry with new modes of geometric transformations which, although comparable in generality to the formula-type transformations of algebra, would nevertheless grant it with a character of easiness and universality setting it apart from ancient Geometry. Chasles presented these modes of transformation as “true instruments,” allowing anyone to produce infinitely many geometrical truths from previously known ones, thus ultimately making genius dispensable in mathematics (Chasles Reference Chasles1837, 269). ^{Footnote 11}

The comparison between ancient and modern learning in the sciences and the arts had been at the center of the Quarrel between the Ancients and the Moderns—the late seventeenth- and early eighteenth-century controversy that polarized the European République des Lettres—in which Fontenelle played an important part. More than a century after the dust had settled, Chasles’ grand opus, the Aperçu historique, resounded with echoes of those earlier concerns. ^{Footnote 12} His historiography of mathematics placed the primary emphasis on the comparison of methods, ancient and modern. As we shall see, his long-lasting interest in restoring the lost meaning of Euclid’s Porisms, and his claim that the Porisms contained the “germ” from which pure geometry should grow in its modern form, owed much to his close reading of the British seventeenth- and eighteenth-century mathematicians, whose reflections on ancient geometry were articulated in the midst of this scholarly whirlwind.

The Quarrel began in 1687 in France, ignited, under the gilding of the Louvre, by the public reading of a poem by Charles Perrault glorifying the century of Louis le Grand. The following year, Fontenelle published his influential Digression sur les Anciens et les Modernes, which stated:

In his series of Parallèles des Anciens et des Modernes (1688–1692), Perrault claimed the superiority of the Moderns over the Ancients in all quarters of scholarship, from mathematics and philosophy to poetry and eloquence. The greater perfection of Huygens’s clock over the clepsydra was largely conceded, the former outmatching the latter owing to the mathematics of the cycloid being inbuilt. However, Cicero and Quintilien still reigned supreme in rhetoric, while architecture would provide a disputed middle ground. ^{Footnote 13}

Although these comparisons had been a common trope of European scholarship since the fifteenth century, in setting out to systematically inventory the strengths and weaknesses of ancient and modern learning, the late seventeenth-century scholars lay bare some of the tensions inherent in the legacy of the Renaissance. As the Quarrel progressively died out on the French arena in the 1690s with the public reconciliation of Boileau and Perrault, the two leading figures of the rival factions, it passed to England, where it was kindled anew by an essay written by Sir William Temple, partly against Fontenelle’s Digression. This soon prompted a reply from William Wotton, a friend of Robert Boyle, Robert Hooke and Isaac Newton, who found support in Perrault to complement an array of arguments derived from the new science. Thus started what Jonathan Swift would call the Battle of the Books. ^{Footnote 14}

Although the Quarrel between the Ancients and the Moderns was fought with different agendas in different contexts across borders, certain shared features, which are significant for our current concern, may be thrown into relief. The main dividing line opposed two kinds of human endeavors, those that were conceived as depending on imitation, such as poetry and the arts, and those that would proceed by accumulation, such as Baconian philosophy and the sciences. The former implied the shaping of taste and sense by dealing with examples, while the latter relied on methods, appliances, devices, and machinery of all ilks, both material and intellectual, supposedly abridging and facilitating learning. Though this tension between imitation and accumulation did not break into open conflict until the late seventeenth century, its sources lay in the achievements of Renaissance humanism. The efforts to reconstruct and elucidate the works of the Ancients had equipped classical scholarship with an impressive arsenal of new techniques of grammatical and textual criticism, which in return would increasingly conflict with classical emulation. By enhancing our understanding of ancient texts, philology progressively instated a critical distance with respect to them that would eventually hinder imitation. ^{Footnote 15}

On the whole, the debate hinged on history, authority, and criticism.

While revisiting a Baconian theme, ^{Footnote 16} Swift vividly, albeit not impartially, captured the gist of the dispute with his well-known prosopopoeia of the spider and the bee, which, jocularly put in the mouth of a fictional Aesop, was to become emblematic of the Moderns opposing the Ancients (Fumaroli Reference Fumaroli and Lecoq2001). In contrast to the wandering bee, visiting flowers in all fields and turning the past into honey, the spider, for all “his great skill in architecture and improvement in the mathematics,” boasts of being obliged to no predecessor, only “drawing and spinning out all from [himself]” (Swift [1704] Reference Swift1824, 237). Mathematics, however, had a history of its own, as ancient texts bear witness, and as such it appealed to both antagonistic dispositions—accumulative as well as imitative. In the context of Restoration Britain, most mathematicians hardly fitted among either the Moderns or the Ancients. On the one hand, they were engaged in the grand advancement of the experimental sciences, but on the other hand, they were also committed to reading and appropriating ancient Greek texts.

Edmond Halley (1656–1742) is representative of such a form of dual allegiance. He was both a modern empirical natural philosopher—establishing the comet’s periodicity, charting the stars or mapping the terrestrial magnetic variation—and a learned classical scholar, who launched into editing, translating or reconstructing Apollonius’ canonical texts. ^{Footnote 17} In 1704, Halley came to the Savilian Chair of Geometry at Oxford, where he succeeded John Wallis, who had just started an edition of Apollonius when he died. Halley agreed to take up this project together with David Gregory. His first foray in the realm of Apollonius was with On the Cutting off of a Ratio, a text only extant in an Arabic manuscript that had been copied and partially translated by his Oxford colleague Edward Bernard. Possibly relying on his knowledge of Hebrew, Halley compared Bernard’s Latin version with the Arabic and completed it with the help of Pappus’ lemmas. ^{Footnote 18} The whole text was published in 1706, together with a translation of Book VII of Pappus’ Mathematical Collection and his own reconstruction of another of Apollonius’ problem books, On the Cutting off of an Area (Apollonius Reference Halley1706). As for the Conics, the original plan was that Gregory would edit the first four books for which there is a Greek text, while Halley would deal with the last four, books V to VII being extant only in Arabic, and book VIII being lost and only known from Pappus’ lemmas and an Arabic epitome. Gregory died in 1708, and Halley alone published all eight books of Apollonius’s Conics in 1710, the first four in both the original Greek and Latin translation, the next three in Latin translation alone, and the last one as a conjectural reconstruction. ^{Footnote 19}

Halley’s scholarship partly grew out of the legacy of Renaissance humanism. His work on Apollonius was based on Federico Commandino’s Latin translations of Apollonius’s first four books of the Conics (Reference Commandino1566), and of Pappus’ Mathematical Collection (Reference Commandino1588), but these were read and augmented in a context in which textual criticism and textual reconstruction had gathered unprecedented momentum. Another trait was the pondered balance Halley maintained between the respective merits of the Moderns and the Ancients. In his preface to On the Cutting off of a Ratio, Halley praised the Algebra speciosa, the Arithmetic of the Infinites, or the doctrine of fluxions, while also insisting that “nothing, however, is thereby removed from the glory of the Ancients, who brought Geometry to such perfection that it would perhaps have been easier for posterity to admire it, than to reach it in its investigations without the writings of the Ancients” (Apollonius Reference Halley1706, i). ^{Footnote 20} In the same vein, Halley dismissed Descartes’ claim that Apollonius had failed to solve the four-line locus problem, and instead pointed out Newton’s alternative geometrical construction in Part V, Book I of the Principia. ^{Footnote 21}

In the next generation, the Scottish mathematician Robert Simson (1687–1768), a professor at the university of Glasgow, made a further step in the path Halley had opened. In 1711, Simson spent a year in London. There he met Halley, who encouraged him to study the ancient geometers and gave him a copy of his 1706 opus, including a translation of Pappus enriched with his own personal handwritten notes. For all his knowledge of ancient mathematics, in this book Halley had explicitly acknowledged that he could not make sense of Pappus’ comments on Euclid’s lost Porisms in Book VII of the Mathematical Collection. In a footnote which Simson (and Chasles after him) would recall, Halley had indeed avowed that “the description of the Porisms given up to this point was neither understood by me nor will it be of use to the reader” (Apollonius Reference Halley1706, XXXVII). ^{Footnote 22} Halley’s remark soon caught the attention of Simson, who embraced the task set forth for him “with the sanguine expectation of a young man” (Robison 1817, 371). ^{Footnote 23} Directing all his efforts to the restoration of Euclid’s lost treatise, he made a first breakthrough in 1723 with a partial interpretation of Pappus’ lemmas to the Porisms, later to be completed by a full treatise published posthumously in 1776 (Simson Reference Simson1723, Reference Simson1776).

In the 1830s, Chasles thoroughly studied Halley’s and Simson’s previous attempts at restoring ancient geometrical texts, but he approached the problem posed by the comparison between ancient and modern geometry from a new perspective, which was connected to the context of early-nineteenth century mathematics. Among the many topics covered in the Aperçu historique, Chasles only dealt with Euclid’s lost work in a provisional or indirect way, in connection with Pappus and ancient geometry, as well as with the tentative seventeenth-century reconstructions thereof from Fermat to Stewart, or from the vantage point of the new geometry that Monge had initiated. ^{Footnote 24} However, the reference to the ancient Porisms played a pivotal role in his reflections on geometry, and underlay his conception of mathematics and history of mathematics as being tightly intertwined. In a well-known page of the Aperçu historique, Chasles distinguishes three branches of geometry: firstly, “ancient Geometry;” secondly, what he called “mixed Geometry,” or “analytic Geometry,” or “Descartes’ Geometry;” and eventually “a third kind of Geometry,” viz. the Geometry “whose first seeds were in Euclid’s Porisms and were preserved by Pappus in his Mathematical collections” (Chasles Reference Chasles1837, 116–117). About the latter, Chasles articulates his point in the following terms:

How this new Geometry, to which Chasles himself contributed in no small measure, was to be conceived as the “continuation” of ancient geometrical analysis will hopefully appear clearly, once the original way in which the French geometer connected to the work of his seventeenth-century British predecessors is made explicit. With regards to Euclid’s lost Porisms, Chasles observed that the interpretive problem on the sole basis of ancient sources (viz. Pappus and to a far lesser extent Proclus) was so perplexing that “the famous Halley, so deeply versed in ancient Geometry, confessed that he understood none of it” (Chasles Reference Chasles1837, 12). Hence Chasles gave full credit to Simson for “discover[ing] the meaning of several of these enigmas, as well as the form of these statements which was characteristic of this kind of propositions” (Chasles Reference Chasles1837, 13). However, he judged Simson’s work largely incomplete for, despite all the ingenuity to which it bore witness, it nevertheless failed to address such crucial issues as the relationship between ancient porisms and the would-be substitutive modern methods into which they presumably transformed (Chasles Reference Chasles1837, 14). Raising this question on Chasles’ part was a barely hidden performative step, since these modern methods were themselves shaped to a great extent by his own instrumented reading of Pappus, owing to the interpretive tools he himself created, a notational technology which proved efficient on both scores, for “restoring” the ancient porisms as well as for advancing the new Geometry. In this sense, carrying on in the direction indicated by Simson was only made possible owing to Chasles’ suitably transposing the ancient practice into another key, in conformity with his resolute stance in nineteenth-century debates over a controverted issue.

3. A nineteenth-century dilemma

In the Aperçu historique, when presenting the work of Desargues, Chasles inserted a significant note, in which he claimed that there were no considerations based on the principles of perspective in ancient mathematics. This statement, however, would open a breach in contemporary consensus, for the question of whether the Ancients ever knew the uses of perspective was generally settled in a different way. As a rule, Chasles first observed, one would be tempted, prima facie, to tip the scale in favor of an affirmative answer, for the method of perspective, once acquired, could not but appear as a “natural” one—a seeming fact apparently supported by the way the Ancients conceived of the generation of conics in a cone with a circular base. But this view, he argued, was not only prevalent among geometers,

Desargues was thus credited with two important innovations in the study of conics: First, (rightly or wrongly), with considering conics in the cone in all possible positions of the intersecting plane, without using the axial triangle, as the Ancients did, ^{Footnote 25} and second, with the conception of the properties of the circle at the base of the cone as being “appropriated” by the conic sections themselves. “This [latter] idea,” Chasles insisted, “which seems to us so simple and so natural today because we are accustomed to the procedures of perspective, and various other modes of transformation of figures, did not come to the minds of the geometers of Alexandria, for no trace of it can be found in their works” (Chasles Reference Chasles1837, 75). While on the one hand Chasles explicitly opposed Poncelet’s view that Euclid’s lost Porisms would attest to the fact that the Ancients had relied on considerations pertaining to perspective in their geometrical analysis, on the other hand he nevertheless embraced a pattern of discourse about the various kinds of geometry that he derived from Poncelet and, to a lesser extent, Gergonne.

