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the parabola as a particular case. His method consists in transforming the differential polynomial, which represents the elementary, elliptical, or hyperbolical arc, into another polynomial negatively similar; from which an algebraic quantity results by subtraction, and the subsequent resolution. The glory of having explored this nook of geometry, if I may so say, has placed Fagnani among the most subtile analysts.
A long time after this, in 1756, Euler having considered the same subject, not only resolved the problems of Fagnani in a new manner, but attained a method of resolving a very extensive class of separate differential equations, the two sides of which, though uot resolvable separately, form a whole perfectly resolvable. How to resolve equations of this kind, when the two sides depend at the same time on arcs of the circle, or on logarithms, was already known; but the new solutions of Euler were much more extensive they form a new, pleasing, and very useful branch of the integral calculus, in which the author displays all the resources of genius, and the profoundest knowledge of analysis.
Viviani's problem of the quadrature of the hemispherical vault, long after, gave rise to another of a similar nature, proposed by a geometrician in other respects little known, Ernest von Offenburg, in the Leipsic Transactions for 1718. This was, to pierce a hemispherical vault with any number of oval windows, with the condition that their circumferences should be expressed by algebraic quantities: in other words, it was required, to determine on the surface of a sphere curves algebraically rectifiable. It is ob
vious at first sight, that the curves sought could not be formed by the intersection of the sphere by a plane, for all these intersections, in whatever direc tion made, must be mere circles: but they belong to the class of curves of double curvature. This problem, though curious and difficult, remained a long time unattempted, and it is not known, whether it were resolved by the author.
Hermann, in a paper on the rectification of spherical epicycloids, in the Petersburg Transactions for 1726, imagined, that these curves generally satisfied the question of Offenburg, or were algebraically rectifiable but this they are only in certain particular cases, the rectification of spherical epicycloids depending in general on the quadrature of the hyperbola. In the Memoirs of the Academy of Paris for 1732, John Bernoulli pointed out Hermann's mistake; and, not content with assigning the true algebraic and rectifiable epicycloid, he directly and a priori resolved the problem of Offenburg; that is, he gave the general method of determining the rectifiable curves, that may be traced on the surface of a sphere. He then proposed the same research to Maupertuis, as the most eminent french geometrician of the time, offering to send his own solution, if it were desired. The offer was accepted; and while the solution of Bernoulli was on the road, Maupertuis likewise resolved the problem: at least he declares he did; adding, that he took great care to authenticate his discovery: a precaution so much the more necessary, as the two solutions are in substance the same.
At the same time Nicole gave the method of finding the general expression of the rectification of spherical epicycloids, and afterward determining the cases, in which these curves become algebraic and rectifiable.
Clairaut, at that time only one and twenty years of age, yet who had already acquired reputation by his Inquiries concerning Curves of double Curvature,' treated the question in the same view as Nicole: but his method, in the Memoirs of the Academy for 1734, bears a peculiar stamp of elegance, which always distinguished his different works.
Geometry in a short time after made another very important acquisition. Clairaut considered a class of problems already sketched out by Newton and the Bernoullis; which was, to find curves, the property of which consists in a certain relation between their branches, expressed by a given equation. In this there would be no difficulty, were we permitted to employ the branches of two curves: but here the branches must belong to one and the same curve, and then the calculus is of a new and delicate kind.
In this research Clairaut made one observation particularly worthy of attention. There are questions of this kind, which admit two solutions, one direct, and independent of the inverse method of fluxions, the other founded on this method. The second, in which we suppose care has been taken to introduce an arbitrary constant quantity, seems as if it must include the first, by giving to the constant quantity all the values of which it is susceptible. But this is not
the case; for whatever value be given to the constant quantity, we never fall into the first solution. This kind of paradox in the integral calculus, remarked by Clairaut, was at the same time observed by Euler, as we see in his Mechanics, which appeared in 1736, as well as in the Memoirs of the Academy of Sciences at Paris for 1734. This was the germe of the celebrated theory of particular integrals, which Euler and several other learned geometricians have completely explained. It does not appear, that Clairaut followed up his ideas on this subject.
Problem of isochronous curves in resisting mediums. General reflections on problems of pure theory. Algebra of sines and cosines. Utility of methods of approximation, and in particular of infinite series.
THE problem of isochronous curves is remarkable in the history of geometry, as well for it's singular nature, as for the difficulties that were to be surmounted in it's solution. It consists, as is well known, in finding a curve of such a nature, that a heavy body, descending along it's concavity, shall always arrive at the lowest point in the same space of time, from whatever point of the curve it may begin to descend.
Huygens, examining the properties of the cycloid, found it that of being the isochronous curve in vacuo. Newton shows in his Principia, that the same curve is likewise isochronous, when the descending body, subject to the action of a uniform gravity, experiences every instant from the air, or whatever medium it moves in, a resistance proportional to it's velocity. Euler and John Bernoulli both separately determined the isochronous curve in a medium resisting as the square of the velocity. Mem. of the Ac. of Petersb. 1729; of Paris, 1730.
These three cases form three different problems, for each of which different methods are employed. In the former two, when the body, after having de