Sketching Science in the Seventeenth Century

Michael S. Mahoney

Princeton University

What constitutes a sketch? We think at first of drawings and diagrams, but there are other forms of sketching. The OED offers several pertinent uses of the term, none of them earlier than the late 17th century. The first refers to the visual arts: "A rough drawing or delineation of something, giving the outlines or prominent features without the detail, esp. one intended to serve as the basis of a more finished picture, or to be used in its composition; a rough draught or design. Also, in later use, a drawing or painting of a slight or unpretentious nature." The second extends it to literary creations: "A brief account, description, or narrative giving the main or important facts, incidents, etc., and not going into the details; a short or superficial essay or study, freq[uently] in pl[ural] as a title." A third, rare usage, makes a sketch simply "[t]he general plan or outline, the main features, of anything". One can find all three sorts of sketches at work in the formation of the new science of the seventeenth century.

1 Sketching to See

To a historian of science, especially one concerned with the Scientific Revolution, it is the first definition that initially springs to mind. For the last half-century at least, historians of science and historians of art have sought the patterns of interaction of the two fields in Renaissance and early modern Europe; one thinks immediately of the essays of Erwin Panofsky and Giorgio Santillana in the late 1950s. Over the past two decades, however, the focus of the historians of science has shifted, first to the role of images and diagrams in the presentation of scientific results and then to their role in the creation of those results. It is a matter less of the relation of science and art and more of the nature of visual thinking in the sciences. One may ask of any text, but especially of early modern texts, when the illustrations were added and by whom they were created. One may go beyond that to ask what role the author played in creating the images and how their creation figured into his thinking. That is where we meet the notion of sketching: not in the marvelous woodcuts of De fabrica humani corporis or the meticulous copperplate engravings of the Micrographia, but in what Vesalius was drawing as he dissected, and Hooke, as he squinted through the microscope. These had to be sketches, if only because their subject was ephemeral. The object changed under the anatomist's knife, the microscopist struggled with moving objects, shifting focal lengths, and varying light. Moreover, although the sketches originated in individuals, the goal was a type: not the body of this executed criminal or of this woman, dead in childbirth, but the human body in its male and female forms. Not this fly or that louse, but the fly, the louse.(1) Hence, it could not be a matter of simply drawing what one saw, but of sketching the essentials and of using sketches to determine just what the essentials are. One sketched to learn what to see, as Galileo did when observing the surface of the moon through his telescope.(2)
1. This question of accurate portrayal of the ideal type arose in particular in botany, where illustrations purportedly drawn "to life" (naar het leven, ad vivam) show the plant simultaneously in stages through which it naturally passes in succession.

2. For the most recent in a long line of discussions of Galileo's drawings, see Horst Bredekamp, "Gazing Hands and Blind Spots: Galileo as Draftsman", Science in Context 13,3-4(2000), 423-462.

But sketching in the Scientific Revolution involved more than drawing, even though the scientists did a lot of drawing, much more(3) than their ancient or medieval predecessors. Taking the OED's second definition and extending it beyond the literary, the notion of a sketch as "a brief account, description, or narrative giving the main or important facts, incidents, etc., and not going into the details" leads us to models. Sketches and models were intimately related in the creation of modern science, both as stages in the development of a working device and as ways of thinking about the way world works at a level beyond the senses, indeed at the level that accounts for sensation itself. On the one hand, one sketched models that served as sketches of full-scale mechanisms; on the other, one sketched by modeling, first with mechanisms and then with mathematical abstractions of mechanisms, which enabled a new form of sketching. The product over the century was the clockwork universe of matter in motion, expressed in mathematical equations and instantiated in experimental mechanisms. It has served as the standard of scientific thought down to this day, albeit a standard now undergoing fundamental change by virtue of a new device for modeling that shifts the mathematics to the computer.

3. We suspect this was the case. Can we demonstrate it other than by the absence of manuscripts containing sketches?)

My touchstone for this sort of science in the seventeenth century is Christiaan Huygens, whose career linked Galileo and Descartes to Newton and who took the crucial steps toward making the clockwork universe work. In some ways he may be, or indeed is, exceptional among his contemporaries, but in other, more important ways he is representative of the new direction science was taking in the seventeenth century. Huygens was a skilled draftsman, who thought with a drawing pen in his hand and hence whose creative thinking we can follow in his sketches. Owing to the decision of the editors of his Oeuvres complètes to publish the sketches as they appear in his manuscripts, rather than redrawing them, we can watch his hand at work. He was also a theoretician who worked toward practical results, most notably toward various versions of clocks precise and reliable enough to measure longitude at sea. So his drawings begin and end in the real world. Yet, in the process they move between that world and an abstract mathematical world underlying it, combining physical configuration and mathematical diagram in a new and fruitful interaction. At the risk of a meta-metaphor, one can say Huygens realized in detail what Galileo had only sketched: a way to draw the physical so as to be able to think about it mathematically. He was not alone. We find the same combination in the diagrams of Mathematical Principles of Natural Philosophy (London, 1687) of Isaac Newton, who freely (and unusually) acknowledged the inspiration of Huygens' Pendulum Clock (Paris, 1673). But those diagrams seem more spare, even in the manuscripts.(4) Newton was writing a mathematical treatise on mechanics, not trying to analyze mechanisms mathematically.

