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  • The Picture Multiple:Figuring, Thinking, and Knowing in Descartes's Essais (1637)

Throughout his Essais (1637), Descartes appropriated the visual language of practical mathematics in order to forge a new natural philosophy. This article argues that by grafting geometric line onto descriptive figure, the philosopher and his illustrator, Frans van Schooten Jr., underscored doubts about a natural philosophy based on qualities, all the while situating his new epistemology in the 17th-century present and exercising a deep attention to the differences between nature seen, nature pictured, and nature understood.


René, Descartes, Practical Mathematics, Early Modern Science, Early Modern Visual Culture, Graphic Epistemology

Why did early modern natural philosophers begin to picture a geometrically analyzed world? The problem finds unique and early expression in two of René Descartes's EssaisDioptrique and Météores—published as compendia to the Discours de la méthode in 1637. With these texts, Descartes rejected the strictures of conventional Aristotelianism taught in schools; re-oriented how one might obtain knowledge; and, by extension, sought new ways of describing his approach. The bulk of the historiography about the Essais remembers the philosopher trading Latinate erudition for the French vernacular; the confident subjectivity that propelled his "je pense donc je suis"; the swagger with which he washed his hands of the ancients; how his proclivity for experiment, intuition, and deduction countered the bookish accretion of the most calcified forms of textbook natural philosophy.1 Radical though his positions were, Descartes circumscribed them with care. Galileo's condemnation in 1633 had startled the French philosopher into [End Page 369] formulating his Essais as a preview of his new philosophy.2 Throughout the new book were glimmers of the old ways—relics of Eustachius a Sancto Paulo, whose logic bore a strong resemblance to Descartes's intuitive reasoning; amendments and qualifications that Descartes's old teachers at the Jesuit collège La Flèche had urged him to make.3 The philosopher adhered to uncontroversial, earthbound topics while leaving the heavens unharmed and immaculate. Despite his caution, Descartes filled his Essais with a most potent means for overturning Aristotelian natural philosophy's forms and qualities: illustrations of descriptive figures negotiated with geometrical construction. These were not the Porphyrian logic trees and cosmological tables to be found in scholastic textbooks, but analytical forms whose schema were familiar to any who had dabbled in mixed mathematics—a practice typically held apart from natural philosophy.

By incorporating such pictures into his Essais, Descartes challenged the presumptions of Aristotelian natural philosophy's forms and qualities, according to which an object's persistent essence—as reflected by its appearance—underpinned its constitution and mutation.4 Descartes's illustrations employed mathematical inscription to penetrate what could be seen in the world, thus encouraging readers to reconsider how the eye saw and what kind of knowledge it could secure. To explain the refraction and reflection of light, Descartes and his illustrator, Frans van Schooten Jr., imaged tennis players plucked from their courts and stamped with circles, diameters, and ninety-degree angles (fig. 1). Rainbows were primed for analysis through well-outfitted gentlemen peering up at arcs-en-ciel plied with circles and angles of refraction and reflection (fig. 2). The movement of light particles was likened to vats full of grapes, which fell and filled out empty space as wine was pressed from their skins (fig. 3). Sight itself was portrayed as a camera obscura (fig. 4), whose aperture was outfitted with a diagram of the eye, while a bearded man sporting a comfortable hood [End Page 370] looked on. With figures grafted by geometry, Descartes invoked a world that could no longer be described through its immediate appearance. An object's visible characteristics failed to offer an unimpeachable account of the nature that had produced it; instead, an object's extension and the geometries extracted from that extension offered a more definitive account of the object's composition and behavior in the world. Descartes's pictures thus instructed readers in a technique that subverted any certainty about deriving natural causes from what could be seen.

Figure 1. Illustration by Frans van Schooten Jr. in René Descartes, Discours de la méthode … (Leiden: J. Maire, 1637), 15. RB 336029, the Huntington Library, San Marino, California.
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Figure 1.

Illustration by Frans van Schooten Jr. in René Descartes, Discours de la méthode … (Leiden: J. Maire, 1637), 15. RB 336029, the Huntington Library, San Marino, California.

The Essais' illustrations evidence the slow burn of Descartes's conceptual innovation and the larger process by which nature was mechanized throughout the seventeenth century. Historians of the Scientific Revolution long assumed that the new philosophy lent itself to a streamlined mathematization, straightforward geometricization.5 Recent work has emphasized [End Page 371]

Figure 2. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 251. RB 336029, the Huntington Library, San Marino, California.
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Figure 2.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 251. RB 336029, the Huntington Library, San Marino, California.

the virtues of a nuanced approach to the imaging of the new physics, especially with reference to Descartes's visual output. Brian Baigrie led the way by characterizing Descartes's pictures as laying bare nature's mechanisms and thereby amounting to a set of tools for the new epistemology.6 Claus Zittel's Theatrum philosophicum masterfully extends Baigrie's remit, and charts how readily Descartes conjoined perceptual figures with conceptual diagrams, the inductive and the deductive, the mathematical and the fictional, throughout his oeuvre. With help from work by Daniel Sepper and Peter Galison, Zittel asserts that the imagination was Descartes's crucible for melding these dialectics together. Zittel identifies Descartes's knowledge [End Page 372] as mutable and shifting—even oscillating—between empiricism and cogitation.7

Figure 3. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 6. RB 336029, the Huntington Library, San Marino, California.
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Figure 3.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 6. RB 336029, the Huntington Library, San Marino, California.

Building on its predecessors, this essay stresses the visual cultures that were in closest proximity to Descartes as he devised his Essais. It attempts to associate Descartes's doubts about vision with a graphic habit: that of practical mathematicians, especially those who were trained at the Nederduytsche Mathematique in Leiden. True, the technique of laying geometry on top of a figure did have other, older sources—in, say, Theodoric of Freiberg's De iride (1305) and Walther Hermann Ryff's Architektur (1547). In considering how Descartes spurned a long history of geometricized figuration, one gains a deeper understanding of Descartes's emphatic orientation to his seventeenth-century present, and how its scientia entailed a process through which the mind must attentively reinterpret what the eye has observed. The pictures that Descartes incorporated into the Essais were not only emblematic of this new epistemology; they demonstrated this approach to knowing and interpolated its various elements so well that [End Page 373]

Figure 4. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 36. RB 336029, the Huntington Library, San Marino, California.
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Figure 4.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 36. RB 336029, the Huntington Library, San Marino, California.

[End Page 374] they forged the new epistemology. The first section of this paper thus puts Descartes's graphic epistemology into relation with the Nederduytsche Mathematique's visual practices, evidencing the close contact that the philosopher had with its practitioners, and how the former appropriated the visual preoccupations of the latter. In part two, I examine how such contact transformed Descartes's thinking about sensation and the mind's apprehension of sensation. Part three makes a strong case for the seventeenth-century accoutrements that Descartes integrated into his pictures, and thus the presentist attitude ideally cultivated by those practicing the new philosophy. To conclude, I briefly sketch why pictures were foundation and condition for the new philosophy. This exercise goes beyond showing that pictures were important to Descartes. It casts the philosopher's epistemology into larger visual relief, and demonstrates how the skeptical foundations of Cartesian knowledge were inscribed into line and figure.


