THE MATHEMATICAL REALM OF NATURE

Michael S. Mahoney
Princeton University

1 Mathematics, Mechanics, and Metaphysics

At the beginning of what we now call the "Scientific Revolution", Nicholas Copernicus (1473-1543) displayed on the title page of the De revolutionibus (1543) Plato's ban against the mathematically incompetent --"Let no one enter who is ignorant of geometry". He repeated the notice in the preface, cautioning that "mathematics is written for mathematicians". Although Isaac Newton (1642-1727) posted no such warning at the front of the Principia a century and a half later, he did insist repeatedly that the first two books of the work treated motion in purely mathematical terms, without physical, metaphysical, or ontological commitment. (1) Only in the third book did he expressly draw the links between the mathematical and physical realms. There he posited a universal force of gravity for which he could offer no physical explanation but which, as a mathematical construct, was the linchpin of his system of the world. "It is enough," he insisted in the General Scholium added in 1710, "that [gravity] in fact exists." No less than the De revolutionibus, the Principia was written by a mathematician for mathematicians.

Behind that common feature of the two works lies perhaps the foremost change wrought on natural philosophy by the Scientific Revolution. (2) For, although astronomy had always been deemed a mathematical science, few in the early sixteenth century would have envisioned a reduction of physics --that is, of nature as motion and change-- to mathematics. Fewer still would have imagined the analysis of machines as the medium of reduction, and perhaps none would have accorded ontological force to mathematical structure. Yet, by 1670 John Wallis (1616-1703) treated mechanics and the science of motion as synonymous, positing at the start of his Mechanica, sive de motu that "we understand [mechanics] as the part of geometry that treats of motion and inquires by geometrical arguments and apodictically by what force any motion is carried out." Newton echoed the definition in the preface to his Principia, concluding that "rational mechanics will be the accurately proposed and demonstrated science of the motions that result from any forces and of the forces that are required for any motions." (3) As his account of gravity shows, the mathematization of nature and the mechanization of nature ultimately went hand in hand, each supporting the other in its claim to provide a truly intelligible account of the physical world.

Converging in the concepts and techniques of infinitesimal analysis, rational mechanics became a branch of mathematics, and mathematics opened itself to mechanical ideas. The convergence occurred by an indirect route. The symbolic algebra and the theory of equations from which infinitesimal analysis took inspiration and form were aimed initially at abstracting mathematics from the concrete world and had the effect of freeing it to create imaginary and counterfactual structures irrespective of their real or even possible instantiation. The new analysis pointed mathematics away from physical ontology by shifting attention from objects and their properties to the structure of combinatorial relations among objects, some of which existed only by virtue of the relations, namely as ideal objects needed to complete the structure. Yet, at the same time, mathematics increasingly turned to the physical world for its problems and for guidance in solving them. Almost paradoxically, mathematics enhanced its explanatory power over nature by moving conceptually beyond the intuitive limits of the physical world.

The changing language of mechanics reflected the shift in mathematical thinking. In 1623 Galileo proclaimed that "philosophy is written in this most grand book ...(I am speaking of the universe) ... [which] is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures ...". (4) In the 1660s and '70s Huygens and Newton maintained Galileo's focus on those shapes, while treating them in the new analytical style. But the universal mechanics of Newton's Principia had its full effect only after mathematicians on the Continent, beginning with Pierre Varignon (1654-1722), recast its geometrical style into the symbolic algebra of Gottfried Wilhelm Leibniz' (1646-1716) calculus. As Bernard de Fontenelle (1657-1757) insisted in retrospect, "[I]t was by the geometry of infinitesimals that M. Varignon reduced varying motions to the same rule as uniform [motions], and it does not seem that he could have succeeded by any other method." (5)

In particular, the new calculus (whether Newtonian or Leibnizian) enabled philosophers to comprehend nature in terms that lay beyond the resources of traditional mathematics. While Galileo spoke of triangles and circles, Willebrord Snel (1580-1626) and René Descartes (1596-1650) determined that the refractive properties of lenses lie in ratios of sines. Galileo could express his law of falling bodies in the simple terms of a ratio of squares, but the pendulum he used to determine that law lay beyond the reach of traditional geometry. Christiaan Huygens (1629-95) found that even its approximate, tautochronic behavior, and that of a growing family of simple harmonic oscillators shown to be at work in the world, required the resources of trigonometic relations embodied in the cycloid, itself defined in terms of the arc-length of a circle. The measure of angular position dictated by Johann Kepler's (1571-1630) first two laws could not be expressed in finite algebraic form. (6) The world of mathematical mechanics at the turn of the 18th century was filled with new curves --cycloid, tractrix, isochrone, caustics, logarithmic spiral, sail curve, etc.-- that eluded the grasp of finite algebra and required what Leibniz called the "hidden geometry" of infinitesimal analysis or the "new calculus of transcendents", which "is properly that part of general mathematics that treats of the infinite, and that is why one has such need for it in applying mathematics to physics, ..." (7)

Philosophical concerns followed, rather than led, this dual process of mathematization and mechanization. During the sixteenth and early seventeenth centuries, panegyrics on mathematics emphasized its certainty and its utility. They were distinct qualities, the former resting on Euclid's Elements as the prime exemplar of an Aristotelian demonstrative science and the second on a range of applications from the so-called "mixed" or "middle" sciences of astronomy, optics, music, and mechanics to areas of practical concern, including commerce, surveying, architecture, and the construction of stage scenery. Over the course of the seventeenth century, mathematics became increasingly useful, in terms both of enhanced problem-solving power and of the transformation of the mixed sciences into natural philosophy itself. Accomplishing that, however, involved new forms of mathematical reasoning that cast its certainty in doubt, or at least called for new criteria of certainty, among them effective practice and intuitive understanding based on experience of the physical world. Thus, mathematical explanations of nature and mathematical reasoning itself were interwoven in a new fabric of natural philosophy. Each depended on the other for conceptual support, which was rooted in the technical practice of the new, combined subject. For closer examination, one can separate the weave into two threads of development: the new science of mechanics and the new algebraic analysis.

2 The Mathematization of Nature

The idea of treating a mathematical object as a representation of a physical phenomenon had its origin in Greek philosophy. Plato, who may have got the idea from the Pythagoreans with whom he studied, expounded it in the Timaeus, first by modeling the daily and annual motions of the sun by means of two spheres turning uniformly on different axes and then by sketching a theory of matter based on the division and combination of two kinds of triangle. (8) In both cases the representation was meant to be analytic in the sense that the properties of the mathematical object match those of the phenomenon being represented and that the deductive relationships among the mathematical properties correspond in some way to the causal relationships among the physical properties. The precise nature of the correspondence between the physical world and its mathematical representation became a standing question. Is the physical world inherently mathematical, as the Pythagoreans maintained? If so, is the lack of fit between model and empirical data a fault of the model, or is matter inherently inexact, as Plato insisted? Is all of nature mathematical, or just portions of it, as Aristotle argued, making it the job of physics to identify the subjects that are essentially mathematical, such as optics, astronomy, and mechanics? How do mathematical models explain physical phenomena? Is a model merely hypothetical, "saving the phenomena" without commitment to the reality of its mathematical elements, or does mathematical coherence carry ontological and metaphysical weight? In short, does mathematics follow physics or guide it?

Debates in the fourteenth century over the reality of epicycles and corresponding arguments in the sixteenth century over the real or hypothetical nature of Copernicus's new system show that these questions were current before the extension of the domain of mathematics during the seventeenth century. The dispute between Cartesians and Newtonians over action at a distance and the nature of force show that the questions remained afterward, though perhaps in more sophisticated form. During the period, one finds mathematicians and philosophers of equal caliber on both sides of the issues, which persist down to the present.

These issues took a decisive turn in the seventeenth century not so much from new metaphysical insights as from the proliferation of successful examples of the application of mathematics to natural philosophy on the model of machines. (9) It is a matter of emphases rather than alternatives, but one will understand Galileo Galilei's (1564-1642) new sciences best by looking not at Plato's Academy, nor even at the Accademia dei Lincei, but at the Arsenal of Venice. That is where Galileo placed his interlocutors, and the opening words of the Discorsi straightforwardly announce the new relation of theory and practice embodied in the mechanical philosophy: what engineers know is worthy of the philosopher's attention. (10)

2.1 From Machines to Mechanics

Nature was mathematized in the seventeenth century by means of its extensive mechanization, which by the end of the century extended, at least programmatically, to the living world of plants and animals. The mathematical models were abstract machines, which in turn were models of the physical world and its components. Kepler spoke in 1605 of the "celestial machine",

...not on the model of a divine, animate being, but on the model of a clock --if you think a clock to be animate, you attribute glory to the work of the craftsman. In [that machine] almost all the variety of motions [stems] from one most simple, physical magnetic force, just as in the clock all motions stem from a most simple weight. And I mean to call this form of reasoning "physics [done] with numbers and geometry..." (11)

Kepler's vision found its realization in Newton's Principia, where universal gravitation played the role of the central weight, and the laws of motion converted its force into the motions of the wheels as described by Kepler. In Query 31, added to the Opticks in 1713, Newton imagined similar forces of attractions and repulsion governing chemical and physiological processes, thus making nature "very conformable unto her Self".

Separating the vision from its realization was the development of a science of mechanics capable of describing mathematically the motion of bodies under constraint, although only in hindsight can the task be phrased so clearly and purposefully. That machines could be the subject of scientific (i.e. demonstrative) knowledge, that they consisted essentially of bodies moving under constraint, and that the constraints and the motions could be related mathematically were in themselves ideas that gradually took shape over the sixteenth and seventeenth centuries in response to a variety of social and conceptual influences. As Alan Gabbey has insisted, Newton's system lies on one line of development in mechanics, Huygens' mechanics of rigid and elastic bodies on another. (12) Nonetheless, they share the common view that the essential workings of any mechanical system can be captured in an abstract mathematical model and hence that mechanics is the job of a mathematician. That view they took in common from Galileo, and with him ultimately from Archimedes.

2.2 Galileo and the New Science of Motion

New to European society in the late Renaissance was the engineer, who emerged from the anonymity of guild practice to take charge of the design and execution of large structures and of the machines necessary to build them. With the new social role came a new literature to give his know-how cultural standing: manuals of engineering, "theaters of machines", editions and translations of classical works on machines, accounts of great feats such as Domenico Fontana's Del modo tenuto nel trasportare l'obelisco Vaticano (Rome, 1589), and compendia based on classical models, such as Georgius Agricola's De re metallica (Basel, 1556). Out of this effort to codify practice emerged what might be called "maxims of engineering experience". Phrased in various ways, they came down to such rules as:

You can't build a perpetual-motion machine. (13)
You can't get more out of a machine than you put into it
What holds an object at rest is just about enough to get it moving.
Things, whether solid or liquid, don't go uphill by themselves.
When you press on water or some other liquid, it pushes out equally in all directions.

Beginning in the 1580s with the work of Simon Stevin (1548-1620) and Galileo Galilei, engineers aspiring to natural philosophy transformed these maxims into the principles of mechanics by translating them into mathematical form. Often, that translation involved devising an abstract mathematical model of the physical mechanism that embodied the maxim.

