François de Foix-Candale

Abstract

This chapter presents François de Foix de Candale’s treatment of genetic definitions in his commentary on the Elements, as well as the connections between this work and his commentary on the Poimandres (Le Pimandre) with regard to the ontological status of geometrical objects. The relationship between the conceptions of Peletier and Foix-Candale concerning the geometrical notions of sphere and circle and the epistemic status of geometrical knowledge are considered at this occasion. The chapter concludes with the use that Foix-Candale made of geometrical motion in his commentary on Df. II.1. The present analysis shows that Foix-Candale’s treatment of genetic definitions is very different from those of the other authors considered in this study, given that he rejected the type of genetic definitions that appeal to the translation or rotation of a point or a magnitude of lower dimension, even when defined in terms of flow. Similarly to Peletier, his treatment of genetic definitions reveals both the convergence and the tension that existed, in ancient and premodern philosophy of mathematics, between the mathematical and the metaphysical representations of geometrical objects and of their geneses.

4.1 The Life and Work of François de Foix-Candale and the Significance of His Commentary on the Elements

François de Foix , count of Candale (Bordeaux, 1512—Bordeaux, 1594),¹ was a French humanist, mathematician,² engineer³ and alchemist.⁴ He was born in a powerful family of the nobility of Guyenne and of the city of Bordeaux and occupied as such several ecclesiastical functions, notably as Bishop of Aire-sur-Adour, in the region of Bordeaux (from 1576 to 1594).⁵ He also inherited a lordship as “Captal” of Buch, in the province of Gascony. He was a relative of the King Henri IV (Henri de Navarre ),⁶ as well as a friend of Michel de Montaigne and of the mathematician Élie Vinet (1509–1587).⁷

Candale was a promoter of the mathematical sciences. He founded a chair of mathematics at the Collège de Guyenne in Bordeaux in 1591⁸ and held a “grand salon littéraire” at his castle of Puy-Paulin, where he gathered mathematicians, humanists and other scholars from the region of Bordeaux and beyond.⁹ He published in 1566 a Latin edition and commentary of the fifteen books of Euclid’s Elements (that is, including the apocryphal Books XIV¹⁰ and XV),¹¹ to which he added a sixteenth book on the regular polyhedra,¹² as well as a short treatise on mixed and composed regular solids.¹³ Several of his successors (among whom Billingsley , Clavius and Isaac Barrow ) drew in part or in totality from these additional books for their edition of the Elements.¹⁴ Foix-Candale ’s commentary was reprinted in 1578. He added at this occasion a seventeenth and an eighteenth book, where he pursued his investigation of the properties and relations of the regular polyhedra.¹⁵ Jean Bodin (ca. 1530–1596) described Foix-Candale as the “French Archimedes” (l’Archimede des François) ¹⁶ and Guy Lefèvre de la Boderie (1541–1598) wrote of him that he “surpassed all scholars in the princely and royal art of mathematics.”¹⁷

Foix-Candale also published in 1574 a Greek-Latin edition,¹⁸ as well as a Greek-French translation the same year,¹⁹ of Hermes Trismegistus ’ Poimandres (or Pimander) and an extensive commentary in French of the same work in 1579.²⁰ These were described by A. Faivre as one of the most representative works of the Neo-Alexandrian Hermetic tradition of the second half of the sixteenth century.²¹

Foix-Candale ’s commentary on Euclid was the first extensive commentary on the entire text of the Elements that was written and printed in France.²² Its additional books on the regular polyhedra, which were taken up in various later commentaries on Euclid, including those of Billingsley and Clavius , earned it an international reputation within the early modern Euclidean tradition. These books, and their treatment of the regular polydra, were notably said to have influenced Johannes Kepler (1571–1630).²³ Within the tradition analysed here, and within its French branch in particular, Foix-Candale’s commentary on Euclid is related to that of Peletier both because of its underlying Platonic and Neoplatonic influences and because Foix-Candale followed Peletier in rejecting superposition as an admissible procedure in geometry.²⁴

Beside the commentary on the Elements, the Pimandre, that is, Foix-Candale ’s 1579 commentary on the Poimandres, represents his most significant work and a crucial contribution to Renaissance Hermeticism. It was, for him, the occasion to display his thought on a variety of issues and domains, from theology, to natural philosophy and mathematics, and therefore constitutes a valuable complementary source to approach his philosophy of mathematics. For this reason, this text will also be considered here insofar as it will allow us to better understand his discourse on the status of geometrical objects in the commentary on Euclid.

After presenting Foix-Candale’s treatment of genetic definitions in his commentary on the Elements, this chapter will thus look at the connections between this work and the Pimandre with respect to the issue at stake. This will be the occasion to explore the relationship between the conceptions of Peletier and Foix-Candale concerning the geometrical notions of sphere and circle and the epistemic status of geometrical knowledge. As in the previous chapters, I will briefly look at the use that was made of geometrical motion in the commentary on Df. II.1.

As we will see, Foix-Candale ’s treatment of genetic definitions is very different from those of the other authors considered here, even in comparison to Peletier, given that he did not take up genetic definitions that convey the translation or rotation of a point or a magnitude of lower dimension, even when defined in terms of flow. Similarly to Peletier , his treatment of genetic definitions reveals both the convergence and the frictions that existed, in this framework, between the mathematical and the metaphysical understanding of geometrical objects and of their geneses.

4.2 Formulation and Distribution of Genetic Definitions in Foix-Candale’s Commentary on Euclid

The treatment reserved to genetic definitions by Foix-Candale, in his commentary on the Elements, stands out among those of other commentators considered here by the fact that he did not describe the generation of the line or of the surface through the flow or motion of a point or of a line, but rather described this process as a forward motion, a stretching or an expansion of the line or of the surface itself. Nevertheless, he still presented the point as the efficient principle of magnitude (“punctum quantitatem agens”) in his commentary on Df. I.1.²⁵

This particular mode of generation of magnitude, for which he used the term progressus, is set forth in the commentary on Df. I.3 and I.6 of the Elements, which respectively define points as the limits of the line and lines as the limits of the surface.

The limits of the line are points. Points are called the limits of lines, since to limit is to stop the progress (progressus) of any quantity, because the length of a line, since it is the only dimension in the line, assumes as a limit the extremity or demarcating point, which surely is also used as such by the other dimensions—these are, of course, breadth and height –, although it is none of them. That is why the progress of the line is stopped or limited by the point.²⁶

The extremities of the surface are lines. We will say that the progress of the surface is stopped or limited by the line for the same reason we said that the progress of the line is stopped or limited by the point.²⁷

This process of expansion, which is applied here to both the generation of the line and the generation of the surface, is also indirectly referred to in Df. I.13, which defines the boundary. In this context, Foix-Candale conveyed this notion through the idea of a figure stretching from its centre to its periphery, a process which is then termed motus.

A boundary is the extremity of something. Since the extremities of any quantity enclose it by its boundaries, we say that the boundaries are these extremities which constrain magnitude. Some certainly call the boundary “end” because, by the motion which is assumed in the displayed magnitude, which goes from the smallest part to the extremities, we reach by this motion any extremity once we have found the end of this motion.²⁸

To a certain extent, the idea of a figure caused by the internal stretching of its magnitude brings on a quasi-physical notion of motion and expansion, that is, as temporally-determined, since the boundary of the figure would appear at the point in time when the motion (i.e. that by which the figure is understood to unravel) ends. However, for Foix-Candale, the temporal factor would not be equivalent to that which is considered in physical substances and motions, as this expansion would only be held to take place in the mind and imagination of the mathematician, where the generation of the defined figure takes place without any determination of quantity, place or velocity. Moreover, as will be shown further, this process also evokes an ontologically superior state of magnitudes, which may be physically represented by the propagation of light, as suggested by the role Foix-Candale attributed to light in the divine creation and communication of intelligible forms in his Pimandre.²⁹

4.3 Progressus Versus Fluxus

In Foix-Candale’s commentary on Euclid, the notion of flow (or motion) of the point, of the line or of the surface to generate the line, surface or solid, respectively, was thus somehow replaced by the idea of a generation of magnitudes through a continuous stretching or expansion of the same type of magnitude within the imagination. In this context, the point, the line or the surface, when they are defined as the extremities or ends of the line, the surface or the body is only what stops the progression of the considered magnitude and not what causes its generation.³⁰ This is confirmed by the fact that, in order to explain the interpretation of the spatial boundary of a figure as an end—terminus expresses both concepts—in the commentary on Df. I.13, Foix-Candale suggested that, in a mathematical context, the end or the boundary (extremum) of a figure coincides with the end (finis) of the motion through which it is assumed to be generated.³¹

The use of the notion of progressus as a means of describing the generation of magnitudes may have aimed at avoiding the difficulties raised by the representation of the extremity of a given magnitude (a point, a line, a surface) as its efficient and material cause, as it was incompatible with the Aristotelian notion of continuum. This is marked by the fact that Foix-Candale wrote, in his commentary on Df. I.1 and in his preface, that the point, although conceived as the cause of quantity, is not a part of it, just as the instant is not a part of time.³²

However, it is not entirely clear whether Foix-Candale conceived that the extremity or boundary is imposed from outside to stop the stretching of the line or figure, or whether it is itself caused by the cessation of the expansion. As a matter of fact, the point or the line, even when conceived as an extremity, would not correspond to an actually discrete element of the line or figure, just as the centre of a figure would not itself be regarded as an actually separate part of the figure’s magnitude.³³ On the other hand, according to the formulation of the commentary on Df. I.3 (“the progress of the line is stopped or limited by the point”), the notion of progressus clearly suggests an exterior intervention of the point or extremity.