Descartes was relatively silent about the sources for his combinatory images. No extant document maps the pictorial collaboration he undertook with Van Schooten Jr., or describes why the Essais' pictures look the way they do. The figures, however, were indispensable. Initially, Constantijn Huygens had counseled Descartes to find a draftsman well versed in philosophy, ultimately leading him to Van Schooten Jr.;8 and he consoled the philosopher when the production of the woodcuts delayed the printing of the Essais.9 Beyond this, Descartes's correspondence little anticipates his approach to imaging the world. On occasion he exchanged diagrams with the artisan Jean Ferrier, his trusted mathematician confidante Marin Mersenne, and possibly mechanist Isaak Beekman; these were either mechanical drawings or purely geometric.10 The few passages in which the philosopher describes any facility with pictures communicate a little more: he believed images to be indispensable to conveying ideas;11 and he expressed regret at his poor abilities as a draftsman.12 [End Page 375]

But, curiously, in January 1619, while volunteering for Prince Maurice of Nassau's army, Descartes began to learn how to draw. As he wrote to Beeckman on January 24, 1619: "Lest, however, you consider me so lazy as to have quite uselessly filled my time, [I can tell you that] I am occupying myself with matters which your ingenium, busy with other things, doubtless pays no attention to, and looks down upon from the elevated heavens of the sciences: namely, drawing, military architecture, and, particularly, Flemish."13 Descartes's sentence reveals much about early seventeenth-century scientia and the place of pictorial representation within its study. The young Frenchman's sharp distinction between transcendental scientiaand the artisanal, applicable, and quotidian was commonplace.14 Descartes, however, angled at an even subtler difference, for Beeckman himself had already broken this "habit." An early promoter of the mechanical philosophy, the Dutch philosopher had little aversion to the manmade and experimental; he was convinced that the marriage of ars, techne, and scientiawould result in the better comprehension of both the world's physical properties and its universal laws.15 In fact, Descartes's early visits to Beeckman's workshop had awakened the Frenchman's fascination with physics precisely because Beeckman put artificial mechanism at the center of his reformation of natural philosophy.16 Descartes's letter makes an even subtler observation still: even the most artisanal of natural philosophers were assumed to give little thought to pictures and words. Ars was integral to the generation of true knowledge, but graphic representation was negligible.

Despite this, Descartes sought out visual practices that supplemented the abstractions he had learned in school. His teachers at La Flèche had guided him through the commentaries on Aristotle's Physica (published in 1592) by the Portuguese Jesuit Coimbrans and Collegio Romano professor [End Page 376] Franciscus Toletus's De physica auscultatione (1580).17 These trained him in the verbal traditions of explanatio and argumentum, disputationes and respondens.18 The use of diagrams in such texts was rare, conceptual, and geometric. The Euclidean gnomon—that L-shaped polygon leftover when a similar parallelogram is extracted from the original parallelogram—was one of the few examples that Descartes would have encountered in Toletus's De physica auscultatione. Figures like those in the Essais were nowhere to be found in the image-poor texts at La Flèche.19 Between 1614, when he left the collège, and before reaching Maurice's station at Breda in 1619, Descartes spent a year studying civil and canon law at the Université de Poitiers. This left him restless. As he later put it in Discours, his curiosity directed him well beyond the scholar's cabinet:

As soon as I was old enough to leave the supervision of my instructors, I completely abandoned the study of letters. And, resolving to seek no longer any other science than the knowledge I could find in myself, or in the great book of the world, I spent the rest of my youth traveling: visiting courts and armies, associating with men of different characters, with different lots in life; collecting myriad experiences; proving myself in the situations that fortune set down before me; and, above all, so reflecting on those experiences that I could gain something [from them]. For it occurred to me that I should find much more truth in the reasoning of each man concerning the affairs that mattered to him [—the affairs] in which he must be swiftly punished if he judged poorly—than in those conducted by a man of letters in his study, regarding speculations that have no effect[.]20 [End Page 377]

Descartes pursued experience with zest. His news to Beeckman of drafts-manship, fortification design, and Flemish further indicates that those new encounters with all sorts of people and in so many places had piqued an interest in how best to represent such experience. For if the abstractions he had learned in school were devoted merely to "speculations," only more descriptive modes could give form to his contact with "the great book of the world."

The circumstances under which Descartes was stationed at Breda were ideal for developing graphic skills. Maurice had won peace for the United Provinces. He had beaten back the threat of civil war ignited by the rift between the Arminians (who had restored free will to human dignity) and the Gomarists (who rejected the doctrine of free will in the name of a strict Calvinism); and the Synod of Dort was well on its way to reconciling those two factions and restoring order across the land. Although the threat of Spain usually loomed large, this was year eight of a twelve-year ceasefire.21 Both time and manpower were abundant. Descartes learned to draw among Maurice's engineers, who had gained their graphic know-how at the Nederduytsche Mathematique. Founded by the prince in 1600, the Mathematique was an engineering school in Leiden whose purpose was to train a reserve of surveyors, rangefinders, and military architects skilled enough to sustain the fight against Spain. Simon Stevin—the man who had tutored the prince in practical mathematics, and whom Maurice kept close at hand as both a personal adviser and as quartermaster general for the United Provinces' army—had designed the Mathematique's curriculum. In addition to his talent for making geometry and calculation useful, the Brugesborn engineer was an avid proponent of the clarity of the Flemish language and had a knack for explaining with pictures.22 With Maurice's support, a practical, vernacular, and visual pedagogy thrived.

The Mathematique's courses took men who were assumed to possess no knowledge of mathematics and transformed them into graphic-savvy practical geometers. Each cohort was comprised of carpenters, masons, schoolteachers, and even curious university students from around the United Provinces.23 Their Mathematique educations began with basic arithmetic and the fundamental principles of Euclidean geometry; they learned [End Page 378] all their lessons in Dutch. Students began by learning the definition of a point; went on to recognizing intricate polygons; and, once they had become comfortable with multiplication and division, were then introduced to the art they would most frequently exercise as engineers: trigonometry. Literally the "measurement of triangles," in which knowledge of one side of a triangle and two of its angles could help the mathematician calculate the length of the triangle's other sides, trigonometry was the key operation through which engineers reduced the world to mathematical constructions. With triangles, they measured the distances of faraway landmarks; sited the location of enemies; plotted parcels of land.24 The Mathematique's students honed their trigonometric acumen by "draw[ing] the outlines of lands which they [had] to measure, and further identifying a place on the same page for drawing a clear landmark."25 Figuration laid the groundwork for trigonometric functions to come. It also revealed that drawing at the Mathematique—or teckenen—encompassed three forms of representation: descriptive drawing;26 the abstraction of a visible, tangible object into a mathematically manipulable shape (or "outline"); and the lines and angles of geometric analysis. When the practical geometer brought these visual modes together, he performed a graphic balancing act in which figural objects were preserved even in the face of geometric construction.