4. Insofar as I can determine from the few pages reproduced in the published Mathematical Papers of Isaac Newton (Cambridge, 1967-1981).

Huygens made drawings, and he built some of his own models. I want to show him sketching and modeling, and point to some of the questions those activities raise. But before doing so I need to back up and consider some of the sources from which he was drawing (I'll leave that ambiguous). They suggest two different, though related, uses of models in scientific thought.

2 Modeling the Large in the Small


Fig. 1. Domenico Fontana, Della trasportatione dell'obelisco vaticano, fol. 8r.

At the beginning of Domenico Fontana's 1590 account of how he moved the Vatican obelisk from behind the old sacristy to its present location in St. Peter's Square, he depicts the obelisk in its original location and displays some of the other proposals for carrying out the task. 
However, in the following first drawing one sees what I have said. In addition, I thought it good to represent around this obelisk eight designs, or models: we mean to say of the best that were proposed in the gathering described above, each of which was based on good arguments. True, they are in small format owing to the narrowness of the folio, but I have made [them] to show visually the various means found by many engineers to [achieve] the same effect, to the end of giving greater satisfaction to curious readers.(5)
It is not clear whether Fontana was reproducing designs and models actually presented by others or was offering his own pictorial interpretation of proposals set forth verbally. What is evident is that, despite his remark about their being well found, none of them is scalable to the full task. For one thing, they require humans to make them work, and humans don't come that big. Nor is it clear what would replace humans in any scaled-up versions. Moreover, the designs are not up to the task. Several would either collapse under their own weight or require more force to move them than to move the object itself. Looking at this picture brings to mind the opening passages of Galileo's Two New Sciences, in which he explains why machines don't scale up. It's not merely a matter of geometry, though it may be demonstrated geometrically: at a certain point, the internal stresses and strains of a material device cancel its mechanical advantage. that ultimately there is necessarily ascribed not only to all machines and artificial structures, but to natural ones as well, a limit beyond which neither art nor nature can transgress; transgress, I say, while maintaining the same proportions with the identical material.(6)
5. Domenico Fontana, Della trasportatione dell'obelisco vaticano et delle fabriche di nostro Signore Papa Sisto V (Rome, 1590), fol. 7v. "Però nel seguente primo disegno si vede, quanto ho detto, e di più m'è parso bene rappresentare intorne ad essa Guglia otto disegni, ò modelli: che vogliamo dire de' migliori, che furno proposti nella Congregatione narrata di sopra, ciascheduno de' quali era fontdata sopra buone ragioni, vero è che sono in forma piccola per angustia del foglio, e ciò ho io fatto per dare à vedere i varij modi trovati da molti ingegnieri per l'effetto medesimo à fine de dar magior sodisfattione à desiderosi lettori."

6. Discorsi, 3: " che ultimamente non solo di tutte le machine, e fabbriche artifiziali, mà delle naturali ancora sia un termine necesariamente ascritto, oltre al quale nè l'arte, nè la natura possa trapassare: trapassare dico con osservar sempre l'istesse proporzioni con l'identità della materia."

The Two New Sciences, you will recall, dates from Galileo's time in Padua, when he was consulting for the Arsenal of Venice; the treatise was all but complete when his attention was distracted by the telescope. That the problem of scaling should have led to the first of the two new sciences, the strength of materials, suggests that models were being used, and misused, widely as a means of exploring or demonstrating what would work in the large. One may describe this perhaps as using the world in the small to think about the world in the large. Although Galileo focused on the strength of materials in the direct scaling of a design, such models proved a way of testing an idea, or indeed a sketch of an idea, to see if the mechanism itself held promise of working, leaving open the question of how the mechanism might be adapted to a larger scale or incorporated in a larger device.

We do not know as much about this activity as we should. The papers of Christiaan Huygens offer a glimpse into the process, one that raises important epistemological and social questions. Together with his brother, Constantijn, Huygens was a highly skilled lens-grinder, capable not only of grinding precision lenses but also of designing machines for the purpose. To test whether one of their designs would work, the two brothers often built a model of it before turning to a master instrument-maker for the machine itself. Nonetheless, according to J.H. Leopold, Huygens lacked as skilled a hand with metal as he had with glass.(7) He was not a trained craftsman, nor did he aspire to be one. Because of the relative novelty of telescopes and microscopes and of their importance as scientific instruments, lens-grinding remained both accessible and acceptable to people of Huygens' class. In the area of clock-making, however, Huygens had to turn to masters of the craft. He had the mechanical ingenuity to invent two of the most important regulating mechanisms in the history of clockmaking: the pendulum and the spring regulator. In both instances his creativity extended to a theoretical account of the mechanisms, which enabled him to embed the accurate measure of time in the laws of nature and in some instances suggested new mechanisms. Yet he had neither the skill nor the know-how to build clocks, or at first even models of his designs.