To analyze the inclined plane, for example, Stevin took a triangle (see Fig. 1), around which he imagined a "wreath of spheres" (clootcrans), consisting of equal weights connected by a weightless, flexible cord. Once placed in position, the wreath will not move one way or the other of its own accord; if it did so, it would retain the same configuration and hence "the spheres would by themselves carry out a perpetual motion, which is impossible (valsch)." (14) By symmetry, the portion below the triangle pulls equally in both directions, and hence may be removed without disturbing the equilibrium. Clearly, then, the portions on the two sides counterbalance one another, and their weights are as the number of spheres, which in turn are as the lengths of the sides. Since the sides have a common height, the weights are inversely as the sines of the base angles, which is the law of the inclined plane.

The sciences in Galileo's Discourses and Mathematical Demonstrations About Two New Sciences Concerning Mechanics and Local Motions were singly new in different ways, as the different languages and typefaces of the original publication suggest. What was new about the first, the strength of materials, was calling it a "science" at all. The vernacular and italic font seem to reflect the artisanal roots of the question why big machines did not always perform as well as smaller ones of the same design. (15) The second, scientia de motu locali, had a philosophical pedigree apposite to both the Latin in which it was couched and the roman font in which it was set. But it had the new form of a thorough-going mathematical theory built on empirical grounds, a prime example of the union of sensata experienza and necessarie dimostrazioni. (16)

That they were two sciences rather than one reflected not only their diverse origins but a compromise born of Galileo's unsuccessful efforts to relate force to motion mathematically. Initially following the lead of Archimedes and of the author of the Mechanical Problems ascribed at the time to Aristotle, Galileo sought models in machines and mathematized the machines by abstraction. That is, he translated physical devices into mathematical configurations by abstracting their geometrical structure from the physical material of which they were constructed. For statics, the approach worked well, facilitating a shift of focus from one machine to another; for example, from the effective moment of a body on a bent-arm balance to its positional weight on an inclined plane perpendicular to the arm (yielding as a by-product the law of the inclined plane), and from there to the pendulum as the body sliding along a sequence of inclined planes tangent to the arc of the its swing (17). By combining such abstractions Galileo arrived at the distinction between gravific and positional weight. Since the latter is zero on a horizontal plane, where the body is indifferent to motion, he concluded that weight accounts not for motion itself, but for change of motion: in free fall, weight produces acceleration. (18)

The Archimedean approach worked less well in exploring how weight changes motion. Studies with the inclined plane and pendulum showed that bodies gain force as they move faster and that the force they gain in free fall from rest is just enough to raise them to their initial height. Although Galileo knew the relation between length of fall and final velocity, he also knew that it was independent of the body's size, while its momento, or impeto, depended on the size of the body as well as its speed, and he could not find a way to disaggregate them. He did point the way, however. An unpublished Fifth Day of the Two New Sciences dealt with the force of impact, and in other's hands the pendulum proved to be the instrument for measuring and modeling that phenomenon. For the moment, he could separate kinematics from dynamics. Experiments with the pendulum showed that the acceleration is the same for all bodies, and on an inclined plane the constant positional weight should produce uniform acceleration, or what the medieval science of motion referred to as "uniformly difform motion". Building what he could on the principle that a constant force produces constant acceleration, he left open the question of variable forces and the resulting motions.

Galileo linked the science of motion to two classes of mathematical problems. The laws of accelerated motion associated the distance traversed by a moving body with the area under the graph of the relation between velocity and time and thus tied kinematics to the quadrature of curves. The analysis of projectile motion in terms of uniform horizontal and accelerated vertical components connected trajectories with curves defined in terms of their axes and ordinates. In the latter case, Galileo could take advantage of Apollonius's Conics to relate the components of motion to the symptomata, or defining properties, of the parabola and then to work from properties of that curve to the kinematic relations of projectile motion. But he did not undertake to develop the mathematics itself or to explore other, more general connections between curves and the motion of bodies along them. That came with the work of Roberval and others on the generation of curves by compound motion (see below, §3). By contrast, Galileo played a central role in the development of the first class of problems, both directly and through his followers. The nature and extent of the influence of the medieval science of motion on his analysis of accelerated motion remains a matter of debate, but it is clear that he knew of Federigo Commandino's (1509-75) work on Archimedes's method of quadrature and later of Bonaventura Cavalieri's (1598?-1647) method of indivisibles: an appendix to the Two New Sciences contains several theorems on centers of gravity of solids using similar methods. (19)

The concept of "uniformly difform motion" had a picture associated with it. In the doctrine of the configurations of qualities, also known as the latitudes of forms, uniformly difform qualities took the shape of a triangle. In moving from statics to kinematics Galileo appears initially to have identified that triangle of motion with the abstract figure of the inclined plane on which the motion took place. The effect was to link velocity acquired in acceleration to the distance along the plane, which is proportional to the vertical distance of free fall. For a short time in 1604 Galileo believed that from such a definition of uniform acceleration it followed that the distance traversed varies as the square of the time, a proposition he had already established through experiments with inclined planes.

Closer examination of the mathematics of the diagram of motion revealed the problem with the definition and the flaw in his deduction. It also confronted him with the problem of reasoning with infinite aggregates. The medieval doctrine referred to configurations of qualities, or latitudes of forms; that is, the extensive representation and measure of intensive properties. The paradigm was a body exhibiting different degrees or intensities of hotness at different points. In a similar sense, the degree of speed measured the intensity of a body's motion, either at different parts of the body in the case of rotation about a fixed point or at different times in the case of motion over a distance. The measure of the velocity as intensity of motion at each point or at each instant was derived from its total effect, or "total speed", over the course of the motion. If the motion was uniform, any degree was representative of the whole; if difform, one sought a particular degree that was representative in the sense that, if the body were to move uniformly at that degree, it would have the same total speed as it did in its difform motion. The famous "mean speed theorem" expressed the rule in the case of uniformly difform motion: the total speed would be the same if the body were to move uniformly at half the final speed.

That theorem originated among the "calculators" at Merton College, Oxford, in the fourteenth century and was justified by appeal to the intuitive notion that every defect on one side of the mean is counterbalanced by a corresponding excess on the other. (20) In the geometric form devised by Nicole Oresme (1320?-82), the measure of total motion became the area of the figure determined by lines, or latitudes, representing the degrees of speed in one dimension and a baseline, or longitude, representing the body or the time in the other. (21) Just how the individual degrees of speed were related to the total speed, that is, how the latitudes were related to the area, did not arise as a question. In the case of two uniform motions over a common interval, it was evident that the areas were to one another as the latitudes, but the heterogeneity of the terms of the proportion precluded taking the cross-product of latitude and longitude to form the area.

Moreover, in the medieval doctrine the meaning of "total speed" remained vague, as did the meaning of "motion" when referring to an end result rather than a process. What, for example, was the "motion" of a body rotated about one of its points, once the motion was completed? In general, both terms were taken to refer to the distance traversed, evidently on the premiss that a body moved from A to B has somehow "acquired" the distance AB, and that the whole effect of the different intensities at which it acquired that distance is the distance itself.

However well Galileo understood the medieval doctrine, it seems clear that he had it in mind when he attacked the problem of the kinematics of falling bodies and that he fell afoul of the doctrine's vagaries. Having identified motion along an inclined plane with the triangle of uniformly difform motion, with the distance traversed from rest as longitude and the speed acquired as latitude, he tried to move to the conclusion that the distance acquired is proportional to the square of the time. To bring time into the picture, he appealed to the mean speed theorem. But, as he soon recognized, only a paralogism could avoid the instantaneous motion that followed from applying the theorem to that configuration of motion. For, if v S, then by the configuration the total distance will be proportional to the final speed and by the mean speed theorem will be proportional to half the final speed. But that can obtain only if the motion occurs in an instant.

In addition to redefining uniform acceleration as the acquisition of equal increments of speed over equal intervals of time, Galileo drew two conclusions from his mistake. Firstly, the graph of motion is a mathematical representation rather than an abstraction from the physical world, and one must distinguish between the two in drawing diagrams. Hence, in the revised version of his theorems on accelerated motion, the triangle of speeds and times appeared alongside a line representing the trajectory of motion (see Fig. 2). Secondly, a properly mathematical science of motion would require confronting directly the relation between speed as an instantaneous intensional quality and distance as an extensional measure of motion over time, which is also an extended magnitude. (22) Through the mediation of the geometrical configuration, that problem was embedded in the larger questions of mathematical atomism, the paradoxes of the infinite, and the nature of continuous magnitude. These questions crop up in various forms throughout the Two New Sciences, and Galileo passed them on to his disciples and their students, foremost among them Cavalieri and Evangelista Torricelli (1608-47).

Galileo was too well trained in scholastic philosophy not to appreciate both the power and the pitfalls of reasoning with infinites. The concept of one-one correspondence that shows that there are as many square numbers as there are numbers also resolves Zeno's paradox and the relation of motion to rest: whatever the fraction of the distance to be traversed, there is a corresponding fraction of the time in which it is traversed; as a body slows down, to every speed, however small, there corresponds an equally small interval of time through which motion at that speed takes place. But, as Galileo pointed out to Cavalieri, the concept as embodied, say, in the method of indivisibles (see below, §4.2) had its paradoxes. One could use it, for example, to argue that a point is equal to a line. Draw (Fig. 3) semicircle AFB about center C, rectangle ADEB around the semicircle, and triangle CDE on base DE, and then imagine the configuration rotated about axis CF to generate a hemisphere, a cylinder, and a cone, respectively. Removing the hemisphere reduces the cylinder to a "bowl". Galileo then asserts that any plane GN parallel to base DE will cut the bowl and the cone in equal cross-sections --that is, the "band" of which GI and ON are opposite parts is equal to the disk HL-- and that the portions of the bowl and cone cut off by the plane are equal to one another. (23) The paradox arises at the upper limits of the configuration, where the equality of cross sections would seem to lead to the conclusion that the point C is equal to the circle AB. As Cavalieri argued in response, the paradox is more semantic than logical: neither the point as the last of the circles nor the circle as the last of the bands constitutes an area, or rather each has zero area, and hence the two are equal only in that sense. (24)

Semantic or logical, the paradox pointed to potential problems. Galileo demonstrated only the first part of the proposition, referring for the full demonstration to Book II, Proposition 12 of Luca Valerio's On the Center of Gravity (Rome, 1604). Nonetheless, the accompanying discussion suggested that the equality of volumes followed from the one-one correspondence of cross-sections, and Galileo went on to speculate about how one might conceive of finite quantities as composed of an infinite number of indivisibles; in this case of the bowl and the cone consisting respectively of all the corresponding cross-sections. The paradox reflected a danger lurking behind the use of one-one correspondences over open infinite sets to apply a shared property to a limiting value not belonging to the set, e.g. to apply to a circle a property shared by all inscribed polygons. That too formed a continuing strategy of the new mathematics of change. It got Galileo into trouble in reasoning from motion along chords in a circle to motion along the subtending arcs, and Newton exercised appropriate caution in shaping the theorem in the Principia on the centripetal force constraining uniform motion in a circle.

