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'sMathematical 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.

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.Joseph-Louis Lagrange, 1788^{(3)}

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 scientificThe 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.andartistic revolutions.^{ (5)}

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.

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. |

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. |

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 |

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 |