Published in D.C. Lindberg and R.S. Westman (eds.), Reappraisals of the Scientific Revolution (Cambridge, 1990), Chap. 12; a shorter, preliminary version appears as "Changing Canons of Mathematical and Physical Intelligibility in the Later Seventeenth Century" in Historia Mathematica 11(1984), 417-23
| It was by means of the geometry of
infinitesimals that M. Varignon reduced varying motions
to the same rule as uniform [motions], and it does not
seem that he could have done so by any other method.. Fontenelle(1) |
During the seventeenth century the world became a machine, and mechanics became the mathematical science of motion. The two developments proceeded in tandem, driven, one generally supposes, by people's desire for greater intelligibility than that offered by the traditional world picture. They understood nature better because it was now mechanical, and mechanics better because it was now mathematical. But what about mathematics itself? It too underwent radical change over the course of the century, not only in its theories and techniques but also in how mathematicians understood their subject. The Ancients would hardly have recognized geometry in the book Descartes published by that name, and what he thought incomprehensible in the 1630s formed part of standard mathematical practice in the 1690s.
Only by taking account of the altered canons of intelligibility in mathematics can one understand how the world machine of the late seventeenth century could be comprehensible and yet transcendental. By the 1690s the mechanics of nature consisted of relations describable only in terms of logarithms and exponential functions, of sines, cosines, and tangents. Planets moved by an area law on curves that were not algebraically quadrable or rectifiable. (2) Pendulums swung in periods only approximately measurable by trigonometric functions, and even then only by assuming a value for . Mathematical and physical space held a host of new curves --caustics, isochrone, brachistochrone, tractrix, catenary, and so on-- many of them expressible only by means of differential equations.
Differential equations themselves belonged to a new conceptual realm. Algebraic in appearance, they symbolized mathematical operations at the upper and lower boundaries of finite quantity. For many thinkers, those boundaries marked the limits of understanding. Moving along them or across them required a point of reference by which the mind could check the course of its operations. In some cases, that point of reference was nothing other than the new mechanics that was leaning on the new mathematics for conceptual support. On the one hand, then, Fontenelle insisted that infinitesimal analysis yielded mechanical insights otherwise unattainable. On the other, proponents of the calculus often rooted its concepts in kinematical experience. In the developmental pattern now familiar as "bootstrapping", mechanics and mathematics helped one another to intelligibility. They did so in part by conspiring to change the canons of intelligibility by which understanding itself was measured. Mathematicians and mechanicians at the end of the century did not understand their subjects better than had their predecessors of the early 1600s; they understood them differently.
Read in light of subsequent developments, a letter from Descartes to Mersenne in 1638 illustrates the nature of the difference. To the consternation of admirers in Paris, Descartes had recently announced his intention to turn from mathematics to other fields of inquiry. He was now responding to the protests conveyed by Mersenne:
M. Desargues obliges me by the concern he is pleased to show for me in owning to be disappointed by my no longer wanting to study mathematics. But I have resolved to give up only abstract geometry, that is, the study of questions that serve only to exercise the mind; and I do so to have that much more leisure to cultivate another sort of mathematics that takes on as questions the explanation of the phenomena of nature. For if he will please consider what have written on salt, on snow, on the rainbow, etc., he will understand that my entire physics is nothing other than mathematics. (3)
In retrospect, given such goals, Descartes left off where he should have begun. For the abstract mathematics in which he had been engaged that spring and to which he would be recalled willy-nilly over the next years is the mathematics to which physics was eventually reduced. But Descartes did not see it that way. Such questions as determining the tangent and area of the cycloid (and of other "special" curves) or of finding a curve given a defining property of its tangent seemed to him merely curiosités, especially since in many cases neither the curves nor the methods to handle them fitted his notion of what was mathematical.
A passing remark in Christiaan Huygens' Horologium oscillatorium of 1673 points up the irony of Descartes' posture. Parts II and III of that work, one of the premier examples of the application of mathematics to "the explanation of natural phenomena", deal primarily with the cycloid, first as the trajectory of a falling body, then as a curve which is its own evolute. After pointing to the curve's rectification, which follows as a corollary to the latter property, Huygens digressed to acknowledge Christopher Wren as the first to determine the arc-length of a cycloid, and he looked back to summarize the work of Blaise Pascal, G.P. de Roberval, John Wallis, and Antoine de Laloubère in analyzing the curve's various properties. He followed Pascal in crediting Marin Mersenne with having been the first to call attention to the curve "in the nature of things". (4) "We have recounted the foregoing", he concluded, "only because it was not seemly to pass by in silence the outstanding findings by which it has come about that, of all curves, none is now known better or more fully than is the cycloid." (5)
In fact, twenty years later it would be even better known, as the Bernoullis found in it several unsuspected physical properties and used it as the basis for new techniques of the calculus. For the moment, however, Huygens' statement in itself is what is important. It is a casual remark, obviously made with no expectation of any reader's demurrer. Hence it captures a consensus. By the 1670s (actually, by at least a decade earlier), the curve that Descartes thought a curiosité and barred from the precincts of mathematics had become for Huygens and his contemporaries the best known curve of all. The nature of the curve had not changed. Huygens used the same definition as had Descartes. Rather, the criteria of understanding had changed; the precincts had been extended. They would be extended even farther over the next several decades.
What follows is an essay at charting the course of that change and that extension. The cycloid will serve off and on as marker along that course, which begins in the finite realm of the new algebra of the early seventeenth century.