In the introduction to the Traité des propriétés projectives des figures (1822), Poncelet produced a string of subtle arguments to delineate the characteristics of what he took to be distinct ways to practice geometry, namely, in his own terms, analytic Geometry, Monge’s “general Geometry,” and “ordinary or ancient Geometry.” Poncelet had already engaged in a public debate with Joseph-Diez Gergonne over the strengths and weaknesses of both pure and analytic Geometry a few years earlier, in the pages of the Annales de mathématiques pures et appliquées. ^{Footnote 26} In an open letter to Gergonne, Poncelet replied to some thoughts by the former on the comparison of the competing geometries, included in two papers published in the seventh issue of the Annales (Gergonne Reference Gergonne1816–1817a, Reference Gergonne1816–Reference Gergonne1817b). Gergonne had indeed offered solutions to some well-known geometrical problems, supposedly demonstrating that when “suitably handled,” analytic geometry surpasses pure geometry in elegance and simplicity (Gergonne Reference Gergonne1816–1817a, 289). Poncelet responded that although he admired the elegant way in which Gergonne was able to bend algebraic analysis to the most difficult questions of geometry, he could not accept the demotion of purely geometric methods. In his view, distinctions had to be made. If pure geometry were to be understood only as that of the Ancients, ^{Footnote 27} Poncelet conceded that he would have sided with Gergonne. But if, by pure geometry, one meant “that geometry in which one simply refrains from using the method of coordinates, or even any kind of computation whatsoever which would allow to momentarily lose sight of the figure one deals with,” or in other words, “that geometry cultivated by the Moderns,” then Poncelet felt compelled to starkly disagree, substantiating his view with a purely geometric proof of a theorem, generalizing that which Gergonne had adduced in support of his own (Poncelet Reference Poncelet1817–1818, 142–143). ^{Footnote 28}

Gergonne wittily made amends by hinting at what he referred to as an apology of exaggeration, a frame of mind for which hitting hard is to be preferred to hitting fair, for a curved stick can only be straightened when forcefully bent in the other direction. Basically, however, Gergonne expressed his full agreement with Poncelet about the equal merits of analytic and pure geometry in their own respective ways, provided the latter refers to Monge’s geometry, rather than to the geometry of the Ancients and their modern followers. Gergonne added that each of these two branches of geometry would in fact benefit from being cultivated in isolation, without the help of the other. For this reason, he expressed the wish that, in pure geometry, one could even “evade all use of proportions, because at bottom these are nothing more than disguised computations” (Gergonne Reference Gergonne1817–1818, 160). He exhorted Poncelet to make his case even more compelling by publishing his results.

Reminiscences of this exchange are clearly perceptible in the 1822 Traité des propriétés projectives des figures, in which Poncelet pursued even further the comparison of the various ways to practice geometry. Not only did Poncelet hold that “ancient or ordinary geometry” lacks generality, but also that it can be reproached for “a too frequent and too extended use of the mechanism of proportions, which at bottom is nothing but a disguised computation, as a knowledgeable geometer, M. Gergonne, observed” (Poncelet Reference Poncelet1822, XIX). In reference to the well-known schemata of inference characterizing the language of ratios and proportions of Euclidean geometry (viz. invertendo, permutando, ex aequali, etc.), Poncelet coined the phrase, “mechanism of proportions.” He always used this phrase as a badge of the inherent limitation of ancient geometry, and turned Gergonne’s point about proportions being “disguised computations” into one of the cornerstones of his criticism of ancient geometry (Poncelet Reference Poncelet1822, XXVII, 17). On the other hand, he pointed out that Monge’s geometry did not develop as pure geometry proper, cut out from analysis, but still owed part of its generality to “the mixing, the inner fusion so to speak, of the two ways to deal with the magnitude in figure” (Poncelet Reference Poncelet1822, XI). It begged the question of how exactly to characterize that in which consisted the power of analysis thus transfused into geometry, why “ordinary or ancient geometry” was constitutively deprived thereof and, eventually, how this defect could be remedied so as to endow it after all with an equivalent power.

The core of Poncelet’s elaborate answer to these questions lies in his working out the notion of implicit reasoning, that is the kind of reasoning that occurs when one reasons with indeterminate magnitudes, as one does in algebra when one uses abstract signs to represent magnitudes which are left entirely undetermined. The essential weakness of “ordinary Geometry, which is often called synthesis,” consists in the fact that,

This being the main reason why such geometry ranked lower than both Monge’s or analytic geometries, Poncelet’s research program aimed at finding ways to implement implicit reasoning in pure geometry by gaining new leeway with respect to the figure.

However, Poncelet noted, this was not wholly unprecedented in the history of geometry, and here Euclid’s porisms began to come into new focus.

Poncelet’s view of ancient Geometry evinced a tension between the essential weakness of traditional synthesis being riveted to the actual figure, and the suggestion that ancient geometrical analysis, to a certain extent, adumbrated modern projective geometry. Indeed, in this connection, Poncelet explicitly assumed that “Euclid had composed a Treatise in three books, on Porisms, whose considerations were probably very close to those of the theory of transversals and the geometry of the ruler” (Poncelet Reference Poncelet1822, XXIV). Referring to Commandino’s edition of Pappus, he observed that the meaning of Euclid’s Porisms had been obscured and lost over the centuries, and, while recalling the many attempts at recovering it, he did not acknowledge Simson’s work as particularly outstanding. However, without having any pretensions to explaining Pappus’ text, he claimed that,

Such emphasis on Euclid’s lost Porisms being an integral part of Poncelet’s whole pattern of discourse (viz. the correlation between implicit reasoning, freedom with regard to figures, and the role these propositions supposedly played), it comes as no surprise that, since Chasles embraced this very pattern, he was also led to pay heed to this ancient text. However, he approached it in a more cautious and reflective way than Poncelet. Unlike Poncelet, Chasles did not ignore the historical distance so to speak, in compliance with some Comtean philosophical insights.

His elder by five years, Chasles had always been in friendly terms with Auguste Comte (Pickering Reference Pickering2009, 415), to whom he referred in print as his “old comrade from the École Polytechnique” ^{Footnote 29} (Chasles Reference Chasles1837, 415). In spite of their offbeat start, they shared the same scientific training and belonged to the same milieu. However, Comte advanced faster than Chasles who, for a few years, spent the leisurely life of a young man attending Parisian theaters while working as a stockbroker, before actively engaging in mathematics with his first publications around 1827. At that time, Comte had already made a name for himself with his Système de politique positive (published in Reference Comte1824), his series of open lecture courses, which began on April 2, 1826, in the presence of such major figures of the scientific elite as Arago, Fourier or Humboldt, and the first volume of his Cours de Philosophie positive (eventually completed in 1830). With regards to innovative views on the problem posed by the comparison between ancient and modern geometries, Comte was also, at first, a good length ahead of Chasles. As early as 1818, young Comte had already translated John Leslie’s Geometrical Analysis (Reference Leslie1809) into French. ^{Footnote 30} This book by the Scottish mathematician was essentially based on the reconstructive work of Halley, Simson, and Playfair, and drew from ancient sources to present a comprehensive view of Greek analysis in the form of a modern textbook. In a three-page footnote at the end of the volume, Leslie listed his sources for the eight ancient treatises, which, according to Pappus, belonged to the domain of analysis, that is mainly Halley’s Reference Halley1706 and 1710 editions of Apollonius and Simson’s work on Euclid’s lost Porisms (Hachette Reference Hachette1818, 439–442). However, all these historical references disappeared in the French translation, which, as Comte confided in his correspondence, would eventually result in a “bad book” (Valat Reference Valat1870, 36). In later years, he would take another approach to this old topic.

In the tenth Leçon of his Cours de Philosophie Positive, Comte defined geometry as the science whose object is the measure of extension (la mesure de l’étendue), but he considered that there were essentially two different ways of dealing with geometrical questions, two different methods, hence two geometries whose philosophical character he set out to accurately define. Ordinarily, Comte observed, one referred to these two geometries by using the expressions synthetic and analytic geometry, but he added that he “would prefer by far the purely historical denominations Geometry of the Ancients and Geometry of the Moderns, which, at least, have the advantage to prevent one from disregarding their true characteristic” (Comte Reference Comte1830, Leçon 10, 383–384). However, he proposed to call them special and general geometry respectively, as these alternative labels were more in step with a dogmatic presentation. As is well known, in Comte’s view, any science is susceptible to two distinct modes of exposition. According to the historical mode, one presents knowledge in succession, in the same order in which the human mind effectively acquired it, and following—as far as possible—the same series of steps. According to the dogmatic mode, by contrast, one presents the system of ideas as it could be conceived today by one single mind, so that the whole science is, so to speak, “remade” from this privileged standpoint.

In striving to organize an ever-increasing amount of knowledge into a coherent whole, the human mind displays a constant tendency to substitute the dogmatic order for the historical one. However, Comte warns, it would not be possible to eliminate the historical mode completely. Indeed, the dogmatic mode alone would not permit a perfectly rigorous exposition. Since, as a rule, it requires a new elaboration of knowledge already acquired, it cannot be suitably applied, in each and every epoch of science, to the most recent parts, which can only be presented according to the historical mode. Thus, “any effective mode of exposition inevitably proves to be a certain combination of the dogmatic order with the historical one” (Comte Reference Comte1830, Leçon 2, 80). In each science, an adequate balance must therefore to be found, although, Comte insisted, any rational classification of the sciences “will always necessarily contain something, if not arbitrary, at least artificial” (Comte Reference Comte1830, Leçon 2, 76). The need to find the right balance between both modes, as well as the recourse to well-chosen artificial devices to perform such balance, will also prove essential in the relationship between ancient and modern geometry. Indeed, it will be seen below that, on this score, Chasles put Comte’s ideas to good use. As for Comte’s own agenda, the previous general considerations would apply to geometry as to any other science, so as to make a perfectly rational exposition of geometry possible. However, for our present concern, only the following salient features of Comte’s doctrine will be of interest.

The fundamental difference between the “Geometry of the Ancients” and the “Geometry of the Moderns” does not consist in the kind of deductive tools that are employed, but in the nature of the questions that are dealt with. Contrary to a common view, Comte claimed, it is not the use of calculus that distinguishes modern geometry from ancient geometry. For one thing, he noted, the Greeks did not totally ignore all kinds of calculus in geometry, for they heavily relied on the theory of proportions which “to them was, as a means of deduction, some sort of actual equivalent of our current algebra, albeit a very imperfect and above all an extremely limited one” (Comte Reference Comte1830, Leçon 10, 384). In this, Comte concurred with Gergonne, without, however, ever mentioning the debate that had unfolded in the Annales more than a decade earlier. Besides, Comte remarked, certain methods exist, such as Roberval’s method for finding tangents, which he deemed essentially modern in character, although they owe nothing to calculus whatsoever. Considered as a whole and in a state of perfection, he explained, geometry should embrace all imaginable figures on the one hand, and on the other hand, it should be able to discover all properties of each figure. Thus, geometry can be organized according to two different plans: either by gathering all questions, however diverse, about one and the same figure, and by separating those that bear on different figures; or by uniting all similar enquiries under one and the same point of view, no matter what figures they apply to, and setting apart those pertaining to different properties, even if they hold for one and the same figure.

In short, geometry can go by the figures, or by the properties. Ancient geometry would adopt the first plan, whereas modern geometry, after Descartes, distinctly favored the second one. The Greeks, for instance, would exhaustively study the properties of one curve before passing to the next, and would then start all over again, despite considering the same properties as before. By contrast, modern geometers tend to develop uniform methods to deal with any single problem, whether of tangents, rectification, curvature, or anything else, whatever the curve considered, whether it be a conic section, a cissoid, or a transcendental curve. This is the original meaning Comte attached to the terms special and general, which he used to refer to these different geometries. What form, then, would the necessary balance between them take, and beyond, why would such a balance be needed in this particular instance?

In his third Leçon, Comte had indeed elaborated a specific notion of “calculus,” which he used as a synonym for “abstract mathematics,” a central notion for his whole dogmatic exposition of the mathematical sciences. However, in spite of its centrality, Comtean “calculus” as such could not unfold its operations out of nothing; it had to start from some input received from outside its own sphere.

Hence the conclusion:

The problem then is to precisely delimit the domain of that “Geometry of the Ancients” when conceived as such an indispensable “preliminary basis” for modern geometry. Comte comes to grips with this problem in his eleventh Leçon, and reaches the conclusion that its perimeter encloses the study of the straight line, the quadrature of polygons and the cubature of polyhedra, to the exclusion of any other topic. The option of including the study of the circle and conic sections is discussed, but eventually rejected as ultimately unnecessary. Descriptive geometry, however, offers an opportunity for an interesting digression on Monge which, by contrast, will help cast light on Chasles’ own take on the relationship between ancient and modern geometry.

