Diagrams and Dynamics

Mathematical Perspectives on Edgerton's Thesis

Michael S. Mahoney

Princeton University

[A] mathematician, however great, without the help of a good drawing, is not only half a mathematician, but also a man without eyes.
Lodovico Cigoli to Galileo Galilei, 1611 (1)
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.
Leonhard Euler, 1736 (2)
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.
Joseph-Louis Lagrange, 1788(3)
In "The Renaissance Artist as Quantifier" Samuel Y. Edgerton sets forth a new defense of Erwin Panofsky's thesis that the technical innovations of Renaissance art, in particular linear perspective and chiaroscuro, laid essential foundations for the Scientific Revolution of the seventeenth century. (4) Exercising some caution at the outset, Edgerton hesitates to assert a causal link between the two achievements; rather, he emphasizes their striking concomitance. "I would like to propose," he says,
that instead of trying to discover elusive, one-on-one connections between individual geniuses of Renaissance art and seventeenth-century science, we concentrate our investigations on the shared uniqueness of the Western European scientific and artistic revolutions. (5)
The two areas of innovation thereby become manifestations of some unspecified, more fundamental change in Europeans' experience or perception of the world. Without the one, the other is unlikely to appear.

But that caution soon gives way to boldness. Edgerton goes on to argue that "the new Renaissance pictorial language allowed ... the ability to invent machines solely by means of drawings...." (6) Then, after presenting evidence that Chinese illustrators could not "read" the pictures of machines in the 1588 edition of Agostino Ramelli's Le diverse et artificiose machine (7) and possessed no equivalent language of their own, Edgerton concludes that the Chinese could not invent machines solely by drawing them and finds the conclusion perplexing: "In fact, by Western standards, it is hard to comprehend how Chinese science and technology was [sic] able to progress at all with so little involvement of artists or pictures." To put the matter that way would seem to make changes in pictorial representation not a concomitant of scientific innovation but a prerequisite to it. (8)

First, he supposes that new techniques for depicting machines led to the invention of new machines, or more precisely, that the machines drawn by Renaissance artist-mechanics were new machines invented on the drawing board. Second, he assumes that these new machines, by virtue of the means by which they were conceived, played an essential role in the Scientific Revolution. This second assumption appears to derive from the more fundamental premise that the Scientific Revolution consisted in essence of the creation of the science of mechanics and of the mechanistic world view. Edgerton's strong thesis thus comes down to the claim that the new pictorial techniques were a prerequisite for the new science of mechanics and for the new world-machine described by that mechanics.

Several arguments militate against this thesis, even if one accepts (as I do) the underlying characterization of the Scientific Revolution. First, quite apart from the difference between invention and design, the machines being drawn in new ways during the Renaissance consisted of components devised in previous ages Renaissance devices may have been larger and more intricate than their ancient or medieval prototypes, but they were not different in kind. They still essentially compounded the five simple machines to transmit the force of weight, wind, water, and animals. (9) The ingenuity of the combinations and the occasional innovations in linkages between components do not suffice to alter the basically derivative nature of the technical content of Renaissance treatises on machines. The disegnatori may have been drawing in new ways, but they were not drawing new things.

Second, neither the things they drew nor the ways they drew them contributed to revealing the working of machines, at least not in the sense of a scientific account. The science of mechanics followed other paths to that revelation. The theoretical discipline began in the late sixteenth century as a science of machines considered as systems of weights in equilibrium. In the early seventeenth century, mechanics expanded to become the science of motion, which conceived of machines as systems of bodies moving under constraint. (10) Fundamental to either line of inquiry was its concept of the machine as an abstract, general system of quantitative parameters linked by mathematic relations. Conceptualization of this sort rested on a small and long-familiar empirical base, namely the five simple machines and the common hydrostatical phenomena, which was expanded in the early seventeenth century by the pendulum considered both as a phenomenon and as an instrument of experiment. (11) Whether new or not, whether real or fancied, Renaissance additions to the inventory of machines added nothing to that empirical base. Moreover, it is difficult to see how more accurate depiction of the basic phenomena as physical objects could have conduced to their abstraction into general systems. For the defining terms of the systems lay in conceptual realms ever farther removed from the physical space the artists had become so adept at depicting. Those terms could not be drawn; at best, they could be diagramed.

That leads to a third argument. The vehicle of abstraction in mechanics was mathematics, which from the outset served as the language for expressing and analyzing mechanical concepts. Neither the earliest forms of mathematical mechanics nor its subsequent development lends support to Edgerton's thesis. The Renaissance search for a science of machines began in emulation of Archimedes' geometrical treatment of statics in terms of spatial arrangements considered in equilibrium. Here the initial steps of abstraction, namely the reduction of a machine's physical structure to a geometrical configuration, followed long-standing forms of mathematical depiction. Galileo took his mathematic diagrams from the classical Greek mathematicians; some he drew from medieval prototypes. Despite the Renaissance pictorial techniques evident in the woodcuts of his Two New Sciences, as mathematical diagrams they remained classical in form. (12)

Further stages of abstraction, together with the mathematics for handling them, placed a strain on the classical form, especially in the transition to a science of motion. The introduction of dynamical parameters, that is, of the forces that determine the laws of machines in action, brought with it quantities that did not fit into three-dimensional space and hence had no place, except by proxy, in a geometrical diagram representing a spatial configuration. But that lack of fit could not be remedied by new modes of representing space pictorially. Temporary adaptations of geometrical technique ultimately gave way to an altogether different form of mathematical representation: to infinitesimal analysis couched in the language of symbolic algebra.

In taking the path to analytical mechanics, mechanicians at the turn of the eighteenth century followed a line of conceptual development on which mathematics itself was by then well embarked. As I have argued in detail elsewhere, the domination of seventeenth-century mathematics by algebraic modes of thought freed it from its ties to physical intuition and opened it to the consideration of abstract structures defined by combinatory relations. (13) Only by reaching into realms for which no physical correlate existed, for example the realm of imaginary numbers, could mathematicians achieve the theoretical generality they claimed for their subject. Mathematicians reached those realms not by looking at the physical world in a new way but by looking beyond it altogether. To the extent that mechanicians followed suit, the science of mechanics that epitomizes the Scientific Revolution manifests modes of thought antithetical to those of Edgerton's inventive disegnatori.