Mathematical for Descartes was what could be grasped in a single intuition or could be reduced in clearly understood steps to such immediate apprehension. In geometry the straight line was immediately intelligible; in arithmetic, the unit and the combinatory acts of addition and subtraction. From these one derived the notion of ratio (or rapport) and equality of ratio (or proportion), and thence the concepts of multiplication and division, raising to a power and finding mean proportionals. By building on these elements, he argued at the beginning of his Géométrie, one could give clear meaning to all finite algebraic expressions and clearly relate those expressions to plane curves, thus supporting one's intuition of the simple figures and providing a vehicle for understanding the complicated ones. (6) pp For Descartes, then, mathematical intelligibility came to rest in closed algebraic expressions. What they did not suffice to express did not participate in the clarity of mathematical knowledge and hence was not mathematics. So it was that, even though Descartes easily determined the area of the cycloid by two different methods --the Archimedean method of exhaustion and a method akin to Cavalieri's-- and went on to find the tangent to the curve by a technique akin to determining its center and radius of curvature at the given point, he could discern nothing of mathematical import in such exercises. He could not express his techniques algebraically. Why people like Roberval, claiming to be interested in mathematics, made such a fuss over such problems was something Descartes could not understand. (7)
To a large extent, Roberval shared Descartes' view of what was properly mathematical, if not of what was mathematically interesting. He admired Fermat's solutions to the problems in question particularly because Fermat used an algebraic method of tangents. Indeed, he saw in Fermat's method a means of remedying what Descartes' method lacked. (8) In a letter to Torricelli written sometime after 1645, he contrasted his method of infinites to Fermat's use of analysis in finding centers of gravity, where by "analysis" he meant "algebra". Fermat's approach seemed to Roberval "most abstruse, most subtle, and most elegant"; his own, perhaps "simpler and more universal". Yet, the latter qualities did not outweigh the mixture of numbers with geometrical magnitudes, a sign that something was amiss. (9)
The standard was clear. It shines forth in James Gregory's Vera quadratura hyperbolae et circuli of 1667 and even in Christiaan Huygens' strenuous criticism of that work. A "true quadrature", i.e. a truly mathematical quadrature, is an analytic quadrature. A problem is analytic if it can be expressed and investigated algebraically, where "algebra" now denotes the theory of equations. Given the foundations of intelligibility in the basic operations of algebra, talking about curves and what could or could not know about them came down to talking about equations and how they could be transformed.
Built into that view shared by the "analysts" were three ambiguous elements that opened the way to small, yet profound shifts in the algebraic grounds of intelligibility. First, Descartes' argument in the Géométrie that polynomial expressions constitute relations (rapports) compounded from basic combinatory operations on simple magnitudes encouraged mathematicians to view the expressions as magnitudes on a conceptual par with their constituent terms and factors. It was here in particular that the recent solutions of the cubic and quartic equations had their theoretical effect. For, like Viète before him, Descartes could show that any cubic or quartic could be reduced to the canonical form for which the solution was known and that the solution itself consisted of a reduction of the equation to a "pure power" of the form un = M, the structure of which made u the first of n-1 mean proportionals between 1 and M. Hence, Descartes and his successors maintained, polynomials of whatever degree can be understood as compound proportions, ultimately resolvable by algebraic means to the simple, directly intuitible rapport that is ratio.
Equations of the fifth degree and higher resisted such a general reduction to the pure form. By the 1670s the question of whether it was in fact possible seems to have grown in urgency. (10) The range of opinions --and in the absence of a proof or a disproof there could be only opinions-- illustrates both the strength of the underlying principle and the latitude it allowed. Cartesians argued, to take a phrase of the time, that the fault lay in the artisan rather than the art. Others were readier to accept that the rules of the art might change at the fifth degree. They had come to find algebraic expressions intelligible in themselves, or at least no less intelligible for not being explicitly resolvable into simpler forms. Equations expressed relations. "A quantitative relation," wrote Leibniz, (11) "is a way of finding one quantity by means of another", and, as will become clear below, an equation of whatever degree constituted a modus inveniendi. He said this in Latin, using relatio. When he talked this way in French, relatio became rapport. (12)
Second, Descartes' focus on algebraic expressions as composite magnitudes directed the attention of algebra away from the solution of equations and toward the analysis of their structure and the transformations that lead from one structure to another. That is, algebra consisted essentially of the theory of equations. Though rooted in the notion of equations as relations among magnitudes, it branched out to the study of relations among equations viewed themselves as mathematical entities. So, for example, the structural affinities among the conic sections, and in particular between them and the circle and straight line as degenerate cases, were the essence of Pierre de Fermat's original memoir on analytic geometry. (13) Here, as in Descartes' presentation of the same subject, the identification of "simple" figures with polynomial equations reinforced the tendency to treat the latter as intelligible in themselves.
Emphasis on the structural analysis of equations gave prominence to the question of solvability rather than of solution and to the idea of determining the nature of a problem without necessarily solving it. So, for example, ancient geometers reduced the trisection of an angle to a neusis and thus classified the former problems, even though at the time they could not resolve the neusis itself. (14) If the reduced problem could not be solved, at least it might show what sort of solution or what limits of solution the original problem involved.
Descartes played heavily on that theme in the Géométrie. Even if he could not solve a given polynomial equation, he had the means of knowing how many solutions it had, how many of them were positive, how many real, how many integer, and so on. He could increase or decrease the roots and thereby recast the terms of the equation, and in some cases thereby reduce it to a form for which a canonical solution was known. (15) The theory of equations revealed a great deal about equations as mathematical entities, both in themselves and in relation to other equations. Hence, it conveyed an understanding of them independent of the "simpler" quantities that were their roots. So it was that both Fermat and Descartes determined the tangent to the curve x3 + y3 = pxy (and Fermat knew it had two tangents for certain values of the abscissa) before either of them drew it correctly. (16)
Third, the drive toward the structural analysis of equations led Descartes in one instance, and Descartes and Fermat both in another, to extend the theory of equations by means of what may be termed "counterfactual quantities". That is, to gain the insights afforded by considering every nth-degree expression as the product of n linear binomials formed by the unknown and the successive zeros of the expression, Descartes postulated "imaginary" quantities. Inaccessible by means of operations on real magnitudes, (17) the imaginaries nonetheless combined according to their own peculiar operations to become real factors in the terms of expressions. The subsequent development of complex numbers should not mask their historical origins. Initially, they were called into being as counterfactual quantities solely for the sake of a more general and powerful theory of equations than would have been possible without them. (18)
In a similar vein, Descartes and Fermat both extended the analysis of equations to encompass the determination of extreme values and tangents by introducing a counterfactual difference between the repeated (and hence equal) roots of an equation. Both men treated the roots as distinct for long enough to derive a general relation linking them (and the hypostasized difference between them) to the parameters of the equation. On the grounds that the relation was fully general for all values of the difference, including zero, they then set it to zero, thus returning to the real equality of the repeated roots. (19) Their operationally successful methods had two sustained effects on mathematical thinking over the next several decades. First, by suggesting that the methods of extreme values and of tangents were essentially analytic techniques, they posed the problem of their application to non-analytic (i.e. "mechanical") curves. Most of the methods of tangents published and touted in mid-century aimed at handling the wider range of curves in a uniform manner. Second, Descartes' and Fermat's methods planted the seeds of the notion that infinitesimal analysis was somehow reducible to finite or "ordinary" analysis.
