-
Fluxions, Limits, and Infinite Littlenesse. A Study of Newton's
Presentation of the CalculusAuthor(s): Philip KitcherSource: Isis,
Vol. 64, No. 1 (Mar., 1973), pp. 33-49Published by: The University
of Chicago Press on behalf of The History of Science SocietyStable
URL: http://www.jstor.org/stable/229868 .Accessed: 06/01/2014
15:48
Your use of the JSTOR archive indicates your acceptance of the
Terms & Conditions of Use, available at
.http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars,
researchers, and students discover, use, and build upon a wide
range ofcontent in a trusted digital archive. We use information
technology and tools to increase productivity and facilitate new
formsof scholarship. For more information about JSTOR, please
contact [email protected].
.
The University of Chicago Press and The History of Science
Society are collaborating with JSTOR to digitize,preserve and
extend access to Isis.
http://www.jstor.org
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
Fluxions, Limits, and Infinite Littlenesse
A Study of Newton's Presentation of the Calculus
By Philip Kitcher*1
I. INTRODUCTION
THE WORK OF ISAAC NEWTON has received a great deal of attention
from historians of mathematics. Why then should there be any need
for another paper on
Newton's calculus? My aim is not to rehearse the familiar
account of how Newton developed his theory but rather to cast light
on the relations among the concepts that he employed, relations
which have not previously been sufficiently explicated.
The data from which we begin are straightforward enough. In his
early papers Newton followed the style of his day and cast his
methods in algebraic terms. When pressed for geometric
interpretation of his algebraic maneuvers, he resorted, as did many
of his contemporaries, to talk of infinitesimals. Later he
abandoned this refer- ence in favor of the theory of fluxions, and
still later that theory in turn came to rest on the doctrine of
ultimate ratios. Many historians have amply filled in the details
of this crude outline. But they have not asked whether the
progression was anything other than a succession of frameworks each
of which Newton regarded for a time as the basis of his
theory.2
Instead of seeing each set of concepts as a candidate for the
true foundation of the calculus, I shall contend that we should
recognize that each occupied a special place in Newton's total
scheme. Putting the matter briefly, the theory of fluxions yielded
the
Received Jan. 1972: revised/accepted Aug. 1972. *Program in the
History of Science, Princeton
University, Princeton, N. J. 08540. 1 I am very grateful to two
anonymous readers
for Isis who helped me to improve substantially an earlier
version of this paper, which received the 1971 Henry Schuman Prize
in the History of Science. My chief acknowledgements must how- ever
be to Patricia Kitcher and to Professor Michael S. Mahoney. I am
indebted to them for considerable help, kindly criticism, and much
patient encouragement.
2 Both Carl B. Boyer and D. T. Whiteside in their studies of
Newton's mathematics seem to accept the idea that Newton devised
his concept of a fluxion as a substitute for the use of in-
finitesimals and that the method of ratios was
also intended to perform an equivalent task. Boyer, for example,
refers to the "fact that Newton could thus present all three views
as essentially equivalent . . ." (The History of the Calculus and
Its Conceptual Development, New York:Dover, 1949, p. 201).
Whiteside makes a similar assumption when he writes: "In the summer
and early autumn of that year (1665). ... he recast the theoretical
basis of his new-found calculus techniques, rejecting as his
foundation the concept of the indefinitely small discrete in-
crement in favour of the 'fluxion' of a variable . . ." (The
Mathematical Works of Isaac Newton, Vol. I, New York: Johnson,
1964, p. x). The approach to Newton's work elaborated below, which
contends that the functions of Newton's three sets of basic
concepts are quite distinct, seems incompatible with these
views.
33
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
34 PHILIP KITCHER
heuristic methods of the calculus. Those methods were to be
justified rigorously by the theory of ultimate ratios.3 The theory
of infinitesimals was to abbreviate the rigorous proof, and Newton
thought that he had shown the abbreviation to be permissible.
Rather than competing for the same position, the three theories
were designed for quite distinct tasks.
Like the natural scientist, the mathematician works in the
context of discovery as well as the context of justification,
making use of different methods to suit the demands of each. One
obvious part of his task is to provide solutions to those problems
which seem important to the community in which he works and to
develop techniques for coping with classes of such problems
wholesale. The seventeenth-century community not only provided many
classes of problems but also challenged its members to tackle
particularly stubborn instances. As an example, problems of finding
the quadrature of "special curves"-such as the conchoid, cissoid,
cycloid, and hyperbola4-occupied the attention of mathematicians,
and vigorous research on these recalcitrant cases led to new
techniques applicable to the general class of quadrature problems.
The ultimate aim was a fully general method, and certainly the
century saw the generation of many partial algorithms toward this
end. Pierre de Fermat's method of maxima and minima and
Rene-Francois de Sluse's rule, familiar to us as the rule for
differentiating powers of a variable, are prime examples. Both were
successes for the new analytic mathe- matics. In Sections II and
III below I shall defend the claim that Newton's theory of fluxions
was an advance at this level.
Quite different are those parts of a mathematician's work which
connect with questions of proof. Rigor may be sacrificed in the
tussle with obstinate problems, and the solution to a puzzle may be
achieved by means which come short of accepted canons of clarity
and exactness. But if those techniques are ultimately to be
accepted as legitimate parts of the discipline, mathematicians will
demand that they be justified according to the standards which the
community accepts. If the techniques prove their power but also
resist attempts to fit them into recognized practice, conflict
ensues. From that conflict a redefinition of what counts as
mathematics may emerge.5 Part of this study will be concentrated on
the effort to make a new piece of powerful mathe- matics-the
calculus-fit a mathematical paradigm which had the authority of
Aristotle and Euclid.
The heuristic triumphs of the "method of analysis"6 derived
directly from the use of the new algebra in reformulating
geometrical problems. Unfortunately some of the
3 Newton's creation and use of the method of first and last
ratios approaches a theory of limits. The method will be examined
in Sec. V.
4 See, e.g., Oeuvres de Fermat, ed. Paul Tannery and Charles
Henry, Vol. III (Paris: Gauthier-Villars, 1896), pp. 238-240. 5
Perhaps the most dramatic example of this is the change in attitude
toward set theory. One need only compare the highly negative
remarks of Leopold Kronecker-committed to an ideal of pure
mathematics as essentially arithmetic- with David Hilbert's battle
cry forty years later that mathematicians should not be driven out
of Cantor's paradise. The staggering reappraisal of the nature of
mathematics which followed the recognition, fostered by Kronecker,
of problems
in the foundations of analysis has yet to be fully
appreciated.
