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Transcendental curves by the reduction of quadratures
CHAPTER 7
Transcendental curves by the reduction of quadratures Die nöthige canones außzucalculiren. —Leibniz (1693k)
7.1. Introduction In the previous two chapters we have studied two methods of representation for transcendental curves, corresponding to the two great desiderata of having Euclid-style geometric constructions and Descartes-style analytic equations. We shall now study an approach to the representation of transcendental curves that starts at the other end: a “work with what you got” approach that sets aside the pursuit of the loftiest of ideals and instead settles for making at least some concrete progress in the mathematical state of affairs such as it is. This is the problem of the reduction of quadratures. Many problems—including all separable differential equations—amount to quadratures, i.e., the finding of areas or integrals.305 Theoretically these quadratures specify the solution unequivocally, but they were not themselves considered a fully satisfactory R √ form of solution since it is a far stretch to simply assume that quadratures such as 1 − x4 dx can be considered known. In the 1690s it was considered fine to assume quadratures as known when they could be expressed in closed algebraic form, or by exponentials and logarithms (which amount to the quadrature of a hyperbola), or by circle measurements (which in modern terms amount to trigonometric functions). As Leibniz (1691e) puts it: When I reduce a transcendental problem so that it depends on logarithms or arcs of circles, and thus canonical tables, or, what amounts to the same thing, bring it back to the quadrature of the circle and the hyperbola, then I consider it complete.306 But the consensus was that more complicated quadratures ought to be reduced in one way or another to more basic problems. Thus Leibniz (1691i) defends quadratures on grounds Transcendental Curves in the Leibnizian Calculus http://dx.doi.org/10.1016/B978-0-12-813237-1.50007-7 Copyright © 2017 Elsevier Inc. All rights reserved.
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of their usefulness in reducing problems, while admitting that they are not complete solutions in and of themselves: I judge it best (as often as one may do so) to reduce an inverse tangent problem to quadratures. . . . For analysis is of two kinds: one by a leap by which we resolve the proposed problem completely into first principles, the other by degrees, in which we reduce the proposed problem to another easier one. And because the first method often requires exceedingly lengthy calculations, it is not rare to have recourse to the second; for although the former is more perfect and requires no other previous knowledge, the latter is nevertheless favourable because it lessens the work that is already to be used for discovery.307 The insufficiency of mere quadratures is stressed again by Leibniz (1694i): I am of your opinion, Sir, in that you believe that the problem is not yet solved well when one has merely reduced it to some quadrature. . . . However, I agree with Bernoulli that it is always great when a problem is reduced to quadratures. It is in my view a great and necessary route towards its true solution.308 So the reduction of a differential equation to quadratures is a useful first step but not yet a complete solution. The same point is stressed by Leibniz (1690d): When I can reduce these problems to quadratures, I believe then to have overcome the greatest difficulty. However, to perfect this method one must also complete the doctrine of quadratures.309 What, then, remains to be done in order to “complete the doctrine of quadratures”? How should one proceed once the problem has been brought back to quadratures? Leibniz devised two general strategies, one analytical and one geometrical. The analytical strategy consisted in the computational reduction of integral expressions by methods such as partial fraction decomposition. The geometrical strategy consisted in the reduction of areas to lengths, which are in a sense geometrically simpler. In this chapter we shall proceed to study the histories of these two strategies in turn.
7.2. Computational reduction of quadratures When faced with a complicated quadrature (i.e., integral) the obvious thing to do analytically is to try to reduce it to a simpler or more well-known form. This serves two related purposes: firstly, it brings the problem back to simpler ones and thereby minimises the assumptions needed for the solution, and, secondly, it leads to a classification as to what kind of quadrature one is dealing with. That is to say, one does not only want to reduce
Transcendental curves by the reduction of quadratures
R √ a complicated integral to, say, 1/ 1 − x2 dx because it is simple in terms of analytical form, but also because it shows that the problem amounts to measuring the arc length of a circle, thereby giving a qualitative characterisation of the solution. The latter gives, as it were, a calculus-independent meaning to the solution, which was crucial at a time when the calculus was not yet the de facto language of the mathematics of curves, but rather a methodological appendix to geometry and thus under obligation to justify itself in classical geometrical terms. Although this was generally recognised, it is not surprising that Huygens was especially concerned with it; Huygens (1691b) was reluctant to embrace the new calculus wholesale but ready to appreciate it insofar as it could deliver such geometrical insights:
It would be a beautiful thing to have a method for recognising, when the [differential] equation for a curve is given, whether its dimension may be reduced to those of the hyperbola or the circle.310 Indeed, Huygens’s solution of the catenary problem was outshone by Leibniz’s being superior to his in this regard, as Leibniz (1692b) boasts: By the Leibnizian method the problem was reduced to its true type, which is of course the simplest that can be obtained, namely to the quadrature of the hyperbola. The solution of Huygens, on the other hand, although completely true, nevertheless supposes a more complex quadrature, of which he does not give the nature and reduction, so that hence from it the nature and degree of the problem is not established.311 Resolving a problem according to its “true type”—that is to say, without needlessly advanced methods or assumptions (as in Huygens’s solution)—had always been a recognised requirement on sound mathematical method, as we saw in Section 3.3.5 and as Huygens of course agreed. In keeping with this requirement, it was a long-standing research problem for Leibniz, and an essential part of his vision of the calculus, to give “the nature and reduction” of all quadratures. As Leibniz (1690a) says: This manner of calculation [the calculus] also gives me great ease in resolving transcendental problems, such as those where a curve is sought from a given property of the tangents. And I even believe that by this route one will be able to improve analysis by reducing all transcendental problems to quadratures; and these quadratures to certain classes.312 Leibniz (1694a) refers to the same goal:
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For analysis I desire the reduction of all quadratures to certain fixed types, which are irreducible to one another, and I want suitable expressions of their values in this regard.313 In other words, Leibniz saw in his calculus the technical potential to impose a hierarchy on quadratures in a natural way, analogous to how Descartes had classified algebraic curves by their degree. Exponential, logarithmic, and trigonometric expressions are a good start to this end, since they can readily be construed as a natural next level beyond the algebraic. As we have seen in Section 6.3.3, Leibniz seemed at one time hopeful that such expressions might be sufficient. At any rate, he envisioned for a long time that a standardised system for the reduction and classification of quadratures was really just a matter of computational work. Thus Leibniz (1690d) writes hopefully: I believe that we could make an enumeration of fundamental quadratures, to which all others should be reduced; . . . and when one will have reduced the problems to quadratures, and the quadratures to certain chief [types], as I project, this type of analysis will have arrived at its perfection.314 Four years later Leibniz (1694i) has only grown more confident if anything: I have no doubt that we will find one day a method of reducing everything to the simplest quadratures. I even believe I see the method, of which I also have examples, but I am not in a state to work on it.315 Leibniz (1693k) too sees the problem as a rather straightforward one “to compute the necessary canons, for which I have no time.”316 In retrospect it is abundantly clear that Leibniz was overly optimistic in these hopes. The “canons” he envisioned did not materialise, of course, and ten years later he had little progress to show for his programme except for a highly imperfect paper on partial fractions. Partial fractions are indeed a crucial method for the reduction of quadratures to standard ones, and Leibniz makes it clear that the general, foundational problem of the reduction of quadratures to canonical ones is his motivation in studying this method. Thus Leibniz (1702) writes in his partial fractions paper for example that: This leads us to a question of the greatest importance: whether all rational quadratures can be reduced to the quadrature of the hyperbola and the circle . . . I have discovered however that to him who thinks this the abundance of nature will be more tightly contracted than is appropriate.317 This enough, but Leibniz’s justification for it is not. According to Leibniz, R is accurate 4 4 “ dx : (x + a ) can be reduced to neither the circle nor the hyperbola by this analysis of
Transcendental curves by the reduction of quadratures
ours [i.e., partial fractions], but establishes a new kind of its own,”318 whereas in reality R 1/(x4 + a4 ) dx can be expressed in terms of logarithms and arctangents. This mistake notwithstanding, it is illuminating to see how Leibniz wishes to give geometrical meaning to the result: R And I have wished . . . that as dx : (x + a), or the quadrature R of the hyperbola, is known to give logarithms or the division of a ratio, and dx : (xx + aa) the division of an angle, so the sequence further, and it ought R could be continued R to be established to which problem dx : (x4 + a4 ), dx : (x8 + a8 ), etc., correspond.319 So still at this late date, and in the context of this eminently analytical line of research, Leibniz remains adamant that the results be anchored in geometry. Indeed, his desire to concoct geometrical problems to match foundational questions seems to have a direct equivalent in Greek geometry, as discussed in Section 3.2.3. As ever, it is not Leibniz’s way to start with concrete problems and address foundational questions as needed for those problems; rather to him foundational questions always come first, and specific problems are merely ways of instantiating them and making them concrete.
7.3. The rectification of quadratures The analytic-computational simplification and classification of quadratures discussed above is easy to relate to from a modern point of view, but at the time a geometrical and decidedly more idiosyncratic method for reducing quadratures was more prominent, namely that of reducing quadratures to rectifications. That is to say, when encountering an integral R √ such as 1 + x4 dx, which cannot be evaluated in closed algebraic form, the pioneers of the Leibnizian calculus preferred to express it in terms of the arc length of an auxiliary R pcurve instead of leaving it as an area, i.e., in effect, to rewrite the integral in the form 1 + (y0 )2 dx for some algebraic curve y(x) concocted solely for this purpose. They were fully aware that the opposite reduction (expressing an arc length as an integral) is the easy and natural one computationally from the point of view of the integral calculus. Nevertheless they insisted on doing it the other way around, because they thought it made more geometrical sense. As Leibniz (1693a) puts it: I would like completely general and short ways of reducing inverse tangent problems in any case at least to quadratures, and then the quadratures to the extension of curves into lines, since it is more natural to measure areas by lines than the other way around.320 There are numerous such examples of problem first being reduced to integrals and then
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being reexpressed as arc lengths. We saw one example already in Johann Bernoulli’s early lectures on the calculus, where he described the catenary in terms of the rectification of a parabola (Section 4.4.3.3). We shall soon have reason to study another instructive example, but first we must look at the reasons for seeking a solution of this kind on the first place.
