De Zeventiende Eeuw. Jaargang 12

[p. 215]

Huygens and Newton on Curvature and its applications to Dynamics
Michael Nauenberg

Introduction

In the preface of his book A view of Sir Isaac Newton's Philosophy, Henry Pemberton, the editor of the third edition of Newton's Principia wrote that

‘... Sir Isaac Newton has several times particularly recommended to me Huygens's style and manner. He thought him the most elegant of any mathematical writer of modern times, and the most just imitator of the ancients ...’^1.

In 1673, when Newton received a copy of Christiaan Huygens's Horologium Oscillatorium^2. from Henry Oldenburg, he promptly responded to him that

‘... I have viewed it with great satisfaction finding it full of very subtile and usefull speculations very worthy of the Author. I am glad we are to expect another discourse of the Vis Centrifuga which speculation may prove of good use in natural Philosophy and Astronomy, as well as Mechanicks ... The rectifying of curved lines by that way which M. Huygens calls Evolution, I have been considering also, and have met with a way of resolving it which seems more ready and free from the trouble of calculation than that of M. Huygens. If he please, I will send it him ...’^3.

Further indication of Newton's high esteem for Huygens can be found in his recommendation to Richard Bentley who in 1691 had asked him for some help to understand the Principia. After giving him a list of books to read in mathematics and astronomy, Newton ended by saying

‘... if you can procure Hugenius's Horologium Oscillatorium, the perusal of that will make you much more ready.’^4.

In his book An Essay on Newton's Principia,^5. W.W. Rouse Ball writes that after the publication of his book, Newton's ‘enhanced celebrity led to the gratification of a long standing desire to make the personal acquaintance of Huygens who in the summer of 1689 came to England in order to meet Newton: and at a meeting of the Royal Society on June 12 Huygens spoke at length on gravitation and, to return the compliment Newton spoke on double refraction and the polarization of light.’

From these quotations it is evident that Newton had great respect and admiration for Christiaan Huygens whose tricentennial we are celebrating at this conference in Leiden.^6. However, it is not generally realized that ‘what M. Huygens calls Evolution’ and its application to the cycloid pendulum, which is the main subject of Huygens's great book, played also an important role in the early development of Newton's dynamics.^7. Over a decade later, when writing the Principia, Newton devoted an entire section (section X) to ‘... the oscillating pendulous motion of bodies ...’.^8. The first non-trivial problem in dynamics (other than circular

1. H. Pemberton, A View of Sir Isaac Newton's Philosophy. Printed by S. Palmer, London 1728.

2. C. Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris, 1673. Reproduced in O.C., vol. 8. English translation by Richard J. Blackwell, Christiaan Huygens' The Pendulum Clock. Ames, 1986.

3. O.C., vol. 7, p. 325.

4. The Correspondence of Isaac Newton, vol. 3 (1688-1694), ed. H.W. Turnbull. Cambridge, 1961, pp. 155-156.

5. W.W. Rouse Ball, An Essay on Newton's Principia. London, 1898.

6. A.R. Hall, ‘Huygens and Newton’. In: The Anglo-Dutch contribution to the Civilization of Early Modern Society. An Anglo-Netherlands Symposium, London 27 and 28 June 1974, under the auspices of the British Academy and the Royal Society in association with the Royal Netherlands Academy of Arts and Sciences. London 1976, pp. 45-59. I am indebted to Professor Hall for calling my attention to his article after the completion of my paper.

7. M. Nauenberg, ‘Newton's Early Computational Method for Dynamics’. In: Archive for History of Exact Sciences, 46 (1994), pp. 221-252.

8. In section 10, book 1 of the Principia, Newton discusses oscillations on the epicycloid which are shown to be isochronous when the external force is a central force depending linearly on the distance from the center. The epicyloid is defined as the curve generated by a point of a circle rolling on another circle.

[p. 216]

motion) which Newton investigated was the cycloid pendulum,^9.10. and he solved it by applying essentially the same physical assumptions and mathematical techniques used by Huygens. Indeed, the further refinement of these techniques led Newton to his first method to calculate the orbits of bodies moving under the action of arbitrary central forces and to the explanation of the physical origin of Kepler's laws of planetary motion.^7.

9. A. Rupert Hall and Marie Boas Hall, Unpublished Scientific Papers of Isaac Newton. London, 1962, pp. 170-180.

10. The Mathematical Papers of Isaac Newton, 8 vols, ed. D.T. Whiteside. Cambridge, 1967-1981. Here: vol. 3 (1969), pp. 420-431.

7. M. Nauenberg, ‘Newton's Early Computational Method for Dynamics’. In: Archive for History of Exact Sciences, 46 (1994), pp. 221-252.

The cycloidal pendulum and the concept of curvature

One of the great mathematical achievements of Christiaan Huygens was his development in 1659 of the theory of evolutes or curvature^11., evidently motivated by his discovery that the cycloid is the isochronous path for a pendulum weight.^12.13.14. Contrary to Galileo's claims,^15. Mersenne had observed that the period of a simple pendulum depends on the amplitude of the oscillations. This dependence led to inaccuracies in Huygens's earliest pendulum clocks which he attempted to correct by installing bend metal strips or cheeks, as he called them, to shorten the effective length of the pendulum for increasing amplitude of the swing (see Figures 1 and 2). The shape of these cheeks were adjusted empirically by requiring the period of the pendulum to be independent of the maximum amplitude, constraining the path of the pendulum weight along a noncircular path.^16. Moreover, applying the relation that the change in the square of the velocity v of a body, falling under the action of gravity, is proportional to the change in its vertical height h, i.e. v²αh for zero initial velocity, called later the principle of energy conservation,^17. Huygens carried out a remarkable calculation for the period of the pendulum oscillations for small amplitudes. Exploring the approximations which he had made in this calculation, he found that his result for the period would apply also for arbitrary amplitudes if the path of the pendulum weight was a cycloid.^18.19. Excited about his discovery, he promptly communicated this result to his former mathematics teacher, Frans van Schooten,^20. and to some of his friends, without giving any proofs until 14 years later^21. with the publication of the Horologium Oscillatorium. During this intervening period, Huygens' problem, to find a proof for the isochronous property of the cycloid path of a pendulum weight, posed one of the most difficult challenges in mathematical physics to his contemporaries. One need only read the repeated false starts made by Lord Brouncker,^22. then President of the Royal Society, to appreciate the mathematical skill and physical insight exhibited by Huygens in solving this problem.^23. Eventually the brilliant French Jesuit Ignacius Gaston Pardies found an elegant scaling proof for the isochronous property of the cycloidal pendulum,^24. and Newton was able to obtain a complete solution to this problem^25. including the analytic expression for the period of the pendulum as well as a demonstration that the evolute of a cycloid is a similar cycloid.^10.