Comte defined the study of the straight line, one of the topics to be ranked in the realm of ancient geometry, as the problem of determining the various parts of an arbitrary rectilinear figure from one another, a problem which ancient geometers solved by graphical means. When the original figure was not planar, projections were needed. In this connection, Comte alluded to Vitruvius’ and Ptolemy’s analemnata, which were systems of projections they devised in their dealings with problems pertaining to the celestial sphere. Allegedly, this supported the view that “the Ancients knew the elements of descriptive geometry, although they did not conceive them in a distinct and general way” (Comte Reference Comte1830, Leçon 11, 411). More generally, Comte noted, descriptive geometry should be granted an intermediary status between the Geometry of the Ancients and the Geometry of the Moderns not only from the viewpoint of the historical order, but also from that of the dogmatic one. While being akin to the former by the genre of its solutions, namely graphical ones, descriptive geometry gets closer to modern geometry by the nature of its questions.

However, in spite of all this, in Comte’s view, descriptive geometry should not be included in the “preliminary basis,” for essential reasons involving the way in which its true philosophical significance should presumably be assessed.

This was a serious point of contention for Comte with Poncelet and Chasles. When contrasted with Comte’s above statement, Chasles’ well-known praise of Monge’s geometrical style “without figures” strikes one in hindsight as being, at least partly, an elegant remonstrance to his philosophically minded comrade.

Chasles thus decidedly distanced himself from Comte’s view of Monge, but at the same time he adopted his lexicon and his semantic oppositions—namely, abstract/concrete, general/special, dogmatic/historical—as for instance when characterizing ancient geometry a few lines below.

Chasles gathered these different threads in an original way. Like Poncelet, he regarded Monge as the initiator of the current of research intent on opening paths to implicit reasoning in pure geometry by granting new freedom with respect to figures. But unlike him, Chasles denied that the Ancients had had projective properties of their own, however embryonic or prototypal. Unlike Comte, he fully perceived how Monge’s geometry paved the way to “a truly distinct order of geometrical speculations,” while he gainsaid that modern geometry should only amount to mere analytic Geometry. Like Comte, however, Chasles thought of the relationship between the “Geometry of the Ancients” and the “Geometry of the Moderns” as one that implied a measure of reworking or rewriting, so that historical continuity and the growth of knowledge could only be correctly assessed if diffracted through the prism of a suitable balance between the historical and the dogmatic modes. In following Comte’s above-mentioned suggestions, Chasles knew that any rational classification of the sciences would involve “something, if not arbitrary, at least artificial,” but that having recourse to such an artificial element would not downgrade the exposition to a mere motley collection of ancient and modern topics. On the contrary, provided it was suitably chosen, it would permit one to achieve a fully rational exposition of geometry, both ancient and modern, without cancelling the historical distance separating them, or projecting unwarrantedly modern concepts onto ancient practices.

Comte had eloquently made clear that there would always be a seam between the “Geometry of the Ancients” and the “Geometry of the Moderns,” an inescapable fact it was both useless and illusory to try to conceal. The best one could do, and this would be no small achievement, was to figure out how both Geometries should best be stitched together so that the true meaning of their respective methods be fully grasped and a perfectly rational exposition of the whole of Geometry be possible. In his third Leçon, after explaining what “abstract mathematics” was, Comte significantly added: “so as to conceive sharply the true nature of a science, one always has to suppose that it is perfect” (Comte Reference Comte1830, Leçon 3, 144). In other words, the “true nature of a science” can only be revealed by an intervention on the expositor’s part. No such disclosure is possible unless one makes a strong assumption, whatever artificial devices may be needed to implement it. One indeed must assume that the science one deals with has been brought to completion, and is hence perfect from the dogmatic viewpoint, although it may still prove to be in an imperfect state from the historical one (Macherey Reference Macherey1989, 92). In the case of geometry, both ancient and modern, Chasles had recourse to such an artificial device, which would qualify as an interpretive tool and an instrument for further research, and thus make it possible to embrace the whole of geometry from an assumedly privileged standpoint: the concept of the anharmonic ratio. In contrast to his seventeenth-century predecessors, Chasles could read Pappus’ lemmas on Euclid’s lost Porisms through the lens of a notational technology of his own devising, supposedly connecting ancient and modern methods. But here again, he drew on previous work.

4. A criterion turned into an interpretive tool

In the first chapter of his Traité des propriétés projectives des figures, Poncelet had indeed suggested a criterion of projectivity for metric relations, which prepared the ground for Chasles’ further step. Poncelet defined the projective properties of a given figure as those which still subsist in all central projections of the said figure.

Among these properties, the “graphic” ones would be those that depend only on the disposition of points and lines, and do not need any criterion, since it is easy to recognize whether or not they are projective, “merely on the basis of their statement, or the inspection of the [corresponding] figure” (Poncelet Reference Poncelet1822, 5). However, for those “which concern the relations of magnitude and which one can call metric, nothing for sure can indicate a priori if they hold in all projections of the figure to which they belong” (Ibid.). Considering an arbitrary spatial figure composed of the points A, B, C, D, … and a center of projection S from which the projecting lines are drawn to the corresponding new points A′, B′, C′, D′, … (see figure 1, on the right), Poncelet asks for a criterion which may grant that any property about the original figure A, B, C, D, … would still hold for the projected one A′, B′, C′, D′, … Since any metric property amounts to a relation involving distances between some of the points composing the figure, Poncelet observes that two straight lines SA, SB, drawn from the center of projection, determine two different segments AB and A′B′ on two arbitrary transversals. Any segment AB may thus be seen as the base of a triangle SAB (see figure 1, on the left) with sides SA = a, SB = b, and perpendicular p, so that

$${\rm{Area\; of}}\,SAB = {{p.AB} \over 2} = {{m.a.b} \over 2},{\rm{ hence}}\,AB = m.{{a.b} \over p}$$

Figure 1.

Poncelet’s criterion of projectivity for metric properties.

where m is constant, depending on the angle $$\widehat {ASB}$$ (viz. the sine of the projecting angle). Now, if one considers a metric relation involving the segments connecting the points of the figure A, B, C, D, …, and if one replaces the distances AB, CD, etc. respectively by their expressions in terms of the sides a, b, c, d, …, the corresponding perpendiculars and constants m, the condition of projectivity imposes that

However, in certain cases there is a way to know beforehand whether a relation is projective, as stipulated in the following:

Poncelet’s criterion of projectivity for metric relations could then be obtained by merely considering their form.

Poncelet contented himself with giving “a very simple example of this sort of relation,” namely the relation of four points on a straight line for which the proportion $${{CA}\over{CB}} = {{DA}\over{DB}}$$ holds, “a property which was known by the Ancients, as Propos. CXLV of the VII^th book of Pappus’ Mathematical Collection shows” (Poncelet Reference Poncelet1822, 12). Remarking that this particular relation belongs to the class of those defined in art. 20, Poncelet characteristically sidestepped the problem posed by a clear-cut delineation of the general class as a whole. While on the one hand, he traced back the above-mentioned particular relation to the harmonic proportion of the Greeks, ^{Footnote 31} curiously enough, on the other hand, he more or less conflated it with a more general property first enunciated by the mathematician Charles-Julien Brianchon. In the hands of Chasles, however, this property would later prove best suited to characterize a whole class of properties encompassing by far that of harmonic proportion.

In contrast to Brianchon, who did not give this last ratio any name and thus failed to identify the corresponding general concept, Chasles made it into a new tool, both interpretive and constructive, the so-called “anharmonic ratio,” by means of which he read in a new light the very same Pappus’ lemmas on Euclid’s Porisms to which Poncelet had paid heed. In so doing, Chasles nevertheless shifted the main emphasis from Prop. 145 of Pappus’ lemmas to Prop. 129 (see figure 2), now endowed in his view with fundamental significance. ^{Footnote 33}

Figure 2.

Proposition 129 of Pappus’ Mathematical Collection.

Yet Chasles’ interpretation of Proposition 129 would definitely imply a significant amount of distortion, for Pappus’ lemma was originally formulated as a proportion between rectangular areas.

Chasles thus couched the ancient geometrical proportion in another language so as to transform it into an equality of anharmonic ratios, that is

$${{\Theta E.HZ}\over{\Theta H.ZE}} = {{\Theta B.\Gamma \Delta }\over{\Theta \Delta .\Gamma B}},\;{\text{hence}}\;{{\Theta E}\over{\Theta H}}:{{ZE}\over{HZ}} = {{\Theta B}\over{\Theta \Delta }}:{{{\Gamma B}}\over{{\Gamma \Delta }}}$$

Besides, no less than eight different configurations are distinguished in Commandino’s Latin translation of Pappus, each case being illustrated by a different diagram, whereas Chasles’ treatment claimed to be uniform and carried out in one stroke. It is further to be noted that neither Poncelet in his 1822 Traité, nor even Chasles at the stage of the Aperçu historique took into account the difference in the orientation of the segments when considering quotients of ratios being equal for four segments on a transversal. It will be seen below that Chasles’ later notational technology would remedy this dearth of expressive means.

Fully aware that Pappus’ lemmas, when rewritten in terms of anharmonic ratios, would inevitably have an artificial ring to them, Chasles nevertheless insisted that having recourse to such a contrived device was essentially justified because of the extent to which it would simplify and unify geometry.

However, this viewpoint was not unanimously shared. Poncelet would later consider the choice by Chasles of the epithet anharmonic as misguided for both linguistic and mathematical reasons. “What logical and grammatical meaning can be attached, [he] wonder[ed], to those words anharmonic, homographic, of which M. Chasles has so skilfully taken advantage” (Poncelet Reference Poncelet1866, §IV, 435). Since the negative prefix a- as in anarchy or anhydrous would not make sense in this case, Poncelet repeatedly insisted that he would have definitely preferred alternative labels such as surharmonic ratio or, to meet the German school half-way, ratio of double section, which would correspond to Möbius’ Doppelschnittsverhältniss (Poncelet Reference Poncelet1866, 360, 408, 411, 435). ^{Footnote 34}

But there were more compelling reasons underlying Poncelet’s discomfort with Chasles’ decision to confer a central importance to this notion. Not only did he admit that the whole Aperçu historique was “one of the books whose reading cost [him] the most, because of the assessments therein contained and the false interpretations to which it gave rise of [his] own most earlier works” (Poncelet Reference Poncelet1862, 493), but he also adduced a few remarks in support of his reluctance to accept the new terminology. By deliberately giving pride of place to the conservation of the anharmonic ratio upon which the whole of the new geometry was intended to be built, Chasles would presumably limit himself, Poncelet retrospectively claimed, to a restricted class of projective properties in the meaning he himself set for this term in his Traité:

The main point here seems to be confirmed by another passage, in which Poncelet later came back to his earlier writings on Euclid’s porisms from the winter of 1817 to 1818, so as to reiterate what his views were long before the publication of Chasles’ Aperçu historique. Without claiming any authority as a Greek scholar, ^{Footnote 36} he reasserted his conviction that any rewriting of Pappus’ lemmas in terms of anharmonic ratios would constrain our understanding of ancient mathematics into an alien format. Referring to some of Pappus’ lemmas rewritten in terms of equations involving ratios, he made the following observation that “these ratios being comprised among those to which the Mœbius, the Steiner, the Chasles have, long afterwards, applied diverse denominations, … which would not have suited the taste of the Ancients, enemies of this modern neologism much more adapted to restrict than to generalize the use and the understanding of mathematics” (Poncelet Reference Poncelet1866, 401). Moreover, Poncelet explicitly pointed out the misrepresentation of ancient mathematics brought about by the notational technology which, as will be discussed below, Chasles would gradually feel compelled to shift to in order to yield a satisfactory account of both Euclid’s lost Porisms and the modern pursuit of pure geometry.

In his 1870 report on the advances of geometry, Chasles, for his part, reaffirmed his opposition to Poncelet’s view that the ancient porisms adumbrated to a certain extent the use of perspective in modern geometry. “Actually, one had remarked,” Chasles conceded, “in Pappus’ lemmas pertaining to the Porisms, some traces of the theory of transversals, such as a few properties about the anharmonic ratio of four points, and a relation of involution in a quadrilateral cut by a straight line” (Chasles Reference Chasles1870, 240). And here Chasles explicitly mentioned in a footnote Poncelet’s analyses in the introduction to his 1822 treatise: “But above all it is another proposition, at first overlooked, which forms the stronger link between Euclid’s work and modern theories, meaning this property that the anharmonic ratio of four points is conserved in perspective. This simple proposition, expressed in other terms, of course, happens to be demonstrated six times, that is in six different lemmas” (Ibid.). ^{Footnote 37}

With hindsight, however, Chasles could now assess that the interpretive task that he had set for himself in the Aperçu historique actually required some preliminary work so as to build the appropriate frame to carry it out. “Thus, the study and the development of the elementary theories of modern Geometry were necessary to achieve the restoration of Euclid’s three books on p orisms. This is what stopped the geometers of the last century from fulfilling the work of which Simson had established the principle” (Ibid.). Indeed, in 1837, on the basis of the mere expressive means available at the time of the Aperçu, Chasles could only roughly delineate the goal to achieve and offer a few hints as to the direction this interpretive work should take and the way it may proceed on a middle ground, so to speak, between ancient and modern mathematics.