7.3.1. Why rectify quadratures? Rectifying a quadrature replaces a very weighty assumption (that certain areas can be found) by a more modest one (that certain lengths can be found), but, to be sure, the latter is still an assumption that is far from trivial. Thus the rectification of quadratures is more of a simplification than a completely satisfactory solution. As Leibniz (1694i) explains: There are several degrees of solutions [to transcendental problems]; the most perfect, without doubt, is that which reduces transcendentals to the area of a circle or hyperbola. In the absence of that I want to be able to describe the transcendental curve by points in imitation of the logarithmica which is described by mean proportionals. And when this is also lacking, I content myself with obtaining my goal by the rectifications of curves. But there are cases so difficult, where all that I can do so far is to give an infinite series.321 But despite the reduction of quadratures to rectifications being some rungs below the most perfect methods, its importance and value is nevertheless attested in both words and deeds by all the major figures involved the early Leibnizian calculus. Indeed, there is perhaps greater universal consent on this issue than on any other scheme for resolving the problem of transcendental curves. For example, Leibniz (1691h) writes to Huygens: I would also like to be able to reduce quadratures to the dimensions of curved lines, which I consider to be simpler. Have you perhaps considered this matter, Sir?322 To which Huygens (1691d) replies: I would also like to be able to reduce the dimensions of unknown spaces to the measurement of some curved line . . . but I think in most cases it will be very difficult.323 Johann Bernoulli (1694a) likewise agrees: I believe you are right to say that it is better to reduce quadratures to rectifications of curves, rather than the other way around.324
Transcendental curves by the reduction of quadratures
But despite this widespread agreement the motivation for reducing quadratures to rectifications is not completely unambiguous. The most common argument is that “the dimension of the line is simpler than that of an area,” as Leibniz repeatedly stressed. Thus Leibniz (1693e) writes: I would much prefer, for example, to reduce the quadratures to the rectification of curves, because the dimension of the line is simpler than that of a space.325 And again Leibniz (1694a): It is better to reduce quadratures to the rectifications of curves than the other way around, as is commonly done. . . . For certainly the dimension of a line is simpler than the dimension of a surface.326 Leibniz (1691c) even traced the pedigree of this principle back to Archimedes’s reduction of the area of a circle to its circumference: I would like to be able to always reduce the dimensions of areas or spaces to the dimensions of lines, since they are simpler. And that is why Archimedes reduced the area of the circle to the circumference, and you [i.e., Huygens], Wallis and Heuraet have reduced the area of the hyperbola to the arc of the parabola. It is easy to reduce arcs to areas, but the converse—that is the task, that is the toil. If you should come to facilitate this research some day, Sir, I would be delighted to benefit thereof.327 But elsewhere Leibniz (1693i) emphasised instead that a rectification “enlightens the mind” more than a quadrature: But among the geometrical constructions I prefer not only those which are the simplest but also those which serve to reduce the problem to another, simpler problem and which contribute to enlighten the mind; for example, I would wish to reduce quadratures or the dimensions of areas to the dimensions of curved lines.328 Then again in other cases rectifications seem to be preferred over quadratures for the sake of greater practical feasibility. Thus Huygens (1694b) writes: It is a strange assumption to take the quadratures of every curve as given, and if the construction of a problem ends with that, apart from the quadrature of the circle and the hyperbola, I would have believed that nothing had been accomplished, since even mechanically one does not know how to carry anything out. It is a bit better to assume that we can measure any curved line, as I see your opinion is also.329
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In light of the diversity of arguments used to justify the rectification of quadratures it is perhaps not surprising that there was, within the unanimous consensus in its favour, room for disagreement as to the specifics of how to carry it out in any given case. One case in particular served to highlight the fault lines hiding beneath the apparent unanimity, thus forcing extramathematical principles to show their true colour. To this illuminating conflict, which pitted brother against brother and philosophy against philosophy, we now turn.
7.3.2. The conflict over the paracentric isochrone
√ The integration of 1/ 1 − x4 was to be a showcase for the rectification of quadratures. This complicated integral cannot be solved by standard means such as logarithmic and trigonometric functions (or in geometric parlance, measurements of hyperbolas and circles), so something innovative is required. As so often, late 17th-century mathematicians did not pose this kind of integral abstractly, but rather came up with a physical equivalent that functioned as a placeholder or excuse to tackle this important and fundamental integral. This was the paracentric isochrone problem, which we shall discuss more fully in Section 8.3. For now we are concerned only with how it pushed the boundaries of the theory of integration. Jacob Bernoulli (1694b) solved the paracentric isochrone problem by rectification, as shown in Figure 7.1. The curve he rectified for this purpose is the elastica, i.e., a shape assumed by a bent elastic beam. We shall postpone a full mathematical discussion of this curve too until later (Section 8.2), but for the purposes of this debate what matters is that the curve is physically simple (and “given by nature”) but analytically complicated (being a transcendental curve that can only be described by a differential equation not explicitly solvable by standard methods). In introducing his solution, Jacob Bernoulli appears quite certain that it will be appreciated. And with good reason: the rectification of quadratures was universally valued, as we have seen, and the use of one mechanically defined curve to construct another also had ample precedent, such as Leibniz’s construction of logarithms by the catenary (Section 6.3.2) and Leibniz’s and Huygens’s use of the tractrix to, e.g., square a hyperbola (Chapter 5). Indeed Jacob Bernoulli (1693b) had noted in another context that a certain quantity “depends on the quadrature of a hyperbola; therefore it is found by means of a logarithm or string.”330 This endorsement of the “string” (i.e., catenary) construction of hyperbolic quadratures suggests that his own mechanical construction is sincere, and not a misguided attempt at promoting his own elastica. Thus, by way of justification of his paracentric isochrone construction, Bernoulli only passingly alludes to the practical feasibility of his solution: For although it is possible to carry out the same by means of the squaring of
Transcendental curves by the reduction of quadratures
Figure 7.1: The paracentric isochrone constructed by rectification of the elastica in Jacob Bernoulli (1694b). The elastica RQA is the shape of an elastic beam attached perpendicularly to the ground at R and weighed down by a weight attached to its other endpoint A. The weight is such that the tangent of the beam at A is horizontal. The construction of the paracentric isochrone goes as follows. Draw the circle iBL with midpoint A and radius AB equal to the horizontal extent of the elastica. Pick any point Q on the elastica, and let E be the point perpendicularly above it on the horizontal diameter of the circle. Find the point g such that Ag = AE 2 /AB. Find the points ε on the circle such that εζ = Ag (the ε in the bottom right quadrant is shown, but one should consider also the ε in the bottom left quadrant). On the radial line Aε, mark off Aα such that > > Aα = AQ2 /AB in the left quadrant or Aα = φRQ2 /AB in the right quadrant, where the arcs are taken along the elastica RA and its mirror image Rφ. Repeat the construction for other choices of Q. The points α are on the paracentric isochrone (with initial velocity given by the vertical fall iA).
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any algebraic area, another method of construction is to be preferred, I judge, since it is generally easier in practice to rectify a curve than to square an area, and especially since nature herself seems to have drawn it [i.e., the elastica].331 Perhaps to his surprise, Bernoulli’s construction was universally condemned. Huygens (1694b), writing to Leibniz, finds it “strange” and would prefer a construction by rectification of an algebraic curve: It seems that you hold for true his construction of your paracentric [isochrone], after having examined, as I believe, the demonstration, as I have not yet done. It’s quite a strange encounter to have there been able to employ his elastic curve; but your construction will assuredly be much better, if you only need to measure a geometric curve, or at least [a curve] for which you know how to find the points.332 Leibniz (1694e) agrees: He makes use of the rectification of a curve which is itself already transcendental, namely his elastica, and thus his construction is transcendental of the second order. In place of which I only make use of the rectification of an ordinary curve for which I give the construction by common geometry. 333 l’Hôpital (1694b) also agrees: Regarding the curve which you call the paracentric isochrone, I am very pleased that one has finally found its solution, but as my remoteness from Paris has prevented me from seeing the Acts of Leipzig, I am not yet able to judge. It seems to me from what you tell me that your own [solution] will be much simpler and more general than that of Mr Bernoulli, since you find that there is an infinity [of solutions] where he only finds one, and since you use the rectification of an algebraic curve while he uses that of a transcendental one.334 The strongest condemnation, however, came from Jacob’s younger brother, Johann Bernoulli (1694b): No one can fail to see that [the paracentric isochrone] can be constructed by quadrature of a curvilinear area [i.e., from the differential equation with separated variables]; but because the squaring of areas is not easy in practice, one attempts to do it by rectification of some other curve; if this curve can be algebraic, he sins against the laws of geometry who has recourse to a mechanical [curve]; especially if this mechanical [curve] itself is no less complicated to describe by the quadrature of areas.335
Transcendental curves by the reduction of quadratures
This attack is issued in a paper where Johann Bernoulli instead constructs the paracentric isochrone by the rectification of an algebraic curve (the “lemniscate”—see Figures 7.2 and 7.3). But before this attack went to print Jacob Bernoulli (1694c) had already arrived at the same rectification himself. However, he did so without altering his extramathematical views. In response to criticisms he instead elaborated on his original justification for his construction: There are three main methods for constructing mechanical or transcendental curves. The first is by areas of curvilinear figures, but it is ill-suited for practice. It is a better [method] to employ a construction by rectification of an algebraic curve; for curves can be more quickly and accurately rectified, using a string or small chain wrapped around them, than areas can be squared. I hold as equally good such constructions as are carried out without rectification and quadrature, by means of a single description of some mechanical curve, whose points, though not the whole curve, can be found geometrically in infinite number and arbitrarily close to each other; such is the usual Logarithmica, and perhaps others of the same type. The best method, however, wherever it is applicable, is that which uses a curve that Nature herself, without artifice, produces with a quick motion, almost instantaneously at the will of the geometer; for the preceding methods require curves whose construction, whether by continuous motion or by the finding of many points, is usually either slow or exceedingly difficult to carry out. Thus constructions of problems that assume the quadrature of a hyperbola or the description of the Logarithmica, other things being equal, I consider to be inferior to those which are carried out using the Catenary, as a suspended chain assumes this shape of its own accord more quickly than you will have moved the first hand for the rest to be described.336 Thus the construction of the paracentric isochrone by the elastica “would without a doubt be the best,” he continues, if the assumption regarding the laws of tension made in the derivation of the elastica was truthful. But “it is safer not to trust” this assumption, and instead “have recourse to the second mode of construction and seek an algebraic curve whose rectification achieves the result.”337 The fact that both Bernoullis found the construction by rectification of an algebraic curve almost immediately following the initial construction using the elastica speaks to the credibility of Jacob Bernoulli’s professed preferences when he first introduced the elastica construction. For had he not truly felt that the rectification of the elastica was preferable to the rectification of an algebraic curve, he would surely have sought—and thus found rather easily, as subsequent history shows—the solution by algebraic curves, rather than allowing
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Figure 7.2: The paracentric isochrone constructed by rectification of the lemniscate in Johann Bernoulli (1694b). We seek the0ͅ paracentric isochrone the (ͅ free fall30ͅ through is the $ͅ circle with #& Lj = ͅ !1ͅABC with Lj initial !$ͅ velocity lͅ & Ljdetermined !.ͅ lͅby¶Lj lͅPA6 Lj= a.0ͅHFPE #6 Lj 2 2 2 radius a and centre A. AMONA is (part of) the lemniscate (x2 + y2 )2 = 2a a point E on the bottom half of the p (y − x ). Pick & #6 the corresponding lemniscatic circle. Determine its vertical positionͅ AG =l y. Mark the distance AP =< < ͅ0 ay+ (ͅy2Eͅ . Determine >ͅ! (ͅ! Lj ͅ > > > arc AM (or AMON when E is in the bottom left quadrant). Extend AE and mark on it theLjpoint B such that AB = AM 2 /2a. Repeat for other choices of E to generate more points B on the paracentric isochrone.
