master them.

As the mind proceeds, it perceives the imperative call for an extension of the processes of the subordinate science, and it is placed in a position to appreciate the evidence on which the truth of the extension is based. In the contrary system, the mind has no previous ground-work on which it can gradually rise to view the field on which it has to enter; there is laid before it simply a series of statements, the consistency of which it must take for granted, and if subsequent examination produce doubts of the accuracy of the princples, there is no hope of its finding any argument by which they may be removed.

I feel an apology due for the apparently controversial tone which the latter part of this lecture has assumed; but I trust that the importance of a fair discussion of principles by no means settled will not be thought irrelevant or useless. That I have selected one particular writer in order to discuss the grounds of his system will, I hope, satisfy him and the world of the high value I attach to his work.

LECTURE VI.

ON THE DIFFERENTIAL CALCULUS.

The Differential Calculus a subject which requires time for the mastery of its principles-1. necessity for extension of methods of comparison -2. development of methods-that of Exhaustions-of Indivisiblesof Infinitesimals-of Fluxions-prominent characteristic, the introduction of a subsidiary quantity-Residual Analysis-Derived Functions -3. importance of notation-in arithmetic-in algebra-in science generally-it indicates processes to the eye-extends them-classifies forms-and is a ready instrument for operation-4. methods actually in use-that of Limits the most to be recommended.

of

IN the foregoing Lectures is contained an outline of the principles of the more simple methods of comparison; in that which follows, I purpose to take a brief survey the nature of the refined processes of the Differential Calculus. It is impossible, within the limits assigned to me, to present you with more than a hasty sketch. It requires both time and patience fully to comprehend this interesting subject, and I do not expect, in the course of a single lecture, to open the method so clearly that no further difficulties will occur. On the contrary, I am quite sensible of the fact, that the familiarity acquired by long experience is absolutely indispensable to the right understanding of the very principles of the science. I do not now, therefore, enter on my labours with the impression that I shall carry you along with me at the first. I anticipate, that, in reviewing your progress, I shall find you often hesitating to accept, or, at least, un

able fully to appreciate some of the fundamental doctrines. But let me assure you of one fact, that, however difficult may be the entry on this study, the barriers which are now presented to your mind will in time be effectually removed. You will in the end obtain as clear an insight into the nature of this method as you now possess of the simple processes of elementary geometry.* If plausible arguments have been urged against it, they have been brought forwards, either for the purpose of display, or because their authors took a false view of the nature of the principles of the science. Perhaps, indeed, in some cases, the objector has purposely seized on an incorrect statement to aid his own peculiar line of argument.† Be assured that the principles of the science. you are about to enter on are as firm and as real as those of any other branch of human knowledge. Let us, then, proceed to show,

1. The necessity for an extension of methods of comparison beyond those of superposition and proportion. The simple mode of demonstration which constitutes plane geometry consists in mentally applying surfaces, lines, &c. to one another, so as to cause them to fill up the same space. By this means the areas of triangles, squares, and all figures bounded by straight lines, as well as straight lines themselves, can be compared, and their properties determined. The same mode of proceeding serves also to develope some of the more simple properties of circles, an example or two of which may be

* D'Alembert's advice to beginners was, " allez en avant et la foi vous viendra."

+ Berkeley's Analyst, Sect. 14, &c.

found in Euclid's third and sixth Books. But, in general, when we would investigate the properties of curved lines, their length, the area of the figure bounded by them, or the volume of the solid which they describe by revolution; when we want, in fact, to compare curved lines with straight ones, it is clear that we cannot proceed by a similar process. It is evidently impossible, for instance, to take the area of a circle, and by unfolding it, or by cutting it in any way, to make it exactly fill up a square, or cover a space bounded by straight lines. If we are to compare a curvilinear area with a rectilinear one at all, we must of necessity have recourse to some process other than direct superposition.

We may further evince the necessity for some refinement in the processes of reasoning, by reference to the doctrines of motion. The illustration afforded by this means is, perhaps, more simple and more easily understood than any other. When a body moves uniformly, there is no difficulty in telling how much space it has travelled over in a given time. All we require to know is its speed, that is, the space through which it moves in one interval, say one second; the rest is an operation of pure arithmetic. But if the motion be not uniform; if it be that of a body falling by the action of gravity, the case is very different. The peculiar feature in this species of motion is a continuous change of velocity. We cannot, in strictness say, that the body ever moves with a specific velocity at all. At whatever point of its descent. we view it, no sooner has it quitted that point than it has changed its velocity. At whatever instant of time we reflect on it, the next has witnessed a difference in its rate

of motion. This, then, is the characteristic of that class of questions which the elementary processes of comparison are unable to grapple with,-a continuous change; not merely sudden and repeated transitions from state to state, but changes whose period is inappreciable, and which are always going on.

Now, it is unnecessary to say that the method of superposition, or of comparison by finite units of measure, is inadequate to the solution of questions which present this aspect. But neither does proportion offer anything to aid us in this case. For in Proportion, as we have shown in Lecture IV., comparison is effected by a reductio ad absurdum. Although the magnitudes which are to be compared are not directly made to occupy the same space, they are made to do so indirectly. Congruity is in reality the test of equality. Two magnitudes are equal, when equimultiples of them in all cases together exceed or together fall short of a multiple of some other magnitude. The multiples of both are superimposed on some multiple of a third, and although they do not coincide with that multiple in any one case, it can be easily shown that they do coincide with each other. The method of comparison by proportion is consequently a branch of the method of congruity. Now, cases such as those which I have mentioned are not susceptible of congruity. No multiple of a circle can be congruous with a multiple of a square. Nor can we establish their equality by any refinement on this doctrine. We have, therefore, to seek for some new process essentially different in its character from either simple congruity or proportion.