ever that by numerous observations of the successive positions of the moving body, its motion during any instantaneous period of time (i.e., its motion from point to point) has been ascertained; in other words (see preceding chapter) that the angle which the curve makes with the axis at any point in general has been determined by observation. Then the form of the orbit evidently is fixed, and may be calculated in the following manner:— от M N Taking the same points (P and Q,) and the same co-ordinates (OM, MP, and ON, NQ,) as before, we have now the angle PTM, and therefore the angle QPR, which is equal to it, of definite size. And as before, when the relation between QR and PR was given, the angle QPR could be calculated, so now, by virtue of the same proposition, the angle QPR being known, the relation between QR and PR can be determined. Thus we obtain a relation between the infinitesimal increments of the co-ordinates due to the motion of the body from point to point along the course. And this relation can be no other than that which exists between the co-ordinates of every point in the curve, modified only by the fact that PR and RQ are infinitesimal. In other words, it must be the derived equation to the curve. What we require, therefore, will be the exact converse of the method described in the last chapter. Then the relation which existed between the coordinates of every point in the orbit was given, and the form which that relation assumed when the motion was for an infinitesimal portion of time, was deduced. Now the relation being given when the motion lasts for an infinitesimal portion of time, it is required to ascertain what it is throughout the whole orbit; and the rules for effecting this are supplied by the Integral Calculus, which is the exact converse in every respect of the method of infinitesimals. Thus as by the principles of the last chapter we were endeavouring, on the one hand, when the law is given according to which a body moves, to trace its course from point to point— i.e., to calculate its motion for an infinitesimal period of time; so can we, on the other hand, by those of the present chapter, determine the law which regulates the whole orbit, when its motion from point to point has been duly ascertained by observation. The latter is, of course, the method we are obliged to pursue in analysing the phenomena of nature, for the examination of which the path now begins to be clear. For, when the physical laws which cause a body to assume a particular shape or to move in a particular curve are known, the results of such a form or such a motion may, to a certain extent at least, and in some few cases, be calculated. In some few cases, I say, for the variety of forms which every day meet the eye, and might, could the mind attain to it, be reduced to measurement, have been already shown to be well nigh infinite. Our knowledge, on the contrary, is so limited in extent and so slow in growth that the curves which can be put to any practical use have almost all been mentioned in these pages, and even these have taken thousands of years for their examination. Little by little, however, natural science does progress: and who shall say to what limit it may attain, especially when we compare the advances made since the days of Kepler and of Newton with those of former generations? With regard to both these chapters, the Treatise of Mr. Price on Infinitesimal Calculus contains the full elucidation of the principles contained in them and their application to Geometry, as well as to Statics, Dynamics, &c. Gregory's Examples will give a great number of suitable Problems; but Rögner's "Materialien aus der höheren Analyse," published at Leipsic, is fuller, and requires but a very slight knowledge of German. There are many French books also, which are said to be clearer than those by English authors, but I am only acquainted with one or two of them. But Mr. Price's Treatise appears to have the special advantage that the preliminary subjects of Trigonometry and Algebraic Geometry being mastered, the student may read the Treatise, comprehensive as it is, almost without a guide. Of course it will require some Mathematical talent, and a good deal of application; but it is astonishing to mark in how many students the taste and power for Mathematics are developed, when once the dreary commencement is over. The terrible mind of Euclid—that vultus instantis tyranni—so far removed from all sympathy, from all dulness and error, keeps a perpetual watch, awes us at the very threshold of the temple, and not till we have penetrated some way within its chambers, and have become familiar with some of its riches, do we feel the full consciousness of our own powers. Yet after all, Euclid and Newton were once students puzzled at the first problems of Geometry. What then should prevent the student now puzzled at the first problems of Geometry, but that he may himself become a Newton or an Euclid? THE sketch we have given in the preceding chapters, slight as it is, contains all that a work so elementary as the present can endeavour to describe with regard to the means possessed by modern mathematicians, of tracing the laws of motion in general. But in presenting to the public a work which deals with the difficulties of the first processes of reasoning, it may not be out of character to give some account of the steps by which it was itself thought out, and at the same time to offer to those readers who may be already acquainted with mathematics, the reasons which have induced me to follow methods and principles which sometimes, in appearance at least, differ considerably from those ordinarily pursued. It would, doubtless, be more usual to place such remarks at the beginning than the end of the treatise; but it appeared strange to preface a work the essence of which is, that it should not require any mathematical knowledge whatever, with observations which would demand a tolerably familiar acquaintance with the subject. The first germ of the idea, if I recollect rightly, sprang up when, as a boy, I was studying the fifth proposition of Euclid's first book, and found that the meaning which had been very obscure to me when merely illustrated in the usual manner, became at once quite clear when diagrams were drawn by a friend, showing each step of the process. A question then suggested itself as to what difficulty had been removed by the diagrams. Upon examination, I found that three difficulties had been removed. First, I had not understood the meaning of the expression "the angles at the base of an isosceles triangle; secondly, I had not perceived how the same lines and angles could form part of two different triangles; and thirdly,I had fancied that the base of a triangle must be level. In the further course of study, similar difficulties occasionally presented themselves, and it occurred to me that it might be useful to others were I to enumerate the errors into which I myself had fallen, and thus make a list, as it were, of the weeds which spring up of themselves in the mind, and which must be eradicated before any seed can bear fruit. But before long it turned out that the list was by no means so easy to make as it had appeared at first. Principles were involved in the explanation of some of the difficulties which extended far and wide; sometimes into the regions of Metaphysics, as for example in the idea of Extension, sometimes into those of Logic, as in the case of Definition, but far more frequently into the wide spreading department of Infinitesimal Calculus. To take a single instance of the latter: The number of straight lines which can be drawn through any one point is infinite. But from the ordinary diagrams and from the use of the expression, "tangent at a point," a notion was created in my mind that a tangent had but one point in common with a circle, thereby differing from a secant, which had two. And yet, as Euclid proves, there can be but one tangent to a circle at any given point, and not an infinite number. How could these apparent contradictions be reconciled? The principles of motion and infinitesimal growth alone can give the explanation. For by them the definition of a tangent becomes "the straight line which has two consecutive points in common with the circle. And as not more than one straight line can pass through two given points, the difficulty is at once removed. The principles upon which this little treatise is based, are as follows:-I take it for granted that whatever the truth may be with regard to " innate ideas," that is to say, whether we obtain our notions, of space for example, from "intuition;" or again as to abstract ideas, as, whether we can form a conception of "length" independently of " breadth" or not; yet that as it is certainly possible to apply geometrical notions to natural phenomena, so it must be possible also from natural phenomena to conceive geometrical notions. In other words, Reason is the Father, but Observation the Mother, of Mathematics. Accordingly I have been led to sketch the process which I should conceive to have taken place were a man at the dawn of geome. trical science to have thought out for himself, alone, and in a natural sequence, what in reality numbers of thinkers have only attained by the labour of centuries. For a child may run now where Socrates could with difficulty walk. Thus the senses have been made the primary source of every geometrical idea and principle; and I have endeavoured never either to define the one or to enunciate the other until it has already sprung up in the mind from the effects of reasoning upon what has been observed. But in carrying out the principle just laid down, I found that the sole way in which Geometry could appeal to the senses was by |