An understanding of the above three elements of algebraic thought in the seventeenth century illuminates the language and the mathematical style of Leibniz' early articles on the calculus. He knew he was moving into new conceptual territory, even if he himself did not always appreciate just how new it was. His strategies for convincing his readers to follow him and for guiding their thinking along the way depend heavily on what he thought to be the shared canons of intelligibility derived from algebraic analysis.
Leibniz waited for some nine years after devising his calculus to publish it. When he did, in a short article in the May 1684 number of the Acta eruditorum, he couched it in terms that emphasized its links to ordinary algebra. The title spoke of "A new method for maxima and minima, and also for tangents, which stops at neither fractions nor irrational quantities, and a singular type of calculus for these." (20) In the language of the time, describing it in those terms made it appear not so much a new method as another method, the most recent in a line stretching back to Fermat's and including those of Descartes, Hudde, Barrow, de Sluse, and Roberval. All alike in supplementing the algebraic analysis of curves in terms of the properties investigated by Apollonius in the Conics, the various techniques differed in the ease with which they accommodated new curves expressed by equations involving algebraic fractions and surds. There, according to Leibniz' title, lay the virtue of his latest version. It was generally applicable, owing to a specially adapted mode of calculation.
Leibniz set forth the rules of the calculus without reference to infinitesimals and without demonstration or derivation, as if continuing an conversation already underway. Given a set of curves, the ordinates y, v, w, etc. of which are referred to a common axis x,
let some straight line taken at will (21) be called dx, and let some straight line that is to dx as v (or w, or y, or z) is to [the respective subtangents] XB (or XC, or XC, or XE) be called dv (or dw, or dy, or dz), or the difference of these v (or w, or y, or z). (22)
Leibniz' later writings suggest that he was concerned here to link differentials to finite quantities via equality of ratios. (23) For any point on the curve y(x), (24) dy:dx = y:subtangent. But he was meeting his concern at the price of circularity, for the subtangent is in fact what a "method of tangents" was supposed to determine; it located the point on the axis from which to draw the tangent to the point on the curve. It was a curious way to introduce the "characteristic triangle" that was to become a linchpin of infinitesimal analysis. (25)
Leibniz scarcely went beyond symbolic formulas in setting forth the rules for finding the differentials of sums, differences, products, quotients, powers, and roots. While he digressed to consider how the signs prefixed to differentials affected their geometrical interpretation, he provided no discussion of the rules themselves. They constituted an "algorithm, as I would say, of this calculus, which I call differential". Having called it that, he immediately linked it to the algebra (now called "common" or "ordinary") familiar to his readers and emphasized its improvement over previous methods. Given a knowledge of the algorithm,
all other differential equations can be found by means of the common calculus; maxima and minima, and also tangents, can be had without the necessity of removing fractions or irrationals or other bonds, (26) which has been necessary in the methods hitherto published.(27)
The demonstration of all this he left to those "versed in these matters", adding as a final hint that the differences of magnitudes are proportional to their momentary increase or decrease. Noted in passing, the point did not appear to be essential.
When Leibniz finally did come to speak of infinitely small quantities, he did so in the context of showing how his new calculus went beyond previous methods of tangents in handling "transcendent (28) lines, which cannot be reduced to algebraic calculation or which are of no determinate degree." Such application of the method called for no particular assumptions other than that
to find the tangent is to draw a straight line which joins two points of the curve which have an infinitely small distance [between them], or [to draw] the extended side of the infinitangular polygon that for us is equivalent to the curve. (29)
Only at this point did Leibniz link his differentials to such infinitely small segments. As his first example of a curve amenable to such treatment, he cited that best known of curves, the cycloid. After several detailed examples striking only for their computational complexity, he concluded by showing that the solution of Debeaune's problem is the logarithmic curve. (30)
By his language and by the structure of his article, Leibniz created the impression that the new calculus grew out of ordinary algebra and that it entered the realm of the infinitely small only when it was applied to "transcendent" curves. Hence, the reader might well gather, the methods of the calculus became "singular" by virtue of the singular curves to which they were applied, rather than being so for inherent reasons. But that too was a longstanding theme of algebraic analysis.
After introducing the calculus in a guise that masked its novelty and scope, Leibniz waited two years before saying more about it publicly. A review of John Craig's De dimensione figurarum (London, 1683) for the Acta eruditorum provided the occasion. Craig had not only taken critical note of Leibniz' new method, but had also given a glimpse of what Newton had achieved in the quadrature of curves. Although Craig spoke well of Leibniz's calculus and showed by his own use of it that he understood its main features, his discussion of the quadrability or non-quadrability of certain curves revealed --to Leibniz at least-- a lack of appreciation of the full range of its power. The title of Leibniz's essay review suggests what he thought Craig had missed: "On a hidden geometry and the analysis of indivisibles and infinites". (31) The calculus was more than just another method of maxima and minima; it was a form of analysis that opened up new realms of mathematics.