Evidence from his own writings suggests that he did not appreciate the nature of that skill nor the extent to which he lacked it. For example, some notes show that on 20 and 23 Jan 1675 he devised two mechanisms for regulating a clock by means of a coiled spring.(8) On the 20th, he first attached the spring directly to the frame, on the side of the escapement, and to a dumbbell balance above and then revised the design, moving the spring above the frame, placing it in a drum and attaching it to a balance wheel. On the 23rd, he modified the latter arrangement to add a second dumbbell (similarly changed to two balance wheels) to counteract the effects of turning the clock. Or rather I should say he sketched pictures of these arrangements. Here we see his drawings: on the left, the basic mechanism and his original sketch of an escapement; on the right an improved design that may well reflect an intervening encounter with a leading Parisian clockmaker, Isaac Thuret.(9)

7. John H. Leopold, "Christiaan Huygens and his instrument makers", in Hendrik J.M. Bos, et al., eds. Studies on Christiaan Huygens (Lisse, 1980), 221-233

8. Oeuvres Complètes de Christiaan Huygens (La Haye, 1888-1950; hereafter HOC), 7: 408ff

9. Leopold, "Huygens and his instrument makers", 227-29. For a discussion of the practical difficulties Huygens faced in applying his invention to watches, see Leopold's "L'Invention par Christiaan Huygens du ressort spiral réglant pour les montres", in Huygens et la France: Table ronde du Centre National de la Recherche Scientifique, Paris, 27-29 mars 1979 (Paris, 1982), 153-157.

Fig. 2. Christiaan Huygens' drawings of a spring balance and escapement, Oeuvres complètes (hereafter HOC), 7:408-409.

For Huygens did not build the mechanisms himself; rather, he related that on the 21st he sought out Thuret but did not find him until the morning of the 22nd, when he had Thuret construct a model of the mechanism while Huygens waited. The model was completed by 3 PM, and Huygens took it with him. The following day, Thuret built a model for himself and then on the 24th and 25th undertook to apply it to a watch. He subsequently claimed part of the credit for the invention of the spring-regulated watch. Huygens vehemently rejected the claim and ended their longstanding collaboration. Several aspects of this episode seem to me pertinent to our concerns.

First of all, as we shall see below, we have here a case of the application of a physical principle expressed first in a drawing and then in a working mechanism. What was the relation between the two? When Huygens found Thuret, how did the scholar explain to the craftsman what he had in mind, and how did the craftsman understand him? Or perhaps we should ask whether Huygens and Thuret needed to understand it in the same way. Did Huygens show Thuret the sketch? One would assume so. How else would he explain what he wanted? Indeed, John Leopold maintains that the drawings of the 23rd reflect Thuret's suggestions for improvement. Did Thuret work from the sketch to build the model? Huygens said of his first encounter with Thuret on the 22nd that "In explaining it to him, he said (as yet barely understanding it), 'I find that so pretty that I fear it may not be so.'"(10) What did he need to understand to translate the sketch into a mechanism? Whatever it was, he must have already understood it, because he built a working model from scratch in a couple of hours. Huygens then departed with the model and, one assumes, the sketch as well. Whereupon Thuret built another model, evidently from memory (or did he make his own sketch?) and at once set about to incorporate a working version into a watch. How much did he have to understand to do that? What did the second model look like, and what changes did Thuret make in adapting it to the watch? In this case, it is not a matter of scaling the model up, but rather of making it smaller and fitting it in which the other parts of the watch. Thuret thought that the know-how involved in that process entitled him share in the privilege for the clock on the grounds that he had helped invent it. Huygens argued that Thuret had simply been following his sketch, which contained the essence of the invention. Huygens understood the principle; Thuret had simply put it into practice.

The rights and wrongs of this dispute need not concern us here.(11) What is more interesting is the intersection of craft knowledge and high science, of what one knew from building mechanisms and what one knew from analyzing their dynamics. Regrettably, only the latter knowledge was self-consciously set down in words. We have only Huygens' account of the affair. For the former we must reason indirectly from the artifacts, something I must leave to the antique horologists who know enough about clockmaking to do it critically.(12)

10. HOC, 7:410: "En la luy expliquant il dit, ne l'entendant encore qu'a peine, je trouve cela si beau que je me défie tousjours qu'il ne soit ainsi."

11. The proprietary aspects of this episode are not germane to the present discussion, but Huygens had grounds for being angry. For details, see my "Drawing Mechanics", in Wolfgang Lefevre (ed.), Picturing Machines, 1400-1700 (MIT Press, 2004), 281-306, esp. 297-301. Although the two men eventually reconciled, they never resumed their active collaboration, which constituted a powerful creative force in timekeeping.

12. Several of the papers in William J.H. Andrewes, ed. The Quest for Longitude (Cambridge, MA., 1996) show how insightful their readings can be, as I pointed out in the conclusion to my own contribution to the volume, "Longitude in the History of Science", pp. 63-68.