In this regard, it is important to note that the discourse Foix-Candale held on these matters evokes that of Proclus, in his own commentary on Df. I.3, as he wrote: “when Euclid says that the line is limited by points, he is clearly making the line as such unlimited, as not having any limit because of its own forthgoing (πρόοδος)”.³⁴ Indeed, Foix-Candale’s notion of progressus seems to have a similar meaning to Proclus’ πρόοδος or forthgoing of the line, for which the point represents the agent of its interruption. Now, in Proclus’ commentary, which Foix-Candale very likely knew³⁵ and positively received in view of his strong adhesion to Platonic and Neoplatonic ideas, the notion of an indefinitely progressing line interrupted by a point is founded on the admission that all beings, and all mathematical objects for that matter, participate in the two metaphysical principles of the Unlimited (ἄπειρον or ἀπειρία) and the Limit (πέρας).³⁶ While the Unlimited provides infinite potentiality to the universe and to all the things it contains, the Limit endows them with a definite essence and unity, allowing them to participate in the perfection of the divine One.³⁷ Thus, in geometry, the Unlimited provides magnitudes with the capacity to extend indefinitely, while the Limit bounds this progression and endows it with determination and unity.

In Proclus ’ commentary, the operation of these principles is however expressed differently depending on the ontological level at which geometrical objects are taken. Indeed, as he wrote in his commentary on Df. I.3, “in imagined and perceived objects the very points that are in the line limit it, but in the region of immaterial forms the partless idea of the point has prior existence”.³⁸ This means that, in the realm of imaginary objects with which the geometer deals and to which local motion may be attributed, magnitude and its limit are distinct and the point intervenes as the agent of the interruption of the indefinite growth of the line. But, in its intelligible and indivisible form, the point is both what produces magnitude and what limits it,³⁹ conveying the intelligible notion of flow of the point which the imaginary motion of the point intends to imitate, according to Proclus’ commentary on the postulates.⁴⁰

As we will see further, both these meanings may be applied to Foix-Candale ’s conception of the generation of geometrical objects, given that he distinguished a mathematical and a metaphysical state of magnitudes, as had Peletier .⁴¹ Hence, it seems that, in the context of these Euclidean definitions, Foix-Candale conceived the terminating point or the boundary as agents of the interruption of the forthgoing of magnitudes. These would have a proper existence only in the geometer’s imagination, where such imaginary representations are used to study the relation between magnitudes and their parts.

However, Foix-Candale did not go as far to admit, in this context, that the point itself is the agent of the generation of the line.⁴² He may have intended thereby to avoid the problems raised by this notion with regard to the composition of the continuum. For, if one should leave aside the issue of the composition of magnitude, the conception of the generation of magnitudes as occurring according to the mode of a progressus is not incompatible, on a mathematical level, with the concept of line or surface as produced by the flow of a point or of a line, respectively.⁴³ These two perspectives are actually found in Proclus’ commentary,⁴⁴ as well as in other sixteenth-century commentaries on Euclid’s Elements, as those of Fine , Peletier and Billingsley . Indeed, these authors all explicitly defined the line as resulting from the flow of the point (in Df. I.1 and I.2), but also hinted at the generation of the line in their commentary on Df. I.3 in a way which suggests that the line is the active principle of its own generation (“The line starts from a point and ends in a point”) and that the two extreme points determine the beginning and end of this process.⁴⁵ Some commentators even presented the connection between the two perspectives, as did Clavius in his commentary on Euclid’s first postulate, where he defined the line as progressing from one point to another through a direct flow (“linea recta fluxus directo omnino itinere progrediens”).⁴⁶

Nevertheless, Foix-Candale ’s approach on the matter is relatively different from that of the other commentators considered here. In fact, most of those who aimed at the time to account for the generation of the line when commenting on the Elements followed what was philosophically considered as the natural order of causation of substances (from the simplest to the most complex), which is also the order followed by Euclid for the definitions of Book I. Therefore, these other commentators generally placed the genetic definition of the line in their commentary on Df. I.1 or Df. I.2, showing how the lines derives from the point. But Foix-Candale insisted rather on the relation between the generation of the line (conceived in terms of progressus) and the cause of its interruption or limitation (which he called the demarcating point) and thus presented a genetic definition of the line only in his commentary on Df. I.3. The situation is the same in Df. I.6 and in Df. I.13, in which the line and the boundary are defined as the limit of the surface and of the figure, respectively.⁴⁷

The contrast with Fine is, in this regard, compelling. Beside Foix-Candale , Fine was the only other commentator of Euclid, among those considered here, who connected generative process and the notion of boundary in his commentary on Df. I.13. Now, the generative process to which he referred in this context, as well as in his commentary on Df. I.3 and I.6, is the description of the line, the surface and the solid by the flow or motion of a point, a line and a surface, respectively.⁴⁸ Also, if Fine, like Foix-Candale, presented the line as the active principle of its generation in his commentary on Df. I.3, he also presented it as constituted by the infinite multiplication of the point,⁴⁹ which precisely raised the problem which Foix-Candale would have aimed to avoid by appealing to the notion of progressus of the line instead of that of flow or motion of the point.

4.4 The Two Definitions of the Sphere: Euclid Versus Theodosius

As we have seen, the will to guarantee the homogeneity between magnitude and its parts may have been an incentive for Foix-Candale’s avoidance of genetic definitions which present the line and surface as generated by the motion of a point or a line. But he may also have avoided such generative processes because they were too close to the mode of generation of the sphere in Euclid’s Df. XI.14, which he did not consider as a proper definition. Indeed, in his exposition of this definition, Foix-Candale placed the definition of the sphere proposed by Theodosius in the Spherics ⁵⁰ before Euclid’s definition and justified this by stating that the definition of the sphere commonly ascribed to Euclid did not teach the essence of the sphere, but merely its mode of generation. As in Fine ’s Geometria, the mode of generation of the sphere is then designated as a descriptio, in order to distinguish it from a definitio.

The Sphere is a solid figure, enclosed by one surface, toward which all the straight lines led from one point within it are equal. However, the description of the sphere is the circumduction of a semicircle around its fixed diameter until it returns to the place where it started to move. Since the sphere is the most perfect of all solids, we do not judge unworthy to present its two expositions. And truly, through the first, we express the true definition of its substance, which is fully convertible with the defined term. However, through the second, we have defined its description, clearly demonstrating the rule according to which it should be described. Because Theon and Campanus only expressed the description of the sphere, but not the proper nature of its substance, we placed this definition, which we support by the most powerful discourse of geometry, before the description. Indeed, the perfection of the nature of this solid, by the equality of the lines proceeding from a unique point and, furthermore, by the flexion, everywhere uniform, of its unique and admirable configuration, encloses the solid so skilfully that the regularity of its perfection cannot be corrupted by any split of angles or sides, but sets forth a certain image of its ineffable eternal essence deprived of beginning and of end.⁵¹

Although Euclid’s definition of the sphere was traditionally distinguished from Theodosius ’ definition insofar as it involved motion, it could be said here that, for Foix-Candale, both definitions did in a certain sense imply motion. For, in Theodosius’ definition, the fact that the lines joining the centre of the sphere to the circumference are said to go or advance toward (prodeuntium) its periphery clearly resonates with the notion of progressus introduced by Foix-Candale to describe the generation of the line or of the surface in Df. I.3 and I.6, and most of all in Df. I.13 through the notion of outward expansion of the figure.

Admittedly, Foix-Candale did not formulate this in this manner in his commentary on Df. XI.14. Moreover, the classical formulation of the Theodosian definition of the sphere, as that of the Euclidean definition of the circle, generally contained a reference to lines going or drawn from the centre to the circumference, without this having been interpreted as a generative process which would be the cause of the sphere or the circle.

Nevertheless, based on Foix-Candale ’s commentary on Df. I.13, where the figure in general is conceived as generated by the outward motion or expansion of the magnitude of the figure from its centre until this process is stopped by its boundary, Theodosius ’ definition of the sphere may be interpreted as signifying the outward motion or expansion of lines which uniformily proceed (prodeunt) in all directions from the centre of the sphere to its periphery. Hence, according to this conception, one would be led to assume through Theodosius’ definition of the sphere a motion starting from the centre or smallest and innermost part of the figure to its extremities (“supposito in oblata magnitudine motu, ab intima parte ad extremas”). Through this motion, the tridimensional magnitude of the sphere would expand in all directions until this expansion is stopped by its bounding surface.