Mathematische werken—a manuscript of lessons compiled, circa 1620, by one of the Mathematique's most popular instructors, Frans van Schooten Sr.—attests to the intensity with which a figured geometry guided instruction.27 For example, the problem represented in figure 5 instructs the [End Page 379]

Figure 5. Frans van Schooten Sr., Mathematische Wercken, fol. 83r. BPL 1013, Universiteitsbibliotheken Leiden.
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Figure 5.

Frans van Schooten Sr., Mathematische Wercken, fol. 83r. BPL 1013, Universiteitsbibliotheken Leiden.

student to measure the distance between two "peaks," A and B, with the length of base CD given. The landscape and its most prominent landmarks are outfitted with details that overwhelm Stevin's original advice to "out-line": at the top of the ridge stands a church whose nave has been extended by a few piecemeal additions. Toward the bottom are woolly trees and bushes. To the left of point B is a curious stony absence whose shape seems to resolve into a nose, chin, and neck. Grass spills fluidly down the mountainside, from one promontory to the next. Staffage dots the base of the mountain: in the lower left corner, a couple gazes across the page as a dog leads an energetic boy toward them. To their right is a lone woman, stopped short of the cliff face. Frozen in her tracks and hunch-backed, this figure is a ready measure of the mountain's size and solidity. With descriptive attention to the jagged landscape, with a sensitivity to how that landscape dwarfs its inhabitants, Van Schooten evoked the detailed landscapes of Hendrik Goltzius's early drawings of Haarlem's dunes or Jan van Goyen's later views of Den Haag. Observation structures the triangles necessary for trigonometric calculation: the terrain proves critical to the identification of useful triangles AFC, ABD, and ABC. This landscape occupies most of Van Schooten Sr.'s page, and its scale stresses that the surveyor's numbers and [End Page 380] geometric lines only make sense in terms of the landscapes he has experienced.

If Descartes's few years with Maurice gave him only temporary exposure to such pictorial modes, there was another, more direct channel through which the philosopher likely became versed in them: his illustrator, Frans van Schooten Jr., consummate child of the Mathematique. During his youth, Van Schooten Jr. had seen his father in his pedagogical prime, and had sharpened his mathematical attentions under Jacob Golius at the Universiteit Leiden. Upon graduation, the younger Van Schooten was hired as a temporary instructor at the Mathematique. After his father's death in 1645, the Universiteit's curators named Van Schooten Jr. to the Mathematique post.28 By then, junior had become known to Constantijn Huygens. Huygens then brought him to the philosopher's attention, believing that Van Schooten Jr. could bring both mathematical instinct and fine drafts-manship to a revised conception of nature.29 Under Descartes's guidance, the young Leidener married those talents, transposing them into a critical apparatus for the new science.


How did Descartes and Van Schooten Jr. import engineering's visual language into the rarefied world of natural philosophy? In the second section of the Discours, Descartes identified mathematics as the basic tool of all the sciences. But he scoffed at the notion of mastering all mathematical operations. His new science would be based on the "relations" and "pro-portions" between objects. Descartes had certainly learned about these from Euclid's Elements, and he had delved into their implications for music with his Musicae compendium, but note his wish to draw lines between things, and how this bears a kinship with the engineer's pictorial task:

Observing that, although [the various sciences'] objects may be different, they all agree in considering only the various relations or proportions which can be found among those objects, I thought it [End Page 381] best that I examine only proportions in general, without assuming that they were present solely in those objects that made themselves most readily available to me. In other words, I analyzed [such relations and proportions] without binding them to these objects alone, so as to be better able to apply them to all other similar objects. … [I]n order to understand these relations I should sometimes have to consider each one in particular and sometimes only have to recall them, or apprehend many of them together, [and] I thought that, in order to treat them best individually, I should view them as [manifesting] in straight lines, since I could find no simpler object to present to my imagination and my senses; and that, in order to recall them, or reflect on many at the same time, I should explain them by a few characters as brief as possible.30

In moving between the abstractions of "general form[s]" and the vagaries of particulars, Descartes outlined a commitment to invisible lines and economic symbols similar to the geometric concerns of the Nederduytsche Mathematique's curriculum. For both Maurice's school and Descartes's new school of thought, epistemic relationships were underwritten by representing the visible surround and calculating only with reference to representations of what was seen. But if Leiden-trained engineers measured the distance between one landmark and another for eventual use on the battle-field, Descartes and Van Schooten Jr. replaced these landmarks with gentlemen playing tennis or looking at rainbows, with anatomies of the eye and brain, with vats full of grapes. Then Descartes used these geometrically entangled pictures to examine the world's laws. He leveled these pictorial claims at the Aristotelian understanding that qualities such as hot, cold, brilliant, dull, soft, hard, thick, or thin dwelled in an object, or were inherent to each object by itself. Qualities were to be understood as the result of distances between objects or between one part of an object and another. Merely looking at or sensing such qualities would not help to decipher their causes.

No combination of image and text better demonstrated this than Descartes's lesson about eyes sensing objects at a distance (fig. 6). To undermine the Aristotelian theory that characterized sight as the transmission of images from objects to the eyes, Descartes and Van Schooten Jr. coordinated four different forms of visual representation: anatomical description (the brain and that grasping, disembodied hand), reductive diagramming [End Page 382]

Figure 6. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 49. RB 336029, the Huntington Library, San Marino, California.
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Figure 6.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 49. RB 336029, the Huntington Library, San Marino, California.

[End Page 383] (the eyes), geometric analysis (alternately converging lines that denote light traveling from one point to another), and letters and numbers that signified location (M, Y, X, V at the top; r, s, t and R, S, T at the back of the eyes; and the paired sets of 7, 8, 9 at the optic nerve). Much like Van Schooten Sr. in his landscapes, Van Schooten Jr. lavished descriptive detail onto the folds and recesses of the brain's anterior parts, exhibiting an adept facility with chiaroscuro and an esteem for pictures derived from empirical investigation. Here, the anatomy of the brain and the eye dictated mathematical work. Descartes reasoned that the visual perception of an external object depended on the part of each eye that received the line of sight from the object (the line of sight, according to Descartes, depended on a chain of sensations that reverberated from the object to the eye and was then refracted by the pupil). Such perception was further complicated by the distance between the eyes. Descartes and Van Schooten Jr. were essentially guiding their readers through a triangulation between an external object, the distance between the eyes, and the lines of sight between the eyes and the object. Though the same operation had appeared in Kepler's Ad Vitellionem paralipomena (1604) and its medieval predecessors, Descartes and Van Schooten Jr.'s picture stressed the negotiation of figure and line.