Just as the title of the 1684 paper placed it in a familiar category and so shaped readers' expectations of what it contained, so too the new title linked the calculus with the second of the themes discussed in Section I above, here in the guise of the "analytic program", which had made "analysis" synonymous with symbolic algebra and the latter, with the theory of equations. (32) In ways just set forth above, analysis (or "the analytic art") had thereby become a symbolic language for talking about mathematics as well as for doing it. By calling the calculus "analysis", Leibniz embedded it in that language and assigned to it both the heuristic and the theoretical powers of algebra.
By calling the calculus an "analysis of indivisibles and infinites", Leibniz emphasized the special nature of the methods that were necessary to elicit from equations properties and relationships that ordinary algebra could not reach, for example the defining properties of the tangent or normal of a curve. Neither Fermat nor Descartes had fully realized that their methods entailed the existence of quantities different from those symbolized by the variables and parameters of the curve. That in fact both the methods and the quantities extended beyond the algebraic realm became evident only as mathematicians tried to apply algebraic analysis to curves that could not be expressed by ordinary equations. The various methods of quadrature introduced around the middle of the century sharpened that perception, as they showed that even areas bounded by algebraic curves cannot always be measured by finite algebraic means. (33) Extending the powers of algebraic analysis to problems of that sort meant moving into new domains of quantity and of relations among quantities.
It was on this point that Leibniz turned from Craig's work to the question of "hidden geometry" and the new analysis. For what differentiated the calculus from other methods of tangents and quadrature was precisely its capacity to operate beyond the realm of ordinary algebra. To make that clear, Leibniz thought it opportune "to lay bare the source of transcendent quantities; why, that is, some problems are not plane or solid or sursolid or of any determinate degree, but rather transcend every algebraic equation." (34) In fact, Leibniz offered not so much an explanation of these quantities as a legitimation of them as proper objects of mathematical study. His argument followed the lines of earlier defenses of new quantities or relationships in mathematics: without them one could not solve problems that were well defined in terms of familiar quantities.
He began with the theory of equations and its capacity for talking about the nature of solutions without expressly finding them. He did not, he noted, need calculus to show that the circle and hyperbola did not have algebraic quadratrices, that is, that the areas of these curves or of any portion of them could not be expressed in closed algebraic form. For, were that possible, the problems of dividing a given angle into any number of equal parts and of finding any number of mean proportionals between two given quantities would both have solutions of a fixed degree. (35) But, for any n, the n-section of an angle and the finding of n mean proportionals are nth and (n + 1)st degree problems respectively. Hence, their general solutions must encompass all degrees; that is, they must be "of indefinite degree and transcend every algebraic equation."
But, he continued, being transcendent does not make the quadratrices of the circle and hyperbola any less real than the curves they square:
Nonetheless, since such problems can in fact be posed in geometry -indeed, should count among the basic problems-- and are determinate, then surely it is necessary that we also accept into geometry those lines by which alone [the problems] can be constructed. And, since they can be drawn exactly and by a continuous motion, as is clear for the cycloid and others like it, they should in fact be considered not mechanical, but geometrical, especially since in their usefulness they leave the lines of common geometry (except for the circle and straight line) many leagues behind and they have properties of great importance that are directly capable of geometrical demonstration.
Therefore, Leibniz concluded with special emphasis, "Descartes was no less in error for excluding them from geometry than were the Ancients, who rejected solid and linear loci as less geometrical." (36)
Coupled with the associations called forth by the article's title, this critical reference to Descartes suggests that transcendent geometry was mathematics as Leibniz would have had it, not as it was for most mathematicians at the time. Leibniz was anticipating criticism of what he knew were violations of the traditional canons of intelligibility and the accepted scope of mathematics. By the standards of most of his readers, curves such as the quadratrix of the circle or of an hyperbola, which cannot be constructed by finite geometrical or algebraic means, (37) were mathematically unintelligible and hence did not constitute solutions of problems, however clearly the problems may be stated.
To recast those standards, Leibniz first shifted attention from the quadratrices to the cycloid and curves like it, which result from the clearly intuited motion of familiar lines and curves --in the case of the cycloid, the rolling of a circle along a straight line. Generated in a wholly intelligible way, the curves are nonetheless transcendental. Hence, transcendence in itself does not preclude intelligibility.
But that argument begged the question, at least as Descartes had posed it and those after him had interpreted it. Understanding the imagination's picture of a motion (or motions) generating a curve required a mathematical description of the motion (or motions in relation to one another). Being mechanically intelligible did not make a curve geometrical. Yet, that is what Leibniz wished to assert. So he subtly shifted ground again. Why should such curves be considered geometrical? First, they are heuristically useful. They constitute the solutions to otherwise unsolvable problems. Second, they often have essential properties demonstrable by ordinary geometrical or algebraic means. Those properties form bridges from the geometrical to the transcendental realm.
The first reason rests on a premiss that the curves are acceptable as solutions. But, to be acceptable they must be intelligible and lie within the scope of mathematics. Thus, the appeal to utility brought Leibniz back to his starting point. It could have led him nowhere else. The curves he was talking about simply did not belong to classical mathematics either in its original geometrical form or in the extended algebraic form given it by Descartes. Leibniz could escape the circle only by changing the criteria of acceptability of solutions, that is, by altering the canons of intelligibility and redrawing the boundaries of mathematics. To make the quadratrices or the cycloid legitimate solutions required revising what mathematicians understood by "legitimate", and that in turn required not proof but persuasion. Leibniz' article, along with several to follow, aimed to persuade his fellow mathematicians (especially the Cartesians) to accept a redefinition of their subject. He meant to show that doing so brought rewards worth the price of change and that the new canons of intelligibility did not overly stretch the mathematical imagination.
