Suppose for the
circle (x-r)2 + y2 =
r2 that ydx = pm(x)
= b0 xm + b1xm-1
+ ... + bm for some fixed
m. Then area of sector 
and area of sector 
for all n. Hence, finding u is a
problem of degree m, fixed for all n. 1. Histoire de l'Academie Royale des Sciences 1700, p.87. "[C]e fut par la Geometrie des Infiniment Petits que M. Varignon reduisit les Mouvemens variés à la même Règle que les Uniformes, & il ne paroît pas que par toute autre methode on eût pû y parvenir."
2. Isaac Newton, Philosophiae naturalis Principia mathematica, (London, 1687), Book I, Sect.6, Lemma 28: "There is no oval figure of which the area cut off by arbitrary straight lines can be found generally by equations of a finite number of terms and dimensions." See The Mathematical Papers of Isaac Newton, ed. D.T. Whiteside, 8 vols. (Cambridge, 1966-81), 6:302-8. (Unless otherwise noted, all translations are my own.)
3. Descartes to Marin Mersenne, 27 July 1638, in Oeuvres de Descartes (hereafter AT), ed. Charles Adam and Paul Tannery, 12 vols. (Paris, 1897-1913), 2:268: "Mr des Argues m'oblige du soin qu'il lui plaît avoir de moi, en ce qu'il témoigne être marri de ce que je ne veux plus étudier en géométrie. Mais je n'ai résolu de quitter que la géométrie abstracte, c'est à dire la recherche des questions qui ne servent qu'à exercer l'esprit; & ce afin d'avoir d'autant plus de loisir de cultiver une autre sorte de géométrie, qui se propose pour questions l'explication des phainomènes de la nature. Car s'il lui plaît de considérer ce que j'ai écrit du sel, de la neige, de l'arc-en-ciel &c, il connaîtra bien que toute ma physique n'est autre chose que géométrie."
4. Wallis was so infuriated by what he took to be Huygens' francocentric view of the history of the cycloid that he severed correspondence. Cf. J.E. Hofmann, Die Entwicklungsgeschichte der Leibnizschen Mathematik während des Aufenthalts in Paris (1672-1676) (München, 1949), sec. 8.
5. Christiaan Huygens, Horologium. oscillatorium. (Paris, 1673), p. 69.
6. For a discussion of Descartes' treatment of curves, see A.G. Molland, "Shifting the Foundations: Descartes' Transformation of Ancient Geometry," Historia Mathematica 3(1976), 21-49, and H.J.M. Bos, "The Representation of Curves in Descartes' Géométrie", Archive for History of Exact Sciences 24(1981), 295-338.
7. Cf. Descartes' correspondence with Mersenne for the spring and summer of 1638 in AT.II passim. When Roberval pointed out that Descartes' treatment of the cycloid did not conform to the method of tangents set out in the Géométrie, Descartes revealed his basic commitment: "Il faut aussi remarquer que les courbes descrites par des roulettes sont des lignes entierement mechaniques, & du nombre de celles que i'ay reiettés de ma Geometrie; c'est pourquoy ce n'est pas merveille que leurs tangentes ne se trouvent point par les regles que i'y ay mises." Mathematical methods could be tested only by mathematical curves.
8. Cf. Roberval to Fermat, 4.VIII.1640, Oeuvres de Fermat (ed. Ch. Henry and P. Tannery), II, 200.
9. Roberval to Torricelli, [post 1645], Memoires de l'Academie Royale des Sciences, 1666-1699 (hereafter MARS), X, 440-478; at 449.
10. E.g.John Collins to HenryOldenburg for Leibniz, ca. 1 April 1673 (The Correspondence of Henry Oldenburg, ed. A.R. and M.B. Hall, Vol.IX, 551): "Dulaurens, a late writer, in the preface to his specimina [mathematica, Paris, 1667] promised a method of takeing away all the midle powers in any aequations, and consequently leave none but the highest and lowest power aequall to an absolute (of which doubtlesse Monsr. Frenicle can give an account). If this could always be done, we should confesse, that the Logme. Curve might serve for the construction of all aequations; if you [sc. Leibniz] can procure this Notion, or any other, of Monsr. Hozanna [Ozanam], about the divideing of aequations into their components &c; they will come as a seasonable supplement of our designes [in completing Kersey's Algebra], and the Author shall have an honorable mention with thankes."