The measurement of invisible lines between tangible figures featured regularly in subsequent phases of their explanation. First, Descartes illustrated a blind man who "sees"—that is, feels—the world with a pair of batons (fig. 7). This analogy, which stretched back to Aristotle and medieval commentators on perspectiva,31 reified sight as tactile sensation, rather than immaterial emission of an image. But Descartes then used Van Schooten Jr.'s image to reason that even if the blind man did not know the lengths of his batons, provided that he knew the distance between his arms (that is, between A and C) and the degree of the angles ACE and CAE, he would still be able establish the location of point E. The solution would simply be derived "by a natural Geometry."32 Then, when switching back to the eyes themselves (fig. 6) and encouraging his readers to import (men-tally) the model of those sensation-giving batons, Descartes proposed that if one knew the length of Ss and the degrees of angles SsX and XsS, one would still be able to figure out the position of point (or object) X. He soon defined his "natural Geometry": "this, by an action of thought, which, although of the simplest imagining, does not fail to involve reasoning [End Page 384] entirely similar to that which is made by Surveyors, when, by means of two different stations, they measure distant sites."33 Descartes thus elevated the surveyor's practice and its forms of representation; these were a collective mode of "reasoning," "an action of thought," that possessed natural philosophical stakes. The bond between the surveyor's art and the new science signaled a sharp inversion of ars and scientia. It required the investigation of nature to place unprecedented stock in figurative descriptions that framed geometrical analysis: to take experience and mathematize its contours.

Figure 7. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 56. RB 336029, the Huntington Library, San Marino, California.
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Figure 7.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 56. RB 336029, the Huntington Library, San Marino, California.

This was not an obvious train of thought; to join surveying with natural philosophy was novel even for Descartes. Recall 1619 once more. While stationed with Maurice's regiment at Breda and learning to draw, Descartes [End Page 385] was also writing a manuscript, Regulae ad directionem ingenii. Included among his guidelines for intellectual comportment was a working definition of perception. Remarks about the labor of the surveyor were nowhere to be found. Instead, representation first surfaced as a phenomenon endemic to the cognitive processing of sensation. In rule twelve, Descartes described optical perception vis-à-vis Aristotle's (Platonic) wax seal: "You must not think this expression is just an analogy; the external shape of the sentient organ must be regarded as really changed by the object, in exactly the same way as the shape of the surface of the wax is changed by the seal."34 But the philosopher did not stop there. He further conceived that this initial, literal impression triggered a chain of similar impressions, where negative and positive casts of the external object were relayed from one sensing body part to the next. At the end of this process, an image of the external object—without a corporeal equivalent—would be relayed to the common sensibility. The common sensibility would deliver its own representation of the object to the imagination. That is, Descartes was beginning to juxtapose a chain of sense impressions and physical resemblance to the task of cognition. Yet he was still puzzled by how the mind made sense of the myriad impressions it had received:

Sometimes [cognition] receives images from the common sensibility at the same time as the phantasy does; sometimes it applies itself to the images preserved in memory; sometimes it forms new images, and these so occupy the imagination that often it is not able at the same time to receive ideas from the common sensibility, or to pass them on to the locomotive power in the way that the body left to itself would. In all these processes the cognitive power is sometimes passive, sometimes active; it plays the part now of the seal, now of the wax; here, however, these expressions must be taken as merely analogical, for there is nothing quite like this among corporeal objects.35

In this scheme, cognition worked as a command post that negotiated sense impressions, as well as the imagination's a-corporeal reconfigurations of sense impressions and the common sensibility. It read the images it received [End Page 386] while simultaneously generating its own images for the imagination to consider and then transform into a-physical images. Descartes's early understanding of cognition featured the mind juggling multiple representations, some old, some new, others that seemed drawn from the senses, and others that were not. But none of these images were direct replicas or pictures of the outside world; all had undergone a process of sensational and cognitive alteration.

Sensed impressions alone, Descartes implied, clouded thought. Reason divided problems—such as the senses in general and optical perception in particular—into their simplest terms. Each division, he insisted, "should be pictured in our imagination entirely by bare figures. Thus it will be perceived much more distinctly by our intellect." "Bare figures" disciplined the mind more readily than descriptive pictures; streamlined images would ensure a smoother, more direct translation of the external, sensed object to the mind's interpretation of that object. In rule sixteen, Descartes offered further counsel for how to keep the mind focused: "As for things which do not require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures."36 Together, these passages prop up abstract figures—diagrams and letters—as cognition's essential help-meet.37 Such reduction ensured a mind free of distractions.

But during the eighteen years between the Regulae's first precepts and Joannes Maire's publication of the Discours and Essais, such distractions constituted Descartes's search for knowledge. The philosopher was confronted with the stubborn fact of the senses, their figuration in pictures, and the mind's ability to throw any manner of representation into disarray. (For instance, during his travels in Dordrecht, he watched a professor of anatomy strive to put words to the dissection of muscle and bone.38) He refined his ideas about the sequence of physical, physiological transformations through which the body sensed an object and the mind then received information about it. He replaced the analogy of cognition as wax with the blind man's batons, and thus eliminated the visual impression that could be conveyed from the external object to the eye, from the eye to the mind. (The mind would then rework the sense impression back into an image.) Descartes began to chart an even more pronounced series of representations between media, which started with the external object bending light and [End Page 387] that bent light then hitting the eyes, and proceeded to how the eyes continued to refract those rays until they suddenly met at the retina. At the retina, the rays triggered a vibration of the nerves that would travel to the pineal gland, which would then be translated into immaterial thought for the mind.39 This concatenation of mechanisms produced representations that were suffused with materiality until they reached the center of cognition. As Descartes would soon describe in the sixth part of Discours, the mind wove a web of empiricism and deduction:

it is necessary also to confess that the power of nature is so ample and vast … that I have hardly observed a single particular effect which I cannot at once recognize as capable of being deduced by man in many different ways, and that my greatest difficulty usually is to discover which of these modes the effect is dependent upon; for out of this difficulty I cannot otherwise extricate myself than by again seeking certain experiments, which may be such that their results are not the same if we explain the phenomenon in one mode and then explain it in the other.40

The entirely "bare figure" would no longer suffice and the graphic possibilities for the new natural philosophy abounded. Descartes used the sixth part of the Discours to confess that nature's laws required a process of knowledge-making, rather than the tired set of categories that had long guided Aristotelian investigation. The mind disciplined by the Regulaeplayed but a part in this process. The visible world was the matter with which the mind had to contend; and pictures that figured that world became integral to pointing up the struggle to know. Because the mind was now set to weigh figural information alongside intellectual distillation, Descartes and Van Schooten Jr. filled the Essais with pictorial variety. These figures were the products of moment-to-moment assessment—the traces of a process that considered the sensed as altered by reason. They mirrored the flux of contingent analysis; a mind firmly situated in the world; and a method that preserved the complexity of encountering, processing, and thinking nature.


