DIFFERENTIAL (1815) — Encyclopaedia Britannica Historical Editions

DIFFERENTIAL

Volume 7 · 1,465 words · 1815 Edition

(Differentiale), in the higher geometry, an infinitely small quantity, or a particle of quantity so small as to be less than any assignable one. It is called a differential, or differential quantity, because frequently considered as the difference of two quantities; and, as such, is the foundation of the differential calculus: Sir Isaac Newton, and the English, call it a moment, as being considered as the momentary increase of quantity. See Fluxions.

DIFFERENTIAL Equation, is an equation involving or containing differential quantities; as the equatio Differential \(3x^2dx-2axdx+aydx+axdy=0\). Some mathematicians, Equation, ans, as Stirling, &c. have also applied the term differential equation in another sense, to certain equations defining the nature of series.

DIFFERENTIAL Method, a method of finding quantities by means of their successive differences.

This method is of very general use and application, but especially in the construction of tables, and the summation of series, &c. It was first used, and the rules of it laid down, by Briggs, in his construction of Logarithms and other Numbers, much the same as they were afterwards taught by Cotes, in his Constructio Tabularum per Differentialis.

The method was next treated in another form by Newton in the 5th Lemma of the 3d book of his Principia, and in his Methodus Differentialis, published by Jones in 1711, with the other tracts of Newton. This author here treats it as a method of describing a curve of the parabolic kind, through any given number of points. He distinguishes two cases of this problem; the first when the ordinates drawn from the given points to any line given in position, are at equal distances from one another; and the second, when these ordinates are not at equal distances. He has given a solution of both cases, at first without demonstration, which was afterwards supplied by himself and others: see his Methodus Differentialis above mentioned; and Stirling's Explanations of the Newtonian Differential Method, in the Phil. Tranl. N° 362.; Cotes, De Methodo Differentiali Newtoniana, published with his Harmonia Mensurarum; Herman's Phoronomia; and Le Seur and Jacquier, in their Commentary on Newton's Principia. It may be observed, that the methods there demonstrated by some of these authors extend to the description of any algebraic curve through a given number of points, which Newton, writing to Leibnitz, mentions as a problem of the greatest use.

By this method, some terms of a series being given and conceived as placed at given intervals, any intermediate term may be found nearly; which therefore gives a method for interpolation. Briggs's Arith. Log. ubi supra; Newton, Method. Differ. prop. 5.; Stirling, Methodus Differentialis.

Thus also may any curvilinear figure be squared nearly, having some few of its ordinates. Newton, ibid. prop. 6.; Cotes De Method. Differ.; Simpson's Mathematical Differ. p. 115. And thus may mathematical tables be constructed by interpolation: Briggs, ibid. Cotes Canonotechnia.

The successive differences of the ordinates of parabolic curves, becoming ultimately equal, and the intermediate ordinate required being determined by these differences of the ordinates, is the reason for the name Differential Method.

To be a little more particular.—The first case of Newton's problem amounts to this: A series of numbers, placed at equal intervals, being given, to find any intermediate number of that series, when its interval or distance from the first term of the series is given.

—Subtract each term of the series from the next following term, and call the remainders first differences, then subtract in like manner each of these differences from the next following one, calling these remainders 2d differences; again, subtract each 2d difference from the next following, for the 3d differences; and so on: Different then if A be the 1st term of the series,

\( d' \) the first of the 1st differences, \( d'' \) the first of the 2d differences, \( d''' \) the first of the 3d differences,

and if x be the interval or distance between the first term of the series and any term sought, T, that is, let the number of terms from A to T, both included, be \( =x+1 \); then will the term sought, T, be:

\[ A + \frac{x}{1} d' + \frac{x(x-1)}{1 \cdot 2} d'' + \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} d''' + \text{&c.} \]

Hence, if the differences of any order become equal, that is, if any of the diffs. \( d''', d'''' \), &c. become \( =0 \), the above series will give a finite expression for T the term sought; it being evident, that the series must terminate when any of the differences \( d''', d'''' \), &c. become \( =0 \).

