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sait is the common catenarian; but if the sail, always considered as perfectly flexible, be filled by a fluid gravitating vertically on itself, as water presses on the sides of a vessel in which it is contained, it would form a curve known by the name of linteaire, and the nature of which is expressed by the same equation as the common elastic curve, in which the extensions are supposed to be proportional to the forces applied at each point. The identity of the two curves not being easy to recognize, James Bernoulli displayed profound sagacity in this question, as well as in some others of the same kind.

While he was busied in his first meditations on the curvature of a sail, he communicated his progress from time to time by letters to his brother, who was then at Basil. It is clear, that these communications led John Bernoulli to the solution he published of the same problem in the Journal des Savans, in 1692; which equally showed, that the curve of the sail is a catenary. By the manner in which he exhibits the facts, he himself furnishes us with a proof of the assistance he had borrowed: of course we have some reason to be a little surprised, at finding here the first appearances of that jealousy toward his former master, which he afterward too publicly displayed.

The theory of curves which produce others by rolling on similar curves was a rich field of discovery for James Bernoulli. A. D. 1692. He supposes, that, any curve being given, and considered as immovable, an equal and similar curve is made to roll upon it: he determines the evolute and the caustic of the epicycloid described by a point of the

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moving circle; and he derives from it two other analogous curves, which he calls the antiecolute and pericaustic. All these curves display a number of properties, well worthy of exciting the curiosity of geometricians, particularly at a time when they had yet but little experience in the new analysis. On applying his methods to the logarithmic spiral, James Bernoulli found, that this curve is it's own evolute, caustic, antievolute, and pericaustic: a singular character, at which he was so astonished, that he could not avoid declaring with warmth, that, if it were still the custom, as in the time of Archimedes, to place mathematical figures and inscriptions on the tombs of geometricians, he would have desired a logarithmic spiral to be engraved on his tomb with these words: Eadem mutata resurgo.

The cycloid possesses properties analogous to those just mentioned of the spiral. James Bernoulli made them known in a supplement to his former memoir; and at the same time mentioned, that his brother had separately found the same results.

I ought not to omit a work of Leibnitz, on the curves which are formed by an infinite number of straight or curved lines terminating in a series of points subject to a given law. This tract, which does not go into the subject very minutely, contains general hints for the solution of several problems, such as those of caustics, of curves cutting a series of other curves at a given angle, &c. Leibnitz seldom entered into minuti: as soon as he found himself in possession of a method, he gave it up, leaving

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to others the pleasure of extending it, and carrying it to perfection.

Among this multitude of problems a very curious one was proposed in 1692 by Viviani, a celebrated italian geometrician, under the following title: Enigma geometricum de miro Opificio Testudinis quadrabilis Hemisphæricæ. The author feigns, that among the monuments of ancient Greece there still exists a temple of a hemispherical figure, pierced by four equal windows with such art, that the remainder of the dome was capable of being perfectly squared; and hopes, that the illustrious analysts of the age, so he styles the geometricians skilled in the new calculus, will easily divine this enigma.

His hope was not disappointed. The very day on which Leibnitz and James Bernoulli received Viviani's challenge, they solved the problem: and other geometrical analysts would no doubt have solved it also, had it reached them in time. Viviani was profoundly skilled in the ancient geometry; and had particularly distinguished himself by divining or restoring the five books of conic sections of the ancient Aristeus, which are lost: but when the geometry of infinites appeared, he was too much advanced in years, to study and make himself master of it. He was however a truly modest man, and had no intention to perplex the illustrious analysts.' At the same time it must be acknowledged, that his own solution, founded on the synthetical method of the ancients, is eminent for it's simplicity and eleHe demonstrated, that the question might be solved by placing parallel to the base of the

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hemispherical dome two right cylinders, the axes of which should pass through the centres of two radii constituting a diameter of the circle of the base, and piercing the dome each way.

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A problem belonging to the method of maxima and miniina long employed the two Bernoullis without success. This was, to find the day of the shortest twilight for a place of which the latitude is given. This question, treated in the analytical way, leads to an equation of the fourth order, in which it is embarrassing to separate the useful roots from those which ought to be rejected: but by employing the synthetic method, each of them separately obtained a very simple analogy, and very convenient for astronomical computation.

The place of mathematical professor in the university of Basil, which was occupied by James Bernoulli, procured his pupils and the public an excellent treatise on the summation of series. The first part had appeared in 1689, the second was published in 1692.

Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years.

The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of the methods for summing up differences. To find a curve such that the tangents terminating at the axis shall

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be in a given ratio with the parts of the axis comprised between the curve and these tangents.' This was resolved by Huygens, Leibnitz, James Bernoulli, and the marquis de l'Hopital.

On this occasion Huygens passed on the new methods an encomium so much the more honourable, as this great man, having made several sublime discoveries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld with surprise and admiration the extent and fertility of this art; that, wherever he turned hiş eyes, it presented new uses to his view; and that it's progress would be as unbounded as it's speculations.' How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services!

Tschirnhausen had some years before made known the celebrated curves designated by the name of caustic, which are formed by the concurrence of the rays of light, either reflected or refracted by some other curve. By the aid of common geometry alone Tschirnhausen had discovered several elegant properties of them: as, for instance, that they are equal to right lines, when they are produced by geometrical curves. The geometry of infinites greatly facilitated all these researches, and in 1693 James Bernoulli carried them very far, particularly the theory of caustics by refraction.

The copiousness of the subject, and the limits of this essay, compel mé to pass over in silence several other papers, which James Bernoulli gave to the public in the same year, on the different subjects of

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