There is no evidence that Huygens responded to Newton's 1673 letter quoted above, in which he had offered to send him his method of solution. However, contrary to some views,^26. Newton's solved this problem by virtually the same mathematical techniques that Huygens had used in 1659, and therefore it is unlikely that Huygens would have learned anything new from him. This is not surprising, because Newton learned many of the contemporary mathematical methods of

11. Although Huygens did not discuss explicitly the concept of curvature in the Horologium Oscillatorium, the idea of approximating a small segment of a curve by the arc of a circle, later named the osculating circle by Leibniz, was central to his method of evolution. Curvature or ‘crookedness’, as Newton first named it in 1664, is the measure for the bending of a curve given by the inverse of the radius of this circle also called the radius of curvature. In his lectures to the Marquis de L'Hospital in 1691, Johann Bernoulli devoted lesson number 15 to Huygens' method of evolution and its connection to the osculating circle (J. Bernoulli, Die erste Integralrechnung. Ostwald Klassiker nr. 194, Leipzig und Berlin, 1914, pp. 63-67). His derivation of a general formula for the radius of curvature (lesson 16) is the same as that given in prose by Huygens in Proposition 11, part 3, of the Horologium Oscillatorium. Incidentally, when Bernoulli showed an application of this formula to L'Hospital, the Marquis was greatly impressed and hired Bernoulli as his tutor (see Der Briefwechsel von Johann Bernoulli. Basel, 1955, p. 136). In 1696 L'Hospital published the first textbook on calculus, Analyse des infinitement Petits..., based mostly on Bernoulli's lectures, and he too devoted and entire section entitled ‘Usage des calcul des differences pour trouver les Dévelopées’ to Huygens' theory of evolutes, reproducing essentially lesson 15 of Bernoulli. For a further discussion of Johann and Jakob Bernoulli's application of Leibniz's differentials to curvature see H.J.M. Bos, ‘Differentials and Derivatives in Leibniz's Calculus’. In: Archives for History of Exact Sciences, 14 (1974), pp. 1-90.

12. H.J.M. Bos, ‘Huygens and Mathematics’. In: Studies on Christiaan Huygens, ed. H.J.M. Bos, M.J.S. Rudwick, H.A.M. Snelders, R.P.W. Visser. Lisse, 1980, pp. 136-146.

13. M.S. Mahoney, ‘Christiaan Huygens: The measurement of time and longitude at sea’. In: Studies on Christiaan Huygens (n. 12), pp. 234-270.

14. J.G. Yoder, Unrolling Time. Cambridge, 1989; O.C., vol. 16, p. 304.

15. G. Galilei, Dialogues concerning Two New Sciences. Northwestern University Press, 1968, p. 95. During a discussion on musical harmony, Salviati states ‘... to the question whether one and the same pendulum really perform its vibrations, large, medium, and small, all in exactly the same time, I shall rely upon what I have already heard from our Academician. He has clearly shown that the time of descent is the same along all chords ...’ (Theorem VI, Proposition VI, pp. 188-191). However, he continues, ‘... If now we consider descent along arcs instead of their cords then, provided these do not exceed 90^o, experiments show they are all traversed in equal times; but these times are greater for the chord than for the arc’ (Scholium to Theorem XXII, Proposition XXXVI, pp. 239-240). (In this Scholium Galileo also surmised that the circular arc is the curve of quickest descent, but the correct curve is the cycloid, as shown first by Johann Bernoulli in 1696 and later by Newton, Leibniz and L'Hospital and others in response to Bernoulli's famous challenge to mathematicians in 1697.) It is surprising that Galileo reached such an erroneous conclusion, particularly because he presented some experimental data (on the Fourth Day): ‘Suspend two equal lead balls from equal threads four or five braccia long. The thread being attached above, remove both balls from the vertical, one of them by eighty degrees or more and the other by no more than four or five degrees, and set them free ... if two friends shall set themselves to count the oscillations, one counting the wide ones and the other the narrow ones, they will see that they may count not just tens but even hundreds, without disagreeing by even one; or rather by one count alone [my italics] (as translated by Stillman Drake in ‘New light on a Galilean claim about Pendulum’. In: Isis 66 (1975), pp. 92-95). Galileo's words at the end of this paragraph are ambiguous: ‘... senza discordar d'una sola, anzi d'un sol punto’. Under similar conditions one expects a difference of one count after about 30 oscillations, and because air damping decreases the amplitude many more oscillations pass before there is a discrepancy of another count. However he said nothing about the fact that after only a few swings one readily observes that the two pendulum are not oscillating synchronously. When writing his book, Galileo was already an old man confined to a villa in Siena by decree of the Holy Office, and his manuscripts had to smuggled outside Italy by Ludwig Elzevier who in 1638 published it in Leiden. It is unlikely that Galileo repeated again the experiment mentioned by Salviati. Instead, it is more likely that he assumed that his theorem for equal times of descent along the chords applied also to the arcs of a circle, and that small observed differences in the period as a function of amplitude were due to the effects of air resistance.

16. Since Huygens knew from Galileo and Mersenne that the period of a simple pendulum depends on its length, (more precisely, on the square root of the length) he installed a sequence of nails, as shown in Fig. 1, to shorten the length of the suspension when the amplitude of the swing increased above certain values. Such an arrangement corresponds to a cheek shaped as a polygon, which determines a trajectory for the pendulum weight as a sequence of joined arcs of circles of different radii. The centers of these circles are located at the vertices of the polygonal cheek, and the radius of a given circular arc is determined by the length of the suspension string minus the total length of the sides of the polygonal cheek which are in contact with the string. A pendulum cheek made of a continuous curve can be considered as the limit of a polygonal cheek with and infinite number of vertices, thus elucidating the origin of some of Huygens' fundamental ideas on evolutes. Thus, Huygens's early concept of the osculating circle, as an approximation to an arc of a general curve becomes evident. A similar description based on a polygonal evolute was used by Bernoulli and by L'Hospital to explain Huygens' method of evolution.

17. The relation v²/h was given first by Galileo for the case of inclined planes. In the Two New Sciences, p. 184 (n. 15) Salviati states that ‘We must recall the fact that on a plane of any inclination whatever a body starting from rest gains speed or momentum in direct proportion to the time, in agreement with the definition of naturally accelerated motion given by the Author. Hence, as shown in the preceding proposition, [Theorem II, Proposition II, third day] the distances transversed are proportional to the square of the times and therefore to the square of the speeds.’ In the first Scholium of the Principia, Newton attributed this relation to Galileo. In proposition 8-10, part 1, of the Horologium Oscillatorium, Huygens demonstrated the validity of this relation for a weight descending along a constrained path, by considering this path as a sequence of inclined planes, similar to Galileo's discussion for the corresponding motion along the arc of a circle in the Scholium to theorem 12 in his Two New Sciences (n. 15). Huygens' proof was criticized by Gregory and by Newton who pointed out that Huygens had neglected to consider the effects at the junctions where two planes with different inclinations meet. In his June 23, 1673 letter to Oldenburg (Correspondence (n. 51), p. 290), Newton remarked ‘In the Demonstration of the 8^th proposition, De Descensu gravium, there seems to be an illegitimate supposition, namely that the flexures at B and C [two junctions] do not hinder the motion of the descending body. For in reality they will hinder it ... if this supposition be made because a body descending by a curve line meets with no such opposition, ... it should have been shown that though rectilinear flexures do hinder, yet the infinitely little flexures which are in curves, though infinite in number, do not at all hinder the motion.’ This passage gives further evidence that by 1673 Newton had given careful consideration to the principle of energy conservation. Newton's own derivation of this principle was based directly on the component of the acceleration tangent to the curve, as given in Proposition 9 of the Principia (see Eq. 3), and thus he avoided Huygens' difficulty.

18. O.C., vol. 16, pp. 392-413. Some properties of the cycloid were discussed in van Schooten's commentaries to his Latin translation of Descartes' Géométrie. 1649 edition, pp. 224-230, and 1659 edition, pp. 264-270) which was important to Huygens' and Newton's work, but is absent in Descartes' original book. It is interesting to note that already in the first edition, one finds a construction and clever heuristic proof for its tangent (p. 226). Descartes' discussion of this problem appeared in a letter to Mersenne, August 23, 1638, and it may have also been communicated to van Schooten (Oeuvres de René Descartes, ed. Ch. Adam and P. Tannery. Paris, 1898).