This understanding of the Cartesian method calls into question a longstanding tendency: to identify Descartes's "clear and distinct ideas" with a [End Page 388] geometric minimalism and epistemic clarity that resulted from total retreat from the world. This interpretive custom has stubborn evidence to recommend it. In what has become one of the Discours' most famous passages, the philosopher reports having holed himself up in a Bavarian cabin and experiencing three vivid dreams whose lucidity guided him toward his new epistemology.41 Descartes's quasi-nomadic restlessness—which saw him move from Leiden to Friesland, Utrecht to Amsterdam, and Groningen to the Hague within ten years—has been interpreted as further proof of his eagerness to avoid the distractions of society.42 However, more recent histories of the philosopher have attributed Descartes's remarkable independence of mind—and even the cogito itself—to a life lived fully in seventeenth-century society. To recover Descartes's contemporary surround, these scholars have pointed to the philosopher's enthusiastic contact with artisans and his pursuit of the crafting of things. Descartes has been caught in the thankless pursuit of a perfectly ground, perfectly curved lens;43 he has been spotted at Saint-Germain-en-Laye admiring the grottoes full of automata and mechanical fountains and wondering at how they worked;44 he has been characterized as being so frustrated with the chaos of the artisanal workshop (and his apparent protégé, Jean Ferrier) that he penned the fourth part of Dioptrique to get the artisanal house, body, and mind in order.45 Most recently, Harold Cook has dispelled the image of the sickly, bed-bound French philosopher and built in its place a new Descartes—a highly social creature who wore flourishes of green silk cut in the fashion of his late king, Henri IV.46 Descartes the man and Cartesianism the philosophy were made from a thorough engagement with the world.47

What follows is a similar probing of some of the Essais' images and an argument for their semiotic bearing. If previous pages have historicized the [End Page 389] graphic mode that Descartes put to work, those that remain examine the men and objects that Van Schooten Jr.'s figures signified. Here, I attend to traces of seventeenth-century culture that Descartes and Van Schooten Jr. incorporated into the new philosophy. As Descartes himself put it: "I thought that books—at least those whose reasons are only probable, and which have no demonstrations, being composed of and bloated by the opinions of so many diverse men—do not apprehend the truth as much as the simple reasoning that a man of good sense can make of the things which present themselves to him."48 Reading the Essais' pictures alongside this statement results in a loose portrait of the sorts of men who could capably and skeptically import a figured geometry into natural philosophy; indeed, such a combination of text and image amounts to a prescription of the cosmopolitan objects that would further the new philosophy's cause.

Figure 8. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 268. RB 336029, the Huntington Library, San Marino, California.
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Figure 8.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 268. RB 336029, the Huntington Library, San Marino, California.

Descartes and Van Schooten Jr.'s rainbow landscapes offered up a ready, if vague, portrayal of the "man of good sense" (figs. 2, 8, 9). Between the three pictures, the contemplative gent and his accessories varied, from skirted doublets to capes, from broad-brimmed hats to bare heads. But one feature remained consistent: the side sword—a weapon that demarcated a wide range of fighters. In the Netherlands, the side sword might be the sign of a member of the French noblesse de l'épée—a mercenary of the sort [End Page 390]

Figure 9. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 270. RB 336029, the Huntington Library, San Marino, California.
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Figure 9.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 270. RB 336029, the Huntington Library, San Marino, California.

Descartes had been to Maurice—or of a regular foot soldier called to do his part.49 It could designate a nobleman's son who may have been trained at one of the many military academies that had begun to sprout up across Europe.50 It was liable to signal that he had witnessed—or even participated in—the brute depravity of war, as Jacques Callot had pictured it unrelentingly in his cycle of prints Grandes misères et malheurs de la guerre(1633).51 But it was just as likely that the sword accompanied the sort of soldier who had been the subject of kortegaarde, or guardroom, scenes, a genre of painting made most popular by a circle of Amsterdam-based schilderij during the 1620s and 1630s. Many such pictures caught soldiers making merry at one inn or another while playing cards, throwing back steins of beer, gambling, or reveling in enemy plunder. But Simon Kick, who was active in Amsterdam between 1624 and 1652, gave form to a more refined type of warrior. His Company of Soldiers in a Guardroom Preparing for Battle (undated) (fig. 10) revealed a middle-class taste for [End Page 391]

Figure 10. Simon Kick (1603–52), A Company of Soldiers in a Guardroom Preparing for Battle (oil on panel). Johnny van Haeften Gallery, London, UK / Bridgman Images.
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Figure 10.

Simon Kick (1603–52), A Company of Soldiers in a Guardroom Preparing for Battle (oil on panel). Johnny van Haeften Gallery, London, UK / Bridgman Images.

soldierly refinement and wonder. Having abandoned large tomes and weaponry to the floor, Kick's soldiers marvel at the glint of a glass goblet; in their eagerness to plot their next siege, they grab the nearest drum and use it as a surface to sketch a map.52 Van Schooten Jr.'s and Descartes's admirers of rainbows were cut from a similar cloth. Equipped to engage with the enemy, "men of good sense" also indulged their curiosity about the world. It was this flexibility—this independence of mind—that characterized the new natural philosopher. [End Page 392]

Figure 11. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 127. RB 336029, the Huntington Library, San Marino, California.
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Figure 11.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 127. RB 336029, the Huntington Library, San Marino, California.

As Van Schooten Jr.'s pictures indicated, this seeker of knowledge would continue expanding his store of experience. In addition to reaching out to anatomists who dissected the brains of oxen and sheep (fig. 6), and circulating among artisans who ground glass and constructed telescopes (fig. 11),53 he would use the enhanced techne around him to revive older analogies about the behavior of light. Foremost among such devices was the camera obscura. Earlier in the century, in Ad Vitellionem paralipomena, Kepler had likened the darkened chamber to ocular mechanics. The cornea acted as a hole drilled into the walls of such a chamber; and the retina, much like the back wall onto which the camera obscura's image was cast, [End Page 393] received (and thereby perceived) an upside-down image of the external object. For most men, direct encounters with these machines were rare. The construction of such a darkroom was no menial task; it required practitioners to devote an entire room to the enterprise, piercing a hole through one of its walls and darkening it completely.54 Though the German mathematician had elected not to picture such a chamber in his own text, his analogy proliferated. Christoph Scheiner confirmed this in a plate for Rosa ursina (1626–30) (fig. 12): the analogy worked because it maintained a distance between the art of the camera obscura and the nature of the eye. But the philosopher that Descartes envisioned would take such thinking one step further and picture the analogy anew (fig. 4). He would embed the eye in one of the camera obscura's walls, and declare it a mechanical aperture. In this play of light and shadow, the boundaries between art and artifice were erased.55

Similarly, Descartes's new philosopher would remake old Aristotelian warhorses, for instance by comparing the refraction and reflection of light to the bounce of a ball.56 The twist was to outfit the correspondence with a popular seventeenth-century activity: tennis. In earlier optical treatises, scholars—including Friedrich Reisner, who had compiled Alhazen's treatment of vision alongside that of thirteenth-century scholar Erazmus Witelo's in an Opticae thesaurus in 1572, as well as Kepler in his responses to Witelo—had visualized the analogy as a geometrical abstraction, thus reducing physical cause to line and angle. When Descartes's new philosopher took on the familiar trope, he would recast graphically the analogy as a series of one-sided tennis matches at the edge of a lake. Reflection would feature a tennis player about to hit a ball onto a solid, highly reflective surface (fig. 1); refraction was another man of sport launching a ball toward a body of water, which would then refract the ball's path (fig. 13); and, finally, a kind of muted refraction was embodied by a third player hitting the ball across a taut piece of fabric (fig. 14). All this as tennis flourished across Europe. Among students at Leiden, it was a choice study break, and a prime opportunity for swindling freshmen less adept with the racket. [End Page 394] Amsterdam had twenty courts for its townsmen and bugermeesters; Paris had as many as 250.57 By casting the analogy in such contemporary terms, the new natural philosopher mitigated any sense that he was reiterating some weak nugget of ancient wisdom. These pictures reinforced that this analogy—and this graphic method for recuperating natural knowledge—operated in the present tense.