These guidelines were to become effective in later years when geometrical resources were augmented enough with the shaping of a new notational technology.

5. Porisms as geometrical equations: A working hypothesis

Chasles published his first interpretation of Euclid’s lost Porisms in Note III of the Aperçu historique, which he wrote in 1835. At this stage, his interpretation did not entail an exhaustive survey of Pappus’ propositions, a task to which he would turn in the late 1850s when preparing his book on Euclid’s porisms. Instead, Chasles sought to tackle the questions which he deemed unanswered by Simson’s previous restitution of the meaning of Pappus’ fragments, namely those of “the doctrine of porisms, its origin, or the philosophical thought which created it, its destination, its uses, its applications, and its transformation in modern doctrines” (Chasles Reference Chasles1837, 275). Chasles’ putative answer to this series of questions, which at this stage he could not or would not support with a complete discussion of the extant sources, was to frame Euclid’s porisms as “the equations of curves;” and, more precisely, as a doctrine whose chief purpose was to introduce in (pure) geometry “changes of coordinates” (Chasles Reference Chasles1837, 276). In other words, for Chasles, the doctrine of porisms was the art of passing from one mode of description of a figure to another. The porisms thus interpreted, Chasles claimed, therefore constituted a crucial junction between ancient geometry and modern methods.

Indeed, according to this description, all of Descartes’s geometry consists of “one continual porism” (Chasles Reference Chasles1837, 278), expressed in the language of algebra. As an example of this claim, Chasles considers the following (Cartesian) equation:

$${x^2} + {y^2} + ax+by= {c^2}$$

defined along two perpendicular axes, and from one single origin. This equation can just as well be “translated” into geometrical language as the following local problem:

Using a simple change of coordinates, ^{Footnote 38} that is to say a transformation of the axes and the origin by means of which the locus was theretofore defined, the former equation can be transformed into

$${X^2} + {Y^2} = {A^2}$$

This equation, in turn, can also be translated into the language of pure geometry: it means that “there exists a point in the plane, which can be determined, and which is always at a constant distance, which can also be determined, from all the points being sought” (ibid.). Through this description, one recognizes easily that the locus whose description has been transformed was a circle all along—a conclusion to which one might have also arrived by means of the algebraic transformation of the first equation. In particular, the point mentioned in the second description turns out to be the center of the circle, and the constant distance is its radius.

For Chasles, Euclid’s doctrine of porisms was the art of navigating between these modes of description of loci without the help of Descartes’s instrument. What precisely this navigation might have entailed for ancient geometers remains, at this stage, partly unclear in Chasles’ exposition. However, Chasles seems to have had in mind (among other things) geometrical transformations of the components involved in the description of a locus (such as the squares and straight lines involved in the above description of the circle).

Of course, Chasles acknowledged that the generality and uniformity of Descartes’ geometry was absent from Ancient geometry. Where, for Euclid or Pappus, two propositions which differ only in that a point is either to the left or to the right of another point require two different proofs, modern geometry can identify one single theorem, and prove it in “a single stroke of the quill” (Chasles Reference Chasles1837, 143). Furthermore, the transformation of an algebraic equation (for instance through a change of coordinates) was deemed much more systematic and uniform than the means of transformation of geometrical figures at the disposal of ancient geometers—a state of affairs which, in Chasles’ narrative, changed dramatically at the onset of the nineteenth century, in large part owing to Monge’s and Carnot’s contributions. ^{Footnote 39} And yet, in Chasles’ reconstruction, the scope of the doctrine of porisms goes considerably beyond Descartes’ analytic geometry proper. The example above shows how Descartes’ conceptions could have arisen from what Chasles took to be Euclid’s porisms, had the Greeks possessed Algebra. ^{Footnote 40} In fact, Cartesian equations at large were only one of the many by-products of the doctrine of porisms in Chasles’ understanding thereof.

This appears clearly in Chasles’ presentation of “two very general propositions,” whose many corollaries were deemed to include Pappus’ list of propositions (Chasles Reference Chasles1837, 279). The first of these two general propositions is the following (see figure 3):

$${{Oa}\over{Ea}} + \lambda {{O'a'}\over{E'a'}} = \mu $$

Figure 3.

Chasles’ first porism (general case).

Furthermore, Chasles noted, the converse statement is true. If, drawing the straight lines joining the points m of a given locus to P and P′, one intersects EO and E′O′ at two points a, a′ which satisfy the equation above, then said locus is a straight line.

While Chasles did not provide any proof for either part of this proposition, we shall see below that it derived easily from his work on the theory of homographic divisions—a theory itself based on the aforementioned concept of the anharmonic ratio (see section 6). In the course of his 1837 analysis of the porisms, however, Chasles made no mention of this ratio, of which he would later say that it was the keystone of Euclid’s lost doctrine (Chasles Reference Chasles1870, 239–240). In Note IX of the Aperçu historique, devoted entirely to this mathematical concept, Chasles put forth certain equations which make the connection with the general porism more palpable. Remember that the anharmonic ratio of four aligned points A, B, C, D was defined by Chasles as the quantity:

$${{AC}\over{AD}}:{{BC}\over{BD}}$$

For Chasles, the characteristic feature of this quantity as a function of the segments formed by four points was the one he had read in Proposition 129 of Book VII of Pappus’ Mathematical Collection (see section four above). However, Chasles noted, one could just as well have considered the quantities $${{AD}\over{AB}}:{{CB}\over{CD}}$$ or $${{AD}\over{AC}}:{{DB}\over{DC}}$$ , which are also functions of the segments formed by A, B, C, D and which could serve a similar role in Pappus’ lemma. These two quantities, which Chasles defined as the second and third anharmonic ratios (of the same four points) are in fact the only such functions satisfying this criterion, and they are easily expressible in terms of the first ratio. For instance, denoting r ₁ and r ₂ the first and second ratios, Chasles showed that, for instance,

$${r_1} =1\, - \,{{1}\over{r_2}}$$

Therefore, Chasles noted, if the first anharmonic ratios r ₁ and r ₁ ′ of two systems of four points A, B, C, D and A′, B′, C′, D′ are equal, then so are their second and third ratios.

This was particularly useful to Chasles because it implied that one could express the equality between anharmonic ratios $${r_1}$$ , $${r_1}'$$ in the following manner:

$${r_1} =1\, - \,{{1}\over{r_2}'}$$

that is to say:

$${{AC}\over{AD}}:{{BC}\over{BD}} + {{A'B'}\over{A'D'}}:{{C'D'}\over{C'B'}}=1$$

This is what Chasles called a three-term equation for the equality of two anharmonic ratios (Chasles Reference Chasles1837, 304).

Chasles’ first general porism above derives from precisely such an equation. Let us denote L the straight line being described by the porism; and, to any point a on OE, let us associate the point m on L obtained as the intersection of L and aP. In such a manner, we can also associate to the point a a unique point a′ on O′E′, namely the intersection of O′E′ and mP′. Conversely, to each point a′ on O′E′, we can associate, in the same way, a unique point a on OE, by forming m′, the intersection of a′P′ and L, and then a, the intersection of m′P and OE. Note that the points E and E′ correspond to one another via this construction, and let us denote Q and Q′ the two points (on OE and O′E′ respectively) corresponding to O′ and O. The key property of this correspondence between the points of OE and O′E′, in connection with the discussion above, is that it “preserves the anharmonic ratio.” In other words, for any variable point a on OE, the anharmonic ratios of a, E, O, Q and a′, E′, O′, Q′ are always equal.

Indeed, the first anharmonic ratio can be viewed as formed on OE by a pencil of rays drawn from P. But the rays Pa, PE, PO, PQ also intersect L at four points m _a , m _E , m _O , m _Q . The anharmonic ratio of the four points thus defined on L is equal to that of the four points on OE, precisely owing to Pappus’ lemma. The same can be said of the anharmonic ratio of a′, E′, O′, Q′, which is defined on O’E′ by a pencil of rays drawn from P′. However, the points m _a and m _a′ , as well as the points m _E and m _E′ , coincide per construction. Furthermore, m _O = m _Q′ and m _Q = m _O′ , per definition of Q and Q′. Thus, the anharmonic ratios on OE and O’E′ are both equal to the same anharmonic ratio on L, and so are equal to one another. Using the three-term equation for the equality of the corresponding anharmonic ratios, one can therefore write:

$${{Oa}\over{OQ}}:{{Ea}\over{EQ}} + {{O'a'}\over{P'Q'}}:{{E'a'}\over{E'Q'}}\,=\,1$$

which is exactly the equation associated to Chasles’ general porism, with $$\lambda = {{OQ}\over{EQ}} \cdot {{E'Q'}\over{O'Q'}}$$ and $$\mu = {{OQ}\over{EQ}}$$ .

We can now understand the analogy which Chasles drew between the components of this proposition and the terminology of Cartesian geometry. In this proposition, for Chasles, the points O and O′ therefore act as two origins, and the straight lines OE, O’E′ as axes; such that the abscissa and ordinate of a locus (in this case, a straight line) can be defined as the ratios $${{Oa}}\over{{Ea}}$$ and $${{O'a'}\over{E'a'}}$$ . Note that they might as well have been defined simply as the magnitudes Oa and O’a′, but the introduction of E and E′ in the equation gives Chasles more flexibility in future transformations of this description of the straight line. For instance, one can consider particular descriptions of a locus in which E and E′ coincide or lie on another given curve. The points P and P′ serve to define the way in which points of the locus are projected onto the axes. Denoting these ratio x and y, this equation states merely that a straight line is a locus whose abscissa and ordinate are in a linear relation, and conversely. This equation (and the geometrical setting from which it derives) does indeed constitute a general equation of the straight line, in the same way as the Cartesian equation ax + by = c does.

The epistemic strength of Cartesian equations, for Chasles, was that one could submit them to all sorts of transformations, thereby obtaining all possible properties of the figure they represent. The same can be said of this first porism: by applying to it various kinds of transformations, one can obtain as many descriptions of the figure it represents, namely the straight line. For instance, by taking O and O′ to coincide, and by sending P and P′ at infinity on the transversal lines OE and OE′, one obtains the Cartesian equation of the straight line. Indeed, the two lines OE and OE′ serve as two axes (i.e. as the abscissa and the ordinate), and O as the origin. Since P and P′ are at infinity, the straight lines drawn from m are the parallel lines to OE and OE′ (cf. figure 4). More generally, if the equation above is of degree m in $${{Oa}\over{Ea}}$$ and $${{O'a'}\over{E'a'}}$$ , then the locus is a curve of order m. ^{Footnote 41} Chasles takes note of this fact, but contends that Euclid’s porisms were only used to describe rectilinear figures (as well as circles).

Figure 4.

Chasles’ first porism (special case).

For this reason, Chasles claimed that from this single general porism, “one will draw a multitude of porisms, whose number could rise to two or three hundred without exaggeration” (Chasles Reference Chasles1837, 281). This assertion echoes many others made by Chasles throughout the Aperçu historique, such as his praise of Pascal’s mystical hexagram: in a single theorem, prone to take many forms, “an immense number of properties of conics were comprised” (Chasles Reference Chasles1837, 73). This sort of geometrical achievement was the preserve of the Moderns. The general porism, in that sense, could not possibly be one of the Ancients’ porisms proper, but rather a more general theorem, encompassing all of them.

Chasles’ interpretation of Euclid’s porisms also supplied him with an alternative model of modern geometrical practice to that of analytic geometry. For all its fecundity and efficiency, the use of Cartesian equations had one major drawback for Chasles: geometrical knowledge derived from the transformations thereof rested on “an auxiliary and artificial system of coordinates,” whereas truths obtained by pure geometry alone always required “a more direct reason, borrowed from the very nature of the thing [under study]” (Chasles Reference Chasles1837, 119). In other words, modern analytic geometry could let practitioners bypass some of the intermediary truths between data (of a problem or a theorem) and conclusion by hiding them amidst uniform, algebraic computations. Practitioners could thereby be rewarded with speed and efficiency, but only at the cost of full knowledge of “why and how a thing is true, [and of] its place in the order of truths to which it belongs” (Chasles Reference Chasles1837, 114). Chasles famously rejected such a trade-off on the grounds that it was possible to elevate geometrical methods to the same level of simplicity and generality as that of analysis without reneging on epistemic completeness and clarity (Michel Reference Michel2020, 125–129).