Admittedly, what is said to proceed or expand from the centre of the figure, according to the definition of Theodosius , is not the tridimensional magnitude of the spherical body, but merely the lines joining the centre of the sphere to its bounding surface. This could be held as problematic with regard to the composition of the continuum, as the lines would then be thought to constitute the whole volume of the sphere. This definition would therefore not be better than a definition of the line as generated by the motion of a point. However, the conflict between the terms used in the Theodosian definition and the mode of generation of the sphere interpreted according to Foix-Candale’s commentary on Df. I.13 is only apparent. For what this definition aims to state in the first place (as Foix-Candale would have acknowledged) are the quantitative and spatial properties of the sphere, which are determined by the equality of all the lines situated between the centre of the sphere and its boundary.

But although Foix-Candale referred to this process of expansion of magnitudes in a geometrical context, where it would be held as mathematically relevant, it remains that it would also bear a philosophical meaning, relating to the ontological status of geometrical objects, even if only underlyingly expressed. Indeed, as stated at the end of the above-quoted passage, through the forward motion of the lines from the centre to the circumference of the sphere, Theodosius ’ definition is able to account for the fact that the sphere possesses no angles and is deprived of beginning and end, conveying thereby an image of its ineffable and eternal essence.⁵²

By contrast, Euclid’s definition, which defines the sphere as generated through the rotation of a semicircular surface on its axis and which Foix-Candale interpreted as a later addition by Theon of Alexandria (ca. 335–ca. 405) taken up by Campanus , would not be able to express this suprasensible essence of the sphere. And this would not be because it appeals to motion, but because the type of motion it involves does not convey the essential mode of generation of the sphere. This definition would rather express an accidental or extrinsic mode of generation, which is not dictated by the true and essential mode of being of the sphere. As Foix-Candale wrote here, the descriptio of the sphere expresses the “rule” or precept instructing how the sphere should be produced or, more literally, drawn out (lex describenda) by the geometer, and not the mode of generation that would be conform to its essence.

Thus, by distinguishing in this manner the Euclidean and Theodosian definitions of the sphere, Foix-Candale did not only distinguish two modes of definition of geometrical figures, but also two modes of generation of geometrical figures: one that would be intrinsic and essential to the figure and one that would come about only through the will and action of the geometer, be it carried out instrumentally or in the imagination. In other words, Foix-Candale ’s motivation for privileging a mode of generation of geometrical objects through the expansion of magnitude in one, two or three dimensions, rather than through the translation of a magnitude of lower dimension, would also be related to the instrumental, extrinsic and non-essential character of the latter, as it would be unsuited to the ontologically higher status and origin of geometrical objects. In the case of Euclid’s definition of the sphere, the figure would indeed correspond to a solid generated by the translation, or for that matter the rotation, of the surface of a plane figure around one of its sides. This position would be confirmed by Foix-Candale’s rejection of geometrical superposition in view of its alleged mechanical character, since he took it to be instrumentally performed and to subvert, because of this, the purely rational and abstract nature of mathematical demonstrations.⁵³

4.5 The Commentary on Euclid and the Pimandre

In the commentary on the Elements, Foix-Candale remained rather laconic on the topic of the ontological status of geometrical figures. Yet, the fact of attributing to the sphere, such as expressed by Theodosius ’ definition, an “eternal and ineffable essence” resonates with what he would later write about this figure in the Pimandre, that is, his commentary on the Poimandres, which corresponds to the first part of the Corpus hermeticum.

This commentary was certainly published by Foix-Candale in 1579, that is, thirteen years after the publication of his commentary on the Elements, but there are reasons to think that he already adhered to this doctrine, at least in part, when he published his commentary on the Elements in 1566, as will be shown further.⁵⁴ It is important at least to note that the Pimandre was actually written earlier then 1579, since the preface indicated that it had been completed by 1572.⁵⁵ Its publication would have been delayed because of the political and social unrest resulting from the massacre of St Bartholomew’s day. Moreover, although it is unclear when Foix-Candale started to work on this commentary, it appears to have taken him quite a long time to complete,⁵⁶ which means that he may have started working on the Pimandre a few years before 1572, that is, during the time when or immediately after he was working on his commentary on the Elements. Furthermore, the fact that Foix-Candale published an augmented edition of his commentary on Euclid in 1578, with an additional book on the regular polyhedra, shows that he continued the geometrical work he had started in his 1566 commentary while he was preparing his works pertaining to the Hermetic tradition, confirming the temporal and conceptual continuity between these two aspects of his intellectual work. Moreover, when he founded his chair of mathematics in 1591 at the College of Guyenne, he stated that one of the conditions required of the candidates to the position of professor of mathematics was to demonstrate a new proposition on the topic of regular polyhedra.⁵⁷ Kepler , who connected the Pythagorean theory of numbers to Hermetic theses,⁵⁸ quoted Foix-Candale’s work on regular polyhedra in his Mysterium cosmographicum,⁵⁹ hinting at the fact that he himself saw a continuity between the French humanist’s commentary on Euclid and his commitment to Hermeticism.

Although it is difficult to determine the extent to which Foix-Candale adhered to Hermeticism when he wrote his commentary on Euclid, in particular as little is known of Foix-Candale’s life before 1570,⁶⁰ J. Harrie , in her thesis dedicated to Foix-Candale’s Pimandre, considered that his interest in Hermeticism held a central role in the whole of his intellectual life. His involvement in alchemy, mathematics and theology, as well as his inclination toward the Platonic, Neoplatonic and Pythagorean doctrines, would therefore be related to this chief interest,⁶¹ as it was for other Renaissance humanists who preceded him and seem to have influenced him in this regard. These were Marsilio Ficino (1433–1499) and Jacques Lefèvre d’Étaples ,⁶² but one may also count Adrien Turnèbe (1512–1565),⁶³ as well as John Dee, to whose 1550 Parisian lectures on Euclid Foix-Candale may have attended.⁶⁴ Harrie considered furthermore that all of Foix-Candale’s works presented common philosophical views, shaped by his Platonic, Neoplatonic, Pythagorean as well as Hermetic influences, and that these conceptions are properly displayed in their complexity and richness within the Pimandre.⁶⁵

On a more general note, it is important to bear in mind that the texts belonging to the Hermetic tradition circulated widely from the late fifteenth century thanks to the Latin translation of the Corpus hermeticum by Marsilio Ficino , published in 1471,⁶⁶ after the recovery by Leonardo da Pistoia (fifteenth c.) of a Greek manuscript containing fourteen treatises belonging to the Hermetic tradition.⁶⁷ These texts prompted indeed great interest on the part of Renaissance intellectuals, both because of their philosophical and theological content and because of their association with related scientific and esoteric practices in the fields of astrology, medicine, botany, alchemy, magic and divination.⁶⁸ The Hermetic doctrine, which was reinterpreted in the light of Christian faith in the West from late Antiquity, was notably held in the Renaissance as both the foundation and synthesis of several ancient philosophical and theological traditions (among which Pythagorism, Platonism, Orphism and Neoplatonism).⁶⁹ It was thus considered by many as a means of retrieving what Ficino called a prisca philosophia, or a philosophia perennis, as Agostino Steuco (ca. 1497–1548) would later call it,⁷⁰ representing a means of reconciliation between the various philosophical and theological doctrines developed from Antiquity in the pre-modern West. The treatises of the Corpus hermetica were therefore edited and commented on many times up to the seventeenth century.

In France, many scholars contributed to this tradition before Foix-Candale ,⁷¹ starting with Jacques Lefèvre d’Étaples , who published in 1494 a commented edition of Ficino ’s Latin translation,⁷² reprinted again in 1505 with the Asclepius and the Crater Hermetis of Lodovico Lazzarelli (1447–1500).⁷³ These works were followed in 1507 by the Liber de quadruplici vita: Theologia Asclepii Hermetis Trismegisti discipuli cum commentariis by Symphorien Champier (1471–1539)⁷⁴; the Mercure Trismégiste, de la puissance & sapience de Dieu by Gabriel du Préau (1511–1588) in 1549⁷⁵; the edition of the Greek text of the Corpus hermeticum by Adrien Turnèbe in 1554, which was based on Ficino ’s manuscript and which included the Ficinian translation in Latin⁷⁶; the Deux discours de la nature du monde et de ses parties by Pontus de Tyard (ca. 1521–1605) in 1578⁷⁷ and the French translation of the De Harmonia Mundi totius Cantica tria of Francesco Giorgio (1466–1540) by Guy Lefèvre de la Boderie .⁷⁸ Foix-Candale, as said,⁷⁹ published in 1574 his own edition of the Greek text of the Poimandres on the basis of Turnèbe ’s edition⁸⁰ corrected by Joseph Scaliger (1540–1609),⁸¹ together with a Latin translation, as well as a Greek-French edition of this text. In 1579, he published in French his comprehensive commentary of the Poimandres entitled Le Pimandre de Mercure Trismegiste de la philosophie Chretienne, Cognoissance du verbe divin, et de l’excellence des œuvres de Dieu. ⁸² These works by Foix-Candale fully belong to the above-described tradition, and the Pimandre actually came forth, as said, as one of the most representative texts of French Christian Hermeticism.⁸³