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ͅ $ l his brother the opportunity to immediately undermine Lj his work with what the latter calls a “more excellent” solution. = ͅ ͅ ͅ
ͅ ͅ ͅ ͅ ͅ ͅ ͅ ͅ !ͅ Thus I believe that$we have here a genuine conflict of extramathematical preferences, ͅ $ͅ ͅ ͅͅ$ͅͅ as ͅͅ ͅͅ ͅ$ͅͅ$ͅ opposed to a mere attempt to save face. Whereas some enthusiastic phrase casually 1!7ͅa grain of ͅ ͅ!ͅ >#ͅ ͅ ͅ Eͅ dropped by Leibniz inͅ a personal letter to a friend may have to19ͅͅ be taken with salt, the raging sibling rivalry between the Bernoullis suggests that they would have taken these matters with the utmost seriousness and left no room for error when they put their FGH
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-5$ "5. 0 +5 $5 5-."5&/#-."5 % $0$ $-.(5&.$,5 +%(%$5%$0$ $4 a2 (x2 − y2 ). Figure from Jacob Bernoulli (1695). -)5 %#+,(.- %$ 5 "5 "#* +5 &-"5 %5 &,%5 ."-%5 $$ ,5 &/$,%5 $5 .(05 "# +,5 +.-$-.)5 5 15 $%%5 ),5 ,(, 5 &(%&%(, %$ +5 5 '.5 + ,5 &&! -.(5 +" (. (-+5 % extramathematical preferences on record in these articles.%5 % .$,.)5 $5 '.5 + 5 + $-.(5 -(- 5&(%&%(, %$ *55 % .,"5 For this reason I shall consider this conflict as "# +-5&%(- %$"5 the key to evaluating extramathematical " #.-"05%#+-#-5'."5 %# ,. #5 % '.5" 5+ -5&)%&%(- %# *5 $-(5 motivations for rectifying quadratures. So what does this episode tell us? In part it concerns % 5 ( ,5 &.$-/"5 $5 '."5 +%(%$5 '."5 &&(-5 ,($+ -.)"5 &(5 ,."5
the legitimacy of using physically given curves in mathematics, an issue which we must set aside for our present purposes. But it also casts some light on the motivations for the "%!% $# % " % !!%% % problem of rectification of quadratures. In particular, Jacob Bernoulli’s idea that a rectification is preferable to a quadrature since it can be effected by placing “a string or small chain” along the curve and then pulling it taut has been treated by several scholars as more or less interchangeable with the Leibnizian dimensionality argument.338 However, the quotation from Jacob Bernoulli (1694c) above is, to my knowledge, the first explicit mention of it,339 despite the numerous discussions of the problem of rectification of quadratures predating this paper. And, as we have seen, Jacob Bernoulli stood alone against the rest of the establishment in this conflict. In opposition to this concrete argument rooted in practice we saw Johann Bernoulli argue a more abstract case, namely that using a mechanical curve where an algebraic one will do is to “sin against the laws of geometry.” To be sure, Johann also refers to practical ease as a motivation, but practice plays a different role in his argument. To him, it seems, practical simplicity is merely a suggestive justification for the “laws” of geometry, not an
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ultimate arbiter in and of itself. This point of view is certainly consistent with Leibniz’s views cited above. Leibniz’s appeals to a dimensional hierarchy, though initially suggested by simplicity considerations, seem to go beyond whatever partial justification such considerations can confer upon them and take on an absolute, legislative stature akin to Johann’s “laws.” This is reminiscent of the hierarchy of degrees in Cartesian geometry, or the distinction between “plane,” “solid,” and “linear” problems in ancient Greek geometry. As in these cases, so in ours: simplicity, practical feasibility, or, for that matter, properties of “mind”—a favourite with Descartes as well as Leibniz—are invoked to justify the hierarchy, but once in place it is the hierarchy itself that is used to evaluate mathematics, not the underlying reasons originally used to justify it. In this way I think the conflict over the paracentric isochrone suggests a useful framework for imposing some order on the multitude of arguments thrown about to motivate the problem of rectification of quadratures. This point of view squares well with Leibniz’s reproof of Jacob Bernoulli’s construction by rectification of the elastica as “transcendental of the second order”340 : the construction is judged by its hierarchical classification rather than on the basis of simplicity, the enlightening of minds, or what have you. This also agrees with our argument in Section 3.3.5.2 about the role of a hierarchy of methods more generally. I propose that the need for such a hierarchy of methods was the fundamental force underlying the principled preference for rectification over quadratures. In this way some cohesion emerges in the variety of arguments presented for reducing quadratures to rectifications. In particular, the numerous arguments alluding to simplicity in various forms speak only to what I called pre facto justifiability, which explains to some extent the indefinite nature of these arguments and their weak force in an actual moment of conflict. Thus, as we have seen above, the various arguments raised by Leibniz are readily interpreted as alternately addressing these desiderata, but at the moment of truth, when the elastica conflict cut to the heart of the matter, he phrased his judgment in terms of the hierarchy of methods itself rather than its subsidiary desiderata. Again, this explains also why Jacob Bernoulli’s simplicity arguments were unanimously opposed despite their prima facie similarity to previous arguments by his opponents: he did not recognise the subordinate role of such arguments as addressing pre facto justifiability only. In this way I believe that cohesion and rationale can be brought out in the apparent diversity and disparity of extramathematical arguments regarding the rectification of quadratures by considering them as subsidiary to more fundamental principles, namely the need for a hierarchy of methods being both retroconsistent and justifiable pre facto. Admittedly, the precise foundational status of the rectification of quadratures remained somewhat elusive. They were certainly foundational in the general sense of pertaining to underlying principles, as they did not concern specific results or problems but rather ad-
Transcendental curves by the reduction of quadratures
dressed the underpinnings of all work on transcendental curves. It is debatable to what extent they were also foundational in the stricter sense of pertaining to the certainty of mathematical knowledge and the delineation of which objects and methods are acceptable in mathematics. I believe our protagonists deliberately left this question open, and that they did so with good reason. On the one hand, to rectify quadratures is to build up the complicated from the simple—arguably the premier safeguard of certainty and exactness in Euclid and Descartes alike, as well as a time-honoured principle of methodological purity. Thus the motivation for elevating the requirement that quadratures be reduced to rectifications to a “law of geometry” akin to the foundational principles of Euclid and Descartes is readily apparent. On the other hand, such a move would have been premature given the lack of general methods for actually performing this reduction in practice and the exceptional state of flux and rapid expansion of the field at this time. Indeed, as we have seen, Leibniz often spoke of the rectification of quadratures as a kind of research programme rather than an absolute law, though at the same time recognising its foundational potential. If this research programme had been conclusive, it may very well have led to definitive proclamations on the foundational status of the rectifications of quadratures, just as Descartes’s foundational programme was the conclusion of his geometrical research rather than its starting point.341 But things did not turn out that way, and the programme never advanced beyond its exploratory, pre-legislative stage.
7.3.3. The motivation for Leibniz’s envelope paper of 1694 The importance of the problem of rectification of quadratures in guiding the direction of mathematical research can be seen in a historical episode where a quirk of history affords an opportunity to study Leibniz’s attitude towards a certain mathematical result just before and just after he realised that it had important implications for the problem of transcendental curves. The result in question is Leibniz’s method of finding envelope curves, i.e., curves determined by their being tangent to a given family of curves; for example, in Figure 7.5, C(C) is the envelope of the family of lines TC, (T )(C), etc. Leibniz’s method may be stated thus in modern terms: to find the envelope of the family of curves f (x, y, α) = 0, combine d f (x, y, α) = 0 so as to eliminate α.342 the two equations f (x, y, α) = 0 and dα Envelopes are central in optics (as they define caustic curves), and quite possibly Leibniz discovered his envelope rule in this context. But with this as its only merit he treated the rule rather disparagingly. Leibniz (1692c) published his rule in an inconspicuous little article of just over three pages, and a good part of it is spent discussing matters of vocabulary that has no direct bearing on the envelope rule. Optical problems are mentioned as motivation, and the rule is then alluded to in abstract terms without a single formula or figure appearing in the entire paper. The article also appeared under the pseudonym “O.V.E.”
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which “are the second letters of my name after the initials G.L.L.” as Leibniz later explained privately.343 Indeed, it seems to me that Leibniz used this pseudonym based on the second letters of his names for works of secondary importance.344 After having realised the importance of the envelope rule for foundational questions, however, Leibniz’s tone is markedly different. Leibniz (1694f) now publishes a five-page paper with detailed calculations and figures devoted entirely to his envelope rule, calling it a “new” application of his calculus “of no small importance for the development of geometry.”345 The bearing of envelope determination of the rectification of quadratures can be seen in Section 5.4, for, in the figures of that section, the desired curve C(C) is the envelope of the family of tangents TC. Indeed, manuscript evidence shows that Leibniz originally drafted the paper of Section 5.4 and his envelope paper as a single treatise,346 and Blåsjö (2012) has shown how each of the five envelope problems mentioned by Leibniz in his paper is relevant to the problem of rectification of quadratures.347 In short, viewing this episode through the lens of foundational concerns regarding the representation of curves explains the timing, the specific content, and the rhetoric of the envelope paper. The same cannot be said for the traditional lens of taking mathematical advances as driven by the need to solve “applied” problems—which is indeed the approach taken by Scarpello and Scimone (2005) in arguing that optical problems regarding caustics were a key motivation for Leibniz’s paper. Yet again, thus, we see the all-eclipsing importance of the problem of transcendental curves to Leibniz.
7.3.4. Technical results on the rectification of quadratures In 1693, Leibniz wrote to Newton asking for “something big,”348 namely a solution to the problem of rectification of quadratures: “I would very much like to see how squarings may be reduced to the rectifications of curves, simpler in all cases than the measurings of surfaces or volumes.”349 In reply, Newton (1693) offered with considerable indifference the solution “which you seem to want.”350 The solution he offered was a construction by envelopes based on his 1666 notes, where this problem occurs inconspicuously as one among many possible permutations of geometrical problems of the form “given this, find that,” without any indication that this problem has a special foundational status. This is yet another clear manifestation of the contrasting views on the foundations of geometry in Newton and Leibniz outlined in Chapter 3. Newton’s solution as sent to Leibniz goes R B as follows (Figure 7.4). A curve y(x) is given and we seek to express its quadrature D y(x) dx in terms of arc lengths. For each point on the x-axis from x = D to x = B we draw a ray whose angle φ with the x-axis is defined by cos φ = y(x) (or, more generally, a cos φ = y(x), where a is a constant large enough so that y(x)/a does not exceed 1; for the moment I limit my discussion to the case
Transcendental curves by the reduction of quadratures
ds
y dx
L(x)
φ dx Figure 7.4: Left: Newton’s figure from his 1693 letter to Leibniz. Right: the same configuration with my notation.
a = 1 for clarity). Next we find the curve FG enveloped R B by all these rays, where F is the point corresponding to x = D and G to x = B. Then D y(x) dx = BG − (GF + FD), so the integral has been expressed in terms of arc lengths, as required.351 But this general result is not very feasible for practical use. Some problems present themselves when one tries to apply this method in specific instances using Leibnizian reasoning. The obvious thing to do from Leibniz’s point of view would be to attempt to find Newton’s envelope curve FG using his envelope method. For this purpose it is necessary to translate Newton’s condition cos φ = y(x) into an algebraic equation for the family of enveloping √ lines. An easy calculation shows that Newton’s condition translates 1−y2
into a slope of − y , so the line in the family having x-intercept α has the equation √ 1−(y(α))2 Y = − y(α) (X − α) (I use capital letters for the variables as y(x) already has a meaning). To go further we must specify the curve whose area is to be rectified. Let us consider the √ 1−α4 2 case y(x) = x . In this case the family of enveloping lines is Y = − α2 (X − α). Leibniz’s method for finding envelopes tells us to eliminate α by combining this equation with ! √ d 1 − α4 −Y − (X − α) 0= dα α2 ! √ √ 2α 1 − α4 2 1 − α4 =− √ (X − α) + − α3 α2 1 − α4 =−
α5 + α − 2X √ , α3 1 − α4
i.e., 2X − α5 − α = 0—a formidable task, and this was the very easy case y(x) = x2 , which
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Figure 7.5: The main figure of Leibniz’s 1694 paper on envelopes (as reproduced in Leibniz (1908)). Below is a complete list of the problems considered by Leibniz in the order in which they appear in the paper, followed by a note on how they can be interpreted in the context of Newton’s rectification method. (1) Given the tangent line CT as a function of its axis intercepts T and ϑ, find the curve C(C). This is equivalent to Newton’s construction with φ = ∠AT ϑ. (2) Given the point ε on the curve ε(ε) as a function of T , find the curve C(C). This variant of the first problem is analogous to uncovering the hidden involute curve not mentioned by Newton in his letter and using this curve in place of the earlier angle condition, as Newton did in his 1666 notes. (3) Given the normal PC as a function of its axis intercepts P and π, find C(C). As Leibniz remarks, this problem is reducible to the above since the normals also define the evolute F(F) (equivalent to the C(C) of the first problem), from which C(C) is given by evolution. This is Newton’s construction with φ = ∠AπP and the evolute–involute pair explicitly indicated. (4) Given the length of the normal PC as a function of the coordinate AP, find the curve C(C). This is the problem of finding the involute curve needed for Newton’s construction for a given f (x) (= PC). (This is the only problem Leibniz actually works out in the paper.) (5) Given the length of the tangent TC as a function of the coordinate AT , find the curve C(C). This is the problem of finding the evolute curve given l(x). This is how Newton in fact finds the evolute in his 1666 notes (using a curvature-style formula for l(x); see below). (As Leibniz notes, this problem is not solvable by his envelope method; instead he refers to his method of construction by tractional motion of Section 5.4.)