Once again he played on the theme of structural analysis via the theory of equations. As the factoring of a polynomial equation revealed the number and nature of its real (and imaginary) roots, so transcendent analysis had its own theoretical power.
Furthermore, the method of investigating indefinite quadratures or their impossibility is for me only a special (and indeed simpler) case of a much greater problem, which I call the method of inverse tangents. In it is contained the greatest part of all transcendent geometry, and, if it could always be solved algebraically, all [questions] would be taken as answered. But I see nothing yet that satisfies it. Hence, I shall show how it can be solved no less than quadrature itself. (38)
He offered only a sketch of the method he had in mind; he was moving, he claimed, along well trod algebraic paths:
Since, then, in the past algebraists have taken letters or general numbers to stand for quantities being sought, in such transcendent problems I have taken general or indefinite equations to stand for the curves being sought. (39)
Basically, Leibniz had one indefinite equation in mind, namely the general expression of an algebraic curve in two unknowns: 0 = a + bx + cy + ex2 + fxy + gy2 + hx3 + ix2y + ... , where a,b,c, ... are real numbers.
To follow how Leibniz used such equations to find the quadratrices of curves or to show they were not algebraic, the reader had to be familiar both with the new method of tangents set out in 1684 and with the technique of term-by-term comparison of coefficients, which dated back to the earliest analysts. (40) Here Leibniz thought it enough to say
... with the help of this proposed indefinite equation ... I seek the tangent of the curve and, comparing what I find with the given property of the tangents[!], I find the value of the assumed letters a,b,c, etc. and thus I define the equation of the curve sought. At times some [letters] remain arbitrary, in which case innumerable curves can be found satisfying the problem, which was the reason why many people looking at a problem insufficiently defined a posteriori thought they could not solve it. The same things emerge also by means of series. I have many ways of shortening the calculations, of which elsewhere. If, however, the comparison does not go forward, I pronounce the curve sought not to be algebraic, but transcendent. (41)
Leibniz expected his readers to understand this description without examples. They will help here.
To find the quadratrix of the parabola x2 = py, take its equation as the equation of the tangent of the indefinite curve given above. That is, rewrite the parabola in the form x2 = 2p(dy/dx), or x2dx - 2pdy = 0, where dy and dx are the differentials of the variables of the indefinite equation 0 = bdx + cdy + 2exdx + fxdy + fydx + 2gydy + ... . On a term-by-term comparison with the given tangent, however, that equation reduces to 0 = cdy + 3hx2dx, whence c = -2p, 3h = 1, and all other terms are 0, except for a, which remains indeterminate. Substitution of those values into the original indefinite equation yields 0 = a - 2py + x3/3, the equation (or, rather, family of equations) of the sought quadratrix.
How does substitution fail to "move forward"? Consider the circle x2 + y2 = r2, and recast it in the form x2 + (dy/dx)2 = r2, or x2dx2 + dy2 - r2dx2 = 0. The corresponding indefinite form, obtained by differentiating and squaring, is 4e2x2dx2 + c2dy2 + b2dx2 = 0, whence 4e2 = 1, or e = +1/2; c2 = 1, or c = +1; and b2 = -r2, which produces no real value for b. The absence of the latter means that no algebraic equation exists for the quadratrix of the circle.
After suggesting, again in the sketchiest terms, how the assumption of a third variable v, standing for a transcendent quantity such as an arc or a logarithm enables the method at times to uncover "the special nature of that [quantity]", Leibniz reemphasized his theme of extending the analytic program.
By this method applied to quadratures or to the finding of quadratrices (in which, moreover, the property of the tangents is always given) it is clear how to find whether indefinite quadrature is algebraically impossible but also how, once this impossibility has been recognized, the transcendent quadratrix can be found. That has not yet been handed down, whence it seems to me not vain to have asserted that by this method geometry has been furthered an immense distance beyond the boundaries set by Viete and Descartes. For by this argument a sure and general analysis may be applied to those problems that are of no fixed degree and therefore are not comprehended by algebraic equations. (42)
What Viete and Descartes had achieved through algebra for "ordinary" quantities, Leibniz now offered through the calculus for transcendent quantities: reliable, general techniques of analysis for determining whether a given problem can be solved and, if so, for providing its solution.
He rested the conceptual extension on the same sort of appeal to the symbolic imagination as Descartes had made.
I prefer moreover to set out dx and similar [expressions], rather than letters for them, because this dx is a sort of modification of x itself. Thus with its help, when it is necessary, only the letter x with its powers and differentials enters the calculation, and the transcendent relations between x and some other [quantity] is expressed.
Here Leibniz offered as examples "algebraic"
expressions of the arc a of the versed sine x,
, and of the cycloid, 
, the latter of which "perfectly expresses the
relation between the ordinate y and the abscissa x; from
it all the properties of the cycloid can be demonstrated." (43) Again, he pointed to the broad
implications of his method. It moved "analytic calculus
forward to those curves [lit. lines] that have
hitherto been excluded for no other cause than that they were
believed not to be subject to it."
But where Descartes had combined symbolic quantities according to clearly intuited "arithmetical" operations, (44) Leibniz spoke only of a "sort of modification" (quaedam modificatio) of x, which he did not further specify, as if a reference to "the small ... specimen or corollary ... published in the Acta of October 1684" sufficed. But the 1684 article, though it set out the algorithms of differentiation, did not derive or demonstrate them. Some sort of operation, or sequence of operations, was involved, but it was far from clear just what it consisted of. Leibniz was asking the reader to accept the algorithms as a substitute for that insight.