We may now begin to grasp the importance of this interpretation of the doctrine of porisms for Chasles. If these forgotten statements acted as equations of loci, and if their doctrine taught us how to transform such equations (qua modes of description of a locus), they could accomplish the same task as Descartes’ system whilst retaining geometrical clarity. Chasles’ selection of examples clearly indicates that, indeed, the Cartesian equation is but a special form of the porism. Moreover, the system of coordinates from which a locus is described in a porism is totally undetermined. One could select, transform, tailor the choices of the origins and axes to a specific problem or proof, and thus enshrine it into the “chain of truths” without introducing artificial auxiliaries, foreign to the question at hand.

However, within the Aperçu historique, this tentative reading of the porisms and of their epistemic importance was but one of many attempts by Chasles to characterize the form that modern geometrical theories should adopt. One, in particular, deserves a mention here. In his retelling of the modernization of geometry—that is to say in the second stage or “epoch” of his narrative for the development of geometrical methods—Chasles had highlighted the importance of Desargues’s involution theorem for the theory of conics. Like Pascal’s aforementioned mystical hexagram, Desargues’s theorem could yield a large number of other propositions via simple transformations. However, Chasles saw a greater purpose in this theorem: by rewriting it so as to highlight the role of the anharmonic ratio therein, he claimed that it could be made into the adequate foundation for a properly geometrical, yet fully general theory of these curves (Chasles Reference Chasles1837, 77–81). ^{Footnote 42} In Note XV, which was written at a later date, Chasles went back to this point, and sought to derive the basis for a methodical theory of conics from these historical analyses. The modified version of Desargues’ theorem which Chasles thought so promising stated the following:

In the rest of this note, Chasles showed how all classical properties of conic sections (including Pascal’s mystical hexagram, Newton’s organic description, or Pappus’ ad quatuor lineas problem) could be obtained by simply transforming this general proposition, for example by moving the pencils around or specifying their correspondence further. Since all truths pertaining to conics can be obtained at the end of a short “chain” of propositions starting from it, Chasles had effectively shown that this theorem ought to be taken as the “center” of the theory of these curves (Chasles Reference Chasles1837, 338). ^{Footnote 44}

Whilst Chasles makes no explicit connection between the two, this proposition has much in common with the first general porism described above. ^{Footnote 45} Both describe a locus as the intersection of corresponding elements of two pencils of straight lines satisfying a specific property, which Chasles would later define as the homography of the two pencils (see section four). Furthermore, both are called to play a similar role in the organization of a geometrical theory as a general proposition, involving as many indeterminate elements as possible, to which one can apply a variety of transformations (such as those given by the principles of duality and homography) and specifications, thereby producing the entire body of properties of these loci in a systematic and effortless manner. And in both cases, the centrality of these propositions is linked to their intimate connection with the concept of the anharmonic ratio. This connection is such that the equation defining the first porism (i.e. the homographic description of the straight line) is reproduced verbatim in Notes XV and XVI, albeit with different interpretations: in these two notes, this same equation now characterized the (anharmonic) correspondence between two pencils whose intersection generated a conic instead of a straight line (Chasles Reference Chasles1837, 339, 345).

These interpretative strands, by 1837, remained largely untied. The center of the theory of conics was not thought of as an equation of these curves, and the role of anharmonic ratios and homographic correspondences in the doctrine of the porisms was not made explicit. In what follows, we shall see that this would change considerably when, throughout the 1850s, Chasles would return to the porisms from the vantage point provided by his new language for the writing of geometrical propositions.

6. On the need to rewrite the propositions of Greek geometry

Chasles’ Aperçu historique was explicitly geared towards epistemological ends as much as historical ones. Through the study of past methods and theories, Chasles aimed to understand what had enabled geometers to develop ever more general methods and results, and to further these investigations on that very same path of increasing generality. But this was not the final form to which Chasles destined his geometrical research. Borrowing from Comte’s terminology, Chasles stressed in the Aperçu historique already that this historical exposition was to be completed by a “dogmatic exposition,” wherein he would “coordinate these partial and isolated truths [which rational Geometry until then had produced], to make them all derive from only a few of them, taken amongst the most general” (Chasles Reference Chasles1837, 234). ^{Footnote 46} Such was the project he took up starting in 1846, as a chair of higher geometry was created for him at the Faculté de Paris. ^{Footnote 47}

In both the inaugural lecture of the very first year of this course’s existence, delivered on 22 December 1846, and the opening lesson for the following academic year (1847–1848), Chasles came back to his historical studies in order to present and motivate the content of the courses of the year to come. ^{Footnote 48} In both texts, Chasles framed his upcoming lectures as the continuation of “the Methods of the Ancients,” but through the lens of new notational dispositifs and methods created by Monge, Carnot and other “Moderns.” The works of the Ancients were not devoid of interest because they were somehow outdated or obsolete, Chasles argued, but their study alone would be insufficient if it was not turned into a unified body of doctrine and set of methods (Chasles Reference Chasles1852, XXXVII). The first step in that direction, in Chasles’ teaching, would involve the construction of a new technology for the writing of geometrical propositions.

In a particularly telling passage of his 1846 lecture, Chasles described the chief purpose of Carnot’s works as “the extension of the Geometry of the Ancients,” but expressed in the garb of the theory of transversals and of the principle of correlation. ^{Footnote 49} Carnot’s methods, Chasles continued, lead to “easiness and strength, as well as brevity and generality”—in other words, the core epistemic values which structured the main narratives of the Aperçu historique. The main factor behind this virtuous geometrical practice, Chasles went on, lay in Carnot’s “way of writing Geometry,” a way which he claimed had come to “characterize Modern Geometry.” Chasles explained that one underlying reason for the benefits of Carnot’s notational innovations is that they serve to represent curves in a variety of ways, and to move from one such representation to another. Connecting this diagnosis of modern successes with his interpretation of ancient techniques, Chasles commented that, as such, Carnot’s “beautiful research…belongs essentially to the doctrine of Euclid’s porisms” (Chasles Reference Chasles1847, 34–35). In keeping with this tradition, Chasles set out to construct a new technology for the expression of geometrical propositions—that is to say, a “way of writing geometry”—partly modelled on his understanding of the nature of Euclid’s porisms. In doing so, Chasles would deepen his working hypothesis on the equation-like nature of these ancient propositions by centering them more explicitly around the concept of anharmonic ratio, thereby finalizing the construction of an edifice from which Pappus’ text could be read anew.

Chasles’ rewriting of geometry begins with the introduction of what he called the “principle of signs,” with which letters are used to denote segments (or angles) instead of the numerical values of their lengths (or magnitudes). ^{Footnote 50} For any two points a, b on a straight line, if one denotes the segment delineated by these points as ab, then a will be taken to be the “origin” of the segment; conversely, b is the origin of the segment ba. These two ways of denoting what seems to be the same segment naturally correspond to two different directions; and in the course of computations, these directions will be denoted by + and – signs. ^{Footnote 51} This is obviously not to say that Chasles assumes direction to be an absolute property of segments, but rather that when multiple segments occur in a single proof or proposition, one will merely have to arbitrarily select one as the basis for comparison with all others. For this reason, the following equation always holds:

$$ab\; = \; - \;ba$$

This equation merely translates the fact that, if one takes the direction of ab as a basis for comparison, then ba is obviously a segment of the opposite direction, and therefore should be denoted with a “minus” sign next to it. More generally, for any three points a, b, c on a straight line, the following holds (and, mutatis mutandis, for any n points on a straight line): ^{Footnote 52}

$$ab + bc + ca=0$$

Among the many reasons for the introduction of this principle, Chasles stressed that it allowed one to write, in a single formula, the “abstract form” of a theorem, thereby encompassing what could otherwise appear as many “concrete expressions” (Chasles Reference Chasles1852, 17–18). One crucial example is the following relation between four points a, b, c, d on a straight line, or rather between the six segments that they determine: ^{Footnote 53}

$$ab \cdot cd\; + \;ac \cdot db + ad \cdot bc=0$$

If this equation is read as a relation between the lengths of six segments (or the areas of the three rectangles they determine), then its geometrical meaning varies with the relative configuration of the four points. Each permutation of the four points yields a different property of these rectangles. All of these properties, however, essentially derive from the same unique theorem, which the principle of signs brings to the fore as a general property of six points on a line. Besides the use of Comte’s semantic opposition (concrete/abstract) to frame the modernity brought about by his notational innovation, Chasles here stressed a virtue of his mode of representation that is crucially absent in Ancient geometry. The propositions of the Greeks, Chasles had argued at length in the Aperçu historique, were limited in that they could not deal with the multiplicity of cases generated by the various configurations of a same figure (here for instance, the “figure” being the system of six points on a straight line). Through the principle of signs, Chasles brought into geometry the same sort of generality and abstraction that Descartes brought through the introduction of algebraic equations, but in a different manner.

The Géométrie supérieure was to be built upon three central theories, namely that of the anharmonic ratio, the homographic division, and the involution of segments. The last two theories, as we shall see, are in many ways correlative to the first one. Chasles had already contended, at many points in the Aperçu historique, that anharmonic ratios could serve as the appropriate basis upon which to found a large part of modern (pure) Geometry (Chasles Reference Chasles1837, 35, 72, 255). In his courses, he did turn them into the main resource through which to express relations between figures (or elements thereof). Furthermore, anharmonic ratios would now be written using the principle of signs, thereby gaining new features in terms of generality and independence from specific configurations that they did not have in Chasles’ earlier works. More specifically, the anharmonic ratio of four points a, b, c, d on a straight line could now be defined as the expression:

$${{ac}\over{ad}}:{{bc}\over{bd}}$$

the segments involved in this expression being oriented in the way described above (Chasles Reference Chasles1852, 7).

Amongst the many properties of this notion which Chasles stated and proved, let us mention the fact that given a number λ and three points a, b, c on a straight line L, it is always possible to construct a fourth point d on L such that the anharmonic ratio of a, b, c, d is equal to λ. Chasles’ construction also has the important merit of being general, a term which here means that the operations stipulated in the procedure are the same regardless of the positions of the given points and given ratio. This generality also extends to configurations that are absolutely foreign to ancient geometry. An important example is the case of points at infinity, which Chasles was well equipped to tackle at this stage. Suppose, indeed, that the point a is at infinity on the straight line on which lie b, c, d, then the anharmonic ratio defined above is equal to $${{bd}\over{bc}}$$ . More broadly, the arithmetic of segments introduced by Chasles allowed for the cancellation of ratios of “infinite segments.” ^{Footnote 54}

From the anharmonic ratio, Chasles derived his second fundamental notion, namely that of “homographic divisions” (divisions homographiques). ^{Footnote 55} These are defined as two divisions of two straight lines in a one-to-one correspondence such that the anharmonic ratio of any four points of one division is always equal to that of the four corresponding points (Chasles Reference Chasles1852, 67). Rather than an abstract object to be studied in its own right (like a family of mappings), homographic divisions serve as the basic grammar for Chasles’ geometrical language. Such correspondences between series of points occur often in geometrical practice, as Chasles had noted. For instance, the fact that the anharmonic ratio is projective means that the transversal lines of a pencil intersect any two fixed straight lines L, L′ respectively at points a, b, c, … and a′, b′, c′, … in a way that forms a homographic division. Another typical example is the following: consider a ray of straight lines turning about a fixed point and intersecting a fixed conic section at two (variable) points P, Q. The tangent lines to the conic at P, Q will intersect an arbitrary fixed straight line L in the plane, at two (variable) points a, a′. This defines two divisions of the line L, among which the construction previously described yields an obvious one-to-one correspondence. One can easily show that this is in fact a homographic correspondence. In fact, given a division a, b, c, d, … of any straight line L, one can construct an infinity of homographic divisions of any other straight line L′, by choosing arbitrarily three points a′, b′, c′ as the homologues (that is to say, corresponding points) to a, b, c. The other points d′, e′, … of the second division will then be uniquely determined, and can be constructed as the points whose anharmonic ratios with a′, b′, c′ are respectively equal to the (given) anharmonic ratios of a, b, c, d; of a, b, c, e; and so on.

Upon defining this concept, Chasles immediately explored its uses with an example that would have sounded familiar to anyone who had read his previous discussions of Euclid’s porisms. Indeed, Chasles showed how homographic divisions could serve as the basis for the writing of a geometrical equation of the straight line.

Chasles had noted that if the rays turning about two points intersect at points on a straight line, these rays form two homographic pencils, with the line joining the centers of both pencils being its own homologue. ^{Footnote 56} Conversely, if two homographic pencils are such that the line joining their centers is its own homologue, the intersection points of homologous rays will all lie on a line (which they entirely describe). ^{Footnote 57} This provided Chasles with the following “general mode of description of a straight line” (see figure 5):

Figure 5.

The general mode of description of a straight line (adapted from Chasles Reference Chasles1880, 68).

