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scended, reascends through the second branch of the cycloid, it passes through the ascending are in the same time as through the descending; so that all the oscillations, each of which consists of a descent and an ascent, are performed in the same time. But in the hypothesis of the resistance being as the square of the velocity, the isochronous descending arc is not the same as the isochronous ascending arc, so that they must be sought out separately. They are found, however, exactly in the same manner, and consequently it is sufficient to consider either of them.
Fontaine made a great step in this theory. He invented a method of a truly original cast, by which alone he resolved the three cases proposed; and he even added a fourth, in which the resistance shall be as the square of the velocity plus the product of the velocity by a constant coefficient. It is very remarkable, that the isochronous curve is the same in the fourth case as in the third. The spirit of this method is, to consider the variable quantities both with respect to the difference of two proximate arcs, and with respect to the elements of one and the same arc: the author employs the differentials of Leibnitz for the variations of the first kind, and the fluxions of Newton for those of the second. Dr. Taylor had given an opening to this fluxio-differential method; and Fontaine also resembled him in the defect of being obscure: but they were both profound geometricians.
Euler, who, not content with enriching geometry from his own stores, has sometimes remoulded the works of others, and always for their advantage, developed the method of Fontaine, and placed it in a clearer
clearer light, in the Petersburg Memoirs for 1764, bestowing on it at the same time all the praises it deserves. He goes over each of the cases already resolved; and adds another, which includes them all: that in which the resistance is composed of three terms, the square of the velocity, the product of the velocity by a given coefficient, and a constant quantity. The method of Fontaine extends no farther: besides, as it gives the isochronous curve independantly of the consideration of the time, the expression of the time, which the body employs in passing through any arc of the curve, remained still to be determined. Euler has solved this new problem, which depended on the resolution of a very complicated differential equation.
Fontaine imagined he had so completely exhausted the theory of isochronous curves, that in the collection of his works, published in 1764, he says, when speaking of his solution of 1734, the problem ceased to be a subject of discussion soon after it's appearance.' Happily, however, it continued a subject of discussion still. It was not enough, to have found the isochronous curves on certain hypotheses of accelerating forces it was necessary, by inverting the problem, to point out the means of determining, what are the hypotheses of accelerating forces, that admit isochro nism. Two great geometricians have made this discovery, and thereby opened a new store of problems on the subject. Memoirs of the Ac. of Berlin, 1765.
When the medium is rare, or of little resistance, the investigation of the isochronous curve is more easy. Euler has resolved several cases of this nature
with great simplicity and elegance in his Mechanics, to whatever powers of the velocity the resistance may be proportional.
The enemies of geometry, or even those who are imperfectly acquainted with it, consider all these difficult theoretical problems as mere amusements, consuming that time and reflection which might be better employed. But they do not consider, that nothing is more capable of arousing and unfolding all the powers of the human intellect; that the mind, to use an expression of Fontenelle's, has it's wants, as well as the body; and finally, that a speculation, which appears sterile at first sight, ultimately finds it's application, or sometimes, when least expected, gives rise to new views respecting objects of public utility. Let us give genius a free wing: let the geometrician seek and contemplate intellectual truths, while the poet depicts the passions of the heart, or the beauties of nature.
The more attraction those who cultivate the sciences find in this faculty of invention, the more does it incline them to avoid working on the dead matter, which would lead to no new and useful result. Thus, when such a one has surmounted the analytical difficulties of an abstract problem, he rarely completes the calculation: commonly he contents himself with pointing it out clearly; or, in cases that require it, reduces it to certain formulæ, which evade a strict analysis, as the quadrature of the circle, the equation of Riccati, &c.; and when we come to these, the problem is considered as resolved. But when we would
apply analytical formula to the physico-mathematical
sciences, where all the quantities must be ultimately expressed in numbers, we cannot dispense with a completion of the analytical calculation; and then we have frequent occasion for methods that serve to abridge it, whether it terminate in algebraical results, or contain expressions of which the values can be given only by approximation.
The algebra of sines and cosines, for which we are principally indebted to Euler, is one of the means of abbreviation, to which all parts of mathematics, and physical astronomy in particular, have inestimable obligations. By the combination of arcs, sines, and cosines, we obtain formulæ, which in many cases readily submit to the methods of resolution: and thus we are enabled to solve a number of problems, which we should be forced to abandon from the length or difficulty of the computations, if we were to employ the arcs, sines, and cosines, in their ordinary form, or even in the exponential one.
In defect of strict solutions we are obliged to recur to methods of approximation, and to these in great measure we owe the success of practical mathematics. The theory of infinite series is the chief basis of all these methods. Frequently there is much difficulty, and much art is required, to form the series proper to resolve promptly, and sufficiently near to the truth, the questions which require them. The english, as Newton, Wallis, Stirling, &c., greatly cultivated this beautiful part of analysis: but no person has carried it so far as Euler; no one has summed up so many curious series, or applied this mean to the so