The discussion to follow will concentrate on the changing nature of the diagram in seventeenth-century treatises on mechanics, that is, on the third argument just outlined. By way of transition into that subject, however, some remarks concerning the second argument seem in order.

The first treatises of machines captured in pictorial techniques many (but not all) of the tactile methods of the traditional engineer, whose job it had long been to devise ingenious mechanisms to overcome the difficulties of a particular situation. (14) The treatises collected various mechanisms, depicted them in action to show both what they did and how they were put together, and explored ways in which new combinations might be assembled to carry out new but related tasks. During the fifteenth and sixteenth centuries, the methods of depiction clearly improved, as Edgerton shows. The machines represented on the two-dimensional page looked increasingly like their three-dimensional models as seen in action, even as the artist exploded them, bored through to their internal parts, and twisted and turned their components. (15)

But to show what machines do or how they are assembled is one thing; to show how they work is quite another. However accurately and fully a complex mechanism may be portrayed, an understanding of its operation as a whole rests ultimately on familiarity with the operations of its basic components. Treatises of the genre under discussion took that familiarity for granted. Their authors could not do otherwise, given the nature of their medium. A picture of a windlass, or of a system of pulleys, cannot in and of itself set forth the laws that define the device's mechanical advantage. A drawing of a closed tube standing in a pool of water and having a piston with a valve that opens in one direction only will still not explain a water pump until the readers know the laws (or at least the rules of thumb) that link the reduction of air pressure to the rise in the head of a column of liquid. (16) Readers must bring knowledge or experience of such matters to the illustrations in order then to appreciate or profit from the ingenuity with which the basic machines are combined or adapted to particular circumstances.

Such active participation by the knowledgeable reader is especially required when the machine depicted is extrapolated from the world of experience to the realm of fantasy. Several of Ramelli's devices, if actually constructed, would surrender all mechanical advantage to internal friction. To know that does not detract from an appreciation of their ingenuity, but it does place that ingenuity in a different light. If Vittorio Zonca's Teatro nuovo di machine ed edificii (Padua, 1607) holds closer to the line of actual mechanical practice, the author nonetheless caps his presentation with a perpetual-motion machine. (17) Again, the experienced reader will admire the ingenuity but not be misled by it. Yet, discernment lies in the eye (or rather the mind) of the beholder, not in the depiction of the machine. The picture itself cannot distinguish between the feasible and the fantastical. For that, one needs a different medium.

Treatises on mechanics first appeared about a century after the earliest collections of drawings began to circulate. (18) Although the authors of these treatises may have been motivated to their subject by the mechanical activity around them, their manner of treating it drew inspiration from more distant sources, namely from the Mechanical Problems attributed to Aristotle, from the Pneumatics and the other mechanical fragments of Hero of Alexandria, from Book 8 of Pappus of Alexandria's Mathematical Collection, and from the statical and hydrostatical works of Archimedes. Following these sources, the sixteenth-century treatises on mechanics began with an inventory of basic devices, usually the five simple machines. But, rather than assuming their mechanical action as known and then compounding them, as the theaters of machines did, the treatises of mechanics analyzed them as compound phenomena to discover what principle or principles explained their action in general.

From the outset, then, mechanics as the science of machines aimed not at variety through ingenious adaptation to specific tasks, but at uniformity through derivation of the general laws that defined the limits of adaptation and ingenuity. While treatises on machines displayed the products of the engineer's craft, treatises on mechanics probed the basis of his know-how. Behind the design and application of machines lay the engineer's experience of the physical world, and it was in that experience, rather than in the machines, that writers on mechanics sought understanding of how machines-and the worldworked. Relatively little of the experience could be captured, much less examined, within the confines of pictorial space, however sophisticated the techniques for organizing it.

What did engineers know that writers on mechanics found noteworthy? Engineers knew that getting something from a machine meant putting something into it; usually more went in than came out. For example, to move a heavy weight through a short distance a small force had to travel over a great distance. Engineers knew that, if a given force sufficed to hold a given weight in equilibrium, the slightest additional force would move the weight; they also knew that some additional force was necessary in practice. Engineers knew that a small weight moving quickly could have a greater effect on an object than did a large weight simply resting on it. Engineers knew that making machines bigger did not necessarily increase or even maintain their mechanical advantage. Engineers knew that there was a limit to the height to which water could be pumped under the best of conditions and that the practically attainable height lay below that. Engineers knew that, when one presses on a fluid, it spurts out in all directions. And so on.

Engineers knew such things and others like them-call them "maxims or precepts of engineering experience"-in many different forms and at many levels of specificity. (19) Transforming them into the principles of a science of mechanics meant analyzing them into a body of concepts susceptible to mathematical expression and manipulation. At first, the mathematics for such an enterprise came from the same classical sources as the notion of the enterprise itself: from Archimedes, Hero, Pappus, and the corpus of geometrical knowledge on which they had drawn. (20) From the same sources came also the models for the diagrams in which writers like Galileo abstracted physical bodies and processes; they were the same sorts of diagrams that had lain on the leaves of geometrical manuscripts for almost a millenium, be the language Greek, Arabic, Latin or Italian. If, as Edgerton claims here and, even more forcefully, in his Renaissance Rediscovery of Linear Perspective, Galileo and his contemporaries were seeing the spatial world in new ways, they did not alter the geometrical patterns by which they represented and analyzed the metrics of that space. (21) As far as spatial abstraction is concerned, Galileo's world looked the same as Archimedes'.

Yet, as far as physical theory is concerned, Galileo's world looked quite different from Archimedes', and the crux of the difference lies precisely in Galileo's conclusion that the mechanical world has other than simply spatial parameters. For Archimedes, the science of mechanics was the science of bodies held in equilibrium, either by simple machines like the balance or by fluids. (22) His approach to such statical phenomena was similarly statical and, for that reason, also spatial. It rested on the principle that materially homogeneous bodies balance (that is, have no sufficient reason to move from their resting position) about points, lines, and planes of spatial symmetry. In each instance, he began with a system in equilibrium; his diagram represented a cross-section of the system by a plane of symmetry, and his various theorems followed from the ways he could rearrange the elements of that cross section while maintaining the initial symmetry.

