DIFFERENCE ; DIFFERENCES; CALCULUS OF FINITE DIFFERENCES. The word dif ference means usually the excess of one quantity over another of the same kind, and this is its meaning in arithmetic. In the higher branches of mathematics, however, it has a peculiar meaning, which we shall briefly explain. When we have a series of num bers connected by a regular, though not obvious law,. the character of that law may be detected by forming a new series of the dt:breremees between each term of the original series and the next, and then treating the new series (which we may call the series of " first differences") in the same way; and so on, till we reach a series of differences the law of which is manifest. Thus, Given series, . . 4, 7, 11, 18, 31, 54, 92, 15'1 First differences, . 3, 4, 7, 13, 23, 38, 59 Second differences, . 1, 3, 6, 10, 15, 21 Third differences; . . 2, 3, 4, 5, 6 The law of the series of third differences is manifest; we see that its next term must be 7, which gives 28 as the next term of the series of second differences, 87 as the next of the first differences, and so 238 for that of the original series, which we can thus con tinue to any number of terms. To take a simpler case. Let the series be
43, 47, 53, 61, 71 First differences 4, 6, 8, 10 Second differences 2, 2, 2 Here the law is manifest in the first differences, and we should be able to calculate the series if we knew the first terms of the three series; viz., 43, 4, 2. It is on this princi ple that calculating machines (q.v.) can be constructed to compute tables of logarithms, etc. Out of the method of differences sprung the calculus of finite differences, first treated by Dr. Brook Tayloll. under the name of the method of increments. This calcu lus has nothing to do with the transcendental analysis. See CALCULUS DIFFERENTIAL, etc. It deals with the changes of functions when definite increments are made to the variables; while the transcendental analysis considers only their changes when indefi nitely small or infinitesimal additions are made to the variables. It would be out of place here to attempt an account of the calculus of finite differences. See Fuxcnoxs.