Also interesting is the direction Huygens took subsequently. The spring balance, it turned out, had its deficiencies. The practical goal of the enterprise was an accurate sea-going clock for determining longitude. While the spring was mechanically more stable than a pendulum, it was sensitive to changes in temperature and humidity to a degree that undermined its accuracy. So Huygens pursued other mechanisms, all of which had in common the underlying mechanical principle of the spring: the force driving them varied as the displacement from equilibrium. They were all what subsequently came to be called simple harmonic oscillators. Sketches for such mechanisms run for pages in his works, most accompanied by mathematical demonstrations of their workings. In some instances, it is not clear how they would be incorporated in a clock. In other cases, the designs were in fact realized. In the case of two of the best known of them, his tricordal pendulum and his "perfect marine balance", Huygens built a model before turning to a clockmaker to produce a working timepiece.(13) In both cases, the model seems to have been an necessary proof of concept before attempting a full-scale clock.

13. Huygens seems to have drawn a lesson from his experience with Thuret. In his "Application de Décembre 1683" he wrote: "Le 17 dec. 1683 j'ay portè [sic] a Van Ceulen l'horologer le modelle que j'avois fait de ce mouvement de Pendule Cylindrique, pour changer de cette facon les 2 horologes que je luy avois fait faire pour la Compagnie des Indes Orientales. J'avois prié mon frere de Zeelhem de venir avec moy: parce que ledit horologer s'imaginoit d'avoir trouvé la mesme chose que moy, apres m'en avoir ouy dire quelque chose en gros. Mais ayant vu le modelle il avoua que ce qu'il avoit modelé n'y ressembloit nullement." (HOC, 18:532) Here a comparison of models seems to have prevented confusion between what Huygens and Van Ceulen had in mind respectively and suggests that a drawing such as the one Huygens had earlier made for Thuret might have been too vague or (literally) sketchy to do so.

But here as earlier in his work on clocks, the mechanisms served another purpose. They were also his means of probing the mathematical structure of the world. Here he took his lead from Descartes, who provides us with an example of the literary meaning of "sketch". Huygens learned Descartes' cosmology and laws of motion from the Principles of Philosophy, published in Latin in 1644 and in French in 1647. Preceding the Principles, but withheld from publication when Descartes learned of Galileo's condemnation in 1633, was The World, or Treatise on Light.(14) It was meant quite self-consciously as a sketch in the OED's sense of "[a] brief account, description, or narrative giving the main or important facts, incidents, etc., and not going into the details."

14. René Descartes, The World, or Treatise on Light, translation and introduction by Michael S. Mahoney (New York, 1979). Following the decision not to publish the work, Descartes took some of the chapters and published them in amplified form in 1637 as the Optics and Meteorology, two of the Essays for with his famous Discourse on Method served as prologue. The bulk of the work then became the basis for the pertinent sections of his Principles of Philosophy.

3 Modeling the Small in the Large

Composed between 1628 and 1630, The World brought together a decade of Descartes' explorations in mechanics, optics, and natural philosophy. Begun as an explanation of an meteorological phenomenon, a "double sun", it grew to encompass a new cosmology based on a radically reduced ontology of matter in motion. By a path we need not follow here, Descartes had reached the conclusion that there existed in reality only space-filling matter broken into pieces of varying size pushing up against one another and moving around one another, combining and recombining mechanically according to mathematically expressible laws. What we see happening in the world, the actions and properties that we perceive, even our perception itself: all these could be explained in terms of the size and relative motion of those particles. It was the goal of The World to persuade readers of the plausibility of that cosmology, in part by showing how it could account for the phenomena of light and vision, and indeed for light itself.

After laying some epistemological foundations in the opening chapters, Descartes set up his World by inviting the reader to listen to a story - and thus unintentionally inviting his critics to condemn his physics as no more than a belle romance (Huygens). "For a short time, then," he wrote at the beginning of Chapter Six containing the "Description of a New World", "allow your thought to wander beyond this world to view another, wholly new one, which I shall cause to unfold before it in imaginary spaces."(15) It was meant as a sketch. After laying out in Chapter Seven the "laws of nature" governing this world, Descartes assured the reader that they accounted for all that occurred in it. "Nonetheless," he concluded,

in consequence of this, I do not promise you to set out here exact demonstrations of all the things I will say. It will be enough for me to open to you the path by which you will be able to find them yourselves, whenever you take the trouble to look for them. Most minds lose interest when one makes things too easy for them. And to compose here a setting that pleases you, I must employ shadow as well as bright colors. Thus I will be content to pursue the description I have begun, as if having no other design than to tell you a fable.(16)
15. Ibid., 49.

16. Ibid., 77.

While calling it a fable, Descartes nonetheless reminded his readers time and again of how closely this imagined world of particulate matter interacting according to the laws of motion resembled in its behavior the world they saw around them. Beginning with a chaotic soup of particles, it will develop a sun and stars, planets and comets, which will all move in the same patterns as ours. Its sun and stars will generate light, which will propagate through the heavens, reflect from planets, and refract through lenses according the same laws as ours. It will strike our eyes as does the light in our world and produce the same sensations. Containing no more than undifferentiated matter in motion, it will look and feel the same as the world that Aristotle and his followers have filled with forms and qualities, spirits and intelligences, potential and actualization.