Looking for possible traces of Hermetic influences in the commentary on Euclid, one may turn to a verse written in the honor of Foix-Candale by Arnaud Pujol (Arnoldus Puiolius) from the “Bordeaux Academy” (in academia Burdegalensi). In this verse, the author mentions the name of Hermes (Mercurius) as the soul’s guide through its journey from body to body⁸⁴ when evoking the ancient doctrine of the transmigration of the souls, in reference to Pythagoras ’ alleged incarnations, notably in Aethalides, the son of Hermes, and in Euphorbus.⁸⁵ Although the God Hermes, in classical Greek mythology, was already attributed the function of guiding the souls of the dead to the underworld, this function was transferred to the Egyptian-Greek syncretic figure of Hermes Trismegistus .⁸⁶ The Hermetic tradition also attributed a non-negligeable place to the theory of the transmigration of the souls.⁸⁷ A reference to this theory in association with the name of Hermes could therefore indicate that philosophical and theological ideas stemming from the Hermetic tradition were being circulated in Foix-Candale ’s intellectual circle at the time when he wrote his commentary on Euclid.⁸⁸ This intellectual circle could notably correspond to what Arnaud Pujol referred to as the academia burdigalensis.

In the epistle to Charles IX which prefaced his commentary on Euclid, Foix-Candale referred to the important place of geometry, and of mathematical disciplines in general, among the knowledge required of the priests of ancient Egypt,⁸⁹ which was immediately followed by an assertion of the importance of geometry and philosophy for the Church fathers.⁹⁰ Although these topics are not specific to the Hermetic tradition, since they are first and foremost related to the topos of the origins of geometry, in particular as told by Flavius Josephus (ca. 38–100 AD) in the Jewish Antiquities, they strongly resonate with the representation of Hermes trismegistus as an ancient Egyptian priest-king or prophet, who invented writing (in the form of hieroglyphs), who received God’s revelation and whose teaching represents the common foundation of all Western philosophical and theological doctrines.⁹¹ The alleged history and representation of Hermes Trismegistus as the inventor of mathematical sciences, who would have transmitted this knowledge to Moses and thereby to the rest of humanity,⁹² is also recalled in these terms under the authority of Plato , Iamblichus and Josephus ⁹³ in the preface to Foix-Candale’s Pimandre ⁹⁴ written by the humanist Jean Puget de Saint Marc (fl. 1579).⁹⁵

There is, admittedly, no explicit statement in the commentary on Euclid that would indicate with certainty that Foix-Candale fully adhered to the philosophical doctrine found in the Corpus hermeticum when he wrote this commentary. But it is at least certain that he then already adhered to some of the Pythagorean, Platonic, Neoplatonic ideas that were propounded by the Hermetic doctrine, and which had been held by Renaissance scholars to stem from the teachings of Hermes Trismegistus . This appears in particular through Foix-Candale’s references to the Pythagorean and Platonic conception of mathematics in the epistles and prefaces of his commentary on the Elements.⁹⁶ Indeed, among the doctrinal elements common to Hermetism, Pythagoreanism, Platonism and Neoplatonism, which (as said) many Renaissance philosophers, such as Ficino and Giovanni Pico della Mirandola (1463–1494), regarded as historically and conceptually connected,⁹⁷ are the transcendence of forms or essences, an ontology of participation linking intelligible and sensible realms, as well as the representation of the Sun as a physical image of a divine principle governing the existence and intelligibility of all things. To this adds the representation of God as a demiurge, who created the cosmos on the basis of the intelligible essences eternally present in his mind. In Foix-Candale ’s commentary on Euclid, such conceptual elements come forth in particular through the assertion of the existence of transcendent intelligible essences in the divine mind, which God would have communicated to the material realm,⁹⁸ as well as through his conception of mathematics as a propaedeutic to the contemplation of theological truths.⁹⁹ He notably made repeated references to the Platonic theory of reminiscence,¹⁰⁰ which not only presupposed the transcendence of intelligible forms and the participation of sensible beings in the essences of the intelligible realm, but also the immortality and reincarnation of the soul.

As expected, most of these doctrinal elements are also found in one way or another in Foix-Candale’s Pimandre.¹⁰¹ This commentary was indeed more an occasion to set forth his conceptions on a range of issues pertaining to natural philosophy, cosmology and theology, among other domains, than a mere project of exegesis of the first treatise of the Corpus hermeticum.¹⁰² According to J. Dagens , one of the chief aims of this treatise would have been to develop a form of natural theology, whereby Foix-Candale intended to show the compatibility and even the coincidence of the conclusions of philosophy and of theology, notably between pagan philosophy and Christian theology, against Paduan Averroism.¹⁰³ To J. Harrie , a key thesis of the Pimandre was the interpretation of Christian redemption and salvation according to Hermes’ doctrine of regeneration. This doctrine prescribed a gnostic process of purification of the soul through knowledge and piety, that is, as a detachment of the spirit from the corruption of matter, which takes up the scalar epistemological model advocated by the Platonists and the Neoplatonists, as by Christian ascetics and mystics, from Origen (ca. 184–ca. 253) to Ficino .¹⁰⁴

Hence, the Pimandre is important to consider here insofar as it offers complementary information to better understand the ontological status Foix-Candale attributed to geometrical objects in general (and to the sphere in particular) and the conceptual system that motivated his dismissal of translational or rotational generations of geometrical objects as part of their definitions. It also provides keys to understand the epistemological status of geometrical definitions, and, more generally, of geometrical knowledge as a whole in his philosophy of mathematics. As will be shown further, it also echoes certain conceptions held by Peletier concerning the epistemological status of geometrical knowledge.

4.6 The Ontological Status of the Sphere in the Pimandre

Foix-Candale ’s statement that Theodosius ’ definition of the sphere sets forth an image of its “ineffable eternal essence” is clearly coherent with the doctrinal basis common to the Hermetic, Platonic and Neoplatonic traditions, according to which geometrical objects, and most of all the geometrical circle or sphere, correspond to spatialised images of an ontologically higher substance, of divine and eternal essence, and devoid of spatiality and divisions.

In the Pimandre, Foix-Candale asserted the divine origin and perfection of the sphere over all figures,¹⁰⁵ notably on account of the fact that it may rotate on its axis while always occupying the same space, for which the universe was given a spherical shape and was made the cause of the motion of all material beings, from the celestial bodies to the elementary substances.¹⁰⁶ The sphere of the universe, which is the most perfect of all created things, is then compared to the human intellectual faculty, which corresponds to the noblest and most divine part of man¹⁰⁷ and which is deprived of local motion while setting all parts of the human body in motion.¹⁰⁸

Admittedly, in this context, the rotation of the spherical universe on its axis and the circular motion of the celestial bodies would evoke Euclid’s definition of the sphere rather than that of Theodosius . Yet, Foix-Candale’s aim then was not to describe the mode of generation and essence of the geometrical sphere, but rather to assert the ability of the sphere, which is deprived of any angle and perfectly uniform in all its parts, to remain in the same space while rotating on its axis. It was indeed for this reason, in addition to the fact that it was the most capacious of all geometrical solids, that it was considered the most perfect and divine of all geometrical figures and was therefore used to shape the cosmos, as the material, mobile and finite expression of its intelligible, immobile and omnipotent divine principle.