Differentiation and indefinite equations lay at the opposite extremes of the range of magnitude. Behind the algorithms of the former lurked the infinitesimal, while the number of terms and hence the degree of the equation could grow to infinity. Yet, in Leibniz' hands the two methods complemented one another. Taking algebraic equations as themselves basically intelligible, he moved to widen the concept of the equation by shifting focus from the notion of a compound relation to that of a modus inveniendi, a way of finding (one quantity by means of others), thereby encompassing the new, transcendent relations that were so interesting both mathematically and physically. The two new modes of mathematical expression each reduced non-algebraic relations to quasi-algebraic forms susceptible of the sort of structural analysis that had brought understanding of algebraic relations. What a differential equation did to reveal the properties of the cycloid worked as well for the logarithmic function or for the trigonometric relations, which also responded in revealing ways to analysis via infinite series. (45) In less tractable cases, the mathematician's inability to sum an infinite series or to integrate a differential equation no more detracted from the usefulness of these tools of analysis than his inability to solve generally equations of the fifth degree or higher diminished the power of the theory of equations. Understanding the meaning of an equation did not depend on being able to solve it.
Nonetheless, the new forms of expression, especially those containing the symbols d and , harbored operations and relations that were not readily reducible to intuitible finite forms, and hence Leibniz' argument could at best persuade his readers, not compel their understanding. He went into no further detail in the article, but later on in an unpublished treatise he tried at some length to show that the rapports among differentials can be reduced to those among finite quantities and hence that differential equations are finite equations at heart. (46)
The goal of the effort and the language in which it is couched are more important than its ultimate futility. Accustomed to understanding complex mathematical relations and intricate geometric configurations by means of their equations, Leibniz sought to include new relations and configurations by introducing new sorts of equations into analysis. For him the scope of mathematics was set by the methods of solving ordinary and differential equations, which thereby became themselves the proper and central object of mathematical understanding. In this, followers of Descartes saw a triumph of technique over intelligibility.
Not only mathematics was at stake. As mentioned at the outset, the new curves and techniques of solution Leibniz was bringing under the aegis of analysis pertained largely to physical problems. As he put it in concluding his 1694 article, "Considérations sur la différence qu'il y a entre l'analyse ordinaire et le nouveau calcul des transcendantes", "Finally, our method is properly that part of general mathematics that treats of the infinite, and that is why one has such need for it in applying mathematics to physics, because the character of the infinite Author usually enters into the operations of nature."(47) At heart, nature itself was transcendent in Leibniz' sense, and a mathematics of nature would have to be similarly so. Examples from the work of Leibniz' early mentor, Huygens, from that of his proponent, Johann Bernoulli, and finally from his own writings show what was involved mathematically and where it could lead.
Huygens, too, knew how to use infinitesimals and counterfactual configurations as tools of analysis, albeit an analysis of geometrical figures. His work on the pendulum over the period 1659-64 documents his increasing confidence and sophistication in that mode of mathematical inquiry, especially where he could rely on physical intuition to hold him on course. In deriving the period of a simple pendulum in 1659, Huygens began with a geometric picture of the bob's circular arc over a short swing and then superimposed on it first the parabolic gradient of its speed as a function of distance fallen and then an auxiliary curve, the quadrature of which would yield him his answer. (48) But to carry out the quadrature --no, carry out the transformation of the curve's equations into a form Huygens could square-- he had to replace the bob's trajectory by a parabolic arc quite close to it. (49) Asking himself then for what trajectory the substitution would be exact rather than approximate, he recognized a recently derived property of the cycloid. (50) In fact, with the cycloid in place he no longer needed to assume that the arc be small. It could be of any size. The cycloid is the tautochrone. Since it is also its own evolute, cycloidal leaves constrain a pendulum to follow a tautochronic path.
Of the accuracy of his results Huygens had no doubt. Of the means of persuading others he was less sure. Helped by mechanical intuition, he had gradually developed the skill of jumping back and forth in his mind between a finite configuration and its infinitesimal reduction while keeping track of what was exact and what was approximate. But how to describe his analyses so as to jump without leaving his reader behind --that is, how to engage the reader in shared intuitions-- evidently weighed on his mind as he thought about publishing his work on the pendulum clock. A worknote for Part II of the Horologium oscillatorium reveals both the sorts of leaps he himself was making and the difficulty he envisaged in sharing them.
"In treating motion along the cycloid," he wrote in an intended prologue to his study of that topic, (51)
I shall consider ... that curve as if made up of an infinite multitude of tangents. Further, for the time of accelerated descent along all these tangents I shall consider the sum of the times in which the individual tangents are traversed by uniform motion at the velocity acquired by fall from the start of descent to the point of tangency. That this be made clearer and that it be apparent that the sum of these times does not differ from the time of natural descent along the infinite tangents, let the infinite tangents of which the curve consists be AB, BC, CD, etc.
Huygens took the segment CD as instar omnium and set the upper and lower limits on the time of accelerated motion through it. The time is greater than that of uniform motion at the final speed acquired by acceleration (which, given its initial speed, would be the same as the speed acquired in accelerating along arc AD ), and the time is less than that of uniform motion at the initial speed (acquired by acceleration through AC).
But since I posit the tangent to be infinitely small with respect to the whole AE, therefore the difference of the speed that the body acquires, whether it falls from A to C, or from A to D or in particular from A to the point H where CD is tangent to the curve, is to be taken as nothing.
Hence the longer and shorter times should not be thought to differ, either from each other or from the time of natural descent. Hence the mean time between them, the time of uniform motion at the speed acquied by fall through AH, should not be thought to differ from the time of natural descent. Therefore, Huygens will use that mean time. "Skilled geometers," he concluded, "will easily see that this can be safely done, and they will not, I think, wish me to pursue by a long circuit what would be required for writing out a demonstration more veterum."
It is not clear from this argument, read strictly, what Huygens thought his audience would find missing from his treatment of the cycloid as an infinite concatenation of infinitesimal tangential segments, nor how his prolegomenon would fill the lacuna. It takes no appeal to negligible differences to demonstrate the equality between the total time of accelerated fall over the succession of tangents and the total time of uniform traversals at the respective mean speeds. That relationship is exact whatever the size and number of the segments, as Huygens knew full well from Galileo. By the same token, the argument does not seem to address the real question of why motion along the tangents can be substituted for motion along the curve itself.