Calling his account a fable served several purposes for Descartes. First, it gave him what we now refer to as "deniability". A variation on the then popular ploy of proceeding ex hypothesi (as Urban VIII had urged Galileo to do), it enabled Descartes to set forth an account of the world that contradicted Aristotle (and possibly Scripture) at several points, while avoiding any commitment to its reality. Second, it required only plausibility. Descartes could point in the direction of a true account without having to set forth that account in full formal detail. And that was to his advantage because Descartes did not have all the details in hand. While claiming the virtue of mathematical precision, he could in fact achieve it only in the limited realm of optics, and even there only at an analogical level above that of the underlying real mechanism. For the rest, and in particular for the theory of light on which his optics rested, he could argue only by analogy. He could do no more than sketch the outlines, highlighting the things he knew and leaving the others in shadow.

And so one finds in The World a variety of models of light chosen to explain its real behavior in terms of the readily available or intuitable experience of the reader. In the drawings here, taken from the text, we see the rectilinear propagation of light as a pressure through a medium portrayed as balls tending toward an opening at the bottom of a container, which is then pictorially likened to the motion of sand in an hourglass and to the push one feels while leaning on a curved walking stick. The mechanical action of light is likened to the tension on ropes attached to a common pulley, and the capacity of rays to cross without interference to the passage of air through tubes meeting at a common center. Other such models abound in The World and in the Essays and Principles that followed it. Descartes did not intend them as exact descriptions but as pointers, as suggestions, as sketches. 
Fig. 3. Models of the action of light in Descartes' World.

4 Mathematical Modeling, Mathematical Sketching

In retrospect, Descartes' world needed other models to make it work in detail. In particular, it needed mechanical models that linked the real mechanism of light to mechanisms working at a measurable level. It required a measure of centrifugal force that tied it to the force of gravity. It needed laws of impact that could be tested by the collision of billiard balls and pendulum bobs. Over the career of Christiaan Huygens, model and mechanisms merged in the pendulum clock as the first of a series of precision devices in which the laws of matter in motion met the measure of time. In responding to the agenda of his predecessors and shaping that of his successors, Huygens' work formed a crucial point of passage in the development of mechanics in the seventeenth century. More importantly for present purposes, as I'd like to show in a series of examples, we can follow Huygens' thinking through his sketches.

Sketching motion

I'd like to come back again to a drawing about which I've already written an article, which, however, few here are likely to have encountered.(17) Here it is in all its messiness, as it emerged from Huygens' own pen, first on the page as a whole and then by itself. Let me step you through its genesis, so as to separate the three different spaces it interactively combines: the physical, the physico-mathematical, and the mathematical. Huygens will help in that task, since he himself separated them as he thought his way through the problem. The creative moment, I have argued, lay in Huygens' ability to keep them separate even as he used them interactively. 

Huygens began with a drawing of the pendulum with the bob displaced through an angle which he described as "very small" but which he drew large to leave room for reasoning. On that drawing of the physical configuration he then overlaid a semi-parabola representing the increasing velocity of the pendulum as it swung down toward the center point. On that graph of motion, he then drew another curve representing the times inversely proportional to the speeds at each moment of the bob's fall. Finally, he drew a series of curves of a purely mathematical sort to facilitate the summation of the instantaneous times, that is to find the area under the curve RLXNYHV (which is quite roughly sketched). The mathematics led to the introduction of two important elements into the diagram. First, if the circular arc of the bob's trajectory were a semiparabola congruent to the graph of its speed, the products of the ordinates to the two curves could be expressed as a constant times the ordinate to a semicircle drawn on their common base. Huygens knew from earlier work on centrifugal force that a circle and a parabola with the right common parameters coincided in the immediate neighborhood of their point of mutual tangency, so he took the intersecting curves as congruent over the "very small" interval of the bob's fall and drew in the semicircle. Second, from another source Huygens knew that the same circle also served the purpose of finding the area under the curve of times, and so he again shifted his gaze in the diagram.

Fig. 4.Huygens' determination of the period of a simple pendulum, Leiden University Library, Collectio Hugenii, Chartae mechanicae, fol. 72 recto.

17. For full mathematical details, see M.S. Mahoney, "Huygens and the Pendulum: From Device to Mathematical Relation", in H. Breger and E. Grosholz, eds. The Growth of Mathematical Knowledge (Amsterdam, 2000), 17-39.

Fig. 5. The main diagram of Huygens' derivation with those of related lemmas, HOC, 16:392, 398-399.