Galileo's demonstration of the law of the lever in Day Two of the Two New Sciences captured this Archimedean style of analysis and demonstration. (23) Consider a uniform solid suspended from a balance beam by cords attached at either end. By symmetry the beam balances at its midpoint C. Now imagine the solid cut perpendicularly at some point D and the two resulting pieces temporarily supported by the common cord DE. Each thereby becomes a smaller version of the original system and hence balances about its center, G on the left side and F on the other. Indeed, if portion AD were supported at its midpoint by a single cord LG and portion DB by a similarly placed cord MF, the other cords could be removed and the equilibrium maintained. It is a straightforward matter to show that CG: CF = EI: EH = DB:AD, the last two quantities being proportional in turn to the weights of the segments.
Note that nothing ever moves in the space of Galileo's diagram, either actually or virtually. Rather, space is simply redivided and the points and lines of symmetry correspondingly relocated. As an abstract depiction of the physical system, the diagram contains elements that directly represent the parameters of the theoretical explanation, namely weight and distance from the fulcrum. Those elements can be manipulated geometrically to reflect the various combinations of those parameters; at each stage the transformed diagram represents immediately an actual state of the physical system. One can "see" what is going on.
Galileo knew of an alternative to the Archimedean approach. Stemming originally from Aristotle's Mechanical Problems, it informed the treatises on weight ascribed to jordanus de Nemore and dating from the thirteenth century. (24) The alternative mode of analysis defined equilibrium by the exactly countervalent effects of any assumed, or virtual, disequilibrium. Let bodies A and B be placed at opposite ends of a balance beam and suppose that B were to descend, causing A to ascend. A and B would then each move in any given time through arcs proportional to their respective distances from the fulcrum. On the assumption that they move at uniform speeds, it follows that those speeds are inversely proportional to the respective arcs and hence also to the respective distances of the bodies from the fulcrum. On the further assumption that the speeds are directly proportional to the forces moving the bodies, it follows that the force causing B to descend is to the force causing A to rise inversely as the respective distances of those bodies from the fulcrum. If, then, the weights of A and B were likewise inversely proportional to their distances from the fulcrum, the weight of A would precisely counteract the force exercised to raise it by the weight of B acting through the beam. Hence, no motion would occur, and A and B would be in equilibrium. 

Clearly, the diagram of the system plays in this analysis of its workings a role quite different from that of the diagram in Archimedean statics. The rearrangement recorded by the former takes place not in space but in time. The arcs connect the endpoints of the beam in two positions separated by an interval of time and themselves represent the trajectories of those endpoints. The weights remain unchanged throughout and their magnitudes play no operative role in the diagram; hence, they are reduced to dimensionless points. In fact, little in the diagram plays any operative role, once we have ascertained that the arcs are proportional to the distances from the fulcrum. From then on, the reasoning takes place off the diagram, and the determinative parameters have no spatial representation, at least not in the same space as the system depicted. Speed, force, and weight can be located in the diagram only by transformation of their various relations into relations among its elements. The rules of transformation, that is, the laws of dynamics and kinematics used to link weight and distance, correspond to no geometrical operations executable directly on the diagram. Only when the rules turn out to be successively, and hence compositely, linear can one then take as surrogates for the weights the corresponding arms of the balance or the arc-lengths of any putatively incipient motion of the system.

In the research carried out during the 1590s in Padua, Galileo explored both the Archimedean and the Aristotelian/Jordanian approaches to statics-and hence to mechanics understood as the science of weight-lifting machines. (25) Indeed, in his On Motion composed just before moving from Pisa to Padua in 1592 he used Jordanus's analysis of the bent-arm balance to derive the law of the inclined plane. Here, having embedded the magnitudes of the weights in those of the balance's arms, or segments thereof, he could move mathematically from the ratios among these lengths to those between the heights and lengths of the inclined planes coincident with the lines of action of the weights in different positions. 
But by the time Galileo composed his Mechanics, sometime around the turn of the century, he was favoring the statical Archimedean approach, terming the dynamical Jordanian merely "in agreement [with fact]" and "probable." (26) The reason for his preference seems evident: the dynamical approach relied more on "physical" than on "mathematical" reasoning. Within a few more years, he had even more cause to suspect the Jordanian mode, for he had learned that the dynamics on which it (and his own use of it) rested was not correct. As he pressed forward with his investigations into the science of motion, (27) he found that the theory that accounted for the mechanical advantage of an inclined plane did not lead to an explanation of the motion that occurred when a body at rest began to roll down the plane, picking up speed as it went. Nor did the combination of bent-arm balance and inclined plane account for the regular motion of the pendulum, which by something akin to a gestalt-switch appeared to the receptive eye in the diagram of the bent-arm balance when several positions of the arm were depicted together. Bodies falling by virtue of their weight do not move uniformly. They accelerate, and the relation between motive force and rate of acceleration proved an elusive problem for Galileo. "Good drawings" did not provide much help until he figured out what to draw.

For various reasons not pertinent to the present argument, Galileo postponed the search for the parameters of dynamics until he had worked out mathematically the kinematics of uniform and accelerated motion. Experiments with the pendulum had convinced him that all bodies, regardless of weight and size, fall at the same rate in the empty space of a vacuum, that the speeds acquired in falling depend only on the height from which the bodies fall, and that those speeds suffice to impel the bodies back up to their initial positions, again independently of the paths taken. On the basis of these empirically determined principles, Galileo sought first to work out the mathematical patterns of the ensuing motions.

As is well known, his mathematics initially got in his way. (28) Moving from the pendulum to the inclined plane as the machine that served both as phenomenon and as experimental apparatus, (29) he determined that the distances traversed from rest varied as the squares of the times elapsed, and in a famous letter written to Paolo Sarpi in 1604 he asserted that this law of falling bodies followed from the principle that in uniformly accelerated motion the speed acquired is proportional to the distance traversed from rest. (30) Many explanations have been offered for this error, which Galileo himself almost boastfully acknowledged in his Two New Sciences.(31)Spatial intuition, as reflected in Archimedean diagrams, must take some share of the blame. For those diagrams depicted the apparatus as a spatial object and located the operative parameters in its constituent elements. So too Galileo began the mathematics of motion on an inclined plane-that is, motion according to the times-squared rule-with a picture of the plane and sought to locate the parameters of acceleration in its elements. Since the speed acquired depends only on the height of fall and is independent of the path taken, what would be more natural than to identify the base of the plane as the geometrical measure of the speed reached over the entire plane? Since acceleration is uniform, the speed at any intermediate point on the plane would then be measured by the line through that point and parallel to the base. That line is directly proportional to the height through which the body has descended vertically.