While Chasles did not draw this connection explicitly in his dogmatic exposition of his geometrical methods, this mode of description of the straight line is strongly connected to the general porism he had given in 1837, as the points P and P′ serve to project the points of a straight line on two axes, in a way that results in the corresponding points having a certain relationship (here, them forming a homographic division). This connection appears even more clearly in the following paragraphs of Chasles’ Traité, as he examined the different ways in which homographic divisions can be expressed. To that end, Chasles formed several kinds of equations by introducing variable (or, rather, indeterminate) as well as fixed points on the two lines that are homographically divided.

For instance, given two straight lines, the equation for a homographic division thereof can be obtained by fixing three points a, b, c on the first one, and three points a′, b′, c′ on the second one. For any point m on the first line, its homologue m′ is determined by the two-term equation:

$${{am}\over{bm}}:{{ac}\over{bc}} = {{a'm'}\over{b'm'}}:{{a'c'}\over{b'c'}}$$

which can be rewritten as $${{am}\over{bm}} = \lambda {{a'm'}\over{b'm'}}$$ , where $$\lambda $$ is a constant quantity, depending only on a, b, c, a′, b′, c′. Conversely, two variable points m, m′ on two lines ab, a’b′, which satisfy an equation like this one (for any fixed $$\lambda $$ ), form homographic divisions on these lines. Chasles then noted that $$\lambda $$ has a “very simple geometrical expression,” which he obtained by sending m′ at infinity. If I is the homologue (in the first division) of the point at infinity, then $$\lambda = \;{{aI}\over{bI}}$$ . This, in turn, allowed Chasles to choose a, b, a′, b′ such that $$\lambda = - 1$$ , and to simplify the equation into $${{am}\over{aI}} = {{a'm'}\over{J'm'}}$$ , where J′ is the homologue of the point at infinity (viewed as a point of the second division).

The equations we have discussed so far are what Chasles called “two-term equations” (équations à deux termes), in a transparent analogy to the equations relating the various anharmonic ratios of the same four points (Chasles Reference Chasles1852, 81). Chasles also showed how to obtain three-term equations by substituting to an anharmonic ratio r the second or third such ratios, which, as we saw previously, are simple functions of r. Subsuming the fixed quantities depending only on a, b, c, a′, b′, c′ into two constants, he obtained, for instance, the following equation for homographic divisions (Chasles Reference Chasles1852, 87):

$$\lambda {{am}\over{bm}} + \mu {{c'm'}\over{b'm'}}=1$$

Then, in a similar fashion, he derived the following four-term equation (Chasles Reference Chasles1852, 93):

$$am \cdot b'm' + \lambda am + \mu b'm' + \nu =0$$

In fact, Chasles even showed how to create an equation of an arbitrary number of terms, even though uses of equations with more than five terms are scarce in his entire body of work. What must be stressed here is that these equations are all equivalent, in the sense that they all represent the same fundamental configuration. This work on the many possible equations representing one identical configuration both illustrates the fruitfulness of the concept of anharmonic ratio and constitutes a reservoir of equations, from which the most suitable will be picked by Chasles when actively solving a problem—including that of “restoring” the content of Euclid’s lost Porisms.

At this stage, one can understand how Chasles’ Reference Chasles1837 general porism for the straight line easily derives from the fundamental theories of higher geometry. Indeed, the equation at the heart of the general porism was merely a specific kind of three-term equation: one way of describing the configuration in which two homographic pencils of rays intersect along a straight line. This connection between homographic divisions and Euclid’s lost porisms would, in the years following the publication of the Traité, run deep. In fact, the very statements that Chasles had given in 1837 are reproduced in the third section of this book, without any reference to the Aperçu historique (Chasles Reference Chasles1852, 339–344). In his 1870 Rapport, Chasles would explicitly state that all of Pappus’ porisms are cases of this general equation (Chasles Reference Chasles1870, 239). And, in his 1860 book on porisms, in order to classify and interpret Pappus’ propositions, Chasles explicitly mobilized this notational technology, which he had crafted for the description of homographic divisions. ^{Footnote 58}

Before doing so, and in keeping with his theses on the limitations of Ancient geometrical methods (in particular in terms of their generality), Chasles had to carry out a meticulous adaptation of his abstract equations to the multiplicity of concrete cases with which he imagined Euclid had busied himself. The scope of this task, however, would be transfigured throughout the 1850s, as Chasles adapted his notational technology not only to his anterior interpretation of the porisms, but also to the dissenting opinions of a growing community of classicists.

7. Conflicting norms

While Chasles delayed publishing the thorough analysis of Euclid’s Porisms he had heralded in the Aperçu historique until the very end of the 1850s, the so-called “question of porisms” was the focus of heated scholarly debates throughout the decade. In particular, a string of controversies raged about the proper way in which a few sentences by Pappus should be read. A microhistorical analysis of this dispute reveals how the expectations of the academic community became increasingly polarized by conflicting norms within a gradually splintering scientific field, in which classicists and mathematicians would eventually struggle to find common ground.

The main controversy burst into the open as a priority claim in 1859. It set the towering Michel Chasles, Academician and Professor at the Sorbonne, against Paul Émile Breton de Champ (1814–1885), former Polytechnicien (1834), Ingénieur des Ponts et Chaussées, and a minor figure occupying a rather peripheral position in the Paris scientific community. This controversy had been smoldering, however, since Breton’s initial clashing claims about ancient porisms were released in a series of publications, first in the Comptes rendus and then in Liouville’s journal (Breton Reference Breton de Champ1849, Reference Breton de Champ1853, Reference Breton de Champ1855). These immediately elicited skirmishes on two distinct fronts. Charles Pierre Housel (born in 1817) and Alexandre Joseph Hydulphe Vincent (1797–1868) were both trained in mathematics at the École Normale Supérieure, albeit twenty years apart. They occupied different positions in the scholarly field, and therefore read and questioned Breton’s contributions from different perspectives. With hindsight, both these minor quarrels appear to have been nested within the main structuring controversy, as side-effects of the fracture line that was progressively setting apart two distinct kinds of agendas. Thus, the controversy casts light on the process of social differentiation that underlay the growing estrangement between two different groups of actors, who were embracing different scholarly methodologies.

Ever since his 1849 note, Breton had suggested that Simson had been wrong to replace the original definition of porisms given by Pappus, however obscure and intractable, with his own made-up formulation: ^{Footnote 59} an utterly different phrasing, allegedly designed to match Pappus’ incidental comment that porisms were neither theorems nor problems, but instantiating some sort of mean between them. In Breton’s view, Simson had misconstrued external features in the porisms’ statements as the elements of their possible definition. However, Breton only offered his alternative reading of Pappus in 1855. He fully acknowledged that Simson’s recast definition of porisms, as well as his reconstruction of the only two complete propositions that came down to us in Pappus’ text as instances of Euclid’s porisms, ^{Footnote 60} made perfect mathematical sense. But he also pointed out that the whole approach was entirely misguided, for it was not based upon an accurate and updated account of the original sources themselves.

By contrast, Breton insisted that one should start with the Greek text, which he set himself the task of editing and translating on the basis of new and enriched evidence. He thus went back to Halley’s 1706 printed edition (Apollonius Reference Halley1706, XXXIII–XXXVI), which he amended by comparing it with two different manuscripts he could consult at the Bibliothèque impériale. ^{Footnote 61} His primary goal was to provide a more faithful edition than Halley and a better translation than either Commandino or Simson. His main contention concerned the difference between two kinds of utterances, namely between Pappus’ twenty-nine statements, which were listed verbatim in the Greek text, and Euclid’s 171 lost porisms, of which only two are extant and explicitly enunciated in the original. Pappus’ statements are all framed after the same pattern and phrased in the form of an object clause introduced by the Greek preposition ὃτι, as for instance in “that this point touches a line given in position,” or “that the ratio of this (line) to this (line) is given” (Jones Reference Jones1986, 100). Obviously, these were incomplete statements and the whole point was to understand the exact way they would connect with Euclid’s 171 supposedly complete propositions.

In a much commented upon passage, Pappus seemed to offer a hint with respect to this interpretive problem, by indicating that various hypotheses may be adjusted to one and the same ending, stipulating the result and that which is sought. ^{Footnote 62} The common view held by generations of scholars from Halley and Simson to Chasles (at least in the 1830s) was that there were lacunae in Pappus’ original Greek text, an assumption which was reflected, from Halley’s Latin translation onwards, by the presence of typographical ellipses in the way Pappus’ statements were rendered. ^{Footnote 63} Breton strongly denied the reality of these lacunae, which in turn made it compelling to interpret Pappus’ text as it stood, that is, to acknowledge each of these statements as constituting a porisma in its own right, as something to be discovered or invented in the context of various hypotheses. Breton, moreover, claimed that Simson, and Chasles after him, had failed to ascertain the exact nature of Pappus’ statements, for they had incorrectly thought of these as defective propositions, whereas, in Breton’s view, they were neither defective nor propositions. Besides, Simson and Chasles would frequently jumble together the two kinds of utterances, namely Pappus’ statements and Euclid’s porisms proper, which Breton insisted should be strictly distinguished as being different types of linguistic expression. ^{Footnote 64}

The controversy emerged publicly in 1859, when Chasles presented his forthcoming book on porisms to the Paris Academy. In a note published in the Comptes-rendus, which would be reprinted in the book as §§ I–II of the introduction, he sounded much more specific with respect to the textual particulars than in the Aperçu historique. The difference between Pappus’ statements and Euclid’s porisms was now made explicitly. In contrast to the latter, assumed to be of a propositional nature, the former were now thought of as genera, that is to say, the “twenty-nine statements, transmitted by Pappus, which, in their concise and enigmatic style, summarize Euclid’s many propositions” (Chasles Reference Chasles1859, 1038; Chasles Reference Chasles1860a, 8). Chasles only acknowledged his debt to Simson and expressly warned that he would leave aside any work on porisms later than 1835, when his ideas came to be fixed on this topic. Hence, he would “not [even] allude to the most recent research, especially that which gave rise to an ongoing polemic” (ibid.). This was enough to infuriate Breton, who claimed priority for introducing this very notion of “summarizing” so as to highlight the way the relationship between the two kinds of utterances should be understood. Thus, most of the dispute hinged on words, and Chasles soon had the upper hand by dismissing Breton’s priority claim as mere quibble about nothing, a vain equivocation about phrasing and commas. As Breton reiterated and substantiated his priority claim in a series of notes submitted to the Academy, Chasles eventually delivered a detailed answer. ^{Footnote 65} With the icy tone of outraged pride, he reaffirmed the consistent continuity of his own independent train of thought, from the Aperçu historique to the 1860 book. To settle the case, a committee was appointed, which was to include Bertrand, Serret, Lamé and Chasles himself. The last two declined, and in the end the report was written by Serret alone. After having dissected Simson’s Latin to decide whether it meant what Breton and Chasles contradictorily imputed to it, Serret concluded that, by and large, Breton’s claim was to be rejected, and only conceded an imprecise use of language on Simson’s and Chasles’ part.

However, one aspect of this controversy is of interest for our present concern, as it sheds light on the contrasting ways in which the investigation’s very goal was to be conceived on both sides. In Breton’s view, solving the question of porisms solely amounted to establishing Pappus’ text so as to make it “explicit enough” (Breton Reference Breton de Champ1860b, 996). What he dubbed the “conjectural restitutions of Euclid’s one hundred and seventy-one propositions,” could not in the least pretend to solve the question, since setting up reconstructions was essentially secondary to the primary task of providing a satisfactory edition and translation of the text. Moreover, it would prove an open-ended endeavor, as “the field open to restitutions is inexhaustible” (ibid.). Therefore, Breton underlined, the priority claim did not encroach upon Chasles’ reserved area—the mathematical content of the presumably reconstructed porisms—but only on matters pertaining to the text itself and the way Greek locutions should be understood.

Although trained as a mathematician, Breton concurred here with another group of scholars, those who gathered at the Académie des Inscriptions et Belles Lettres rather than the Académie des Sciences, such as the well-known lexicographer Émile Littré, with whom Breton was in contact. ^{Footnote 66} Originally trained as a physician, Littré entered the former of these Académies in 1839, authored monumental translations of Hippocrates and Pliny the Ancient, and devoted most of his life to the famous Dictionnaire, in which, significantly enough, the entries Philologie and Philologues both referred to Charles Rollin’s definition of this subgenre within the Belles-Lettres:

The term “philology” thus had a different resonance in France than in Germany at the time. Associated with the tradition of the grammaire générale and rhetorics, the philological practice was never imbued with the overbearing social and cultural significance it acquired in the German context. ^{Footnote 68} In return, the value attached to the kind of composite and universal knowledge required of those engaging with ancient texts may account for greater social porosity, allowing newcomers to cross apparent boundaries between disciplines and draw on their original training to contribute to the common knowledge of our literary past. Since the turn of the century, a specificity of the French higher education system was that the preparatory training to the “grandes écoles” should be entrusted to the lycées and collèges royaux, most of which, “for lack of a proper statutory setting, improvised locally, on the fringes of the classical curriculum, a special program for these ‘mathematical pupils’” (Belhoste Reference Belhoste2001, 9). This provided a distinct social context for the shaping of a public interest in combining classical and scientific backgrounds. For all these structural reasons, Breton could claim to consistently occupy a middle ground between mathematics and classical philology, despite not being a professional classicist. ^{Footnote 69}

In his reply to Breton before the Paris Academy, Chasles proposed a counter-definition of the so-called “question of porisms” by putting a premium on understanding the mathematics of the Greeks, rather than on faithfully translating the language in which it was (supposedly) originally couched.