Figure 12. Illustration in Christoph Scheiner, Rosa Ursina sive sol ex admirando facviarvm & macularum suarum phaenomenovarivs … (Bracciani: Andreas Phaeus, 1626–29), 123. RB 751846, the Huntington Library, San Marino, California.
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Figure 12.

Illustration in Christoph Scheiner, Rosa Ursina sive sol ex admirando facviarvm & macularum suarum phaenomenovarivs … (Bracciani: Andreas Phaeus, 1626–29), 123. RB 751846, the Huntington Library, San Marino, California.

With other analogies came worldliness. Not only was Descartes's a-scholastic thinker likely to be caught admiring wine in a glass, he might [End Page 395] have also glimpsed a wine press in Bourdeaux, La Rochelle, or the Loire Valley, only to realize that the trickle of grape juice through half-pressed grapes could serve as analogy for the movement of light through a world full of subtle particles (fig. 3). Such encounters would have marked him as more than a consumer of wine—and, especially if Dutch, a wealthy consumer at that. He may have had more direct contact with the viniculture that was prospering across Europe and even in its colonies. Perhaps he was a Dutchman who had invested in distilleries in France. Or, he may have seen business boom as he imported casks of Canarian Malvasia into the United Provinces from Tenerife. Or perhaps, as a French vintner himself, he had chuckled at Lord Delaware and his vignerons' failed attempts to plant hundreds of vines in Virginian soil.58 Whatever his lot in life, he did not stop at enjoying the fruits of his world; he was attuned to the process by which they came to his table.

Figure 13. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 11. RB 336029, the Huntington Library, San Marino, California.
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Figure 13.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 11. RB 336029, the Huntington Library, San Marino, California.

[End Page 396]

Figure 14. Illustration by Van Schooten Jr. in Descartes, Discours (1637), 19. RB 336029, the Huntington Library, San Marino, California.
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Figure 14.

Illustration by Van Schooten Jr. in Descartes, Discours (1637), 19. RB 336029, the Huntington Library, San Marino, California.

All this was to insist on a present juxtaposed to the past. More conspicuous indictments of the ancients were latent in illustrations about sight that we have already discussed. Recall the blind man with his batons (fig. 7). That figure, with his loose-fitting toga and coarse bare feet, was a relic of an old world. Passing from one place to another with only his impoverished senses and his hunchbacked dog, he embodied the ancients' blindness to reason. Blindness extended, too, to those philosophers who sought to emulate the ancients. When, for example, unthinking proponents of a fallen age—marked by their supposedly sagacious, unshorn beards and stoic hair in disarray59—stepped into the interior of a camera obscura (fig. 4), their eyes were revealed to have been gouged out; their wits unaware of the mechanisms of sight that a considerable investment in techne could reveal. An absence of insight was all the ancients had to offer.

Descartes's natural philosophy required a form of representation in which semiosis and mimesis could unite in mutual dependence. These pictures expressed the new philosophy's admixture of reason and experiment, observation and thought. In order to upend the now-suspect assessment of the world's qualities, new pictures—and emended understandings of such [End Page 397] pictures' capacities—were imperative. This re-education would have to start from the basic constitution of any picture—a point that Descartes relays in the fourth discourse of Dioptrique:

As we see in engravings, a little ink here and a little ink there on a piece of paper seem to us forests, towns, people, and even battles and storms; and although they make us think of countless different qualities in these objects, it is only due to their shapes that there is any real resemblance. And even this resemblance is very imperfect, since engravings represent bodies of varying relief and depth on a surface which is entirely flat. Moreover, following the rules of perspective, they often represent circles by ovals better than by other circles, squares by rhombuses better than by other squares, and similarly for other shapes. Thus it often happens that in order to be more perfect as an image and to represent an object better, an engraving ought not resemble it.60

It may seem that this characterization of pictures as brute matter would diminish Van Schooten Jr.'s figures. Such a materialist view was liable to bleed the Essais' tennis players into mere inkblots; dissolve its rainbows into black confetti; unravel that cross-section of the brain into filament after filament. But sensitive makers of images could grasp that the pictorial description of an object found in the world required the artist to effect a radical distortion of that object on the flat page.61 Such transformation spoke to the physical incommensurability between the two-dimensional confines of the picture and the three-dimensional world. Nature and its illustrations were to be understood as absolutely different in kind and power. Thus a figurative picture was a gesture toward the experience of nature, but, at base, evidenced how an artist had wrought a series of mental reconfigurations to ready nature's objects for pen and ink. This fact—that the most mimetic picture is a drastic reformulation of the outside world—allowed Descartes to rely on the role of illustrations as he forged his new philosophy. He no longer had to worry that pictorial description would invoke the Aristotelian dependence on qualities. Every figure was the product of a mental alteration, and, once on the page, could also become a [End Page 398] condition—a synoptic device—for other modes of analysis. Descriptive forms were vessels produced by thought about the world that then conditioned further probing of that world. Such devices had to be tempered with analytical equipment such as the sub-visible particle, the surveyor's line, and the geometer's construction—each of which had been held apart from the old natural philosophy. These graphic combinations bound corresponding varieties of thought together, rendering the new physics into a multiva-lent art—a picture multiple.62

Melissa Lo
Southern California Institute of Architecture


All translations are mine unless otherwise indicated.

1. A. G. Balz, Descartes and the Modern Mind (New Haven: Yale University Press, 1952); Desmond Clarke, Descartes' Philosophy of Science (Manchester: Manchester University Press, 1982), 22–23; Daniel Garber, Descartes's Metaphysical Physics (Chicago: University of Chicago Press, 1992); Neil M. Ribe, "Cartesian Optics and the Mastery of Nature," Isis 88 (1997): 42–61; Garber, "Descartes and Experiment in the Discourse and Essays," in Descartes Embodied: Reading Cartesian Philosophy through Cartesian Science (Cambridge: Cambridge University Press, 2000), 85–110; Jed Buchwald, "Descartes's Experimental Journey Past the Prism and through the Invisible World to the Rainbow," Annals of Science 65, no. 1 (2008): 1–46.

2. René Descartes, Oeuvres de Descartes, vol. 1, ed. Charles Ernest Adam and Paul Tannery, 3rd ed. (Paris: Vrin, 1996), 270–73, 280–84. Adam and Tannery's edition of Descartes's Oeuvres will hereafter be abbreviated AT, followed by volume number and page number.