The theorem Galileo was trying to prove linked distance to time. But time as yet had no representation in the diagram, which lacked any element by which to express that parameter directly. For an indirect measure, Galileo appears to have resorted to a medieval notion of "total velocity" (velocitas totalis), which transformed velocity conceived of as the intensive measure of a body's motion from instant to instant into an extensive quantity. In his notes:

because the velocity with which the moving body has moved from a to d is composed of all the degrees of velocity [it] had at every point of the line ad, and the velocity with which it has traversed the line ac is composed of all the degrees of velocity [it] had at every point of the line ac, so the velocity with which it has traversed the line ad has to the velocity with which it has traversed the line ac the proportion [read ratio] that all the parallel lines drawn from all the points of the line ad up to ah has to all the parallels drawn from all the points of the line ac up to ag; and this proportion is the one that the triangle adh has to the triangle acg, to wit, the square ad to the square ac. Thus the velocity with which line ad is traversed has to the velocity with which the line ac is traversed the double proportion of da to ca. (32)
At this point only a paralogism based on the ambiguity of the term "inversely proportional" could save the argument. For the "total velocity" of medieval theory was equivalent to nothing other than the distance traversed. That distance, conceived of physically as the sum over time of the instantaneous degrees of speed, found mathematical representation in the sum of the lines corresponding to those degrees, i.e., in the area of the triangle of which one side represented the time of motion and the other the final speed attained. But Galileo summed the degrees of speed over distance, not time; as a result, he ended up with two dimensionally incompatible measures of the same physical parameter: the distance traversed was expressed both by the segments of the side ab and by the segments of the triangle abk.

Galileo soon recognized the contradiction and the means of correcting it. The speed acquired in uniform acceleration is proportional to the time elapsed, not to the distance traversed. The diagram did not seem to change much in accommodating that correction. The height of the plane became a measure of the elapsed time, the lines parallel to the base remained the measure of the velocity, and the area now measured the distance traversed. But those are essential changes, for they make the diagram not a picture of the inclined plane, but a graph of the relation between time and speed in a mathematical space wholly divorced from the physical space in which the motion itself is taking place. (33)

In the kinematics of accelerated motion no perspective construction maps the three dimensions of space, time, and velocity onto a two-dimensional picture. The diagram that reveals the structure of a kinematical process cannot at the same time hold a representation of the moving body itself. To capture the changing velocity of a ball rolling down an inclined plane, Galileo had to convert the plane into a gradient of velocities and the ball into a point sliding along the gradient at a uniform rate. To depict the body's motion through space over time required yet another shift of mathematical space, to one in which the moving point now traces a parabola. That parabola, in turn, should not be confused with the trajectory of a projectile launched into physical space, as found in the theorems in the Fourth Day of the Two New Sciences. There again the direct representation of a physical occurrence comes at the cost of removing the effective mathematico-mechanical parameters from the picture and thus of reasoning off the diagram.

If, then, Galileo started with diagrams that looked classical, he ended with a new sort of representation altogether: a configuration tracing the relation between two quantities in a mathematical space of which they define the dimensions-time and speed, time and distance, speed and distance, and so on. The relations between such configurations, that is, the rules for moving from one space to another, lay in the laws of motion. For example, by the law S = vt (and by the mathematics of infinitesimals that is necessary to such reasoning but not to the exposition of the present argument) distance may be represented as an area in the space of speed and time.

Whatever the mathematician's eye is seeing here, it has little to do with new pictorial techniques for the accurate representation of physical objects in threedimensional space. It is the mind's eye that is looking here, and it is peering into the structural relations among quantities belonging to many different conceptual (rather than perceptual) spaces. The more abstract those quantities and their relations become, the less helpful or revealing it is to model them in a graphic space analogous to the pictorial space of the body's eye.

Descartes saw this situation developing and argued in his Rules for the Direction of the Mind (ca. 1628) that the exploration of a new mechanical world would require a new mathematics, universal in its scope. For what the mind must see before it is the path of its reasoning, and it should restrict its picture of the objects under consideration to the features essential to the reasoning itself. Indeed, the mind needs not so much a picture as a set of symbols, a conventional notation recording no more nor less than what the mind requires for the operation at hand and, by the notation's very economy, allowing the mind to maintain its bearings, as it were, by a mere glimpse at the symbolic record. (34) The mathematics Descartes had in mind was his version of symbolic algebra, meant as a language into which to cast all quantitative relations, whatever their particular physical manifestation.

Despite Descartes' admonition, for much of the seventeenth century Galileo's geometrical model set the style for mechanics. The result was remarkable success, combined with revealing episodes of an impending limit to the model's resources. Christiaan Huygens's derivation in 1659 of the period of a simple pendulum and the corollary he drew from the derivation regarding the trajectory that would render the pendulum's period independent of the amplitude of its oscillations display the mixture of pictorial and graphic representation inspired by Galileo. (35)

Huygens began with a schematized pendulum having its bob (at rest at) K supported by cord TK anchored at T. In swinging to its centerpoint Z, the bob follows the circular arc KEZ. In doing so, it accelerates over the arc, picking up speed with the increase of its vertical distance from rest, as measured by successive segments of line AZ; e.g., at point D the velocity is a function of segment AB. That function is not linear; by Galileo's law of fall, BD, representing the velocity, varies as the square root of AB, and hence D lies on a parabola ADΣ. But note that the parabola itself is not a trajectory, but a gradient. It belongs not to a picture of the physical system, to the pictorial space of the pendulum, but to the conceptual space of its kinematics.

