17.] Let us now investigate these properties with greater precision, and with reference to symbolized number; and let the first subjects of our consideration be continuous functions of continuous variables; and let us take the simple case of an explicit function of one variable, of the form

let us consider it in two successive states, and first at a finite

interval apart.

Let ▲ be used as an abbreviation of difference, and represent a finite increment of a function or of a variable; so that ax* and ay represent the finite increments which x and y receive, and af(x) represents the finite change in f(x) due to the finite augment of the independent variable x; whence we have,

sf(x) = ▲y = f(x+▲x) −ƒ(x).

(2)

Thus af(x) is the quantity by which f(x) is increased, as the variable on which it depends is increased, and is therefore called the difference of f(x).

Now suppose these increments to become infinitesimal, in which case we shall use d, the abbreviation of differential or small difference, to symbolize them: so that dr and dy represent the infinitesimal increments which x and y receive, and df(x) the infinitesimal increment which f(x) receives, owing to the infinitesimal increment of its independent variable; so that (2) becomes

df(x) = dy = f(x+dx) − f(x).

(3)

Thus df(x) is the infinitesimal quantity by which f(x) is increased, by reason of the infinitesimal increase of the variable on which it depends, and is therefore called the differential of f(x).

As the symbols d and ▲ will be employed throughout the treatise in the meanings here assigned to them, let the difference between them be noticed. Also let it be observed, that when they are prefixed to a or y or f(x), they have not the effect of multiplication: that is, da is not d times a; but their

▲ may be negative; in which case it might perhaps be more properly called a decrement; but in the following treatise we shall use the words augments and increments to express the variations in the values of the variables, whether such variations cause them to increase or decrease.

power is that of an operation performed on the subject to which they are prefixed: thus do is the differential of x, the infinitesimal increment of x, or the infinitesimal quantity by which a grows so df(x) is the infinitesimal increment of f(x) when x has been increased by the infinitesimal quantity dx.

From the form (3), which the most convenient mode of determining such infinitesimal increments takes, has the name "Differential Calculus" arisen.

The operation of determining the values of the differentials of the functions due to the differential increase of the variable is called Differentiation, and is the first work of the Calculus; and we are said to differentiate the function with respect to that variable, owing to the change of which the function changes. Hereby the materials will be formed, which are subject to the axioms previously enuntiated, and the other laws of which will be subsequently developed. We shall generally differentiate directly and without the intervention of any other symbols than those introduced above, but sometimes it is convenient to put the result in the following form.

18.] Since the left-hand member of equation (3) is an infinitesimal of, say, a first order, the right-hand member must be an infinitesimal of the same order; an infinitesimal must therefore be a factor of it. But the only infinitesimal that it involves is da, therefore we may reasonably presume that de will be the factor; the presumption however must be verified by subsequent investigations. Dividing both sides by dx, we have

of which the last member is of the form, but will be a finite quantity, if our presumption is correct. Let us represent it by f'(x), so that

dy
dx

=

d.f(x)
dx

f'(x) is called the derived function of f(x), and represents the

(6)

ratio of the differential of the function to the differential of the variable; and therefore if it is known, the absolute change of the function due to the change of the variable is also known. Also f'(x) is called the differential coefficient, because it is the coefficient of de in the equation d.f(x) = ƒ'(x) dx; and f'(x)dx is by (6) the excess of the function when its subject variable is infinitesimally increased over its value when the subject is not increased.

It is also manifest, that generally

ƒ(x+▲x)−ƒ(x)

= f'(x)+R,

(7)

Ax

when R is some residual quantity, which must be neglected when a becomes dx.

The process by which derived functions are determined is called Derivation: and evidently by it a function becomes the parent of some other function. On this principle Lagrange, one of the most eminent mathematicians of the eighteenth century, has constructed his works on the Infinitesimal Calculus, viz. "Théorie des Fonctions Analytiques," and its sequel "Leçons sur le Calcul des Fonctions." It is however beside my present purpose to unfold the process, because it is so different to that which I have made fundamental; and it is impossible adequately to discuss the reasons why the preceding principles are preferred, because fully to do so requires a knowledge of all the Calculus. On the latter subject however a few words are said in the Preface.

19.] Similarly, in considering a function of many variables, such as

u = F(X, Y, Z, ...),

we shall use ▲ and d to symbolize respectively finite and infinitesimal changes: so that au, Ax, Ay, ... represent finite, and du, dx, dy, dz... infinitesimal quantities; wherefore

▲U = F(X+▲x, y + ▲Y, ...) — F (X, Y, ...)

du = F(x+dx, y + dy, z + dz, ...) — F(x, y, z, ...).

(8)

(9)

It is, however, more convenient to reserve the complete inquiry into the variations of functions of many variables for a later part of our treatise.

PRICE, VOL. I.

F

20.] Hence we may describe the Differential Calculus as follows:

The Differential Calculus is a general method and system of rules by which are determined the corresponding changes of functions and variables, when the variations of the variables are infinitesimal; and the code of laws to which they are subject, and conformably to which they may be applied to questions of Geometry and Physics.

Before however we proceed to give either general rules for the differentiation of functions, or to illustrate the process by particular examples, it is necessary to determine the values of two functions for certain values of the variables, which will be frequently applied hereafter.

SECTION 2.-Fundamental Lemmas of the Infinitesimal Calculus.

21.] LEMMA I. To evaluate (1+x), when x is an infinitesimal.

Let x be some small positive fractional number; then it is plain that each factor in the numerators of the several terms of the series is less than 1; and therefore, no term being negative, the whole series is greater than its first two terms, that is, is greater than 2. Also since

each term of this after the second is greater than the corresponding term in the series above, and therefore the whole series is greater; and therefore, when x is a small positive fractional number, (1+x) is equal to some number greater then 2 and less than 3.

1

Let x be an infinitesimal, and thus be symbolized by 0, in which case by virtue of Theorem VI, Art. 9, we have

(1 + 0)* = 1 + 1+

+

+

+

1.2 1.2.3 1.2.3.4 1.2.3.4.5

which must be summed arithmetically as follows:

which is the correct sum of the series to seven places of decimals. This arithmetical number, which is incommensurable * with any digit of the decimal scale of notation, and which is identical with the base of the Napierian logarithms, is symbolized by e; so that we have

(1+x) = 2.7182818

= e,

(10)

when x is an infinitesimal; and wherever e is met with in the course of this work, it is used as the symbol for this arithmetical quantity.

Again, if x were a small negative fractional quantity, then

* Peacock's Algebra, Vol. I, Art. 203, Cambridge, 1842.