3. Frederick P. Van de Pitte, "Some of Descartes' Debts to Eustachius a Sancto Paulo," The Monist 71, no. 4 (1988): 487–97; Roger Ariew, "Descartes and Scholasticism: The Intellectual Background to Descartes' Thought," Cambridge Companion to Descartes, ed. John Cottingham (Cambridge: Cambridge University Press, 1992), 58–90, 61–63; Ariew, "Descartes and the Scotists," in Descartes and the Last Scholastics (Ithaca: Cornell University Press, 1999), 39–57.

4. Garber, "Physics and Foundations," Cambridge History of Science, vol. 3, Early Modern Science, ed. Katharine Park and Lorraine Daston (Cambridge: Cambridge University Press, 2006), 21–69.

5. Michael Mahoney, "Diagrams and Dynamics: Mathematical Perspectives on Edgerton's Thesis," Science and the Arts in the Renaissance, ed. John W. Shirley and F. David Hoeninger (Washington: Folger Shakespeare Library, 1985), 198–220, 209; Mahoney, "Drawing Machines," in Picturing Machines, 1400–1700, ed. Wolfgang Lefèvre (Cam-bridge, Mass.: MIT Press, 2004), 281–308.

6. Brian Baigrie, "Descartes's Scientific Illustrations and 'la grande mécanique de la nature,'" Picturing Knowledge: Historical and Philosophical Problems Concerning the Use of Art in Science, ed. Baigrie (Toronto: University of Toronto Press, 1996), 86–134.

7. Claus Zittel, Theatrum philosophicum: Descartes und die Rolle a¨sthetischer Formen in der Wissenschaft (Berlin: Akademie Verlag, 2009); Daniel Sepper, Descartes's Imagination: Proportion, Images, and the Activity of Thinking (Berkeley: University of California Press, 1996); Peter Galison, "Descartes's Comparisons: From the Invisible to the Visible," Isis 75, no. 2 (1984): 311–26.

8. Huygens to Descartes, 15 June 1636, AT 10:607–8.

9. Huygens to Descartes, 30 October 1636, AT 10:613–15.

10. Descartes to Mersenne, 8 October 1629, AT 1:22–32; Descartes to Ferrier, 8 October 1629, AT 1:32–38; Ferrier to Descartes, 26 October 1629, AT 1:38–52; Descartes to Ferrier, 13 November 1629, AT 1:53–69; Descartes to Mersenne, [18 March 1630], AT1:128–35; Descartes to [Beeckman], 22 August 1634, AT 1:308–12.

11. AT 10:214. Christoph Lüthy, "Where Logical Necessity Turns into Visual Persuasion: Descartes' Clear and Distinct Illustrations," in Transmitting Knowledge: Words, Images, and Instruments in Early Modern Europe, ed. Ian Maclean and Sachiko Kusukawa (Oxford: Oxford University Press, 2006), 97–133, at 97.

12. Descartes to Mersenne, [5 April 1632], AT 1:243.

13. "Neque me tamen ita desidiosum existimes, ut plane tempus inutiliter conteram; immò nunquam utiliùs, sed in rebus quas ingenium tuum, altioribus occupatum, haud dubie contemnet, & ex edito scientiarum caelo despiciet: nempe in Picturâ, Architecturâ militari, & praecipue sermone Belgico." Descartes to Beeckman, 24 January 1619, AT10:151–53. In translating this passage, I have benefitted from a summary of it in William Shea, The Magic of Numbers in Motion: The Scientific Career of René Descartes (Canton, Mass.: Science History Publications, 1991), 10.

14. Daston and Park, Wonders and the Order of Nature, 1150–1750 (New York: Zone Books, 2006), 265.

15. Klaas van Berkel, Isaac Beeckman (1588–1637) en de mechanisering van het wereldbeeld (Amsterdam: Rodopi, 1983); Steven Gaukroger, Descartes: An Intellectual Biography (Oxford: Oxford University Press, 1995): 70–72; Jean-François Gauvin, "Artisans, Machines, and Descartes's Organon," History of Science 44 (2006): 187–216, at 196–97.

16. Descartes to Beeckman, 23 April 1619, AT 10:162–63; Shea, Magic of Numbers, 11.

17. Francisco de Toletus, Commentaria vna cum quaestionibus in octo libr. (Lyons: Alexand. Marsil. Lucensem, 1580): 308–22, at 311.

18. François de Dainville, L'éducation des jésuites: XVIe–XVIIIe siècles (Paris: Minuet, 1991), 27. For more on genres of natural philosophy: Ann Blair, "The Problemata as a Natural Philosophical Genre," in Natural Particulars: Nature and the Disciplines in Early Modern Europe, ed. Anthony Grafton and Nancy Siraisi (Cambridge, Mass.: MIT Press, 1999), 171–204.

19. Camille de Rochemonteix, Un collège de Jésuites aux XVII e et XVIII e siècles: Le Collège Henri IV de La Flèche, vol. 4 (Le Mans, 1889), 1–148. Natural philosophical broad-sheets operated primarily on allegory: Véronique Meyer, L'illustration des thèses à Paris dans la seconde moitié du XVII e siècle: Peintres, graveurs, éditeurs (Paris: Paris-Musées, 2002); Susanna Berger, "Martin Meurisse's Theater of Natural Philosophy," The Art Bulletin 95, no. 2 (2013): 269–93.

20. AT 6:9–10. Translation with assistance from The Philosophical Writings of René Descartes, trans. John Cottingham, Robert Stoothoff, and Dugald Murdoch (Cambridge: Cambridge University Press, 1984–85), Kindle edition, 1:9–10. Hereafter this translation will be designated as CSM.

21. Jonathan Israel, The Dutch Republic: Its Rise, Greatness, and Fall, 1477–1806(Oxford: Clarendon Press, 1995).

22. Simon Stevin, Wisconstige gedachtenissen … (Leiden: Jan Bouwensz, 1605–8).

23. P. C. Molhuysen, "Resoluties van Curatoren: 338," Bronnen tot de Geschiedenis der Leidsche Universiteit, vol. 1 (The Hague: Martinus Nijhoff, 1913), 389*–398*, 392*.

24. Ibid., 391*–392*. Also see: Maniere ende ordre die sijn Ex tie verstaen heeft dat sal worden achtervolght int doen van de Nederduytsche Lessen in de universiteyt tot Leyden tot offeninghe van het ingenieurschap (Leiden: Jan Paedts Jacobsz, 1600); Van Berkel, "Simon Stevin et la fondation de l'école militaire de Leyde en 1600," in De la vie civile, 1590, ed. Catherine Secretan and Pim den Boer (Lyon: 2005), 95–108; Charles van den Heuvel, "Le traité incomplet de l'Art Militaire et l'instruction pour une école des ingénieurs de Simon Stevin," in Simon Stevin (1548–1620): L'émergence de la nouvelle science, ed. Hossam Elkhadem and Wouter Bracke (Turnhout: Brepols, 2004), 103–13; Fokko Jan Dijksterhuis, "The Golden Age of Mathematics: Stevin, Huygens and the Dutch Republic," Nieuw Archief voor Wiskunde 5, no. 9 (2008): 100–107; Jennecke Krüger, "Lessons from the Early Seventeenth Century for Mathematics Curriculum Design," BSHM Bulletin: Journal of the British Society for the History of Mathematics25, no. 3 (2010): 144–61; Jantien Dopper, "Duytsche Mathematicque," Euclides 85, no. 6 (2009–10): 230–33.