For the time, the derivation so far was already a mathematical tour de force but Huygens was only getting started. Having determined that the time of the bob's descent bore a constant ratio to the time of its fall through the length of its cord, Huygens noted that the result was only approximate. He knew that empirically, because others had shown that, contrary to Galileo's claim, the period of a simple pendulum increases with increasing amplitude. But he now knew it mathematically, because he had explicitly made an approximation in deriving his result. He had substituted an arc of a parabola for an arc of a circle. For what curve instead of a circle would that substitution be exact? To determine the answer, Huygens had to unpack his drawing to see how the mechanics of the body's motion would generate a parabola to match that of its speed of free fall. It came down to the relations among the tangent, normal, and ordinate to the curve: for any point B on the trajectory, the normal BD is to the ordinate BC as some constant line EC to CF. For what shape of AB would the points F determined by CF lie on a parabola? Without evident hesitation, Huygens noted "I have found this to agree with the cycloid by a known way of drawing the tangent."
Cycloid? Where did the cycloid come from? Well, it was on Huygens' mind; he had been involved recently in a debate over the curve. But, I want to maintain, it was also before his eyes. It, or rather its Gestalt, had crept into his diagram when he drew that semicircle for auxiliary purposes. With the parabola streaming off from the top and the trajectory of the pendulum swinging up from the bottom, the semicircle now looked like generating circle of a cycloid in the then standard diagram of the curve, which indeed appears lower down on the same page. Huygens need no more than a hint; note how the semicircle in the original diagram has become a generating circle in the above diagram. Once Huygens had the hint, the details quickly followed. 

Fig. 6. Unpacking the diagram: Huygens' derivation of the cycloidal path; HOC, 16:400.

From the cycloid to incitation parfaite

Once Huygens had determined that a pendulum swinging along a cycloid would beat out the same time independently of the length of its swing, he faced the immediate problem of determining a mechanism by which to constrain a pendulum to that path. The answer lay in the mathematics of the cycloid, which turned out to have the property of being its own evolute. That is, if one wraps a cord around a cycloid and then pulls it away, always remaining tangent to the curve, the endpoint will trace another cycloid, congruent to the first [animation]. Hence the mechanism in question needed to be no more complex than a pair of cycloidal leaves surrounding the cord of the pendulum at the suspension point.

Having drawn the mechanical consequences, Huygens later turned to the mathematical principles. How was that dynamic property of the cycloid related to the laws of motion that governed the fall of bodies? Apparently, it took some time for him to discern the underlying principle, perhaps while putting the finishing touches on the Horologium oscillatorium for its publication in 1673. Sometime in 1673 or '74 he sketched a demonstration that the effective weight(18) of a body at rest or moving on an inverted cycloid is proportional to its distance along the cycloid from its vertex. To persuade himself and others that this was indeed the essential property, he then turned to apply this insight to a vibrating string, which was generally acknowledged to be tautochronous.(19) The drawings tell the story: he analyzed the string in terms of the cycloid. That is, he used the cycloid to model the vibration of the string and find where in its configuration the tautochronic property lies. One can see him moving back and forth between the drawings. The sketches don't do all the work for him. He still has to make the mathematics work in detail. But the sketches help him in a crucial way, here by creating counterfactual inequalities, that is, by letting him stretch the string without lengthening it.(20) The sketches help him to find the cycloid in the vibrating string. <fig 7> 

18. That is, the weight as measured by the force necessary to restrain it from moving along the tangent to the curve at the point at which the body is located.

19. As Marin Mersenne had pointed out, a stretched string sounds the same tone, that is, vibrates at the same rate, no matter how hard it is plucked.

20. The string is inextensible, yet he needs a right triangle so he can have a sine-component of the effective motive force.

Fig. 7. The principle of incitation parfaite, derived from the cycloid and applied to a vibrating string; HOC, 18:489-490.

With the principle thus confirmed, Huygens subsequently shifted his focus from mapping systems onto the cycloid to analyzing them directly in terms of displacements and forces. Over the next years, that shift of focus gradually took him away from both the cycloid and the pendulum. We can watch him undertake a search for other mechanisms, other models of which he has begun to call incitation parfaite. As noted earlier, he first returned to something Robert Hooke and others had suggested in the early '60s, namely a spring.(21) Hooke did not publish his now famous law, ut tensio sic vis, until 1678, but Huygens apparently already knew it by the early '70s, perhaps through his own investigations dating back to 1659. Identifying tension with displacement made the spring an instantiation of his new principle, which served as the basis for the drawings of the regulatory mechanism described above.

Fig. 8. Incitation parfaite illustrated for springs; HOC, 18:497.

21. Writing at the end of July 1665 (HOC, 7:427), Robert Murray, Secretary of the Royal Society, informed Huygens of a lecture by Hooke in which he had proposed applying a spring, rather than a pendulum, to the balance of a clock. He offered no details, saying that Hooke would be publishing them soon. Huygens replied (Ibid., 486) that the idea was not new; it had been proposed by a clockmaker in Paris in 1660. However, the spring was too sensitive to disturbances to be of any practical use. The discussion in the two letters makes clear that neither side was aware of the spring's tautochronic property.
A bit later he found the principle expressed in a mechanism he called a tricord pendulum, a ring suspended on three cords and oscillating rotationally about its center. Since the cords are inextensible, the ring rises and falls, whereby each of the suspension points follows the arc of a curve on the surface of the cylinder determined by the interior radius of the ring. Here Huygens could show [in the diagram on the right] that, if that curve were a certain parabola wrapped around the cylinder, then the horizontal force on each point would be as the displacement from the vertex. It remained then to show that the actual curve closely approximately that parabola in the neighborhood of the vertex. The drawing that accompanies that analysis looks very much like the drawings for the cycloidal pendulum and thereby suggests the configuration in general made visible a way of thinking about curves and their relations. 