19. A. Ziggelaar, ‘Les Premières Démonstrations du Tautochronisme de la Cycloide’. In: Centaurus 12 (1967), pp. 21-37; Mahoney, ‘Measurement’ (n. 13), pp. 234-270; Yoder, Unrolling Time (n. 14), pp. 48-64.

20. J.E. Hoffman, Frans van Schooten der Jungere. Wiesbaden, 1962.

21. Mahoney, ‘Measurement’ (n. 13) writes that ‘... For two years in the mid-1660's, while colleagues eagerly awaited the long promised treatise, Huygens worked instead to secure patents and privileges from the Dutch, French and English governments not to mention the reward offered by the Spanish throne for a reliable method of [measuring] longitudes ...’.

22. See the Jan. 1662 letter of Brouncker to Huygens, ‘A Demonstration of the Equality of Vibrations in a Cicloid Pendulum’, O.C., vol. 4, p. 28, and Huygens' response on Feb. 10, 1662 in a letter to his friend R. Moray, O.C., vol. 4, p. 50.

23. In a letter to Oldenburg on June 24, 1673, Huygens observed ‘... ce n'est pas de grand chose ... d'avoir fait la demonstration d'une proposition deja trouvé ...’, O.C., vol. 7, p. 312.

24. O.C., vol. 18, p. 487, n. 5; Ziggelaar, ‘Démonstrations’ (n. 19). A very promising scientist, Pardies was a professor of mathematics and physics at the College de Clermont when he died at the age of 37. In a letter to Oldenburg, Huygens commented that ‘... je regrette extremesment le perte ...’, O.C., vol. 7, p. 314.

25. On a visit to Newton in Cambridge in May 1694, David Gregory reports that ‘I saw a manuscript [written] before the year 1669 ... the principle of equal times of a pendulum suspended between cycloids, before the publication of Huygens' Horologium Oscillatorium. (Correspondence (n. 4), p. 332). D.T. Whiteside dates this manuscript, found among the unpublished scientific papers of Newton, to about 1671. On July 20, 1671 Newton had written to John Collins that “... I began a new methodise that discourses of infinite series ...”, referring to his treatise De Methodis Serierum et Fluxionum. Indeed, one finds that this treatise also contains an application of his new fluxional methods to the theory of evolutes. He derives the differential formula for the evolute of a curve and applies it also to the case of the cycloid (see Mathematical Papers, vol. 3 (n. 10), pp. 161-163). Later, when Collins heard rumors that Huygens intended to bring out a book entitled De Dioptrices et de Evolutione Curvarum he urged Newton to publish his De Methodis, but Newton responded that it was altogether improbable that “their Hypotheses or Deductions can be the same.” (Yoder, Unrolling Time, (n. 14), p. 106). However, in the first publication of Newton's manuscript, Gravia in Trochoide Descendentia, A.R. Hall and M. Boas Hall state their opinion that it is “more reasonable to suppose that he composed this short piece after 1673.” (see Unpublished Scientific Papers (n. 9), p. 170). Likewise in his biography of Newton, Westfall states, without giving any evidence, that “... it was the receipt of Huygens' Horologium Oscillatorium in the summer of 1673 which prompted Newton to develop a demonstration of the isochronous property of the cycloid.” (R.S. Westfall, Never at Rest. Cambridge, 1980, p. 257). Finally, in a commentary, the editors of Huygens's collected works and correspondence claim that “Il est de toute évidence que c'est le lecture de l''Hor. Osc.” qui a amené Newton à considérer l'oscillation cyclodal ...’ O.C., vol. 16, p. 484.

10. The Mathematical Papers of Isaac Newton, 8 vols, ed. D.T. Whiteside. Cambridge, 1967-1981. Here: vol. 3 (1969), pp. 420-431.

26. For example, in an article in the present issue entitled ‘The Melancholic Genius’, C.D. Andriesse claims that Newton had a ‘simpler proof’ which he obtained by ‘the calculus which he had invented eight years before but had kept secret.’ However, Newton's technique and proof was essentially the same as that of Huygens unpublished proof in 1658.

[p. 217]

1. An early sketch by Huygens showing the path KGEL of a weight oscillating a the end of a string HG. This is intercepted at points F and B corresponding to a polygonal cheek HFB. (O.C., vol. 17, p. 17)

calculus by reading Descartes' Géométrie in the second edition (1659)^27. translated from the Latin by Frans van Schooten,^28. who had been Huygens's private tutor and professor of mathematics at the University of Leiden. In this translation, van Schooten wrote extensive commentaries, and included new results obtained by Huygens and by some of his other gifted students Jan Hudde, Hendrik van Heuraet and Jan de Witt. In 1653 Huygens had solved Apollonius' problem of determining the normals to a parabola from a given point by applying Descartes' algebraic geometry to construct and solve graphically a cubic equation (see Fig. 3). At the end of his short paper he pointed out that, depending on the location of this point, there were three possible cases corresponding to the existence of one, two or three normals. The condition for the existence of only two normals to the parabola from a fixed point determines a curve which separates the plane into a domain where there is only a single normal from another domain where there are three normals. This curve corresponds to the evolute of the parabola, but it was not given in this paper. In his original Géométrie (1637), Descartes had used the tangency of a circle to a given point on a curve to obtain the normal to the curve at that point, but apparently the connection of normals to tangent circles was not considered by Huygens at this time. However, a year later Huygens called van Schooten's attention to a special case of the general solution of the intersections of a circle with a parabola,^29. which van Schooten had discussed in a commentary to the third book of Descartes' Géométrie in the first edition (1649) of his Latin translation

27. Francisi à Schooten, Geometria à Renato Des Cartes, Anno 1637 Gallice edita: Cum Notis Florimonde de Beaune, LVGDVNI BATAVORVM. Amsterdami, Ludovicum & Danielem Elzevirios. The first edition, published in 1649, contained extensive commentaries by van Schooten that nearly tripled the length over Descartes' original book. The second edition, which appeared in 1659, included contributions by Huygens and articles by Johannes Hudde and Jan de Witt which lengthen the book by another factor three.

28. An account of Newton's self-education in mathematics is given by Whiteside in the Introduction to the Mathematical Papers of Isaac Newton (n. 10), vol. 1, pp. 3-24).

29. O.C., vol. 1, p. 305. In an appendix to a letter written on October 29, 1653, Huygens includes various comments and corrections to specific pages of van Schooten's first edition (1649) of Descartes' Géométrie. For p. 289 one encounters the following: ‘Hoc annotavi. Potest etiam fieri ut tres hae, DE, CP, NM inter se aequales sint atque in unam rectam conveniant, quae tertia pars erit rectae HI. Quo casu hoc animadversione dignum es, quod in puncto eo quo circulus Parabolam secat, eadem tamen linea recta et circulum et Parabolam contingat.’ (‘In this case the following is worth considering, namely, that although the circle cuts the parabola, yet in this point of intersection one and the same line is tangent to both the circle and the parabola.’ I am indebted to H.J.M. Bos for this translation). Here DE, CP, and NM represent also the magnitudes of the lines normal to the axis of the parabola from three adjacent intersections of a general circle with the parabola, while HI corresponds to the fourth intersection on the opposite branch of the parabola. Van Schooten had demonstrated that if DE, CP and NM lie on one side of the axis, then HI = DE + CP + NM. Hence, for the special case that DE = CP = MN, HI = 3 DE, as Huygens pointed out. Imposing this condition on the relations for the coordinates for the center of the circle given by van Schooten immediately yields the parametric form of the equation for the evolute of the parabola (see footnote 31). It seems likely that this calculation would have been done at some point by van Schooten, but this result does not appear his book.