is by no means atypically complicated. For almost any other choice of y(x) the situation is just as bad if not worse.352 Having failed to find the envelope, one may gone on to seize on the idea that Newton’s rectification is based on evolutes, as the “unwrapping” of GFD into GCH in Figure 7.4 clearly suggests. Recall from Sections 4.3 and 4.4.4 that evolutes enable us to rectify curves whose involutes are known. While Huygens was able to give a general method for finding
Transcendental curves by the reduction of quadratures
f(x) (x, 0) (-C, 0) (0, 0) s(x)
l(x)
Figure 7.6: The hidden evolute geometry of Newton’s rectification method, not mentioned in the letter to Leibniz but treated in Newton’s 1666 tract on fluxions. (The L(x) from Figure 7.4 would be obtained by extending l(x) to the y-axis in this figure. Thus the angle φ is not the angle this line makes with the y-axis, which corresponds to the x-axis in the previous figure.)
the evolute of a given curve, he had no general method for finding the involute for a given evolute. Thus he was able to rectify a great many curves by starting with various involutes, but the general problem of rectifying any given curve remained unresolved. Newton has an ingenious trick for circumventing this problem, as we know from his more complete account of his construction in his October 1666 tract on fluxions,353 namely to consider the involute in terms of what Whiteside calls an “unusual semi-intrinsic system of coordinates.” It is quite easy to reconstruct this trick from Newton’s letter. One only needs to add the hidden evolute to the diagram to obtain Figure R7.6 and proceed as above. In the exact same way as we found the expression L = s + y dx + c above, we find that RX RX l(X) = s(X) − 0 y dx + C in this diagram. Thus the quadrature 0 y dx can be expressed in terms of arc lengths as C + s(X) − l(X). Applying an analogous argument for f (x) gives f (x) + l(x) + ds = f (x + dx) + l(x + dx), i.e.,R df = ds − dl. Integrating and substituting the X known expression for l, we obtain f (X) = 0 y dx. This uncovers Newton’s fundamental RX idea: to find the involute needed to rectify z(X) = 0 y dx, form the curve with this z as the radial coordinate f (x) in the “semi-intrinsic” coordinate system (x, f (x)) of Figure 7.6. Once this idea is in place everything about cos(φ) can be forgotten and the rectification can be restated purely in terms of evolute and involute.
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To put this idea to use one must first find the hidden curve determined by the z’s. This is another envelope problem: to find the envelope of the families of circles with centres at x and radius z. But contrary to the previous envelope problem this one is solvable by Leibniz’s method. In fact, it is the √ very problem solved in Leibniz’s envelope paper (see Figure 7.5) in the case z(X) = X. Leibniz’s solution is as follows. We seek the curve (x, y) enveloped by a family of circles centred on the x-axis whose radii are z. The √ circle centred at the point (X, 0) has the equation (x − X)2 + y2 = z2 . Since z(X) = X this equation reduces to x2 + y2 + X 2 = 2Xx + X. Following Leibniz’s general envelope method, we differentiate this expression with respect to X to get X = x + 21 . Then we use this equation to eliminate X from the equation we have before differentiating, which leaves us with the equation for the envelope, y2 = x + 14 . √ RX Thus Newton’s method for rectifying z(X) = 0 y dx = X involves finding the evolute of this other parabola, y2 = x + 41 . In this case the evolute is the semicubical parabola 16(x − 14 )3 = 27y2 , as was well known and could easily be determined using one of the several available general methods for finding evolutes. Indeed, the rectification is easily confirmed in this case. See Figure 7.7. However, this method of finding the involute is not very powerful. It fails, prima facie, as soon as y(x) does not have an explicit algebraic antiderivative, since then it is typically not possible to eliminate the parameter as required by Leibniz’s method. But Newton claims in his letter that the evolute can be found “geometrically” (i.e., is an algebraic curve) whenever y(x) is “geometrical.” In fact, Newton is right, as we know from his 1666 notes. There he derives an expression for l(x) equivalent to a radius of curvature calculation, namely l(x) = −
1 − y2 , y0
where y = f 0 is again the integrand of the integral being rectified. From here it follows that the evolute has the parametrisation354 y − y3 (1 − y2 )3/2 x− , − . y0 y0 Thus the evolute is indeed geometrical whenever y is—at least in the sense of having an algebraic parametrisation—so Newton’s claim is correct (as he knew from this derivation).355 Leibniz quite probably did not consider Newton’s construction a fully satisfactory solution to the problem he had in mind. It seems that Leibniz was thinking of the more direct
Transcendental curves by the reduction of quadratures
2
1
1
-1
2
3
4
-1
-2
R √ . The dashed parabola has a Figure 7.7: The evolute form of Newton’s rectification method applied to the quadrature 2dx x R dx √ y-value of f (x) = 2 √ x = x. For each point on the x-axis, a circle is drawn (shown dashed) with this point as its centre and f (x) as its radius. These circles envelope the involute needed, the solid parabola. The evolute of this curve is the semicubical R 0.65 √ . For this case we have parabola also shown solid. Drawn thin is the tangent line needed for the rectification of 0 2dx x R 0.65 dx C + s(0.65) − l(0.65) ≈ 0.5 + 1.60 − 1.29 = 0.81 ≈ 0 2 √ x , verifying that this quadrature has been expressed in terms of these three lengths. Note that the coordinate system of this figure is shifted over 0.25 in the x-direction compared to the coordinate system of Figure 7.6 since in that figure the origin was defined as the starting point of the evolute.
problem: given a quadrature
R
y dx, find a curve g(x) whose arc length equals it, i.e., Z Z p y dx = 1 + (g0 )2 dx.
For when Leibniz next brings up the problem R √ of rectifying quadratures in an article in the Acta the following month, he claims that a4 + x4 dx can be rectified by a hyperbola,356 and this is certainly not the result of using Newton’s construction, which would give a much more complicated curve. Thus when Leibniz says that he wants to “reduce squarings to the rectifications of curves” he means that he wants to transform a quadrature problem into a rectification problem. Newton, on the other hand, takes the problem to be about actually rectifying the quadrature, that is to say, to find a straight line segment with a length equal to the given area. No wonder, then, that Newton’s method gives a more complicated solution than Leibniz desires, since it in effect solves two problems at once: it both reduces the quadrature to
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a rectification problem and solves the rectification problem at the same time. Nevertheless, it seems plausible that Leibniz would have considered Newton’s construction as important, not only because it provides one very general and powerful way of rectifying quadratures (although perhaps too indirectly for Leibniz’s tastes), but also since it in a way solves the problem of rectifying a curve by evolutes when the involute is unknown, which had been a recognised lacuna in the theory of evolutes since its introduction by Huygens. Newton’s method for rectification of quadratures remained unpublished until Hermann (1723), apparently unaware of Newton’s work, published an essentially equivalent method. This caused Johann Bernoulli (1724) to take a renewed interest in the problem. Apparently the three decades that had passed since the problem’s heyday had not faded Bernoulli’s estimation of its importance, for he writes: In the constructions of transcendental problems, it is a matter of great importance to separate the variables of differential equations, so that their effections can be obtained by quadratures of algebraic curves. But it is of greater importance to reduce those quadratures to the lengths of algebraic curves, and so indeed that the parts of the areas to be squared are simply proportional to the arcs of these curves.357 In light of the importance of the problem “it is pleasing what the celebrated Hermann wrote about this matter,”358 according to Bernoulli, but ultimately he is critical of Hermann’s approach: It would be desirable that the approach he uses in this matter were simpler and better suited for use. For to have recourse to the nature of evolutes, and also to require the assistance of the mutual inclinations of lines upon themselves, seems to me an indirect and scarcely natural approach, by which we are led astray among elaborate calculations, as often happens if we mix pure analysis beyond necessity with geometry.359 Privately too Johann Bernoulli (1738) laments in a letter to Euler that Hermann’s quadrature rectification has the character of a chance discovery, though it does what it should. Euler (1738b) agrees. The title of Bernoulli’s paper promises his own “natural and convenient method for reducing transcendental quadratures of any degree to lengths of algebraic curves,” but his method is far from universal and not as free from “elaborate calculations” as his critique R of Hermann might suggest. Faced with the problem of rectifying f (x) dx, Bernoulli
Transcendental curves by the reduction of quadratures
considers the parametric curve
(1 − f 2 )2/3 ( f − f 3 ) − x , f0 f0
.
It follows that Z 0
X
f (x) dx = L(X) +
1 − f (X)2 + C, f 0 (X)
where L(X) is the arc length of the parametric curve from x = 0 to x = X. This equality can be checked by direct differentiation. If f (x) is algebraic then so is the right-hand side of this equation, so the quadrature has been reduced to an algebraic expression involving arc lengths but no areas, as desired. Euler (1738a) later derived essentially the same result in essentially the same way, and may be consulted for a shorter exposition. Bernoulli suggests some minor ways in which this basic idea can be elaborated (slight generalisations and the use of integration by parts) and expresses the hope—which seems highly unrealistic to me—that these little tricks could somehow be made to absorb the auxiliary term: “Perhaps it will Rnot be difficult,” he speculates optimistically, to apply these manipulations to arrive at p dx = L, “that is, so that an algebraic curve can be R found, whose arcs . . . are proportional to the parts of the transcendental area p dx.” But he provides no example or workable strategy for accomplishing such a thing, hiding instead behind the declaration that “it suffices for me to have opened the gate through which the road is spread out to higher things, and to those especially, which until now have been considered among geometers most abstruse.”360 His rather overly optimistic rhetoric notwithstanding, it is clear that Bernoulli’s method falls well short of settling the general problem of the rectification of quadratures. By his own admission, the truly desired form of solution is a direct equality of an area integral with an arc length integral, but to this end he has nothing to offer but a certain form of trial and error, which is hardly an improvement on where the problem had already stood for decades. Despite these lingering dissatisfactions this was perhaps a time for closure. After three decades of efforts by two generations of mathematicians the problem of the rectification of quadratures could not have seemed likely to receive any very satisfying resolution. Johann Bernoulli (1726), for one, seemed willing to draw a line under the matter: When some differential equation of the first degree is reduced to p dx = q dy, where p is given by x, and q by y, . . . the construction of the equation is burdened with no difficulty, certainly if quadratures are allowed; and these very [quadratures] were reduced in general to the extensions [i.e., rectifications] of
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algebraic curves not so long ago. . . . To such a degree that in this area nothing further seems to be desired for greater perfection, except only that, among the infinitely many methods, by which we have shown that the same thing can be furnished, that one is chosen which produces the simplest curve to construct, whose extension one pleases to use for the determination of the quadrature.361 Here, then, the recent general methods for the reduction of quadratures to rectifications are accepted as sufficient. But despite the optimistic language, one cannot quite shake the feeling that the battle was called off more due to exhaustion with a futile fight than outright triumph. Such, indeed, was the fate of many attempts at resolving the problem of transcendental curves.