Fig. 9. Huygens' analysis of his tricord pendulum; HOC, 18:527.

Indeed, in another drawing we find the new tricord pendulum next to what appears to be a cycloid but is in fact the curve on the cylinder approximated by the parabola. At the top of one of the suspension cords on the left, we see leaves meant to constrain the cord to the parabolic path. 

Fig. 10. Tricord pendulum; HOC, 18:531.

Still later he looked at chains and weights suspended in mercury as tautochronic mechanisms, and here again it is useful to juxtapose sketches to show recurring patterns. On the left is the sketch of the spring balance, on the right a balance around which a chain is draped. As the balance swings to one side it lifts a portion of the chain, the weight (and thus counteracting force of which) will be proportional to its length, which in turn is proportional to the angle through which the balance has turned. 

Fig. 11. Replacing the spring with a chain; HOC, 7:408 (see above, Fig. 2), 18:536.

Linking the physical world to the mathematical

In his analysis of the pendulum, Huygens had been able to avoid the problem of getting force into the picture. It was well established that the period of a pendulum is independent of the weight of the bob, and the same is true of the Galilean law of free fall that provided the parabolic graph of motion. Once Huygens established the principle of tautochronism, however, he faced the problem of introducing into his spatial configurations a parameter that has no spatial representation. How does one draw, or sketch, force in its various manifestations? It is a question which mathematicians of the period ultimately decided did not have a geometrical answer.(22) It is instructive to see how Huygens addressed it.
22. For details, see Michael S. Mahoney, "Diagrams and Dynamics: Mathematical Reflections on Edgerton's Thesis", In John W. Shirley and F. David Honiger, ed. Science and the Arts in the Renaissance (Washington, 1985), Chap. 10.

In order to build a working clock of requisite accuracy to be used for determining longitude at sea, Huygens had to solve a series of problems related to its physical structure. Most important, his original analysis of the pendulum rested on a fiction, namely that all the weight of the bob was concentrated in its center and that the string by which it was suspended was weightless. A real clock would involve a bob with a solid shape mounted on a metal rod suspended by a cord. The compounded object would behave like a pendulum, but the center of oscillation would like somewhere in the middle of the configuration. No one had as yet solved that problem, and Huygens again sketched his way to a solution. 

Fig. 12. The compound pendulum with two bobs; HOC, 16:418.

To determine it, Huygens began with two bobs B and C joined by a common(weightless) rod AC and drew a simple pendulum HP swinging through the same angle. Under the constraints of the pendulum, the speeds of B and C will be directly proportional to the speed of P at corresponding points of their swings. The speed of P can be measured by the square root of the height QP through which it falls to K, and that height is proportional to the heights BO and CS through which B and C fall toward E and D, respectively. But the speeds of B and C are constrained by their rigid connection and hence do not correspond directly to the heights though which they individually fall. To get a measure of their speeds, Huygens imagined them impacting with equal bodies G and F, respectively, which are then directed upward. Each will climb to a height proportional to the square of its velocity, which can be expressed as a function of the height CS and of the ratio of the distance of the bob from A to the unknown length. 

At this point, Huygens invoked the principle that the center of gravity of G and F will rise to the same height as that of the compound pendulum at the beginning of its swing(23). What it means here is that Huygens had an algebraic equation in which the unknown is the length of the simple pendulum and the knowns the bobs and their distances from A. If HK = x, AB = b, AD = d, and B and D denote the weights of the respective bobs, then the centers of gravity before and after will be  and  respectively. Setting the two expressions equal and solving for x yields the length HK of the center of oscillation.

23. On this principle, associated with the name of Evangelista Torricelli, see Michael S. Mahoney, "The Mathematical Realm of Nature", in Daniel Garber and Michael Ayers, eds. The Cambridge History of Seventeenth-Century Philosophy, 718.
Note the use of algebra here when dealing with a discrete system; the mathematics moved away from the diagram. It returned, however, when Huygens analyzed the center of oscillation of a uniform rod. Here he imagined the rod as consisting of contiguous small bodies, swinging down under the constraints of a rigid body and then freed to rise individually to the heights corresponding to their acquired speeds. But, rather than measuring their heights vertically, Huygens drew them horizontally, thus forming a parabola. By reasoning mutatis mutandis from the case of two bodies, he showed that the equality of the heights of the centers of gravity before and after corresponded to the equality of areas of the triangle on the left and of the parabola on the right. The solution of the center of oscillation now came down to the quadrature of the parabola.

Note that the triangle is simply an overlay on the physical picture of the pendulum, while the parabola is a mathematical configuration, graphing the height attained against the velocity as a function of the distance from the point of suspension. The centerline forms an interface between the two realms. The physical configuration is pictured on the one side, the mathematical structure of the mechanics is pictured on the other. Transition from the one to the other takes place at the centerline. 

