HUYGENS AND INFINITESIMALS Herman Erlichson Department of Engineering Science and Physics The College of Staten Island The City University of New York Staten Island, NY 10314, USA E-mail: erlichson@postbox.csi.cuny.edu ABSTRACT. Christiann Huygens did not want to use infinitesimals in his published book, the Horologium Oscillatorium. The reason for this is that he felt he had to go the route of a double absurdity proof when he was producing a formal proof for publication. However it is interesting to note that following Proposition XXI of the Horologium there is an addendum proposition which is a "passage to the limit" proof which is tantamount to the use of the forbidden infinitesimals. We explore these matters in this paper. 1. Introduction In this paper we investigate Huygens' use of infinitesimals in the corollary-like statement following Proposition XXI of his Horologium Oscillatorium. This book was published in Latin in 1673. It has been translated into English as The Pendulum Clock. We shall see in what follows that Huygens in his Proposition XXI used an "implicit infinitesimals" proof. The Pendulum Clock is in 5 parts- I. A Description of the Pendulum Clock, II. The Falling of Heavy Bodies and Their Motion in a Cycloid, III. The Evolution and Dimension of Curved Lines, IV. The Center of Oscillation, and V. Another Construction, Based on the Circular Motion of Pendula, and the Theorems on Centrifugal Force. This paper concerns itself only with Part II. Huygens' discovery of the isochronism of the cycloid was made while he was pursuing the expression for the limiting period of a simple pendulum. Huygens knew that for very small oscillations a simple pendulum was very close to isochronous; that is, for small angles the period of the pendulum is essentially independent of the angle. He sought the limiting value of the pendulum period as the angle became vanishingly small. The date Huygens wrote on the first page of his manuscript concerning his work on this question was December 1, 1659. Since his work on this problem led to his fortunate discovery of the cycloid as the isochronal curve, this is also the date he announces for that discovery [ 1, vol. 16, p. 392]. Joella Yoder, in her book ,Unrolling Time, [3], presented a careful and comprehensive account of Huygens' discovery of the isochronism of the cycloid. An excellent introduction to the English translation of the Horologium was written by the Huygens scholar Henk Bos. With respect to the subject matter of the present paper Bos wrote " after having found that the cycloid is the path of isochronic oscillation, Huygens proceeded to prove that result directly. That proof, with the preliminary mechanical and geometrical theory, forms the second part of the Horolgium Oscillatorium. It is formulated with utmost precision,meeting all standards of rigorous geometrical and mechanical argument." [2, p. xiii]. The double absurdity proof of the Horologium was done to avoid infinitesimals and passage to the limit of a sum of infinitesimals. Part II of the Pendulum Clock has 26 Propositions in it. Our attention will be focussed on Proposition XXI. 2. Proposition XXI The wording of the proposition and the diagram which accompanied it are shown below: "Let a body descend by a continuous motion through any number of contiguous planes, and later let it descend from the same height through another series of an equal number of contiguous planes. Let the latter series be constructed in such a way that each plane corresponds in height to another plane in the first series, but let the planes in the second series have a larger inclination than those in the first series. Now I say that the time of descent through the less inclined planes will be less than the time of descent through the more inclined planes" [2, p. 58]. By "inclination" Huygens means the angle between a plane and the vertical. We are used to 'inclination" meaning the angle between a plane and the horizontal. In Huygens' diagram the planes of larger inclination are those in the second series on the right.The proof of the proposition is straightforward. The acceleration on each plane is constant and the difficult matter of a continuously varying acceleration is not present. After the proof Huygens continues, "Next, if we consider curved lines to be composed of innumerable straight lines, then from the above it is clear that if there be two inclined surfaces which correspond to curved lines of the same height, and if at every point of the same height the inclination of the one surface is always larger than the other, then a body will fall through the less inclined surface in a shorter time than through the greater inclined surface" [2, p. 59]. We shall refer to this proposition as the "addendum proposition". The addendundum proposition shows what may be called Huygens' 'implicit' use of infinitesimals. What he has in mind is a division of the two curves into N parts, as shown in Fig. 2. Each section of each curve is approximated by a straight line as shown in the figure. Say that AB is the steeper curve. Then Huygens has by Proposition XXI that tAB < tCD, where the times are descent times along the two sets of contiguous planes. In the limit NÆ*, the inequality continues to hold for the limit of the contiguous planes, i.e., for the two curves. One would have thought that Huygens would have applied this technique to the result of Proposition XXIII, and quickly have achieved the cycloid descent time by summing small descent times and passing to the limit. But he was obviously not satisfied by the rigor of such a passage to the limit. It is noteworthy that the corollary-like statement following Proposition XXI was not given any formal status by Huygens in his Pendulum Clock. Passage to the limit of a sum was not accepted in a formal proof by Huygens. Thus he went the long and cumbersome route of his double absurdity proof. Huygens simply wasn't ready to sum infinitesimals in a formal proof. 3. The Double Absurdity Proof and Passage to the Limit Another point which should be raised is that Huygens' carefully constructed double absurdity proof has within it, not surprisingly, the idea of a limit. Let us consider that proof. Figure 3 is taken from The Pendulum Clock. It is Huygens' diagram for his first absurdity proof of the Horologium. The cycloid is curve BA. The point N must be permitted to come arbitrarily close to the point B. The distance from F to G is divided by the equally spaced parallel lines shown, with the distance from between these lines less than the height of N above B. If the width of the parallel strip spacing FP, QR, ... is less than the height of N above B, when the point N approaches arbitrarily close to the point B, the strip width becomes vanishingly small. Under these circumstances the sum of the infinitesimal cycloid tangents is identical with the cycloid itself. Hence, in this sense Huygens' ingenious double absurdity proof is equivalent to a simple direct argument where one does a sum and passes to the limit NÆ*. Of course, this is exactly what Huygens was trying to avoid by his double absurdity proof. Huygens, in the Horologium, does not deal with this point. He merely says: "it is clear that the point N can be taken so close to B that the difference of these times is as small as one wishes, and hence would be less than the time by which Z is exceeded by the time through BE. Let the point N be so chosen" [2, p. 65]. With the infinitesimally thin strips we have the use of infinitesimal methods here. The addendum proposition of the Horologium is a passage to the limit proposition which stands in sharp contrast to the classical reductio ad absurdum proof. References [1] Huygens, Christiaan, Horologium Oscillatorium sive De Motu Pendulorum ad Horologia aptato Demonstrationes Geometricae, originally published in Paris in 1673. The original Latin of the 1673 edition, with translation into French, can be found in Huygens, Christiann, 'uvres Complètes de Christiann Huygens, published by La Société Hollandaise des Sciences, 22 vols., The Hague: Martinus Nijhoff, 1988-1950. 11