11. In his Dynamica de potentia et legibus naturae corporeae (ca.1690), Sect.1, Cap.1 (LMS.VI.295).
12. One can trace a shift from the narrow to the extended sense of rapport over the course of Malebranche's later career.
13. See Michael S. Mahoney, The Mathematical Career of Pierre de Fermat (Princeton, 1973; 2nd ed. 1994), Chap.III.
14. See Michael S. Mahoney, "Another Look at Greek Geometrical Analysis", Archive for History of Exact Sciences 5(1968), 318-348.
15. See Part III of the Géométrie, which sets out Descartes' theory of equations; cf. Michael S. Mahoney, "The Beginnings of Algebraic Thought in the Seventeenth Century", in S. Gaukroger (ed.), Descartes. Mathematics, Physics, Philosophy (Totowa, NJ/Brighton, Sussex: Barnes & Noble/Harvester Press, 1980), Chap.5.
16. See the discussion in Mahoney, Fermat, p.181.
17. I use the term here in the sense current today. In Descartes' theory, only positive magnitudes were "real"; negative ones were "false".
18. As Euler would continue to insist in his Vollständige Einleitung zur Algebra (1770), I, 1, par.151: "We must finally drop our concern that the doctrine of impossible numbers might be viewed as useless fantasy. This concern is unfounded. The doctrine of impossible numbers is in fact of the greatest importance, since problems often arise in which one cannot know immediately whether they demand somethingpossible or impossible. Whenever their solution leads to such impossible numbers, one has a sure sign that the problem demands something impossible." ( "Endlich muss noch das Bedenken behoben werden, dass die Lehre von den unmöglichen Zahlen als nutzlose Grille angesehen werden könne. Dieses Bedenken ist unbegründet. Die Lehre von den unmöglichen Zahlen ist in der Tat von grösster Wichtigkeit, da oft Aufgaben vorkommen, von denen man nicht sofort wissen kann, ob sie Mögliches oder Unmögliches verlangen. Wann dann ihre Auflösung zu solchen unmöglichen Zahlen führt, hat man ein sicheres Zeichen dafür, dass die Aufgabe Unmögliches verlangt.")
In the long run, it was the intelligibility of equations as composite relations that made complex numbers intelligible. The latter took on precisely the operational meaning that, when combined according to specifiable rules, they formed real magnitudes denoted by expressions consisting of real terms and factors.
19. For an extended discussion of this concept, see Mahoney, Fermat, Chap.IV.
20. "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur & singular pro illis calculi genus," Acta eruditorum (1684), 466-473; LMS.V.220-226.
21. Fermat used a similar term in French, pris à discrétion, to finesse the question of the precise size of such a segment; cf. Mahoney, Fermat, Chap.IV, pp.204-205.
22. LMS.V.220
23. See his "Justification du calcul des infinitesimales par celuy de l'Algèbre ordinaire," LMS.IV.104-106, and his manuscript work, "Cum prodiisset atque increbuisset Analysis mea infinitesimalis ..." [post 1701], published by C.I. Gerhardt in Historia et origo calculi differentialis a G.G. Leibnitzii conscripta (Hannover, 1846), trans. J.M. Child, Early Mathematical Manuscripts of Leibniz (Chicago, 1920), 145ff, and discussed by H.J.M. Bos, "Differentials, Higher-Order Differentials, and the Derivative in the Leibnizian Calculus", Archive for History of Exact Sciences 14(1974), 1-90; at 56-64.
24. The notation is, of course, anachronistic, as is the precise concept of function it represents. It is used here as a shorthand for the seventeenth-century notion of a curve, of which the ordinates y are computable from the corresponding value x of the abscissa.
25. Unless, that is, Leibniz assumed his readers would recognize the configuration, accept the subtangent as given, and thus overlook the circularity.
26. Note the operational, combinatory view of mathematical quantities implicit in this term (Lat. vincula). Fractions and roots are relations that link their operands, either directly to one another or indirectly to mean proportionals. It was this view Leibniz had in mind when he spoke of dx as "some sort of modification of x" (see below).
27. LMS.V.222-223
28. Here, and throughout this article, I shall follow Leibniz' technical language as closely as translation allows. Although he distinguished between "differences" and "differentials", he used transcendens rather than transcendentalis when referring to non-algebraic curves.
