not grasp and wield such an aggregate of conditions, however certain might be our knowledge of each. In the simplest cases in which we desire to approximate the abstract to the concrete conditions, with any completeness, -as in the phenomenon of the flow of a fluid from a given orifice, by virtue of its gravity alone,-the difficulty is such that we are, as yet, without any mathematical solution of this very problem. The same is the case with the yet more simple instance of the movement of a solid projectile through a resisting medium.

To the popular mind it may appear strange, considering these facts, that we know so much as we do about the planets. But in reality, that class of phenomena is the most simple of all within our cognizance. The most complex problem which they present is the influence of a third body acting in the same way on two which are tending towards each other in virtue of gravitation; and this is a more simple question than any terrestrial problem whatever. We have, however, attained only approximate solutions in this case. And the high perfection to which solar astronomy has been brought by the use of mathematical science is owing to our having profited by those facilities that we may call accidental, which the favourable constitution of our planetary system presents. The planets which compose it are few; their masses are very unequal, and much less than that of the sun; they are far distant from each other; their forms are nearly spherical; their orbits are nearly circular, and only slightly inclined in relation to each other; and so on. Their perturbations are, in consequence, inconsiderable, for the most part; and all we have to do is usually to take into the account, together with the influence of the sun on each planet, the influence of one other planet, capable, by its size and its nearness, of occasioning perceptible derangements. If any of the conditions mentioned above had been different, though the law of gravitation had existed as it is, we might not at this day have discovered it. And if we were now to try to investigate Chemical phenomena by the same law, we should find a solution as impossible as it would be in astronomy, if the conditions of the heavenly bodies were such as we could not reduce to an analysis.

MATHEMATICS: FINAL CONSIDERATIONS.

49

In showing that Mathematical analysis can be applied only to Inorganic Physics, we are not restricting its domain. Its rigorous universality, in a logical view, has been established. To pretend that it is practically applicable to the same extent would be merely to lead away the human mind from the true direction of scientific study, in pursuit of an impossible perfection. The most difficult sciences must remain, for an indefinite time, in that preliminary state which prepares for the others the time when they too may become capable of mathematical treatment. Our business is to study phenomena, in the characters and relations in which they present themselves to us, abstaining from introducing considerations of quantities, and mathematical laws, which it is beyond our power to apply.

We owe to Mathematics both the origin of Positive Philosophy and its Method. When this method was introduced into the other sciences, it was natural that it should be urged too far. But each science modified the method by the operation of its own peculiar phenomena. Thus only could that true definitive character be brought out, which must prevent its being ever confounded with that of any other fundamental science.

The aim, character, and general relations of Mathematical Science have now been exhibited as fully as they could be in such a sketch as this. We must next pass in review the three great sciences of which it is composed,—the Calculus, Geometry, and Rational Mechanics.

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CHAPTER II.

GENERAL VIEW OF MATHEMATICAL ANALYSIS.

ANALYSIS.

THE

'HE historical development of the Abstract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics. The Concrete portions of the science depend on the Abstract, which are wholly independent of them. We will now therefore proceed to a rapid review of the leading conceptions of the Analysis.

True idea of

First, however, we must take some notice an equation. of the general idea of an equation, and see how far it is from being the true one on which geometers proceed in practice; for without settling this point we cannot determine, with any precision, the real aim and extent of abstract mathematics.

The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others. It is only by forming a true idea of an equation that we can lay down the real line of separation between the concrete and the abstract part of mathematics.

It is giving much too extended a sense to the notion of an equation to suppose that it means every kind of relation of equality between any two functions of the magnitudes under consideration; for, if every equation is a relation of equality, it is far from being the case that, reciprocally, every relation of equality must be an equation of the kind to which analysis is, by the nature of the case, applicable. It is evident that this confusion must render it almost impossible to explain the difficulty we find in establishing the

FUNCTIONS OF A TWO-FOLD NATURE.

51

relation of the concrete to the abstract which meets us in every great mathematical question, taken by itself. If the word equation meant what we are apt to suppose, it is not easy to see what difficulty there could be, in general, in establishing the equations of any problem whatever. This ordinary notion of an equation is widely unlike what geometers understand in the actual working of the science.

According to my view, functions must themselves be divided into Abstract and Concrete; the first of which alone can enter into true equations. Every equation is a relation of equality between two abstract functions of the magnitudes in question, including with the primary magnitudes all the auxiliary magnitudes which may be connected with the problem, and the introduction of which may facilitate the discovery of the equations sought.

This distinction may be established by both the à priori and à posteriori methods; by characterizing each kind of function, and by enumerating all the abstract functions yet known,—at least with regard to their elements.

Abstract func

Apriori; Abstract functions express a mode of dependence between magnitudes which may tions. be conceived between numbers alone, without

Concrete func

the need of pointing out any phenomena in which it may be found realized; while Concrete functions are those whose expression requires a specified tions. actual case of physics, geometry, mechanics,

etc.

Most functions were concrete in their origin,—even those which are at present the most purely abstract; and the ancients discovered only through geometrical definitions elementary algebraic properties of functions, to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers. There is another example which well exhibits the distinction just made-that of circular functions, both direct and inverse, which are still sometimes concrete, sometimes abstract, according to the point of view from which they are regarded.

A posteriori; the distinguishing character, abstract or concrete, of a function having been established, the question of any determinate function being abstract, and there

fore able to enter into true analytical equations, becomes a simple question of fact, as we are acquainted with the elements which compose all the abstract functions at present known. We say we know them all, though analytical functions are infinite in number, because we are here speaking, it must be remembered, of the elements-of the simple, not of the compound. We have ten elementary formulas; and, few as they are, they may give rise to an infinite number of analytical combinations. There is no reason for supposing that there can never be more. We have more than Descartes had, and even Newton and Leibnitz; and our successors will doubtless introduce additions, though there is so much difficulty attending their augmentation, that we cannot hope that it will proceed very far.

It is the insufficiency of this very small number of analytical elements which constitutes our difficulty in passing from the concrete to the abstract. In order to establish the equations of phenomena, we must conceive of their mathematical laws by the aid of functions composed of these few elements. Up to this point the question has been essentially concrete, not coming within the domain of the calculus. The difficulty of the passage from the concrete to the abstract in general consists in our having only these few analytical elements with which to represent all the precise relations which the whole range of natural phenomena afford to us. Amidst their infinite variety, our conceptions must be far below the real difficulty; and especially because these elements of our analysis have been supplied to us by the mathematical consideration of the simplest phenomena of a geometrical origin, which can afford us à priori no rational guarantee of their fitness to represent the mathematical laws of all other classes of phenomena. We shall hereafter see how this difficulty of the relation of the concrete to the abstract has been diminished, without its being necessary to multiply the number of analytical elements. Thus far we have considered the Calculus as a whole. We must now consider its divisions. These divisions we must call the Algebraic Calculus, or Algebra, and the Arithmetical Calculus, or

Two parts of the Calculus.