DIFFERENTIAL

, an indefinitely small quantity, part, or difference. By some, the Differential is considered as infinitely small, or less than any assignable quantity; and also as of the same import as fluxion.

It is called a Differential, or Differential Quantity, because often considered as the difference between two quantities; and as such it is the foundation of the Differential Calculus. Newton used the term moment in a like sense, as being the momentary increase or decrease of a variable quantity. M. Leibnitz and others call it also an infinitesimal.

Differential of the 1st, 2d, 3d, &c degree. See Differentio-Differential.

Differential Calculus, or Method, is a method of differencing quantities. See Differential Method, Calculus, and Fluxions.

Differentio-Differential Calculus, is a method of differencing differential quantities.

As the sign of a differential is the letter d prefixed to the quantity, as dx the differential of x; so that of a differential of dx is ddx, and the differential of ddx is dddx, &c; similar to the fluxions x., x.., x, &c.

Thus we have degrees of differentials. The differential of an ordinary quantity, is a differential of the first order or degree, as dx; that of the 2d degree is ddx; that of the 3d degree, dddx, &c. The rules for differentials, are the very same as those for fluxions. See Fluxions.

Differential

, in Logarithms. Kepler calls the logarithms of tangents, differentiales; which we usually call artificial tangents.

Differential Equation, is an equation involving or containing differential quantities; as the equation . Some mathematicians, 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. This method 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; as I have shewn in the Introduction to my Logs. pa. 69 & seq. See Briggs's Arithmetica Logarithmica, cap. 12 and 13, and his Trigonometria Britannica.

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 2d, 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 Explanation of the Newtonian Differential Method, in the Philos. Trans. N° 362; Cotes, De Methodo Differentiali Newtoniana, published with his Harmonia Mensurarum; Herman's Phoronomia; and Le Seur & 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 interpolations. Briggs's Arith. Log. ubi supra; Newton Meth. 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. pa. 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, &c; 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+(x/1)d′+((x/1).((x-1)/2))d″+((x/1).((x-1)/2).((x-2)/3))d‴ &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 diffs. d″, d‴, &c, become = 0. |

It is also evident that the co-efficients x/1, (x/1).((x-1)/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″ 62/300, or 5′ 1″ .2066 &c.

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

Tang.d′d″
5′0″7.162696414453-48
517.164141714404-49}
527.165582114357-47
537.1670178

Here A = 7.1641417; x = 62/300; d′ = 14404; and the mean 2d difference d″ = -48. Hence

A7.1641417
xd2977
(x/1).((x-1)/1)d4
Theref. the tang. of 5′ 1″ 12‴ 24′′′′ is7.1644398

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 , its , its , 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, 0, 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 + (x/1)d′+(x/1).((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 0 instead of A, A instead of d′, d′ instead of d″, d″ instead of d‴, &c, also x+1 instead of x; which being done, the series expressing the sums will be 0 + ((x+1)/1)A + ((x+1)/1).(x/2)d′ + ((x+1)/1).(x/2).((x-1)/3)d″, &c. Or, if the real number of terms of the lines be called z, that is, if , or , the sum of the series will be &c. See De Moivre's Doct. of Chances, pa. 59, 60; or his Miscel. Analyt. pa. 153; or Simpson's Essays, pa. 95.

For ex. to find the sum of six terms of the series of squares 1 + 4 + 9 + 16 + 25 + 36, of the natural numbers.

Termsd′d″d‴
132
4520
9720
169
25

Here A = 1, d′ = 3, d″ = 2, d‴ &c = 0, and z = 6; therefore the sum is 6 + (6/1).(5/2).3 + (6/1).(5/3).(4/3).2 = 6 + 45 + 40 = 91 the sum required, viz. of 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. See Stirling's Method. Differ. pa. 97. This ingenious author has treated very fully of the differential method, and shewn its use in the solution of some very difficult problems. See also Series.

Differential Scale, in Algebra, is used for the scale of relation subtracted from unity. See Recurring Series.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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DIASTYLE
DIATESSARON
DIATONIC
DIESIS
DIFFERENCE
* DIFFERENTIAL
DIFFRACTION
DIGBY (Sir Kenelm)
DIGGES (Leonard)
DIGGES (Thomas)
DIGIT