[p. 218]

2. A picture of Olaf Romer at his transit instrument (from P. Horrebow's Basis Astronomiae ..., Ch. VIII. Copenhagen, 1735) showing on the left side his indispensable Huygens' pendulum clock. Romer's 1676 discovery of a delay in arrival on the earth of the light signals coming from the eclipse of Io, a moon of Jupiter, was made possible by the accuracy of one of Huygens' clocks.

of this work.^30. In this commentary, van Schooten had given the parameters of this circle in terms of the coordinates of three adjacent intersections (see Fig. 4), and Huygens emphasized the special case when these intersections coalesce at a single point.^29. Huygens called special attention to the interesting fact that in this case the circle cuts the parabola while it has a common tangent with the parabola

30. See van Schooten (n. 27), first edition (1649), p. 289.

29. O.C., vol. 1, p. 305. In an appendix to a letter written on October 29, 1653, Huygens includes various comments and corrections to specific pages of van Schooten's first edition (1649) of Descartes' Géométrie. For p. 289 one encounters the following: ‘Hoc annotavi. Potest etiam fieri ut tres hae, DE, CP, NM inter se aequales sint atque in unam rectam conveniant, quae tertia pars erit rectae HI. Quo casu hoc animadversione dignum es, quod in puncto eo quo circulus Parabolam secat, eadem tamen linea recta et circulum et Parabolam contingat.’ (‘In this case the following is worth considering, namely, that although the circle cuts the parabola, yet in this point of intersection one and the same line is tangent to both the circle and the parabola.’ I am indebted to H.J.M. Bos for this translation). Here DE, CP, and NM represent also the magnitudes of the lines normal to the axis of the parabola from three adjacent intersections of a general circle with the parabola, while HI corresponds to the fourth intersection on the opposite branch of the parabola. Van Schooten had demonstrated that if DE, CP and NM lie on one side of the axis, then HI = DE + CP + NM. Hence, for the special case that DE = CP = MN, HI = 3 DE, as Huygens pointed out. Imposing this condition on the relations for the coordinates for the center of the circle given by van Schooten immediately yields the parametric form of the equation for the evolute of the parabola (see footnote 31). It seems likely that this calculation would have been done at some point by van Schooten, but this result does not appear his book.

[p. 219]

3. This page from Huygens' notebook, dated 17 September 1653, shows three graphical evaluations of the normals to a parabola from a fixed point labelled C. Huygens' method to obtain the figures is described in n. 31. The figure on the left hand side and in the middle show a single normal from C to a point A on the parabola, while the right hand figure shows three normals from C. A case for only two normals is not shown, although the existence of this special case is indicated in the text. To find the locus of the points C for this case requires an analytic argument (coalescence of two adjacent normals) which is not presented in the manuscript (Leiden University Library, ms Huyg. 12, f. 122; O.C., vol. 12, pp. 81-82.

at the intersection point (see Fig. 3).^31. This time it may not have escaped Huygens' attention that this limit corresponds to his previous solution when two of the three possible normals from a fixed point to a parabola coalesce. This case determines the center and radius of the osculating circle to the parabola, as Leibniz was to name it 32 years later.^32. As had been pointed out by Descartes in his Géometrie, the coincidence of two intersections of a circle and a parabola determines a tangent circle, that is, at the intersection this circle and the parabola have a common tangent. Then if the radius of this circle is varied until a third intersection coincides with the first two, the center of this circle is also the intersection of two coalescing

31. Referring to Fig. 3, given a point C with x and y coordinates b and a respectively, and a parabola y = x²/r, Huygens determined the intersections of a line from C perpendicular to this parabola by the solutions of the cubic equation x²/r + x(r/2 - a) - br/2 = 0. Huygens obtained this equation by using Descartes' result that the subnormal of a parabola (corresponding here to xdx/dy) is r/2. It is of some historical interest that his cubic equation is the same equation obtained by requiring that the distance from C to the parabola is a minimum (or maximum), corresponding to Apollonius original formulation of the problem. To solve this equation, Huygens gave a geometrical constructio for the solution taken from Descartes. This method solves a cubic equation by obtaining graphically the intersections of a circle with a parabola for the special case that one of these intersections occurs at the vertex. In this case the coordinates of the center of the circle are x_c = b/4 and y_c = a/2 + r/4, while the radius is given by . The condition for a double root of this cubic equation leads to the relation a = -4x³/r² and b = r/2 + 3y, which corresponds to the parametric equations for the evolute, the locus of the center of the osculating circle of the parabola. Substituting y = x²/r gives the evolute in the form of a cubic parabola b² = (16/27)(a - r/2)³. Huygens obtained this result by a different method in (Proposition 11, part 3 of the Horologium Oscillatorium, 20 years after his first letter to van Schooten about this problem. Notice that the auxiliary circle centered at x_c,y_c, which Huygens constructed to find graphical solutions of his cubic equation, is not the osculating circle, although at a double root it also becomes tangent to the parabola while the normal goes through its center.

32. In Unrolling Time (n. 14), p. 104, Yoder reached the surprising conclusion that ‘... Huygens's early work on evolutes also [like Apollonius] had nothing to do with curvature, and only hindsight turns his derivations of companion curves for the cycloid and conics into a method for determining curvature ...’. On the contrary, Huygens's theory of evolutes stems directly from his early effort to approximate segments of the path of an isochronous pendulum weight by the arcs of circles of varying radius (see n. 11). In the limit of a continuous curve these circles become the osculating circles. When Leibniz called Huygens attention to the osculating circle as a measure of curvature (O.C., vol. 10, p. 156), Huygens replied ‘... Vous me parlez à propos de la courbure de la chainette de votre discours de angulo Contactus et Osculi. Vous pouvez bien croire qu'en ce lisant je ne trouvay pas cetre considération nouvelle, parce que le fortes de contact entrent naturellement dans mes Evolutions des Lignes courbes. Je me souviens aussi que longtemps devant que de publier ce Traité [Horologium] j'avois communique a van Schoten quelque remarque la dessus, scavoir de la circonference que coupant une parabole, semble la toucher au mesme point ...’ (O.C., vol. 10, p. 183-184). Indeed, already in 1654 Huygens had indicated to van Schooten how to construct the osculating circle to a parabola by considering it as the limiting case when three of the four possible intersections of a circle with a parabola coalesce (see n. 29). Ironically, Leibniz defined the osculating circle incorrectly as the limiting case when four intersections of a circle and a curve coincide. This definition is valid only at special points of minimum or maximum curvature (see Leibniz, La naissance de calcul differential. Introduction, traduction et notes par Marc Parmentier (Paris 1989) p. 123).

[p. 220]

4. A page of the second edition of van Schooten's Latin translation of Descartes' Géométrie showing the diagram for the intersections of a circle and a parabola, and the insertion (on top of p. 339) of Huygens' condition for an osculating circle.

normals to the parabola. Van Schooten promptly included Huygens' results in his extended commentaries of the second edition (1659) of his book (see Fig. 4).^33. Newton, who in 1664 began his studies of tangents and curvature by reading this edition,^34. would have paid special attention to these results.^35. Another important result included in van Schooten's book was Hudde's method to evaluate double and higher order roots of algebraic equations. Indeed, one of Newton's techniques to obtain the osculating circle, was to apply Hudde's method to obtain the triple roots of the quartic equation for the intersection of a circle with a parabola.^35.36.