In the demonstration of the mean speed theorem in Theorem 1 on uniform acceleration in the Third Day, Galileo moved from speculation to assertion. Taking AB as the time of motion from rest at C, and BE as the final velocity, he drew AE and claimed that "all the lines drawn parallel to BE from each of the points of line AB will represent the increasing degrees of speed after instant A." Having constructed parallelogram AGFB on FB = BE/2, he argued that the triangle and the parallelogram would be equal, because

if the parallels of triangle AEB are extended to IGF, we will have the aggregate of all the parallels contained in the quadrilateral equal [aequalem] to the aggregate of those contained in the triangle AEB, for those in triangle IEF are equal [paria] to thoe contained in triangle GIA, and those in trapezium AIFB are common.

Each of the parallels then became a "moment of speed" [momentum velocitatis], and the respective aggregates of moments became the distances covered. The demonstration did not proceed by ratios of distances and speeds with time held constant, but by summation of indivisible distances traversed at instantaneous speeds over instants of time. (25)

Although the argument appeared to rest on the principle of correspondence, logical difficulties lurked in the equating of the indivisibles of triangle IEF with those in triangle GIA. For there was no rule of correspondence that tied the cross-sections to a common base. Arguing that lines GI and IF contained the same number of points with corresponding cross-sections led to an immediate counterexample. Consider (Fig. 4) rectangle ABCD and on the diagonal AC construct rectangle AEFC with AE = AB. Now, drawing "all the parallels" to AC establishes a one-one correspondence between all the points on AD and all those on AC. Using those points to draw "all the parallels" to AE should establish a one-one correspondence of equality between the two aggregates of parallels, whence rectangle ABCD is equal to rectangle AEFC. But that clearly is not the case.

It was a mathematical rather than a physical puzzle, and Galileo left it for his pupils. The solution lay in the notion of indivisibles varying in thickness according to the bases on which they stood. Fully articulated in the technique of transmutation of areas based on infinitesimals (see below, §4.2), it came back to bear on mechanics in the work of Huygens, who followed Galileo's mathematical lead in dealing with questions posed by Descartes's physics.

2.3 Descartes: Mathematics and the Cosmology of Light

In a sense, Descartes picked up where Galileo had left off, having arrived there independently along a shorter path. Although interested in practical devices, Descartes was more a philosopher than an engineer like Galileo. Taking Kepler more seriously and persuaded that a physical account of the laws of optics, especially refraction, would open up larger questions of cosmology, he could not avoid dynamics. His radical scepticism allowed the physical reality only of matter and motion, and the latter could be defined only relatively. Both experience and reason told him that bodies continue to move at the same speed in the same direction, and hence along a straight line, unless other bodies push them in new directions at new speeds. Thus impact became the central dynamical phenomenon of Cartesian physics.

In applying mathematics to physical questions, Descartes had his greatest success in optics. Since he strove in his later writings to make his results look methodical, one must reconstruct his heuristic path. Evidence suggests that his independent determination of the sine law of refraction in the mid-1620s emerged from measurements made with a refractometer (Fig. 5), which he generalized by applying the "image rule" traditionally used to account for magnification of refracted images. (26) By abstracting the refractometer to a circle and then adjusting the radius of the lower half by means of the image rule applied to a single pair of incident and refracted rays, he arrived at a mathematical configuration that allowed the construction of any other pair. The conjectured original configuration, which in essence incorporated the sine rule into a geometrical calculating device, presupposed no particular mechanism for the phenomenon and implied no derivation of the law.

In the years immediately following, Descartes combined his optical research with his work in mechanics to construct a derivation of the laws of reflection and refraction. Believing on both empirical and metaphysical grounds that light is a force (a tendency to move) transmitted instantaneously by a medium and proportional in magnitude to its density, he likened the behavior of light rays at an optical interface to the static forces counterbalancing one another at a point at rest; equal forces cancel one another along the same line, unequal forces counteract one another at a compensating angle. To make that model more accessible to general experience, Descartes translated it into kinematical terms, likening a pulse of light to a tennis ball and equating force with velocity. To this model he added the notion of impact, assumed to affect only the component of motion normal to the surface. The analogy to tennis worked best for reflection: should a body moving at a given speed strike an unyielding surface at a given angle (Fig. 6), it would be reflected at an equal angle, retaining its speed. In the case of refraction, one must imagine the ball breaking through the surface, thus losing some of its speed along the normal and being deflected from it. However, the real model behind the analogy required that, on entering a denser medium, the ball be deflected toward the normal, and hence that the ball gain speed by an extra stroke along the normal on impact. By such seemingly ad hoc adjustments, the final version of Descartes's argument as published in La dioptrique (1637) posed conceptual difficulties that have been the subject of extensive commentary, but they have little to do with mathematics. In the shift from abstract instrument to support for a derivation, the diagram ceased to be constructive or operational. It simply exemplified kinematical relations based on dynamical principles that could not be located in the mathematical configuration.

As Book II of the Géométrie shows, Descartes could apply the laws of optics mathematically to derive the reflective and refractive properties of curved lenses, and in that sense his theory of optics was fully mathematical. Yet, the laws themselves did not follow mathematically from the mechanical cosmology meant to explain them. In describing that cosmology in Le monde, ou Traité de la lumière (1633), and later in his Principles of Philosophy (1644), Descartes spoke of forces and the motions that arose from them, but he could not relate them to one another mathematically. He could not convert his analysis of centrifugal force into a mathematical relationship between velocity and radius, and his principle of conservation of quantity of motion, measured by the product of magnitude and speed, did not suffice to characterize the interaction of two unequal bodies striking one another. Indeed, as Huygens would show, if conceived in absolute rather than relative (that is, vectorial) terms, it could not account for two equal bodies striking at unequal speeds.

2.4 Huygens and the Pendulum: From Instrument to Relation

Huygens lacked Descartes's philosophical sophistication, but his continuing engagement with physical mechanisms, especially the pendulum and the mechanical clock, enabled him to make the mathematical connections Descartes had missed. Pursuing the full implications of relativity of motion (see IV.6) and modifying the law of conservation of motion to include the relative direction of bodies, Huygens established laws of impact consonant both qualitatively and quantitatively with experiments carried out with pendulums. Using infinitesimal quantities to trace change of motion at a point, he identified centrifugal acceleration with the acceleration of free fall and determined a measure of the former. Taking advantage of new techniques of quadrature and rectification via transmutation of areas (see below, §4.2), which he himself enhanced, he derived the approximately constant period of a simple pendulum for very small oscillations and, by analyzing the nature of the approximation, found that the measure is exact for any system in which the force moving the body is proportional to the displacement from equilibrium, in particular for a pendulum tracing a cycloidal (27) path and for a spring obeying Hooke's Law. The same body of mathematical techniques underlay his derivation of the center of oscillation of a compound pendulum. (28)

As the touchstone of Huygens's mechanics, the pendulum embodies the main pattern of development of mathematical physics in the seventeenth century. What began as a physical system became an experimental apparatus and then an abstract model ultimately expressed in mathematical terms and thus divorced from its original physical configuration. (29) Huygens's use of the pendulum as a means of experimenting with and analyzing the impact of bodies rested on its abstraction from a single object to a system. The isochrony of two pendulums of the same length swinging over small arcs from the center provided a means of measuring the initial and final speeds of impact of two bodies by means of their initial and final heights. More importantly, swinging two impacting pendulums from a common suspension suggested a crucial generalization of a principle of mechanics first enunciated by Torricelli but surely drawn from engineering practice: Two heavy bodies joined together cannot move on their own unless their common center of gravity descends. (30) In applying the principle, Huygens dissolved the physical link between the bodies. Two pendulums starting from initial heights have a common center of gravity. As they descend to the point of impact, so too does the center of gravity; as they rise again after impact, so too does the center. If no motion is lost on impact, the bodies will continue indefinitely to bounce back and forth against one another. But they can do that only if the center of gravity rises to its original height each time, that is, only if it acts in the manner of a pendulum. Since the time of rise and fall is the same, it follows that the speed of approach is equal to the speed of separation, and from Galileo's law relating height of fall to the speed acquired it follows that

If two bodies collide with each other, that which results from multiplying the magnitudes of each by the square of their velocities, added together, is found to be equal before and after collision; if, that is, the ratios of both the magnitudes and the velocities are posited in numbers or lines. (31)

The proposition became a staple of Huygens's mechanics, as he took advantage of the techniques of infinitesimal analysis to apply it to continuous as well as discrete systems. Yet the central parameter, mv2, remained a mathematical construct for which he hypostatized no physical correlate. Only in the hands of Leibniz did it become vis viva, the "live force" of a moving body.

To free the laws of collision from the experimental apparatus, Huygens took the center of gravity as fixed and placed the bodies in two moving frames of reference. In the version intended for publication, these were presented in terms of a man in a boat moving past a colleague on the shore, handing the pendulums over at the moment of impact. Hence the central role of the center of gravity receded behind the concept of relativity of motion, as gravity itself disappeared from the mathematical space in which the bodies moved and collided in accord with abstract relationships. Gravity reentered the space as a mathematical relationship in Huygen's derivation of the center of oscillation of a solid bar in Chapter IV of his Horologium oscillatorium (Paris, 1673). Dividing the bar into an arbitrarily large number of equally weighted segments, he imagined it to swing rigidly from its initial position, and, as it passes through the vertical, to dissolve into its individual components, each of which then rises vertically to a height determined by the velocity it acquired over the downswing. The velocity of each depends on the center of oscillation of the bar, which is located by setting the heights of the centers of gravity of the constrained and the unconstrained systems equal to one another. Equating the "actual descent" and "potential ascent" of bodies in motion proved to have broad application, perhaps most impressively in Daniel Bernoulli's (1700-82) Hydrodynamica (1734). Its effectiveness as a physical principle ultimately depended on the mathematical resources available to carry out the quadratures and cubatures (i.e. integrations) involved.

As experimental apparatus the pendulum only approximated the essential property that gave it power as an analytical model, namely a period independent of amplitude, or, anachronistically, simple harmonic oscillation. Huygens's discovery that a cycloidal pendulum is exactly tautochronic relocated the property from the pendulum to the cycloid, that is, from a physical system to a mathematical curve. In the process it stimulated the development of the theory of evolutes, the basis for the later theory of curvature. Further analysis showed that motion along the cycloid is tautochronic because the tangential component of the force on a body sliding along its concave surface is proportional to the distance along the curve from the vertex at the bottom. Thus, the property was again relocated from the cycloid as a particular mathematical curve to any curve or system in which the motive force is proportional to the displacement from equilibrium, and Huygens's later notebooks abound with such systems, motivated by the search for a robust sea-going clock. Although Huygens himself did not embrace Leibniz's calculus, the generality embodied in the equation ddS = -kSdt2 is fully consonant with the level of abstraction reached in those investigations. In turn, the plethora of mechanisms instantiating the abstract relationship lent intuitive support to the concepts underlying its mathematical expression.