Nonetheless, Huygens operated on both curves as if they belonged to the same space and, at a later point in his derivation and for mathematical reasons, substituted for the circular trajectory arc ZK of a parabola congruent to parabola ADΣ. That substitution, made under the assumption of very small oscillations (i.e., under the assumption that K lies quite close to Z), enabled Huygens to eliminate the measure of angle ZTE from the determination of the bob's motion at E and hence from the determination of the period as a whole. When, later, he explored how the substitution as an approximation brought about the elimination, he recognized that the approximation would become an exact relation if the bob's trajectory were not a circle but a cycloid. (36) Again, the diagram superimposed on the picture of the physical pendulum a graph of its accelerated motion, and again one moved by relations among segments BD, BE, BF, and BG from the pictorial space to the kinematical. The geometrical mode of mechanics worked, but it demanded considerable flexibility in interpreting what the eye was seeing in the diagram. 
Further consideration of Huygens's first diagram shows that it called for three levels of interpretation. It not only superimposed a kinematical graph onto a physical picture, but also added to the graph auxiliary curves needed for the solution of the mathematical problem posed by the combination of the first two curves. That solution necessarily rested on manipulation of infinitesimal elements, which the diagram could not accommodate directly, but only through transformations of relations between them into relations between finite elements. For example, Huygens began his analysis by taking infinitesimal segments of line AZ and of arc KZ at B and E respectively. Assuming that the segment B were traversed uniformly at the speed acquired by the bob at point Z and the segment E at the speed acquired in fall from K to E (or from A to B), he knew from Galileo's kinematics that .
The first ratio on the right-hand side could be readily transformed via the newly developed method of tangents, which mapped it onto the ratio of the finite lengths TE and BE. Since the ratio of the speeds was already given by finite components of the parabola of acceleration, Huygens had or (by setting BG = TE) = . BG and BF are constant magnitudes, but BE and BD vary with the pendulum's motion, that is, with the changing position of E on arc KZ.

From point to point, then, or rather from instant to instant, the value of the ratio changes. From this Huygens inferred that the sum of all the time over B values taken by that ratio over the distance AZ would yield the ratio of the time of fall over arc KZ to the time of uniform motion over the segment AZ. (37)

The inference posed both technical and conceptual problems, which the geometrical mode of analysis tended to exacerbate rather than mitigate. Conceptually, aside from the question of how an infinite number of finite quantities might sum to a single finite quantity, the transition from an infinite sum of ratios to the ratio of two infinite sums required a hardy leap of the imagination. Although the geometry helped in the first question, suggesting an area under a curve as the sum of all the ordinates to the curve, it could throw no light on the steps beneath the leap. As will be clear presently, these conceptual problems took on dynamical as well as mathematical significance in Newton's Principia.

Technically, Huygens's inference required that he determine the sum of the values taken on by the expression for every point on AZ. To carry out that determination, he transformed the expression containing two variables into an expression containing only one by constructing BX such that . The curve RXNS on the far right side of his first diagram is the graph of the values of BX over the interval AZ, and the sum of all BX is represented by the area AZSNR. Similarly, the sum of all BF is the rectangle AZEK. But how to measure the first area? To do so, that is, to carry out what we now term an integration, he undertook a series of further constructions which amount to what again we now term transformations of variable. It was these transformations and the possibility of determining the curvilinear areas they produced that led Huygens to the approximation referred to above. That is, the mathematics dictated the approximation that ultimately led to the cycloidal pendulum.

But that mathematics does not appear in the diagram, which records not the transformations themselves, but the resulting curves only. For example, nothing in the diagram suggests visually (or operationally) that or that, once E is placed on parabola ZK, where BI, the sole variable on the right side, lies on semicircle AIZ. Even setting aside the inadequacy of the diagram for representing the infinitesimal elements of Huygens's solution, one still cannot "see" what is going on among its finite elements, which give no sign of their structure in terms of the original spatial, kinematical, and dynamical parameters.

Hence, Huygens's diagram requires another sort of flexibility besides that needed to separate physical from kinematical space. It demands that one recognize in the lines and areas of mathematical space traces of a mathematical argument taking place in another conceptual realm altogether. The body's eye looking at the diagram gains little insight into that realm, which is open only to the mind's eye and which, Descartes had long since argued, requires a different system of representation, to wit, symbolic algebra. That is the system in which Pierre de Fermat had ultimately couched his theory of the transformation of areas understood as sums of infinitesimal elements, the theory from which Huygens was working here. (38) Only in algebraic form did the transformations become analytically transparent, so that one could "see" the ongoing process by which individual parameters shifted their structural roles in successive configurations.

As already noted above, the use of infinitesimal arguments only emphasized the inadequacies of the geometrical modes of representation and analysis. Quite apart from the paradoxical behavior of infinite sums of infinitesimal quantities, the latter lacked the dimension to have more than mere position in the space of a configuration of finite elements. Representing infinitesimal quantities geometrically required moving to yet another mathematical space dimensionally incompatible with the pictorial space of physical objects. Thus, to the extent that dynamics concentrated increasingly on relations involving infinitesimals and their sums, the relation of its mathematical space to the physical space of machines became ever more complicated, and the geometrical representation of dynamics ever more opaque.

The early theorems of Book 1 of Isaac Newton's Mathematical Principles of Mathematical Philosophy (London, 1687), or Principia, show by their very sequence the analytical difficulties posed by the geometrical mode of mechanics. To demonstrate in Theorem 1 that a body moving under the constraint of a centripetal force sweeps out with respect to the center of force areas proportional to the times, Newton divided the time into equal infinitesimal intervals and let line AB represent the distance covered during the first such interval by a body moving at an initial velocity. Imagining the continuous force broken up into a regular sequence of impulses acting on the body at the end of each interval of time, he let the body's motion change at B and determined its resultant path BC by means of Bc, the path it would have followed inertially if not subjected to the impulse, and cC drawn parallel to SB and proportional in length to the impulse delivered. From there, it is a matter of elementary geometry to show that SAB = SBc = SBC and by repetition of the argument to continue the equality: SAB = SBC = SCD = . . . . Allowing the interval of time to decrease indefinitely will not affect that property of equal areas traversed in equal times. But the evanescence of the interval makes the sequence of impulses coalesce into a continuous force, while the rectilinear segments AB, BC, CD, etc. approach ever more closely a continuous curve.
Leaving aside for the present the legitimacy of Newton's move to the limiting case, (39) note that it means a shift to a space of another dimension and that in the shift two parameters disappear from the configuration: the rectilinear segments AB, BC, CD, . . . , which had served as measures of the velocity at points A, B, C, . . . , and the segments cC, dD, eE, . . . , which had measured the force (or rather the average force concentrated in the impulse). Getting these important parameters back into the limit-configuration became a task in itself for Newton. The velocity at any point was easily restored; it can be represented by the perpendicular drawn from the center of force to the tangent to the trajectory at that point. (40) Expressing the force as a combination of elements of the configuration proved trickier, and the first six theorems culminate in the solution (or rather, solution-schema) of that problem. In general, the force at P is given by the limiting value of the expression , where QP represents an infinitesimal segment of the trajectory, RPZ is tangent to the trajectory at P, and QR is parallel to SP, the distance from P to the center of force at S. The precise relations among these infinitesimal elements depend on the nature of the trajectory; for example, if P lies on an ellipse of which S is a focus, the value of  is constant, and hence the force is proportional simply to , that is, to the inverse square of the distance. 
Like Huygens's diagrams, Newton's, too, superimpose process on object by adding to the picture of the trajectory lines and areas representing not only kinematical but also dynamical parameters that dimensionally have no place in the space of that picture. Also like Huygens's diagrams, Newton's cannot accommodate the mathematical transformations that constitute the body of his reasoning, especially when the transformations move in and out of the infinitesimal realm. The diagrams hold, at best, traces of those transformations in the elements of the configuration. Yet, tied to the configuration, even those traces mask, rather than reveal, their structure in terms of the parameters of the problem. For example, as line lengths in the last figure above, QT and QR give no hint of the nature of their dependence on the trajectory. That is why each application of Newton's measure of force to a new trajectory required its own demonstration. Before the variables could be rearranged, they had to be extricated from their geometrical setting.