Now, in other passages of the Pimandre, the notion of sphere as associated with a higher ontological level of being and as caused by a process of expansion from a single source is conveyed by the representation of light as the divine virtue and essence¹⁰⁹ that is most accessible to man’s senses.¹¹⁰ The primordial light, first emanating from God’s Holy Word,¹¹¹ is then said to fill all things, allowing both material and intelligible entities to exist, as well as to be seen and known.¹¹² The light of the Sun, which corresponds to the physical manifestation of the divine intelligible light dispensed by Jesus Christ and the Holy Spirit,¹¹³ is said to illuminate all parts of the universe, its rays reaching the whole celestial realm as well as the most intimate parts of the earth.¹¹⁴

When discussing the position and role of the Sun in the universe, Foix-Candale wrote that it is however not situated in the geometrical centre of the cosmos, since he explicitly rejected the cosmological model provided by Copernicus , although he acknowledged its value as a mathematical model.¹¹⁵ Following the Ptolemaic system, he asserted that, in relation to the planets, the Sun is situated above the Moon, Mercury and Venus (the inferior planets) and below Mars, Jupiter and Saturn (the superior planets).¹¹⁶ And in relation to the entire sphere of the universe, it is situated between the earth, as the centre of the cosmos, and the eighth sphere (the Firmament or sphere of the fixed stars), which corresponds to the boundary of the celestial realm. The Sun would thus move between these two extremes according to a perfectly circular motion. As J. Dagens formulated it, this conception represents a form of “mystical heliocentrism”, which was further developed and explicitly related to the divine status of the Sun in ancient Egypt by Pierre de Bérulle (1575–1629) in the seventeenth century.¹¹⁷

In this framework, light (both intelligible and physical) was defined as a divine and primordial principle of God’s creation, whereby the figure of the sphere as resulting from a uniform expansion of space from a single source is associated with a higher ontological state. In this regard, it is significant that, in the preface addressed to Charles IX within the commentary on Euclid, Foix-Candale frequently appealed to terms relating to light and illumination when talking about the truth procured by philosophical or geometrical knowledge and their objects. Moreover, in the Platonic representation of truth and knowledge, the image of the Sun and of its light to represent the divine principle (the Idea of the Good) and its primordial role as first cause and source of knowledge of all things held a central place,¹¹⁸ and Foix-Candale  repeatedly referred to this conception in his preface to Charles IX . Now, this conception also clearly relates to the divine status and causal role of light in the Pimandre insofar as it allows all things to be known.¹¹⁹

4.7 Foix-Candale and Peletier on the Ontological Status of the Sphere and of the Circle

The foundational place and divine status attributed to light in Foix-Candale’s Pimandre, as well as the description of truth in terms of light and luminosity in the paratext of his commentary on Euclid, also indirectly recalls elements of the discourse held by Peletier in his commentary on Euclid and in his scientific poetry. While, in his commentary on Euclid, geometrical objects are compared to luminous phenomena,¹²⁰ in the Louange de la Sciance, Peletier offered a metaphysical representation of the primordial point-unit as instantaneously expanding in all directions to create the universe.¹²¹ As we have seen, this representation evoked, by many aspects, the role of God’s primordial light in the medieval metaphysics of light.

These common elements between the philosophical and theological conceptions of Foix-Candale and of Peletier are chiefly related to the fact that both were influenced by the Platonic and the Neoplatonic doctrines.¹²² And, in this regard, a further connection may be established between the ontological perfection and suprasensible origin which Foix-Candale attributed to the geometrical sphere, both in the commentary on Euclid and in the Pimandre, and the discourse Peletier held concerning the circle in his commentary on Euclid, where he asserted the limits of the human mind faced with the ontological perfection and divine origin of geometrical objects:

For what can we understand regarding the things which have emanated in a divine manner, when we judge them in a human manner? The circle taking therefore its origin from itself, seems to come from the rectilinear; it is infinite, and however similar to what is finite: it contains all, as it is the most capacious, but appears however to admit something exterior to itself.¹²³

This passage, which conspicuously displays the influence of Cusanus on Peletier ’s thought,¹²⁴ sets forth the contradictions inherent to the nature of the circle, as it is simple and uniform, yet able to contain all other figures; infinite, yet similar to what is finite; self-caused and self-sufficient, yet appearing to stem from something exterior to it (i.e. the rectilinear, as the straight line produces the circle by rotation). Such “coincidence of the opposites” within the circle was also expressed by Peletier in his commentary on Prop. III.1, as he said that, “in the circle, affirmation and negation come together, as do action and privation, and generation and corruption within the universe.”¹²⁵ As such, the circle may be taken as divine and as similar to the universe, and ultimately to God himself, insofar as it concentrates the properties of all things. It may thus only be defined through a series of oppositions (as described by Cusanus through his concept of coincidentia oppositorum), which place it beyond the grasp of human discursive thought.¹²⁶

These oppositions inherent to the nature of the geometrical circle, and its similarity with the divine Creator and with the universe as its material image, are made clearer in a passage of the De usu geometriae, Peletier ’s treatise of practical geometry dating from 1572:

The excellence of the circle is such that it may be rightfully regarded as the first and the last of the figures.¹²⁷ The first because it is enclosed by a single line. And for this reason it is the simplest and most beautiful of all figures. The last because it is the most capacious and largest of all, enclosing all figures in itself, the triangle, the square, the pentagon up to the infinite number of remaining figures, to which it provides rule, measure and proportion, as if all were carved out and cut off from it. And although it appears to have no angles, nor sides, it can however be said [to be composed] of an innumerable number of angles and sides, as the line may be said [to consist in] an infinite number of points, and the surface, in an infinite number of lines, in the manner we imagine God to be, infinite and immense, containing and governing all things.¹²⁸

This passage explains in particular that, if the circle is infinite and similar to God, it is because its circumference may be regarded as composed of an infinite number of angles and because an infinite number of lines join its centre to its circumference, just as the line may be held as composed of an infinite number of points, and the surface of an infinite number of lines. As was shown in the previous chapter,¹²⁹ the connection between the discrete and the continuous was only admitted by Peletier on a metaphysical level, since the mathematician may only admit the line as composed of lines and the surface as composed of surfaces. Hence, this coincidence of opposites within the figure of the circle, which allows Peletier to compare the essence of the circle to the nature of God, is to be situated on a metaphysical level. For, in a mathematical context, as Peletier added in the corresponding passage of the French translation of his De usu geometriae (published in 1573 as De l’usage de geometrie), this infinity, as expressed by the infinite number of lines joining the centre to the circumference of the circle, is only potential or virtual (infiny en puissance).¹³⁰

When dealing with the angle of contact, in the De contactu linearum from 1563, Peletier asserted again the divine nature of the circle. This assertion was then based on the fact that, within the circle, opposites coincide, but also on the foundational role of the circle in the constitution and understanding of other figures, as well as on its relation to the structure of the universe. In this context, Peletier showed that, if the circle is the first of all figures, it is ultimately because the straight line itself derives from the circle, either by being led from its centre to its circumference or by resulting from the motion of the circle on a plane, and also because the proper motion of the straight line necessarily results in a circle.

Therefore all the figures are contained and enclosed within the circle, whose circumference is made stable by the perpetual and invariable flow of the points, so that nothing may escape. All other figures have visible angles aspiring to lead to that perfect sum. Within it, the straight lines that go from the centre and that end at the periphery bend back again toward the innermost part. Others are transversal and led crosswise, so that the sight of all actions and operations appear in the greatest capacity, whose points dispense in their infinity as they are dispersed through the constitution of the wheel-shaped figure, their limitless powers being self-sustained from within while sustaining everything. For this reason, the circle is the last figure but also the first, since it is brought about by the revolution of the straight line. Indeed, the straight line cannot create through its own motion any other figure than a circle. And we take it according to this second meaning, so that we may have an art that exercises us. As it happens, the straight line does not come before the circle, since it may be understood to be created by the driving of the circle in a straight line on a plane. And that same plane is once again a circle, that which is God, one and infinite, all embracing, rendered visible by the beautiful orb.¹³¹

Peletier does not only show here that the straight line finds its foundation in the circle, but also that the straight line and the circle are ultimately one in their mode of generation, mutually causing each other.¹³² This is the reason why the circle, yet the simplest of figures, may contain all other figures and ressemble both the universe in its absolute capacity and God in its omnipotence and supreme perfection. The coincidence of opposites in the circle is also marked by the comparability of the infinite-sided polygon to the circle, although this identity remains again only virtual.

Although this is not made as clear here as in the commentary on Euclid,¹³³ this passage also suggests that the proper mode of generation of the straight line, and with it, of all rectilinear figures and even of the circle itself, may not be determined by the rational and discursive thought proper to the human mind. For, in geometry (i.e. the “art that exercises us”¹³⁴), it is equally possible to define the motion of the straight line as the cause of the circle or the rectilinear motion of the circle on a plane as the cause of the straight line.¹³⁵

Foix-Candale did not so much aim, in his commentary on Euclid’s definition of the sphere, to display the oppositions inherent to the definition of the sphere, but rather to set forth the qualities that display the divine essence and origin of this figure, that is, its simplicity, its uniformity, as well as the perfect equality of all the lines that proceed from its unique and indivisible centre to its circumference. Yet, both Peletier and Foix-Candale saw in the perfect simplicity and uniformity of the circle and sphere, which is determined by the equality of the infinite number of lines joining the centre to the circumference, the mark of the ontological superiority and divine origin of these figures. Moreover, as will be shown later,¹³⁶ Foix-Candale , in his Pimandre, asserted the coexistence of all things within God according to the mode of complication, whereby all opposites exist within him in a state of coincidence, and compared God to the unitarian and indivisible principle of quantity, both of which remain beyond the grasp of the human intellect. One finds therefore, in the philosophies of mathematics of Peletier and of Foix-Candale, common doctrinal elements shared by Christian Hermeticism and by the theological tradition of Neoplatonic inspiration stemming from the works of pseudo-Dionysius Areopagitus ,¹³⁷ and later developed by Cusanus .¹³⁸ J. Harrie actually considered that Foix-Candale followed Cusanus , at least through the intermediary of Lefèvre d’Étaples ,¹³⁹ in attributing a key role to mathematics in the theological process of salvation.¹⁴⁰ In this context, the way mathematicians consider the line, the circle or the sphere to have been generated maintains a conjectural character, though some genetic definitions (such as the definition of the sphere by Theodosius according to Foix-Candale) would be more proper than others to hint at the divine nature of these objects.

