INCREMENT, is the small increase of a variable quantity. Sir Isaac Newton calls these increases" moments," and ob serves, tkat they are proportional to the velocity or rate of increase of the flowing or variable quantities, in an indefinitely small time. The notation of increment is different by different authors. The method of increments is a branch of analytics, in which a calculus is founded on the proper ties of successive values of variable quan. tities, and th eir differences, or increments. It is nearly allied to the doctrine of fluxions, and, in truth, arises out of it. Of the latter the great Newton was the inventor ; of the former we have different treatises by Dr. Taylor, Mr. Emerson, and others. We shall give Mr. Emerson's observations on the distinction between the method of increments and fluxions. "From the method of increments," he says, "the principal foundation of the method of fluxions may be easily derived; for, as in the method of increments, the increment may be of any magnitude, so in the method of fluxions it must be sup posed infinitely small; whence all preced ing and successive values of the variable quantity will be equal, from which equa lity the rules for performing the principal operations of fluxions are immediately deduced. That I may give the reader," continues he, " a more perfect idea ofthe nature of this method : suppose the ab scissa of a curve be divided into any num ber of equal parts, each part of which is called the increment of the abscissa, and imagine so many parallelograms to be erected thereon, either circumscribing the curvilineal figure, or inscribed in it; then the finding the sum of all these pa rallelograms is the business of the method of increments. But if the parts of the ab scissa be taken infinitely small, then these parallelograms degenerate into the curve; and then it is the business of the method of fluxions to find the sum of all, or the area of the curve. So that the method of increments finds the sum of any num ber of finite quantities ; and the method of fluxions the sum of any infinite num ber of infinitely small ones: and this is the essential difference between these two methods."
Again : " There is such a near relation between the method of fiuxions and that of increments, that many of the rules for the one, with little variation, serve also for the other. And here, as in the method of fluxions, some questions may be solved, and the integrals found, in finite terms ; whilst in others we are forced to have re course to infinite series for a solution. And the like difficulties will occur in the method of increments, as usually happen in fiuxions. For whilst some fluxionary quantities have no fluents but what are expressed by series, so some increments have no integrals but what infinite series afford; which will often, as in fluxions, diverge and become useless." By means of the method of increments, many curi ous and useful problems are easily re solved, which scarcely admit of a solu tion in any other way. As, suppose seve ral series of quantities be given, whose terms are all formed according to some certain law which is given; the method of increments will find out a general series, which comprehends all particular cases, and from which all of that kind may be found. The method of increments is also of great use in finding any term of a series proposed : for the law being given by which the terms are formea, by means of this general law the method of incre ments will help us to this term, either ex pressed in finite quantities, or by an in. finite series. Anotheruse of the method of increments is to find the sum of series, which it will often do in finite terms. And when the sum of a series cannot be had in finite terms, we must have recourse to infinite series ; for the integral being expressed by such a series, the sum of a competent number of its terms will give the sum of the series required. This is equivalent to transforming one series into another, converging quicker ; and some times a very few terms of this series will give the sum of the series sought. See Emerson's Increments.