In solving Huygens's challenging cycloid pendulum problem, both Pardies^24. and Newton^37. considered a fundamental physical question which seems to have been overlooked by Huygens at that time, and was not discussed in his Horologium Oscillatorium. The question concerns the underlying reason for the isochronous property of the oscillations of a body moving on a cycloidal path under the action of the constant force of gravity. They found that the component of the force of gravity tangent to a cycloidal path is proportional to the distance along this path.^38. This result implies that other physical systems where the force is linear with the distance leads to isochronous or harmonic oscillations. The best known example is the restoring force of a spring. This property of springs was first pointed out by Pardies,^24. and later by Huygens^39. who was apparently aware of Pardies' work, and

33. Without reference to Huygens, van Schooten added his comment (n. 29) to the second edition (1659) of his book, in essentially the same form. ‘Hence it proves correct (and it is certainly worth our attention) that if the straight line [tangent] which touches the parabola in any point outside its vertex is in that very place also touched by a circle, which does not pass through the vertex, and which intersects the parabola at the same point, it is the case that the straight lines NM, CB and DE should all three be equal among themselves: indeed at that moment HI will be triple of NM, CB or DE.’ (n. 27, p. 339, see Fig. 4). (I am indebted to N. van Saarloos Smits for the translation of the original Latin version).

34. Whiteside states that ‘there can be no doubt that Newton read de Geometrie in Schooten's second edition. All his explicit references to it in particular are quoted by page from the latter.’ (Mathematical Papers (n. 10), vol. 1, p. 7, n. 17).

35. Referring to a figure similar to those found in van Schooten's commentary on this problem (see Fig. 4), Newton describes his calculation dated February 1664 as follows: ‘There is another way of finding the crookedness [curvature] in lines and that is by not supposing two perpendiculars [Newton's and Huygens' preferred method], but by three intersections of a circle with the figure [parabola] ... Supposing it to have all three equall rootes, by Huddenius method ... that equation produceth ... d = r/2 + 3x.’ (Mathematical Papers (n. 10), vol. 1, p. 262).

35. Referring to a figure similar to those found in van Schooten's commentary on this problem (see Fig. 4), Newton describes his calculation dated February 1664 as follows: ‘There is another way of finding the crookedness [curvature] in lines and that is by not supposing two perpendiculars [Newton's and Huygens' preferred method], but by three intersections of a circle with the figure [parabola] ... Supposing it to have all three equall rootes, by Huddenius method ... that equation produceth ... d = r/2 + 3x.’ (Mathematical Papers (n. 10), vol. 1, p. 262).

36. Newton also learned probability theory from an article written by Huygens, ‘Treatise on calculations for games of chance’, written in Dutch, but translated into Latin in an appendix of van Schooten's book Exercitationum mathematicorum libri quinque, published in Leiden, 1657. The original text and a French translation appears in O.C., vol. 14, pp. 1-179.

24. O.C., vol. 18, p. 487, n. 5; Ziggelaar, ‘Démonstrations’ (n. 19). A very promising scientist, Pardies was a professor of mathematics and physics at the College de Clermont when he died at the age of 37. In a letter to Oldenburg, Huygens commented that ‘... je regrette extremesment le perte ...’, O.C., vol. 7, p. 314.

37. Mathematical Papers (n. 10), vol. 3, p. 422-423.

38. Later, when writing the Principia, Newton included in book 1 Proposition 53 which reads: ‘Granting the quadrature of curvilinear figures, it is required to find the forces with which bodies moving in a curvilinear line always perform their oscillations in equal times.’

24. O.C., vol. 18, p. 487, n. 5; Ziggelaar, ‘Démonstrations’ (n. 19). A very promising scientist, Pardies was a professor of mathematics and physics at the College de Clermont when he died at the age of 37. In a letter to Oldenburg, Huygens commented that ‘... je regrette extremesment le perte ...’, O.C., vol. 7, p. 314.

39. O.C., vol. 18, p. 497. Here Huygens discussed briefly his general concept of accelerating force which he called ‘incitation’.

[p. 221]

also by Robert Hooke^40. (it is now known as Hooke's law). Both Huygens and Hooke applied this property to develop a spring balance to regulate a clock as a substitute for the pendulum, and presumably for this reason Huygens withheld his result from publication, while Hooke delayed writing his account. Thus, the discovery of one of our most fruitful results in physics, harmonic oscillations, appeared first through the study of Huygens' cycloid pendulum (an early example of applied research leading to fundamental physics).

After discovering mathematically that the cycloid is the isochronous path of a weight moving under the action of gravity, Huygens proceeded to study, also mathematically, the shape of the cheeks restricting a weight suspended by a string to move along this curve. This study led him to develop further mathematical methods to obtain the evolute or center of curvature of a given curve. The evolute corresponds precisely to the curve for the cheeks which restricted the motion of the pendulum weight on a given path.^16. For the cycloidal path, he found that its evolute is a similar cycloid.^14. Seven years later, unaware of Huygens' result, Newton also developed the theory of evolutes along precisely the same lines. Given a curve, which Huygens calls the evolute, he considered another curve now called the evolvent or involute which is described, in his own words, as follows:

‘If a string or flexible line is understood to be stretched around a curve in one direction, and if one end of the string remains fixed while the other end is pulled away such that the freed part of the line always remains taut, then it is clear that some other curve is described by this end of the string. This latter is called the curve “described by evolution.”’^2.

The mechanical origin of this definition, based on the construction of a pendulum with cheeks, is further in evidence as he continues:

‘The curve around which the string has been stretched is called the evolute.’^2.

Since the length of the string at a given point on the evolute is equal to the corresponding arc length of this curve, to obtain the evolvent requires the length of an arc (rectification) of the evolute, which is a problem of integration.^41. However, in Proposition 1 of the Horologium Oscillatorium, Huygens proves that

‘Every straight line tangent to an evolute meets a line described by evolution at right angles.’

Hence Huygens considered the problem of obtaining the evolute given an evolvent by finding the envelope of the lines perpendicular to the evolvent. A point on the evolute is obtained by considering the intersection of two adjacent lines each of which is perpendicular to the evolvent in the limit that these lines approach each other. The length of the limit line corresponds to the radius of curvature of the evolvent, although the term curvature was not used by Huygens in his book.^11.32. Huygens' construction, given in Proposition 11 corresponds precisely to one of the methods developed by Newton in 1664 to determined the curvature or evolute of a parabola and of several other curves.^42. In subsequent years, Newton extended his technique further, and by 1671 he even obtained the evolute of a curve in polar coordinates.^43. In his tract on methods of series and fluxions problem 5, to find the curvature of any curve at a given point, begins with ‘the problem has the mark of

40. R. Hooke, De Potentia Restitutiva or of Spring: Explaining the Power of Springing Bodies. London, 1678. Reprinted in R.T. Gunther, Early Science in Oxford, vol. 8. Oxford, 1931, pp. 331-356. Hooke writes that ‘About two years ago, since I printed this Theory in an Anagram at the end of my Book of the Description of Helioscopes, (...), id est, Ut tensio sic vis; That is, the Power of any Spring is in the same proportion with the Tension thereof ...’. While Hooke deduced correctly the energy conservation principle for a spring (shown prominently in the frontispiece of his book as diagram of an ellipse in phase space), he was unable to obtain the correct harmonic time dependence for the oscillations.

