The Differential Calculus was embarrassed for many years after its first appearance, by considerations of Motion, Limits, and Infinitesimals. Our countryman, Landen, having made an ineffectual attempt to remove these difficulties, the subject was taken up by Arago, who completely new-modelled the Calculus, and founded it upon a pure analytical basis. The genius of La Grange first established the new system, but left the theory obscure and the practical application inconvenient; by attempts to demonstrate matters of definition; by innovations in the notation; and by the universal substitution of the differential coefficients for the differentials themselves. Many of these defects were pointed out by Mr Woodhouse; and the following propositions present a systematic view of the calculus in its latest form, with some farther modification of the principles, and some generalization of the formulæ.

Definitions.

1. If \( f(x + h) \) can be expanded in the form

\[ f(x) + f'(x) \cdot h + f''(x) \cdot h^2 + \ldots \]

Then \( f'(x) \) is called the differential function of \( f(x) \), and its relation to \( f(x) \) is thus expressed, \( f'(x) = Df(x) \).

2. If \( (x + h) \) be put under the form \( (x) + D(x) \cdot h \), it appears that \( Dx = 1 \). This, however, is entirely arbitrary, and unity is merely selected for convenience, since by defining the differential function to be the coefficient in the second term of \( f(x + a \cdot h) \) expanded, we should have had \( Dx = a \). Similarly \( c = (c) + D(c) \cdot h \), and \( D(c) = 0 \).

3. In different functions of the same principal variable \( (x) \), we may avoid a perpetual reference to \( (x) \), and have the advantage of considering such functions as independent variables, by means of another symbol, which shall denote expressions proportional to the differential functions. Thus \( dz(x) \), \( d\varphi(x) \) are called differentials, and are merely limited to satisfy the equation, \( Dz(x) \times d\varphi(x) = D\varphi(x) \times dz(x) \).

4. Hence \( du, dz, dv, \&c. \) are expressions proportional to the differential functions of \( u, z, v, \&c. \), considered as functions of some variable, the same in all of them.

5. If in the equation \( Dz(x) \cdot d\varphi(x) = D\varphi(x) \cdot dz(x) \), we take \( \varphi(x) = (x) \), then \( D(x) \cdot d\varphi(x) = D\varphi(x) \cdot dx \), or \( d\varphi(x) = D\varphi(x) \cdot dx \).

6. If there are several characteristics, \( D \) only refers to the first, thus \( Df(x) = Df_1(x) \), \( Df_2 \) being always regarded as a single symbol. Prop. I.—To find the Differential of a Function with successive Characteristics.

Let the function be \( f_1 \ldots f_n(x) \), then

\[ df_1 \ldots df_n(x) = Df_1 \ldots df_n(x) \times df_1 \ldots df_n(x) \]

\[ = dx \times Df_1 \ldots Df_n(x) \times \ldots Df_1 \ldots Df_n(x) \text{ by continuing the same process.} \]

Cor. If \( f_1 = f_2 = f_n = f \), Then

\[ df_n(x) = dx \times Df_1(x) \times Df_2(x) \times \ldots Df_{n-1}(x). \]

Prop. II.—To find the Differential of an Inverse Function.

If \( f(x) = z \), and if \( x \) found in terms of \( z \), give \( x = F(z) \), then \( F \) is represented by \( f^{-1} \); the inverse form of \( f \), and \( f^{-1}(x) \) is called the inverse function of \( f(x) \). Now, as we may frequently know the differential of \( f \), our object is to determine the differential of \( f^{-1} \) by means of it. Since

\[ x = f^{-1}(x), \quad dx = Df^{-1}(x) \times df^{-1}(x) \]

\[ \therefore df^{-1}(x) = \frac{dx}{Df^{-1}(x)} \]

Prop. III.—To find the Differential of a Function involving independent Characteristics.

