three equations: but Lagrange, in a paper, printed in the second volume of the Journal of the Polytechnic School, has simplified the system by reducing it to one fundamental equation. This reduction completes the analytical theory of spherical triangles: for, in analysis, perfection consists in employing the smallest possible number of principles, and in deducing from these all the truths they contain by the power of the analytic method alone. On the contrary, in the synthetic method perfection consists in demonstrating separately every proposition, in the most simple manner, by means of other propositions previously established.

Although Lagrange's analysis of spherical triangles is, perhaps, not capable of any very considerable improvement, I propose to alter it slightly, with a view of rendering it more uniform and easy to beginners. It will also be an important part of my duty to apply the formulas of plane and spherical trigonometry to some of the most interesting problems of practical astronomy, and to certain cases that occur in the trigonometrical surveys.

The mathematical subjects which are to occupy the second academic year, are Analytical Geometry and the Differential Calculus. Although numerous simple applications of algebra to geometry have appeared in the works of various analysts, from Regiomontanus to Vieta, the science of analytical geometry, in its modern form, is unquestionably the discovery of Des Cartes. His discovery appears to have originated in his solution of the following problem, which had been attempted by Euclid, Apollonius, Pappus, and other ancient geometers, but without success:-To find a point such, that if four straight lines be drawn from it to make given angles with four given lines, the rectangle under two of these lines shall be equal to the rectangle under the remaining two. Des Cartes discovered that this problem was indeterminate, and his researches led him to the conclusion, that every equation between two variables represents a geometrical locus; and, conversely, that every

geometrical locus has such an equation corresponding to it. This equation is, in reality, a compendious formula, which contains all the properties of the curve; and from which they may be deduced more easily than from any other definition. By this fortunate discovery the whole science of geometry was completely revolutionized, and it burst instantaneously beyond the narrow limits within which it had been previously confined to an extent which is literally infinite. The great advantage of this new method is, that its investigations proceed according to general rules, and require comparatively small talent and ingenuity in the student: while, in the ancient geometry, the solution of one, or even of a thousand problems, afforded no clue to the solution of the next; and each new proposition, if not discovered by chance, required a proportional expenditure of intellectual energy.

These we shall find to comThese curves, (the properties talents of geometers from that

In this infinite variety of researches, our attention will be chiefly confined to curves represented by the general equations of the first and second degree. prehend all the conic sections. of which have employed the time till the present) were invented by the mathematicians of the early Platonic School: and what has given them considerable additional interest is, that they have been discovered, by Kepler, to be the paths of the planets and comets in their revolutions about the sun.

The other subject to which it is my duty to call the attention of this class is the Differential Calculus-perhaps the most splendid of the pure conceptions which the human mind has ever entertained. The glory of its discovery has been contested by Newton and Leibnitz, but, by minute observation, we may perceive the germ of this science in the ancient method of exhaustions, which was so successfully used by Archimedes, the Newton of the ancients.

The principles of this calculus have been established by three different methods, due respectively to Newton, Dalem

bert, and Lagrange. Newton's method, commonly called the method of fluxions, has now been universally abandoned: partly on account of the inferiority of its notation; but chiefly because it involves the consideration of motion;—an idea foreign to the spirit of pure analysis. In the brevity of its demonstrations also, and in the facility of its applications, it is inferior to all the other methods; and we are utterly unable to understand the connection between the different orders of fluxions and their primitive function.

The method of limits seems at present to have acquired a decided preference. This has probably arisen from the adoption of this method by the late eminent mathematician Poisson, who has conferred such extensive benefits to every department of pure and mixed science. It is, however, beset with considerable metaphysical difficulties, which to some students are quite insurmountable. Our notion of a ratio whose terms are evanescent (called by Berkeley the ghosts of departed quantities) is necessarily obscure; however rigorously its existence and magnitude may be demonstrated: and, its introduction into all the reasonings by which the principles of this calculus are established, tends to throw an air of mystery over all its operations, which can only be removed by showing us the origin of the differential calculus in the principles of common algebra.

The method of Lagrange, therefore, which is founded on the principles of common algebra, seems to me best adapted to elementary instruction. The principal objection to this method is that it is incomplete; the differentiation of circular functions having never been accomplished independently of limits. I propose, however, to obviate this difficulty, and to establish the principles of the calculus by the powers of this method alone; not only independently of limits, but even without any assistance from common algebra, except in the performance of simple algebraical operations.

The applications of the differential calculus to the theory of

logarithms, the method of tangents, the theory of maxima and minima, the determination of singular points of curves, and the theory of osculating curves, each claim a considerable portion of our time and attention. The differentials of ares and areas, and the differential equations of motion, shall be investigated, in order to reduce all problems of rectification, quadrature, and motion, to purely geometrical questions; and to bring them at once under the dominion of the integral calculus. The relations between the variations of the sides and angles of plane and spherical triangles shall also be discussed, and applied to the corrections necessary in the reduction of astronomical observations. At this stage of our progress I will attempt to explain the improvements of Bessel, the first astronomer of the present day, whose recent labours have completely changed the state of astronomical calculations, and rendered all the work of an observatory almost capable of being performed by a mere machine.

The mathematical course of the third year is the Integral Calculus. A very minute portion of this surprising and infinite science, is all that can be attempted in the space of nine months. But that portion, small as it must necessarily be, will enable you to accomplish, by a mere dash of the pen, problems which foiled the genius of Archimedes ; eluded the sagacity of Appolonius, and defied the power of Des Cartes' analysis. We shall probably find it impossible to extend our investigations beyond the methods of integrating explicit functions of one variable. This, however, will enable us to solve many problems of rectification and quadrature, and to integrate the equations of motion in many of the simplest and most interesting cases. As examples in integration may be selected, the equations of projectiles, in vacuo and in a resisting medium; the theory of the pendulum; central forces, and such other parts of physical astronomy as the limits of our time may permit.

The object of the special class is a further developement

of the integral calculus, and its application to some of the most celebrated physico-mathematical theories-in particular to the theories of the figure of the planets, the tides, electricity, heat, light, and sound. This class is not comprehended in the ordinary College curriculum. It will be adapted, as much as possible, to two classes of students. First, such as may wish to obtain a Master's degree from the London University: second, such Cambridge students as may wish to read an extensive course of mathematics and physics, in order to become Wranglers at their degree examination. The calculus of finite differences, and the theory of probabilities, are introduced chiefly for the sake of the first class, although they possess considerable general interest. In treating these two subjects, I shall make extensive use of the calculus of generating functions, one of the most refined instruments of analysis, which we owe to the inventive genius of Laplace.

The integration of differential equations, from its immense importance, will necessarily occupy a large portion of our time; as the student will soon perceive that his progress in physical science must bear an exact proportion to the extent of his knowledge in the integral calculus. The theory of partial differential equations will be introduced, as far as it is necessary in the problem of sound; the theory of vibrating cords, and the longitudinal vibrations of elastic wires: the transversal vibrations of such wires depend on an analysis too complicated and refined to obtain a place in elementary instruction, and must be left almost entirely to the industry of the student.

The geometrical course of the special class is analytic geometry of three dimensions, comprehending the general theory of curve surfaces and of curves of double curvature. Our attention will be directed chiefly to surfaces of the second order; in particular to the ellipsoid and its circular sections; which are so intimately connected with some of the most remarkable properties of polarised light, and with

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