2.5 Newton: Pendulums and Moons

Huygens's success in analyzing centrifugal force and in determining the dynamical basis of tautochronic oscillation did not lead him to a general treatment of forces and the resultant motions. Tautochronic oscillation was a special case, and Huygens found no means of giving to what he called incitation and defined as "the force that acts on a body to move it when it is at rest or to increase or decrease its speed when it is in motion" a mathematical form that would allow its application to other situations.. By 1674, when Huygens set down this definition, Newton had already worked out just such a mathematical formulation, thinking along the same lines as Huygens but focusing on a different problem, namely the motion of bodies acted on by a centripetal force. (32) The trick lay in accounting for both change of speed and change of direction, and an early analysis of uniform circular motion appears to have provided the model. Modifying his diagram and argument slightly to bring out the underlying reasoning (Fig. 7), consider a body moving at speed v along the sides of a polygon inscribed in a circle and reflected by the circle at each vertex. If the body were not reflected, it would continue at the same speed in the same direction by the first of Descartes's laws of motion. However, the body is reflected onto the succeeding side, and the change of motion is the distance between the next point of impact and where the body would have been had it continued unreflected. Denoting that distance as v and the distance along the side as v (in uniform motion, distance is proportional to speed), one has from similar triangles v/v = v/R, where R is the radius; that is, v = v2/R. Nothing in this relation depends on the number of sides or frequency of impact. It holds for any number of sides of a given polygon, and it applies to all inscribed polygons of however many sides. Hence (by an assumed principle of continuity), it holds for all corresponding arcs of the circle that is the limiting figure. (33)

Proposition I,1 of the Principia uses the same mathematics to prove that a body moving under any central force will describe a plane orbit at a speed such that the line connecting it to the center of force sweeps out equal areas in equal times. Again, Newton begins (Fig. 8) with a finite, rectilinear motion from point A over some interval of time at speed v. At point B, he imagines the body pushed instantaneously toward the center S, changing the body's direction toward C. Had the body not been pushed, it would have proceeded over an equal interval of time to c, where Bc = AB, and hence the line cC, drawn parallel to BS, represents the change of motion. The measure of that change is not of immediate concern; rather, the fact that cC is parallel to BS makes triangles SBC and SBc equal, and AB = Bc means that triangle SBc = triangle SAB. Again Newton argues that the mathematical relations hold independently of the number and frequency of the impulses toward the center and therefore hold of the curve that limits the rectilinear cases. The crucial steps in the derivation are the mathematical expressions of the first two laws or axioms of motion with which Newton opened the Principia: the law of intertia and the law of force: "Change of motion is proportional to the impressed motive force and takes place along the straight line in which that force is impressed."

To get a measure of the force in the case of a curvilinear orbit requires several geometrical results that relocate its representation from a hypothetical interval cC to some combination of the finite parameters of the orbit, and the bulk of the first ten sections of Book I is addressed to evaluating that combination for a variety of known shapes, foremost among them the ellipse (v ∝ 1/R2, where R is the distance from the body to the attracting focus), and extending it to the case of an infinitely distant center of force so as to encompass Galileo's laws of motion for bodies close to the surface of the earth. (34) It is by that extension that the pendulum's swing becomes a limiting case of the moon's orbit, and the heavens are tied to the earth in a common mathematical structure, which, Newton asserts, reflects their common physical structure.

While the main argument of the Principia amounts to showing that Kepler's laws of planetary motion entail an inverse-square force, Newton also laid the groundwork for working in the other direction, namely, finding the orbit, given a force law and initial position and momentum. Here the effectiveness of the mechanics depends on one's skill and repertoire as a mathematician, since, as Propositions 39-41 demonstrate, the problem ultimately comes down to the quadrature of curves, for which there is no general algorithm. In the "inverse problem of forces", however, lay the promise of Newton's mathematical mechanics in its application to any system of bodies attracting or repelling one another, whether they be planets acting under gravity or small particles of bodies exhibiting chemical or electrical properties. "And thus Nature will be very conformable to her self," he mused in a "Query" added to his Opticks in 1713, "and very simple, performing all the great Motions of the heavenly Bodies by the Attraction of Gravity which intercedes those Bodies, and almost all the small ones of their Particles by some other attractive and repelling Powers which intercede the particles." (35) Nature's mathematical structure was all-embracing, and Newton's approach to analyzing it held sway through the eighteenth century. (36)

3 Mechanical Mathematics

Despite popular legend, Newton did not create fluxions to accommodate problems involving motion. To the contrary, as the first essay of what would become his technique of fluxions shows, he began with motion, and in that he followed a line of thought rooted in the classical sources but given new vitality by the developments in mechanics just described. Until the creation of the calculus, however, the analysis of curves generated by motion was mathematically suspect. For example, Descartes readily determined (Fig. 9) the tangent to the cycloid at any point by considering its mode of generation. As the circle rolls along the plane, the direction of the fixed point on its circumference is perpendicular to the chord linking the point to the point in contact with the plane, for it is momentarily rotating about that point at the end of the chord. (37) But the perpendicular to that chord is simply the chord from the moving point to the vertex of the circle. That is, from the point on the cycloid draw a parallel to the plane. Where it intersects the generating circle about the center of the cycloid, draw a chord to the vertex. A line through the given point parallel to that chord is tangent to the cycloid.

Yet, Descartes would not admit the cycloid among the curves he considered geometrical, because it could not be described in terms of an algebraic relation among rectilinear segments. Requiring circular motion for its description, it formed a "mechanical" curve and hence required mechanical means of constructing its tangent. However neat and clever the means, he thought them a curiosity, not mathematics. (38) Others among his contemporaries were less particular. In Observations sur la composition des mouvements et sur le moyen de trouver les tangentes des lignes courbes, Gilles Personne de Roberval (1602-75) showed how to express the defining properties of curves both old and recent in terms of the compound motion of points describing them, from which the tangent then followed as the resultant. (39) Roberval based his techniques on an extension of the parallelogram of motions from uniform to non-uniform motion, taking as axiomatic that "the direction of the motion of a point describing a curve is the tangent of the curve at each position of that point." His work was thus of a piece with Galileo's determination of the parabolic trajectory and with Descartes's analysis of circular motion into normal and tangential components, as Roberval's basic terms --mouvement uniforme, mouvement irrégulier ou difforme, puissance, impression, etc.-- make clear. Expounded by Isaac Barrow (1630-77) in his Geometrical Lectures in the mid-1660s, the technique was the basis of Newton's first version of the theory of fluxions in "To resolve problems by motion" in 1666, and it underlay the analyses of curves in the Principia. The various expositions differed largely in the specific means used to express the defining properties of curves in terms of compound motions and to resolve the motions into directional components at the point of tangency.

With the method of fluxions Newton recast the analysis of curves by motion into wholly algebraic terms, set usually in a Cartesian framework. (40) If p is the rate of "flow" of a point in the x-direction and q is the corresponding rate of flow in the y-direction, then q/p determines the direction of the tangent. The rule for finding p and q for an algebraic equation f(x,y) = 0 remained the same in all versions: multiply each term by p/x times the power of x in the term and, similarly, by q/y times the power of y in the term, and add the results. Behind the rule lay the notion of the momentary increase (or decrease) of x and y, whereby over a "moment" o each grows by an infinitely small amount proportional to its velocity at that point. That is, over a "moment" non-uniform motion may be treated as uniform, whence p:q = po:qo. Here, the kinematical model hooked into the algebraic method of maxima and minima and of tangents created by Fermat and expounded as a "rule" by a series of writers (below §4.1). (41)

As used by Newton here, the notion of "moment" was suggestively ambiguous, connoting both an instant of time and the force by which a mechanical system is held in equilibrium or with which it first begins to act. It tied the method of fluxions to the determination of the centers of gravity of curvilinear figures, a problem of increasing interest to mathematicians and mechanicians from the time of its introduction through the works of Archimedes in the sixteenth century. In this literature, plane and solid figures acquired a uniformly distributed "weight", by which portions of them could be balanced against one another with reference to their distance from a point. If one imagined an area sliced into very small sections, then each of them constituted a "moment" of the area with respect to its center of gravity. Generalized to denote the rate by which the area grows when generated by a moving ordinate, the "moment" of the area A(x) under y = f(x) becomes the fluxion of A, which Newton showed is simply y itself. (42) In that relation of area to moment lies the inverse relation of fluxion to fluent, that is, the fundamental relation of the calculus. Leibniz arrived at similar results through an "analysis of quadrature by means of centers of gravity" at roughly the same stage in his path to the calculus. (43)

One need not look hard to find other examples of mechanical thinking in seventeenth-century mathematics. Balances, levers, centers of gravity, velocities, moments, and forces informed creative mathematics while mathematics became the means to express and understand them. In particular, machines stimulated mathematicians' interest in mechanical systems and the curves traced by the motion of their parts. The cycloid was only the first of a host of curves introduced into mathematics from nature construed mechanically; it was soon joined by tractrix and the catenary (respectively the shapes of a flexible cord dragging a weight along a plane and of one hanging freely), by the curve of descent at a uniform vertical rate and that of a sail under a constant wind, and by families of caustics generated by optics. The new science of mechanics legitimated these curves as mathematical objects and spurred the development of mathematical methods for analyzing, transforming, and ultimately constructing them. Although often couched in geometrical terms, these methods increasingly derived from the conceptual resources of a new way of talking about mathematics, namely symbolic algebra.

4 Algebra and the Art of Analysis

Leibniz's calculus and Newton's fluxions arose out of a line of mathematical thought reaching back to François Viète (1540-1603) and passing through Descartes and Pierre de Fermat (1601-1665). It may be termed the "analytic program", and it was aimed at the development of a systematic body of techniques for solving any mathematical problem, or at least classifying it according to the nature of its solution, if it could not be solved explicitly. In particular, the analytic program sought a means of expressing curves in a form that captured all their essential properties and that could be analyzed and transformed to reveal those properties. The properties of particular interest over the period included the tangent and normal to a curve at any point, the area under it, the length of its arc, and its curvature. As these properties of curves acquired significance within a geometrical mechanics, the new analytical methods of determining them became identified with mechanics, which in turn was then couched directly in the language of those methods.

The analytic program rested on the idea of algebra as the symbolic art of analysis. In retrospect one can see adumbrations of the idea in 16th-century discussions of a "universal mathematics", for which the classical reference was Aristotle's Metaphysics, which spoke of a body of concepts and propositions common to all the distinct branches of mathematics, and hence superordinate to them. (44) As specified by Viète, however, the art of analysis was specifically rooted in

... a certain way of seeking truth in mathematics, which Plato is said to have been the first to invent, and which was called "analysis" by Theon and defined by him as "the assumption of what is sought as if admitted [and the passage] by consequences to an admitted truth. Conversely, synthesis [is] the assumption of what is admitted [and the passage] by consequences to the goal and comprehension of what is sought. (45)

Viète took his definitions from the classical discussion at the beginning of Book 7 of Pappus of Alexandria's Mathematical Collection, where it served as introduction to a compendium of treatises providing tools for the working geometer and thus constituting what Pappus called "the field of analysis" (Gr. ho topos analyomenos). As Pappus described the method, one proceeds analytically by assuming that a proposed theorem is true, or a problem is solved, and then chasing out the consequences of that assumption until one arrives at a theorem known to be true, or a problem known to be solved. Synthesis turns the process around by starting with what is known and proceeding deductively to a proof of the theorem or a construction of the problem. Synthesis is necessary because the advantages of analysis as a method of discovery come at the price of logical rigor: A B may suggest a way of proving A by means of B, but one cannot simply reverse the implication. (46)

Viète sought to capture the heuristic power of analysis in a general form common to arithmetic, geometry, and the other branches of mathematics. (47) The practical art of algebra, applied traditionally to numbers, provided the basis. To solve a problem, one expressed it in the form of an equation linking the known number with the unknown, denoted by a symbol. Manipulating the unknown as if it were known, the rules of algebra specified how to reduce the equation so that the unknown stood alone on one side, equated to a known number on the other. Moreover, most of the reductions involve substitution of equivalent forms and hence run logically in both directions. Viète extended the basis through a reformed algebra in which the letters of the alphabet represent general quantities, "the species or forms of things", characterized only by their being subject to the four operations of addition, subtraction, multiplication, and division, suitably defined. Multiple application of those operations results in composite quantities represented by expressions and equations. Taking advantage of a symbolic convention that distinguishes between unknowns denoted by vowels and parameters denoted by consonants, the art of analysis reveals the structures (constitutiones) of those equations and hence the relations among them that provide the means of reducing a problem to a form for which a solution is known.