The whole point was then to frame a system of mathematical propositions which would help bringing out the mathematical content of Pappus’ statements. “What should be understood by this word restoring? Obviously, [Chasles insisted,] what Simson himself understood, namely finding hypotheses that could be applied to Pappus’ statements, while considering these not as individual propositions, defective and mutilated in the manuscripts, but as only expressing assertions emanating from Pappus himself” (Chasles Reference Chasles1860b, 1058–1059). Chasles did not conceal the trial-and-error nature of this interpretive process.

Chasles’ and Breton’s respective methodologies were thus presented on both sides as being incompatible. The unfolding of the controversy indeed implied a hardening of the positions, in which each protagonist became ever more entrenched. A closer inspection, however, reveals the situation to be more nuanced.

In 1856, Charles Housel contributed a paper to Liouville’s Journal, in which he attempted to conciliate, to a certain extent, both approaches. Favorably impressed by Breton’s conjecture that Euclid’s porisms were about geometrical configurations liable to variation with respect to their form, he tried to flesh out this suggestion from the mathematical point of view. Fully endorsing Breton’s point about the distinction between Pappus’ statements and Euclid’s propositions, Housel held that, while the latter comprised both a hypothesis and a result interlocking into one another, the former focused on the result alone (namely the porisma proper, according to Breton). In other words, Pappus’ statements focused on what was to be sought, constructed or determined from what was given at the outset, so that the whole proposition could then be demonstrated. More specifically, the hypothesis would by and large contain the fixed parts on which the moving figure would hinge, together with its “conditions of variability or mobility” (Housel Reference Housel1856, 196–197). ^{Footnote 70} Having done so, Housel was able to perceptively re-interpret Euclid’s so-called Hyptios-porism, by adjusting it to the format in which the result would be separated by an object clause as in Pappus’ statements:

Furthermore, Housel made the connection between Euclid’s porisms, which he interpreted after Breton, and Newton’s organic description of conics by means of rotating angles, which he offered alongside Euclid’s Hyptios-porism as another telling example of porism, albeit one taken from “outside Pappus and Euclid” (Housel Reference Housel1856, 197).

Although both examples were instances of the generation of curves by means of homographic pencils, it is significant that Housel did not attribute this parallel to Chasles, whose printed views on porisms, he pointed out, amounted only to characterizing them as playing, in ancient Geometry, a role analogous to that of Cartesian coordinates in modern Geometry. Indeed, it was a long way from the Aperçu historique to the 1860 book on porisms. Despite Chasles’ claim that his ideas were already fixed on this score in 1835, it seems that much remained implicit and needed to be patiently unpacked and delineated.

With hindsight, however, Chasles retraced the path he had trodden while allegedly unfolding an unequivocal thread:

Chasles’ unpublished drafts reveal that he proceeded in this way so as to frame the equations that would assumedly match Pappus’ statements. However, it remains unclear whether he engaged in this kind of “practical experiment,” consisting in tinkering with his equations to adjust them to the Greek text, on his own initiative, or because of the controversy. In any case, in the 1860 book, the emphasis would shift from coordinates to anharmonic ratios as yielding the key to the restoration of porisms. ^{Footnote 71}

8. A “practical experiment”

Archival evidence sheds light on how Chasles endeavored to bridge the gap between his 1837 “first (and second) porism(s),” and the wealth of porisms adjusted to the variety of cases that Greek geometry supposedly required. Among the many drafts on ancient porisms that have been preserved in the Chasles’ archive at the Paris Académie des sciences, we select here the sheaf titled “Porismes généraux,” ^{Footnote 72} which best illustrates the kind of “practical experiment” Chasles alluded to in his controversy with Breton, and which would admittedly buttress his own attempt at restoring Euclid’s lost porisms. Although undated, these drafts do arguably document the process through which Chasles would transform the consequences drawn from his 1837 general propositions into two-, three- and four-term equations, each of which would presumably fit one of Pappus’ statements.

Starting from the general configuration, corresponding to the 1837 “first porism” (see figure 3 above), Chasles aimed to regain the Greek cases by unfolding a similar procedure in various instances. Recall that, in the general case, one dealt with two straight lines, each joining one of two fixed points P and Q and a variable point m sliding along the guiding line MM. These two straight lines would in return intersect two given transversals in respectively two points a and a′. Since the whole configuration allowed for free variation to an extent, one may also consider different cases by making the two points P and Q coincide, by sending them to infinity, or by making the transversals themselves coincide into one and the same straight line, which may, or may not, be assumed parallel to the guiding line or to the straight line joining P and Q, etc. Chasles methodically enumerated all cases and dealt with them in a parallel way. Here, however, we shall only focus on the generic configuration, consisting of one single transversal SS being parallel neither to the guiding line MM nor to the straight line joining P and Q (see figure 6 and 7).

Figure 6.

Chasles’ diagrams from the sheaf “Porismes généraux,” Cardboard Box 5, Chasles archive, Académie des sciences, Paris.

Figure 7.

Chasles’ generic configuration.

Further points may then be constructed. Let us denote C the intersection of the straight line PQ and the transversal. Drawing the straight line Pm′ parallel to the transversal SS, one determines the point m′ on the guiding line, hence the straight line joining the points Q and m′ intersects the transversal in point I′. Correlatively, the parallel through Q to the transversal determines the point m″, so that the straight line Pm″ intersects the transversal SS in point I. On the basis of this figure (and a few others), Chasles then offers a series of ten equations, each of which is attributed a number in the margin, namely the number of porisms corresponding to the equation at hand (see figure 8).

Figure 8.

Chasles’ list of equations with their associated number of porisms. Sheaf “Porismes généraux,” Cardboard Box 5, Chasles archive, Académie des sciences, Paris.

Let us explain briefly how Chasles obtained these equations by taking full advantage of the theory of homographic divisions. In the figure above (figure 8), the four straight lines PQ, Pm, Pm′, Pm″ meet the four corresponding ones QP, Qm, Qm′, Qm″ in four points D, m, m′, m″ on the guiding line MM. ^{Footnote 73} Since both groups of four lines respectively meet the transversal in the points:

C, a, ∞, I, and C, a′, I′, ∞,

the anharmonic ratio of the first four points is equal to that of the last four, that is

$${{Ia}\over{IC}} = {{I'C}\over{I'a'}}$$

for the segments containing a point at infinity are canceled from the equation according to the practice codified by Chasles in his lecture course, hence,

$$Ia \cdot I'a' = IC \cdot I'C$$

which is the fourth equation in Chasles’ list (see figure 8), allowing in return for only one porism, namely in Chasles’ own terms: [Given the generic configuration with the transversal being parallel neither to the guiding line nor the straight line PQ,] “one will always be able to find two points I and I′ on the transversal so that the product $$Ia \cdot I'a'$$ be constant.”

Now one may take into account two further points, O and O′, which formerly served as origins on both transversals in Chasles’ Reference Chasles1837 first porism. Assuming, at first, that they be given arbitrarily on the now unique transversal SS, one would have:

$Ia = Oa - OI\;{\rm and}\; I'a' = O'a' - O'I',$

hence the fourth equation above can be rewritten in this way:

$$\left( {Oa - OI} \right)\left( {O'a' - O'I'} \right) = IC \cdot I'C$$

hence

$$Oa \cdot O'a' - Oa \cdot O'I' - O'a' \cdot OI + \left( {OI \cdot O'I' - IC \cdot I'C} \right)=0$$

One may now introduce a further point E′ on the transversal SS as the homologue of point O in the above homographic division, that is such that PO and QE′ meet in one and the same point on the guiding line MM, and correlatively a point E as the homologue of O′. Then, owing to the fourth equation previously obtained, one may write,

$IO \cdot I{\rm{'}}E{\rm{'}} = IC \cdot I{\rm{'}}C, \;{\rm hence}\; OI \cdot O{\rm{'}}I{\rm{'}} - IC \cdot I{\rm{'}}C = OI \cdot O{\rm{'}}I{\rm{'}} - IO \cdot I{\rm{'}}E{\rm{'}} = OI \cdot O{\rm{'}}E{\rm{'}},$

so that the above equation can be rewritten as

$$Oa \cdot O'a' - Oa \cdot O'I' - O'a' \cdot OI + OI \cdot O'E'=0,$$

which is the first equation in Chasles’ list (see figure 9).

Figure 9.

How the five general porisms associated to the first equation are obtained. Sheaf “Porismes généraux,” Cardboard Box 5, Chasles archive, Académie des sciences, Paris.

One may now assume (which is still possible) that the point O coincides with the point C. If one then makes further assumptions so as to particularize the case some more, then one obtains all the other equations in succession. ^{Footnote 74} From each of these ten equations, one then obtains the number of the corresponding porisms by considering the various ways in which the given components of the configuration may be distributed in two groups, namely those which are given from the outset and those which are sought so that the property enunciated may hold and which the porism is precisely designed to yield. Owing to his new notational technology, Chasles could now couch the whole configuration into an equation in which the sought components would be represented by Greek letters, playing the role of the indeterminate coefficients of symbolic algebra. Chasles’ interpretive practice thus confirmed his theoretical point about the “deep analogy” between Euclid’s Porisms and his Data (Chasles Reference Chasles1860a, 41–44), a notion which was subtly implied in his very notation, both in the drafts and in the 1860 book (see figure 10). Some letters (for instance a, m) denoted variable points, while others (like O, I, E, etc. or A, B, etc.) denoted fixed points, whether “given actually or virtually.” Lastly, Greek letters “λ, μ represent[ed] segments, (rectilinear) areas, or constant ratios, which are also given, actually or virtually” (Chasles Reference Chasles1860a, 68–69). In Chasles’ parlance, the phrase virtually given would refer here to what, though not being given from the outset, can still be determined, or constructed by means of what is actually given—a distinction already at the core of Simson’s account, which Chasles nevertheless inventively brought to fruition in unprecedented ways. Knowing how to delicately handle this flexible notational apparatus, Chasles could eventually parse his equations so as to mold them into hundreds of presumably Euclidean porisms, thereby resuming and completing his long-standing restoration project.

Figure 10.

Chasles’ 1860 equations matching Pappus’ statements (Chasles Reference Chasles1860a, 68–69).

Chasles’ claim about Euclid’s Porisms being akin to his Data had already been clearly articulated in the Aperçu historique. However, adjustments were made in his later book on porisms, and the emphasis shifted significantly. ^{Footnote 75} In 1837, Chasles first noted that the very practice of explicitly quoting the propositions of Euclid’s Data, in addition to those of the Elements, and thus using them meaningfully in geometry, had been lost between Newton, for whom it still made sense, and the nineteenth-century geometers. Hence the need to recover the original Euclidean meaning attached to the term “given”: namely, in his own rewording, “what immediately results, by virtue of the propositions contained in [Euclid’s] Elements, from the conditions of a question” (ibid. 10). Chasles illustrated this need with the following proposition from Euclid’s Data, Dt 90: “If from a given point a straight line be drawn tangent to a circle given in position, the straight line drawn will be given in position and in magnitude” (Taisbak Reference Taisbak2003, 229). In his Note III on Euclid’s porisms, Chasles made his views somewhat more precise, and averred, however tersely, that porisms were to local theorems, what data were to theorems (Chasles Reference Chasles1837, 275). ^{Footnote 76} This assertion was completed with an interesting footnote about two distinct meanings of the term “given” being associated with only one word in Euclid’s usage:

Interestingly enough, Chasles found support for his interpretation of Euclid’s Data in Louis-Amélie Sedillot’s translation of the treatise On Known Things by the Arab mathematician Ibn Al-Haytham (c. 965–c.1040), based on the so-called manuscript Ms 1104 of the Bibliothèque royale (now registered as Ms 2458 at the BNF) (Sédillot Reference Sédillot1834). ^{Footnote 78} While deeming Al-Haytham’s work “an imitation and continuation” of Euclid’s treatise, Chasles probably took his cue for his own distinction between what is “virtually” and “actually given” from Sedillot’s rendering of a corresponding pair of Arabic terms with the French locutions “connu de fait (realiter notum)” and “connu virtuellement (virtualiter notum)” (see Sédillot Reference Sédillot1834, 9–10). ^{Footnote 79} Chasles always remained indebted to Sédillot for calling his attention on the importance of Arabic sources for the history of mathematics, ^{Footnote 80} and henceforth became an active supporter of fellow scholars who, like Sédillot and later the young Franz Woepcke (1826–1864), combined the rare skills of a both an orientalist and a historian of mathematics. ^{Footnote 81}

In his commentary, Pappus had explicitly averred that Euclid’s three books on Porisms contained twenty-nine general statements, thirty-eight lemmas and 171 theorems (Jones Reference Jones1986, 104). In his 1860 book, Chasles eventually yielded his “restoration” of the ancient doctrine, that is, a complete and systematic presentation of the way in which these three different types of propositions would subtly and precisely interlock. This required an extensive amount of preparatory work of the kind outlined above, so as to gear his abstract equations to the multiplicity of cases characteristic of ancient geometry. Chasles’ opus basically comprises a hundred-page Introduction and three parts. The Introduction authoritatively recapitulates Chasles’ own allegedly longstanding take on the various points at stake in the above-mentioned multi-faceted controversy, but without naming any names. It also offers further evidence in support of his claims with respect to the way Pappus’ and Simson’s texts should be understood. The three subsequent parts unfold the porisms presumably imputed to Euclid in his three lost books, plus forty-eight additional ones, by grouping them by genus. Chasles considered Pappus’ twenty-nine general statements as providing many genera, which he listed (see figure 10), and paired with equations couched in his own geometrical notations “so as to fix the meaning attributed to each statement in which enters a relation of segments” (Chasles Reference Chasles1860a, 68). In Euclid’s first two books, as reconstructed by Chasles, almost all these relations of segments correspond to geometrical configurations consisting of two series of points forming two homographic divisions. Both these segments may be formed either on two different straight lines (which is mostly the case in Euclid’s Book I), or on one and the same straight line (which prevails in Book II). By contrast, in Euclid’s Book III, one still considers configurations of the same kind, but the two straight lines turning around two fixed points now meet on a circle, instead of on a straight line as in the previous two books, and the fixed points are also themselves on a circle.