Let the function be \( F \{ f_1(x), f_2(x), \ldots, f_n(x) \} \) in which the characteristics \( f_1, f_2, \ldots, f_n \) contained under \( F \) are wholly independent of each other. If the differential function be taken on the supposition, that only one of these independent functions contains \( x \), the expression is called a partial differential function,

and \( DF \{ f_1(x), \ldots, f_n(x) \} \times Df_1(x), Df_2(x), \ldots, Df_n(x) \) may denote such partial differential functions, taken with regard to \( f_1(x) \) and \( f_n(x) \) respectively. The general differential function of \( F \{ f_1(x), \ldots, f_n(x) \} \) may be ascertained, if we successively expand \( F \{ f(x + h), \ldots, f_n(x + h) \} \) by means of its partial differential functions; for it will become

\[ F \{ f(x), \ldots, f_n(x + h) \} + DF \{ f(x), \ldots, f_n(x + h) \} \times Df(x) \times h + \ldots \]

and finally, by continuing the same operations, we shall find the coefficient of \( (h) \) to be

\[ DF \{ f(x), \ldots, f_n(x) \} \times Df_1(x) + \ldots DF \{ f(x), \ldots, f_n(x) \} \times Df_n(x), \]

from which it appears, that the general differential function is the sum of all the partial differential functions, and that it is indifferent in what order they are taken, whether \( D \), in respect of \( f_1(x) \), or \( D \) in respect of \( f_n(x) \).

Cor. 1. \( dF \{ f(x), \ldots, f_n(x) \} \)

\[ = DF \{ f(x), \ldots, f_n(x) \} \times Df(x) \times dx + \ldots \]

\[ = DF \{ f(x), \ldots, f_n(x) \} \times df(x) + \ldots \]

Cor. 2. If we substitute for \( f(x), f(x), \ldots \) the independent variables \( u, z, \ldots \), it appears that

\[ dF \{ u, z, \ldots \} = DF \{ u, z, \ldots \} \cdot du + \]

\[ DF \{ u, z, \ldots \} \cdot dz + \ldots \]

Cor. 3.—If the partial differentials be denoted by \( d_1, d_2, \ldots \) and if the characteristics be separated, we have \( d = d_1 + d_2 + \ldots d_n \)

Cor. 4. If we denote the partial differential taken with regard to \( (u) \) by \( \frac{d}{du} F \{ u, z, \ldots \} \cdot du \), and the rest similarly, and if \( Q \) be substituted for \( F \{ u, z, \ldots \} \), \( dQ = \frac{d}{du} Q \cdot du + \frac{d}{dz} Q \cdot dz + \ldots \),

which very elegant notation is used by Mr Babbage in the Transactions of the Royal Society of London.

Cor. 5.—By combining the present proposition with Prop. 1., we are enabled to find the differentials of the intricate expressions in Mr Babbage's Calculus of Functions.

Lemma.—To enumerate and arrange the primary relations of which analysis consists.

1. In order to exhibit the relations subsisting among the successive functions \( x + a, x \times a, x^a \), it will be necessary to present them under a different form of notation. Let

\[ x^1 a = x + a \]

\[ x^2 a = x \times a \]

\[ = x + \{ x + \ldots \text{to } (a) \text{ terms.} \]

\[ x^3 a = x^3 \]

\[ = x \times \{ x \times \ldots \text{to } (a) \text{ terms.} \]

\[ x^4 a = x^4 \]

\[ \text{to } (a) \text{ terms.} \]

Then generally

\[ x^{n+1} a = x^n \{ x^n \ldots \text{to } (a) \text{ terms.} \]

These are called primary direct relations. 2. The primary inverse relations are found by determining \( z \) from the equation \( z^n = a = x \). They are \( x = a, x = a^{\frac{1}{n}}, \sqrt[n]{x}, \ldots \), &c.

3. The primary reciprocal relations are represented by \( a^n x, (a) \) and \( (x) \) being interchanged, and are \( a + x, a \times x, a^n, a^{n+1} \) to \( (x) \) terms, &c.

4. The primary inverse reciprocal relations are found by determining \( (z) \) from the equation \( a^n z = x \), thus from \( x = a^n z = a^{\frac{1}{n}} \) is deduced log \( (x) \) base \( (a) \).

5. By taking alternately the reciprocal and inverse forms of the first three relations, we shall find that they circulate and only introduce known functions. Thus \( x^n \) becomes \( ax, \log (x) \) base \( (a), \log (a) \) base \( (x), a^{\frac{1}{n}}, x^{\frac{1}{n}}, x^{\frac{1}{n}} \).

6. Every finite expression now used, is composed of these primary relations. It cannot therefore be a matter of surprise that analysis should frequently present expressions appearing to pass into each other discontinuously and without a law, when the laws of succession in these fundamental forms have never been stated; and the functions beyond \( x \pm a \) have not even been mentioned.