As Leonhard Euler later complained, then, it took a highly practiced eye to "see" Newton's system of central-force dynamics in the diagrams of the Principia. There was nothing "natural" or evident about them. They formed, rather, intricate patterns of lines analyzable only by a body of sophisticated, at times even counterintuitive, concepts. To succeed at analyzing one such pattern gave no guarantee of succeeding with the next. Continental mathematicians of the generation before Euler's, schooled in a Cartesian tradition, felt that in Leibniz's calculus as articulated by the Bernoullis they had a mathematical language adequate to setting forth those concepts explicitly. Soon after the appearance of the Principia, they undertook to recast Newton's work into that language, which they thought more suited to it than its original geometrical style. For example, in a series of memoirs presented to the Academy of Sciences in the early 1700s, (41) Pierre Varignon showed, among other things, that if y is the central force acting on a planet, r the radial distance of the planet from a fixed center, S the distance along the planet's orbit from a fixed point, and t the time, then y = When translated into his preferred coordinate system of radius r and arc z ( = r, where is the angle between r and the axis linking the center and the fixed point), the general rule becomes y = . Applied to various astronomical hypotheses, it leads by mathematical calculation to the precise relations inherent in them among central force, center of force, orbit, and measure of time. For example, from the assumption that a planet is subject only to a centripetal attraction and to a counteracting centrifugal force, i.e., that , where each term represents a force, it follows that the planet sweeps out areas proportional to the times, i.e., that r dz = k dt, and conversely. From the latter it easily follows that the measure of the resultant force is . If, in addition, the planet travels on an ellipse with the center of force at one focus, the force then varies inversely as the square of the distance from that focus, i.e., y = .

Using algebraic symbolism to represent the mathematical description and analysis of quantitative physical relations thus opened to view the nature of the correspondence that such an interpretation of nature posited between the structure of those relations, or of the mechanisms that embody them, and the structure of the relations among the mathematical quantities themselves. (42) The equation v =as a statement of the calculus relates three domains of quantity in expressing the physical notion that motion is change of distance over time and that the velocity of motion is the relative rate of change of distance with respect to time. The equation a describes a corresponding notion of acceleration as the rate of change of velocity with respect to time. To move from the first equation to the statement that S =to assert a correspondence between the mathematical operation of integration-the formation of a limit-sum-and the physical process of motion over time resulting in traversal of distance. More importantly, to use the first equation to rewrite the second in the form ais to posit a correspondence between the mathematical operation of differentiation and the physical process by which velocity changes from instant to instant of time. Newton's definition of force as mass times acceleration, expressed in the form F = ma, then leads from this group of kinematical relations into the series of dynamical relations presented above, each tied to the others by the operations of algebra and the calculus, each exhibiting its composition out of the basic parameters immediately measurable in the physical world.

In the view of Lodovico Cigoli, an artist alert to the currents of scientific thought in his time, a good diagram gave sight to the mathematician investigating nature. In the view of Euler, a mathematician setting the course of scientific thought in his time, a diagram, however good, formed a curtain hiding the essence of mechanics; it took algebra to lift that curtain. That contrast between late Renaissance artist and early Enlightenment mathematician cannot be resolved into different stages along the same line of progress. The positions lie on different trajectories.

A clear line of development links the treatises of Taccola and Francesco di Giorgio Martini in the early fifteenth century to Diderot's Encyclopedie and the subsequent Description des arts et metiers in the mid-eighteenth. (43) Thereafter, that line branched. Illustrated encyclopedias and manuals of machines continued to catalog an ever-expanding inventory of new devices right through that latter part of the nineteenth century that is commonly referred to (for some mysterious reason) as the "First Machine Age." At the same time the work of Monge and others in descriptive geometry defined a new body of techniques for mechanical drawings which, when coupled with advances in the machining of precision tools, enabled machinists to build machines directly from engineers' drawings.

By contrast, despite the efforts and apparent success of Galileo, Huygens, and Newton, mechanicians at the turn of the eighteenth century described machines-and in particular the great machines of the heavens-not by drawing pictures of them but by writing differential equations for them. Analytic mechanics, that is, mechanics expounded in the language of symbolic algebra and by the methods of infinitesimal analysis, became the premiere science and the touchstone for natural philosophy in the mechanistic mode. As such, it epitomizes the Scientific Revolution of the previous century, at least as an intellectual phenomenon. Historical and cultural explanations for that unique occurrence must, therefore, take account of the conceptual structure of the algebraic, analytic approach to mechanics in particular and to mathematics in general. Part of that structure involves a conscious move away from the visual, tactile world of immediate experience and into abstract, logical worlds of imaginative construction, where mathematical reasoning could operate freed from the constraints of physical ontology, where the mind could summon into mathematical existence whatever composite quantitative relations it required to make systematic sense of the perceived world.

The emergence of the fifteenth and sixteenth centuries of new conventions of pictorial representation may well prove to have developmental themes in common with those of the new mathematics a century later. (44) But to link in a directly causal manner new techniques for the accurate depiction of machines with the emergence of the science of mechanics is to ignore the line of thought that drove the diagram from dynamics.