Commenting on his list of equations (see figure 10), Chasles remarks that four of them, namely genera III, IV, VII and VIII, seem to express the same thing, namely that the ratio of two segments is constant. “One may therefore believe at first sight, Chasles continued, that there is some confusion here as a result of some error in the text. But there are significant differences in Pappus’ expressions, and he certainly had in view propositions that are not identical, especially with respect to the things sought” (Chasles Reference Chasles1860a, 71). Although the equations are either strictly identical (as for genera III and VII), or almost so (as for genera IV and VIII), the genera are nevertheless to be distinguished, depending on whether one considers segments of which both origins are known and only the ratio is sought (genus III), or—alternatively—segments for which only one origin is known and the other is sought together with the ratio (genus VII). Chasles’ use of his notational technology thus affords him enough leeway to connect his equations to the multiple cases of Greek geometry in a both a flexible and insightful way.

In most cases, Pappus’ twenty-nine genera severally give rise to more than one porism, hence the corresponding equations in the above list. The way Chasles proceeds to unpack the multiplicity of cases implied in these genera may be briefly illustrated by an example. The fifteenth genus, for instance, corresponds to a statement by Pappus, namely “that this (line) cuts off from (lines) given in position (abscissas) containing a given” (Jones Reference Jones1986, 102), whose meaning is fixed with the equation $$Im.J'm' = \nu $$ . But then, Chasles stresses, “the things that are sought are multiple and cannot be explicitly indicated, some of them remaining implied [sous-entendues]” (Chasles Reference Chasles1860a, 63). Thus, there is a certain amount of implicitness inherent in the genus, for the statement can be understood in different ways. In the case in point, two different porisms arise depending on the way one makes explicit what, in the genus, is only implicit. According to Chasles’ rephrasing, one may indeed consider a straight line of variable position forming, on two fixed straight lines, two segments whose rectangle is constant, so that one has to determine the value of this rectangle and the position of both origins. But one may also interpret the configuration in a different way by assuming that both origins are “actually given,” and that what is to be deemed as “virtually given,” or what one has to find or to determine, consists of both the directions of the fixed straight lines through these given points and the value of the rectangle. In due course, Chasles would then associate two porisms, namely, the porisms LXXVI and LXXVII, to the fifteenth genus (Chasles Reference Chasles1860a, 174–175), the first of which being none other than a porism already “restored” by Simson as his forty-first Proposition (Tweedle Reference Tweddle2000, 115–116). While serving to fix the meaning of geometrical configurations which they do not aim to exhaustively recapture, Chasles’ equations provide an elaborate scaffolding, structuring a definite textual strategy. Owing to this, Chasles is able to systematically connect and organize the interpretive bits and pieces found Pappus’ and Simson’s texts, as well as to evaluate them.

9. The pitfalls of “archeolatry”

Shortly after its publication, Chasles’ restoration of the porisms was the subject of a lengthy and dithyrambic review in the Bulletin de bibliographie, d’histoire et de biographie mathématiques (de Jonquières Reference de Jonquières1861). This Bulletin, sometimes regarded as the first journal entirely devoted to the history of mathematics, was published between 1855 and 1862 as an appendix to the Nouvelles Annales de Mathématiques. Both journals were then co-edited by the French mathematician Olry Terquem (1782–1862). ^{Footnote 82} The author of this review, the vice-admiral Ernest de Jonquières (1820–1901), had long been a self-identifying disciple of Chasles, despite having scarcely set foot in the latter’s lecture halls. ^{Footnote 83} He was known to take Chasles’ publications with him aboard his vessels, quickly embracing the language and arguments of his remote professor as his own. In his review of the book on the Porisms, de Jonquières strongly reiterated Chasles’ main arguments regarding the necessity of developing modern Geometry in order to uncover the mystery behind Pappus’ statements, especially the sameness of form between Euclid’s porisms and the propositions of modern mathematics. Towards the last section of his review, having faithfully summarized Chasles’ main claims, de Jonquières offered some more personal remarks. This successful “divination” of the porisms, he commented, was a welcome distraction from the hegemonic flow of publications by analysts. More broadly, Chasles’ methods, which at this point de Jonquières had tied to the tradition of the porisms, refuted the widespread claim that analytic methods had any intrinsic superiority over pure geometry.

By connecting the porisms and this broader polemical context, de Jonquières explicitly mentioned a review penned a few months prior by Terquem himself, in which a recent memoir by Charles Méray written in the style of Chasles’ higher geometry was negatively assessed (Méray Reference Méray1860; Terquem Reference Terquem1860). In this memoir, only one (symbolic) equation was written, only to be immediately set aside and followed by geometrical derivations of the properties of surfaces of the second order. For Terquem, such mathematical writing was insufficiently “equational,” a vice he also found, to a lesser extent, in Chasles’ Reference Chasles1852 Traité (Terquem Reference Terquem1860, 70). Perhaps aware of Chasles’ contention that there might be geometrical equations of a non-algebraic nature, Terquem immediately made his complaint more precise: while geometrical treatises such as Méray’s may contain several “spoken equations,” these are always of lesser value than “written equations” (i.e., algebraic ones). To limit one’s expressive resources to verbal and other non-symbolic components, for Terquem, was a typical symptom of a larger epistemic vice he dubbed “archeolatry”: the unrestrained idolization of an invented past. This geometrical practice, he hammered, was akin to “rusting a freshly minted medal, just to give it a gloss of antiquity” (ibid.). Not only was this of no scientific utility, Terquem concluded, but it was not even true to the spirit of the ancient geometers unduly imitated.

Borrowing a well-known line, which he attributed to a courtesan of Frederick the Great, but which had also been used by Stendhal in his defense of romanticism against classicism, Terquem drove a final wedge between Chasles’ modern geometry and the Ancients by arguing that Euclid reborn in the nineteenth century would learn how to use equations just like everyone else, within a few months master them on account of his mathematical genius, and eventually regain his status as an exceptional geometer (Terquem Reference Terquem1860, 70–71; Stendhal 1823, 24). ^{Footnote 84} Of course, this was not admissible for de Jonquières, whose argument rested largely on this hidden yet pervasive connection between Euclid’s porisms (as reinterpreted by Chasles) and the kind of propositions that had successfully reinvigorated pure geometry. In his view, Chasles’ treatises had successfully demonstrated that homographic divisions and other crucial theories were better served by words than symbols, both in terms of concision and clarity (de Jonquières Reference de Jonquières1861, 10). But, more profoundly, de Jonquières rejected Terquem’s normative account of scientific progress and its modalities. In an earlier issue of the Bulletin, whilst discussing the history of logarithms, Terquem had compared the introduction of algebra and logarithm in mathematical practice to that of railroads and telegraphs (Terquem Reference Terquem1855, 1). All three inventions, Terquem explained, had enabled the swift transportation of (respectively) mathematical ideas, goods, and messages over immense swathes of conceptual and material land. As a train connects two remote places, so do algebraic computations connect seemingly disjointed hypotheses and conclusions—and in no time at all. Quoting from Benjamin Franklin’s Almanack, Terquem stressed how essential such celerity had become to modern life, mathematical and otherwise: “Time is the Stuff Life is made of” (Franklin Reference Franklin and Labaree1746, entry for 4 June). To divest oneself of these modern means of transportation in the name of a hypothetical appreciation of the past, in this regard, was simply unreasonable.

On that score, Terquem had already clashed with Chasles himself, albeit only in the form of short comments scribbled on his eminent colleague’s (and friend’s) unpublished manuscripts. For reasons yet unknown, Chasles had written down and shared with Terquem a copy of the opening lecture of his course on higher geometry in the year 1847–1848. ^{Footnote 85} Unlike Chasles’ teaching of the previous year (which he presented in his 1852 Traité), this set of lectures bore solely on the theory of conic sections. In the aforementioned opening lecture, Chasles elected to take a stroll through the history of all geometrical methods that had been employed for the study of these curves; at the end of which, he claimed, it would appear clearly that, of all these methods, “only one, namely the rational and progressive Method of the Ancients [sic] is fit for the establishment of the relations and the connection between all parts of the subject, so as to constitute a true theory [of conic sections]” (Michel and Smadja Reference Michel and Smadja2021b, 288). ^{Footnote 86} Whilst reading this manuscript, Terquem—who was equally erudite and invested in the history of mathematics—had made various annotations in the margins, either to signal faulty grammar, to ask for more precise bibliographical references, or to express his own opinion on Chasles’ historical exposition. Next to Chasles’ conclusion and argument for the necessity of a return to “the method of the Ancients,” Terquem expressed, in no uncertain terms, his disapproval:

De Jonquières faced this argumentative line head-on; and, in his review of Chasles’ Porisms, he turned Terquem’s metaphors on their heads. Even if one were to concede that analysis had the upper hand with regard to speed and invention (which he was not quite ready to do), pure geometry would still be necessary because knowledge imparted on its practitioners has other, fundamental qualities. If the celerity of analysis was like that of a locomotive, de Jonquières countered, the slower geometrical method was like a pedestrian stroll, whereby one gains a more complete “lay of the land,” a fuller understanding of the figures being surveyed. Chasles’ work, in this regard, was a dazzling display of the treasures which can be expected by the geometers who prove to be “patient, observing, and critical” in the face of algebra’s promise of immediate and thoughtless results (de Jonquières Reference de Jonquières1861, 9). In other words, to Terquem’s preference for prompt (and modern) methods, de Jonquières opposed other epistemic norms and virtues. ^{Footnote 88}

This implicit rebuttal did not escape Terquem, who used his editorial discretion to add a short answer at the end of de Jonquières’s review. In his response, Terquem re-asserted that “in the two analytic centuries that have gone by since Descartes, the science of space has made more considerable progress than in the fifty-six geometrical centuries that preceded” (Terquem Reference Terquem1861, 11). Even more crucial to Terquem was the fact that the advances made by pure geometry had only been made possible by earlier discoveries made by analysts. Using the example of Poinsot’s mechanics (which Chasles and De Jonquières had, on many occasions, lauded as one of the leading achievements of pure geometry), Terquem noted that:

Terquem, who had trained in the same schools as Chasles and shared his erudite interest in the history of mathematics as well as in this science itself, mobilized the same categories when discussing Ancient Greek mathematics: the dichotomy between Ancients and Moderns, the concept of equation, the trope of genius etc. However, his position was the exact opposite of Chasles’ (and de Jonquières’s). Where Chasles called for a geometrical practice in which genius was no longer necessary, it remained, for Terquem, the ideal to which mathematicians ought to aspire. Where Chasles defended the slow and patient construction of geometrical understanding, Terquem opposed the value of prompt methods at a time of ubiquitous recourses to science in technological and commercial applications. And where Chasles wished to pursue the methods of the Ancients by reconnecting to them those of the Moderns, Terquem saw an unbridgeable gap, leading to the ultimate necessity of staying firmly rooted in modern methods, whilst relegating the study of the Ancients to the status of an (admittedly important) cultural and erudite endeavor.