**Prop. IV. — To find the Differentials of Functions under the first relation.**

If the expression be \( f(x) + f(x) \),

\[ f(x+h) + f(x+h) = \left\{ \begin{array}{l} f(x) + Df(x) \cdot h + \\ f(x) + Df(x) \cdot h + \ldots \end{array} \right\} \]

\[ \therefore D \left\{ \begin{array}{l} f(x) + f(x) \end{array} \right\} = Df(x) + Df(x) \]

Similarly

\[ D \left\{ \begin{array}{l} f(x) - f(x) \end{array} \right\} = Df(x) - Df(x) \]

Cor. 1. \( d \left\{ \begin{array}{l} x + \ldots x_n \end{array} \right\} = dx + \ldots dx_n \)

Cor. 2. \( D \left\{ \begin{array}{l} f(x) \pm a \end{array} \right\} = Df(x) \pm D(a) \)

\[ = Df(x) \]

Cor. 3. \( d(x \pm a) = dx, d(a \pm x) = \pm dx \).

**Prop. V. — To find the Differentials of Functions under the second relation.**

If the expression be \( f(x) \times f(x) \),

\[ f(x+h) \times f(x+h) = \left\{ \begin{array}{l} f(x) + Df(x) \cdot h + \\ f(x) + Df(x) \cdot h + \ldots \end{array} \right\} \]

\[ \therefore D \left\{ \begin{array}{l} f(x) \times f(x) \end{array} \right\} = Df(x) \cdot f(x) \]

By a process nearly similar,

\[ D \left\{ \begin{array}{l} f(x) \div f(x) \end{array} \right\} = \frac{Df(x) \cdot f(x) - Df(x) \cdot f(x)}{f(x)^2} \]

Cor. 1. \( d \left\{ \begin{array}{l} x \div \ldots x_n \end{array} \right\} = \frac{x \div \ldots x_n}{x_1} \times \left\{ \begin{array}{l} dx \div \ldots dx_n \end{array} \right\} \)

Cor. 2. \( D \left\{ \begin{array}{l} f(x) \times a \end{array} \right\} = Df(x) \times a \).

Cor. 3. \( d \left\{ \begin{array}{l} x \times a \end{array} \right\} = dx \times a \).

\[ d \left\{ \begin{array}{l} a \div x \end{array} \right\} = -\frac{a \cdot dx}{x^2} \]

**Prop. VI. — To find the Differentials of Functions under the third relation.**

If the expression be \( f(x)^a \),

\[ f(x+h)^a = \left\{ \begin{array}{l} f(x) + Df(x) \cdot h + \ldots \end{array} \right\}^a \]

\[ = f(x)^a + a \cdot f(x)^{a-1} \cdot Df(x) \cdot h + \ldots \]

\[ \therefore D \left\{ \begin{array}{l} f(x)^a \end{array} \right\} = a \cdot f(x)^{a-1} \cdot Df(x). \]

If the expression be \( af(x) \),

\[ f(x+h) = f(x) + Df(x) \cdot h + \ldots \]

\[ = a \times \left\{ \begin{array}{l} f(x) + Df(x) \cdot h + \ldots \end{array} \right\} \]

\[ = a \cdot f(x) + L(a) \cdot Df(x) \cdot h + \ldots \]

\[ \therefore D \left\{ \begin{array}{l} af(x) \end{array} \right\} = L(a) \cdot af(x) \cdot Df(x) \]

If the expression be \( \log f(x) \) base \( (a) \) or \( \frac{L_f(x)}{L(a)} \),

\( L \) denoting the Napierian logarithm,

\[ L_f(x+h) = L \left\{ \begin{array}{l} f(x) + Df(x) \cdot h + \ldots \end{array} \right\} \]

\[ = Lf(x) + L \left\{ \begin{array}{l} 1 + \frac{Df(x)}{f(x)} \cdot h + \ldots \end{array} \right\} \]

\[ = Lf(x) + \frac{Df(x)}{f(x)} \cdot h + \ldots \] \[ D \left\{ \log f(x) \right\} = \frac{Df(x)}{L(a) \cdot f(x)} \]

Also, \( D \left\{ \log (a) \text{ base } f(x) \right\} = D \frac{L(a)}{L_f(x)} \) which is known.