6 The seventeenth-century method of analysis was in essence a
combination of what was re- garded as a Greek method (assuming the
un- known as known and proceeding by deduction to known truths) and
the use of the algebra fash- ioned by Francois Vi6te to reformulate
problems. So, for example, to find the maximum and minimum points
of a curve, Fermat supposed that the value of the abscissa had been
found and proceeded to set up the algebraic conditions that it must
meet. His derivation was open to question, because he used a
hypothesis that the formula for the sum of the roots of an equation
will continue to hold when two roots are equal,
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 35
algebraic manipulations performed lacked intuitive validity, and
derivations involving such manipulations ranked poorly when viewed
as proofs. As it was attempted to cast them into the synthetic form
of reasoning favored by Aristotle and Euclid, the absurdi- ties of
certain operations (e.g., dividing by zero) only appeared more
blatantly. The tradition of allowing the use of algebra to thrive
on the existence of a geometric model7 left little room for
dismissing the calculations as "formal and meaningless."8 Hence a
temporary method of justification became popular as mathematicians
came to appeal to infinitesimals or indivisibles to support their
algebraic steps. Yet the question natur- ally arose as to whether
this was enough to meet the demand for synthetic proof in the
manner of Euclid.
Newton's achievement consisted in solving problems in both
contexts. His method of infinite series expansion, with which we
shall not be concerned here, showed how the number of curves
accessible to analytic methods could be increased. Also within the
context of discovery his development of the theory of fluxions
reduced the number of problem classes to just two problems
inversely related. By means of the doctrine of ultimate ratios he
hoped to show that his fluxional calculus could be grounded in the
geometry of "the Ancients"9 and also to demonstrate the validity of
the quicker methods of justification which he used throughout his
career. Although he insisted that this was the aim of his theory of
ultimate ratios, his presentation of it was not clear enough to
satisfy George Berkeley and later critics. Yet Newton's emphasis on
the point illuminates his views on proof and rigor. Writing to John
Keill at the time of the priority dispute with Leibniz, he stated
his view quite plainly:
In demonstrating propositions I always write down the letter o
and proceed by the Geometry of Euclide and Apollonius without any
approximation. In resolving Questions or investigating truths I use
all sorts of approximations which I think will create no error in
the conclusion and neglect to write down the letter o, and this do
for making dispatch.10
and that the geometric interpretation of the equal roots
situation gives the case of the ex- treme values. This rendered his
whole reformula- tion of the problem somewhat suspect. The use of
the derivations of solutions provided by the method could thus be
opposed on a number of grounds. We shall examnine Isaac Barrow's
(negative) response to the idea behind the method in Sec. IV. Yet
the majority of mathematicians felt that allowing the analytic
derivation to stand as proxy for a proof was better than giving no
demonstration at all. The rationale for this will be given below. A
clear account of the Greek sources of the method has been given by
Michael S. Mahoney, "Another Look at Greek Geometrical Analysis,"
Archive for History of Exact Sciences, 1968, 5:318-348.
7 As with both Descartes and Vi6te. Descartes' attitude emerges
quite clearly in the fourth of his Regule. Here he links the true
science of algebra to the supposedly secret methods of Pappus and
Diophantus and remarks that "Arithmetic and Geometry . . . give us
an instance of this [algebra] . . ." (Descartes, Works, trans. and
ed. E. Haldane and G. R.T. Ross, Vol. I, Cambridge:
Cambridge University Press, 1967, p. 10). This passage and many
others indicate that geometry was seen as a model or concrete
instantiation of the algebra. Such an idea helped to raise algebra
from the ranks of the "barbarous arts."
8 Perhaps the most sophisticated presentation of this line
occurs in Bernard Bolzano's Para- doxien des Unendlichen (Prague,
1854), Sec. 37. Of course, in the heyday of algebraic analysis,
when geometry had been displaced as the central mathematical
discipline and algebra had become recognized as the format of
mathematics, this kind of formalism was much more acceptable.
9 I have made no attempt to sort out the con- siderations which
would favor the geometrical paradigm of proof. It is tempting to
view Newton as guided by a different tradition of foundational
studies from that currently in vogue, and the investigation of that
tradition is a task with some significance for the philosophy of
mathematics. In this paper it is only possible for the task to be
defined.
10 Letter to John Keill, May 15, 1714, The Correspondence of Sir
Isaac Newton and Pro- fessor Cotes (Cambridge, 1850), p. 176.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
36 PHILIP KITCHER
Newton failed to convince his contemporaries that he had solved
the problems of justification. In the eighteenth century the
conflict between geometry and algebra for the place of basic
discipline of mathematics resolved in favor of algebra, and New-
ton's faithful successors became the proponents of an outworn way
of pursuing their subject.
II. THE NEED FOR GENERALIZATION AND SIMLIFICATION
The seventeenth-century method of analysis directed its
attention toward a domain of well-defined classes of problems, such
as the construction of tangents to curves, the finding of maxima
and minima, and the computation of quadrature. It aimed to treat
those problems by means of symbolic algebra. The analytic geometry
of Fermat and Descartes enhanced the possibility of treating the
questions, not as individual problems for each curve, but by means
of a general method which would apply to all curves of a certain
type. Classification into types would be undertaken by considering
the alge- braic form of the equation of the curve. It might then be
hoped that by a process of refinement these algorithms could be
combined to form an even more general solution. Descartes, Fermat,
Roberval, and others had used algorithmic methods to handle these
problems before Newton was born. Yet no one had provided the fully
general method which had been hoped for, and there was little
indication of any awareness of relations between the various
problems classes.1' In the wake of the French, mathe- maticians in
the Low Countries endeavored to simplify the methods which had been
proposed, and by the time of Newton's arrival at Cambridge they had
made some pro- gress toward avoiding the tedious computations of
the traditional techniques.'2
This outline of the background against which Newton's work was
set presents three important considerations relevant to
understanding his work. First, we note the exis- tence of a
tradition of mathematical work aimed at the provision of algorithms
for the solution of clearly defined classes of problems. Secondly,
at the time of Newton's arrival at Cambridge the problems to which
mathematicians applied the general approach which they called "the
method of analysis" were still lying loose and separate, and it
would have been impossible to speak of a fully demarcated field of
the calculus. Finally, the existing methods were reckoned
unsatisfactory either on grounds of inconvenience or lack of
generality-in most cases on both counts-and, aside from questions
of justification, there was evident need at the algorithmic level
for extension and simplification of the methods used.
Newton began by accepting the algorithmic challenge. His early
papers on the computation of the Cartesian subnormal13 set out to
investigate individual recalcitrant
11 Fermat's reduction of problems of rectifica- tion to problems
of quadrature is an honorable exception to this charge. Nonetheless
what one expects to see is an awareness of an inverse re- lation
between the problem of constructing a tangent and the problem of
computing quadra- ture. Fermat comes tantalizingly close to this at
times, but the relationship always eludes him just as it does
everyone else before Barrow.
12 The most important simplifications were the rules of Johann
Hudde and Ren&-Fran9ois
de Sluse. De Sluse's rule is effectively equivalent to the
standard rule for differentiating powers of a variable and is thus
a highly convenient tool for tackling tangent problems.