Transcendental curves by the reduction of quadratures Die nöthige canones außzucalculiren. —Leibniz (1693k)
7.1. Introduction In the previous two chapters we have studied two methods of representation for transcendental curves, corresponding to the two great desiderata of having Euclid-style geometric constructions and Descartes-style analytic equations. We shall now study an approach to the representation of transcendental curves that starts at the other end: a “work with what you got” approach that sets aside the pursuit of the loftiest of ideals and instead settles for making at least some concrete progress in the mathematical state of affairs such as it is. This is the problem of the reduction of quadratures. Many problems—including all separable differential equations—amount to quadratures, i.e., the finding of areas or integrals.305 Theoretically these quadratures specify the solution unequivocally, but they were not themselves considered a fully satisfactory R √ form of solution since it is a far stretch to simply assume that quadratures such as 1 − x4 dx can be considered known. In the 1690s it was considered fine to assume quadratures as known when they could be expressed in closed algebraic form, or by exponentials and logarithms (which amount to the quadrature of a hyperbola), or by circle measurements (which in modern terms amount to trigonometric functions). As Leibniz (1691e) puts it: When I reduce a transcendental problem so that it depends on logarithms or arcs of circles, and thus canonical tables, or, what amounts to the same thing, bring it back to the quadrature of the circle and the hyperbola, then I consider it complete.306 But the consensus was that more complicated quadratures ought to be reduced in one way or another to more basic problems. Thus Leibniz (1691i) defends quadratures on grounds Transcendental Curves in the Leibnizian Calculus http://dx.doi.org/10.1016/B978-0-12-813237-1.50007-7 Copyright © 2017 Elsevier Inc. All rights reserved.
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of their usefulness in reducing problems, while admitting that they are not complete solutions in and of themselves: I judge it best (as often as one may do so) to reduce an inverse tangent problem to quadratures. . . . For analysis is of two kinds: one by a leap by which we resolve the proposed problem completely into first principles, the other by degrees, in which we reduce the proposed problem to another easier one. And because the first method often requires exceedingly lengthy calculations, it is not rare to have recourse to the second; for although the former is more perfect and requires no other previous knowledge, the latter is nevertheless favourable because it lessens the work that is already to be used for discovery.307 The insufficiency of mere quadratures is stressed again by Leibniz (1694i): I am of your opinion, Sir, in that you believe that the problem is not yet solved well when one has merely reduced it to some quadrature. . . . However, I agree with Bernoulli that it is always great when a problem is reduced to quadratures. It is in my view a great and necessary route towards its true solution.308 So the reduction of a differential equation to quadratures is a useful first step but not yet a complete solution. The same point is stressed by Leibniz (1690d): When I can reduce these problems to quadratures, I believe then to have overcome the greatest difficulty. However, to perfect this method one must also complete the doctrine of quadratures.309 What, then, remains to be done in order to “complete the doctrine of quadratures”? How should one proceed once the problem has been brought back to quadratures? Leibniz devised two general strategies, one analytical and one geometrical. The analytical strategy consisted in the computational reduction of integral expressions by methods such as partial fraction decomposition. The geometrical strategy consisted in the reduction of areas to lengths, which are in a sense geometrically simpler. In this chapter we shall proceed to study the histories of these two strategies in turn.
7.2. Computational reduction of quadratures When faced with a complicated quadrature (i.e., integral) the obvious thing to do analytically is to try to reduce it to a simpler or more well-known form. This serves two related purposes: firstly, it brings the problem back to simpler ones and thereby minimises the assumptions needed for the solution, and, secondly, it leads to a classification as to what kind of quadrature one is dealing with. That is to say, one does not only want to reduce
Transcendental curves by the reduction of quadratures
R √ a complicated integral to, say, 1/ 1 − x2 dx because it is simple in terms of analytical form, but also because it shows that the problem amounts to measuring the arc length of a circle, thereby giving a qualitative characterisation of the solution. The latter gives, as it were, a calculus-independent meaning to the solution, which was crucial at a time when the calculus was not yet the de facto language of the mathematics of curves, but rather a methodological appendix to geometry and thus under obligation to justify itself in classical geometrical terms. Although this was generally recognised, it is not surprising that Huygens was especially concerned with it; Huygens (1691b) was reluctant to embrace the new calculus wholesale but ready to appreciate it insofar as it could deliver such geometrical insights:
It would be a beautiful thing to have a method for recognising, when the [differential] equation for a curve is given, whether its dimension may be reduced to those of the hyperbola or the circle.310 Indeed, Huygens’s solution of the catenary problem was outshone by Leibniz’s being superior to his in this regard, as Leibniz (1692b) boasts: By the Leibnizian method the problem was reduced to its true type, which is of course the simplest that can be obtained, namely to the quadrature of the hyperbola. The solution of Huygens, on the other hand, although completely true, nevertheless supposes a more complex quadrature, of which he does not give the nature and reduction, so that hence from it the nature and degree of the problem is not established.311 Resolving a problem according to its “true type”—that is to say, without needlessly advanced methods or assumptions (as in Huygens’s solution)—had always been a recognised requirement on sound mathematical method, as we saw in Section 3.3.5 and as Huygens of course agreed. In keeping with this requirement, it was a long-standing research problem for Leibniz, and an essential part of his vision of the calculus, to give “the nature and reduction” of all quadratures. As Leibniz (1690a) says: This manner of calculation [the calculus] also gives me great ease in resolving transcendental problems, such as those where a curve is sought from a given property of the tangents. And I even believe that by this route one will be able to improve analysis by reducing all transcendental problems to quadratures; and these quadratures to certain classes.312 Leibniz (1694a) refers to the same goal:
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For analysis I desire the reduction of all quadratures to certain fixed types, which are irreducible to one another, and I want suitable expressions of their values in this regard.313 In other words, Leibniz saw in his calculus the technical potential to impose a hierarchy on quadratures in a natural way, analogous to how Descartes had classified algebraic curves by their degree. Exponential, logarithmic, and trigonometric expressions are a good start to this end, since they can readily be construed as a natural next level beyond the algebraic. As we have seen in Section 6.3.3, Leibniz seemed at one time hopeful that such expressions might be sufficient. At any rate, he envisioned for a long time that a standardised system for the reduction and classification of quadratures was really just a matter of computational work. Thus Leibniz (1690d) writes hopefully: I believe that we could make an enumeration of fundamental quadratures, to which all others should be reduced; . . . and when one will have reduced the problems to quadratures, and the quadratures to certain chief [types], as I project, this type of analysis will have arrived at its perfection.314 Four years later Leibniz (1694i) has only grown more confident if anything: I have no doubt that we will find one day a method of reducing everything to the simplest quadratures. I even believe I see the method, of which I also have examples, but I am not in a state to work on it.315 Leibniz (1693k) too sees the problem as a rather straightforward one “to compute the necessary canons, for which I have no time.”316 In retrospect it is abundantly clear that Leibniz was overly optimistic in these hopes. The “canons” he envisioned did not materialise, of course, and ten years later he had little progress to show for his programme except for a highly imperfect paper on partial fractions. Partial fractions are indeed a crucial method for the reduction of quadratures to standard ones, and Leibniz makes it clear that the general, foundational problem of the reduction of quadratures to canonical ones is his motivation in studying this method. Thus Leibniz (1702) writes in his partial fractions paper for example that: This leads us to a question of the greatest importance: whether all rational quadratures can be reduced to the quadrature of the hyperbola and the circle . . . I have discovered however that to him who thinks this the abundance of nature will be more tightly contracted than is appropriate.317 This enough, but Leibniz’s justification for it is not. According to Leibniz, R is accurate 4 4 “ dx : (x + a ) can be reduced to neither the circle nor the hyperbola by this analysis of
Transcendental curves by the reduction of quadratures
ours [i.e., partial fractions], but establishes a new kind of its own,”318 whereas in reality R 1/(x4 + a4 ) dx can be expressed in terms of logarithms and arctangents. This mistake notwithstanding, it is illuminating to see how Leibniz wishes to give geometrical meaning to the result: R And I have wished . . . that as dx : (x + a), or the quadrature R of the hyperbola, is known to give logarithms or the division of a ratio, and dx : (xx + aa) the division of an angle, so the sequence further, and it ought R could be continued R to be established to which problem dx : (x4 + a4 ), dx : (x8 + a8 ), etc., correspond.319 So still at this late date, and in the context of this eminently analytical line of research, Leibniz remains adamant that the results be anchored in geometry. Indeed, his desire to concoct geometrical problems to match foundational questions seems to have a direct equivalent in Greek geometry, as discussed in Section 3.2.3. As ever, it is not Leibniz’s way to start with concrete problems and address foundational questions as needed for those problems; rather to him foundational questions always come first, and specific problems are merely ways of instantiating them and making them concrete.
7.3. The rectification of quadratures The analytic-computational simplification and classification of quadratures discussed above is easy to relate to from a modern point of view, but at the time a geometrical and decidedly more idiosyncratic method for reducing quadratures was more prominent, namely that of reducing quadratures to rectifications. That is to say, when encountering an integral R √ such as 1 + x4 dx, which cannot be evaluated in closed algebraic form, the pioneers of the Leibnizian calculus preferred to express it in terms of the arc length of an auxiliary R pcurve instead of leaving it as an area, i.e., in effect, to rewrite the integral in the form 1 + (y0 )2 dx for some algebraic curve y(x) concocted solely for this purpose. They were fully aware that the opposite reduction (expressing an arc length as an integral) is the easy and natural one computationally from the point of view of the integral calculus. Nevertheless they insisted on doing it the other way around, because they thought it made more geometrical sense. As Leibniz (1693a) puts it: I would like completely general and short ways of reducing inverse tangent problems in any case at least to quadratures, and then the quadratures to the extension of curves into lines, since it is more natural to measure areas by lines than the other way around.320 There are numerous such examples of problem first being reduced to integrals and then
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being reexpressed as arc lengths. We saw one example already in Johann Bernoulli’s early lectures on the calculus, where he described the catenary in terms of the rectification of a parabola (Section 4.4.3.3). We shall soon have reason to study another instructive example, but first we must look at the reasons for seeking a solution of this kind on the first place.