**Cor. 1.** \( d \left( x^{\frac{x}{2}} \right) = d \left( x^{\frac{x}{2}} \right) + d \left( x^{\frac{x}{2}} \right) \)

\[ = x \cdot x^{\frac{x}{2}-1} \cdot dx + L_x \cdot x^{\frac{x}{2}} \cdot dx \]

**Cor. 2.** \( d \left\{ x^a \right\} = ax^{a-1} \cdot dx \)

\( d \left\{ \log (x) \right\} = \frac{dx}{L_a \cdot x} \)

The first three propositions will enable us to extend these formulas to every finite function at present used in analysis, but when we attempt to find the differentials in relations of a higher order, the difficulties increase very rapidly. Indeed, the direct calculus, which is generally considered perfect, is only less limited than the inverse. Let the most expert analyst attempt, in analogy to Prop. 2, to express the differential of the reciprocal function \( f(x, a) \), by means of the direct function \( f(x, a) \); let him endeavour to develop \( (x + h)^n \cdot a \) in analogy to Sir Isaac Newton's expansion of \( (x + h)^n = a^n \); or let him search for the differential of \( a^x = a^x \) to \( x \) terms, and he will be forced to admit the great imperfections of the instrument.

**Prop. VII.**

*To find Successive Differentials.*

If we take the differential of \( Df(x) \) considered as a new function, we obtain the second differential function \( D \left\{ Df(x) \right\} \), which is represented by \( D^2f(x) \), and, in the same manner, we may find the value of \( D^n f(x) = D \left\{ D \left\{ \ldots Df(x) \right\} \right\} \). The laws observed by the successive differentials of the functions previously examined are very complex, and can only be expressed by separating the differential characteristics, and operating upon them as independent symbols.

**Prop. VIII.—To find the Coefficients of the Differential Expansion.**

Since \( f \left\{ x + (i + h) \right\} = f \left\{ (x + h) + i \right\} \), we develop these expressions separately, and equate their homologous terms. The first becomes

\[ f(x) + f(x) \cdot (i + h) + f(x) \cdot (i + h)^2 + \ldots \]

which is equal to

\[ f(x) + f(x) \cdot i + f(x) \cdot i^2 + \ldots \]

The second becomes

\[ f(x + h) + f(x + h) \cdot i + f(x + h) \cdot i^2 + \ldots \]

which is equal to

\[ f(x) + f(x) + f(x) + \ldots \]

\[ + Df(x) \cdot h + Df(x) \cdot ih + Df(x) \cdot i^2h + \ldots \]

And now, by equating the same powers of \( i \) and \( h \), in the second terms of each development,

\[ f(x) = Df(x) \cdot 2 = Df(x) \cdot 3 = Df(x) \]

\[ f(x) \cdot n = Df(x) \cdot n \]

\[ \therefore f(x) = Df(x) \cdot \frac{1}{n} \]

\[ = D^n f(x) \times \frac{1}{1 \cdot 2 \cdot \ldots \cdot n} \]

Hence \( f(x + h) \) becomes

\[ f(x) + Df(x) \cdot \frac{h}{1} + \ldots D^n f(x) \cdot \frac{h^n}{1 \cdot 2 \cdot \ldots \cdot n} + \ldots \]

**Cor. 1.** \( f(x) = f(a + x) \)

\[ = f(a) + Df(a) \cdot \frac{x-a}{1} + D^2f(a) \cdot \frac{(x-a)^2}{1 \cdot 2} + \ldots \]

**Cor. 2.** \( D^{-1} f(x) \) is the inverse differential function, and is of such a nature that \( D \left\{ D^{-1} f(x) \right\} = f(x) \).

\[ D^{-1} f(x) = D^{-1} f \left\{ a + (x-a) \right\} = \]

\[ D^{-1} f(a) + f(a) \cdot \frac{x-a}{1} + Df(a) \cdot \frac{(x-a)^2}{1 \cdot 2} + \ldots \]

where \( D^{-1} f(a) \) is the value of \( D^{-1} f(x) \), when \( x = a \), and it is called the Correction.

**Cor. 3.** \( D^{-1} f(a) = D^{-1} f \left\{ x + (a-x) \right\} = \]

\[ D^{-1} f(x) + f(x) \cdot \frac{a-x}{1} + Df(x) \cdot \frac{(a-x)^2}{1 \cdot 2} + \ldots \]

\[ \therefore D^{-1} f(x) = D^{-1} f(a) - f(x) \cdot \frac{a-x}{1} - \]

\[ Df(x) \cdot \frac{(a-x)^2}{1 \cdot 2} - \ldots \]

From these theorems the inverse calculus may be deduced by direct processes.

