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THE INFINITESIMAL CALCULUS 67 differential of a;," BO that "d" does not denote a number but "dx" altogether stands for an " infinitesimal p> ) respectively, and called the collection of rides for working with his signs the "differential calculus." But before the notation of the differential calculus and the rules of it were discovered by Gottfried Wilhelm von Leibniz (1646-1716), the celebrated German philosopher, statesman, and mathematician, he had invented the notation and found some of the rules of the " integral calculus " : thus, he had used the now well-known sign "f" or long "s" as short for "the sum of," when considering the sum of an infinity of infinitesimal elements as we do in the method of indivisibles. Suppose that we propose to determine the area included between a certain curve y=f(x), the #-axis, and two fixed ordinates whose equations are x = a and x = 6 ; then, if we make use of the idea and notation of differentials, we notice that the area in question can be written as "fy.dx," the summation extending from x a to x = 6. We will not here further concern ourselves about these boundaries. Notice that in the above expression we have put a dot between the " y " and the " dx " : this is to indicate that y is to multiply dx. Hitherto we have used juxtaposition to denote multiplica- tion, but here d is written close to x with another end in view ; and it is desirable to emphasize the difference between " d " used in the sense of an adjective and " d " used in the sense of a multiplying number, at least until the student can easily tell the difference by the context. If, then, we imagine the abscissa divided into equal infinitesimal parts, each of length dx, corresponding to the constituents called " points " in the method of indivisibles, y . dx is the area of the little rectangle of sides dx and y which stand at the end of the abscissa x. If, now, instead of extending to x = 6, the summation extends to the ordinate at the indeterminate or " variable " point x, y .dx becomes a func- tion of x. Now, if we think what must be the differential of this sum, that the infinitesimal increment that it gets when the abscissa of length x, which is one of the boundaries, is increased by dx,
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