16. Since Huygens knew from Galileo and Mersenne that the period of a simple pendulum depends on its length, (more precisely, on the square root of the length) he installed a sequence of nails, as shown in Fig. 1, to shorten the length of the suspension when the amplitude of the swing increased above certain values. Such an arrangement corresponds to a cheek shaped as a polygon, which determines a trajectory for the pendulum weight as a sequence of joined arcs of circles of different radii. The centers of these circles are located at the vertices of the polygonal cheek, and the radius of a given circular arc is determined by the length of the suspension string minus the total length of the sides of the polygonal cheek which are in contact with the string. A pendulum cheek made of a continuous curve can be considered as the limit of a polygonal cheek with and infinite number of vertices, thus elucidating the origin of some of Huygens' fundamental ideas on evolutes. Thus, Huygens's early concept of the osculating circle, as an approximation to an arc of a general curve becomes evident. A similar description based on a polygonal evolute was used by Bernoulli and by L'Hospital to explain Huygens' method of evolution.

14. J.G. Yoder, Unrolling Time. Cambridge, 1989; O.C., vol. 16, p. 304.

2. C. Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris, 1673. Reproduced in O.C., vol. 8. English translation by Richard J. Blackwell, Christiaan Huygens' The Pendulum Clock. Ames, 1986.

2. C. Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris, 1673. Reproduced in O.C., vol. 8. English translation by Richard J. Blackwell, Christiaan Huygens' The Pendulum Clock. Ames, 1986.

41. In modern notation, the integral for the arc length of a curve given in cartesian coordinates as a function y(x), is . For a parabola y = x²/r and dy/dx = 2x/r which implies that the integrand is a hyperbola. This result was first obtained by Huygens in 1657 (Yoder, Unrolling Time (n. 14), p. 117). Probably inspired by Hugyens, the general expression for the arc length as an integral was introduced in an equivalent geometrical form by Hendrik van Heuraet who communicated it to van Schooten in a letter written January 13, 1659, and was published in van Schooten, Geometria (n. 27), pp. 517-520. In Unrolling Time (n. 14), p. 129-130, Yoder claims that ‘... with Huygens' theory of evolution the arc length was directly and universally attainable ...’ and that with his technique ‘Huygens had provided rectification for all the important curves of the seventeenth century’. Both of these statements are not correct, and Yoder later on admits (p. 147) that ‘... Huygens could not rectify the circle or any other curve’. Although Huygens, as well as Newton, provided a mechanical approach to measure the approximate arc length of a given curve by wrapping a physical string around a model of the curve, his method of evolution did not provide a mathematical algorithm to evaluate this are length. His mathematical method determined the radius of curvature of a given curve, the evolvent, which corresponds to the arc length of another curve, the evolute. To obtain the arc length of a given curve by Huygens' method of evolution it would be necessary to know in advance its evolvent. However, even for such a simple curve as a parabola this is not the case, and therefore its arc length cannot be calculated by this method. (For a similar critique, see also H.J.M. Bos' review of Yoder's book in: Tractrix 3 (1991), pp. 183-194).

11. Although Huygens did not discuss explicitly the concept of curvature in the Horologium Oscillatorium, the idea of approximating a small segment of a curve by the arc of a circle, later named the osculating circle by Leibniz, was central to his method of evolution. Curvature or ‘crookedness’, as Newton first named it in 1664, is the measure for the bending of a curve given by the inverse of the radius of this circle also called the radius of curvature. In his lectures to the Marquis de L'Hospital in 1691, Johann Bernoulli devoted lesson number 15 to Huygens' method of evolution and its connection to the osculating circle (J. Bernoulli, Die erste Integralrechnung. Ostwald Klassiker nr. 194, Leipzig und Berlin, 1914, pp. 63-67). His derivation of a general formula for the radius of curvature (lesson 16) is the same as that given in prose by Huygens in Proposition 11, part 3, of the Horologium Oscillatorium. Incidentally, when Bernoulli showed an application of this formula to L'Hospital, the Marquis was greatly impressed and hired Bernoulli as his tutor (see Der Briefwechsel von Johann Bernoulli. Basel, 1955, p. 136). In 1696 L'Hospital published the first textbook on calculus, Analyse des infinitement Petits..., based mostly on Bernoulli's lectures, and he too devoted and entire section entitled ‘Usage des calcul des differences pour trouver les Dévelopées’ to Huygens' theory of evolutes, reproducing essentially lesson 15 of Bernoulli. For a further discussion of Johann and Jakob Bernoulli's application of Leibniz's differentials to curvature see H.J.M. Bos, ‘Differentials and Derivatives in Leibniz's Calculus’. In: Archives for History of Exact Sciences, 14 (1974), pp. 1-90.

32. In Unrolling Time (n. 14), p. 104, Yoder reached the surprising conclusion that ‘... Huygens's early work on evolutes also [like Apollonius] had nothing to do with curvature, and only hindsight turns his derivations of companion curves for the cycloid and conics into a method for determining curvature ...’. On the contrary, Huygens's theory of evolutes stems directly from his early effort to approximate segments of the path of an isochronous pendulum weight by the arcs of circles of varying radius (see n. 11). In the limit of a continuous curve these circles become the osculating circles. When Leibniz called Huygens attention to the osculating circle as a measure of curvature (O.C., vol. 10, p. 156), Huygens replied ‘... Vous me parlez à propos de la courbure de la chainette de votre discours de angulo Contactus et Osculi. Vous pouvez bien croire qu'en ce lisant je ne trouvay pas cetre considération nouvelle, parce que le fortes de contact entrent naturellement dans mes Evolutions des Lignes courbes. Je me souviens aussi que longtemps devant que de publier ce Traité [Horologium] j'avois communique a van Schoten quelque remarque la dessus, scavoir de la circonference que coupant une parabole, semble la toucher au mesme point ...’ (O.C., vol. 10, p. 183-184). Indeed, already in 1654 Huygens had indicated to van Schooten how to construct the osculating circle to a parabola by considering it as the limiting case when three of the four possible intersections of a circle with a parabola coalesce (see n. 29). Ironically, Leibniz defined the osculating circle incorrectly as the limiting case when four intersections of a circle and a curve coincide. This definition is valid only at special points of minimum or maximum curvature (see Leibniz, La naissance de calcul differential. Introduction, traduction et notes par Marc Parmentier (Paris 1989) p. 123).

42. Mathematical Papers (n. 10), vol. 1, p. 245-272.

43. Ibid. (n. 10), vol. 3, pp. 432-441.

[p. 222]

exceptional elegance and of being pre-eminently useful in the science of curves’. Moreover, the geometrical and algebraic calculus techniques developed by Newton to evaluate the evolute were virtually identical to those of Huygens. In describing his results for the evolute of a parabola, Newton states

‘... note that the crooked line [evolute] ... is always touched by the perpendicular line [to the evolvent] in such sort as to be measured by it ... By this means the length of as many crooked lines may be found as desired ...’,^44.

which echoes Huygens' formulation quoted above, although Newton could not have seen Huygens's great work at that time.

