(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 equation Differential Method, a method of finding quantities by means of their successive differences. See Differential Calculus, Supplement.

It is of very general use, especially in the construction of tables, and the summation of series. 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 Differentias.

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. Trans. No 362.; Cotes, De Methodo Differentiali Newtoniana, published with his Harmonia Mensurarum: Herman's Phoronoma: 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 Dissert. 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: 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

\[ \Delta + \frac{x}{1} d' + \frac{x-1}{2} d'' + \frac{x-2}{3} d''' + \ldots \]

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 \( \frac{x}{1}, \frac{x-1}{2}, \ldots \)

&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 \( 6' 1'' 38'' \), or \( 5' 1'' 2066 \), &c.

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

| Tan | d' | d'' | |-----|----|----| | 5' 0'' | 7'1626964 | 14453 | | 5' 1'' | 7'1641417 | 14404 | | 5' 2'' | 7'1655821 | 14357 | | 5' 3'' | 7'1670178 |

Here \( A = 7'1641417; x = \frac{6}{8}; d'' = 14404 \); and the mean 2d difference \( d''' = -48 \). Hence

\[ A = 7'1641417; \quad \frac{x}{1} d' = 2977; \quad \frac{x-1}{2} d'' = -4 \]

Theref. the tang. of \( 5' 1'' 12'''' 24'''''' \) is \( 7'1644398 \)

Hence may be deduced a method of finding the sum 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, \( o, A, A+B, A+B+C, \ldots \); 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 \( A + \frac{x}{1} d' + \frac{x-1}{2} d'' + \ldots \),

&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 \( d''' \), &c. of the navy, and governor of the Trinity-house. He granted him letters of reprisal against the Venetians, by virtue of which he took several prizes with a small fleet under his command. He fought the Venetians near the port of Scanderoon, and bravely made his way through them with his booty. He was a great lover of learning, and translated several authors into English; and his "Treatise of the Nature of Bodies and the Immortality of the Soul," discovers great penetration and extensive knowledge. He applied to chemistry; and found out several useful medicines, which he gave freely away to people of all sorts, especially to the poor. He distinguished himself particularly by his sympathetic powder for the cure of wounds at a distance; his discourse concerning which made a great noise for a while. He had conferences with Des Cartes about the nature of the soul.

In the beginning of the civil wars, he exerted himself very vigorously in the King's cause; but he was afterwards imprisoned by the parliament's order, in Winchester-house, and had leave to depart thence in 1643. He afterwards compounded for his estate, but was ordered to leave the nation; when he went to France, and was sent on two embassies to Pope Innocent X. from the queen, widow to Charles I. whose chancellor he then was. On the restoration of Charles II. he returned to London; where he died in 1663, aged 60.

This eminent person, on account of his early talents, and great proficiency in learning, was compared to the celebrated Picus de Mirandola, who was one of the wonders of human nature. His knowledge, though various and extensive, appeared to be greater than it really was; as he had all the powers of elocution and address to recommend it. He knew how to shine in a circle of ladies or philosophers; and was as much attended to when he spoke on the most trivial subjects, as when he conversed on the most important. It is said that one of the princes of Italy, who had no child, was desirous that his princess should bring him a son by Sir Kenelm, whom he esteemed a just model of perfection.