HISTORY OF SCIENCE. 



considerations, as velocity or space traversed in a given time ; Leibnitz 

 resorted to the notion of infinitely small elements. 



Besides the class of problems which the application of the direct 

 differential calculus can solve, there is another class of problems to be 

 dealt with which require the inverse process. This inverse process, 

 which is now termed the Integral Calculus, consisted, according to 

 Leibnitz's view, in finding the sum of elementary parts accordingly to 

 the notion which has been already explained by the method of "/- 

 dinnsibles" (page 183). It does not enter into our design to explain 

 the methods of the infinitesimal calculus further than suffices to illus- 

 trate its objects. Let o p p', Fig. 84, be a curved line, o Y and o x 

 being the axes of the co-ordinates, and let us suppose that the area 

 of the space M p' p M'is required. Let the line M M' be divided into 

 a number of equal parts, i, 2, 3, etc., and lines be drawn parallel to 



o Y, cutting the curve in the points i' 2' 

 3', etc. Through the points P, i', 2', etc., 

 draw lines parallel to o x, as shown in 

 the figure. Now, if we add the areas of 

 all the rectangles of which the divisions 

 of M M' are the bases, and the heights 

 M P, i i', 2 2', etc., their sum will evi- 

 dently fall short of the true area of the 

 figure by the sum of the quasi triangular 

 spaces. The construction of the figure 

 will show that this deficiency must be 

 less than the rectangle P q m n, which 

 FlG - 84> is obviously equal to the sum of all 



the little rectangles of which the several 



triangle spaces are part. Note that if the space M M' were divided 

 into forty or four hundred, or any number of parts, the height of 

 the rectangle P m would be unchanged, but its width m n, and there- 

 fore its area, would diminish as M 3' became a smaller part of M M. 

 Consequently, by dividing M M' into a sufficiently great number of 

 parts, we may find a set of rectangles the sums of whose areas shall be 

 as near as we please to the area of the curved figure M M' p p'. The 

 equation to the curve is the general expression for each ordinate in 

 terms of its abscissa (page 151), and if the sign <p be used to indicate 

 the function in question, and dx the small increment of the abscissa 

 in passing from one rectangle to the next, we shall have for the sum 

 of the rectangles the following series, where a is the value of o M : 



etc.; or 

 dx $a<ba+dx$a+2dx+<t>adx etc. 



The number of terms in this series being equal to the number of parts 