On closer consideration, that indeed appears to be the question on Huygens' mind. Despite the drawing that accompanies the text, he was thinking of the infinitesimal segments as coincident with the corresponding arcs of the cycloid; C and D, for example, are the common endpoints of a segment of the cycloid and of a tangent. When distinguished analytically, however, the corresponding endpoints lie on lines drawn parallel to the axis, with the result that the midpoint of the tangent is not the midpoint of the arc. Nor is the speed reached by acceleration over the arc to that point necessarily the mean speed of acceleration over the whole arc. Hence, while the speeds at the endpoints of tangent and arc are respectively equal, the reduction to mean speeds over the concatenated arcs is approximate in all but the limiting case. That is why Huygens had to "close in" on the reduction by means of negligible differences. The mechanical intuition that was guiding the mathematical analysis needed itself the steadying hand of mathematical justification.
As phrased, the worknote was as opaque as the reasoning it was supposed to clarify. The reader who could see what Huygens was getting at did not need the explanation in the first place. Between writing that note and preparing the final draft of the Horologium, (52) Huygens evidently realized this and changed his mind about how much he could rely on the indulgence of his skilled colleagues. In the published treatment, Proposition II.23 considers motion on an infinitesimal arc of a cycloid as if it were occurring along the tangent to that arc at its midpoint, and the summation to a finite arc via the principle just discussed seems straightforward. However, Proposition II.24, in which the summation is carried out, proceeds by the "long circuit" of a double reductio in the style of Archimedes. To make that circuit, Huygens had to change the manner in which the tangential segments were constructed and the motions on them measured.
In the years immediately following the publication of the Horologium oscillatorium, Huygens became Leibniz's mathematical mentor and looked on as Leibniz thought his way through to the calculus. (53) Familiar with Fermat's and Descartes' algebraic techniques for analyzing curves, Huygens could appreciate both the old and the new elements of Leibniz' method. But, as A.R. Hall has pointed out, Huygens did not then or later view the calculus as a wholly new mathematical enterprise. (54) Yet, Huygens kept Leibniz' mathematics at a distance, preferring the geometrical mode at which he was so adept and with which he felt so comfortable. (55)
J.E. Hofmann thought such conservatism warranted explanation:
The reason probably lies in the fact that the new direction did not entirely please him. Leibniz strove for a technique of representation which is simplified and formalized down to the [detail] by means of appropriate symbols, yet which cannot be immediately grasped but must be learned. Whoever can acquire this has an unimaginable advantage over the uninititated, even when he has no particularly deep insights into the connections: the formalism thinks for him. It was precisely this possibility that Huygens saw as undesirable. He was the last and most important representative of the old school, which arrived at their results on the basis of truly ingenious, but particular arguments that stood in isolation. (56)
In the long run, perhaps, the difference between Huygens' geometrical style of infinitesimal analysis and Leibniz' algebraic style did emerge as starkly as Hofmann portrays it here. But over the twenty years between Leibniz' first sketches of the calculus and Huygens' retirement from the mathematical scene, the two styles were hard to distinguish in practice. They supplemented one another. The operations of the calculus lay hold of relations and dimensions inaccessible to the geometry of finite elements, while the latter (especially when representing mechanical situations) gave intuitive significance to those operations, guiding their application to configurations and preventing them from degenerating into a mere Spielerei of symbolic transformations.
Even with the Bernoullis' articulation of Leibniz' scheme over the decade following its publication in 1684, the calculus as a working tool of mathematical and mechanical analysis still had need of "geniale, aber isoliert stehende Einzelüberlegungen". As Johann Bernoulli observed at one point in the Lectures on the Method of Integrals he wrote for the Marquis de l'Hôpital,
Just as in the method of taking integrals definite rules cannot be given by which one might find the integrals of all differentials, but rather only those rules are set forth that are suitable in very many (even in innumerable) cases, so too a general rule could never be given in the method of inverse tangents. So it is that one must form various rules according as the nature of the case demands. One can even say that almost every example has its own rule, the most suitable version of which depends on the sagacity of the person undertaking the solution of the problem. Hence it is both unnecessary and impossible to prescribe definite rules. (57)
Some of the ad hoc rules would, of course, involve algebraic transformations and recombinations of differential expressions. For example, in using second differentials, the analyst faced the sometimes crucial choice of which first differential to treat as constant. (58) But other rules would demand the geometrical facility and insight of which Huygens was a master. At the level at which Bernoulli was working, there was no room for the "unschooled" in either realm.
Indeed, many of the analytical techniques that made the new calculus work in the first place were rooted in Huygens' style of mathematics. They were crucial in guiding the analyst, giving him a sure sense of what could and could not be ignored as negligible, of which variable it was best to treat as independent, of what sort of coordinates to choose, of how to handle limiting and degenerate cases. Leibniz and some of his supporters may have claimed that the calculus was straightforward and mechanical, but those who used it --or tried to use it-- discovered otherwise. For all the formalism, the calculus was a body of techniques not readily reducible to a few general rules. That is what made it often difficult to learn. The formalism did not think for the analyst. Rather, the analyst had to know, largely through experience, where and how to apply the formalism. For that reason, the way in which mathematicians actually used the calculus provides the historian with insight into the concepts and patterns of thought they were expressing in the new language.
Bernoulli's treatment of osculating circles and evolutes shows the sort of informal flexibility and geometrical insight demanded for the application of the calculus. Both the topic and the approach to it stemmed from Huygens, whose initial stimulus had been the physical problem of constraining a pendulum to follow a cycloidal path. (59) (As he admitted in the Horologium oscillatorium, the theory of evolutes presented there went far beyond his immediate needs.) Except for some counterbalancing mechanism that adjusted the actual length of the cord, such as he used later in his conical pendulum, he could think only of using leaves about which the cord would wrap itself. That physical mechanism became the geometrical configuration of a curve which was "unwound" by drawing successive tangents to it and laying off on them segments equal to the arc length of the curve from some fixed origin to the point of tangency. Huygens showed by Archimedean methods that the resulting locus was perpendicular to the tangent generating it at each point and that it was therefore unique for a given origin. Given the peculiar rectification of the cycloid, it followed as an immediate and special result that a cycloid unwinds into another, equal cycloid.