25. Molhuysen, Bronnen, 390*.

26. Svetlana Alpers, The Art of Describing (Chicago: University of Chicago Press, 1983), 28–29.

27. Universiteitsbibliotheken Leiden, Bijzondere Collectie, BPL 1013: Frans van Schooten Sr., Mathematische werken (ca. 1622).

28. "Schooten, Frans van (2)," Nieuw Nederlandsch Biografisch Woordenboek, ed. P. C. Molhuysen and P. J. Blok, vol. 7 (Leiden: A. W. Sijthoff's Uitgeversmaatschappij, 1911–37), 1110–14, at 1111; Molhuysen, Bronnen, vol. 2 (1916), 407; Joseph Ehrenfried Hof-mann, Frans van Schooten de Jüngere (Wiesbaden: F. Steiner, 1962).

29. Huygens to Descartes, 15 June 1636, AT 10:607–8; and Huygens to Descartes, 30 October 1636, AT 10:613–15.

30. AT 6:20. With help from CSM 1:20.

31. A. Mark Smith, From Sight to Light: The Passage from Ancient to Modern Optics(Chicago: University of Chicago Press: 2015), 232–322.

32. AT 6:137.

33. AT 6:138.

34. AT 10:412; CSM 1:412. On the wax metaphor, see Raz Chen-Morris and Ofer Gal, "Baroque Optics and the Disappearance of the Observer: From Kepler's Optics to Descartes's Doubt," Journal of the History of Ideas 71, no. 2 (2010): 191–217, at 215–17.

35. AT 10:415. Translation aided by CSM 1:415–16.

36. AT 10:454. Translated aided by Descartes, Philosophical Writings, trans. Elizabeth Anscombe and Peter Thomas Geach (Edinburgh: Nelson, 1954): Rule XVI.

37. Garber, "Descartes and Method in 1637," in Descartes Embodied, 33–51, at 34–36.

38. Descartes to Mersenne, 15 April 1630, AT 1:137–46.

39. AT 6:137.

40. AT 6:64–65. With help from CSM 1:64–65.

41. Alan Gabbey, "The Melon and the Dictionary: Reflections on Descartes's Dreams," Journal of the History of Ideas 59, no. 4 (1998): 651–68.

42. Kevin Dunn, "'A Great City is a Great Solitude': Descartes's Urban Pastoral," Yale French Studies 80 (1991): 93–107.

43. Graham Burnett, The Hyperbolic Quest: Lens Making Machines and their Significance in the Seventeenth Century (Philadelphia: American Philosophical Society, 2005).

44. Simon Werrett, "Wonders Never Cease: Descartes's 'Météores' and the Rainbow Fountain," British Journal for the History of Science 34, no. 2 (2001): 129–47.

45. Gauvin, "Artisans, Machines, and Descartes's Organon"; Matthew Jones, The Good Life and the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue (Chicago: University of Chicago Press, 2006).

46. Harold Cook, "The Unexpected Descartes: Exile in the Netherlands?," Lecture at the Mahindra Humanities Center, Harvard University, September 11, 2013.

47. Steven Nadler, The Philosopher, the Priest, and the Painter: A Portrait of Descartes(Princeton: Princeton University Press, 2013).

48. AT 6:12–13.

49. Geoffrey Parker and Lesley M. Smith, "Introduction," in The General Crisis of the Seventeenth Century, ed. Parker and Smith (London: Routledge, 1978), 1–25, at 14.

50. J. R. Hale, "The Military Education of the Officer Class," in Cultural Aspects of the Italian Renaissance: Essays in Honor of Paul Oskar Kristeller, ed. Cecil H. Clough (Man-chester: Manchester University Press, 1976), 440–61.

51. Katie Hornstein, "Just Violence: Jacques Callot's Grandes Misères et Malheurs de la Guerre," The Bulletin of the University of Michigan Museums of Art and Archaeology16 (2005): 29–48.

52. Jochai Rosen, "The Dutch Guardroom Scene of the Golden Age: A Definition," Artibus et Historiae 27, no. 53 (2006): 151–74.

53. Gauvin, "Artisans, Machines, and Descartes's Organon."

54. Sven Dupré, "Playing with Images in a Dark Room: Kepler's Ludi inside the Camera Obscura," in Preprint 333: Inside the Camera Obscura—Optics and Art under the Spell of the Projected Image, ed. Wolfgang Lefèvre (Berlin: Max-Planck-Institut für Wissenschaftgeschichte, 2007), 59–74.

55. Alistair Crombie, "The Mechanistic Hypothesis and the Scientific Study of Vision," in Science, Optics, and Music in Medieval and Early Modern Thought (London: Hambledon, 1990), 139–60; Smith, From Sight to Light.

56. Mary Hesse, Models and Analogies in Science (London: Sheed and Ward, 1963); A. I. Sabra, Theories of Light: from Descartes to Newton (London: Oldbourne, 1967).

57. Fokko Jan Dijksterhuis, "Jeu de Paume & Jeux de la Raison in Seventeenth-Century Optics," Nuncius 28, no. 1 (2013): 115–41, at 131.

58. Tim Unwin, Wine and the Vine: An Historical Geography of Viticulture and the Wine Trade (London: Routledge, 1991), 203–35.

59. Dom. Auguste Fangé, Mémoires pour servir à l'histoire de la barbe de l'homme (Liège:J. Fr. Broncart, 1774), 66–69.

60. AT 6:113; with translation assistance from CSM 1:113.

61. Cf. Lyle Massey, "Anamorphosis through Descartes, or Perspective Gone Awry," Renaissance Quarterly 50, no. 4 (1997): 1148–89, at 1157–59; Rebecca Zorach, "'A Secret King of Charm Not to be Expressed or Discerned': On Claude Mellan's Insinuating Lines," RES 55/56 (2009): 235–51, at 238.

62. Response and resistance to Descartes's visual program comprised a robust strain of the transmission of Cartesian natural philosophy throughout the rest of the seventeenth century. This is the subject of a manuscript-in-progress by the author. See also Rebecca Wilkin, "Figuring the Dead Descartes: Claude Clerselier's L'Homme de Rene´ Descartes," Representations 83 (2003): 38–66; Zittel, "Conflicting Pictures: Illustrating Descartes' 'Traite´ de l'Homme,'" in Silent Messengers: The Material Vehicles of the Circulation of Knowledge in the Low Countries, ed. Sven Dupre´ and Christoph Lu¨thy (Mu¨nster: LitVerlag, 2011), 217–60.

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