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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

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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 reference 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 however 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 infinitesimals 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


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 mathematics. 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 mathematics-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 regarded as a Greek method (assuming the un- known as known and proceeding by deduction to known truths) and the use of the algebra fashioned 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,


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 naturally 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 extreme values. This rendered his whole reformulation 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 considerations 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.


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 algebraic 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, mathematicians 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 existence 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 relation between the problem of constructing a tangent and the problem of computing quadrature. 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.



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 development 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 algorithm 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.


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 conception 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 mathematics. These series had no meaning in traditional 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.



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 quadrature 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 algorithmic 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 algorithms. Indeed, Newton continued to justify his solution of the fundamental problems 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.


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 algorithmic 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.



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 enables 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 solution, and the reversal of the analysis gave synthetic proof of the validity of that solution. Unfortunately, some new seventeeth-century techniques did not have this convenient 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 problems 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 substitute 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 concerned. An exception to this attitude was Isaac Barrow.

28 See, e.g., Descartes, Geometrie, p. 101 (facsim., p. 345).


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 problems 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 geometrical. 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 synthesis 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 relations 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 synthesis. 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 himself 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.



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 manipulations 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 between 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.


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 algebraic manipulation, both of which are dubious and are defended only through the invocation 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 insignificant. 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 algorithm; 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).


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 resolution of the general problems of tangents and curvature for algebraic curves. In a passage 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 infinitesimal. The method relied upon this interpretation-the analytic methods were dependent on the geometry-and Newton concluded, even in 1665, that if his justifications 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.


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.



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 instance 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 distinction 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 infinitely 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 grounding 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 infinitesimals 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 demonstrations 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 algorithms 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 geometrical." The equation of certainty with geometry 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.



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 proceed 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 convenience, 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 achievement 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.


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]

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