29. ibid., 223
30. Originally posed in geometric terms to Descartes and other mathematicians in Paris and Toulouse by Florimond de Beaune in 1639, the problem takes the analytic form of finding a curve y(x) such that y:subtangent = n:(x-y), where n is a given quantity. For Descartes' solution (a logarithmic curve) and other mathematical details, see Jules Vuillemin, Mathématiques et metaphysique chez Descartes, (Paris, 1960), 1-25. Both the original problem and a generalized version became touchstones for subsequent methods of inverse tangents.
31. "De geometria recondita et analysi indivisibilium atque infinitorum," Acta eruditorum (1686), 292-300; LMS.V.226-233.
32. For what follows, see M.S. Mahoney, The Mathematical Career of Pierre de Fermat (Princeton, 1973), esp. Chaps. 1-2.
33. Cf. above, note 2.
34. LMS.V.228
35. Suppose for the
circle (x-r)2 + y2 =
r2 that ydx = pm(x)
= b0 xm + b1xm-1
+ ... + bm for some fixed
m. Then area of sector and area of sector for all n. Hence, finding u is a
problem of degree m, fixed for all n.
36. LMS.V.229; see Molland, "Shifting the Foundations" (above, n. 6).
37. The two come down to the same thing once the operations of the two fields are brought into correspondence. That is the task with which Descartes began his Géométrie; cf. the opening paragraphs.
38. LMS.V.229
39. ibid.
40. For Fermat's varied and inventive use of the technique, cf. the references in Mahoney, Fermat, Index, s.v. "equations, indeterminate". It is also essential to Descartes' method of normals; cf. Géométrie (1637), 341ff.
41. LMS.V.229-230
42. ibid., 230
43. ibid., 231
44. Cf. Géométrie (1637), 297-298.
45. See Leibniz to Bodenhausen, LMS.VII.372: " ... or it will make the equation ady = ydx, or, with a set = 1, it will make dy = ydx, which is a differential equation expressing the nature of the logarithm; it is of the greatest simplicity, just as certainly the logarithm is the most simple of all the transcendents." Jakob Bernoulli echoed the algebraic theme in his "Constructio generalis omnium curvarum transcendentium ope simplicioris tractoriae & logarithmicae" (Acta eruditorum, 1696, 261): "The inverse method of tangents, which without doubt constitutes the highest peak of geometry, is carried out chiefly in three parts, of which the first is aimed at reducing differentials of higher degrees [generum] (or differentio-differentials) to differentials of the first degree; the second in separating from one another the indeterminate letters together with their differentials; and the third in constructing the equations reduced in this way."
46. Cf. above, n. 23.
47. LMS.V.308. "Enfin notre méthode étant proprement cette partie de la mathématique générale, qui traite de l'infini, c'est ce qui fait qu'on en a fort besoin, en appliquant les mathématiques à la physique, parce que le caractère de l'Auteur infini entre ordinairement dans les opérations de la nature." Some thirty years later, when analytic mechanics was flourishing, Fontenelle put an interesting twist on Leibniz' idea: "The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics, where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there, only the clearest ideas that the human mind can form of magnitude, compared with one another and combined in an infinity of different ways, while Nature could well have used in the construction of the universe some mechanics that escapes us entirely. If however mathematics always has some essential obscurity that one cannot dissipate, it will lie uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little, and perhaps also on a too lofty metaphysics, of which we are permitted to perceive only some rays." ( "Le calcul n'est guere en Geometrie que ce qu'est l'experience en Phisique, & toutes les Verités produites seulement par le Calcul, on les pourroit traité[!] de Verités d'experience. Les Sciences doivent aller jusqu'aux premières causes, surtout la Geometrie, où l'on ne peut soupçonner comme dans la Phisique des principes qui nous soient inconnus. Car il n'y a dans la Geometrie, pour ainsi dire, que ce que nous y avons mis, ce ne sont que les idées les plus claires que l'Esprit humain puisse former sur la Grandeur comparées ensemble, & combinées d'une infinité de façons differentes, au lieu que la Nature pourroit bien avoir employé dans la structure de l'univers quelque Mechanique qui nous echape absolument. Que si cependant la Geometrie a toûjours quelque obscurité essentielle, qu'on ne puisse dissiper, & ce sera uniquement, à ce que je crois, du côté de l'Infini, c'est que de ce côté-là la Geometrie tient à la Phisique, à la nature intime des Corps que nous connoissons peu, & peut-être aussi à une Metaphisique trop élevée, dont il ne nous est permis que d'appercevoir quelque rayons.") (Elements de la geometrie de l'infini, Preface, ciiir).