When the Bernoullis took up the question of evolution, they subsumed it under Leibniz' newly introduced notion of the osculating circle (60) and of the radius of curvature, a notion of increasing importance to the analysis of mechanical problems. Yet, Huygens' configuration remained the intuitive touchstone. For, by its very structure it suggested that curves could consist of infinitesimals in two ways. The evolute could be viewed as an aggregate of rectilinear elements, the extensions of which constituted its successive tangents. The curve resulting from evolution (61) could be understood as an aggregate of arcs of circles, the radii of which constituted its successive normals. Since the cycloid is its own evolute, the two modes of infinitesimal analysis are clearly equivalent. Hence, the same curve can consist of infinitesimal elements in two ways.
That the straight and the curved coalesced at the level of the infinitesimal was a premiss of the calculus, indeed its raison d'être. But with the assistance of Huygens' configuration it became a two-edged tool of analysis, for in the Lectiones Bernoulli used it to treat the same infinitesimal element first as a straight line, then as an arc, and again as a line. (62) The sudden shifts of frame could be as unsettling as they were effective. In Figure 3b, let B and O be neighboring points on curve AB, BD and OD the respective normals (which intersect at the center of curvature D), and BM and ON the respective tangents. Draw MPON. Bernoulli points out that, since B and O are infinitesimally close, one may view MN as the infinitesimal increment or differential of the line AM. Although he will use that fact presently, it distracts attention from the next assertion. For, whatever the magnitude of MN, if one first thinks of OB as a straight line, and of ON as its extension (hence of B as lying on it), the proportion MP:NM = OF:OB holds. It follows from the characteristic triangle familiar from the method of tangents (see Figure 3a).
If, by contrast, one thinks of OB as an
arc of the osculating circle centered at D, then tangents ON and
BM become analytically distinct. If, in addition, MP is taken as
the arc of a circle of radius OM, then the proportion MP:OM =
BO:BD holds. But looking at the configuration in this way both
moves B off ON and treats O as the intersection of NO and MB
extended (which is tantamount to making BO at once arc of a
circle and straight segment of a triangle). The first shift would
seem to invalidate the proportion previously derived, while the
second would appear almost self-contradictory. Bernoulli offered
no justification, perhaps expecting his reader to see the
consequence of taking MN as the differential of AM. For then in
Figure 3c BB' is the corresponding differential of FB, which in
itself is the differential of the abscissa. Hence, BB' is a
second difference. It is negligible in a comparison of
first differences and finite quantities.
(63) Similarly, OO' is a second difference and
hence negligible.
Applying the calculus to the determination of the radius of curvature, then, demanded that one recognize in one configuration two others in which differences among elements of apparently the same order are, by the special rules of the calculus that govern how configurations are interpreted, of different orders of magnitude altogether. Nothing in the formalism of the calculus specified either the initial configuration nor its two derivatives. (64) For that one needed precisely the insight Huygens prized.
The process of "bootstrapping" by which mathematics and mechanics assisted and provoked one another to deeper sophistication seems, in fact, to have extended beyond problem-solving to encompass the formation of new concepts, even new physical concepts. Again, Leibniz and Huygens provide revealing examples of mathematical and mechanical thinking embarked on a new course.
In 1675, some years after the initial work on the cycloidal pendulum, Huygens was studying the rectification of the cycloid and recognized that the arc length measured from the vertex of an inverted cycloid was proportional to that segment of the tangent that would correspond to the accelerative force on a body moving along the cycloid at the point of tangency. (65) That is, in an inverted cycloid the distance from the lowest point is proportional to the gravitational force moving a body along it. But, he reasoned, the cycloid is a tautochrone; hence, that relation is itself the tautochronic relation. Moreover, the relation holds for springs. Therefore, springs are tautochrones. Indeed, he could think of a host of mechanisms expressing the relation. His notes of the early 1690s are filled with them. All of them must be tautochrones. As one reads through his studies on this subject, it seems clear that from 1675 the physical pendulum that had once embodied his and his predecessors' understanding of what we now call harmonic motion was replaced, not by another physical instance, but by a mathematical relation. Huygens himself expressed it in words with reference to a diagram. Leibniz' new calculus then provided the symbolic form: ddS + kS = 0. That was the tautochrone; it was an equation, not a device.
Something of the sort may underlie Leibniz' 1689 effort to place the vortex theory on a mathematical footing, "Tentamen de motuum coelestium causis". (66) One reads his definition of "harmonic oscillation" as that in which the linear speed of the circulating body is inversely proportional to its distance from the center of circulation, and one wonders what fluid Leibniz thought he was describing. It may be that in his mind at the time the fluid in question was precisely the fluid he was describing. (67)In the order of things, first the tautochrone, then the tautochronic mechanism; first harmonic circulation, then a harmonically circulating fluid. Again, the way he is thinking, that is, the grounds of his understanding and what he takes to be the grounds of his readers' understanding, is more important historically than whether he was correct in thinking that way.
These examples from the work of Leibniz can be multiplied many times over by means of material from the writings of the Bernoullis, of Varignon, and of similar Continental proponents of the new calculus and its application to mechanics. As important, perhaps, are the examples from the writings of contemporaries, especially Cartesians, whose criticisms of the calculus seem strangely off the mark until one realizes what lies behind them, namely an inability to "see" what is going on or an unwillingness to accept heuristic success as a substitute from clear understanding. Discourse among the "new" analysts rested on new canons of mathematical and physical intelligibility. That people of intellectual standing at the time had difficulty accepting them shows how new they were.