By focusing on structures, the new symbolic algebra directed attention away from the properties of mathematical objects to the relationships among the objects and from techniques of solution to analysis of solvability. Thus, while including in the analytic art the canonical procedures for the numerical resolution of equations and for the geometrical constructions corresponding to them, Viète focused attention (in De aequationum recognitione et emendatione tractatus duo) on the transformations by which given equations were reduced to the canonical forms to which those procedures could be applied. Although he did not introduce the term "theory of equations", Viète laid out the foundations of the subject and made it the core of his "art". (48)

In establishing a new style of mathematics, Viète also set down an agenda for investigation. He called for the recovery of the content of the ancient corpus of analysis reported in varying detail by Pappus and for the discovery of the analysis that lay hidden under the synthetic form of the great works of Apollonius, Archimedes, and others. The algebra Viète had inherited from its Arabic authors extended only to the solution of linear, quadratic, and some cubic equations in one unknown. (49) Those same authors had pointed out the relation of their numerical procedures to theorems in Books II and VI of Euclid's Elements, thus suggesting to Renaissance mathematicians the idea of an algebra underlying Greek geometry -- as Viète put it in his Apollonius Gallus, "the (wholly geometrical) algebra that Theon, Apollonius, Pappus and other ancients handed on". Apollonius's Conics in turn related the defining properties (symptomata) of the conic sections to Euclid's technique of the application of areas; indeed, that was the source of the names "parabola", "hyperbola", and "ellipse". But these and other curves served algebra only as a means of constructing solutions to determinate equations, and algebra in turn offered aid in solving section problems in geometry. Although Pappus' corpus included indeterminate problems for which curves constituted solutions, traditional mathematics offered models neither for the algebraic treatment of loci nor for the geometrical expression of indeterminate equations.

4.1 Curves and the Theory of Equations

Working independently of one another, Fermat and Descartes first devised those models and then extended the techniques of the analytic art to the structural properties of curves. Although Descartes claimed not to have read Viète's work until after composing the Géométrie in the early 1630s, Descartes's thinking developed along remarkably similar lines beginning in the late 1610s. He too sought to recover a hidden art of analysis from the classical Greek texts and from the "barbarous" notation of Arabic and cossist algebraists. (50) He too proposed a new alphabetic symbolism aimed at expressing the combinatory relationships common to all quantities, whatever their specific form. However, he went beyond Viète by reformulating the concept of magnitude to reflect the focus on structural analysis. To maintain subtraction as the inverse of addition, Descartes accepted negative quantities, though he referred to them as fausses. Rejecting the classical view that, in the absence of a common measure, the product of two line segments could only be the rectangle formed by the factors and hence incomparable with either of them, Descartes argued that for algebraic purposes one can choose a common measure at will. Multiplication then takes the form of a proportion, 1:a = b:ab, all the terms of which are homogeneous and comparable. (51) If 1, a and b are lines (Fig. 10), then so too is ab, and the proportion is represented by a pair of similar triangles. By the same means, quotients, powers, and roots can also be represented by simple line segments in parallel with numbers.

Two major developments flowed from this approach. First, Descartes could express a general polynomial of the form xn + a1xn-1 + a2xn-2 + ... + an = 0, where n was a definite number and the ai either numbers or algebraic expressions. Arguing by induction in Book III that every polynomial of degree n could only result from the multiplication of n binomial factors xi, the constant terms of which are the zeros of the polynomial, Descartes derived what are now called the elementary symmetric functions expressing the relationship between the roots of an equation and its coefficients. He also called into existence a new species of quantity necessary to maintain the generality of his analysis, namely the αi he called imaginaires because they could not be reached by any operations on ordinary quantities and yet could be combined with them and with each other to yield real values. For example, expressed in the form (x-1)(x-α)(x-β) = 0, the equation x3 - 1 = 0 has, in addition to the real root 1, two imaginary roots α, and β, the sum of which is -1 and the product, 1. (52) Enhancing the power of his method by the addition of ideal elements was a bold strategy to which Leibniz would later appeal in defense of infinitesimals.

The second development followed from the removal of dimensionality from the degree of an equation. Since all the relations inherent in an equation in one unknown can be expressed as segments of a single line, those of an equation in two unknowns require two lines which, placed at an angle to one another, define a plane. The equation determines the relations among corresponding segments of the two lines and is represented by a curve in the plane. Since the equation captures the metric structure of the curve, the determination of its properties, including the tangent and normal to any point, became a matter of algebraic analysis, as indeed Descartes noted by way of introduction to his method from drawing the normal to a curve.

Simply by knowing the relation that all the points of a curve have to those of a straight line, in the manner I have explained, it is easy to find also the the relation they have to all other given points and lines, and consequently to know the diameters, axes, centers, and other lines or points to which each curve will have some more specific or simpler relation than to others, and thus to imagine various means of describing them and from among those [means] to choose the simpler ones. ... That is why I shall believe I have set out here all that is required for the elements of curves when I shall have given generally the means of drawing straight lines that fall at right angles [to the curve] at any of its points one might choose. And I dare to say that this is the most useful and more general problem, not only that I know but that I have ever wanted to know in geometry. (53)

Thus, by this construction, Descartes extended Viète's analytic program to the classical treatises on loci, foremost among them Apollonius' Conics.

Aiming the Geometry at a specific problem and ultimately at an application to optics, Descartes offered few details of the new system. But Fermat had arrived at the same system and had laid out its fundamentals in his Ad locos planos et solidos isagoge (Introduction to plane and solid loci [ca. 1635]). (54) He posited that equations in two unknowns correspond to curves in the plane determined by two lines of reference: a fixed main axis with a point on it as origin, and a variable ordinate translated parallelly at a fixed angle to the axis. The axial system stemmed from Apollonius' Conics, and Fermat argued for the general proposition by showing how the conic sections, including circle and straight line, accounted for all possible cases of the general quadratic equation in two unknowns, and conversely. The demonstration had two components: linking the defining parameters of the individual curves to their canonical equations, e.g. the center and radius of the circle to the equation x2 + y2 = r2, and reducing equations to one of the canonical forms by steps that correspond to translation, change of scale, and rotation of the axial system.

The Ad locos planos et solidos isagoge essentially reduced the contents of Books I-IV of the Conics to algebraic form, showing how the various structural properties of the conics corresponded to relations among the parameters of their equations. Although Book V was not extant at the time, Fermat and his contemporaries knew it involved the determination of tangents and normals to the conic sections, elements central to their optical properties as reflectors and refractors. Eliciting those elements from the equations led both Fermat and Descartes to another extension of Viète's new analysis. For Fermat, the crucial hint came from Pappus of Alexandria, who insisted on the uniqueness of extreme values. Consider the equation bx - x2 = M. In general, it has two roots, say u and v. By a technique taken from Viète, bu - u2 = M = bv - v2, or b(u-v) = u2 - v2 = (u+v)(u-v), or b = u + v. That, argued Fermat, is a general relationship linking the roots of the equation to one of its parameters. (55) In the case where M is the maximum or minimum value of the expression bx - x2, the equation will have a single, repeated root, that is, u = v = b/2 (whence M = b2/4). If one represents the two roots in terms of their difference, that is, u and u + e, then Fermat's analysis takes a familiar algorithmic form: b(u+e) - (u+e)2 = bu - u2, whence be - 2ue - e2 = 0, or b - 2u - e = 0. That is the general relationship for all pairs of roots u+e and u. In the case of a repeated root, e = 0, whence b = 2u, etc. (56)

To understand the conceptual origins of the calculus, it is essential to recognize that Fermat's difference e is a counterfactual, rather than an infinitesimal, quantity. That is, Fermat treated an equation with a repeated root as if the two roots were unequal, used the theory of equations to derive a relation that is generally true of all such unequal pairs, and then extended the relation to equal roots. The assumption of inequality covered the division by a quantity that in fact is 0. For the method of maxima and minima, at least, he made no appeal to limits or infinitesimals to justify that extension. That is, e carried no connotation of ranging over only very small values, as it later acquired when interpreted as an infinitesimal. Descartes followed a similar line of reasoning in his method of normals in Book III of the Géométrie. It was in keeping with the reasoning that lay behind his assumption of imaginary roots to maintain the full generality of his theory of equations.

Counterfactual reasoning also shaped Fermat's method of tangents, which he claimed to have derived from the method of extreme values, although the derivations he offered seem contrived after the fact. Given a point B on a curve (Fig. 11), assume the tangent to have been drawn, intersecting the axis at E. Let OI be drawn parallel to BC at a distance e from it, intersecting the curve at O'. Except when OI coincides with BC, O and O' will be different points. Assume, however, that they coincide. Then, on the one hand, the subtangent EC is to BC as EI, i.e. (EC - e), is to OI. On the other hand, OI , i.e. O'I, together with DI, i.e. (DC - e), satisfies the conditions of the curve. Expressing the first relation in terms of the second and carrying through the sequence of operations for the method of maxima and minima leads to a determinate expression for the subtangent in terms of the given ordinate and abscissa.

4.2 From Counterfactuals to Infinitesimals

Fermat and Descartes thought of their methods of determining tangents and extreme values in terms of special cases of general algebraic relations. They viewed the increments essential for deriving those relations as counterfactual quantities, which took the value 0 not by being negligibly small nor by convergence on a limit, but by instantiation of the special case. (57) It was a matter of the manipulation of symbolic forms without reference to the particular meaning of their constituent terms. (58) The structure of quadratic equations dictates that they have two roots, even when they appear to have only one. Using symbols for the roots preserves the structural distinction between them, even when they have the same value.

That view avoided infinitesimals only by excluding two classes of problems: curves defined with reference to other curves and the quadrature of curvilinear figures (that is, finding their areas or volumes). The first class included the "special curves" cited by Pappus and other Greek sources, but it expanded rapidly with the addition of curves representing physical phenomena, such as the cycloid, the locus of a point on the circumference of a circle rolled along a line. Since most of the curves could not be represented by an algebraic expression, they were not open to algebraic analysis, including the method of tangents, without assuming that over small intervals their curvilinear elements could be treated as if they were rectilinear. That is how mathematicians, including Fermat, began to treat the curves and were thus drawn by specific instances into the realm of infinitesimal quantities and evanescent differences. Neither they nor their successors over the next half-century felt entirely comfortable there, and algebraic reduction of relations among infinitesimals to finite terms was one of the ways they sought to get out again.