13 This work is collected in The Mathematical Papers of Isaac
Newton, ed. D. T. Whiteside, Vol. 1:1664-1666 (Cambridge: Cambridge
Uni- versity Press, 1967), pp. 216 f. I shall henceforth refer to
this volume as MPN; the volume ofMathe- matical Works, also edited
by Whiteside (n. 2), will be referred to as MWN.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 37
cases, a natural preliminary to the extension of analytic
techniques. From this point on (late 1664) his writings showed a
clear structure in which the provision of an algorithm was a
fundamental component. The work which is considered central to the
develop- ment of the calculus was structured in the following
way:
1. An algorithm or method for solving the problem being
considered was given. 2. The validity of the algorithm was
"demonstrated" by justifying the method for
particular cases. Newton asserted that the method of "proof"
revealed how the algorithm worked for any problem selected from the
original problem class.
Newton typically expressed his algorithms in the form of a set
of instructions to the reader. The mechanical application of his
rules to the problems at hand would then yield their solutions. For
example, in the early paper on "The General Problem of Tangents and
Curvature Resolved for Algebraic Curves" Newton offered a number of
what he called "Universall theorems." Typical in style is the
following method for finding the tangent to an algebraic curve via
the computation of the subnormal v.
Having ye nature of a crooked line expressed in Algebraicall
termes wch are not put one pte equall to another but all of ym
equall to nothing, if each of the termes be multiplied by soe many
units as x hath dimensions in them. & then multiplied by y
& divided by x they shall be a numerator: Also if the signes be
changed & each terme be multiplied by soe many units as y hath
dimensions in yt terme & yn divided by y they shall bee a
denominator in ye valor of v. 14
In order to give an algorithm for the computation of curvature
later in the same paper, Newton was forced to introduce expressions
of even greater complexity.'5 Considera- tion of the algorithm
quoted above reveals a problem confronting mathematicians working
in the algorithmic tradition: there is no indication of where the
rule of thumb has come from, nor is any connection made between the
problem to which this algo- rithm is directed and other analytic
problems. Furthermore, the basis on which the methods rest consists
of several examples in which the result given by the algorithm is
yielded as the solution to the problem when it is set up in terms
of infinitesimals.'6 An example of this type of justification will
be given in Section IV. Here it is enough to note that Newton's
resolution of the general problems of tangents and curvature poses
two questions:
1. How are the methods to be reduced to their simplest and most
convenient form? 2. How are they to be justified?
We shall first see how Newton's theory of fluxions answers the
first question.
IH. THE SIMPLIFYING POWER OF FLUXIONS A search for the way in
which Newton's algorithms developed reveals a twofold
progression. In the first place he extended the available
techniques to make it possible to solve more problems from each
class, accomplishing this by the device of expanding
"1 See MPN, p. 276. The algorithm is clearly related to the
earlier rule of de Sluse.
15 Ibid., pp. 289-290. The relative complexity of the curvature
algorithms compared with the tangent algorithm is much the same as
that of the
modern formula for curvature when compared with the simple
derivative.
16 In ibid., pp. 272-276, there is given a series of examples by
means of which the algorithm is justified.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
38 PHILIP KITCHER
obstinate functions into infinite series.'7 More significant for
our study is the way in which the algorithms are structured in
those papers where Newton presented his method in terms of the
fluxion concept. Here he achieved a quite different advance by
reducing the number of classes of problems. Newton replaced the
straightforward but cumbersome rules (such as that quoted above) by
two fundamental algorithms giving solutions to two general classes
of problems. The solutions to these problems-which are inversely
related-can then be applied to solve any of the old problems. The
method of fluxions enhanced analysis by drawing all the traditional
questions together in a neat and satisfying way.
In Newton's "Method of Fluxions" we find a classic statement of
the effectiveness of the fluxion concept at the algorithmic
level.
Now in order to this, I shall observe that all the difficulties
hereof (the problems traditionally associated with the analytic
art] may be reduced to these two problems only, which I shall
propose, concerning a Space describ'd by local Motion, any how
acceler- ated or retarded. I. The length of the space describ'd
being continually (that is, at all times) given; to find the
velocity of the motion at any time propos'd. II. The velocity of
the motion being continually given; to find the length of the Space
describ'd at any time propos'd.18
Newton went on to offer algorithmic solutions to the first of
these problems and, inso- far as he could, to the second.19 We then
see how the problems confronting the method of analysis are
systematically reducible to the fundamental problems. Tangent and
curvature problems can be formulated in terms of the first problem,
that of finding the "fluxions." The task of computing quadrature
reduces to the second problem.
To appreciate the way in which the reduction takes place we need
to understand Newton's employment of a kinematic conception of
curves. The strangeness for us of Newton's relation of motion to
geometry lies in its clash with our view of curves as given by sets
of points in Cartesian space. Ironically, Descartes himself would
have found Newton's viewpoint more sympathetic than ours, since he,
like Fermat, con- structed geometry on a uniaxial approach.20 That
is, a curve would be seen as generated by the motion of an ordinate
segment as its foot moved along a base line. This concep- tion
loomed large in Isaac Barrow's Geometrical Lectures21-especially in
the second-
17 See De Analysi and "Method of Fluxions" (both in MWN). It is
quite clear that Newton's use of infinite series belongs to
analytic mathe- matics. These series had no meaning in tradi-
tional terms until Newton's development of the method of
ratios.
18 "Method of Fluxions," MWN, pp. 48-49.
19 The first statement of these problems is in the work which
Whiteside entitles "The Calculus becomes an Algorithm." Whiteside
dates this tentatively as having been written in the middle of
1665. For the statement of the problems see MPN, p. 344:
1. If two bodys c,d describe ye streight lines ac, bd in ye same
time, (calling ac = x & bd = y, p = motion of c, q = motion of
d) & if I have an equation expressing ye relation of ac = x
& bd = y whose termes are all put equall to nothing. I multiply
each terme of ye
equation by soe many times py or p/x as x hath dimensions in it
& also by soe many times qx or qly as y hath dimensions in it.
The sume of these products is an equation expresing ye relation of
ye motions of c & d....
It is notable that this algorithm can be stated quite neatly,
and it recurs throughout all the fluxional work.
20 See Ren6 Descartes, Geometrie, facsim. ed. (Chicago:Open
Court, 1925), esp. p. 42: "on n'en doit pas plutost exclure les
lignes les plus compos6es que les plus simples, pourvX qu'on les
puisse imaginer estre descrites par un mouve- ment continu, ou par
plusieurs qui s'entrefuient ...." See also Oeuvres de Fermat, Vol.
III, pp. 86 ff.
21 Isaac Barrow, The Geometrical Lectures, trans. and ed. J. M.
Child (Chicago: Open Court, 1916), esp. pp. 42-46.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 39
and was perhaps one point on which Barrow influenced Newton.
Like Barrow, New- ton used both the kinematic approach and the
static view of a curve as a set of points in certain geometrical or
algebraic relations. Barrow's seeming lack of concern with
comparing the two viewpoints is also similar to the pragmatic
approach of Newton's early work, suggesting that neither man saw
any reason except convenience to prefer one or the other. Later,
however, the kinematic emphasis on continuity led Newton to adopt
this conception in his attempt to provide a synthetic foundation
for the calculus.