1. Quoted by Giorgio de Santillana, The Crime of Galileo (Chicago, 1955), 22.

2. Mecbanica sive motus scientia analytice exposita (St. Petersburg, 1736), Preface, [iv].

3. Mécanique analitique (Paris, 1788), Avertissement.

4. Presented originally to the Folger Symposium out of which the present volume grew, Edgerton's paper appears in Margaret A. Hagen (ed.), The Perception of Pictures (New York, 1980), 1:179-212. Panofsky set forth his thesis at a conference held at the Metropolitan Museum of Arc in 1952; the paper appeared in revised form as "Artist, Scientist, Genius: Notes on the Renaissance Dämmerung," in The Renaissance: Six Essays (New York, 1962).

5. Edgerton, 181.

6. Ibid., 195.

7. Rept. in photo-offset, Westmead, Farnborough, Hants., England, 1970. English translation, The Various and Ingenious Machines of Agostino Ramelli, trans. Martha Teach Gnudi, annot. by Eugene S. Ferguson (Baltimore, 1976). Excerpts in A. G. Kelley, ed., A Theater of Machines (London, 1964; New York, 1965).

8. Edgerton, 210. In the version of the paper read to the Folger Symposium, Edgerton went so far as to ask rhetorically: "Dare we say that Chinese attitudes toward art per se inhibited an indigenous Chinese scientific revolution?" Evidently he has since drawn back from such outright assertion of a direct causal relation.

9. The one essentially new device, for which Taccola's notebooks offer the first documentary evidence, was the suction pump with valved piston. It figures prominently in the theaters of machines, e.g., in Francesco di Giorgio Martini's Trattato d'architettura of 1475, in Leonardo's drawings of about 1480, and in Agricola's De re metallica of 1530. See Sheldon Shapiro, "The Origin of the Suction Pump," Technology and Culture 5 (1964): 566-80.

10. Compare, for example, Galileo's Le meccaniche (ca. 1600), in which mechanics consists of the statics of simple machines, with Newton's Principia (1687), the Preface of which defines mechanics as "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 motions whatever."

11. The pendulum was perhaps the single most important mechanical phenomenon of the seventeenth century. Its mathematical description, in particular the measure of its period, posed a fertile problem for theoretical analysis. It provided a means of both qualitative and quantitative experimentation on falling and colliding bodies. It gave an accurate measure of time, especially when fitted to a weight-driven or spring-driven clock. Its variations, in particular the cycloidal pendulum, gave rise to the mathematical theory of evolutes. Variations in its period at different locations on the earth required new theories of the shape of the terrestrial globe and of the distribution of gravitational force. For an initial orientation see P. E. Ariotti, "Aspects of the Conception and Development of the Pendulum in the 17th Century," Archive for History of Exact Sciences 8 (1972): 329-410. Cf. also below, n. 34.

12. It would be helpful to have a comprehensive, critical study of mathematical diagrams in early modern scientific writings, looking both at the (possibly changing) function they served within the text and at the way in which they were produced for printing. At present only a scattering of case studies exists.

13. M. S. Mahoney, "Die Anfänge der algebraischen Denkweise im 17. Jahrhundert," RETE 1 (1971): 15-31 (Engl. translation in Stephen Gaukroger, ed., Descartes: Philosophy, Mathematics & Physics [Sussex and Totowa, 1981], chap. 5).

14. William B. Parson's Engineers and Engineering in the Renaissance (Baltimore, 1939; rpt. Cambridge, Mass., 1967) concentrates on the actual projects of civil and structural engineering carried out by engineers of the period, while Bertrand Gille's Engineers of the Renaissance (Cambridge, Mass., 1967; French orig., Paris, 1954) focuses on their writings and drawings. What remains largely untreated in the current secondary literature is the structure of the engineering community: who became an engineer, how did he do so, what did he do in carrying out his metier? Until that set of questions is addressed, it will be difficult to know precisely what purpose was served by the literature on machines.

15. Let us set aside for the moment the difficult problem of how much of the "realism" of a depiction lies in the eye of the viewer, or more precisely, in the mind of the viewer (cf., for example, E. H. Gombrich's Art and Illusion [Princeton, 1956] for one classic exploration). Here it suffices to say that from the fifteenth century on the fit between what was portrayed and what was built from that picture by widely scattered practitioners grew increasingly close.

16. Not everyone knew those rules. As late as the 1620s Genoa built a water-supply system that failed because it required raising a head of water greater than 20 braccia (= 36 ft.); cf. Stillman Drake, Galileo at Work (Chicago, 1978), 312.

17. Keller, Theater, 8; cf. Arnold Pacey, The Maze of Ingenuity (London, 1974; Cambridge, Mass., 1976), 108.

18. The drawings apparently circulated in manuscript for about a century before they were published in printed books. Hence, the treatise on mechanics and the theater of machines are roughly coeval.

19. It was, of course, the writers who put them into words. Most of the examples cited are taken, in substance if not verbatim, from Galileo's various works. Other examples from other writers (Stevin, Torricelli, Descartes, etc.) readily spring to mind.

20. Aristotle's Mecbanical Problems, while it expounded the idea of a mathematical science of mechanics (thereby echoing remarks made in passing in other works), offered little by way of specific mathematical techniques. Moreover (to anticipate a point about to arise), when in the midseventeenth century geometers began to take mathematical account of the affine space of the perspectively viewed world, they again took their start from classical sources most notably Pappus's Mathematical Collection.

21. Versions of that claim and assertions of its radical, transforming importance to Western scientific thought are scattered throughout Edgerton's book (New York, 1975); see, for example, the conclusion on pp. 164-65.

22. See in particular his Equilibrium of Planes and On Floating Bodies (English translation in T. L. Heath, The Works of Arcbimedes [Cambridge, 1897; rpt. New York, 1950]). On the transmission, circulation, and influence of these texts up to the mid-sixteenth century, see Marshall Clagett, Arcbimedes in the Middle Ages 1 (Madison, 1964), 2-4 (Philadelphia, 1976-80).

23. Discorsi e dimostrazioni matematiche intorno à due nuove scienze (Leiden, 1638), 110; cf. the Engl. translations by Crew and DeSalvio (New York, 1914), 111, and Drake (Madison, 1974), 112.

24. For the original texts see Marshall Clagett and Ernest A. Moody, The Medieval Science of Weigbts (Madison, 1952). For Galileo's use of the Jordanian approach see I. E. Drabkin and S. Drake, Galileo: On Motion and On Mecbanics (Madison, 1960), 64ff. (De mote) and 156ff. (Le meccanicbe). Thefigure is taken from ibid., 155.

