CALCULUS OF FUNCTIONS.
Sir Isaac Newton, the inventor of the method of fluxions, made its principles depend on the properties of motion, (see FLUXIONS, Art. 20-23); and Leibnitz founded its equivalent, the differential calculus, on the nature of quan tities, which might be regarded as infinitely small in re spect of others. At first, mathematicians were more eager to explore the rich mine which these philosophers had opened, than to call in question the principles which had led to its discovery. But when these came to be critically examined, it was observed, that as motion was an idea fo reign to pure analysis, it could not legitimately be made the foundation of one of its most important theories. Also, that the notion of a quantity infinitely little, was too vague to form the basis of a branch of the most precise of all the sciences. Hence it was thought desirable, that the calcu lus should have an origin purely analytical, and should de pena entirely on the properties of finite quantities.
To accomplish this reform, the late M. Lagrange at tempted to model anew the principles of the calculus. He gave his ideas in the Berlin Memoirs for 1772, also in his Theorie des Fonctions ?nalytiques, (1797,) which, he says, i‘ contains the principles of the differential calculus, disen gaged from all considerations of infinitely small or vanish ing quantities, or of limits or fluxions ;" and again in his Lefons sur le Calcul des Fonctions.
In the calculus of functions, the variable quantities are denoted by the last letters of the alphabet 7., y, &c. and the constant quantities by the first letters a, 6, &c. A func tion of a single quantity, is expressed by placing the cha racteristic letter f or F before it. Thus fx, or F x, means any function of x. To denote a function of a quantity, that is itself composed of a variable quantity x, for example or s-i-c &c, the compound quantity is included in a parenthesis, thus or A functu.n or two independent variable quantities x and y is cxpre sed thus f(x, y); and so of others.
If two functions of two variable quantities x and y are composed exactly in the same manner, and with the same constant quantities, for example a +c, and a yz+b y+c, these are like functions, and may be expressed in the same calculation thus,/ x and f y ; but if the constant quan tities are not the same in both, they cannot be represented by the same characteristic in the same calculation. How
ever, if the constant quantities enter alike into both func tions, and only differ in their absolute values, as in a and b , these in the same calculation may be denoted by f(x, a) andf(y, 6.) The general notation we have used in FLUX IONS, art. 18, 23, 28, 45, &c. and in art. 193, Prob. 4. is almost the very same as that of Lagrange.
The theory of functions depends on the change which takes place in the value of a function, when its variable quantity is increased by some indefinite increment, and on the form of the developement of its new value. In the func when x is augmented by the quantity i, then fx becomes f + 2 x and in the function when x becomes x+i, then fx becomes f (x+i)=(x +3 i+3 x and again, in the function/ x= a when x becomes x+i,f x becomes f (x a a a a . a ---= — — i • By an ex X X amination of any number of particular cases, it will appear that they have a common property, which consists in the developcment off (x+i) the new value of the function hav ing always the form f x+i p+il r+ &c. an expression in which the first term is f x, the original function, and the remaining terms are the successive positive inte ger powers of i, the increment, multiplied by a series of quantities p, y, r, Ste. functions of x, which are entirely in dependent of i, and which have a determinate form, that depends upon the nature of the original function. The truth of this analytic theorem, first particularly noticed by Euler, may be inferred from induction : As however it must result from the principles of analysis, Lagrange has endea voured to demonstrate, that if the function f (x+i) be de veloped into a series of the form f x r+ Sce.