We can now see more easily how Newton reduced the traditional
problems to the fundamental motion problems. The following
statement links the tangent problem to the first motion problem
while also making a more general connection:
In ye description of any Mechanicall line what ever, there may
bee found two such motions wch compound or make up ye motion of ye
point describeing it, whose motion being by them found by ye Lemma,
its determinacon shall bee in a tangent to ye mechani-
callline.22
The reduction works as follows. Suppose that we wish to find the
tangent to the curve f(x,y) -0 at the point (X, Y).23 Consider the
curve as swept out by the motion of a point, and resolve the motion
along the axes. Let the component of velocity along the x-axis be
p(t), that along the y-axis q(t), and let the components when the
point is at (X, Y) be P, Q respectively. Applying the algorithm for
the first fundamental problem, we derive from the general relation
f(x,y) 0 a general relation g(x,y,p,q) = 0. In particular, we have
g(X, Y,P, Q) = 0, giving us a relation between P and Q. Let a be
the angle which the tangent at (X, Y) makes with the x-axis. Since
the instantaneous velocity of the moving point at (X, Y) is in the
direction of the tangent, we know that tan a = Q/P. We calculate
this ratio from the equation g(X, Y,P,Q) - 0, thus solving our
problem.24
This example is typical of the way in which the kinematic
conception of curves enabled Newton to break down such traditional
problems as curvature and quadra- ture into two stages. In the
first stage he showed how the problem could be reduced to one of
the fundamental problems, and the solution to the latter then gave
what was required. The first part of the process represented a
natural and direct resolution of the problem in kinematic terms.
Only at the second stage was there need for algorith- mic work to
be done. A unified domain of study emerged as the connections which
Newton forged with the fundamental problems revealed that the
traditional analytic questions could be seen as naturally
related.
The algorithmic reformulation in terms of fluxions thus solved
our problem 1 which we posed for Newton above. Yet it in no way
helped with question 2, for nothing had been done to eliminate the
crudeness of the infinitesimal justifications for the al- gorithms.
Indeed, Newton continued to justify his solution of the fundamental
prob- lems in terms of infinitesimals. The fluxional approach thus
depended on more tradi-
22 MPN, p. 377. The lemma referred to is the parallelogram of
velocities lenmna.
23 We consider the curve as given in Cartesian coordinates.
Problems for the method arise when other systems are used.
24 So far there is no guarantee that the ratio Q:P can be found
from the equation g(X, Y,P,Q) =0. The fact that it can, only
emerges when we
see how the infinitesimal justification of the algorithm for
deriving g from f works (see n. 37 below).
Gilles Personne de Roberval had used the kinematic conception of
curves to the same effect. For the relation of his method to
earlier techniques, notably that of Evangelista Torricelli, see
Boyer, Ilistory of the Calculus, p. 146.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
40 PHILIP KITCHER
tional considerations. The new algorithms needed vindication:
infinitesimals provided it. Yet, although the new justifications of
the fluxional algorithms parallel those which Newton gave when he
presented his results in the old form,25 someone might still object
that the infinitesimal element is an element of time and no longer
part of a line. True enough-but nothing hangs on this. For to
affirm the thesis that Newton repudi- ated infinitesimal linelets
when he espoused the method of fluxions it would have to be shown
that the infinitesimal linelet was not definable in the later
system ofjustification. The objection collapses when we recognize
that the infinitesimal linelet is just Newton's "moment" of a
fluxion.26 Fluxions and infinitesimals coexist in the same papers.
The methods involving fluxions are supported by infinitesimal
justifications. We can only conclude that fluxions and
infinitesimals are not competitors but play two different
mathematical roles.
Any account which does not make this distinction seems to be
faced with the dilemma of either regarding the coexistence as
anomalous or else of supposing that Newton was colossally confused.
The placing of the function of fluxions at the al- gorithmic level
does more than just resolve this puzzle. For the general structure
of Newton's papers in terms of problems posed, rules for solution,
and infinitesimal vindication is perfectly logical if we give due
place to the importance of work in the context of discovery. The
apparently repetitive nature of these papers can then be seen as a
search for the best way of presenting the analysis so that it is at
the same time as fully general as possible and yet also concise in
its offering of technique. We may also note that the dependence of
the fluxional calculus on the notion of instantaneous velocity
probably gave Newton's readers the "feel" of what was going on,
thus making the algorithms easy to work with when cast in fiuxional
terms.
Yet does not this reading of Newton's development of the method
of fluxions over- look the element of vacillation which previous
views have stressed? Why, if Newton had already fashioned this
powerful tool, did he not use it in the De Analysi which postdates
much of the fluxional work? The answer explains away the apparent
ano- maly of the De Analysi. The simplifying power of fluxions is
to effect a reduction in the number of problem classes. Hence
fluxions are most useful where several analytic problems are being
considered together. Plainly, if Newton were writing a treatise
confined solely to the problem of quadrature, this would not be an
issue at all; for dealing with one problem in isolation the use of
fluxions would only have complicated matters. The point can be
reinforced if we accept Whiteside's suggestion of New- ton's
motives in writing the paper. 27 For, if it is right that Newton,
piqued by Gerhardus Mercator's publication of results which were
already known to him, decided to show how he could deal with the
same problem in greater generality, then a concise treat- ment of
the particular problem of quadrature in a way which was fully
sanctioned by tradition would constitute a more effective reply
than a lengthier treatise embodying a
25 One need only compare the demonstrations given in the De
Analysi with those of the "Method of Fluxions"; see, e.g., MWN, pp.
23-25 and 32-33.
28 Newton himself recognized this. Ibid., p. 52: "Wherefore if
the moment of any one, as x be represented by the product of its
celerity x into an indefinitely small quantity o ...." Also,
Letter to Keill, May 15, 1714 (Correspondence of Newton and
Cotes, p. 176): "ffluxions & moments are quantities of a
different kind. ffluxions are finite motions, moments are
infinitely little parts. I put letters with pricks for fluxions,
& multiply fluxions by the letter o to make them become
infinitely little...."
27 See MWN, p. xii.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 41
unified-but radically different-approach to the whole problem
complex of analysis, with respect to which comparisons with
Mercator's work would have been harder to make. The placing of the
fluxional concept as operative at the algorithmic level en- ables
us to make good sense of the forsaking of fluxional methods in the
De Analysi, where there is no work for them to do. We no longer
have to suppose that this was a temporary and inexplicable
rejection of a foundational concept.
We have seen how Newton resolved question 1-the problem of
simplification. We now turn to his attack on questions of
proof.
IV. ANALYSIS, SYNTHESIS, AND INFINITESIMAL JUSTIFICATION In
investigating problems of maxima and minima and tangents by means
of the
method of analysis, Fermat, for one, believed that he was
employing methods used by the ancient geometers. The method of
analysis had complemented the more familiar synthetic presentation;
that is, the method was applied to a problem to yield the solu-
tion, and the reversal of the analysis gave synthetic proof of the
validity of that solu- tion. Unfortunately, some new
seventeeth-century techniques did not have this con- venient
property, and the basic principles to which analyses of the
problems led did not command an unequivocal status. Descartes'
method of handling analytic questions led to algebraic results
(such as the fundamental theorem of algebra) which could not be
justified in the traditional manner.28 The problems were never
reduced to pure geometrical considerations nor seen to depend only
on well-grounded results. Simi- larly, David Gregory and John
Wallis could fit their algebraic analysis to such prob- lems as
quadrature only by invoking the hypothesis of infinitesimals. The
elusive synthesis was not provided, and the justifications offered
showed little similarity to the derivations from uncontroversial
premises which Euclid had exhibited.