25. For an account of Galileo's work in Padua, especially in the areas of mechanics of concern here, see Maurice Clavelin, The Natural Pbilosopby of Galileo (Cambridge, Mass., 1974; French orig., Paris, 1968), chaps. 3, 6, 7.

26. Probable, as opposed to demonstrative, argument could lend support to a conviction or even shed further light on it, but only demonstrative argument could establish full conviction itself.

27. Scientia de motu, the study of motion as set forth in Aristotle's Pbysics and articulated by the logicians of Merton College and the University of Paris in the fourteenth century; cf. Marshall Clagett, The Science of Mecbanics in the Middle Ages (Madison, 1959). Despite Clagett's title, scientia de motu had nothing to do with machines; nor, except for forms of the balance, did the science of weights, which he also includes in his account.

28. Alexandre Koyré first made the point in his Études galiléennes (Paris, 1939), 2, La loi de la chute des corps. Descartes et Galilee. Both men, but especially Descartes, fell victim to what Koyré termed geometrisation à outrance, a conceptual block that results from neglecting the temporal, i.e., non-spatial, nature of motion.

29. Note the shift here in the function of the inclined plane. Instead of being used to lift weights, it serves to slow down the acceleration of a falling body. Thus it changes from a statical to a dynamical machine.

30. Opere, ed. Favaro, 10:115; quoted by Koyré, Études, 2:4, n. 2: ° ... che il mobile naturale vadia crescendo di velocità con quella proportione che si discosta dal principio del suo moto."

31. Discorsi, 163-65; Crew/DeSalvio, 167-68; Drake, 159-61. For an explanation that aims at exculpating Galileo malgré lui, see Drake, Galileo at Work, 100ff. Drake's account lays emphasis on Galileo's assertion found in the fragment about to be discussed here: "Questo principio mi par molto naturale, e che risponda a tutte le esperienze che veggiamo negli strumenti e machine che operano percottendo, dove il percuziente fa tanto maggiore effetto, quando da più grande altezza casca ...."

32. Galileo, Frammenti attenenti ai Discorsi, Opere, ed. Favaro, 8:373; quoted by Koyré Etudes, 2:21ff., n. 1.

33. Discorsi, Dialogo terzo, Theorem 2. Note the addition to the diagram of line HI, "along which the uniformly accelerated body falls from point H, as from the first beginning of motion." HI is the body's physical trajectory, now divorced from the inclined plane along which the motion takes place.

34. "Those things that do not require the present attention of the mind, but which are necessary to the conclusion, it is better to designate by the briefest symbols [nota] than by whole figures: in this way the memory cannot fail, nor will thought in the meantime be distracted by these things which are to be retained while it is concerned with other things to be deduced .... By this effort not only will we make a saving of many words, but, what is most important, we will exhibit the pure and bare terms of the problem, such that while nothing useful is omitted, nothing will be found in them which is superfluous and which vainly occupies the capacity of the mind, while the mind will be able to comprehend many things together." Regulae ad directionem ingenii, Rule 2, it Oeuvres de Descartes, ed. Adam and Tannery, 10:454; quoted by Mahoney, "Beginnings of Algebraic Thought," 150.

35. For further details of the example to follow, see M. S. Mahoney, "Christiaan Huygens: The Measurement of Time and of Longitude at Sea," in H. J. M. Bos et al. eds., Studies on Christiaan Huygens (Lisse, 1980), 234-70, esp. 239ff.

36. Nothing in the diagram itself suggested the nature of the curve in question. Rather, quite by coincidence, Huygens recognized in one of the relations among elements of the configuration a property of the cycloid. The mathematical reasoning took place yet again off the diagram. Cf. ibid., 241-46.

37. At the terminal speed reached at Z; by Galileo's results, that time was half the time taken by a body accelerating in free fall from rest over the distance AZ. Hence, Huygens ultimately compared the time of the pendulum's swing to that of its free fall over the same vertical distance.

38. Cf. Mahoney, Fermat, chap. 5.

39. On the mathematical difficulties of Newton's transition to the limit, see Eric J. Aiton, The Vortex Theory of Planetary Motions (London and New York, 1972), 103-5; for a more extensive discussion of their pertinence to Newton's concept of force, see Richard S. Westfall, Force in Newton's Physics (London and New York, 1971), chap. 8.

40. Newton did not publish this corollary as part of the text until the third edition in 1726. He had entered it, however, presumably at some early date, into his own interleaved and annotated copies of the first edition; see Isaac Newton, Philosophiae naturalis principia mathematica. Third edition (1726) With Variant Readings, ed. A. Koyré and I. B. Cohen (Cambridge, Mass. 1972), 1 :90-91 (apparatus).

41. The titles of the first three of those memoirs, published in the Mémoires de l'Académie Royale des Sciences for 1700, convey the scope of Varignon's project: "Manière générale de déterminer les forces, les vitesses, les espaces, & les temps, une seule de ces quatre choses étant donnée dans toutes sortes de mouvements rectilignes variés à discrétion" (22-27), "Du mouvement en général par toutes sortes de courbes, & des forces centrales, tant centrifuge que centripètes, nécessaires aux corps qui les décrivent" (83ff.), "Des forces centrales, ou des pesanteurs nécessaires aux planètes pour leur faire décrire les orbes qu'on leur a supposés jusqu'ici" (218-37).

42. See Clifford Truesdell's now classic "A Program Toward Rediscovering the Rational Mechanics of the Age of Reason," Archive for History of Exact Sciences 1 (1960-62): 1-36, for some of the deeper insights that emerged from early eighteenth-century analyses.

43. Eugene S. Ferguson, "The Mind's Eye: Nonverbal Thought in Technology," Science 197 (1977): 827-36.

44. It would be interesting, for example, to explore the connections (if any) between Descartes' criteria and justification for a conventional notation in mathematics-that is, for a symbolic mathematics in the modern sense (above, n. 33)-and contemporary shifts in the nature and purpose of symbolic representation in painting; cf. E. H. Gombrich, "Icones symbolicae: Philosophies of Symbolism and their Bearing on Art," in his Symbolic Images: Studies in the Art of the Renaissance (London, 1972; rpt. 1975).