them to three coordinates perpendicular to each other; and he divides them into orders, classes, and genera, as he had done for the simple curves traced on a plane, &c. All these subjects are treated with a perspicuity and method, which facilitate their study to such a degree, that any tolerably intelligent reader may pursue them by himself, without assistance.

any

Finally Euler has collected into five or six volumes in 4to the complete science of the differential and integral calculi. All the stores of the art before known, and a great number of theories absolutely new, are here exhibited and unfolded in the most luminous and instructive manner, and under that original and commodious form, which the author has given to all parts of the higher mathematics. These different treatises together compose the most beautiful and ample body of analytical science, that the human mind ever produced. Every geometrician, who has had it in his power to read these works, has derived information from them; and some have, even arrogated to themselves the honour of methods, which they contain. If father Reyneau were styled for a time, and by hyperbole, the Euclid of the higher geometry, it may be truly said, that Euler was this Euclid; and it may be added, that in genius and copiousness he was far superiour to the ancient.

Among the benefactors of the new geometry, I must not forget, to mention Cramer with distinction, His Introduction to the Analysis of algebraical Curves' is the most complete treatise, that exists on the subject. The author leaves nothing to be wished respecting the theory of the infinite branches of

curves, their multiple points, and all the symptoms in general which serve to characterise them. He was contemporary with Daniel Bernoulli and Euler; like them a pupil of John Bernoulli; and to all these great men he approached very near. We have an excellent Commentary of his on the works of James Bernoulli.

In 1768, fathers le Seur and Jacquier, of the congregation of minims, published a Treatise on the integral Calculus,' a work somewhat prolix, and oc'casionally defective in method, yet containing several new and interesting things, as for instance a very clear explanation of Newton's treatise on Quadratures.

The art of exterminating the unknown quantities, or reducing the equations of a problem to the least number possible, is an essential part of analysis. To this several geometricians turned their attention; among whom Cramer had already extended and simplified it greatly. But Bezout has made it the subject of a learned treatise, in which he has carried it much farther, than any person had done before him.

In the year 1801 the sciences lost Cousin, who had published several works, particularly a treatise on the integral Calculus. This treatise is charged with a little obscurity and want of order: but it is allowed to be very learned, and to contain several new things, chiefly on the solution of equations with partial differences,

CHAP,

CHAP. XI.

Progress of Mechanics.

THE science of mechanics is founded on a small number of general principles, and when these are once discovered, all the applications, that can be made of them, belong properly to geometry. But these applications, particularly in the problems that relate to motion, frequently require much sagacity, and constitute a particular science, which the moderns have carried very far, through the assistance of the analysis of infinites.

After the foundations of statics were laid by Archimedes, it was not difficult, to discover the conditions of equilibrium in every particular case; and these had guided the genius of invention in a number of machines, but they were not yet reduced to a general and uniform principle. Varignon undertook and accomplished this plan of combining them, by means of the theory of compound motions. He He gave some sketches of this in 1687, in his Project of a new System of Mechanics;' and he in some degree exhausted all the combinations of the equilibrium of machines, in his General Mechanics,' noc published till 1725, after his decease. This work, already quoted, is very prolix and tiresome to read, but it is to be commended for the perspicuity of it's detail.

In the second volume of this work Varignon has

given the first notions of the celebrated principle of virtual velocities, from a letter written to him by Joha Bernoulli in 1717. What is called the virtual velocity of a body is the infinitely small space, which the body, excited to move, tends to pass through in one instant of time; and the principle in question, applied to the equilibrium, may be thus generally expressed: Let there be any system whatever of small bodies, impelled or drawn by any powers, and balancing each other; and let as light motion be impressed on this system, so that each body shall pass through an infinitely small space, which expresses it's virtual velocity then the sum of the products of the powers, each multiplied by the small space, which the body to which it is applied passes through, will always be equal to nought, the motions in one direction being subtracted from those in the opposite direction.' Varignon applies this principle to the equilibrium of all the simple machines.

In 1695 la Hire published a Treatise on Mechanics, the general object of which, like that of Varignon's, is the equilibrium of machines.. Beside this, it con-. tains various applications of machines to the arts, in which the author was well versed. Subjoined to this work is a treatise on Epicycloids, and their use in Mechanics. La Hire demonstrates, that the teeth of wheels, intended to communicate motion by means of cogs [engrenages], should have the figure of epicycloids, the properties and dimensions of which he determines. This theory is very beautiful, and must do the author great honour: but Leibnitz, in his let

ters

ters to John Bernoulli, asserts, that it belongs to Roëmer, who communicated it to him twenty years before the book of la Hire appeared.

Could we suspect Leibnitz of any partiality in favour of Roëmer, we should soon be checked by the little probability, that such a discovery was made by la Hire, a geometrician of very moderate science. In fact no other trait of genius is to be observed in his Mechanics; in which, on the contrary, we find a gross blunder, on the subject of the isochronism of the cycloid, perhaps not the only one there. The author intending to demonstrate, prop. cxx, that a heavy body, descending along a reversed cycloid, always arrives at the lowest point in the same length of time, from whatever point it began to descend, employs a mode of reasoning, from which he concludes, that the time of the descent through the half of the reversed cycloid is double the time of the fall through the vertical diameter of the generating circle. But this proposition is false: for it is known by the incontestable demonstrations of Huygens, and it may be proved in several other ways, that the former time is to the latter, as half the circumference of the circle is to it's diameter. The blunder of la Hire comes from his having assumed as a principle, that, if we have a series of any proportions whatever, the sum of all the first antecedents is to that of all the first consequents, as the sun of all the second antecedents to that of the second consequents: which is true only in the single case where all the proportions, being in other respects what they may, are composed of equal ratios.

A great number of elementary treatises on statics have been published beside these; but my plan does

not