ALGEBRAIC AND ARITHMETICAL CALCULUS.

53

Arithmetic, taking care to give them the most extended logical sense, and not the restricted one in which the terms are usually received.

It is clear that every question of Mathematical Analysis presents two successive parts, perfectly distinct in their nature. The first stage is the transformation of the proposed equations, so as to exhibit the mode of formation of unknown quantities by the known. This

constitutes the algebraic question. Then en

Algebra.

sues the task of finding the values of the formulas thus obtained. The values of the numbers sought are already represented by certain explicit functions of given numbers: these values must be determined; and this Arithmetic. is the arithmetical question. Thus the algebraic and the arithmetical calculus differ in their object. They differ also in their view of quantities,-Algebra considering quantities in regard to their relations, and Arithmetic in regard to their values. In practice, it is not always possible, owing to the imperfection of the science of the calculus, to separate the processes entirely in obtaining a solution; but the radical difference of the two operations should never be lost sight of. Algebra, then, is the Calculus of Functions, and Arithmetic the Calculus of Values.

We have seen that the division of the Calculus is into two branches. It remains for us to compare the two, in order to learn their respective extent, importance, and difficulty.

Arithmetic.

The Calculus of Values, Arithmetic, appears at first to have as wide a field as Algebra, since as many questions might seem to arise from it as we can conceive different algebraic formulas to be valued. But a very simple reflection will show that it is not so. Functions being divided into simple and compound, it is evident that when we become able to determine the value of simple functions, there will be no difficulty with the compound. In the algebraic relation, a compound function plays a very different part from that of the elementary functions which constitute it; and this is the source of our chief analytical difficulties. But it is quite otherwise with the Arithmetical Calculus.

Its Extent.

Thus, the number of distinct arithmetical operations is indicated by that of the abstract elementary functions, which we have seen to be very few. The determination of the values of these ten functions necessarily affords that of all the infinite number comprehended in the whole of mathematical analysis: and there can be no new arithmetical operations otherwise than by the creation of new analytical elements, which must, in any case, for ever be extremely small. The domain of arithmetic then is, by its nature, narrowly restricted, while that of algebra is rigorously indefinite. Still, the domain of arithmetic is more extensive than is commonly represented; for there are many questions treated as incidental in the midst of a body of analytical researches, which, consisting of determinations of values, are truly arithmetical. Of this kind are the construction of a table of logarithms, and the calculation of trigonometrical tables, and some distinct and higher procedures; in short, every operation which has for its object the determination of the values of functions. And we must also include that part of the science of the Calculus which we call the Theory of Numbers, the object of which is to discover the properties inherent in different numbers, in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic. Though the domain of arithmetic is thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which mathematical science essentially consists. This is evident, when we look into the real nature of arithmetical questions. Determinations of values are, in fact, nothing else than real transformations of the functions to be valued. These transformations have a special end; but they are essentially of the same nature as all taught by analysis. In this view, the Calculus of values may be regarded as a supplement, and a particular application of the Calculus of functions, so that arithmetic disappears, as it were, as a distinct section in the body of abstract mathematics. To make this evident, we must observe that when we desire to determine the value of an unknown number whose mode of formation is given, we define

Its nature.

and express that value in merely announcing the arithmetical question, already defined and expressed under a certain form; and that, in determining its value, we merely express it under another determinate form, to which we are in the habit of referring the idea of each particular number by making it re-enter into the regular system of numeration. This is made clear by what happens when the mode of numeration is such that the question is its own answer; as, for instance, when we want to add together seven and thirty, and call the result seven-and-thirty. In adding other numbers, the terms are not so ready, and we transform the question; as when we add together twenty-three and fourteen: but not the less is the operation merely one of transformation of a question already defined and expressed. In this view, the calculus of values might be regarded as a particular application of the calculus of functions, arithmetic thereby disappearing, as a distinct section, from the domain of abstract mathematics. And here we have done with the Calculus of values, and pass to the Calculus of functions, of which abstract mathematics is essentially composed.

Algebra.

very

small

We have seen that the difficulty of establishing the relation of the concrete to the abstract is owing to the insufficiency of the number of analytical elements that we are in possession of. The obstacle has been surmounted in a great number of important cases: and we will now see how the esta blishment of the equations of phenomena has been achieved.

Creation of

new functions.

The first means of remedying the difficulty of the small number of analytical elements seems to be to create new ones. But a little consideration will show that this resource is illusory. A new analytical element would not serve unless we could immediately determine its value: but how can we determine the value of a function which is simple; that is, which is not formed by a combination of those already known? This appears almost impossible: but the introduction of another elementary abstract function into analysis supposes the simultaneous creation of a arithmetical operation; which is certainly extremely diffi

new

cult. If we try to proceed according to the method which procured us the elements we possess, we are left in entire uncertainty; for the artifices thus employed are evidently exhausted. We have thus no idea how to proceed to create new elementary abstract functions. Yet, we must not therefore conclude that we have reached the limit appointed by the powers of our understanding. Special improvements in mathematical analysis have yielded us some partial substitutes, which have increased our resources : but it is clear that the augmentation of these elements cannot proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establishment of equations.

Finding equaThis first method being discarded, there tions between remains only one other. As it is impossible auxiliary to find the equations directly, we must seek quantities. for corresponding ones between other auxiliary quantities, connected with the first according to a certain determinate law, and from the relation between which we may ascend to that of the primitive magnitudes. This is the fertile conception which we term the transcendental analysis, and use as our finest instrument for the mathematical exploration of natural phenomena.

This conception has a much larger scope than even profound geometers have hitherto supposed; for the auxiliary quantities resorted to might be derived, according to any law whatever, from the immediate elements of the question. It is well to notice this; because our future improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only auxiliary quantities habitually substituted for the primitive quantities in transcendental analysis are what are called—

1st, infinitely small elements, the differentials of different orders of those quantities, if we conceive of this analysis in the manner of Leibnitz: or

2nd, the fluxions, the limits of the ratios of the simultaneous increments of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or

3rd, the derivatives, properly so called, of these quanti

THE TWO BRANCHES OF ALGEBRA.

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ties; that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange.

Division of

the Calculus

These conceptions, and all others that have been proposed, are by their nature identical. The various grounds of preference of each of them will be exhibited hereafter. We now see that the Calculus of functions, or Algebra, must consist of two distinct branches. The one has for its object the of functions. resolution of equations when they are directly established between the magnitudes in question: the other, setting out from equations (generally much more easy to form) between quantities indirectly connected with those of the problem, has to deduce, by invariable analytical procedures, the corresponding equations between the direct magnitudes in question;-bringing the problem within the domain of the preceding calculus.-It might seem that the transcendental analysis ought to be studied before the ordinary, as it provides the equations which the other has to resolve. But, though the transcendental is logically independent of the ordinary, it is best to follow the usual method of study, taking the ordinary first; for, the proposed questions always requiring to be completed by ordinary analysis, they must be left in suspense if the instrument of resolution had not been studied beforehand.

To ordinary analysis I propose to give the name of CALCULUS OF DIRECT FUNCTIONS. To transcendental analysis, (which is known by the names of Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc., according to the view in which it has been conceived,) I shall give the title of CALCULUS OF INDIRECT FUNCTIONS. I obtain these terms by generalizing and giving precision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis.