Drawing Mechanics

Michael S. Mahoney
Princeton University

Published in Picturing Machines, 1400-1700, ed. Wolfgang Lefebvre (Cambridge, MA: MIT Press, 2004)

Setting the question

As the preceding chapters show, a variety of practitioners in the Renaissance drew machines for a variety of apparent reasons: to advertise their craft, to impress their patrons, to communicate with one another, to gain social and intellectual standing for their practice, to analyze existing machines and design new ones, and perhaps to explore the underlying principles by which machines worked, both in particular and in general. I say "perhaps", because this last point is least clear, both in extent and in nature. We lack a basis for judging. We have no corroborating evidence of anything resembling a theory or science of machines before the mid-16th century, and what appeared then reached back to classical antiquity through the newly recovered and translated writings of Aristotle, Archimedes, Hero, and Pappus, which came with their own illustrations of basic machine configurations.

The absence of a textual tradition to which the drawings themselves are linked, or to which we can link them, makes it difficult to know what to look for in them. (1) How does one know that one is looking at a visual representation of a mechanical principle? It will not do to invoke what we know from the science of mechanics that emerged over the course of the 17th century. That sort of "ante hoc, ergo gratia huius" identification of a valued feature of modern science or engineering begs the historical question of precisely what relationship, if any, the drawings bear to the emergence and development of that body of knowledge. (2) What did practitioners learn about the workings of machines from drawing them, and how did it inform the later theory?

That is actually several questions, which do not necessarily converge on the same result. What people learn depends on what they want to know, and why. What questions were the practitioners of the 15th and 16th centuries asking about the machines they were drawing? What sorts of answers were they seeking? What constituted an explanation for how a machine worked, and what could one do with that explanation? How were the questions related to one another, and where did the answers lead? What then (and only then) did these questions and answers have to do with the science of mechanics as shaped by Galileo, Huygens, and Newton? What follows addresses the contents of the foregoing chapters only by contrast. It revisits an argument made almost twenty five years ago in response to a claim by Samuel Edgerton about the long-term theoretical importance of Renaissance innovations in the depiction of machines. (3) In essence, he maintained that ever more realistic pictures of machines led to the science of mechanics. Three salient passages from the mechanical literature reveal the difficulties with that thesis. The first is found in a letter written to Galileo Galilei in 1611 by Lodovico Cigoli, who observed that "a mathematician, however great, without the help of a good drawing, is not only half a mathematician, but also a man without eyes". (4) The second comes from Leonhard Euler, who in the preface of his Mechanics, or the Science of Motion Set Forth Analytically lamented that the geometrical style of Newton's Principia hid more than it revealed about the mathematical structures underlying his propositions. (5) Finally, capping a century's development of the subject, Joseph-Louis Lagrange warned readers of his Analytic Mechanics that:

No drawings are to be found 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. (6)

Over the short range, looking from Renaissance treatises on machines to the work of Galileo might support the notion that the science of mechanics emerged from ever more accurate modes of visual representation. However, looking beyond Galileo reveals that the long-range development of mechanics as a mathematical discipline directed attention away from the directly perceived world of three spatial dimensions and toward a multidimensional world of mass, time, velocity, force, and their various combinations. Practitioners ultimately found that the difficulties of encompassing those objects of differing dimensionality in a single, workable diagram reinforced a transition already underway in mathematics from a geometrical to an algebraic mode of expression and analysis. (7) One could not draw an abstract machine; one could not even make a diagram of it. But one could write its equation(s). Since the dynamics could find no place in the diagram, the diagram disappeared from the dynamics. (8) But that did not happen right away nor directly, and I want here to take a closer look at the process. In particular, I want to consider the role of drawings and diagrams in mediating between the real world of working devices and the abstract world of mathematical structures.

Edgerton insisted on the capacity of the new techniques to depict machines as they really appeared, rendering their structure in life-like detail. However, the science of mechanics was created as the science not of real, but of abstract machines. It took the form of the science of motion under constraint; as Newton put it in distinguishing between practical and rational mechanics, "... rational mechanics [is] the science, accurately set forth and demonstrated, of the motions that result from any forces whatever, and of the forces that are required for any sort of motions." (9) The clock became a model for the universe not because the planets are driven by weights and gears, but because the same laws explain the clock and the solar system as instances of bodies moving under constraint with a regular, fixed (and measurable) period. Indeed, as Newton himself pointed out, his mathematical system allowed a multitude of possible physical worlds, depending on the nature of the particular forces driving them and on the nature of the initial conditions. Which of these corresponded to our world was an empirical, not a theoretical question. (10)

What made it an empirical question was that the entities and relationships of the mathematical system corresponded to measurable objects and behavior in the physical world. The fit was meant to be exact, and it grew ever more precise as theory and instrumentation developed in tandem. As will become clear below, that mapping of real to abstract, and conversely, emerged from drawings and diagrams in which representations of the two worlds were joined at an interface. But the drawings acted as more than a conceptual bridge. The principle that mathematical understanding be instantiated in specifiable ways in physical devices -in short, that knowledge work-- entailed the need for a cognitive and social bridge between mathematical theory and technical practice and between the theoretician and the practitioner, and drawings played that role too. These issues emerge with particular clarity in the work of Christiaan Huygens, but they are rooted in earlier developments.

Putting Machines on the Philosophical Agenda

Whatever body of principles might have been created by the designers and builders of machines, the science of mechanics as the mathematical theory of abstract machines was the work of 17th-century thinkers who considered themselves mathematicians and philosophers. Hence, before machines could become the subject of a science, they had to come to the attention of the people who made science at the time. That is, a body of artisanal practice had to attract the interest of the theory class. It has not always done so. Machines evidently did not impress medieval thinkers. The 14th-century philosophers who first compared the heavens to the recently invented mechanical clock lived in a society teeming with mills: windmills, watermills, floating mills, mills on streams and mills under bridges, gristmills, sawmills, fulling mills, smithing mills. And mechanically a clock is simply a version of a mill. Yet, no one before the late 15th century thought mills were worth writing about, much less suggested that the heavens might work along the same principles. Mills were of fundamental importance to medieval society, but as part of the mundane, workaday world they did not attract the interest of the natural philosophers.(11)

That had to change before a science of mechanics could even get onto the philosophical agenda. "It seems to me," says Salviati (speaking for Galileo) at the very beginning of the Two New Sciences (published in 1638 but essentially complete by 1609), "that the everyday practice of the famous Arsenal of Venice offers to speculative minds a large field for philosophizing, and in particular in that part which is called 'mechanics'."(12) That the everyday practice of mechanics should be the subject of philosophy is perhaps the most revolutionary statement in Galileo's famous work. Clearly, something had to have raised the intellectual standing of mechanics for Galileo to feel that the philosophical audience to whom he was addressing the Two New Sciences would continue reading past those first lines.(13) One may take the statement as an exhortation, but Galileo had to have grounds for believing it would receive a hearing and indeed enlist support.

The early machine literature appears to have been the start of that process. The "theaters of machines", Popplow argues, were a form of advertising, through which engineers (in the original sense of machine-builders) sought to attract patronage and to enhance their social status. The pictures portrayed not so much actual working machines as mechanisms and the ingenious ways in which they could be combined to carry out a task. Depicting the machines in operation, often on mundane tasks in mundane settings, the engineer-authors of these books offered catalogues of their wares. Yet, in some cases, they pretended to more. The machines were also intended to serve a means of elevating their designers' status as intellectual workers. Claiming to be not mere mechanics but mathematicians, they purported to be setting out the principles, indeed even the mathematical principles, on which the machines were based.(14)

Yet it is hard to find much mathematics in these treatises. Contrary to Edgerton's claim of the Renaissance artist as quantifier, Popplow points out that it is precisely the dimensions that are missing.(15) These are not measured drawings; in some cases, the elements of the machines are not drawn to the same scale. Meant to show how machines worked and what kinds of machines might be built, the drawings were not intended to make it possible for someone actually to build a machine from its depiction, unless that someone already knew how to build machines of that sort. The need to persuade potential patrons of the desirability and feasibility of a device had to be balanced against the need to conceal essential details that would inform potential competitors.(16) The message seems clear: "These are the machines I can build. If you want one, I shall be glad to build it for you, bringing to bear the knowledge of dimensions, materials, and detailed structure that I have omitted from the pictures." If that is the case, then the drawings, however realistically crafted, were not headed toward a science of mechanics.

Dimensions include scale, and the absence of scale enables pictorial representation to mix the realistic with the fantastic without a clear boundary between the two. It is apparent in one of the pictures in Domenico Fontana's Della trasportatione dell' obelisco vaticano of 1590. Fontana's twin towers for lifting, lowering, and raising the obelisk float above a collection of "eight designs, or models, that we consider among the best that were proposed" for carrying out the task.(17) What is striking is that none of these others is even remotely feasible. All but possibly one fail on the issue of scale. They are essentially human-sized devices that either cannot be made larger or wouldn't make any sense if they were, because the humans necessary to work them don't come in such large sizes. In some cases, if they were scaled up as drawn, they would collapse under their own weight, much less that of the obelisk. One can draw them, as one has done here, and one can make models of them, models that might even work. But, scaled up to the dimensions of the obelisk, none of them would constitute a working or workable machine.(18) It is noteworthy that Galileo begins the Two New Sciences with precisely this problem, aiming to provide a theoretical account of why machines do not 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.

...so 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.(19)

Several important things are going on in this statement. First, the task of a philosophical account, or a theory, of machines is to set limits.(20) There are certain things that machines cannot do, for both mathematical and physical reasons. Second, machines in the hands of the natural philosopher have become part of nature, and nature in turn has been made subject to the limits of machines. The same laws govern both.

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

Figure 2. Galileo, Discorsi e dimostrazioni matematiche intorno à due nuove scienze (Leiden, 1637), 110.
In "Diagrams and Dynamics" I pointed to the difficulty that Galileo had in adapting his geometrical techniques from statics to kinematics. Taking his cue in statics first from Archimedes, he transformed his figures while keeping them in equilibrium. For example, in demonstrating the law of the lever, he took a realistically appearing beam suspended at its midpoint and then cut it at various points, adding support at the midpoint of each segment. The (geo)metrics of the situation coincided with the object under consideration. In shifting then to the so-called "Jordanian" tradition of medieval statics to treat the bent-arm balance, he could continue to work with a picture of the apparatus by superimposing directly on it the geometry of virtual displacements, relying on the similarity of arcs traversed in the same time. Moving from there to the inclined plane by way of the circle, he again could overlay the statical configuration on a picture of the object. But, when it came to tracing the motion of a body accelerating down the plane, the spatial configuration at first misled him into thinking that the triangle that was the picture of the plane could also serve as the triangle made up of the instantaneous velocities of the body. Only by separating the graph of motion from the physical trajectory could he get the mathematics to work.

It worked only for kinematics. As is well known, a mathematical account of the dynamics of motion escaped him completely, as it did Descartes. Picking up where they had left off, Christiaan Huygens devised a way of moving back and forth between the physical and mathematical realms and thus to get some of the dynamics into the diagram.

Before looking at specifics, it is perhaps worth emphasizing their context, if only to make clear what the second part of this paper has to do with the first. Huygens' design of a pendulum clock in 1657 marked the beginning of a line of research that continued until his death and that in many respects formed the central theme of his scientific career.(21) In seeking to make his new clock accurate enough to serve for the determination of longitude and durable enough to continue working aboard a ship at sea, Huygens undertook a series of investigations in mechanics that led to fundamental results in the dynamics of moving and rigid bodies. In almost every case, those results led in turn to practical mechanisms that improved either the accuracy or the reliability of working timekeepers: the pendulum clock itself, the cycloidal pendulum, the conical pendulum the sliding weight for adjusting the period, the balance-spring regulator, the tricord pendulum, the "perfect marine balance". In the end, the complete solution eluded him, in part because it was a matter of metallurgy rather than mechanics. However, subsequent efforts picked up where he had left off, culminating in the success of John Harrison a half-century later.(22)

Huygens thus embodied the union of head and hand that is characteristic of the new science of early modern Europe. His work on the clock and on the determination of longitude at sea are a prime example of what happened when machines did attract the attention of philosophers. In his hands, the clock constituted an interface between the mathematical and the physical world, between theory and practice, and indeed between the scientist and the artisan. Huygens not only derived and proved his results in theory, he also designed mechanisms to implement them in practice. He made his own sketches and, in some cases, built working models. But for the finished product he had to turn to master clockmakers and establish productive relations with them. In this he was less successful, in large part because of his inability or unwillingness to recognize the knowledge they brought to the collaboration along with their skill.

Three aspects of Huygens' work warrant closer attention: his use of diagrams in his mechanical investigations, his use of sketches in designing mechanisms, and his relations with his clockmakers. On close examination, his drawings reveal a subtle overlaying of three levels of pictorial representation and establishment a visual interface between the physical and the mathematical. That same interface can be found in Newton's Principia, and its disappearance in later treatises marks the transition to an algebraic mode of analysis. His sketches range from the roughest outline to detailed plans, as they show him moving back and forth between theoretical inquiry and practical design. His famous dispute with his Parisian clockmaker, Isaac Thuret, shows what Huygens deemed to count as intellectual property and who could lay claim to possess it.

The Pendulum and the Cycloid

In December 1659 Huygens undertook to determine the period of a simple pendulum(23) or, as he put it, "What ratio does the time of a minimal oscillation of a pendulum have to the time of perpendicular fall from the height of the pendulum?" He began with a drawing of the pendulum with the bob displaced through an angle KTZ 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 ADΣ representing the increasing velocity of the pendulum as it swung down toward the center point. Next to the graph of motion, he then drew another curve ΣΣG representing the times inversely proportional to the speeds at each moment of the bob's fall. If the bob were falling freely, the area under that curve would represent the time over AZ. But the bob's motion is constrained along the arc KEZ of the circle, so that at, say, point E, one would have (from Galileo):  .(24) Huygens then constructed the curve of time over the arc by the relation  , sketching it roughly as RLXNYH. Note that the curves of speeds and times introduce non-spatial parameters into a drawing that began as a spatial configuration.

Figure 3. Huygens' original diagram; HOC.XVI.392.

The mathematics of these "mixed" curves led Huygens then to the introduction of two important elements into the diagram. First, to simplify the expression, he drew on a mathematical result of unknown provenance: if BE were the ordinate to a parabola congruent to the one to which BD is an ordinate, then the product of BE and BD would be a constant times the ordinate BI to a semicircle of radius AZ drawn on the common base of the two parabolas. 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 circular arc of the bob's trajectory be a parabolic arc congruent to the graph of its speed the intersecting curves as congruent, thus fixing crucial parameters, 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 adjusted curve of times, and so he again shifted his gaze in the diagram and arrived at the result that the area, and hence the time of motion over the arc, varies as π times the product of the radius of the circle and the length of the pendulum. But, in comparing the time over the arc to the time of fall through the length of the pendulum, the radius of the circle cancels out, making the swing of the pendulum a function of the length of the pendulum only.

For the time, the derivation so far was already a mathematical tour de force but Huygens was only getting started. The result was only approximate over a minimal swing of the pendulum. 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 taken BE as the ordinate to a parabola rather a circle. He now asked for what trajectory of the bob would BE in fact lie on that parabola? To determine the answer, he 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. He appears to have found the answer by separating BE as ordinate to the physical trajectory from "another BE" as ordinate to the desired parabola, which was the mirror image of a mathematical (mechanical) curve. As he put it:

... I saw that, if we want a curve such that the times of descent through any of its arcs terminating at Z are equal, it is necessary that its nature be such that, if as the normal ET of the curve is to the applicate EB, so by construction a given straight line, say, BG is to another EB, point E falls on a parabola with vertex Z.

Figure 4.Unpacking the relation of physical trajectory and mechanical graph.

To determine that "other BE", he seems to have asked how the original BE entered into the expression BE BD. It came from the ratio  , which expresses the ratio of the "length" of point E on the circle to that of point B on the centerline. Huygens had simply stated the relation without explanation and without drawing the lines in the diagram from which it derives. To see it, one draws the tangent at E, intersecting the centerline at, say, W. By similar triangles,  . Now, if one looks at the lines that Huygens did not draw but clearly had in mind in forming the ratio  , one more or less readily sees that  . If one takes TW as a "given straight line", then the first proportion maps the ratio into the form  , and the second pair makes  . That is, if BE were extended to F such that BF = EW, then F would lie on the parabola  , with vertex at W and with latus rectum 2TW. Now, if W coincided with Z, then  , which is precisely the parabola of interest related to the length of the pendulum.

At this juncture, one needs a shift of focus. The original circle, centered at T, is no longer of interest as the trajectory of the pendulum. One is looking for another curve, with vertex at Z, the properties of whose normal and applicate can be mapped onto the relations just discussed in the circle of diameter TZ. That is, if BE extended intersects the curve at H, one wants  . That will be the case if the normal to the curve is parallel to TE, and hence the tangent at H is parallel to ZE. But this last relation, as Huygens says, is precisely, "the known method of drawing the tangent" to a cycloid.

Figure 5. The revised diagram with the cycloid now drawn; HOC.XVI.400.

Cycloid? Where did the cycloid come from? Well, it was on Huygens' mind; he had been involved recently in a debate over the curve and so was well aware of the property of its tangent. But, I want to maintain, the curve 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. Huygens needed no more than a hint; note how the semicircle in the original diagram has become a generating circle in the diagram Huygens drew to show that the cycloid is indeed the curve in question. Once he had the hint, the details quickly followed.

It will not have escaped notice, but I want to emphasize the composite nature of Huygens' main drawing here. It contains three spaces: the physical space of the pendulum, the mechanical space of the graphs of speeds and times, and the mathematical space of the auxiliary curves needed to carry out the quadrature of the curve of times. Huygens combined them without conflating them. That is how he was able so readily to modify the trajectory so as to make an approximate solution exact. He knew not only at what step he had made the approximation, but also in what space he had made it and how it was reflected in the other spaces. He needed a curve in physical, the properties of whose normal and ordinate could be mapped by way of a mathematical curve so as to generate another mathematical curve congruent to a graph of velocity against distance.

Physical and Mathematical Space

Huygens conjoined physical and mathematical space in another configuration in his work on the compound pendulum.(25) The problem itself, it should be noted, arose out of a quite practical concern, namely, that in the physical world pendulums have neither weightless cords nor point masses as bobs. Rather one is dealing with swinging objects whose weight is distributed over three dimensions. The task is to find a point in the pendulum at which it acts as if it were an ideal simple pendulum, its center of oscillation.(26)

Figure 6. The two-bob pendulum; HOC.XVI.415.
To determine that point, 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 in the same time. 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 imagines them impacting with equal bodies G and F, respectively, and imagines G and F then directed upward by reflection off a plane inclined at 45o. 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. (27)

At this point, Huygens invokes 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. Let me come back to the source of that principle in a moment and focus here on its application. It requires that Huygens move away from the geometrical configuration, which lacks the resources for determining the unknown length HK. That is, he cannot construct it directly by manipulation of the lines of the diagram, because the weights have no quantitative representation. Rather, he turns to algebra, translating the elements of the drawing into 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. That is,  , which again cannot be exactly located on the diagram. 


The same thing happens when Huygens then turns to extend this result, by generalization of an n-body pendulum, to the oscillation of a uniform rod.(28) In the continuous case, he imagines 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 measuring their heights vertically, Huygens draws them horizontally, thus forming a parabola. By reasoning mutatis mutandis from the case of two bodies, he shows that the equality of the heights of the centers of gravity before and after corresponds to the equality of areas of the triangle on the left (where BS = OV) and of the parabola on the right. The solution of the center of oscillation now comes down to the quadrature of the parabola. However, since the parameters of the parabola include the weight of the rod, that solution must again be couched in algebraic terms.(29)

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 inclined planes that initially rendered that interface mechanically intelligible by directing the motion of the weights upward following collision disappear after the first construction. Thereafter, 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 by a transformation corresponding mathematically to Galileo's law relating velocity to height in free fall.

Figure 7.A rod resolved into contiguous elements; HOC.XVI.421.

Figure 8. Newton's diagram.

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."(30) 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 are 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, at least 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.
For quite independent reasons, the reduced problem of quadrature took 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. Pierre Varignon's adaptation of Newton's configuration shows the result.(31) The left side remains the same, but complex of curves on the right is reduced to a single curve representing a general expression of the central force determining the orbit. Except for the two ordinates to the curve, there are no auxiliary constructions in the diagram to render graphically the transformations by which integration of the curve fixes the angle and radius of the corresponding position on the orbit. The curves thus play no operational role in the argument, which instead takes place entirely off the diagram in the Leibnizian notation of the infinitesimal calculus.(32) Indeed, the orbit is expressed in differential form, reflecting the generality of the problem and its solution.

I come out here where I came out in "Diagrams and Dynamics", albeit along a different route. Here I have placed more emphasis on the importance (and the challenges) of graphical modes of thought in the early development of the science of mechanics, even though they were later abandoned. It is important to see how drawings functioned for Huygens, if only to discern what was involved if one really set out to use pictorial means to analyze the workings of machines and to quantify the underlying principles. Those principles were not to be found in the pictures, precisely because the pictures erased the boundary between the real and the fantastic; there has been no shortage of elegant pictures of perpetual motion machines. (33) What enabled Huygens equate the areas on the right and the left of his diagram of the compound pendulum was a principle for which Torricelli is often given credit, but which surely predates him in the form of what I have referred to as a "maxim of engineering experience".(34) It comes down to this: bodies do not rise of their own accord, or, as the author of the treatise attributed to Juanelo Turriano puts it for a particular case, "water cannot go upward on its own ... because of its heaviness and weight."(35) In the case of interest to Huygens, a swinging pendulum winds down, or at best it keeps swinging at the same rate. It certainly does not swing more widely. Huygens makes this a quantitative principle by applying Galileo's law of free fall to the center of gravity of a system of bodies, moving at first under constraint and then freed of constraint. In doing so, he translates experience of the physical world into measurable behavior expressible in mathematical terms.
Figure 9. Varignon's diagram.

Theory and Practice, Knowledge and Know-How

What is particularly striking, and perhaps unusual, about Huygens' work on the clock is the close interplay between theory and practice. As noted above, the ultimate task was the reliable determination of longitude at sea, which is a matter of keeping time accurately. His abiding goal was to design a device accurate to within seconds a day and durable enough to withstand the rigors of service aboard ship under all conditions. In his own mind, the relation between theory and practice was seamless. A famous dispute surrounding his invention of a spring-regulated clock suggests otherwise and raises a question of interest to this workshop, namely, of what kinds of knowledge drawings contain and of how they serve as means of communication.

Shortly after publishing his Horologium Oscillatorium (Paris, 1673), Huygens uncovered the property of the cycloid that accounted for its tautochronicity: the accelerative force on a body sliding down the inverted curve is proportional to its displacement from the vertex at the bottom, the point of equilibrium. He quickly generalized the property into a principle he called incitation parfaite décroissante: in any situation in which the force acting on a body is proportional to its displacement from equilibrium, the body will oscillate with a period independent of its amplitude. By a series of experiments he then confirmed that the regular vibrations of springs rested on that principle and immediately sought to take advantage of the result.(36)

One of Huygens' worknotes shows that on 20 January 1675 he devised a mechanism for regulating a clock by means of a spiral spring(37). Or rather I should say he sketched such a mechanism. For he did not build the mechanism himself, or even a model of it. Rather, he later related that on the 21st he sought out his clockmaker, Isaac 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. Evidently, the model was completed by 3pm, 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, and hence of the profits, for the invention of the spring-regulated watch. Huygens vehemently rejected the claim, accused Thuret of violating his trust, and ended their longstanding collaboration.(38)

Given the ensuing dispute, a question arises: How did Huygens express or record his invention of the balance spring? That question seems to depend on another, to wit, when did he make the sketches accompanied by "Eureka 20 Jan. 1675"? Was it on that day, or was it some two weeks later, when after learning of Thuret's preemptive visit to Colbert he felt the need to defend his ownership of the invention by means of a day-by-day account of what had transpired in the meantime? That the invention occurred on that day seems clear from the evidence. By Huygens' account, uncontradicted at the time by any of several people in a position to do so, he had the idea on the 20th, spoke of it to Pierre Perrault on the morning of the 21st and described to Isaac Thuret around midday on the 22nd. Rather, the question is how he described it to Thuret. Did he make one or more drawings, and, if so, are they the drawings bearing the date? As John H. Leopold has observed, a look at the manuscript itself suggests the answer.(39) [Fig. 10]

Surrounding the "eureka" are two drawings and some notes. One drawing shows only a coil spring attached to a dumbbell balance, seen in top view. The other is a side view of the dumbbell mounted on an escapement, with the spring in a cylindrical housing mounted underneath a mounting plate.

At the top is a descriptive heading, "Watch balance regulated by a spring". To the left underneath the coil and balance are two notes:

le ressort doit se tenir en l'air dans le tambour et estre rivé au costè et a l'arbre (the spring should be held in the air in the drum and be riveted at the side and at the arbor)

le balancier en forme d'anneau comme aux montres ordinaires (the balance in the shape of a ring, as in ordinary watches)

Figure 10. Codices Hugeniani (HUG) 9, fol. 18r (= Ms. E, p. 35), Bibliotheek der Universiteit, Leiden (with permission).

The note to the right of the drawing of the escapement reads:

le tambour dessus la platine et grand comme le balancier, comme a la pag. suivante (the drum above the plate and as large as the balance, as on the following page [Fig. 11])

Finally, beneath the drawing, Huygens noted, "ressort de cuivre battu pourroit servir peutestre (spring of beaten copper could serve perhaps)."

Figure 11. The improved escapement, Ms E, 36; HOC.VII.409.

Clearly, some of the notes describe not the drawings but rather changes to be wrought on the designs. Leopold suggests that we may have here the record of what transpired when Huygens visited Thuret. Huygens showed Thuret his sketches, described what he had in mind, and then jotted the notes as Thuret made suggestions based on his experience as a clockmaker. Except for the first note, that seems right. Thuret looks at the dumbbell balance and says "Let's use an ordinary balance wheel, the same size as the spring." As to material, he thinks beaten copper might work. He looks at the escapement and suggests moving the spring from under the plate to above it, so as not to interfere with the 'scape wheel. Huygens makes a note, but waits until getting home to make a sketch, perhaps drawing it from the model he has brought home with him.

By contrast, the first note may not reflect the clockmaker's experience but rather have served to reinforce what Huygens was telling Thuret in describing the basic design, perhaps in response to Thuret's uncertainty about it. The spring must be fixed at both ends but move freely in between. Huygens insisted on that as the core of the design in both the privilège and the description published in the Journal des Sçavans. The idea of using a spring as regulator had been floating around for more than ten years, and several people had tried without success to make it work.(40) Most had worked from the model of the pendulum, replacing its swing with the vibrations of a metal strip or the bouncing of a coil spring, in either case leaving one end free. Quite apart from the difficulties of transmitting those vibrations to the escapement, such a direct quotation of the pendulum led to difficulties in adjusting the period of the spring's oscillations. Huygens had approached the spring as he had the pendulum. The goal was to leave the oscillator swinging freely and to keep it separately adjustable, while communicating its swings to the escapement and feeding back just enough force to keep the oscillator from running down. In Huygens' design for the spring regulator, the balance constitutes the freely swinging, adjustable weight, held steady in its pivots and communicating its motion to the escapement, either directly by pallets on the arbor or indirectly by means of a pinion. The spring, independent of the escapement and subject only to the coiling force of the arbor, regulates the swing, and its tension may be separately adjusted.

Toward the end of the dispute with Thuret, Huygens reported having reminded Thuret of just this point about the design. In an effort to explain why he had sought some part of the credit for the invention, Thuret claimed to have been thinking of such a mechanism but to have held back from doing anything because he thought that lateral vibrations of the spring would vitiate its regular oscillations. "I responded," said Huygens,

that what he said of the trouble with these vibrations was something contrived to make it appear that he knew something about the application of the spring, but that this itself showed that he had known nothing about it, because, if he had thought of attaching the spring by its two ends, he would have also easily seen that these vibrations were of no concern, occurring only when one knocked or beat against the clock and even then not undercutting the effect of the spring.(41)

Hence, the first note may account for something Huygens said on several occasions in his dispute with Thuret. "In explaining it to him," reported Huygens of his initial conversation with Thuret, "he said (as yet barely understanding it), 'I find that so beautiful that I still can't believe it is so.'"(42) As Huygens later reminded Thuret, he had said nothing about any investigations of his own prior to Huygens' visit. Yet, if Thuret had been thinking along the lines he later claimed, then he may well have expressed wonder about precisely the way in which the spring was mounted.

And it may well have been wonder born of having wrestled with the problem without seeing a solution. For Thuret evidently understood what Huygens showed him well enough both to make suggestions for its improvement and to build a working model from scratch in a couple of hours. Indeed, when Huygens then departed with the model, Thuret built another, 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 with the other parts of the watch. Irrespective of whether Thuret had in fact been exploring the problem independently of Huygens, he might have thought that the know-how involved in that process alone 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.


Figure 12.The escapement of a Thuret clock, 1675 (from Reinier Plomb, "A Longitude Timekeeper by Isaac Thuret with the Balance Spring invented by Christiaan Huygens", http://www.antique-horology.org/_Editorial/thuretplomp/thuretplomp.htm [accessed 15 July 2002]).

Huygens' response to another challenge of his priority, this time from a gadfly named Jean de Hautefeuille is revealing in this regard. Shortly after Huygens announced his invention to the Academy, Hautefeuille contested the claim both intellectually and legally, arguing that he had earlier proposed replacing the pendulum of a clock with a thin strip of steel, the uniform vibrations of which would have the same effect.(43) It would have the further advantage of working irrespective of the position of the clock and hence lend itself to use in watches. Hautefeuille admitted that he had not succeeded in getting his mechanism to work, but he insisted on having established the principle of using a vibrating spring as a regulator. The principle lay in the equal vibrations of a spring, whether straight, helical, or spiral. The particular shape and configuration of the spring was incidental, a matter to be determined by a workman (ouvrier).

Huygens responded by noting that Hautefeuille was far from the first to suggest using a spring as a regulator. It had been proposed repeatedly, but none of the earlier designs had actually worked in practice. The trick lay in transforming the spring's equal vibrations into the uniform advance of an escapement, and more than the spring was involved. Thinking that one could simply replace a pendulum by a metal strip, even to the point of adding a weight at the end of the strip to strengthen the vibrations, reflected a basic misunderstanding of the problem.

Huygens' claim to the invention of the balance-spring regulator rested on two different sorts of knowledge. First, he knew in principle that a spring, whatever its form, was a tautochronic oscillator. That meant more than knowing for a fact that springs vibrate at the same rate no matter how much they are stretched or compressed. It meant knowing why that was the case. His letter to the Journal des Sçavans announcing the invention spoke of the movement of the clock as being "regulated by a principle of equality, just as that of pendulums is corrected by the cycloid."(44) That piece of knowledge stemmed from his researches of 1673-74. As a result he knew that the relation between a spring and a cycloidal pendulum lay in their both instantiating the principle that the driving force is directly proportional to the distance from rest. They are different manifestations of the same kind of motion.

Second, Huygens had determined the particular arrangement that translated the principle into practice for the spring. As in the case of the pendulum, it was a matter of letting a weight swing freely, driven ever so slightly by the force of the driving weight or mainspring, communicated through the escapement. Hanging the pendulum by a cord attached to the frame and connecting it to the escapement by means of a crutch had been the key to using it to regulate the clock while using the clock to keep the pendulum swinging. In the case of the spring regulator, the weight was the balance wheel, pivoting about the arbor as axis and governed by the winding and unwinding of a spiral spring attached at one end to the arbor and at the other to the plate or support on which the arbor was mounted. While his original drawings showed the arbor with pallets directly in contact with the crown wheel, the version for the Journal des Sçavans showed a pinion moving a rack wheel, which turned the pallets connected with the crown wheel. That design quoted the arrangement in his original clock of 1658.

What Hautefeuille thought was a mere "accident", a technicality to be left to artisan, Huygens considered essential. It was the weight of the balance wheel that held back the advance of the escapement. Since the wheel was swinging about its center of gravity, its motion was independent of its position in space. The spring maintained the tautochrony of that motion. Neither component could do the job alone, yet they could be independently adjusted for the force of the mainspring and the period of oscillation. The secret of a "regular, portable" timepiece lay in their combination.

For present purposes, the rights and wrongs of Huygens' dispute with Thuret are the least interesting aspect of the episode. 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 this 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.(45) In the case of Renaissance machines we lack the high science, but we have the drawings and must rely entirely on modern craft expertise to reconstruct from the artifacts the conceptual framework of their predecessors a half-millennium ago.

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 tricord pendulum and his "perfect marine balance", Huygens built a model before turning to a clockmaker to produce a working timepiece.(46) In both cases, the model seems to have been a necessary proof of concept before attempting a full-scale clock.

What does this all have to do with the topic of this volume? Two things, I think. First, it shows that just drawing a picture of a device, however realistic the rendering, does not suffice to explain how it works. What it conveys depends to a large extent on what knowledge the viewer brings to it. As Huygens tells it, at least, Thuret built a model of the spring balance from Huygens' drawing without understanding how it worked. But having built one model and having heard Huygens' explanation, he evidently knew enough about the workings of the device to adapt it to a watch. That is, Thuret knew what he could change and what not; he knew how to scale it to his own needs. But he went beyond that. He claimed a share in the credit for the invention on the grounds that he had transformed Huygens' idea into a working device. If the interpretation above of what occurred between the two men is correct, Huygens' drawing did not suffice for that; it required a clockmaker's knowledge of the field of application. To Huygens' way of thinking, the invention lay in the idea. For all Thuret's expert knowledge, he had not understood at first why it worked.

Mathematical Models

That difference of perspective leads to the second point. By the late 17th century, two kinds of people were emerging from the machine literature of the Renaissance, those concerned with machines and those concerned with (mathematical) mechanics. What Thuret, the craftsman, built was a model of a specific mechanism, a coiled-spring balance. From that point on, he was interested in the various ways in which it could be used in watches and clocks. As was the case for his counterparts in England, his clockmaking skills recommended him as an instrument maker for the new scientific institutions. In the following century, clockmakers built the first textile machinery. In short, artisans like Thuret became machine builders, for whom a new genre of machine literature would develop in the 19th century.(47)

What Huygens, the mathematician, showed Thuret was a model of a general principle, "perfect incitation". For Huygens the spring balance was one of a series of models that began with the cycloidal pendulum and included the vibrating string, the spring, and the variety of forms of the "perfect marine balance". Once he had discovered the principle, he began to look for the ways in which it was instantiated in physical systems. His drawings of those systems all aimed at bringing out the proportionality between the accelerative force and the displacement from equilibrium. In some cases, it emerged more or less directly, as in the case of chains being lifted from a surface or of cylinders being raised and lowered in containers of mercury. In other cases, it took some sophisticated and ingenious manipulation to relate the geometry of the mechanism to that of the cycloid. Later mechanicians would seek it symbolically by reducing the equation of the system to the form  .
To work that way is to build a model of another sort, namely a mathematical model, in which one seeks to map a physical system onto a deductive structure. As Newton showed, using the same mechanics as Huygens but positing another kind of incitation, one in which the accelerative force varies as the inverse square of the distance, unites Kepler's empirically derived laws of planetary motion in a mathematical structure, of which Galileo's laws of local motion are limiting cases. In a second set of "Queries" added to his Opticks in 1713, he turned his attention to the chemical and electrical properties of bodies and wondered rhetorically whether they might not be explained in terms of small particles of matter attracting and repelling one another by central forces of a different sort from gravity. "And thus Nature will be very conformable to her self," he mused, "and very simple, performing all the great Motions of the heavenly Bodies by the Attraction of Gravity which intercedes those Bodies, and almost all the small ones of their Particles by some other attractive and repelling Powers which intercede the particles."(48)

His suggestion became the agenda of mathematical physics for the next two centuries, as practitioners sought to apply the increasingly sophisticated resources of analytic mechanics to mechanical models of natural phenomena. Following that agenda into the late twentieth century and to its encounter with complexity leads to a new problem of scaling and to what appears to be a redefinition of the relation of mathematics to nature. But that issue really would take this contribution beyond the question of the pictorial means of early modern engineering.


1. Cf. Marcus Popplow's observation in his contribution to this volume (above, p. ??): "The analysis of early modern machine drawings is often confronted with such problems of interpretation. To narrow down the possibilities of interpretation, descriptive texts, text fragments on the drawing and textual documents preserved with the drawings again and again prove most useful. Where such additional material is missing -which is often the case due to the frequent separation of pictorial and textual material practised in a number of European archives some decades ago- interpretation is often confronted with considerable difficulties."

2. Popplow has pointed to the dangers of such retroactive identification: "Die wenigen Blicke, die von Seiten der Wissenschaftsgeschichte auf die Maschinenbücher geworfen wurden, suchten in erster Linie zu beurteilen, inwiefern ihre Diskussion mathematischer und mechanischer Prinzipien bereits auf die wissenschaftliche Revolution des 17. Jahrhunderts verweist. Wiederum ist damit die Tendenz zu erkennen, die Maschinenbücher also noch defiziente 'Vorläufererscheinung' späterer, wissenschaftlich exacterer Abhandlungen zu betrachten. Wie im folgenden deutlich werden wird, entsprechen die dieser Betwertung zugrundeliegenden Maßstäbe jedoch nicht den Intentionen hinter der Abfassung der Werke." (Marcus Popplow Neu, nützlich und erfindungsreich: Die Idealisierung der Technik in der frühen Neuzeit (Münster, 1998), 69) As Edgerton's example shows, it is not only the historians of science who have measured these works by later standards.

3. See Michael S. Mahoney, "Diagrams and Dynamics: Mathematical Perspectives on Edgerton's Thesis", in Science and the Arts in the Renaissance, ed. John W. Shirley and F. David Honiger (Washington: The Folger Shakespeare Library, 1985), Chap. 10; an online version is available at www.princeton.edu/~hos/Mahoney/articles/diagdyn/diagdyn.html. Edgerton set forth his position in "The Renaissance Artist as Quantifier", in The Perception of Pictures, ed. Margaret A. Hagen (New York, 1980), Vol. 1, 179-212. See also the contributions of Bert Hall and David Topper in Brian S. Baigrie (ed.), Picturing Knowledge (Toronto), the former of which refers to the "Edgerton-Mahoney Debate" (pp. 21-28). Edgerton and I have indeed debated the issue on several occasions, but never in a public forum.

4. Quoted by Giorgio di Santillana, The Crime of Galileo (Chicago, 1955), 22, in a rather free translation. Cigoli made the remark in a letter to Galileo dated 11 August 1611 in a perplexed effort to understand why Christoph Clavius continued to resist Galileo's telescopic evidence of the moon's rough, earth-like surface (Opere di Galilei, ed. A. Favaro, Vol. 11,168): "Ora ci ò pensato et ripensato, nè ci trovo oltro ripiegho in sua difesa, se no che un matematico, sia grande quanto si vole, trovandosi senza disegnio, sia non solo un mezzo matematico, ma ancho uno huomo senza ochi." It is not clear just what Galileo's drawings of the moon's surface have to do with mathematics.

5. Mechanica sive de motu scientia analytice exposita (St. Petersburg, 1736), Preface, [iv]: "Newton's Mathematical Principles of Natural Philosophy, by which the science of motion has gained its greatest increases, is written in a style not much unlike [the synthetic geometrical style of the ancients]. But what obtains for all writings that are composed without analysis holds most of all for mechanics: even if the reader be convinced of the truth of the things set forth, nevertheless he cannot attain a sufficiently clear and distinct knowledge of them; so that, if the same questions be the slightest bit changed, he may hardly be able to resolve them on his own, unless he himself look to analysis and evolve the same propositions by the analytic method."