It is also evident that the co-efficients \( \frac{x(x-1)}{1 \cdot 2} \), &c. of the differences, are the same as to the terms of the binomial theorem.

For ex. Suppose it were required to find the log. tangent of \( 5' 1'' 12'' 24''' \), or \( 5' 1'' \frac{65}{180} \), or \( 5' 1'' .2666 \), &c.

Take out the log. tangents to several minutes and seconds, and take their first and second differences, as below:

Here \( A = 7'1641417; x = \frac{65}{180}; d' = 14404 \); and the mean 2d difference \( d'' = -48 \). Hence

\[ \begin{align*} A &\quad 7'1641417 \\ xd' &\quad 2977 \\ \frac{x(x-1)}{1 \cdot 2} d'' &\quad -4 \end{align*} \]

Theref. the tang. of \( 5' 1'' 12'' 24''' \) is \( 7'1643998 \)

Hence may be deduced a method of finding the sums of the terms of such a series, calling its terms A, B, C, D, &c. For, conceive a new series having its 1st term \( =0 \), its 2d \( =A \), its 3d \( =A+B \), its 4th \( =A+B+C \), its 5th \( =A+B+C+D \), and so on; then it is plain that assigning one term of this series, is finding the sum of all the terms A, B, C, D, &c. Now since these terms are the differences of the sums, o, A, A+B, A+B+C, &c.; and as some of the differences of A, B, C, &c. are \( =0 \) by supposition; it follows that some of the differences of the sums will be \( =0 \); and since in the series \( A + \frac{x}{1} d' + \frac{x(x-1)}{2} d'' \), &c. by which a term was assigned, A represented the 1st term: \( d' \) the 1st of the 1st differences, and x the interval between the first term and the last; we are to write o instead of A, A instead of \( d' \), \( d' \) instead of \( d'' \), \( d'' \), differential \( d'' \) instead of \( d''' \), &c. also \( x+1 \) instead of \( x \); which Method being done, the series expressing the sums will be

\[ 0 + \frac{x+1}{1} A + \frac{x+1}{1} \frac{x}{2} d' + \frac{x+1}{1} \frac{x-1}{2} d'', \text{ &c.} \]

Or, if the real number of terms of the lines be called \( z \), that is, if \( z = x+1 \), or \( z = x-1 \), the sum of the series will be \( Az + \frac{z+1}{1} \frac{z}{2} d' + \frac{z+1}{1} \frac{z-1}{2} d'' \), &c. See De Moivre's Doct. of Chances, p. 59, 60; or his Mifcel. Analyt. p. 133.; or Simpson's Essays, p. 95.

For ex. To find the sum of fix terms of the series of squares \( 1+4+9+16+25+36 \), of the natural numbers.

<table> <tr> <th>Terms</th> <th>d'</th> <th>d''</th> </tr> <tr> <td>1</td> <td>3</td> <td>2</td> </tr> <tr> <td>4</td> <td>5</td> <td>0</td> </tr> <tr> <td>9</td> <td>7</td> <td>0</td> </tr> <tr> <td>16</td> <td>9</td> <td>0</td> </tr> </table>

Here \( A = 1 \), \( d' = 3 \), \( d'' = 2 \), \( d''' = 0 \), and \( z = 6 \); therefore the sum is \( 6 + \frac{6+1}{1} \frac{6}{2} + \frac{6+1}{1} \frac{6-1}{2} = 6 + 45 + 40 = 91 \) the sum required, viz. \( 1+4+9+16+25+36 \).

A variety of examples may be seen in the places above cited, or in Stirling's Methodus Differentialis, &c.

As to the differential method, it may be observed, that though Newton and some others have treated it as a method of describing an algebraic curve, at least of the parabolic kind, through any number of given points; yet the consideration of curves is not at all essential to it, though it may help the imagination. The description of a parabolic curve through given points, is the same problem as the finding of quantities from their given differences, which may always be done by algebra, by the resolution of simple equations. Hutton's Math. Dict.