The second class of problems had classical origins, notably in the works of Archimedes, who, following Eudoxus's "method of exhaustion", proved his results by containing a curve between two rectilinear figures differing from one another by an arbitrarily small amount. Archimedes hinted at a more direct, heuristic form of the technique, and the spread of his works in the sixteenth century combined with the revival of atomism to shape the method of indivisibles, or infinitesimals. Although traditionally associated with Cavalieri, various forms of the technique emerged in several places at about the same time. (59)

Cavalieri stated as a principle that if two figures are bounded by the same parallel lines or planes, and the cross-sections generated by any line or plane parallel to the boundaries are equal, then the figures are equal. In that form, the method offered a means of comparing the figures, not calculating the area of either one. While "all the lines" of one figure might be equal to all those of another, or indeed might be a multiple of all those of another, one could not add up the lines to constitute an area. However, imagining the cross-sections as slices of indefinitely narrow width, Torricelli tied Cavalieri's indivisibles to infinite series, and thus the areas to the sums of those series. In France, Fermat and Roberval independently took a similar approach, differing from one another in the range and variety of series they could handle. (60) By the mid-1640s, when Torricelli communicated his and Cavalieri's results to the French, Fermat had already established the general quadrature of curves of the form ym = pxn and xnym = p, the so-called "higher parabolas and hyperbolas." The achievement lay in determining the sum of ik from i = 1 to N for any k, integer or fraction, and success on that front derived more from number theory and the theory of equations than from any new concept of the infinite or infinitesimal.

When coupled with the method of tangents, however, the notion of infinitesimal slices of an area or volume did suggest a means of comparing areas on an element-by-element basis, rather than in the aggregate as Cavalieri's method required. The result was a technique of transformation or, as it came to be called, "transmutation" of areas, which became then the basis of the integral calculus. The Torricelli-Roberval correspondence offers one of the earliest examples, which the two authors treat in slightly different ways.

Let ADB be a curve (Fig. 12). From each point of the curve, draw a line segment parallel to the axis and equal to the length of the subtangent to that point, thus generating another curve AO'Z', or COZ, depending on the direction in which the segments are drawn. In the first case, Torricelli shows that the area between ADB and AO'Z' is equal to that under ADB; in the second, Roberval establishes that the area between ADB and COZ is equal to twice that under ADB. Both arguments rest on the division of the two areas into corresponding infinitesimal segments, which bear to one another a relation that holds only in the limiting case and, short of that, rests on the assumption that a very small arc of a curve coincides with the tangent. The length of the tangent then mediates between the two segments. In Roberval's case, it becomes the common base of a triangular segment of ADB and a rhomboidal segment of ABZC, both contained between the same parallel lines. In Torricelli's case, it establishes that the bases of corresponding rectangular slices of the two areas are inversely as their heights, whence the slices are equal.

If, now, one shifts focus from the generated curve to the any ordinate of the original curve, it is evident that the tangent maps any division of the axis of a curve into a corresponding division of its final ordinate Consider, that is, the parabola y2 = px (Fig. 13), and imagine the area under it sliced into infinitesimal rectangles by parallel ordinates y erected on axis x over the interval [0,a]. If for brevity's sake the rectangles are designated by the ordinates, the area under the curve corresponds to "all the y over a". (61) But one can also erect a set of segments x on axis y over the interval [0,b]. In that case the area under the curve with respect to the y-axis will be "all the x over b". The area under the curve with respect to the x-axis is then rectangle ab - (all x over b). Hence,

all y over x = ab - all x over y.

That relation becomes productive by taking account of the differing widths, albeit infinitesimal, of the segments drawn one way and the other. From some point P on the curve, draw the corresponding slices, PQ and PR, the bases of which correspond to one another through the medium of their common infinitesimal element P of the parabola. Construct (Fig. 14) the tangent PT intersecting the x-axis at T. Then, on the premiss that P coincides with the tangent, P is to infinitesimal Q as PT:QT and to infinitesimal R as PT:PQ; that is, R:Q = PQ:QT, or PQ x Q = QT x R for each pair of corresponding slices Q and R. In the case of the parabola, QT = 2OQ. Hence all PQ over OQ is equal to twice all OQ over OR (= PQ); or, in the symbolic terms Leibniz will soon establish, ydx = ∫subtangent dy = ∫2xdy

Used to reduce unknown figures to known ones, the transmutation of areas took various forms in mid-century. Fermat attached it to his analytic geometry, and hence to the analytic program, by adapting it for application directly to the equations of curves and algebraically transforming, for example, the curve b3 = x2y + b2y into b2 = u2 + v2 by means of the auxiliary curves by = u2 and bv = xu to show that its quadrature involves the quadrature of the circle. (62) By contrast, Barrow and James Gregory (1638-75) retained its geometrical formulation while expanding the means of transformation to include the normal and the subtangent. (63) It is precisely in this work that historians have perceived anticipations of the calculus and sought the source of Newton's and Leibniz's inspiration. Yet, none of the writers on transmutation of areas tied the way in which tangents and normals were being used there to the method of determining them. In their minds, the problem of drawing the tangent to a curve apparently remained separate from the problem of measuring the area under it. The credit for linking them remains with Newton and Leibniz.

Barrow's Geometrical Lectures suggests by contrast the nature of their insight. Although Barrow embraced the new notion of symbolic magnitude as a relation, he ultimately distrusted the abstractive power of algebra, refusing in particular to accept the notion of ratio as quantity. Hence, while he was willing to extend the concept of relation to include equations, he did not see the method of tangents as an operation on an equation yielding another, derived equation and hence a relation of the same sort. Barrow viewed Fermat's algorithm as a means of determining the finite ratio of the unknown subtangent to the known ordinate by means of the ratio of infinitesimal increments of elements of the curve, usually but not always the abscissa and the ordinate. The elimination of the infinitesimals in the limiting case fixed a value for the latter ratio and hence a value for the subtangent, understood always as a line segment on the axis, rather than as a variable bearing a relationship to another variable expressed by an equation. Barrow inherited from classical geometry the notion of the quadratrix of a curve, namely a curve of which the ordinate is proportional to the area of the base curve on the same abscissa, and his treatment of these "squaring curves" has invited credit from historians for adumbrating the calculus. But Barrow never thought to reverse the relationship between the curves, seeing the base curve as proportional to the tangent of the quadratrix. (64)

Leibniz first presented his differential calculus in 1684 as "A new method for maxima and minima, and also for tangents, which stops at neither fractions nor irrational quantities, and a singular type of calculus for these," thus suggesting that he was simply improving earlier methods rather than offering something quite new. (65) Yet, he began where Barrow had left off.

Let AX be an axis (Fig. 15) and let there be several curves, such as VV, WW, YY, ZZ, of which the ordinates, perpendicular to the axis, are VX, WX, YX, ZX, which shall be called respectively v, w, y, z; and AX itself, the abscissa on the axis, shall be called x. Let the tangents be VB, WC, YD, ZE, meeting the axis at points B, C,D, E, respectively. Now let some straight line taken at will (pro arbitrio) be called dx, and let the straight line which is to dx as v (or w, or y, or z) is to VB (or WC, or YD, or ZE) be called dv (or dw, or dy, or dz) or the difference of these v (or of these w, or y, or z).

Speaking of dx as line of arbitrary length misled some readers who, like the Marquis de l'Hôpital, saw at first merely a change in notation from Fermat's and Barrow's a and e. They missed the significance of Leibniz's notation, which in labeling the differences by a common prefix marked them as the result of an operation (he would later refer to it as "a certain modification") on quantities, presenting the rules --indeed, he used the term algorithm-- that governed its application to their sums, differences, products, quotients, powers, and roots; that is, to the ordinary operations by which equations are constructed. Thus, he defined, rather than derived, the differential of a product dxy as xdy + ydx without raising the question of its relation to the form xdy + ydx + dxdy, which results from defining dxy as (x + dx)(y + dy) - xy in line with the general notion of difference introduced at the start of the article.

The method of tangents, too, was a matter of definition: dy:dx = y:subtangent. The definition formed two bridges. While tying back to the earlier method, it also thrust forward into the new realm. The reason why the new method "did not stop" lay in the special properties of differentials, for "to find the tangent is to draw a straight line which joins two points of the curve which have an infinitesimally small distance [between them], or [to draw] the extended side of the infinitangular polygon that for us is equivalent to the curve." (66) That is, in differentiating the curve's equation to determine the relation of dx and dy, one also transformed the curve into the infinite number of rectilinear sides ds that joined the endpoints of the differentials. The relationship ds2 = dx2 + dy2 meant that at the level of differentials all curves were algebraic because, in a sense, all lines were straight.

Thus, infinitesimal analysis served to open the "hidden geometry" of transcendental curves, which Descartes had labeled "mechanical" and excluded from mathematics proper. (67) In Leibniz's calculus, differential equations enjoyed the same status as algebraic equations in representing curves and their properties, and a suitable theory of differential equations would provide means of eliciting from them the same kind of structural information as did the theory of ordinary equations. "It is true, as you very well note," Leibniz wrote to Huygens in 1691,

that what is best and most convenient in my new calculus is that it offer truths by a species of analysis and with no effort of imagination, which often succeeds only by luck, and it gives us all the advantages over Archimedes that Viète and Descartes have given us over Apollonius. (68)

The d denoted a symbolic operation that provided a path from the finite to the infinite and back. The algorithm of the differential calculus linked the realms of the algebraic and the transcendental symbolically, while the method of tangents tied them together metrically. Expressed symbolically, differentials played the same role in infinitesimal analysis that imaginary quantities did in ordinary analysis; as Leibniz explained to Varignon in 1702:

...if someone will not admit infinite or infinitely small lines as metaphysically rigorous or as real things, he can use them surely as ideal notions which shorten reasoning, similarly to what one calls "imaginary roots" in common analysis (as for example -2), which, for all that they are called "imaginary", are no less useful, and even necessary, for expressing real magnitudes analytically. (69)

Speaking of "well founded fictions", Leibniz continued the theme of counterfactual reasoning on which Fermat had originally based his method of maxima and minima. Pressed later on how to move from fiction to reality, Leibniz tried to show in some detail that, although differentials are infinitely smaller than finite quantities and hence had no ratio to them, the ratios of infinitesimals to one another are determinate and equal to ratios between finites; the relation dy/dx = y/subtangent was the touchstone. The ratios establish a correspondence between the two realms, much as combinations such as √[1 + √(-3)] + √[1 - √(-3)] = √6 link imaginary to real roots. (70)

In Leibniz's mind, ideal elements simply gave mathematicians purchase on real processes. There is no last term of an infinite series, but one can imagine the form of that term as it grows smaller and appeal to the "law of continuity" to preserve that form as the term reaches the limit, just as by that law "it is permitted to consider rest as an infinitely small motion (i.e. as equivalent to its contradictory in a sense) and coincidence as an infinitely small distance, and equality as the last of the inequalities, and so on." For that matter, continuity itself could be considered an ideal object, for nothing in nature corresponded to it. Yet,

in recompense, the real does not cease to be governed by the ideal and the abstract, and it happens that the rules of the finite succeed in the infinite, as if there were atoms (i.e. assignable elements of nature), even though there is no matter actually divided without end; and conversely the infinite succeeds in the finite, as if there were metaphysical infinitesimals, even though one does not need them and the division of matter never reaches infinitely small pieces.