7.3.1. Why rectify quadratures? Rectifying a quadrature replaces a very weighty assumption (that certain areas can be found) by a more modest one (that certain lengths can be found), but, to be sure, the latter is still an assumption that is far from trivial. Thus the rectification of quadratures is more of a simplification than a completely satisfactory solution. As Leibniz (1694i) explains: There are several degrees of solutions [to transcendental problems]; the most perfect, without doubt, is that which reduces transcendentals to the area of a circle or hyperbola. In the absence of that I want to be able to describe the transcendental curve by points in imitation of the logarithmica which is described by mean proportionals. And when this is also lacking, I content myself with obtaining my goal by the rectifications of curves. But there are cases so difficult, where all that I can do so far is to give an infinite series.321 But despite the reduction of quadratures to rectifications being some rungs below the most perfect methods, its importance and value is nevertheless attested in both words and deeds by all the major figures involved the early Leibnizian calculus. Indeed, there is perhaps greater universal consent on this issue than on any other scheme for resolving the problem of transcendental curves. For example, Leibniz (1691h) writes to Huygens: I would also like to be able to reduce quadratures to the dimensions of curved lines, which I consider to be simpler. Have you perhaps considered this matter, Sir?322 To which Huygens (1691d) replies: I would also like to be able to reduce the dimensions of unknown spaces to the measurement of some curved line . . . but I think in most cases it will be very difficult.323 Johann Bernoulli (1694a) likewise agrees: I believe you are right to say that it is better to reduce quadratures to rectifications of curves, rather than the other way around.324
Transcendental curves by the reduction of quadratures
But despite this widespread agreement the motivation for reducing quadratures to rectifications is not completely unambiguous. The most common argument is that “the dimension of the line is simpler than that of an area,” as Leibniz repeatedly stressed. Thus Leibniz (1693e) writes: I would much prefer, for example, to reduce the quadratures to the rectification of curves, because the dimension of the line is simpler than that of a space.325 And again Leibniz (1694a): It is better to reduce quadratures to the rectifications of curves than the other way around, as is commonly done. . . . For certainly the dimension of a line is simpler than the dimension of a surface.326 Leibniz (1691c) even traced the pedigree of this principle back to Archimedes’s reduction of the area of a circle to its circumference: I would like to be able to always reduce the dimensions of areas or spaces to the dimensions of lines, since they are simpler. And that is why Archimedes reduced the area of the circle to the circumference, and you [i.e., Huygens], Wallis and Heuraet have reduced the area of the hyperbola to the arc of the parabola. It is easy to reduce arcs to areas, but the converse—that is the task, that is the toil. If you should come to facilitate this research some day, Sir, I would be delighted to benefit thereof.327 But elsewhere Leibniz (1693i) emphasised instead that a rectification “enlightens the mind” more than a quadrature: But among the geometrical constructions I prefer not only those which are the simplest but also those which serve to reduce the problem to another, simpler problem and which contribute to enlighten the mind; for example, I would wish to reduce quadratures or the dimensions of areas to the dimensions of curved lines.328 Then again in other cases rectifications seem to be preferred over quadratures for the sake of greater practical feasibility. Thus Huygens (1694b) writes: It is a strange assumption to take the quadratures of every curve as given, and if the construction of a problem ends with that, apart from the quadrature of the circle and the hyperbola, I would have believed that nothing had been accomplished, since even mechanically one does not know how to carry anything out. It is a bit better to assume that we can measure any curved line, as I see your opinion is also.329
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In light of the diversity of arguments used to justify the rectification of quadratures it is perhaps not surprising that there was, within the unanimous consensus in its favour, room for disagreement as to the specifics of how to carry it out in any given case. One case in particular served to highlight the fault lines hiding beneath the apparent unanimity, thus forcing extramathematical principles to show their true colour. To this illuminating conflict, which pitted brother against brother and philosophy against philosophy, we now turn.
7.3.2. The conflict over the paracentric isochrone
√ The integration of 1/ 1 − x4 was to be a showcase for the rectification of quadratures. This complicated integral cannot be solved by standard means such as logarithmic and trigonometric functions (or in geometric parlance, measurements of hyperbolas and circles), so something innovative is required. As so often, late 17th-century mathematicians did not pose this kind of integral abstractly, but rather came up with a physical equivalent that functioned as a placeholder or excuse to tackle this important and fundamental integral. This was the paracentric isochrone problem, which we shall discuss more fully in Section 8.3. For now we are concerned only with how it pushed the boundaries of the theory of integration. Jacob Bernoulli (1694b) solved the paracentric isochrone problem by rectification, as shown in Figure 7.1. The curve he rectified for this purpose is the elastica, i.e., a shape assumed by a bent elastic beam. We shall postpone a full mathematical discussion of this curve too until later (Section 8.2), but for the purposes of this debate what matters is that the curve is physically simple (and “given by nature”) but analytically complicated (being a transcendental curve that can only be described by a differential equation not explicitly solvable by standard methods). In introducing his solution, Jacob Bernoulli appears quite certain that it will be appreciated. And with good reason: the rectification of quadratures was universally valued, as we have seen, and the use of one mechanically defined curve to construct another also had ample precedent, such as Leibniz’s construction of logarithms by the catenary (Section 6.3.2) and Leibniz’s and Huygens’s use of the tractrix to, e.g., square a hyperbola (Chapter 5). Indeed Jacob Bernoulli (1693b) had noted in another context that a certain quantity “depends on the quadrature of a hyperbola; therefore it is found by means of a logarithm or string.”330 This endorsement of the “string” (i.e., catenary) construction of hyperbolic quadratures suggests that his own mechanical construction is sincere, and not a misguided attempt at promoting his own elastica. Thus, by way of justification of his paracentric isochrone construction, Bernoulli only passingly alludes to the practical feasibility of his solution: For although it is possible to carry out the same by means of the squaring of
Transcendental curves by the reduction of quadratures
Figure 7.1: The paracentric isochrone constructed by rectification of the elastica in Jacob Bernoulli (1694b). The elastica RQA is the shape of an elastic beam attached perpendicularly to the ground at R and weighed down by a weight attached to its other endpoint A. The weight is such that the tangent of the beam at A is horizontal. The construction of the paracentric isochrone goes as follows. Draw the circle iBL with midpoint A and radius AB equal to the horizontal extent of the elastica. Pick any point Q on the elastica, and let E be the point perpendicularly above it on the horizontal diameter of the circle. Find the point g such that Ag = AE 2 /AB. Find the points ε on the circle such that εζ = Ag (the ε in the bottom right quadrant is shown, but one should consider also the ε in the bottom left quadrant). On the radial line Aε, mark off Aα such that > > Aα = AQ2 /AB in the left quadrant or Aα = φRQ2 /AB in the right quadrant, where the arcs are taken along the elastica RA and its mirror image Rφ. Repeat the construction for other choices of Q. The points α are on the paracentric isochrone (with initial velocity given by the vertical fall iA).
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any algebraic area, another method of construction is to be preferred, I judge, since it is generally easier in practice to rectify a curve than to square an area, and especially since nature herself seems to have drawn it [i.e., the elastica].331 Perhaps to his surprise, Bernoulli’s construction was universally condemned. Huygens (1694b), writing to Leibniz, finds it “strange” and would prefer a construction by rectification of an algebraic curve: It seems that you hold for true his construction of your paracentric [isochrone], after having examined, as I believe, the demonstration, as I have not yet done. It’s quite a strange encounter to have there been able to employ his elastic curve; but your construction will assuredly be much better, if you only need to measure a geometric curve, or at least [a curve] for which you know how to find the points.332 Leibniz (1694e) agrees: He makes use of the rectification of a curve which is itself already transcendental, namely his elastica, and thus his construction is transcendental of the second order. In place of which I only make use of the rectification of an ordinary curve for which I give the construction by common geometry. 333 l’Hôpital (1694b) also agrees: Regarding the curve which you call the paracentric isochrone, I am very pleased that one has finally found its solution, but as my remoteness from Paris has prevented me from seeing the Acts of Leipzig, I am not yet able to judge. It seems to me from what you tell me that your own [solution] will be much simpler and more general than that of Mr Bernoulli, since you find that there is an infinity [of solutions] where he only finds one, and since you use the rectification of an algebraic curve while he uses that of a transcendental one.334 The strongest condemnation, however, came from Jacob’s younger brother, Johann Bernoulli (1694b): No one can fail to see that [the paracentric isochrone] can be constructed by quadrature of a curvilinear area [i.e., from the differential equation with separated variables]; but because the squaring of areas is not easy in practice, one attempts to do it by rectification of some other curve; if this curve can be algebraic, he sins against the laws of geometry who has recourse to a mechanical [curve]; especially if this mechanical [curve] itself is no less complicated to describe by the quadrature of areas.335
Transcendental curves by the reduction of quadratures
This attack is issued in a paper where Johann Bernoulli instead constructs the paracentric isochrone by the rectification of an algebraic curve (the “lemniscate”—see Figures 7.2 and 7.3). But before this attack went to print Jacob Bernoulli (1694c) had already arrived at the same rectification himself. However, he did so without altering his extramathematical views. In response to criticisms he instead elaborated on his original justification for his construction: There are three main methods for constructing mechanical or transcendental curves. The first is by areas of curvilinear figures, but it is ill-suited for practice. It is a better [method] to employ a construction by rectification of an algebraic curve; for curves can be more quickly and accurately rectified, using a string or small chain wrapped around them, than areas can be squared. I hold as equally good such constructions as are carried out without rectification and quadrature, by means of a single description of some mechanical curve, whose points, though not the whole curve, can be found geometrically in infinite number and arbitrarily close to each other; such is the usual Logarithmica, and perhaps others of the same type. The best method, however, wherever it is applicable, is that which uses a curve that Nature herself, without artifice, produces with a quick motion, almost instantaneously at the will of the geometer; for the preceding methods require curves whose construction, whether by continuous motion or by the finding of many points, is usually either slow or exceedingly difficult to carry out. Thus constructions of problems that assume the quadrature of a hyperbola or the description of the Logarithmica, other things being equal, I consider to be inferior to those which are carried out using the Catenary, as a suspended chain assumes this shape of its own accord more quickly than you will have moved the first hand for the rest to be described.336 Thus the construction of the paracentric isochrone by the elastica “would without a doubt be the best,” he continues, if the assumption regarding the laws of tension made in the derivation of the elastica was truthful. But “it is safer not to trust” this assumption, and instead “have recourse to the second mode of construction and seek an algebraic curve whose rectification achieves the result.”337 The fact that both Bernoullis found the construction by rectification of an algebraic curve almost immediately following the initial construction using the elastica speaks to the credibility of Jacob Bernoulli’s professed preferences when he first introduced the elastica construction. For had he not truly felt that the rectification of the elastica was preferable to the rectification of an algebraic curve, he would surely have sought—and thus found rather easily, as subsequent history shows—the solution by algebraic curves, rather than allowing
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Figure 7.2: The paracentric isochrone constructed by rectification of the lemniscate in Johann Bernoulli (1694b). We seek the0ͅ paracentric isochrone the (ͅ free fall30ͅ through is the $ͅ circle with #& Lj = ͅ !1ͅABC with Lj initial !$ͅ velocity lͅ & Ljdetermined !.ͅ lͅby¶Lj lͅPA6 Lj= a.0ͅHFPE #6 Lj 2 2 2 radius a and centre A. AMONA is (part of) the lemniscate (x2 + y2 )2 = 2a a point E on the bottom half of the p (y − x ). Pick & #6 the corresponding lemniscatic circle. Determine its vertical positionͅ AG =l y. Mark the distance AP =< < ͅ0 ay+ (ͅy2Eͅ . Determine >ͅ! (ͅ! Lj ͅ > > > arc AM (or AMON when E is in the bottom left quadrant). Extend AE and mark on it theLjpoint B such that AB = AM 2 /2a. Repeat for other choices of E to generate more points B on the paracentric isochrone.