During their sharp controversy over priority of method for the
determination of maxima and minima, Descartes accused Fermat of
proceeding par hasard. The charge was unjust, for Fermat could
vindicate his results as well as Descartes could his own. In the
absence of fully cogent proof, the solutions of both authors could
be construed as the favors of luck. A stopgap method became
appropriate: the derivation of the solution to a problem-even if it
involved dubious algebraic manipulations which were not susceptible
of full geometric interpretation-was permitted to stand as sub-
stitute for a full proof of the validity of the solution.
Justification of this kind could not count as proof under the
Euclidean paradigm. Two options were thus open to the
mathematician: he might relax the standards which a proof must
meet, effectively dethroning Euclid, Aristotle, and Archimedes, or
he could refuse to accept the results and methods of analysis as
fully mathematical. Some mathematicians, such as Wallis, turned
away from the Euclidean norm, castigat- ing it as too restrictive.
But on the whole, although Euclid's influence waned, neither of the
extreme choices found many devotees. The ideal of science as
deductive and infallible, stemming from Aristotle and Archimedes
and seemingly finding its expres- sion in geometry, was not to be
lightly forsaken. Yet the success of the new methods militated
against their outright rejection as far as most mathematicians were
con- cerned. An exception to this attitude was Isaac Barrow.
28 See, e.g., Descartes, Geometrie, p. 101 (facsim., p.
345).
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
42 PHILIP KITCHER
Barrow had no doubt about the viability of our second
alternative. Yet, although the methods of analysis might not be
mathematical, Barrow allowed that the prob- lems to which they were
addressed had a clear mathematical pedigree. Archimedes had shown
how to find the quadrature of the parabola in a way which was truly
geo- metrical. Barrow attempted to follow directly in the synthetic
tradition, solving the problems of tangents and quadrature by
purely geometrical means. His approach led not to an impressive
battery of methods for the working mathematician but to the
perception of relationships among the various classes of problems
which his analyti- cally minded predecessors had missed. Barrow's
geometrical formulation prevented the relations from standing out
as perspicuously as they do in Newton's work,29 but des- pite this,
J. M. Child's eulogy of Barrow is not altogether inappropriate.30
For the combination of the analytic approach with Barrow's drive
toward geometrical syn- thesis and relation of problems created the
calculus. This combination was Newton's.3'
The disrespect which Barrow felt for the method of analysis must
be opposed, not to a feeling of confidence on the part of the
analysts, but rather to a vague unease. Even in the work of the
foremost practitioners of the analytic art we find genuine
conscious- ness of the ultimate need for a synthetic foundation.
The provision of algorithms, the justification of those algorithms
by means of infinitesimals, even the relating of the problems-all
this still left analysis short of the Euclidean ideal. In order to
bring the discipline into line with mathematics, geometrical proof
had to be given.32 Newton took the job of providing such proof
seriously. 33
The analyst, like the physicist, is interested in methods which
are as general and powerful as possible. Qua analyst or qua
physicist, Newton was concerned with our problem
1-simplification-and he resolved it as we have described. Qua
synthe- sist, however, he wanted to give those methods a firm
mathematical underpinning. That is not to say that he was sensitive
to the kind of foundational studies which a twentieth- century
philosopher might favor. It is unclear whether there is any
evidence of Newton
29 Thus, although Barrow formulated what is known as the
fundamental theorem of the calculus, his geometrical version cannot
be seen as having anything to do with the calculus as developed by
his contemporaries and by Newton.
30 Geometrical Lectures, ed. Child, pp. vii ff. 31 I do not want
to suggest anything more than
the possibility that Newton knew about some of the very general
features of Barrow's approach. He may have learned that Barrow
regarded the problems of the calculus as being related. The
difference in the modes of exhibition of those re- lations favored
by the two men belies a stronger connection. In any event, it now
seems clear that Barrow's influence on Newton was nowhere near as
great as was once believed.
32 In equating the synthetic method of proof with reduction to
geometry I am following a conflation which Newton seems to make.
Newton may have been impervious to the distinction between the form
of a proof and the content of that proof, a distinction which we
make with ease, and it may thus be that his foundational
drive was just a desire to reduce the proofs to geometry without
any consideration of the supposed certainty offered by traditional
synthe- sis. In other words "synthetic proof" may have been equated
with "geometrical proof" and, referred to in either way, upheld as
paradigmatic of cogent proof. This speculation trespasses on ground
already defined as beyond the scope of this paper.
33Although Christiaan Huygens and Barrow had both undertaken
geometrical constructions of parts of the calculus before Newton,
their attitudes must be differentiated from Newton's. Huygens and
Barrow wished to extend geometry by building a geometrical calculus
from square one. Newton's goal was the quite different one of
wishing to bring algebraic analysis within the scope of orthodox,
respectable geometry. To the best of my knowledge he is the only
seventeenth-century figure to have devoted him- self to this
particular task. Newton seems to be the genuine compromise between
the analysts and the synthesists, a man who is able to see the
merits and demands of both positions.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 43
asking himself what o denotes (i.e., what an infinitesimal is).
Indeed, in the light of his De Quadratura with its instrumentalist
attitude toward infinitesimals, the question would seem to be
meaningless for him. In pursuing his own brand of foundational work
Newton shows instead the respect for geometry which had
traditionally been common but which in his lifetime was beginning
to be questioned.
The extension of the analytic art was necessarily Newton's first
concern. Once the method had been elegantly formulated, Newton
turned to the problem of providing rigorous proof. The need for
intermediate justification emerges from an understanding of the
temper of the times: the Descartes/Fermat controversy is
symptomatic of the fact that jealous rivals had to be convinced.
Newton was not required to invent this temporary substitute for
himself, however, for there was already a flourishing tradition of
justifying results using infinitesimals. By such means certain
algebraic manipula- tions could be rendered more comprehensible and
perhaps a little less implausible. The status of such justification
is that of a warranty offered by a company we do not quite trust-it
provides an extra safeguard against trouble, but it fails to give
complete assurance.
At an early stage of his career Newton gave several hints that
he was not satisfied to let justification rest with the method of
infinitesimals. It was obviously natural for him to ignore these
worries until analysis had reached its goal. Unlike Barrow, Newton
was able to see what could and should be done with analysis; for
him the synthetic pull was not as urgent. That pull did not drag
him into the puristic complexities of Barrow's work, but, as we
shall see, neither did it leave him unmoved, even while analysis
and the development of algorithms were his main concerns.
To show the way in which the method of infinitesimal
justification falls short of the geometric paradigm of proof, and
thus how our question 2 (How are the methods to be justified?)
arises, we must examine a case in which Newton used this method.