44. Ibid. (n. 10), vol. 1, p. 263.

De Vi Centrifuga

In dealing with the dynamics of the pendulum, Huygens was led to consider the force or tension restricting a body to circular motion. For uniform velocity v (in the absence of external forces) he obtained the celebrated result that this force, which he named centrifugal force, is proportional to the acceleration a,

(1)

where ρ is the radius of the circle. Although Huygens had obtained this result already in 1659, an announcement of this result first appeared at the end of the 1673 Horologium Oscillatorium, and his calculations and proofs where published only posthumously in 1704 in the Vi Centrifuga.^45. Newton's Waste Book reveals that by 1664 he had also found the same relation, Eq. 1.^46. It is interesting that for one of the first application of this result, both consider the effect of the rotation of the earth on the net gravitational attraction at the equator. Since ancient times the main argument against the rotation of the earth as an explanation of the apparent daily rotation of the sun and the stars had been that objects would then fly off its surface. Galileo had pursued this question earlier, but gave an incorrect solution. Applying Eq. 1 with ρ equal to the radius of the earth, and v = 2πρ/T, where T is the period of rotation, Huygens^14. calculated for the ratio of gravity to centrifugal acceleration g/a = 280, while Newton^47. obtained the value 350, thus putting finally to rest the objections against the rotation of the earth (the difference in values of g/a was due mainly to different estimates of the radius of the earth). However, Newton went a step further in applying Eq. 1 to obtain also the radial dependence of the gravitational force. From the observation that the period of the planets, which rotate approximately in circular orbits around the sun, obeyed Kepler's third (harmonic) law (that the square of the period varied as the cube of the radius), he could deduce from Eq. 1 that the gravitational force or acceleration must depend on the inverse square of the distance between the sun and a planet. There is no evidence that Huygens also made this crucial deduction which was obtained later by Wren, Hooke and Halley, as acknowledged by Newton in the Scholium to Proposition 4 of Book 1 of the Principia.

Huygens considered also the tension or force on the string suspending a pendulum weight due to the combined effect of gravity and motion along a circular

45. O.C., vol. 16, pp. 256-328.

46. J.W. Herivel, The Background to Newton's Principia. Oxford, 1965, pp. 1-13.

14. J.G. Yoder, Unrolling Time. Cambridge, 1989; O.C., vol. 16, p. 304.

47. Ibid., p. 196.

[p. 223]

arc. In particular, if the suspension point for the oscillation of a simple pendulum is abruptly changed by means of an intervening nail, as envisioned originally by Galileo, Huygens found the correct condition (missed by Galileo) for which the pendulum weight would wrap itself around the nail.^48. Although he applied his method to obtain the evolute for several curves including conic sections, there does not appear to be evidence that he considered the corresponding tension for bodies moving on these curves under the action of gravity. While Huygens restricted his dynamical investigations to the motion of bodies constrained (by strings and cheeks) to move along prescribed trajectories under the action of a constant (gravitational) force, as early as 1664 Newton began to consider the unconstrained motion of bodies under the action of central forces. One of his motivations appears to have been to understand Kepler's law that planetary motion occurs along elliptical orbits with the sun at one of the foci. At that time Newton had written in his notebook (Waste book)^49. that

‘If the body b moves in an Ellipsis, then its force in each point (if its motion in that point be given) may be found by a tangent circle of equal crookedness with that point of the Ellipsis.’

As Newton's mathematical papers reveal, the term crookedness was his early expression for curvature, and tangent circle corresponds to the osculating circle found by Huygens earlier^30. as named by Leibniz 32 years later. If ρ is the radius of curvature at a given point of an ellipse on which a body moves with varying velocity (motion) v, then the result which he, and independently Huygens, had obtained for circular motion, Eq. 1, applies in this case to the component of the acceleration or force perpendicular to the orbit. For central forces this component is a sin(α), where α is the angle between the tangential and the radial direction, leading to a straightforward generalization of Eq. 1,

(2)

Evidently this is the meaning in mathematical form of Newton's cryptic remark. However, there remains the question of determining the velocity v along the ellipse which Newton takes as given. For motion along an ellipse or along any other noncircular orbit this velocity is not a constant because there is also a component of the force or acceleration which is tangential to the orbit, and is given for central forces by a cos(α). This tangential component of the acceleration determines the time rate of change in velocity, dv/dt = acos(α), and therefore the velocity can be evaluated from given initial conditions. Since the radial component of the velocity determines the time rate of change of radial position, dr/dt = - v cos(α), eliminating the time variable gives immediately the relation

(3)

This equation is the differential form for central forces of the principle of conservation of energy which both Huygens and Newton originally had applied to solve the dynamics of the pendulum for constant gravitation acceleration, a = g. Newton also applied it to harmonic oscillator motion, where a = cr and c is a constant and

48. Galileo thought that the pendulum would wrap itself around the nail if it was located at an altitude half way from the initial height of the pendulum weight, although an experiment would have shown that this is not the case. The correct answer given by Huygens, is that the nail should be at least a factor 3/5 below the initial height, in order to maintain the string taut when the weight reaches maximal height. This calculation required a knowledge of Huygens's vis centrifuga relation for the acceleration, a = v²/r, which in this case must be equal or greater than the pull of gravity, measured by g, at the top of the swing.

49. Herivel, Background (n. 46), p. 130.

30. See van Schooten (n. 27), first edition (1649), p. 289.

[p. 224]

5. (left) This page from Newton's Waste book, dated February 1664 and entitled ‘The crookedness [curvature] in lines may be otherwise found as in the following examples’, shows at the top of the left hand margin a diagram of an osculating circle intersecting a parabola. The parameters of this circle are evaluated, as described in the text, by solving for the triple root of a quartic polynomial obtained from the intersections of a circle with a parabola. The diagram below shows the four intersections a general circle with a parabola which bears a close resemblance to diagrams in van Schooten's Geometria (see Fig. 4). (By permission of the Syndics of Cambridge University Library; Mathematical Papers (n. 10), vol. 1, pp. 259-262.)

[p. 225]

later he gave an equivalent geometrical derivation of this principle in Proposition 39, book 1 of the Principia. Given the radial dependence of the central force a, Eq. 3 can be integrated to determine the radial dependence of v² up to a constant corresponding to the total energy E, where v²/2 = - f adr + E. Substituting this expression for v² in the centrifugal force relation, Eq. 1, determines the radial dependence of ρsin(α), a purely geometrical quantity, in terms of v²/a which depends only on the dynamics, i.e. the force law and the initial conditions. Orbits satisfying these relations can be easily computed approximately by a straightforward generalization of Huygens' and Newton's method of evolution (see Fig. 5), and there is considerable circumstantial evidence that this method was indeed applied by Newton in his earliest computations of orbital motion for central forces.^7. In this case neither the evolute, which corresponds to the orbital curve, nor the evolvent is known in advanced, but both curves are obtained from the condition that the radius of curvature ρ and the relative angle α between the tangential and radial directions, satisfy Eqs. 2 and 3.

Assuming a given curve for orbital motion under the action of central force, Newton could also evaluate the curvature ρ and angle α and then apply Eqs. 2 and 3 to solve for the force. However, in practice only a few simple curves, e.g. conic sections and spirals, provide an analytic solution, while other cases can be solved only approximately.^50. The simplest case is the spiral (equiangular) orbit which Newton first mentioned in his December 13, 1679 letter to Hooke,^7.51. without however giving the corresponding force law. By 1671 Newton had shown that for such a spiral the quantity ρsin(α) = r. Hence Eq. 2 for the force or acceleration implies that v² = ar, or in differential form, that dv²/dr = rda/dr + a. Combining this result with the differential form of the principle of energy conservation, Eq. 3, gives rda/dr = - 3a, which implies that ar³ is a constant. Thus, for this spiral orbit the central force or acceleration depends inversely with the cube of the distance. Newton first revealed his remarkable result, without however giving any proof, in a Scholium of his 1684 De Motu,^52. the earliest known draft of some of the fundamental propositions of the Principia. Later a geometrical scaling proof appeared in Proposition 9 of the first book of the Principia. A similar calculation for conic sections leads to Newton's fundamental result that the corresponding force varies inversely as the square of the distance,^53. forming the basis of his theory of gravitation.