"That is how everything is governed by reason," he concluded, "otherwise there would be neither knowledge nor reason, and that would not conform to the nature of the sovereign principle." (71)

4.3 Proof, Truth, and Utility

Mathematics extended its utility in part by incorporating into theory what had up to then been treated as craft practice. Pressure to do so came in part from the practitioners, as they sought new status for their craft. Persuaded that the "barbarous" art of algebra, inherited from the Arabs, contained traces of the method of analysis the Greek geometers used to find solutions to problems and proofs of theorems and then masked in their demonstrations, Viète created a new symbolism and elevated algebra to the "art of analysis". Descartes followed suit, redefining "geometry" as the class of problems subject to algebraic expression and treatment, and subsequent developments in the methods of series and infinitesimals extended that class. By the end of the century, "analysis" covered most of mathematics beyond the elementary subjects. Each step moved analysis farther away from what Greek mathematicians took to be its inseparable counterpart, synthesis: the rigorous demonstration from first principles or from theorems already derived from first principles. In working by hypothesis from the unknown to the known, analysis had heuristic power but lacked demonstrative force. In the absence of proof that each of the steps of an analysis could be inverted, analysis could not compel assent.

The mathematicians who created and used the new analysis were fully aware of its weakness, and they offered two related responses. First, they argued that any analysis could be reversed to form a synthetic demonstration, albeit not always directly. At worst, the result determined analytically formed the starting point of a double reductio ad absurdum, the steps of which would also follow from the analysis. Fermat often recognized the need for a synthetic demonstration, even if he then waived it as "easy" or "not worth the effort" to carry out in detail. In this, he leaned toward the second response to critics of analysis, expressed perhaps most clearly by Descartes in his Responses to the second set of objection to the Meditations. Analysis, he argued, made clear to the attentive reader how the result had been achieved and hence conveyed intuitive understanding even if it did not constitute conclusive proof.

Descartes's identification of geometry with algebraic polynomials kept derivations close enough to demonstrations for practitioners to believe that the inversion from analysis to synthesis was straightforward. Rather than posing questions of interpretation, imaginary roots by their very impossibility indicated the absence of a solution to a problem. Descartes and Fermat could persuade themselves that their respective methods for drawing tangents rested on finite algebraic foundations, and, indeed, the pseudo-equalities used to find the tangent served as the inequalities needed to demonstrate its unique contact with the curve at the given point. But attention soon shifted to the extension of the method of tangents to non-algebraic curves, requiring assumptions about the negligibility of differences between, say, arcs and their chords over infinitesimal intervals, and it was less obvious how to invert those assumptions in a synthetic demonstration.

Similarly, methods of quadrature and rectification also rested on assumptions about differences over small intervals, in particular when they can be neglected. Once dropped during analysis, they could not readily be recalled during synthesis. But, just as Archimedes inspired the methods of analysis, he also provided the model of synthesis in the form of double reductio ad absurdum, examples of which abounded in his works. Again, the apparently close relation between the pseudo-equalities or limit-sums of infinitesimal analysis and the inequalities on which the reductions rested lent intuitive confidence that proof was a matter of detail. "I have set out these lemmas beforehand," wrote Newton in a scholium to Book I, Section I of the Principia, dealing with the "method of first and last ratios, with the aid of which what follows is demonstrated",

I have set out these lemmas beforehand so that I may avoid the tedium of carrying out involved demonstrations ad absurdum, in the manner of the ancient geometers. For demonstrations are rendered more concise by the method of indivisibles. But since the hypothesis of indivisibles is harder, and for that reason that method is deemed less geometrical, I wanted to reduce the demonstrations of the following matters to the last sums and ratios of evanescent quantities, and to the first [sums and ratios] of nascent [quantities], and for that reason to set out beforehand demonstrations of those limits with all possible brevity. (72)

Newton intended the lemmas to define his meaning even if he subsequently spoke in terms of ratios and sums of indivisibles or took curved "linelets" for straight lines. Through the lemmas, the language of indivisibles translated into that of limits, which could be used "more securely" as "demonstrated principles".

Leibniz used similar terms in asserting safe passage between the realms of the infinitesimal and the finite. The fact that later generations found the passage more hazardous than he, Newton, and their immediate followers portrayed it is less important historically than the fact that they were aware of the difficulties of fitting their concepts and techniques to the reigning standards of rigor as set by Aristotle and Euclid and were attempting to resolve those difficulties by showing that the paths of the calculus, or at least the results reached by them, could be retraced in classical steps and by introducing new canons of intelligibility and criteria of effectiveness as warrants of the soundness of their methods. (73) That dual strategy had been laid down over the century and was evidently persuasive to the audience the practitioners of the new methods were addressing. That is worth bearing in mind, lest nineteenth and twentieth-century concerns with formal rigor be projected back onto the seventeenth century, investing the original concepts of the calculus retrospectively with meanings they did not have for their creators and consequently overlooking the meanings they did have.

Those meanings depended in significant part on shared practice. (74) Barrow's Mathematical Lectures show how much mathematicians' understanding of the philosophical issues depended on their knowledge of mathematics itself. At several points he found it difficult, even impossible, to explain to an unskilled undergraduate audience a concept such as "possible congruence" which underlay Cavalieri's technique for determining that one curved figure was equal in length or area to another. One could not understand the concept without using it. As Bacon had insisted of scientific knowledge as a whole, so too in mathematics truth and utility were "one and the same thing". (75) Intuitive confidence in the new mathematical techniques derived from knowing that they worked, and that knowledge came from knowing how to make them work.

Conversely, philosophical discussions of mathematics that were not rooted in practical experience had little bearing on the developments that would prove of philosophical importance. Barrow dismissed Andreas Tacquet's critique of Cavalieri's method because Tacquet showed he did not know how to apply it to simple problems. Thomas Hobbes's criticism of mathematicians suffered the same fate. In 1695 Bernhard van Nieuwentijdt wrote in the Acta eruditorum of his perplexity over second differentials, and Leibniz tried to explain. But by then Jakob (1655-1705) and Johann (1667-1748) Bernoulli had shown the vast range of problems --some old, some new-- that second differences opened to analysis, thus placing them beyond debate among practitioners. Fontenelle spoke for the majority of the Académie des Sciences when he emphasized the new canons of intelligibility by which they measured Leibniz's calculus:

Although the mathematical infinite is well understood, its principles quite unshakeable, its arguments fully coherent, most of its investigations a bit advanced, it does not cease still to cast us into the abyss of a profound darkness, or at the very least into realms where the daylight is extremely weak. ... a bizarre thing has happened in higher mathematics [haute géométrie]: certainty has undermined clarity. One always holds onto the thread of the calculus, the infallible guide; no matter where one arrives, one had to arrive, whatever shadows one finds there. Moreover, glory has always attached to great discoveries, to the solution of difficult problems, and not to the elucidation of ideas. (76)

The new mathematics belonged to those who knew how to do it.

This "proof-of-the-pudding" approach to what by 1700 was viewed as the twofold field of ordinary and infinitesimal analysis drew support and inspiration from the application of mathematics to mechanical problems. The center of gravity was only one of several foci of mechanical action locatable only by determining the areas and volumes of curved figures. Conversely, by concentrating varying degrees of change in one point, such centers of action suggested a strategy for capturing change mathematically by reduction to a mean value. Although mechanics did not create the problems of drawing tangents to curves and measuring their areas, it did offer intuitive support for the means of solving those problems. The notion of speed made sense of change over an interval, and acceleration gave meaning to a change of change. Viewing speed as an intensional quality made extensional by imagining it counterfactually to be held for a period of time or by summing up its effect over a finite interval gave substance to the notion of "indivisibles" and of their transition into infinitesimals. As Barrow summed it up in his Geometrical Lectures,

To every instant of time, or to every indefinitely small particle of time; (I say "instant" or "indefinite particle" because, just as it matters nothing at all whether we understand a line to be composed of innumerable points or of indefinitely small linelets [lineolae], so it is all the same whether we suppose time to be composed of instants or of innumerable minute timelets [tempusculis]; at least for the sake of brevity we shall not fear to use instants in place of times however small, or points in place of the linelets representing timelets); to each moment of time, I say, there corresponds some degree of velocity which the moving body should be thought to have then; to that degree corresponds some length of space traversed (for here we consider the moving body as a point and thus the space only as length); ... (77)

Thus the intuition of motion, of its continuity, of the speed of motion at a given moment, and of the reducibility of variations of that speed to some mean measure provided a touchstone for the new techniques of analysis, whether algebraic or geometric in style.

What did algebra have to do with mechanics in the 17th century? The common factor was analysis, understood as resolution or reduction into constituents. Algebra was called analysis initially because it embodied Pappus' description of the process of moving from a problem to its solution. But Viète put a new twist on it by introducing the notion of the structure of equations (constitutio aequationum) and making algebra, or the analytic art, the body of techniques by which that structure is analyzed into its basic parts or transformed into equivalent structures. Descartes and Fermat built from there, applying the art to curves and adding techniques for drawing from the structure of equations of curves the properties of their tangents and areas. Those new techniques involved infinitely small quantities and considerations of limiting values, setting the basis for the calculus as devised by Newton and Leibniz. But underlying the new quantities and techniques for calculating with them lay the original themes of the analytic art: a method of heuristic that proceeds by resolution into parts. Infinitesimals allowed the art to analyze motion and the continuum.

It may sound like a truism, but mechanics was linked to analysis through the notion of a machine. What counteracts the truism is the identification of mechanics as the science of motion, canonized by the title of Wallis's treatise; nothing in the concept of a science of motion entails resolution into parts. Machines, however, are quintessentially analytic: one understands their working by taking them apart and seeing how the parts go together. Machines are nothing more or less than the sum of their parts. Mechanizing the world meant making it a machine, that is, conceptualizing it as a structure resolvable into constituents which, understood individually, combine to explain the action of the whole. Francis Bacon (1561-1626) was talking mechanistically when he said,

But to resolve nature into abstractions is less to our purpose than to dissect her into parts; as did the school of Democritus, which went further into nature than the rest. Matter rather than forms should be the object of our attention, its configurations and changes of configuration, and simple action, and law of action or motion; for forms are figments of the human mind, unless you will call those laws of motion forms. (78)

Whether or not Bacon expected those laws to be expressed mathematically, Descartes certainly did. The "laws of nature" by which God created and conserves the world are statements about parts of matter in motion according to quantitative relations.

I could set out here many additional rules for determining in detail when and how and by how much the motion of each body can be diverted and increased or decreased by colliding with others, something that comprises summarily all the effects of nature. But I shall be content with showing you that, besides the three laws that I have explained, I wish to suppose no others but those that most certainly follow from the eternal truths on which mathematicians are wont to support their most certain and evident demonstrations; the truths, I say, according to which God Himself has taught us He disposed all things in number, weight, and measure. (79)

Understanding an "effect of nature", then, comes down to analyzing it into its constituent parts of matter and expressing the effect in terms of their interaction by the laws of motion.

