siderable, not in attacking any of the general principles of the calculus, but in searching out with incredible diligence a number of particular examples, in which he contended the rules of the calculus led to incorrect results. But Varignon, Saurin, and others clearly pointed out that the errors were entirely his own, and arose from his misapplication of the rules. This discussion was brought before the Academy of Sciences in 1701.

The Abbé Gallois joined with Rolle in these attacks, and kept the field after the latter had retired. His objections chiefly arose from too prejudiced a partiality for the ancient geometry. Here the controversy seems to have dropped for the time, though we shall find it afterwards renewed.

Newton's Dynamical Discoveries.

The improvements which Newton effected in the sciences of mechanics and dynamics abstractedly, may be viewed as quite separate from the application he made of them to the great problem of the forces actually concerned in the motions of the planetary system.

We have before noticed the advances in statics made by Stevin, Galileo, and subsequently by Wallis. The two former, in fact, generalised the property of the lever, and showed that an equilibrium takes place whenever the sums of the opposite momenta are equal: meaning by momentum the product of the force or weight into the velocity of the point at which it is applied. This was extended much further by Wallis, who, in the researches we have referred to, collected in his "Mechanica," published in 1669, founded an entire system of statics on the same principle.

To the mention of these we must add that of Varignon, who, in 1687, published his "Projet d'une Nouvelle Mécanique," in which he had the merit of deriving the whole theory of the equilibrium of the mechanical

powers from the single principle of the composition of forces.

The "Principia" of Newton, published in the same year (considered at present merely in reference to its abstract mechanical and dynamical parts), must unquestionably be regarded in that light alone as one of the most extraordinary productions of human genius: it effected an entire revolution in the mechanical sciences. In its introductory part the same general principles of equilibrium, of the centre of gravity, and of the mechanical powers, as those of Varignon and Wallis, are adopted; but they are merely noticed as introductory to the dynamical enquiries which follow.

In this important department, in which the first advances had been made by Galileo in examining the motion of projectiles and of falling bodies, Newton reduces to settled principles the more comprehensive laws on which the whole system depends. These, which he designates as the "axioms" of the science (not necessarily implying the notion of self-evident truths), are the three laws of motion." They had been already employed by preceding writers, though, perhaps, never yet explicitly laid down, and distinctly proved as the foundation of all the subsequent reasoning. Of late years, some discussion has arisen as to the precise cha-, racter of these elementary truths, and the philosophical nature of the evidence on which they are supported : and a considerable difference has existed in the manner of viewing the subject by the English and the French writers. We shall not, of course, in a work like the present, attempt to enter upon such a subject. We shall merely observe that the questions which have been raised, and the critical improvements which have been suggested in the mode of exhibiting these first principles, do not in the least affect Newton's credit, as he does not state them with any claim to originality, nor will the important reasonings which he founds upon them be affected by the somewhat different point of view in which they have been since regarded.

As a corollary to these laws of motion, he gives the demonstration of the composition of forces, which is, in fact, the essential principle on which every system of dynamical reasoning must proceed. We must briefly refer to this fundamental doctrine, in order to preserve the train of reasoning by which his ulterior conclusions were established.

According to Newton's view of the matter, which (like the view taken of the laws of motion) has been canvassed by the French mathematicians, a body is supposed to have an impulsive force acting on it, by virtue of which alone it would continue moving uniformly in a straight line, while at the same time another force acts upon it, having, at every moment of its course, an equal tendency to draw it in a direction inclined to the former at a given angle. It consequently obeys neither impulse entirely, but takes an intermediate course, called the 66 resultant" of the two former, and which is determined by drawing the diagonal of the parallelogram the two sides of which represent the two former forces. The truth of this is made to depend on the laws of motion simply, by virtue of which the body retains each of its first motions unaltered as to its quantity, and at the end of a given time is found at precisely the same distance, measured on a parallel, to which it would have reached in the original line, by virtue of either of the first forces alone. Of the truth of the result there can be no doubt but the French writers adopt a different mode of proving it.

Central Forces.

It is on the above simple principle that Newton proceeds to ground the whole theory of central forces. He arrives at this doctrine by a very striking application of the ancient idea of limits. He supposes a body to be projected in a given straight line, as in the last instance, but to take a resultant course from the action of a new force. At the end of a given time, it has described the diagonal of the parallelogram, constructed as before,

with sides proportional to those two forces. It would now continue in the diagonal. At this point let it be again deflected by a new force, making it describe a new diagonal at the end of another interval,. let, again, a new deflection take place in the same way, and so on successively then, if all these deflections be towards the same side, it is obvious that the successive diagonals will form the successive sides of a polygon, regular or irregular; and the forces which at each point caused the deflection, will have directions, which, if produced, will cut one another at some points within the polygon. They may cut in the same point: if so, it becomes easy to show, that, supposing the portions of time equal, the triangles formed by these lines so meeting, and the bases or sides, will be all equal in area. This equality of areas, then, in equal times, is produced when a body describes in equal times the successive sides of a polygon, by virtue of an original impulsive force which alone could carry it on for ever in a given straight line, and the action f another force which acts at equal intervals of times, and at each time changes both its amount and direction, all its directions at successive intervals converging to one point within the polygon.

This, abstractedly, would have been a curious theorem, but Newton investigated it with a view to a higher use; and, by following up the same idea to its extreme case, by a beautiful application of the principle of a limit (though represented by him under a different form of expression), he succeeds in making this theorem the basis of the whole system of central forces.

The limit of such a polygon as we have described would be a curvilinear periphery; and the limit of motion uniformly continued along each of the sides, but varying from one side to another, would be a perpetually varying motion in a curvilinear path. Few ideas are really more simple than this, when divested of technical obscu rity. In the case of the finite quantities, and finite uniform action, we can subject the forces to mathematical estimation. The relations which we establish remain unal

tered, when we pass from the elementary quantities to their limit. This is the only part of the reasoning which involves any sort of difficulty, and this it is which Newton had abundantly provided for, in previously establishing those various cases of limits which he denominated "prime and ultimate ratios." It was thus that he was led to the demonstration of motion in a curvilinear orbit, within which a certain point was so situated that portions of the orbit described in equal times being assigned, though those portions themselves might be very unequal, yet lines or radii from them to that point would intercept equal sectors or areas; or, in other words, a radius, supposed to revolve about this point, with the variable velocity belonging to the body in the orbit, would sweep over equal areas in equal times. The point with respect to which such a property held good was denominated the centre of force. As the curve, then, formed the limit of the uniformly described rectilinear increments, so the action of a force directed to this centre, but varying in intensity at each successive point of the orbit, was the limit of the successive unequal forces acting throughout each of the rectilinear portions.

In any such polygonal orbit at the end of any one side, the other side of the parallelogram was the measure of intensity of that part of the force by which the body tended to the centre. It easily followed from Newton's lemmas, that the limit of this, in a curvilinear orbit, measures the amount by which the arc, at its further end, has deviated from the direction of the tangent, or the deflexion of the body from a rectilinear course; and this will evidently be, at any point, the measure of the intensity of the central force. The limit of the other side of the parallelogram (which measures the impulsive force) is the same as that of the tangent to the curve. Thus the limiting ratio between these two forces, is that which determines the curvilinear path which the body will pursue. To some idea of this kind, it would seem that we must refer the extraordinary expression of