**Cor. 4.** There is nothing in the nature of analysis which necessarily limits expansions to the differential form. The series may be conceived to proceed by other functions of \( x \) and \( h \); it even may not consist of a succession of sums, but may ascend by products, or according to any other relation. Thus we may find the successive terms of \( (x \times h) \) when developed in the form For, let \( f(x) \) be called the factorial function of \( f(x) \), and let its relation to \( f(x) \) be thus expressed, \( f(x) = P f(x) \); then by expanding

\[ f(x \times i^{h+k}) = f(x \cdot h \times i^k) \]

and by equating the like powers of \( i \) and \( h \), we shall find \( f(x \times i^h) \) become

\[ f(x) \times P f(x)^{i} \times P^2 f(x)^{i^2} \times \cdots \]

a theorem entirely analogous to that of Taylor, and presenting a factorial calculus on principles similar to the differential. It is not, however, necessary to deduce the values of \( P f(x) \), by making it the subject of distinct investigations, since Mr Herschel, on seeing the formula, has discovered an expression for \( P f(x) \) by means of \( D f(x) \), and observes, that the factorial calculus so harmonizes with the differential, that either may be established, and the other deduced from it.

Differentials depend upon the development of \( f(x + a + b) \), factorials upon \( f(x + a + b) \), and it is obvious that a wide field is open for \( f(x + a + b) \).

Mr Whewell further suggests the idea of an expansion, in which the relation of the successive terms shall be expressed by a functional characteristic.

The object of the calculus is not to show that every function of the form \( f(x + h) \) must necessarily proceed by integer powers of \( h \). The legitimate design is to exhibit the relation subsisting among the successive terms, and to find the value of \( D f(x) \) in complex expressions, by means of the values in the primary functions. It supposes the general form of development given, and leaves the determination of the second terms, in the expansion of the primary functions, to the artifices of the ordinary algebra.

**Prop. IX.—To find the Differentials of Continuous Quantities.**

Let \( x \) be a known discrete variable, and let any continuous quantity, which varies with \( x \), be represented by \( \Psi(x) \), an unknown function of \( x \). If then \( x \) becomes \( x + h \), \( \Psi(x) \) becomes \( \Psi(x + h) \), and we obtain the following equal ratios:

\[ \Delta x : \Delta \Psi(x) \] \[ (x + h) - x : \Psi(x + h) - \Psi(x) \] \[ h : D \Psi(x) \cdot h + \cdots \] \[ 1 : D \Psi(x) + D^2 \Psi(x) \cdot \frac{h}{1!} + \cdots \]

Let \( Q \) and \( Q' \) be discrete known variables, one always greater, and the other always less than \( \Delta \Psi(x) \). Also, let the ratios

\[ \Delta x : Q \text{ and } \Delta x : Q' \]

be such, that both may be made to approach nearer than by any assignable difference, to the ratio

\[ 1 : f(x). \]

Then, since \( Q \) and \( Q' \) are one greater and the other less than \( \Delta \Psi(x) \), the ratio of

\[ 1 : f(x) \]

cannot possibly differ from that ratio to which

\[ \Delta x : \Delta \Psi(x), \text{ or } 1 : D \Psi(x) + \cdots \]

approaches nearer than by any assignable difference. It may further be proved, that the only ratio which can be true is,

\[ 1 : f(x) :: 1 : D \Psi(x) \] \[ \therefore D \Psi(x) = f(x), \text{ and } \Psi(x) = D^{-1} f(x). \]

If, therefore, we are investigating any curve, or the motion of any body, where the analytical value is unknown, Newton lays it down as a practical rule for finding the differentials, that they will be in the limiting ratio of the corresponding increments, viz., in the ratio to which the increments approach nearer than by any assignable difference. This practical rule, offered by the great inventor, may be demonstrated ex absurdo, with all the rigour of the ancient geometry. The formula of Taylor not having been distinctly expressed, Newton could not lay down the analytical theory of his calculus, but he left a practical rule, which is the best that has yet been proposed, and he left an illustration, the most beautiful in the circle of the sciences. He compared the differentials to the velocities of a body in motion. Many modern writers still define differentials by means of the limiting ratio; but this is as inconsistent, as it would be to define multiplication by the mechanical rule for finding the product. The method of infinitesimals is now universally exploded, having been originally proposed by Leibnitz, who, perceiving that Newton's results depended solely on the second terms, supposed that his calculus must consist in making the subsequent terms vanish. The results were here happily correct, though a similar attempt to appropriate the theory of gravitation was wholly unsuccessful. The latest English work on this branch of analysis, is the Translation of La Croix's Treatise, where Mr Peacock gives a luminous view of the different theories on the subject.