The following example, taken from the "Method of Fluxions," was
intended to vindicate Newton's algorithm for finding the relation
between velocities from the relation be- tween distances by
deriving the result yielded by the algorithm in a particular
case.34
Now since the moments, as xo and yo are the indefinitely little
accessions of the flowing quantities x and y, by which those
quantities are increased through the several indefinitely small
intervals of time; it follows that those quantities x and y after
any indefinitely small interval of time become x +ko and y + yo, as
between x and y; so that x +Jo and y + 'o may be substituted in the
same equation for those quantities, instead of x and y.
Therefore let any equation x3 -ax2 + axy - y5 zr0 be given,35
and substitute x +*o for x and y +y'o for y, and there will
arise
X3 + 3Rox2 + 3R2oox +303 - ax2 - 2a*ox - a*2oo + axy + axoy +
ayox + ayxoo -y3 - 3yoy2 - 320ooy -303 ?O
34 As also in "The General Problem of Tangents and Curvature
Resolved for Algebraic Curves," MPN, pp. 272-276.
85 Presumably the fifth power was mistakenly written instead of
the cube in this equation.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
44 PHILIP KITCHER
Now by supposition x3 - ax2 + axy - y3=0; which therefore being
expung'd, and the remaining terms divided by o, there will
remain
3XX2 + 3k2ox ?x3oo - 2axk -a*2o + axy + aSx + axy'o - 3~'y2 -
3~2oy -3oo =00
But whereas o is supposed to be indefinitely little, that it may
represent the moments of quantities, consequently the terms that
are multiplied by it will be nothing in respect of the rest;
therefore I reject them, and there remains
3xx2 - 2akx + axy + ayx - 3yy2 =0
as above in Example I.36
Here it may be observed, that the terms which are not multiplied
by o will always vanish; as also those terms which are multiplied
by more than one dimensions of o ;37 and that the rest of the terms
being divided by o, will always have the form that they ought to
have by the foregoing rule.38
This method of justification clearly thrives on an algebraic
assumption and an alge- braic manipulation, both of which are
dubious and are defended only through the in- vocation of
infinitesimals. The assumption is contained in the statement that x
+?xo and y + 'o will stand in the same relation as x and y. Its
plausibility derives from the idea that if a particle moves along
the curve, we may regard its velocity as remaining constant through
an infinitesimal interval of time of length o. At the beginning of
the interval the particle is at a point (x,y) of the curve. At the
end of the interval it has moved along the curve to the point (x
+xo, y + ?o) (by the assumption that the velocity remains constant
throughout the interval). Thus (x +xo, y+ jo) is a point of the
curve. Yet strictly the constancy assumption holds only if o = 0.
Otherwise there will be an error involved. To point out that this
error can be made insignificant is to forsake the method of
infinitesimals for something like the method of first and last
ratios.39 At this stage of his career, as the passage shows, Newton
was still working with the presup- positions of the analytic use of
infinitesimals; later he was to declare that errors are not to be
neglected in mathematics.40 For purposes of algorithmic vindication
in this paper he not only neglected them but explicitly "rejected"
them. As well as tolerating the inexactitude of the constancy
assumption, Newton summarily dropped all terms still containing o
once he had allowed himself to divide through. Yet o cannot be
treated as zero, for that would render the division illegitimate.
But if o is allowed to differ from zero, it may be questioned
whether the sum of the supplementary terms is really in-
significant. The appeal to infinitesimals glosses over the
algebraic difficulties and by sketching a geometric picture of what
is going on-the minute details being left shadowy-helps to block
the charge that the algebra is absurd.
The criticism of the last paragraph is not however answered, and
it was brought forward forcefully by Berkeley and later critics of
Newton.4' Newton himself was not
36 This is the result given by Newton's algo- rithm; see MWN, p.
50.
37 Perhaps Newton realized that the fiuxions will always be of
the same power as the powers of o, and thus that the resulting
equation will be of the first degree in * and y. This ensures that
the important ratio y': x can always be found.
38 MWN, p. 52.
39This advance consists in treating the in. finitesimal o as a
variable.
40 In the 1693 "Treatise on Quadrature," MWN, p. 141.
41 The Analyst, in The Works of George Berkeley, ed. A. A. Luce
and T. E. Jessop, Vol. IV (London:Nelson, 1950).
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITITLENESSE 45
blind to defects in the method which he used. In the remainder
of this section we shall see how his toleration of the method of
infinitesimals could coexist with a realization of its
limitations.
The opening remarks of the "Method of Fluxions" make it clear
that Newton was aware of the distinction between what was
permissible in the context of discovery and what was required for
proof. He also showed that he regarded his fluxional calculus to
belong to the former context, to analytic rather than to synthetic
mathematics. There is an implicit value judgment in the way in
which he drew the distinction:
Having observ'd that most of our modern Geometricians neglecting
the synthetical method of the Ancients, having applied themselves
chiefly to the analytical Art, and by the help of it have overcome
so many and so great Difficulties, that all the Speculations of
Geometry seem to be exhausted, except the Quadrature of Curves, and
some other things of a like Nature which are not yet brought to
Perfection; To this end I thought it not amiss, for the sake of
young students in this Science, to draw up the following Treatise;
wherin I have endeavoured to enlarge the Boundaries of Analyticks,
and to make some Improvements in the Doctrine of Curve Lines.42
Like Barrow, Newton recognized the ancestry of the problems of
the calculus and noted also the distinction between two ways of
approaching them. Here he distin- guished his aim: he proposed to
present an extension of the new analytic methods.43 With his
acknowledgement of a difference between his own enterprise and that
of "the Ancients" we begin to see that Newton could work in the
analytic tradition, using its methods and obeying its code, while
allowing the question of how those methods were to be justified to
await completion of his analytic work. His careful description of
his chosen task certainly suggests this.
To find a clear connection between Newton's recognition of the
pedigree of the calculus and the method of infinitesimals we must
go back to his work on the resolu- tion of the general problems of
tangents and curvature for algebraic curves. In a pas- sage
featuring the use of infinitesimals Newton commented, "wch operacon
cannot in this case bee understood to be good unless infinite
littlenesses may bee considered geometrically."44 This illustrates
the synthetic pull on Newton's thought. Recognizing the
shortcomings of his algebra, he saw the need for a clear
interpretation in geometric terms and saw that this was only
partially accomplished by the invocation of the in- finitesimal.
The method relied upon this interpretation-the analytic methods
were dependent on the geometry-and Newton concluded, even in 1665,
that if his justifica- tions were to have the status of full
proofs, "infinite littlenesse" must be construed in geometrical
terms.
For further evidence of his view of "Analyticks" as growing out
of geometry, we may turn to passages in Newton's early work where
he introduced the concept of the indefinite integral and where he
hinted darkly at the difficulty of integrating differen- tial
equations.