That view of nature as analytic in the same sense as a machine does not in itself entail an algebraic description. Clearly, as most of the literature of seventeenth-century mechanics shows, one can understand both the parts and their motions in geometrical terms. Yet, as the mechanics probed deeper, the many dimensions of bodies in motion --their position, velocity, acceleration, momentum, force-- strained the capacity of geometrical configurations to accommodate them operationally rather than just illustratively. Couching the parameters of motion in algebraic terms made explicit their structure and the structure of the relations between them, and it made those structures accessible to manipulation. As a calculus of motion, analytic mechanics thus made motion a form of machine to be taken apart and reassembled. In that calculus, created at the turn of the eighteenth century, the new mechanics and the new mathematics met to form a new metaphysics.

4.4 Analytic Mechanics: Mathematics, Motion, and Metaphysics

Galileo and Descartes both wrestled with the continuum and its implications for a mathematical account of nature. They knew from Aristotle the logical inconsistencies that attend any geometry based on atomism or actual infinities. (80) In the former, all magnitudes are commensurable; in the latter, all magnitudes are equal. Potentially infinite divisibility sustained the continuity necessary for incommensurability and ordering, as Eudoxus showed in his theory of proportions and the method of exhaustion based on it. As mathematicians Galileo and Descartes were wary of infinites and infinitesimals on the one hand and indivisibles on the other. Both notions courted mathematical incoherence. Acceptable perhaps as shortcuts and temporary expedients for problem-solving, they constituted problems in themselves, to be controlled if not resolved by formal demonstration of the results reached by them. Mathematicians throughout the century shared this view, even as they developed the new methods of infinitesimals and infinite series. They differed over what constituted proper grounding of those methods, not on the need for grounding. Without an unambiguous correspondence between the domain of the infinitesimal and that of the finite, mathematicians could talk only by analogy. However different in form, Newton's lemmas concerning first and last ratios and Leibniz's principle of continuity based that correspondence on the possibility of a definite ratio among indefinite magnitudes.

Equally persuasive to Galileo, Descartes, and their successors was the notion that at some level of fineness the physical world must consist of atoms. (81) To make sense mechanically, matter at some point has to resist division and push back. Whatever the differences in their metaphysics, mechanicians shared the intuitive model of small balls bouncing against one another, whether suspended on strings or rolling along the ground. In that model their interaction was also discontinuous: they met at certain speeds and separated at new speeds, and the change had to be instantaneous; as atoms, they had no substructure to explain the lag required by deceleration and acceleration. Hence, whatever problems discontinuous matter posed for mathematical description, the mechanical model of impressed forces ran headlong into the continuity of time and motion.

Galileo emphasized that continuity as a means of incorporating rest into the state of motion, so that a body might pass through all degrees of motion to zero and then acquire speed again, without ever stopping. Descartes's insistence on the relativity of motion entailed the same continuity, even if his laws of impact violated it, as both Huygens and Leibniz pointed out. It posed a problem for Newton in giving his laws of motion mathematically effective form. In Definition VIII he posited the motive quantity of centripetal force as "its measure proportional to the [quantity of] motion it generates in a given time", that is, to the rate of change of momentum over time. In the second of the laws of motion that served as axioms for the theorems to follow, he asserted that "the change of [quantity of] motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed." Time had disappeared from the process. The difference between the two statements has raised the much-debated question of whether Newton was thinking of continuous force or discrete impulse.

Phrasing the question as an alternative, however, overlooks the interdependence of mathematics and metaphysics in Newton's system. When Newton applied the axiom in Theorem I, he was reasoning on the basis of the orbit as a geometrical object. Drawn in two-dimensional space, the configuration allowed the representation of velocity and time only indirectly by means of lines and areas proportional to them. To show change in velocity, therefore, he needed first to show velocity as a finite segment traversed over an interval of time. Time itself remained off the diagram at the start, divided into a succession of equal intervals, while the orbit began as a concatenation of straight lines each proportional to the distance traversed uniformly during each interval at the current velocity, and thus to the velocity. Change in velocity could be represented only by comparing the path traversed during the next interval at the new velocity and the path that would have been traversed had the body continued during that interval at the old velocity. Hence the force was applied at the end of each interval and it was measured at the end of the next interval by a line segment parallel to the radius drawn to the previous endpoint and bounded by the two paths. In short, the mathematics required that the force act impulsively at discrete intervals. Newton could approach continuity, both of orbit and of applied force, only by shortening the intervals and increasing their number. In assuming that relations that remain unchanged however small the intervals of time between impulses are preserved when time flows as a continuum and the force is continuously impressed, the demonstration of Proposition 1, Theorem 1 echoes Leibniz's principle of continuity.

Newton himself evidently believed that his limit argument reconciled the mathematics of discrete impulses to the metaphysics of continuous forces. Whether consciously or not, he assumed that the variation in the direction of the force acting continuously over a small interval can be ignored. To phrase the second law in terms of rate of change would have required a different body of mathematical tools, which enabled one to articulate the nature and implications of that assumption. In recasting the geometrical analysis of the Principia into the infinitesimal analysis of Leibniz's calculus, Varignon showed what such tools might look like.

In 1700 Varignon sketched a general theory of motion determined by central forces and in a series of memoirs rendered into the language of the calculus the mechanical substance of Book I, Sections 2 - 10, of the Principia. His first memoir, "Manière générale de déterminer les forces, les vitesses, les espaces, & les temps, une seule de ces quatre choses étant donnée dans toutes sortes de mouvement rectilignes variés à discrétion", aimed at capturing Newton's theorems on rectilinear centripetal motion in two "general rules", from which all else followed by the techniques of ordinary and infinitesimal analysis. His modification of the configuration of Proposition 39 of the Principia reveals both the different form of mathematics Varignon was working with and the different ends to which he was applying it.

All the rectilinear angles in the adjoined figure (Fig. 16) being right, let TD, VB, FM, VK, FN, FO be any six curves, of which the first three express through their common abscissa AH the distance traversed by some body moved arbitrarily along AC. Moreover, let the time taken to traverse it be expressed by the corresponding ordinate HT of the curve TC, the speed of that body at each point H by the two corresponding ordinates VH and VG of the curves VB and VK. The force toward C at each point H, independent of [the body's] speed (I shall henceforth call it central force owing to its tendency toward point C as center) will be expressed similarly by the corresponding ordinates FH, FG, FE of the curves FM, FN, FO.

The axis AC, with the center of force at C, stemmed from Newton. The six curves were inspired by Leibniz. They represent graphically the various combinations of functional dependency among the parameters of motion: the "curve of times" TD represents time as a function of distance; the "curves of speed" VB and VX, the velocity as functions of distance and time respectively; and the "curves of force" FM, FN, and FO, the force as functions of distance, time, and velocity respectively. To translate those designations into defining mathematical relations, Varignon turned to algebraic symbolism. At any point H on AC set the distance AH = x, the time HT = AG = t, the speed (HV = AE = GV) = v, and the central force HF = EF = GF = y. "Whence," Varignon concluded from the perspective of the calculus,

one will have dx for the distance traversed as if with a uniform speed (comme d'une vitesse uniforme) v at each instant, dv for the increase in speed that occurs there, ddx for the distance traversed by virtue of that increase in speed, and dt for that instant.

The first of the two general rules simply expressed symbolically the basic assumption of uniform motion over infinitesimal intervals. Since "speed consists only of a ratio of the distance traversed by a uniform motion to the time taken to traverse it", v = dx/dt, whence, by the rules of differentiation, (82) dv = ddx/dt. The second rule took account of the change of speed and of the increment of distance that results from it.

Moreover, since the distances traversed by a body moved by a constant and continually applied force, such as one ordinarily thinks of weight, are in the compound ratio of that force and of the squares of the times taken to traverse them, ddx = y dt2, or y = ddx/dt2 = dv/dt.

The rule appears to have stemmed in the first instance from the Principia. The first half of the measure expresses the second law, and the second half translates into the language of the calculus Lemma 10, which in turn sets out a principle also found in Huygens' analysis of centrifugal force. Varignon's version of the rule literally brings a new dimension to it, however, by capturing through the second differential dds that the effect is a second-order variation of the motion of a body.

Those two rules, v = dx/dt and y = dv/dt sufficed, Varignon maintained, to give a full account of forced motion along straight lines. For, given any one of the six curves set out above, one can use the rules to carry out the transformations necessary to produce the other five. That central proposition reduces the mechanics in question to a matter of mathematics, and for the remainder of the memoir Varignon pursued an essentially mathematical point, echoing in the style and direction of his discussion two articles published by Leibniz in 1694 (83). The solution of the differential equation v(x) = dx/dt yields the curve DT determined by HT = t(x), and, if VB = v(x), then v'(x)dx = dv = ydt will produce y(x,t), which can take two forms, depending on how the curve DT is expressed. Either FG = y(x(t),t) or FH = y(x,t(x)). The other curves emerge by similar transformations. As Varignon noted at the outset, the general claim rests on the dual assumption of complete solvability in the two realms of analysis: the resolution of any algebraic equation (i.e. getting x(t) from t(x)) and the integration of any differential equation. The limits of the mechanics in question were those of the calculus. Varignon pushed further toward those limits in later memoirs. In particular, he directed his analyses of toward expressions which presupposed no differential as constant, that is, no variable as independent. Depending on that choice, the expression took on several different forms. Put another way, the expression determined a family of differential equations, each transformable into the others by a change of variable. In articulating Leibniz's calculus, Johann Bernoulli had shown that set of transformations led to the solution of various integrals and differential equations. By linking it now to the mechanics of central forces, Varignon meant to extend its power even further, first by using it to free his mechanics from dependence on any particular choice of coordinates and second to bring mechanics to bear on mathematical problems.

The details of his argument are of less concern here than the direction in which it took mechanics as the prime expression of nature understood mathematically. Generalizing the expression of mechanical relations to the point of rendering them independent of the choice of independent variable brought to a culmination the trend away from diagrams and toward symbolic expressions that began with Galileo's analysis of acceleration. Varignon's analyses proceeded by manipulation of symbols according to the rules of finite and infinitesimal algebra supplemented by those of kinematics and dynamics expressed symbolically, and the resulting combinations of variables took their meaning from those operations. Varignon, at least, had pointed mechanics toward Euler and Lagrange.

While vastly extending the effective range of mechanics, the symbolic approach brought into sharper focus the question of the relationship between the structure of the mathematical model and the structure of the physical system it is meant to represent and thereby to explain. Put most succinctly, what do the intermediate steps linking two mathematical statements about a physical system have to do with the processes that tie the two physical configurations together? Huygens had felt no need to posit the existence of something in nature corresponding to the measure mv2, even though he used its conservation as a primary tool for analyzing dynamical problems. By contrast, Newton confronted the dilemma of a real force which he could not explain mechanically but which he needed mathematically, and he chose the mathematical horn. An old problem, of which the medieval debate over the reality of epicycles is one form, it assumed new importance with the use of mathematics to analyze the nature of motion and its relation to force. The widening empire of mathematical physics over the next two hundred years would carry the question into new realms of nature.