LôLj#ŊLj Ʊ &&Lj#6ùLj
ͅ $ l his brother the opportunity to immediately undermine Lj his work with what the latter calls a “more excellent” solution. = ͅ ͅ ͅ
ͅ ͅ ͅ ͅ ͅ ͅ ͅ ͅ !ͅ Thus I believe that$we have here a genuine conflict of extramathematical preferences, ͅ $ͅ ͅ ͅͅ$ͅͅ as ͅͅ ͅͅ ͅ$ͅͅ$ͅ opposed to a mere attempt to save face. Whereas some enthusiastic phrase casually 1!7ͅa grain of ͅ ͅ!ͅ >#ͅ ͅ ͅ Eͅ dropped by Leibniz inͅ a personal letter to a friend may have to19ͅͅ be taken with salt, the raging sibling rivalry between the Bernoullis suggests that they would have taken these matters with the utmost seriousness and left no room for error when they put their FGH
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Figure 7.3: Alternative view of the construction of paracentric isochrones by rectification of the lemniscate (x2 + y2 )2 =
-5$ "5. 0 +5 $5 5-."5&/#-."5 % $0$ $-.(5&.$,5 +%(%$5%$0$ $4 a2 (x2 − y2 ). Figure from Jacob Bernoulli (1695). -)5 %#+,(.- %$ 5 "5 "#* +5 &-"5 %5 &,%5 ."-%5 $$ ,5 &/$,%5 $5 .(05 "# +,5 +.-$-.)5 5 15 $%%5 ),5 ,(, 5 &(%&%(, %$ +5 5 '.5 + ,5 &&! -.(5 +" (. (-+5 % extramathematical preferences on record in these articles.%5 % .$,.)5 $5 '.5 + 5 + $-.(5 -(- 5&(%&%(, %$ *55 % .,"5 For this reason I shall consider this conflict as "# +-5&%(- %$"5 the key to evaluating extramathematical " #.-"05%#+-#-5'."5 %# ,. #5 % '.5" 5+ -5&)%&%(- %# *5 $-(5 motivations for rectifying quadratures. So what does this episode tell us? In part it concerns % 5 ( ,5 &.$-/"5 $5 '."5 +%(%$5 '."5 &&(-5 ,($+ -.)"5 &(5 ,."5
the legitimacy of using physically given curves in mathematics, an issue which we must set aside for our present purposes. But it also casts some light on the motivations for the "%!% $# % " % !!%% % problem of rectification of quadratures. In particular, Jacob Bernoulli’s idea that a rectification is preferable to a quadrature since it can be effected by placing “a string or small chain” along the curve and then pulling it taut has been treated by several scholars as more or less interchangeable with the Leibnizian dimensionality argument.338 However, the quotation from Jacob Bernoulli (1694c) above is, to my knowledge, the first explicit mention of it,339 despite the numerous discussions of the problem of rectification of quadratures predating this paper. And, as we have seen, Jacob Bernoulli stood alone against the rest of the establishment in this conflict. In opposition to this concrete argument rooted in practice we saw Johann Bernoulli argue a more abstract case, namely that using a mechanical curve where an algebraic one will do is to “sin against the laws of geometry.” To be sure, Johann also refers to practical ease as a motivation, but practice plays a different role in his argument. To him, it seems, practical simplicity is merely a suggestive justification for the “laws” of geometry, not an
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ultimate arbiter in and of itself. This point of view is certainly consistent with Leibniz’s views cited above. Leibniz’s appeals to a dimensional hierarchy, though initially suggested by simplicity considerations, seem to go beyond whatever partial justification such considerations can confer upon them and take on an absolute, legislative stature akin to Johann’s “laws.” This is reminiscent of the hierarchy of degrees in Cartesian geometry, or the distinction between “plane,” “solid,” and “linear” problems in ancient Greek geometry. As in these cases, so in ours: simplicity, practical feasibility, or, for that matter, properties of “mind”—a favourite with Descartes as well as Leibniz—are invoked to justify the hierarchy, but once in place it is the hierarchy itself that is used to evaluate mathematics, not the underlying reasons originally used to justify it. In this way I think the conflict over the paracentric isochrone suggests a useful framework for imposing some order on the multitude of arguments thrown about to motivate the problem of rectification of quadratures. This point of view squares well with Leibniz’s reproof of Jacob Bernoulli’s construction by rectification of the elastica as “transcendental of the second order”340 : the construction is judged by its hierarchical classification rather than on the basis of simplicity, the enlightening of minds, or what have you. This also agrees with our argument in Section 3.3.5.2 about the role of a hierarchy of methods more generally. I propose that the need for such a hierarchy of methods was the fundamental force underlying the principled preference for rectification over quadratures. In this way some cohesion emerges in the variety of arguments presented for reducing quadratures to rectifications. In particular, the numerous arguments alluding to simplicity in various forms speak only to what I called pre facto justifiability, which explains to some extent the indefinite nature of these arguments and their weak force in an actual moment of conflict. Thus, as we have seen above, the various arguments raised by Leibniz are readily interpreted as alternately addressing these desiderata, but at the moment of truth, when the elastica conflict cut to the heart of the matter, he phrased his judgment in terms of the hierarchy of methods itself rather than its subsidiary desiderata. Again, this explains also why Jacob Bernoulli’s simplicity arguments were unanimously opposed despite their prima facie similarity to previous arguments by his opponents: he did not recognise the subordinate role of such arguments as addressing pre facto justifiability only. In this way I believe that cohesion and rationale can be brought out in the apparent diversity and disparity of extramathematical arguments regarding the rectification of quadratures by considering them as subsidiary to more fundamental principles, namely the need for a hierarchy of methods being both retroconsistent and justifiable pre facto. Admittedly, the precise foundational status of the rectification of quadratures remained somewhat elusive. They were certainly foundational in the general sense of pertaining to underlying principles, as they did not concern specific results or problems but rather ad-
Transcendental curves by the reduction of quadratures
dressed the underpinnings of all work on transcendental curves. It is debatable to what extent they were also foundational in the stricter sense of pertaining to the certainty of mathematical knowledge and the delineation of which objects and methods are acceptable in mathematics. I believe our protagonists deliberately left this question open, and that they did so with good reason. On the one hand, to rectify quadratures is to build up the complicated from the simple—arguably the premier safeguard of certainty and exactness in Euclid and Descartes alike, as well as a time-honoured principle of methodological purity. Thus the motivation for elevating the requirement that quadratures be reduced to rectifications to a “law of geometry” akin to the foundational principles of Euclid and Descartes is readily apparent. On the other hand, such a move would have been premature given the lack of general methods for actually performing this reduction in practice and the exceptional state of flux and rapid expansion of the field at this time. Indeed, as we have seen, Leibniz often spoke of the rectification of quadratures as a kind of research programme rather than an absolute law, though at the same time recognising its foundational potential. If this research programme had been conclusive, it may very well have led to definitive proclamations on the foundational status of the rectifications of quadratures, just as Descartes’s foundational programme was the conclusion of his geometrical research rather than its starting point.341 But things did not turn out that way, and the programme never advanced beyond its exploratory, pre-legislative stage.
7.3.3. The motivation for Leibniz’s envelope paper of 1694 The importance of the problem of rectification of quadratures in guiding the direction of mathematical research can be seen in a historical episode where a quirk of history affords an opportunity to study Leibniz’s attitude towards a certain mathematical result just before and just after he realised that it had important implications for the problem of transcendental curves. The result in question is Leibniz’s method of finding envelope curves, i.e., curves determined by their being tangent to a given family of curves; for example, in Figure 7.5, C(C) is the envelope of the family of lines TC, (T )(C), etc. Leibniz’s method may be stated thus in modern terms: to find the envelope of the family of curves f (x, y, α) = 0, combine d f (x, y, α) = 0 so as to eliminate α.342 the two equations f (x, y, α) = 0 and dα Envelopes are central in optics (as they define caustic curves), and quite possibly Leibniz discovered his envelope rule in this context. But with this as its only merit he treated the rule rather disparagingly. Leibniz (1692c) published his rule in an inconspicuous little article of just over three pages, and a good part of it is spent discussing matters of vocabulary that has no direct bearing on the envelope rule. Optical problems are mentioned as motivation, and the rule is then alluded to in abstract terms without a single formula or figure appearing in the entire paper. The article also appeared under the pseudonym “O.V.E.”
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which “are the second letters of my name after the initials G.L.L.” as Leibniz later explained privately.343 Indeed, it seems to me that Leibniz used this pseudonym based on the second letters of his names for works of secondary importance.344 After having realised the importance of the envelope rule for foundational questions, however, Leibniz’s tone is markedly different. Leibniz (1694f) now publishes a five-page paper with detailed calculations and figures devoted entirely to his envelope rule, calling it a “new” application of his calculus “of no small importance for the development of geometry.”345 The bearing of envelope determination of the rectification of quadratures can be seen in Section 5.4, for, in the figures of that section, the desired curve C(C) is the envelope of the family of tangents TC. Indeed, manuscript evidence shows that Leibniz originally drafted the paper of Section 5.4 and his envelope paper as a single treatise,346 and Blåsjö (2012) has shown how each of the five envelope problems mentioned by Leibniz in his paper is relevant to the problem of rectification of quadratures.347 In short, viewing this episode through the lens of foundational concerns regarding the representation of curves explains the timing, the specific content, and the rhetoric of the envelope paper. The same cannot be said for the traditional lens of taking mathematical advances as driven by the need to solve “applied” problems—which is indeed the approach taken by Scarpello and Scimone (2005) in arguing that optical problems regarding caustics were a key motivation for Leibniz’s paper. Yet again, thus, we see the all-eclipsing importance of the problem of transcendental curves to Leibniz.
7.3.4. Technical results on the rectification of quadratures In 1693, Leibniz wrote to Newton asking for “something big,”348 namely a solution to the problem of rectification of quadratures: “I would very much like to see how squarings may be reduced to the rectifications of curves, simpler in all cases than the measurings of surfaces or volumes.”349 In reply, Newton (1693) offered with considerable indifference the solution “which you seem to want.”350 The solution he offered was a construction by envelopes based on his 1666 notes, where this problem occurs inconspicuously as one among many possible permutations of geometrical problems of the form “given this, find that,” without any indication that this problem has a special foundational status. This is yet another clear manifestation of the contrasting views on the foundations of geometry in Newton and Leibniz outlined in Chapter 3. Newton’s solution as sent to Leibniz goes R B as follows (Figure 7.4). A curve y(x) is given and we seek to express its quadrature D y(x) dx in terms of arc lengths. For each point on the x-axis from x = D to x = B we draw a ray whose angle φ with the x-axis is defined by cos φ = y(x) (or, more generally, a cos φ = y(x), where a is a constant large enough so that y(x)/a does not exceed 1; for the moment I limit my discussion to the case
Transcendental curves by the reduction of quadratures
ds
y dx
L(x)
φ dx Figure 7.4: Left: Newton’s figure from his 1693 letter to Leibniz. Right: the same configuration with my notation.
a = 1 for clarity). Next we find the curve FG enveloped R B by all these rays, where F is the point corresponding to x = D and G to x = B. Then D y(x) dx = BG − (GF + FD), so the integral has been expressed in terms of arc lengths, as required.351 But this general result is not very feasible for practical use. Some problems present themselves when one tries to apply this method in specific instances using Leibnizian reasoning. The obvious thing to do from Leibniz’s point of view would be to attempt to find Newton’s envelope curve FG using his envelope method. For this purpose it is necessary to translate Newton’s condition cos φ = y(x) into an algebraic equation for the family of enveloping √ lines. An easy calculation shows that Newton’s condition translates 1−y2
into a slope of − y , so the line in the family having x-intercept α has the equation √ 1−(y(α))2 Y = − y(α) (X − α) (I use capital letters for the variables as y(x) already has a meaning). To go further we must specify the curve whose area is to be rectified. Let us consider the √ 1−α4 2 case y(x) = x . In this case the family of enveloping lines is Y = − α2 (X − α). Leibniz’s method for finding envelopes tells us to eliminate α by combining this equation with ! √ d 1 − α4 −Y − (X − α) 0= dα α2 ! √ √ 2α 1 − α4 2 1 − α4 =− √ (X − α) + − α3 α2 1 − α4 =−
α5 + α − 2X √ , α3 1 − α4
i.e., 2X − α5 − α = 0—a formidable task, and this was the very easy case y(x) = x2 , which
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Figure 7.5: The main figure of Leibniz’s 1694 paper on envelopes (as reproduced in Leibniz (1908)). Below is a complete list of the problems considered by Leibniz in the order in which they appear in the paper, followed by a note on how they can be interpreted in the context of Newton’s rectification method. (1) Given the tangent line CT as a function of its axis intercepts T and ϑ, find the curve C(C). This is equivalent to Newton’s construction with φ = ∠AT ϑ. (2) Given the point ε on the curve ε(ε) as a function of T , find the curve C(C). This variant of the first problem is analogous to uncovering the hidden involute curve not mentioned by Newton in his letter and using this curve in place of the earlier angle condition, as Newton did in his 1666 notes. (3) Given the normal PC as a function of its axis intercepts P and π, find C(C). As Leibniz remarks, this problem is reducible to the above since the normals also define the evolute F(F) (equivalent to the C(C) of the first problem), from which C(C) is given by evolution. This is Newton’s construction with φ = ∠AπP and the evolute–involute pair explicitly indicated. (4) Given the length of the normal PC as a function of the coordinate AP, find the curve C(C). This is the problem of finding the involute curve needed for Newton’s construction for a given f (x) (= PC). (This is the only problem Leibniz actually works out in the paper.) (5) Given the length of the tangent TC as a function of the coordinate AT , find the curve C(C). This is the problem of finding the evolute curve given l(x). This is how Newton in fact finds the evolute in his 1666 notes (using a curvature-style formula for l(x); see below). (As Leibniz notes, this problem is not solvable by his envelope method; instead he refers to his method of construction by tractional motion of Section 5.4.)
