Princeton University

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.

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)}

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.

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

AEBare extended toIGF, we will have the aggregate of all the parallels contained in the quadrilateral equal [aequalem] to the aggregate of those contained in the triangleAEB, for those in triangleIEFare equal [paria] to thoe contained in triangleGIA, and those in trapeziumAIFBare 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.

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.

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, *mv*^{2}, 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* = *-kSdt*^{2} 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.

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*
= *v*^{2}/*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/*R*^{2}, 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)}

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.

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.

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*
x*^{n} + a_{1}*x*^{n-1} + *a*_{2}*x*^{n-2}
+ ... + *a*_{n} = 0, where *n* was a
definite number and the *a*_{i} 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 *x*-α_{i},
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 α_{i} 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
*x*^{3} - 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 *x*^{2} + *y*^{2} = *r*^{2},
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* - *x*^{2} = *M*. In
general, it has two roots, say *u* and *v*. By a
technique taken from Viète, *bu* - *u*^{2}
= *M* = *bv* - *v*^{2}, or *b*(*u*-*v*)
= *u*^{2} - v^{2} = (*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* - *x*^{2}, the equation
will have a single, repeated root, that is, *u* = *v*
= *b*/2 (whence *M* = *b*^{2}/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* - *u*^{2},
whence *be* - 2*ue* - *e*^{2} = 0,
or *b* - 2*u* - *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*
= 2*u*, 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.

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 *y*^{m} = *px*^{n}
and *x*^{n}*y*^{m}
= *p*, the so-called "higher parabolas and
hyperbolas." The achievement lay in determining the sum of
*i ^{k}* from

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 *y*^{2} = *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* = 2*OQ*. 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* = ∫2*xdy*

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 *b*^{3}
= *x*^{2}*y* + *b*^{2}*y*
into *b*^{2} = *u*^{2} + *v*^{2}
by means of the auxiliary curves *by* = *u*^{2}
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 *ds*^{2} = *dx*^{2}
+ *dy*^{2} 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)}

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.

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, letTD,VB,FM,VK,FN,FObe any six curves, of which the first three express through their common abscissaAHthe distance traversed by some body moved arbitrarily alongAC. Moreover, let the time taken to traverse it be expressed by the corresponding ordinateHTof the curveTC, the speed of that body at each pointHby the two corresponding ordinatesVHandVGof the curvesVBandVK. The force towardCat each pointH, independent of [the body's] speed (I shall henceforth call itcentral forceowing to its tendency toward pointCas center) will be expressed similarly by the corresponding ordinatesFH,FG,FEof the curvesFM,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 dt

^{2}, or y = ddx/dt^{2}= 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.