4.8 Proclus on the Properties and Constitution of the Intelligible Circle

A further connection between Peletier and Foix-Candale on this issue may also be found in the fact that their respective discourses on the circle and on the sphere both resonate with what was written by Proclus, in his own commentary on Euclid, concerning the properties and constitution of the intelligible circle, that is, the suprasensible and indivisible circle of the intellect, as opposed to the divisible circle of the imagination.¹⁴¹ In Proclus ’ commentary, the intelligible circle was shown to be both absolutely simple and containing plurality, finite and infinite, caused by an exterior principle (the straight line) and yet self-sufficient. It was also defined as resulting from a process of expansion or procession (πρόοδος) from its centre, which would be interrupted by the urge of the figure to imitate the simplicity and self-identity of its centre, causing thereby the perfect uniformity of the circle. Thus, according to Proclus , once the geometer has studied the properties of the extended and divisible circle of the imagination and has gone beyond its extendedness and spatiality to contemplate its proper essence, what will be discovered is:

the truly real circle itself—the circle which goes forth (προϊόντα) from itself, bounds itself and acts in relation to itself; which is both one and many; which rests and goes forth and returns to itself; which has its most indivisible and unitary part firmly fixed, but is moving (κινούμενον) away from it in every direction by virtue of the straight line and the Unlimited that it contains, and yet of its own accord wraps itself back into unity, urged by its own similarity and self-identity towards the partless center of its own nature and the One that is hidden there. Once it has embraced this center, it becomes homogeneous with it and with its own plurality as it revolves about it. What turns back imitates what has remained fixed; and the circumference is like a separate center converging upon it, striving to be the center and become one with it and to bring the reversion back to the point from which the procession (πρόοδος) began.¹⁴²

According to Proclus , the unfolding of the intelligible circle—an unfolding which is neither spatial nor temporal—would take place thanks to the metaphysical principle of the Unlimited and its determination, through the interruption of this process by the complementary principle of the Limit.¹⁴³ Through these two principles, the intelligible circle, in which the centre, the surface and the circumference coincide, would expand from itself and bound itself through the circumference’s desire to imitate the simplicity and self-identity of the centre, revolving around it and becoming homogenous with it. Hence, in Proclus ’ description of the properties and of the (non-spatial) generation of the intelligible circle, we find both Peletier ’s assertion of the coincidence of opposites within the essence of the circle and Foix-Candale ’s representation of the generation of the geometrical figure as a uniform expansion and bounding of its own quantity. We also find a justification for Foix-Candale’s dismissal of Euclid’s definition of the sphere in favour of a definition that, to him, properly displays the true cause of the simplicity and uniformity of the sphere, as well as its eternal and divine condition and origin.

Moreover, Peletier ’s assertion of the similarity between the circle and God himself, just as Foix-Candale’s representation of the Theodosian sphere as an image of the ineffable and eternal essence of this figure, resonate with Proclus ’ words, when he wrote that the circle, as it enfolds on itself, is urged to return back to its indivisible centre and to “the One that is hidden there”. According to the Neoplatonic doctrine derived from the philosophical teaching of Plotinus (ca. 204–270)¹⁴⁴ and on the basis of which Proclus developed his metaphysics,¹⁴⁵ the One (τὸ Ἕν) would correspond to the first principle of all things and would produce everything according to a hypostatic and atemporal mode of causation.¹⁴⁶ The Intellect (νοῦς), as the second hypostasis, would proceed or emanate from the One, and the Soul, as the third hypostasis, would proceed from the Intellect, and so forth until all lower degrees of reality have come to existence, each of them ultimately owing their essence and existence to the One.¹⁴⁷ In Proclus ’ version, this was held to take place at each stage through a three-fold process consisting in rest (μονή), procession or progression (πρόοδος) and return or reversion (ἐπιστροφή),¹⁴⁸ which is applied to the generation of the intelligible circle in the above-quoted passage of his commentary on Euclid.

In Peletier ’s characterisation of the circle, this three-fold process is therefore expressed through the series of oppositions inherent to the nature of the circle, whereby the simple, the finite and the self-caused is conjoined with the multiple, infinite and the caused. And in Foix-Candale ’s commentary on Euclid, it is chiefly expressed through the geometrical notion of progressus, as well as through the notion of sphere as generated by the expansion of its magnitude from its centre and by the uniform interruption of this expansion (as based on his notion of figure in his commentary on Df. I.13).

4.9 The Centre of the Sphere as a Divine Principle

As Proclus did for the intelligible circle, both Peletier and Foix-Candale granted a privileged role to the circle and the sphere, respectively, among geometrical figures, given that it corresponds to the first expression of the divine principle. Now, this role was also granted to these figures in view of their ability to express the unity of the centre, as they would both result from its uniform expansion in all directions in the plane or in three dimensions. For this indivisible centre ultimately represents the first image of the unity of an indivisible and all-encompassing God, from which the universe and all it contains has proceeded. Indeed, while Peletier posited, in the Louange de la Sciance, a unitarian divine principle from which the universe instantaneously emanated,¹⁴⁹ this conception was expressed in Foix-Candale’s Pimandre through the representation of the divine Creator as a unitarian principle, from which all things would have proceeded according to the mode of multiplicity, conforming to the Hermetic theological doctrine.¹⁵⁰

Thus, if Foix-Candale chose to express the mode of generation of magnitudes through the notion of progressus rather than through that of fluxus in his commentary on Euclid, it is not only because it allowed him to avoid representing the generation of geometrical figures in an instrumental and non-essential manner and because it could to a certain extent solve the issue of the composition of the continuum. But it is also, more fundamentally, because it offered a more adequate representation of the true mode of procession of magnitude from the point, which mirrors the procession of the multiple from the divine One.

Therefore, when Foix-Candale wrote, in his commentary on Euclid’s first definition, that the point is the efficient cause of quantity (signum quantitatem agens),¹⁵¹ it would not be because it would have generated the line through its spatial translation and, from there, would have enabled the surface and all other magnitudes to be caused. It was rather because it represented the undivided principle of all quantity, which compares to God and to his creative operation. Foix-Candale’s discourse on this issue should then be taken on a philosophical rather than on a mathematical level. This is again confirmed by what he later wrote in the Pimandre, as he asserted that the undivided unit conceived by mathematicians is what allows us to understand the unity of God in the most adequate manner:

There is nothing among us that shows us the nature and divine essence to a greater extent than this unit, which we have said is the one and only beginning of all things, which truly belongs to none other than to the supreme God. It is indivisible, continuous (as geometers say) as opposed to number, which is discrete or divided. In the same manner, we understand God as one, indivisible and whole in its entire essence, different from all its creatures composed of differents units, all taking their beginning in this sole divine unity […]. Thus, as all numbers and composed things start from the unit, God is the one and unique beginning of all things, since he is the only being whose essence is true, firm and stable, and from which all things that have an essence necessarily receive their essence, as he takes his beginning from nowhere else than from himself.¹⁵²

It is important to note that, in this text, Foix-Candale does not make a clear distinction between the unit as the indivisible principle of magnitudes, that is, the point, and the unit as the principle of numbers. The unit, taken in this sense, is not either clearly distinguished from God as the divine One, nor numbers from created beings, which are themselves said to be divided into a multiplicity of units, setting forth the Pythagorean element of the Hermetic cosmogonic doctrine.¹⁵³

Now, in the same context, Foix-Candale referred to his previous geometrical discourse, which is undoubtedly his commentary on Euclid’s Elements,¹⁵⁴ and designated the indivisible principle of quantity as “confused and undetermined”.¹⁵⁵ In the last preface of his commentary on Euclid, as he presented Aristotle ’s concept of quantity, the geometrical point is indeed said to join the parts of continuous magnitude in a “confused manner” (confusè), making it one and undivided.¹⁵⁶ Geometrical magnitude is then also often designated, as it also was in the Pimandre,¹⁵⁷ as a confused and continuous quantity by opposition to the discrete quantity dealt with in arithmetic.¹⁵⁸

While this description chiefly referred to the fact that geometrical magnitude cannot be defined as actually composed of discrete indivisible parts, Foix-Candale also asserted, in the same preface to his commentary on Euclid, that the unit corresponds to the divine principle common to both arithmetic and geometry.¹⁵⁹ By saying this, he was indirectly pointing to the metaphysical understanding of the relation between number and magnitude at an ontologically superior level.