6. Mécanique Analitique (Paris, 1788), Avertissement: "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."

7. See Michael S. Mahoney, "“The Mathematical Realm of Nature”, in The Cambridge History of Seventeenth-Century Philosophy, ed. Daniel Garber and Michael Ayers (Cambridge: Cambridge University Press), I, 702-55.

8. In "The Semantics of Graphs in Mathematical Natural Philosophy" (Renato G. Mazzolini (ed.), Non-Verbal Communication in Science Prior to 1900 [Firenze, 1993], 197-233), John J. Roche notes that Lagrange did not go unchallenged over the next century, pointing in particular to Louis Poinsot's complaint in 1834 that the "solutions are more lost in the analytical symbolism than the solution itself is hidden in the proposed question (225)."

9. Isaac Newton, Philosophiae naturalis principia mathematica, Praefatio ad lectorem, [i]. John Wallis had already spoken in similar terms at the beginning of Chapter 1 of [Wallis 1670]. In contrast to traditional definitions of mechanics that emphasized its artisanal origins or material focus, he insisted that, "We speak of mechanics in none of these senses. Rather we understand it as the part of geometry that treats of motion and investigates by geometrical arguments and apodictically by what force any motion is carried out (p. 2)." ("Nos neutro dictorum sensu Mechanicen dicimus. Sed eam Geometriae partem intelligimus, quae Motum tractat: atque Geometricis rationibus & apodeiktikôs inquirit, Quâ vi quique motus peragatur.")

10. See ibid., the corollaries to Proposition 4 of Book I and the hypotheses (phaenomena in the 3rd edition) and opening propositions of Book III.

11. On the place of the mill in medieval society, see Richard Holt, The Mills of Medieval England (Oxford, 1988) which significantly revises the long standard interpretation of Marc Bloch in his classic"Avènement et conquête du moulin à eau", Annales d'histoire economique et sociale 36[1935], 583-663), at least for England. Marjorie Boyer's "Water mills: a problem for the bridges and boats of medieval France", History of Technology 7(1982), 1-22, calls attention to the urban presence of mills and makes it all the more curious that medieval scholars could talk of the machina mundi without mentioning them.

12. Discorsi e dimostrazioni matematiche intorno à due nuove scienze (Leiden, 1638), 1: "Largo campo di filosofare à gl'intelletti specolativi parmi che porga la frequente pratica del famoso Arsenale di Voi Sig. Veneziani, & in particolare in quella parte che Mecanica si domanda." Cf., most recently, Jürgen Renn and Matteo Valleriani, "Galileo and the Challenge of the Arsenal", MPIWG Preprint 179 (2001) [pdf].

13. The use of the vernacular in the first two days should not mislead us about Galileo's intended audience. The same Salviati who protested when Simplicio began to dispute in Latin in the Two World Systemsslips unapologetically into the language of the schools in the third and fourth days of the Two New Sciences when setting out Galileo's new scientia de motu, a subject long part of the university curriculum. That part, at least, would not need translation to be understood by philosophers elsewhere in Europe.

14. In addition to the chapters in this volume by David Magee and Mary J. Voss, see Mary J. Voss, Between the cannon and the book: Mathematicians and military culture in 16th-century Italy (Ph.D., Johns Hopkins, 1995); Serafina Cuomo, "Niccolò Tartaglia, mathematics, ballistics and the power of possession of knowledge", Endeavour: Review of the Progress of Science 22(1998), 31-35.

15. Popplow, 72ff.

16. Ibid.: "Der Königsweg der Darstellung lag darin, auf der einen Seite die Umsetzbarkeit der vorgestellten Entwürfe sowie die dabei angewandten wissenschaftlichen Prinzipien überzeugend zu vermitteln und gleichzeitig auf der anderen Seite tatsächlich entscheidende Konstruktionsprinzipien zu verschweigen. Die Tendenz der Autoren der Maschinenbücher, 'unrealistische' Entwürfe zu präsentieren, scheint aus dieser Sicht durchaus vernünftig."

17. Domenico Fontana, Della trasportatione dell'obelisco vaticano et delle fabriche di nostro Signore Papa Sisto V (Rome: Domenico Basso, 1590), fol. 7v. The plate shown above is on fol. 8r.

18. A similar juxtaposition of real and fantastic occurs in Georgius Agricola's De re metallica, in which the inventory of machines used in mining begins with the wholly practicable windlass, moves through ever more complicated combinations, and concludes with a water-driven crane that strains credulity once one takes into account the forces involved in reversing its direction in the times necessary to place loads at the desired level. At a certain point Agricola seems to be no longer reporting actual machines but rather imagining potential machines. Another example is Jean Errard de Bar-le-Duc's Le premier livre des instruments mathematiques mechanique (Paris, 1584). Few of the devices are drawn to scale, nor would they work if they were actually built, in some cases again because humans would not be able to drive them.

19. Discorsi, 3: "...si 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."

20. That is what conservation laws do, most notably in thermodynamics. In computer science, theory similarly sets limits on computability, decidability, and complexity.

21. See M.S. Mahoney, "Christiaan Huygens, The Measurement of Time and Longitude at Sea", in H.J.M. Bos et al. (eds.), Studies on Christiaan Huygens (Lisse: Swets, 1980), 234-270.

22. William J.H. Andrewes (ed.), The Quest for Longitude: The Proceedings of the Longitude Symposium, Harvard University, Cambridge, Massachusetts, November 4-6, 1993 ( Cambridge, MA: Collection of Historical Scientific Instruments, Harvard University, 1996).

23. For a detailed account of what follows, 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: Kluwer Academic Publishers, 2000), 17-39.

24. Relying on the well known mean-speed theorem, Huygens takes the speed at B as the constant speed reached at Z, since the time over the interval AZ at that speed will be twice the time of uniform acceleration over the interval.

25. For details, see M.S. Mahoney, "Christiaan Huygens, The Measurement of Time and Longitude at Sea", in H.J.M. Bos et al. (eds.), Studies on Christiaan Huygens (Lisse: Swets, 1980), 234-270.

26. The worknotes dated 1661 (HOC.XVI.415-34) became Part IV, De centro oscillationis, of Horologium oscillatorium

27. Huygens first used this technique in his derivation of the laws of elastic collision in De motu ex percussione in 1659. There, however, the bodies fall vertically from initial positions to acquire the speeds at which they collide, and then the speeds after collision are converted upward to resting positions. The basic is the same: the center of gravity of the system neither rises nor falls in the process. These drawings confirm what one suspects was the physical setup behind the diagrams in the earlier work, namely experiments on impact using pendulums.

28. The indeterminacy of n adds another reason for moving to algebra, where it can be treated operationally as a magnitude. There is no way to represent pictorially an indeterminate number of bodies.

