means of Motion; and consequently Motion has been the fundamental idea of the treatise. An example is easily found in the case of the circle. The readiness with which its definition as a Locus adapts itself to the physical method of drawing it with a pair of compasses, or whirling round a lighted stick, is at once evident; while the definition of Euclid is illustrated only by a property of the figure when drawn, and throws no light on its mode of generation. Other instances will be found in chapters ix. and xxxvii. But as the introduction of Motion and Loci at the outset is certainly somewhat novel, I will quote the authority of Mr. Price in the Treatise on Infinitesimals, to which reference has been already made. "If," says he, "motion will be found to illustrate, as it does, our geometrical results, it is only a servile adherence to an ordinary though inferior method which prevents us from introducing it." And if any one should still urge that motion was never contemplated by Euclid, as an element of geometrical reasoning, let me urge that though indeed he never alludes to it with regard to plane figures, yet he makes use of it often in defining Solids of Revolution. The idea of Motion then, and by consequence its antithesis, Rest, being admitted into Geometry, it became necessary to examine as far as possible in what order it were more natural to the mind to form the two conceptions-another question closely bordering upon Metaphysics. And somewhat contrary perhaps to the usual order, I have treated Motion as the primary, Rest as the secondary, or derived idea: Rest indeed being considered only the particular case, when the motion is zero. Hence, the treatise divides itself into two parts-the first on the Motion, the second on the Rest, or Arrest of a Moving Point. In the First Part a principle has been chiefly used, which readily suits, and I had almost said necessarily flows from, the principle of Motion. It is that of Limits, and was given by Leibnitz as the fundamental principle of Infinitesimal Calculus. M. Comte, indeed, in his "Philosophy of Mathematics," takes exception to it as being unphilosophical, and reducing the whole Calculus to a mere system of approximation; but his own words in regard to those magnitudes with which we have been chiefly concerned, appear founded upon the very same principle which he reprehends. "Although it is evidently impossible," he says, "to conceive any extension absolutely deprived of any one of the three fundamental dimensions, it is no less incontestable that in a great number of occasions even of immediate utility, geometrical questions depend only on two dimensions considered separately from a third, or on a single dimension considered separately from the other two." Now, as wherever a dimension is appreciable, there it must form an element in the calculation, these words can only apply when the dimension is inappreciable. Hence it would appear that M. Comte is content to consider a dimension which is inappreciable as having no existence whatever when compared with an appreciable dimension. The result of this method would seem to be, that whereas hitherto our conceptions have been confined to a system of numbers lying (at perceptible intervals,) between positive and negative infinity, and appreciable in a concrete form to our senses; now they are extended to an infinite series of systems, each of infinite extent, and all either too great or too small for our senses to appreciate, wherein the individuals of each system bear distinct relations to the other individuals of the same system, similar to the relations between the individuals of the ordinary system, and subject only to those modifications which it is the province of Infinitesimal Calculus to ascertain. The enormous range of Calculation which such a method must afford is of course evident at sight; and the invention is perhaps the most lofty idea, numbers and letters alone a Comte's "Philosophy of Mathematics," p. 182. excepted, which was ever conceived by the mere intellect of man. And to these exceptions some have assigned an origin not human but divine. The leading idea of the Second Part, in which the Point considered in the First to be moving in any continuous line is arrested by the introduction of a fresh geometrical condition, was suggested by the analogy of Virtual Velocities. For as by that beautiful device the laws of Equilibrium, or Statics become only a particular case (viz., when the Force ceases) of the more simple and general notion of Dynamics; so here the law of motion in a straight line having been ascertained, the further idea of arrest is introduced by considering what condition is necessary that the motion may become zero. By this means the three straight lines which bound a Triangle may be considered as one of those lines which Descartes called "Mechanical," as distinct from "Geometrical," and Leibnitz more accurately "Transcendental," as distinct from "Algebraical." That is to say, the three straight lines form but one 66 Transcendental" line, the law of which is the result of the two laws, one by which it moves along a straight line, and the other by which it departs from and returns to it. Thus, simple as the notion of a Triangle seems, we find no less than four laws involved in it, besides the idea of a Point as the limit of geometrical magnitude. 