Hence, if it may be striking for the indivisible principle of magnitude to be described as continuous (as it is in the above-quoted passage), since it was defined as deprived of parts in Euclidean geometry, Foix-Candale aimed, through this characterisation, to present the point as both indivisible and as containing all magnitudes virtually. Through this, it could be simultaneously regarded as the proper cause of all magnitudes and as the expression of the oneness and absolute capacity of God, in which all things lay complicated. This omnipotent unit recalls the centre of the circle in Peletier ’s De usu geometriae, which was said to be infinite in the sense that it virtually connects the infinite number of lines proceeding to the circumference.¹⁶⁰ The same may be said of the centre (or innermost part) of figures in Foix-Candale’s commentary on the Elements, which is conceived as the starting point of the expansion of magnitude.

Therefore, if Foix-Candale related the point to the arithmetical unit, both in his commentary on Euclid and in the Pimandre, the condition of the point-unit itself, being characterised as continuous and confused in its oneness in the Pimandre, would be closer to the condition of magnitude than to that of number.¹⁶¹ Now, it is the oneness that is common to both magnitude and to the divine principle of all quantity that allows geometrical objects, and notably the sphere, to express a higher state of being, which is eternal and ineffable.

4.10 Foix-Candale and Peletier on the Epistemological Status of Geometrical Definitions

As was suggested in the previous section, Foix-Candale also joined Peletier concerning the epistemological status of geometrical definitions, and of genetic definitions in particular, by admitting their conjectural character. Indeed, as was shown earlier, Peletier asserted, both in his commentary on the Elements and in his De contactu linearum,¹⁶² that the causality posited by the geometer between different geometrical objects (the point and the line, or the straight line and the circle) has no relevance on a philosophical level, since it cannot capture the simultaneousness of their causation and their proper essence as entities of divine origin. As such, he advised us to express geometrical concepts in the manner that is most useful for the study and practice of geometry, while keeping in mind their conjectural character.

Foix-Candale first suggested the conjectural character of geometrical definitions in his commentary on Euclid’s definition of the sphere, by stating that the essential definition of the sphere (that of Theodosius ), even if it is more appropriate than Euclid’s definition to express the nature of the sphere, only offers an image of its eternal essence. Given the Platonic background of Foix-Candale ’s representation of mathematical knowledge in the prefaces of his commentary on Euclid, and the ontological conceptions he presented in the commentary on the Elements and in the Pimandre, ¹⁶³ the notion of image would entail a distinction between the true state of the geometrical objects within God’s mind and the way they are apprehended and defined in the context of geometry. This would also be the case of essential definitions, even if they offer a more faithful expression of the true essence of geometrical objects than mere descriptions, to which belonged Euclid’s definition of the sphere.

In particular, the fact for the sphere to be, as any other geometrical figure, represented spatially in the imagination would make it, for Foix-Candale, improper as such to display its intelligible essence, since he wrote in the Pimandre that: “ideas that are in themselves a depiction or a figure in the mind only depend on corporeal things, since incorporeal things have no figure or depiction presented to the senses”.¹⁶⁴ What is more, within God, the ideas that are spatially represented to the human mind would have no corporeality, as they are only manifested to the divine mind in their intelligibility and deprived of any spatiality.¹⁶⁵ In other words, the process of expansion the geometer unravels in his imagination as he studies the properties of a figure (to which Foix-Candale refers in his commentary on Df. I.13) would itself correspond to an improper conception of a geometrical object’s true state. Indeed, even if it is not taken to occur in physical space and time, it nevertheless implies a succession of states of the figure from the beginning to the end of the process, through which the centre, the surface (or volume) and the boundary of the figure may be distinguished.

Hence, while corporeal objects or figures are images of intelligible principles contained in God, one must not, according to Foix-Candale , linger on them and seek within them the truth to which they refer.¹⁶⁶ In this regard, the properties and mode of generation displayed by the Theodosian definition of the sphere would point, for Foix-Candale, to the essential condition of the sphere without however directly representing the latter in its true state, that is, as an intelligible and eternal idea within God’s mind. This is precisely why this eternal essence of the sphere, in the commentary on Euclid’s Elements, is said to be “ineffable” (ineffabilis), that is, beyond all expression.

The conjectural character of geometry is even more clearly asserted by Foix-Candale when dealing with the distinction between numbers and magnitudes, both in the last preface of the commentary on Euclid and in the Pimandre, as the nature of magnitude is then said to exceed the human mind because of the indistinction of its parts.¹⁶⁷ For this reason, its properties as quantity may only be understood by the human reason through numbers, which are divisible into units and therefore commensurable. Indeed, for Foix-Candale, the term “confused” as applied to the point-unit and to magnitude is both understood in the mereological and in the epistemic sense, that is, as denoting, on the one hand, mingling, continuity and complication and, on the other hand, intellectual confusion. As such, numbers were regarded by Foix-Candale in both commentaries as a means of unravelling or explicating what remains complicated and undivided, making magnitude, but also the unit common to all quantity, and God himself thereby, more accessible to the human mind. Just as God is apprehended by the human mind through the multiplicity of his creatures, the nature of quantity in general, and of its undivided principle, is apprehended through numbers, whose mode of composition may, on the other hand, be fully apprehended and understood by the human intellect.¹⁶⁸

In other words, if the Theodosian definition of the sphere expressed the true essence and genesis of the sphere more adequately than the Euclidean definition, this definition would remain however, for Foix-Candale, an imperfect expression of an essence and mode of being that is fundamentally beyond the grasp of the human intellect.

Now, the fact that this geometrical representation possesses a conjectural character (whether it is or not more suitable to represent the ontologically superior state of the sphere) should allow us, in principle, to place the motion implied by the progressus of the line or the expansion of figures on the same ontological and epistemological level as the genetic definition of the line as resulting from the flow of the point. This latter definition was indeed regarded by Peletier , in the Louange, as a mathematical expression of the divine process of emanation of the cosmos from the primordial One. And he also regarded it as an acceptable expression of the mathematical properties of the line in his commentary on the Elements.

Thus, if Foix-Candale avoided using genetic definitions which attributed a translational or rotational motion to geometrical objects, it was for other reasons than merely its conjectural character. It was, first of all, because such definitions raised philosophical issues that would impede the properly rational and human understanding and investigation of geometrical notions, notably with regard to the composition of continuous quantity. It was also because it set forth an accidental and quasi-mechanical mode of generation of geometrical objects, rather than one that would be conform to its essential condition and properties. In effect, as we have seen through Fine’s case, the genetic definition of the line through the flow or motion of a point could be interpreted as an abstract representation of an instrumental process, such as the dragging of the pointed edge of a stylus on a wax tablet. This definition of the line would therefore have corresponded, for Foix-Candale, to a descriptio, which would teach the rule or modus operandi followed by the geometer in order to produce a line concretely or imaginarily. Hence, the notion of flow of the point would be closer, in the way it represented the generation of magnitude, to the rotation of the semicircle producing the sphere in Df. XI.14 than to the expansion of the sphere’s magnitude from its centre to its circumference. As said, the dismissal of descriptiones as proper definitions on account of their instrumental character would be confirmed by Foix-Candale’s rejection of mechanical processes in geometry.¹⁶⁹ His rejection of superposition on account of its interpretation as an instrumentally-performed procedure would corroborate this.¹⁷⁰

Thus, if, for both Foix-Candale  and Peletier , the definitions through which geometers express the essential properties of the circle and of the sphere may only be regarded as an attempt to capture, in spatial terms, the proper essences and geneses of these figures, which are in their true state complicated in God’s mind and which exceed the limits of human understanding, it remains that, while Foix-Candale dismissed the Euclidean definition of the sphere by rotation of a semicircle in favour of the Theodosian definition, for Peletier the conjectural character of mathematical definitions precisely legitimated the use in geometry of a diversity of genetic definitions. As Peletier wrote in his commentary on Euclid’s Elements, since all geometrical forms, in their divine state, have appeared simultaneously, the geometer would be perfectly free to admit the straight line as the cause of the circle or, on the contrary, to assume the circle as the cause of the straight line according to the needs of his research and practice.

4.11 Motion in Foix-Candale’s Commentary on Book II

On a very different note, I now turn to Foix-Candale ’s commentary on Euclid’s book II, and more specifically on Df. II.1, where he found useful, as most of the commentators considered here, to appeal to motion. Now, despite the fact that he defended the use of numbers to clarify the principles of geometry in the paratext of his commentary on the Elements,¹⁷¹ he did not however appeal to an arithmetical interpretation of this definition, nor of any other part of Book II for that matter.¹⁷²

In his commentary on Df. II.1, he appealed to motion in order to explain why the parallelograms considered by Euclid in the context of Book II are necessarily rectangular.