7. M. Nauenberg, ‘Newton's Early Computational Method for Dynamics’. In: Archive for History of Exact Sciences, 46 (1994), pp. 221-252.

50. Given an initial velocity v and position r, the tangent to the orbit at r is taken along this velocity and therefore the angle α between the velocity and radial direction is determined. Then the radius ρ of the osculating circle at r can be evaluated from Eq. 1, ρ = ν²/asin(α), and its center located at a distance ρ along the normal to the direction of the initial velocity. Rotating a ‘string’ of length ρ about this center through a small angle δφ determines a small arc of length δs = ρδφ of the orbit or evolute. At the end of this arc new values of the radial distance r' = r + δr and the angle α' = α + δα can be obtained geometrically or analytically, where δr = ρcos(α)δφ and δα = (1-(ρ/r)sin(α)δφ, while the new velocity ν' = ν + δν is then given by the conservation of energy relation, Eq. 3. The corresponding new value of the radius of curvature ρ' = ρ + δρ is then determined analytically by substitution in Eq. 1 which to first order gives δρ = ρcos(α)(1/sin(α) + ρ/r + (ρ/a)da/dr)δφ. Taking the new center of curvature at a distance ρ' along the normal to the arc at its end point, this ‘evolution’ can then be repeated as before. These steps can be iterated to obtain an approximation to the orbit or evolute as a sequence of arcs of circles with centers at the vertices of a polygon which approximates the evolvent. As expected, the length of an edge of this polygonal evolute is given by the change of the radius of curvature δρ of a corresponding arc of the evolvent. The magnitude of the changes in parameters at each step of the evolution are proportional to the magnitude of the value of δφ which can be chosen to obtain a resulting orbit to any degree of approximation. The exact evolute and evolvent is obtained in the limit that the angle δφ vanishes (see ref. 6).

7. M. Nauenberg, ‘Newton's Early Computational Method for Dynamics’. In: Archive for History of Exact Sciences, 46 (1994), pp. 221-252.

51. The Correspondence of Isaac Newton, vol. 2 (1676-1687), ed. H.W. Turnbull. Cambridge, 1960, pp. 304-306.

52. Mathematical Papers (n. 10), vol. 6, pp. 30-74; Herivel, Background (n. 46), pp.

53. For a conic section ρ sin(ρ) = 2r - cr², where c is a constant (c = 0 for a parabola, c>0 for an ellipse and c<0 for a hyperbola) and the corresponding application of Eqs. 2 and 3 gives ar² is a constant (see n. 7).

Summary

Huygens' development of cheeks to construct an isochronous pendulum clock, led him to his method of evolution which determines one curve, the evolvent, by the

[p. 226]

mechanical operation of unwinding a taut string wrapped about another curve, the evolute (cheeks). In Proposition 2, part 3 of the Horologium Oscillatorium he presented a mathematical method for the converse construction:

‘Given a curve line [evolvent], find another curve [evolute] whose evolution describes it ...’

His mathematical procedure, developed also by Newton, was to determine the locus of the intersection points of two adjacent normals to a curve in the limit that these two normals coalesce. This limit corresponds to a special solution of Apollonius's problem, which Huygens had obtained for the parabola in 1653, and again in 1654 for a special limit of the intersection of the parabola with a circle, which he had brought to van Schooten's attention. The dynamics of the pendulum also led him to consider the force or tension on the string, and he obtained the fundamental relation for centrifugal acceleration for circular motion with uniform velocity, Eq. 1.

Almost a decade before the publication of Huygens' great work, the Horologium Oscillatorium, Newton followed an analogous path of discovery to develop concepts of curvature and of centrifugal force, applying similar mathematical methods and physical insights to problems that Huygens had solved earlier. This was not a coincidence, but is due in great part to the publication of a translation into Latin of Descartes' Géométrie by Frans van Schooten, a mathematics professor at the University of Leiden. In this book, van Schooten included extensive commentaries and original contributions by some of his students which included Huygens himself. The influence of this book on Newton's early development in mathematics cannot be overestimated.^54.

Newton generalized the concept of centrifugal force to non-circular orbits, and the energy conservation principle to arbitrary central forces. This enabled him to obtain, for the first time, the geometrical properties of orbital motion for general central forces and to solve the fundamental problem of celestial mechanics in the seventeenth century concerning the physical origin of Kepler's law of planetary motion. In the first edition (1687) of the Principia, Newton took a different approach to describe orbital dynamics, based on suggestions made in 1679 by Hooke,^7.55. and his concepts of curvature and of centrifugal force were hidden in some lemma's and corollaries.^56. This has made it quite difficult to see the connection of his major work to Huygens' work and to his own earlier concepts of curvature and centrifugal force. Already by 1687 Newton had applied these principles, although without much explanation, in some of his calculations of the perturbations of the lunar motion by the solar gravitation force, in corollaries of Proposition 66 in book 1, and Proposition 28 in book 3 of the Principia. However, only in the two later editions (1713 and 1726) of the Principia, did Newton apply his approach based on curvature and generalized centrifugal force as an alternate method of solution of his fundamental propositions. For this reason it had been thought that what appears now to be his earliest dynamics 7 represented instead his later more mature dynamics.^57.58.

54. D.T. Whiteside writes that ‘... we now know that Huygens and Newton, pursuing research into the same problem of tangents, curvature and inflexion points and alike inspired by a reading of Descartes' Géométrie in Schooten's Latin version, but neither aware of the others existence, made many identical discoveries ...’ (Mathematical Papers (n. 10), vol. 1, p. 147-148, n. 14). In his advise to Richard Bentley quoted in the introduction about which mathematics books to read in preparation for an understanding of the Principia, Newton also told him that ‘... Next after Euclids Elements the Elements of the Conic sections are to be understood, and to this end you may read the first part of the Elementa Curvarum of John de Witt ... peruse such Problems as you will find scattered up and down in the Commentaries on Cartes's Geometry and other Algebraic writings of Francis Schoten ... (Mathematical Papers (n. 10), vol. 1, p. 24).

7. M. Nauenberg, ‘Newton's Early Computational Method for Dynamics’. In: Archive for History of Exact Sciences, 46 (1994), pp. 221-252.

55. M. Nauenberg, ‘Hooke, orbital motion and Newton's Principia’. In: American Journal of Physics 62 (1994), pp. 331-350.

56. In lemma 11 of book 1 of the Principia Newton discusses the ‘evanescence subtense of the angle of contact ...’ and gives a mathematical construction for the diameter of the osculating circle, without defining this circle explicitly or the concept of curvature. During his bitter dispute with Leibniz on the development of the calculus, Newton remarked ‘... Leibniz gives names to concepts which I invented.’

57. J.B. Brackenridge, ‘The critical role of curvature in Newton's dynamics’. In: P.M. Harman and A.E. Shapiro eds., The investigation of difficult things. Essays on newton and the history of the exact sciences in honour of D.T. Whiteside. Cambridge, 1992, pp. 231-260.

58. J.B. Brackenridge, The Key to Newton's Dynamics. University of California Press, 1995. The author concludes that ‘if any single measure deserves the title of the key to Newton's dynamics it is the curvature measure’.

[p. 227]

Acknowledgements

I would like to thank H.J.M. Bos and D.T. Whiteside for many useful comments, and my colleagues in the Physics Departments of the University of Leiden and the University of Amsterdam where this work was completed for their kind hospitality.