1. The general law of motion, viz. A line is the Locus of a Point. Ch. viii. 2. The particular law of motion in a straight line, viz. A straight line is the Locus of a Point which moves in such a manner that the angle between each three consecutive points is 180°. Ch. ix. and xvi. 3. The first law of continuity, as we have, perhaps improperly, called it, viz.—That there should be three points the sum of the angles at which should be equal to 180°. Ch. xvii. And 4. The second law of continuity, viz.-That no point in the enclosing lines should be at an infinite distance; in other words, that the sum of the angles at every two of the boundary points should be less than 180°. (Ch. xxi.) But perhaps to this latter law alone should be assigned the name of Continuity, and to the former that of Discontinuity. And when the triangle is to be of definite shape, a fifth law must be added, viz. that some one set of the conditions mentioned in chap. xxxi., must be fulfilled. Hence, clearly the notion of rest is more complicated than that of motion, since in the former case five laws are involved, to only two of which the point is subject in the latter. Finally, in Section V., it is conceived that every curve is made up of a series of infinitesimal straight lines; and each of these straight lines is considered to be the hypothenuse of an infinitesimal right-angled triangle. Now in every triangle thus formed, the sides which contain the right angle are found to be the increments of the co-ordinates due to the motion of the point along the hypothenuse; and the other angles are those which the tangent at the point (being the hypothenuse produced) makes with the axes of x and y respectively. And we have seen in Section IV., what set of parts in any triangle must be definite, in order that the other part may be definite also. That is to say, since one angle of the triangle is 90°, and therefore of course definite, we are enabled, either if the increments of the co-ordinates due to the motion from point to point, or, in other words, if the sides containing the right angle be given, to determine the hypothenuse or infinitesimal arc of the curve, and the angles which the tangent makes with the axes. Or, if the infinitesimal arc of the curve, and the angle which the tangent makes with one axis be given, we can determine the sides of the infinitesimal triangle, that is, the increments of the co-ordinates due to the motion during an instantaneous period of time. The former is the problem in ch. xxxix., where the law for the whole motion being given, that for the motion from point to point is deduced by means of Differential Calculus. The latter sometimes called the method of Tangents, is the converse problem discussed in ch. xl., and performed by the processes of Integration. And this is the meaning of the assertion which we made at the outset, that the forty-seventh proposition of Euclid's first book is the foundation of all geometrical measurement. Such is the outline which I have endeavoured to trace, and such the means I have employed. In many cases I can scarcely hope to have been successful, and I advance much, particularly in the third and fifth sections, with no small diffidence. The difficulties of the subject have been many, not the least among them being the ease with which in such a treatise one may fall into arguing in (as well as on) a circle, an error which it is almost impossible for the author himself to detect. There remains but one word to be said. It may appear to the student that the point we have reached is, after all but a slight and unsatisfactory progress, and that the goal he would attain seems as far or further than ever. If so let me remind him in parting that such is ever the case with regard to scientific advancement. Ergo vivida vis animi prorupit et extra In the philosopher, in the discoverer, nay in the child who has tasted the delight of grasping some new principle, the "living force of Mind" is as potent now as in the days of Lucretius; and not "the glowing barriers of the universe" itself can stay it. "The eye is not satisfied with seeing, neither is the ear filled with hearing." When Newton was asked whether he were not content with the vastness of the knowledge he possessed, "I feel," said he, "like a child playing upon the sea-shore." Step by step we have endeavoured to lead the student to a spot whence he may catch a glimpse of that ocean of science; Et nunc tempus equûm fumantia solvere colla. Nor let him complain that by this course he has been led |