Fig. 13. A swinging bar treated as an n-body compound pendulum; HOC, 16:421.


Fig. 14. Newton's diagram in the Principia mathematica (London, 1687), 128.

In retrospect, Huygens' drawing represents the start of an important means of moving back and forth between the physical and the mathematical world. One finds similar configurations in Newton's Principia, for example in Proposition 41 of Book I: "Assuming any sort of centripetal force, and granting the quadrature of curvilinear figures, required are both the trajectories in which the bodies move and the times of motions in the trajectories found."(24) On the left is a picture of the orbit VIK of the body revolving about the center of force at C, together with a circle VXY superimposed as a measure of time; the angles in the drawing correspond to measurements that can be made by an observer. On the right a variety of curves which represent the measures of various dynamic parameters such as force and velocity. They are mathematical structures with which one calculates, albeit only in principle, since in this diagram they are general curves drawn arbitrarily to demonstrate the structure of the problem, rather than any specific law of force. The lines connecting the two sets of curves at the centerline AC map areas under the mathematical curves on the right to sectors of the circle and orbit on the left, thereby determining the position of the planet on its orbit at any given time. The solution of the inverse problem of forces thus becomes a question of quadrature, of finding the areas under the curves on the right for particular laws of force.
24. For an extended discussion of Newton's mathematical methods, see M.S. Mahoney, "Algebraic vs. Geometric Techniques in Newton's Determination of Planetary Orbits", in Paul Theerman and Adele F. Seeff (eds.), Action and Reaction: Proceedings of a Symposium to Commemorate the Tercentenary of Newton's Principia (Newark, Del., 1993), 183-205.
For quite independent reasons, the reduced problem of quadrature took on a new form with the development of the calculus. Geometry gave way to algebra as the language of analysis, and the construction of curves was supplanted by the manipulation of symbols. The mathematical audience that could read Newton's Principia critically found its geometrical style distracting. It masked the underlying structures by which one discerned the relation of problems to one another, and it made a fresh task of problems that were mere variants of others already solved. At the very start of the eighteenth century, Pierre Varignon began recasting Newton's mechanics into the new mathematical language of Leibniz' calculus. Varignon's treatment of Newton's configuration shows the reversed and truncated result. On the right is the familiar orbit and circle from the left of Newton's diagram. But the curves on the left have shrunk to two arcs. While they represent graphs of times and forces, they play no operational role in the argument, which takes place entirely off the diagram in the Leibnizian notation of the infinitesimal calculus. Indeed, Varignon discarded them altogether in later papers.

What emerged from the recasting was a new form of mathematical sketching by manipulation of symbols, one that eliminated the need for diagrams altogether, as Lagrange pointed out at the beginning of his Mécanique analitique (Paris, 1788): 

One will find no drawings in this work. The methods I set out there require neither constructions nor geometric or mechanical arguments, but only algebraic operations subject to a regular and uniform process. Those who love analysis will take pleasure in seeing mechanics become a new branch of it and will be grateful to me for having thus extended its domain.(25)

Fig. 15. Varignon's diagram (Mémoires de l'Académie Royale des Sciences, 1700, 236).

25. "On ne trouvera point de Figures dans cet Ouvrage. Les méthodes que j'y expose ne demandent ni constructions, ni raisonnemens géométriques ou mécaniques, mais seulement des opérations algébriques, assujetties à une marche régulier et uniforme. Ceux qui aiment l'Analyse, verront avec plaisir la Mécanique en devenir une nouvelle branche, et me sauront gré d'avoir étendu ainsi le domaine." Avertissement.
This new medium of sketching also has a history reaching back to Descartes, who saw in symbolic algebra a means of rendering visible the progress of the mind at work as it moved by steps through the solution of a problem. Algebraic and analytic transformations became a way of view objects from different vantage points, as for example in the chain relating work to energy by way of force and momentum.(26) I think I can make an argument for a form of sketching in algebra, but I shall not attempt to do so here.

26. For the importance of those relations and of the capacity to express them analytically, see M. Norton Wise, "Work and Waste: Political Economy and Natural Philosophy in the Nineteenth Century (I), History of Science 27(1989), 263-301.

In the work of Huygens and then Newton one finds a new mode of science suited to the new conception of the world first sketched by Descartes. The mechanistic world of matter in motion, the clockwork universe, was to be understood as a complex of mechanisms analyzable in mathematical terms and visualizable as mechanical models. At heart, it was a world stripped bare of the forms and qualities, the accidents and essences, the panoply of differences that constituted the world of Aristotle. In the second sense of the OED's definition, it was a sketch, or a collection of sketches, which delineated the basic structures of reality and left the details to further articulation. That world was to be explained by compounding the mechanisms, converging on the complexity of experienced reality by adding terms to the equations.

This now classical model of the world, and the mode of science that went with it, has a long history stretching down to the present. Only in recent decades has it come under challenge by researchers who believe that the complexity of the world can be captured only by models of commensurate complexity, models that cannot be assembled analytically but must be generated computationally. Those models remain sketches, but they are sketches in a new medium. How they compare with the sketches discussed above is a matter of current research.