That a line may be squared Geometrically tis required yt its
area may be expressed in generall by some equation in wch there is
an unknowne quantity ....45
42 MWN, p. 36. 43 He achieved this, as we have remarked, in
two ways. First, with the method of series he opened up new
problems in each problem class. Secondly, with the method of
fluxions he pre-
sented a convenient method for solving the problems
altogether.
44MPN, p. 282. 4Ibid., p. 344.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
46 PHILIP KITCHER
If an equation expressing ye relation of their motions bee
given, tis more difficult & sometimes Geometrically impossible,
thereby to find ye relation of ye spaces described by these
motions.46
The only way to account for the seemingly irrelevant occurrence
of references to geometry in these passages is to construe them as
further indications that Newton considered analysis to be parasitic
upon geometry for its problems and ultimately to be justified in
geometrical terms.
Newton was further conscious of using the method of
infinitesimals within a definite tradition. The De Analysi provides
an example of this: "Neither am I afraid to speak of Unity in
points, or Lines infinitely small, since Geometers are wont now to
consider Proportions even in such a case, when they make use of the
Methods of Indivisibles."47 The first clause of the sentence
indicates that Newton was aware of criticisms of and perhaps
defects in the method of infinitesimals. He did not undercut
objections by careful statement and counterargument; instead he
dismissed them on pragmatic grounds, since there was a recent
tradition-the fashion of the geometers of the day- which used such
methods, and Newton regarded his work as falling therein. In the De
Analysi, as in the "Method of Fluxions," since the work was
intended for circulation, Newton sagaciously guarded himself by
defining his aims in the sphere of the analytic art. His doubt
about the geometric construal of "infinite littlenesse" was
confided to his notebook. Indeeed, Newton's synthetic worries only
became public once he had appeased them.
As we have seen, Newton was quite conscious of the distinction
between "Analy- ticks" and the old synthetic mathematics. Carefully
distinguishing his own work, he remained aware that the method of
infinitesimals was only appropriate as a temporary backing for
algebraic maneuvers in the context of discovery. In writing about
"infinite littlenesse" he pointed the way in which to answer
criticism and correct defects. Once the analytic program was
accomplished he was to bring the new results within the old
paradigm. Newton's synthetic conscience went on view in the
Principia. It seems that it was developing twenty years
earlier.
V. FIRST AND LAST RATIOS AND RIGOROUS JUSTIFICATION In the
Principia and in the 1693 "Treatise on Quadrature" Newton proposed
to
found the calculus on a firm geometrical basis. He advanced the
method of first and last ratios with this in mind. That method will
be the final object of our study. Conclu- sions drawn earlier can
only be reinforced as we see the culmination of the synthetic
strain in Newton's thought.
Book I ofthePrincipia opens with a discussion of the method of
first and last ratios.48 Lemma I carries the essence of the new
approach: "Quantities and the ratios of quantities, which in any
finite time converge continually to equality, and, before the end
of that time approach nearer to one another by any given difference
become ulti-
46 Ibid., p. 302. 47 Ibid., p. 18. 48 Sir Isaac Newton's
Mathematical Principles
of Natural Philosophy and his System of the World, Florian
Cajori, ed., based on Andrew
Motte's translation of the 3rd Latin ed. of 1726 (Berkeley:
University of California Press, 1934), Vol. I, p. 29. All further
references to the Principia will be to this translation and to this
volume.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 47
mately equal."49 Using more modern terminology, we might
rephrase this as "If as time goes on, X is continually closer to Y,
and X is eventually closer to Y than by any given difference, then,
in the limit, X- Y."'0 In Lemma XII Newton utilized an in- stance
of this (Lemma VI) to prove the important result that the ultimate
ratios of chord, arc, and tangent to one another are ratios of
equality. I1
In the 1693 "Treatise on Quadrature" Newton was more informative
about how this gives a geometric grounding for the method of
fluxions. His justification begins with a new emphasis on the
kinematic conception of curves: "I don't here consider Mathematical
Quantities as composed of Parts extreamly small, but as generated
by a continual motion."52 The new stress is on continuity. Newton
characterized the distinc- tion between the use of discrete
elements and the new kinematic considerations by direct reference
to the theory of fluxions.
Fluxions are very nearly as the Augments of the Fluents,
generated in equal, but in- finitely small parts of Time; and to
speak exactly, are in the Prime Ratio of the nascent Augments: Tis
the same thing if the Fluxions be taken in the ultimate Ratio of
the Evanescent Parts.58
Combining the latter characterization of the fluxion concept
with Lemma I of Book I of the Principia, we obtain a result which
may be used to give a synthetic ground- ing for the method of
fluxions.
If dx(t)and dy(t) are small increments in x and y, and x and y
are the fluxions of x andy, then the difference between the ratios
dy(t): dx(t) (N) and y': x can be made as small aswe like
bychoosing t sufficientlysmall.
In effect it is this result, to which he had supplied a firm
geometrical underpinning, that Newton used to justify his analytic
theorems. His proof of de Sluse's rule runs thus:
Let the Quantity of x flow uniformly, and let the Fluxion of xn
to be found. In the same time that the Quantity x by flowing
becomes x + o, the Quantity of xn will become (x + O)n, that is, by
the method of Infinite Series's x + noxP-' + nn -n/2 ooxP-2 +
&c. and the Augments o and noxn- + nn - n/2 oX0:2 + &c. are
to one another as 1 and nxn-l + nn -n/2 ox"-2 + &c. Now let
those Augments vanish and their ultimate Ratio will be the
Ratio of 1 to nxn-l; therefore the Fluxion of the Quantity x is
to the Fluxion of the Quantity xn as 1 to nx1-1 54
It is now clear how the method of first and last ratios enables
the method of fluxions to be brought into line with the geometrical
paradigm of mathematics. Lemma I has been given proof which accords
with traditional standards. Newton's elucidation of the notion of a
fluxion was motivated by the kinematic intuition that instantaneous
velocities stand to one another in the limiting ratios of small
increments. These were combined to form the result (N). The proof
then proceeded bytreating the "Augment" as a variable dependent
upon time and applying (N). By doing this, Newton showed that it
was possible to give synthetic proofs for analytic theorems. 5
49 Ibid. 60 The logical form of this is:
Lim X(t) = Lim Y(t)if and only if t-* oo t-* oo (t)(t') (t >
t'- I X(t) - Y(t) I< I X(t') Y(t') 1) & (z)(Et')(t)((z> 0
&t> t')-*IX(t)-Y(t)I
-
48 PHILIP KITCHER
He was quite aware of what he had achieved. Immediately after
his proof of de Sluse's rule he pointed out that his method "is
agreeable to the Geometry of the An- cients."56 In the same way, in
the second Scholium to Book I of the Principia he em- phasized the
geometrical essence of the calculus.57 His purpose in the Principia
was however slightly different from that in the later Treatise. In
the earlier work he used the method of first and last ratios to
give a geometrical vindication of the method of infinitesimals.
Newton showed in a series of lemmas that all justifications using
in- finitesimals could be replaced by exact geometrical proofs in
terms of ultimate ratios. The method of ratios acts here as a
meta-level principle which supports the fluxional calculus by
demonstrating that the traditional analytic means of vindication is
a short- hand for a true mathematical proof. The same theme appears
in the Treatise, where we have the direct demonstration as
well.