48. For details of the derivation see M.S. Mahoney, "Christiaan Huygens: The Determination of Time and of Longitude at Sea", in H.J.M. Bos, et al., Studies on Christiaan Huygens (Lisse, 1980), pp.238-242.
49. The circle was the osculant of the parabola.
50. Huygens had been working on the cycloid at the time, stimulated by the controversy provoked by Pascal's challenge of 1654.
51. Huygens, Oeuvres complètes (hereafter HOC), XVI, 411-412. In the ms. it follows a demonstration of cycloidal tautchrony dated 15.XII.1659, but begins, "This is to be read before the preceding theorem." It seems to be an afterthought added on review for publication.
52. Huygens held off publication for some nine years. For the reasons, see Mahoney, "Christiaan Huygens ... ", 252-254.
53. Cf. Hofmann, Die Entwicklung der Leibnizschen Mathematik, cited above, n. 4.
54. Philosophers at War, (Cambridge, 1980), p.90.
55. On Huygens' style of mathematics, see H.J.M. Bos, "Huygens and Mathematics", in Bos et al., Studies on Christiaan Huygens, (Lisse, 1980), 126-146; esp.141-143.
56. Hofmann, Entwicklung, 197-198. "Die Ursache ist wohl darin zu erblicken, dass ihm die neue Tendenz nicht ganz zusagte. Leibniz strebte eine durch zweckmässige Symbole bis ins letzte vereinfachte und logisierte Darstellungstechnik an, die sich jedoch nicht unmittelbar erfassen lässt, sondern erlernt werden muss. Wie sich dies anzueignen vermag, ist dem Ungeschulten in ungeahntem Mass überlegen, selbst dann, wenn er keine besonders tiefgehenden Einsichten in die Zusammenhänge hat: der Formalismus denkt für ihn. Gerade diese Möglichkeit war es, die Huygens für unerwünscht ansah. Er war der letzte und bedeutendste Vertreter der alten Schule, die zu ihren Ergebnissen auf Grund wahrhaft genialer, aber isoliert stehender Einzelüberlegungen gekommen war."
57. Lectiones mathematicae de methodo integralium aliisque conscriptae in usum ill. Marchionis Hospitalii, Opera omnia (Lausanne, 1742), III, 386-508; at 414.
58. Cf. the discussion in Bos, "Differentials" (above, n. 23).
59. The idea of attaining tautochronicity by constraining the trajectory predated the discovery of the cycloid as tautochrone.
60. An interesting error reveals the influence on Leibniz' thinking of the algebraic themes discussed above. To "see" analytically what was happening at the point of osculation, Leibniz tried to "pull it apart" into counterfactually distinct points. Reasoning by analogy from earlier such approaches, he argued that, since the tangent to a curve had two points in common with it and the point of inflection three, the osculating circle had four. In fact, as the Bernoullis showed, the analogy was misleading; three points suffice to determine the osculant.
61. The term "involute" had not yet been coined.
62. Lectio 16, in Lectiones mathematicae, esp. 436ff.
63. Note the implicit return to a configuration introduced by Fermat in his method of tangents; cf. Mahoney, Fermat, 165ff.
64. Once the relationships MP = MN×OF:OB = OM×OB:BD had been established by geometrical analysis, another set of techniques came into play for translating the diagrammatic elements into the symbolic quantities of the calculus. Translated straightforwardly, OF = dy, OB = (dx2 + dy2), and MN = d(subtangent - x) = d(ydx/dy - x) Bernoulli set OM = MB = (subtangent2 + y2), simply ignoring the segment OB. At this point in his Lectiones perhaps, he could assume his reader would see that retaining OB would lead in the end to a finite expression plus a differential expression, the latter of which would then be clearly negligible. The juxtaposition of the geometrical analysis and the infinitesimal calculus reveals a curious twist. What is the center of attention in the former becomes negligible in the latter.
65. For greater detail on what follows, see Mahoney, "Determination of Time ... ", 254-7.
66. Acta eruditorum (1689); LMS.VI.144-161
67. [added 2001]Leibniz would have many successors in this regard, for example, James Clerk Maxwell. "Light, [Maxwell] concluded, was an electromagnetic wave. It propagated through the ether by electric and magnetic oscillations perpendicular to each other as well as to the direction of travel. These travsverse waves involved no attendant longitudinal waves. In this respect, electromagnetic waves in ether behaved like waves in no known, or apparently imaginable, elastic solid." (M. Norton Wise, "Mediating Machines" in Science in Context 2,1(1988), 96)