is by no means atypically complicated. For almost any other choice of y(x) the situation is just as bad if not worse.352 Having failed to find the envelope, one may gone on to seize on the idea that Newton’s rectification is based on evolutes, as the “unwrapping” of GFD into GCH in Figure 7.4 clearly suggests. Recall from Sections 4.3 and 4.4.4 that evolutes enable us to rectify curves whose involutes are known. While Huygens was able to give a general method for finding
Transcendental curves by the reduction of quadratures
f(x) (x, 0) (-C, 0) (0, 0) s(x)
l(x)
Figure 7.6: The hidden evolute geometry of Newton’s rectification method, not mentioned in the letter to Leibniz but treated in Newton’s 1666 tract on fluxions. (The L(x) from Figure 7.4 would be obtained by extending l(x) to the y-axis in this figure. Thus the angle φ is not the angle this line makes with the y-axis, which corresponds to the x-axis in the previous figure.)
the evolute of a given curve, he had no general method for finding the involute for a given evolute. Thus he was able to rectify a great many curves by starting with various involutes, but the general problem of rectifying any given curve remained unresolved. Newton has an ingenious trick for circumventing this problem, as we know from his more complete account of his construction in his October 1666 tract on fluxions,353 namely to consider the involute in terms of what Whiteside calls an “unusual semi-intrinsic system of coordinates.” It is quite easy to reconstruct this trick from Newton’s letter. One only needs to add the hidden evolute to the diagram to obtain Figure R7.6 and proceed as above. In the exact same way as we found the expression L = s + y dx + c above, we find that RX RX l(X) = s(X) − 0 y dx + C in this diagram. Thus the quadrature 0 y dx can be expressed in terms of arc lengths as C + s(X) − l(X). Applying an analogous argument for f (x) gives f (x) + l(x) + ds = f (x + dx) + l(x + dx), i.e.,R df = ds − dl. Integrating and substituting the X known expression for l, we obtain f (X) = 0 y dx. This uncovers Newton’s fundamental RX idea: to find the involute needed to rectify z(X) = 0 y dx, form the curve with this z as the radial coordinate f (x) in the “semi-intrinsic” coordinate system (x, f (x)) of Figure 7.6. Once this idea is in place everything about cos(φ) can be forgotten and the rectification can be restated purely in terms of evolute and involute.
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To put this idea to use one must first find the hidden curve determined by the z’s. This is another envelope problem: to find the envelope of the families of circles with centres at x and radius z. But contrary to the previous envelope problem this one is solvable by Leibniz’s method. In fact, it is the √ very problem solved in Leibniz’s envelope paper (see Figure 7.5) in the case z(X) = X. Leibniz’s solution is as follows. We seek the curve (x, y) enveloped by a family of circles centred on the x-axis whose radii are z. The √ circle centred at the point (X, 0) has the equation (x − X)2 + y2 = z2 . Since z(X) = X this equation reduces to x2 + y2 + X 2 = 2Xx + X. Following Leibniz’s general envelope method, we differentiate this expression with respect to X to get X = x + 21 . Then we use this equation to eliminate X from the equation we have before differentiating, which leaves us with the equation for the envelope, y2 = x + 14 . √ RX Thus Newton’s method for rectifying z(X) = 0 y dx = X involves finding the evolute of this other parabola, y2 = x + 41 . In this case the evolute is the semicubical parabola 16(x − 14 )3 = 27y2 , as was well known and could easily be determined using one of the several available general methods for finding evolutes. Indeed, the rectification is easily confirmed in this case. See Figure 7.7. However, this method of finding the involute is not very powerful. It fails, prima facie, as soon as y(x) does not have an explicit algebraic antiderivative, since then it is typically not possible to eliminate the parameter as required by Leibniz’s method. But Newton claims in his letter that the evolute can be found “geometrically” (i.e., is an algebraic curve) whenever y(x) is “geometrical.” In fact, Newton is right, as we know from his 1666 notes. There he derives an expression for l(x) equivalent to a radius of curvature calculation, namely l(x) = −
1 − y2 , y0
where y = f 0 is again the integrand of the integral being rectified. From here it follows that the evolute has the parametrisation354 y − y3 (1 − y2 )3/2 x− , − . y0 y0 Thus the evolute is indeed geometrical whenever y is—at least in the sense of having an algebraic parametrisation—so Newton’s claim is correct (as he knew from this derivation).355 Leibniz quite probably did not consider Newton’s construction a fully satisfactory solution to the problem he had in mind. It seems that Leibniz was thinking of the more direct
Transcendental curves by the reduction of quadratures
2
1
1
-1
2
3
4
-1
-2
R √ . The dashed parabola has a Figure 7.7: The evolute form of Newton’s rectification method applied to the quadrature 2dx x R dx √ y-value of f (x) = 2 √ x = x. For each point on the x-axis, a circle is drawn (shown dashed) with this point as its centre and f (x) as its radius. These circles envelope the involute needed, the solid parabola. The evolute of this curve is the semicubical R 0.65 √ . For this case we have parabola also shown solid. Drawn thin is the tangent line needed for the rectification of 0 2dx x R 0.65 dx C + s(0.65) − l(0.65) ≈ 0.5 + 1.60 − 1.29 = 0.81 ≈ 0 2 √ x , verifying that this quadrature has been expressed in terms of these three lengths. Note that the coordinate system of this figure is shifted over 0.25 in the x-direction compared to the coordinate system of Figure 7.6 since in that figure the origin was defined as the starting point of the evolute.
problem: given a quadrature
R
y dx, find a curve g(x) whose arc length equals it, i.e., Z Z p y dx = 1 + (g0 )2 dx.
For when Leibniz next brings up the problem R √ of rectifying quadratures in an article in the Acta the following month, he claims that a4 + x4 dx can be rectified by a hyperbola,356 and this is certainly not the result of using Newton’s construction, which would give a much more complicated curve. Thus when Leibniz says that he wants to “reduce squarings to the rectifications of curves” he means that he wants to transform a quadrature problem into a rectification problem. Newton, on the other hand, takes the problem to be about actually rectifying the quadrature, that is to say, to find a straight line segment with a length equal to the given area. No wonder, then, that Newton’s method gives a more complicated solution than Leibniz desires, since it in effect solves two problems at once: it both reduces the quadrature to
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a rectification problem and solves the rectification problem at the same time. Nevertheless, it seems plausible that Leibniz would have considered Newton’s construction as important, not only because it provides one very general and powerful way of rectifying quadratures (although perhaps too indirectly for Leibniz’s tastes), but also since it in a way solves the problem of rectifying a curve by evolutes when the involute is unknown, which had been a recognised lacuna in the theory of evolutes since its introduction by Huygens. Newton’s method for rectification of quadratures remained unpublished until Hermann (1723), apparently unaware of Newton’s work, published an essentially equivalent method. This caused Johann Bernoulli (1724) to take a renewed interest in the problem. Apparently the three decades that had passed since the problem’s heyday had not faded Bernoulli’s estimation of its importance, for he writes: In the constructions of transcendental problems, it is a matter of great importance to separate the variables of differential equations, so that their effections can be obtained by quadratures of algebraic curves. But it is of greater importance to reduce those quadratures to the lengths of algebraic curves, and so indeed that the parts of the areas to be squared are simply proportional to the arcs of these curves.357 In light of the importance of the problem “it is pleasing what the celebrated Hermann wrote about this matter,”358 according to Bernoulli, but ultimately he is critical of Hermann’s approach: It would be desirable that the approach he uses in this matter were simpler and better suited for use. For to have recourse to the nature of evolutes, and also to require the assistance of the mutual inclinations of lines upon themselves, seems to me an indirect and scarcely natural approach, by which we are led astray among elaborate calculations, as often happens if we mix pure analysis beyond necessity with geometry.359 Privately too Johann Bernoulli (1738) laments in a letter to Euler that Hermann’s quadrature rectification has the character of a chance discovery, though it does what it should. Euler (1738b) agrees. The title of Bernoulli’s paper promises his own “natural and convenient method for reducing transcendental quadratures of any degree to lengths of algebraic curves,” but his method is far from universal and not as free from “elaborate calculations” as his critique R of Hermann might suggest. Faced with the problem of rectifying f (x) dx, Bernoulli
Transcendental curves by the reduction of quadratures
considers the parametric curve
(1 − f 2 )2/3 ( f − f 3 ) − x , f0 f0
.
It follows that Z 0
X
f (x) dx = L(X) +
1 − f (X)2 + C, f 0 (X)
where L(X) is the arc length of the parametric curve from x = 0 to x = X. This equality can be checked by direct differentiation. If f (x) is algebraic then so is the right-hand side of this equation, so the quadrature has been reduced to an algebraic expression involving arc lengths but no areas, as desired. Euler (1738a) later derived essentially the same result in essentially the same way, and may be consulted for a shorter exposition. Bernoulli suggests some minor ways in which this basic idea can be elaborated (slight generalisations and the use of integration by parts) and expresses the hope—which seems highly unrealistic to me—that these little tricks could somehow be made to absorb the auxiliary term: “Perhaps it will Rnot be difficult,” he speculates optimistically, to apply these manipulations to arrive at p dx = L, “that is, so that an algebraic curve can be R found, whose arcs . . . are proportional to the parts of the transcendental area p dx.” But he provides no example or workable strategy for accomplishing such a thing, hiding instead behind the declaration that “it suffices for me to have opened the gate through which the road is spread out to higher things, and to those especially, which until now have been considered among geometers most abstruse.”360 His rather overly optimistic rhetoric notwithstanding, it is clear that Bernoulli’s method falls well short of settling the general problem of the rectification of quadratures. By his own admission, the truly desired form of solution is a direct equality of an area integral with an arc length integral, but to this end he has nothing to offer but a certain form of trial and error, which is hardly an improvement on where the problem had already stood for decades. Despite these lingering dissatisfactions this was perhaps a time for closure. After three decades of efforts by two generations of mathematicians the problem of the rectification of quadratures could not have seemed likely to receive any very satisfying resolution. Johann Bernoulli (1726), for one, seemed willing to draw a line under the matter: When some differential equation of the first degree is reduced to p dx = q dy, where p is given by x, and q by y, . . . the construction of the equation is burdened with no difficulty, certainly if quadratures are allowed; and these very [quadratures] were reduced in general to the extensions [i.e., rectifications] of
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algebraic curves not so long ago. . . . To such a degree that in this area nothing further seems to be desired for greater perfection, except only that, among the infinitely many methods, by which we have shown that the same thing can be furnished, that one is chosen which produces the simplest curve to construct, whose extension one pleases to use for the determination of the quadrature.361 Here, then, the recent general methods for the reduction of quadratures to rectifications are accepted as sufficient. But despite the optimistic language, one cannot quite shake the feeling that the battle was called off more due to exhaustion with a futile fight than outright triumph. Such, indeed, was the fate of many attempts at resolving the problem of transcendental curves.