Two quotations from Newton make clear the motives for his
vindication of analysis. First,
These Lemmas are premised to avoid the tediousness of deducing
involved demonstra- tions ad absurdum, according to the manner of
the ancient geometers. For demonstrations are shorter by the method
of indivisibles; but because the hypothesis of indivisibles seems
somewhat harsh, and therefore that method is reckoned less
geometrical, I chose to reduce the demonstrations of the following
Propositions to the first and last sums and ratios of nascent and
evanescent quantities, that is, to the limits of those sums and
ratios, and so to premise, as short as I could, the demonstrations
of those limits. For hereby the same thing is performed as by the
method of indivisibles; and now those principles being
demonstrated, we may use them with greater safety. Therefore if
hereafter I should happen to consider quantities as made up of
particles, or should use little curved lines for right ones, I
would not be understood to mean indivisibles, but evanescent
divisible quantities; not the sums and ratios of determinate parts,
but always the limits of those sums and ratios; and that the force
of such demonstrations always depends on the method laid down in
the foregoing Lemmas.58
The lemmas of Book I-especially Lemma VII-do indeed vindicate
the analytic approach. They show how the method of infinitesimals
may be used to justify algo- rithms so long as it is recognized as
a substitute for geometrical proof. Infinitesimals flout the
important considerations of continuity; yet, because they are
sanctioned by the method of limits, they may be used instead of the
laborious geometrical proofs. A parallel with modern mathematics
presents itself: in doing formal logic or arithmetic we extend our
vocabulary by means of definitions and our methods of inference by
derived rules. These procedures are theoretically eliminable, but
they are legitimized by our system and are used for
simplification.59 Infinitesimals ultimately find an ana- logous
function in Newton's theory of the calculus.
5s MWN, p. 143. 57 Principia, pp. 37-39. Note esp. p. 39:
"And
since such limits are certain and definite, to determine the
same is a problem strictly geo- metrical." The equation of
certainty with geo- metry gives an interesting further elaboration
of Newton's attitude toward the geometrical paradigm of
mathematics.
58 Ibid., p. 38. 69See, e.g., Alonzo Church, Introduction to
Mathematical Logic (Princeton: Princeton Univ-
ersity Press, 1956), pp. 75-76: "As we have said, these
abbreviations and others to follow are not part of the logistic
system P, but are mere devices for the presentation of it. They are
concessions to the shortness of human life and patience such as in
theory we disdain to make. The reader is asked, whenever we write
an abbreviation of a wif. to pretend that the wif. has been written
in full and to understand us accordingly." The reader is asked to
compare this with the passage quoted from the Scholiunm to Book I
of the Principia.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
-
FLUXIONS, LIMITS, INFINITE LITTLENESSE 49
The Treatise of 1693 reflects similar views: By like ways of
arguing, and by the method of Prime and Ultimate Ratio's, may be
gathered the Fluxions of Lines, whether Right or Crooked in all
Cases whatsoever, as also the Fluxions of Surfaces, Angles and
other quantities. In Finite Quantities so to frame a Calculus, and
thus to investigate the Prime and Ultimate Ratio's of Nascent and
Evanescent Quantities, is agreeable to the Geometry of the
Ancients; and I was willing to show, that in the Method of Fluxions
there's no need of introducing Figures infinitely small into
Geometry. For this analysis may be performed in any Figures
whatsoever, whether finite or infinitely small, so they are
imagined to be similar to the Evanescent Figures; as also in
Figures which may be reckoned as infinitely small, if you do but
pro- ceed cautiously.60
Newton's aim in the later Treatise was thus twofold. The direct
proofs of the filuxional algorithms were to show how "infinite
littlenesse" was to be interpreted geometrically. In doing this,
Newton pointed out the role of infinitesimal justifications.
Although these could not be allowed as full mathematical
justifications-for errors cannot be ignored when we are doing
mathematics-they might be used with confidence and con- venience,
so long as we "do but proceed cautiously."
In this way we see how fiuxions, limits, and "infinite
littlenesse" all have a part to play in Newton's presentation of
the calculus. Perhaps the substance of his achieve- ment appears
more clearly in Newton's use of the three sets of concepts to meet
different and important demands. Once we have unearthed his
problems, Newton's solutions seem more impressive than his
eighteenth-century critics took them to be.
60 MWN, pp. 142-143.
This content downloaded from 66.77.17.54 on Mon, 6 Jan 2014
15:48:36 PMAll use subject to JSTOR Terms and Conditions
Article
Contentsp.33p.34p.35p.36p.37p.38p.39p.40p.41p.42p.43p.44p.45p.46p.47p.48p.49
Issue Table of ContentsIsis, Vol. 64, No. 1 (Mar., 1973), pp.
1-146Front Matter [pp.1-3]Motion in the Chemical Texts of the
Renaissance [pp.5-17]Mersenne and Copernicanism [pp.18-32]Fluxions,
Limits, and Infinite Littlenesse. A Study of Newton's Presentation
of the Calculus [pp.33-49]The American Scientist as Social
Activist: Franz Boas, Burt G. Wilder, and the Cause of Racial
Justice, 1900-1915 [pp.50-66]Medieval Ratio Theory vs Compound
Medicines in the Origins of Bradwardine's Rule
[pp.67-77]"Quinquevalent" Nitrogen and the Structure of Ammonium
Salts: Contributions of Alfred Werner and Others [pp.78-95]loge:
Richard Harrison Shryock, 1895-1972 [pp.96-100]Notes &
CorrespondenceThe First Carbon-14 Dating in China [pp.101-102]The
1972 George Sarton Medal Awarded to Kiyosi Yabuuti [pp.103-104]
News [pp.105-107]Book ReviewsAstrology and Astronomy in the
Ninth Century [pp.108-110]A Thirteenth-Century Textbook of
Ptolemaic Astronomy [pp.110-112]James Bernoulli [pp.112-114]
Philosophy of Scienceuntitled [pp.115-116]
Scientific Institutionsuntitled [p.116]
Social Relations of Scienceuntitled [pp.117-118]
Physical Sciencesuntitled [p.118]untitled [pp.118-119]
Biological Sciencesuntitled [pp.119-120]
Medicineuntitled [pp.120-121]
Classical Antiquityuntitled [pp.121-122]
Islamic Culturesuntitled [pp.122-123]untitled [pp.123-125]
Renaissanceuntitled [pp.125-126]
Seventeenth & Eighteenth Centuriesuntitled
[pp.126-127]untitled [pp.127-128]untitled [pp.128-129]untitled
[pp.129-130]untitled [p.130]untitled [pp.130-131]
Nineteenth & Twentieth Centuriesuntitled [p.132]untitled
[pp.132-133]untitled [pp.133-134]untitled [p.134]untitled
[pp.135-136]untitled [pp.136-137]
Contemporary Sciencesuntitled [pp.137-138]untitled
[p.138]untitled [pp.138-139]
Back Matter [pp.140-146]