The right angle, as it consists in a unique inclination of lines, expresses the same breadth on account of its length, admitting no other measure. This surely is not the case of the acute or obtuse angle, since there could be, for each, infinite differences in the inclination of the lines. Euclid says therefore that the rectangular parallelogram is formed under the two [lines] containing the right angle, so that one of them, that is, its length or its breadth, expresses the other without any error. […] If we wanted however to understand this through motion, we would say that, as far as the point A will move along the line AB until [it reaches] the point B, as much each part of the straight line AD will simultaneously be produced until [it reaches] the straight line BC; and this motion will describe the rectangular parallelogram ABCD, since the motion of the length takes place at right angles. If however the point A is moved along AE according to an oblique angle, the motion of the line AD will similarly describe a rhombus or a rhomboid parallelogram (Fig. 4.1).¹⁷³

Fig. 4.1

François de Foix-Candale, Euclidis Megarensis mathematici clarissimi Elementa, Libris XV, 1578, p. 33, Df. II.1. Diagram illustrating the extent of the variation of the angle DAE. It represents the angle BAD in the cases when AD does not stand at right angles with AB and shows the extent of the variability of the quadrilateral generated thereby by the motion of the line AD along AE. Courtesy Max Planck Institute for the History of Science, Berlin

The motion introduced here is not one that would properly generate the surface of a parallelogram, but rather one that would measure a parallelogram ABCD that is already given, since the given point A is said to move along the given line AB while the given line AD is said to move along the given line AB to reach the given line BC. This use of the motion of geometrical figures may be interpreted according to the Greek terms διεξέρχομαι, διέρχομαι and δίειμι, such as used by Autolycus of Pitane (c. 360—c. 290 BC) in his treatises On the moving sphere and On risings and settings and by Pappus , in the Collections, to designate the motion of a point on a curve, since then the point or the line is said to move along an already given line or surface.¹⁷⁴ The intention of the geometer when appealing to this notion may of course differ, as it relates in Autolycus to the motion of a celestial body along a circle, which represents its path in the firmament.

Also, if what is conceived as properly moved, in Foix-Candale’s commentary on Df. II.1, is only one of the extremities (the point A) of the moving line-segment (AD), this point is also understood to carry with it the whole line-segment AD along the length of the line-segment AB, which is situated at right angles from AD. In this way, the motion of the point A could be imagined to draw out the whole area of the parallelogram ABCD. This procedure would then be understood as a reconstitution of the generative process through which the parallelogram would have been initially drawn out.

Hence, the motion through which the parallelogram is said to be measured in this context, given that it also takes place through the transversal motion of a line-segment across a given area or surface, would be quite similar, as for its ontological status, to the motion by which the semicircle is said to generate the sphere in Euclid’s Df. XI.14 and which Foix-Candale considered as accidental. If this process were formulated as a genetic definition, it would be equivalent to the definition of the rectangular parallelogram as generated by the transversal flow or motion of a line, then corresponding to a descriptio.¹⁷⁵ Thus, for Foix-Candale, this motion would be different in its nature and ontological status from the progressus or expansion referred to in the first definitions of Book I, and which was presented as intrinsic to the generated magnitude, even if it only corresponds to a humanly apprehensible image of a metaphysical process. And that would be the case whether it is the point A or the whole line-segment AD that is conceived as the mobile element and whether this motion is conceived as a properly generative process or rather as a recreation of the figure in the aim of measuring its area.

As was noted above, even if Foix-Candale explicitly presented the two sides of the parallelogram as measuring its area, as well as mutually measuring each other (as would be expected in the context of Book II), he did not go as far as to compare, as did Fine and Peletier , the generation of the rectangular parallelogram to the multiplication of two numbers. Indeed, if he stated the usefulness of arithmetic to understand the principles of geometry in the paratext of his commentary, for which he justified Euclid’s inclusion of arithmetical books in the Elements ¹⁷⁶—he considered indeed the Elements first and foremost as a geometrical treatise¹⁷⁷ –, he did not however consider it appropriate to directly express the propositions of Book II in arithmetical terms, as some of commentators did.¹⁷⁸ For that matter, according to Foix-Candale, the arithmetical content of the Elements only aimed to help the reader gain a full understanding of the properties and mutual relations of continuous quantities, which are by themselves source of confusion for the human intellect.¹⁷⁹ Therefore, if, at a philosophical and theological level, numbers and magnitudes are thought to have sprung from the same principle and are conjoined in God’s mind, it remains that, for the human mind and in a mathematical context, discrete and continuous quantities themselves are to be clearly distinguished. This, by itself, confirms that one of the reasons why Foix-Candale avoided the notion of flow of the point, beside its quasi-instrumental and non-essential character, is that it would overthrow the distinction between discrete and continuous quantities, as for their proper modes of composition.

As we have seen, Foix-Candale provided, in his commentary on Euclid, genetic definitions of the line, of the surface and of figures in general, but not as generated by the flow or motion of a point or a line, despite his assertion of the causal role of the point in relation to continuous quantity. Instead, he defined geometrical objects as originating from the progressus or expansion of magnitude in one, two or three dimensions and by the interruption of this process by the corresponding extremity or boundary (point, line or surface).

If Foix-Candale did present the point as the efficient principle of magnitude in his commentary on Df. I.1, it would be linked to his philosophical conception of the point-unit as a common principle of geometry and arithmetic, to which he attributed a divine status. This was expressed in more explicit terms in his commentary on the first treatise of the Corpus hermeticum, the Poimandres. In this context, the unit, presented as the principle of quantity in general, is what allows us to understand, to the best of our abilities, the nature of God, which exceeds the limits of the human mind. Just as all things are indistinctly present in God in an intelligible manner, the unit virtually contains all magnitudes, as well as all numbers, within it.

Although this discourse is to be placed on a metaphysical rather than on a mathematical level, it resonates with the notion of expansion of the figure that was presented in the commentary on Euclid. As Foix-Candale expressed it when commenting on Euclid’s definition of the sphere, Theodosius ’ definition, which suggests the expansion of the sphere’s magnitude from its centre, would offer an image of the eternal and ineffable essence of this geometrical figure, which, as made clear in the Pimandre, surpassed all geometrical figures in perfection. According to this Hermetic text, the universe would have been modelled by God in the shape of a sphere in view of its ability to express omnipotence and unity. Within the universe, the process of expansion of the sphere from a point would be represented by the expansion of light from the Sun, which represented the physical expression of the divine light that causes and makes all things known.

Foix-Candale ’s representation of the sphere echoed in various ways Peletier ’s representation of the circle, which stemmed from their common Platonic and Neoplatonic influences, as related to Renaissance Christian Hermeticism and Cusanian theology and epistemology. Hence, in line with Proclus ’ characterisation of the circle, Peletier and Foix-Candale ultimately asserted the coexistence, within the intelligible circle or sphere, of the centre, the figure and the boundary, as well as the procession of the figure from itself according to a non-temporal and non-spatial mode. In this framework, the power of the point to unravel into the circular or spherical figure and to enclose its own magnitude into a uniform whole would be identified with the power of the divine One from which all things would have proceeded and in which all things coexist indistinctly according to the Neoplatonic doctrine.

While most of the medieval and Renaissance scholars who compared the two definitions of the sphere by Euclid and Theodosius distinguished these in view of the former’s appeal to motion, considering that the latter did not imply any motion, Foix-Candale would have considered both as involving motion. And while the motion implied by the Theodosian definition of the sphere would only provide a conjectural representation of this ontologically superior state and mode of causation, it would still be situated at a greater degree of truth compared to Euclid’s definition of the sphere. Foix-Candale actually did not regard this definition as a proper definition given that it would only present the accidental and externally-determined properties and mode of generation of the sphere, for which it was designated as a descriptio. The same would have been held by Foix-Candale of the more traditional genetic definitions of the line, surface and solid as generated by the flow of a point, line or surface, respectively, as these could all be taken to evoke their mechanical generations.

Thus, while it could be conceded that a descriptio may be useful or even necessary to account for the spatial properties of the line, as was asserted by Fine and Peletier , Foix-Candale did not generally appeal to these notions in this aim, save in Book II, where he presented the area of the rectangular parallelogram as generated by the transversal motion of one of its sides. Yet, in this context, the fact of presenting the rectangular parallelogram as produced by the motion of a line-segment, and not as resulting from the expansion of magnitude in two dimensions, would be acceptable insofar as Foix-Candale did not intend then to provide a universal definition of this figure, but rather to express the operation performed by the geometer when assessing the quantitative relation of its sides to its area.

Yet, contrary to Fine and Peletier , and to the other commentators considered here, Foix-Candale did not go as far as to directly compare the generation of the rectangular parallelogram to the multiplication of two numbers in his commentary on Df. II.1. Although he considered that the understanding of geometrical properties required the study of arithmetical principles, he mostly avoided appealing to arithmetical notions in the context of Euclid’s geometrical propositions.

Therefore, like Peletier , Foix-Candale acknowledged the ontological and epistemological boundary between the mathematical treatment of geometrical objects and their philosophical consideration while attempting as much as possible to make them mutually compatible. In doing so, he acknowledged the difference between properly essential and accidental, or intrinsic and extrinsic, kinematic processes in order to avoid attributing a concrete and somewhat mechanical type of motion to intelligible beings. He nevertheless diverged from Peletier insofar as the latter considered it perfectly legitimate to define the circle according to the mode of a descriptio in the context of geometry on account of the necessarily conjectural character of the geometer’s knowledge concerning the true essence and geneses of his objects of study.