29. Indeed, Huygens had to retain the algebra even in the finished geometrical form of the Horologium.

30.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: University of Delaware Press; London and Toronto: Associated University Presses, 1993), 183-205.

31. Pierre Varignon, "Des forces centrales inverses ", Mémoires de l'Académie Royale des Sciences [hereafter MARS] (1710), 534-44; at 534. Varignon's solution of the problem of inverse central forces (given the force, to find the curve) rested on his earlier analysis of the direct problem (given the curve, to find the force) in "Des forces centrales, ou des pesanteurs necessaires aux Planetes pour leur fair décrire les orbes qu'on leur a supposés jusqu'icy", MARS (1700), 218-237, which was based on a similar adaptation of Newton's diagram.

32. Indeed, Varignon's statement of the problem of inverse central forces several years later, while echoing Newton's proposition, reflected the shift of mathematical focus: "Problême: Les quadratures étant supposées, & la loy quelconque des Forces centrales f etant donnée à volonté en y & en constantes; Trouver en générale la nature de la courbe que ces forces doivent faire décrire au mobile pendent des tems ou des élemens de tems dt donnés aussi à volonté en y & en constantes multipliées par dx ou par dz variables ou non." ("Des forces centrales inverses", MARS (1710), 533-44; at 536.)

33. David McGee makes a similar point (albeit eschewing the term "fantasy") in his contribution to this volume when he speaks of Taccola's style of drawing enabling him "to work in the absence of real physics". Indeed, he adds in a note, "the whole point of the drawing style is to eliminate the need for physical considerations on the part of the artist."

34. See M.S. Mahoney, "The Mathematical Realm of Nature", in Daniel Garber and Michael Ayers (eds.), Cambridge History of Seventeenth-Century Philosophy (Cambridge: Cambridge University Press, 1998), Vol. I, 702-55; esp. 706-8.

35. Ps.-Juanelo Turriano, Los veintiun libros de los ingenios y de las maquinas, "el agua no puede ir de suyo para arriba ... por causa de su gravedad y peso." For an example of such maxims filtered through a reading of the Mechanical Problems attributed to Aristotle, see A.G. Keller, "A Manuscript Version of Jacques Besson's Book of Machines, With His Unpublished Principles of Mechanics", in Bert S. Hall and Delno C. West (eds.), On Pre-Modern Technology and Science: Studies in Honor of Lynn White, Jr. (Malibu: Undena Publications, 1976), 75-103.

36. Worknotes and sketches in HOC.XVIII.489-98.

37. HOC.VII.408ff; on Huygens invention of the spiral balance, see J.H. Leopold, "Christiaan Huygens and his instrument makers", in H.J.M. Bos, et al. (eds.), Studies on Christiaan Huygens (Lisse, 1980), 221-233; and "L'invention par Christiaan Huygens du ressort spiral réglant pour les montres",in Huygens et la France (Paris, 1979).

38. Huygens appears to have had grounds for being angry. According to his version of the facts, which Thuret's supporters did not contest, Huygens had pledged Thuret to secrecy before explaining the mechanism he wished to have built. Over the following week, Huygens did more work on the design as he planned both its announcement to the scientific community and its presentation to Minister Colbert for the purpose of securing a privilège restricting its manufacture to those licensed by Huygens. On 30 January, he wrote Henry Oldenburg, secretary of the Royal Society, announcing a new mechanism for regulating watches accurately enough to determine longitude and encoding its basic principle in an anagram. The next day he visited Colbert. On 1 February, he learned that Thuret had already attended the Minister on the 24th to show him the second model and that people were now speaking of Thuret as the inventor. Thuret had said nothing of this to Huygens, even though the two had collaborated during the week. Thuret had made hints to Huygens that he desired a share in the credit. But Huygens rejected the idea, even as he pointed out that in enjoying Huygens' license to produce watches with the mechanism Thuret stood to reap the greater monetary gain. On learning of what he took as a betrayal of trust, Huygens cut off all relations with Thuret, excluding him from a license. As the leading clockmaker of Paris, Thuret enjoyed the protection of some powerful patrons, not least Madame Colbert and her son-in law, Charles Honoré d'Albert de Luynes, Duc de Chevreuse. Their intercession led to Thuret's written acknowledgement of Huygens' sole claim to the invention and expression of regret that he might have acted in any way to suggest otherwise. In return he received authorization to produce the new watches. But, then, so too did all clockmakers in Paris, as Huygens decided not to ask the Parlement de Paris to register his privilège. Although the two men eventually reconciled, and indeed Huygens recognized the superiority of Thuret's craftsmanship, they never resumed their active collaboration, which had constituted a powerful creative force in timekeeping. For details of the dispute, which dominated Huygens' attention for six months, see HOC.VII.405-498.

39. Leopold, "Christiaan Huygens and his instrument makers", 228.

40. See, e.g., Huygens' reply of 18.IX.1665 (HOC.V.486) to a report from Robert Moray (HOC.V.427) that Robert Hooke had spoken of "applying a spring to the balance of a clock in place of a pendulum". Huygens said he recalled having heard of the idea on a visit to Paris in 1660 but did not think it would work, at least as proposed. Having himself made the idea work in 1675, he never adverted to those earlier suggestions and to what role, if any, they played in his thinking.

41. Ibid.

42. "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." HOC.VII.410.

43. J. de Hautefeuille, "Factum, touchant Les Pendules de Poche" (a petition to the Parlement de Paris to block the registration of Huygens' privilège, HOC.VII.439 -53; cf. his letter to the Académie des Sciences, 7 July 1674, describing his idea, HOC.VII458-60.

44. Journal des Sçavans, 25.II.1675; repr. in HOC.VII.424-5; at 424: "... leur [sc. les horologes] mouvement est reglé par un principe d'égalité, de même qu'est celuy des pendules corrigé par la Cycloïde."

45. On this point, see M.S. Mahoney "Longitude in the History of Science", in W.J.H. Andrewes (ed.), The Quest for Longitude (Cambridge; Harvard University Collection of Historical Scientific Instruments, 1996), 63-68. The several articles in that volume on the clocks of John Harrison are brilliant examples of such direct "readings" of the artifacts.

46. 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 our 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.XVIII.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.

47. In England, the great clock and instrument makers could aspire to learned status, as Richard Sorrenson has shown in his dissertation, "Scientific Instrument Makers at the Royal Society of London, 1720-1780" (Ph.D., Princeton, 1993); for an example of one such clockmaker, see his "George Graham, Visible Technician", British Journal for the History of Science 32(1999), 203-21.

48.  Isaac Newton, Opticks, or a Treatise of the Reflections, Refractions, Inflections & Colours of Light (4th ed., London, 1730; repr. N.Y.: Dover Publications, 1952), 396.