SEMANTICS OF THE DIFFERENTIAL CALCULUS 583
the ratio of the increment of the function to the increment of the independent variable approaches 6x2 1, true for any value we may arbitrarily assign to x.
It should be noticed that in our function the left-hand side represents the 'whole' as composed of interrelated elements which are represented by the right-hand side. When instead of x we selected a slightly larger value; namely, x+Ax, we performed upon this altered value all the operations indicated by our expression. We thus have in mathematics, because of the self-imposed limitations, the first and only example of complete analysis, impossible in physical problems as in these there are always characteristics left out.
An important structural and methodological issue should also be emphasized. In the calculus we introduce a 'small increment' of the variable; we performed upon it certain indicated operations, and in the final results this arbitrary increment disappeared leaving important information as to the rate of change of our function. This device is structurally extremely useful and can be generalized and applied to language with similar results.
It has been noticed already that the calculus can be developed without any reference to graphs, co-ordinates or any appeal to geometrical notions; but as geometry is an all-important link between pure analysis and the outside world of physics, we find in geometry also the psycho-logical link between the higher and lower orders of abstraction. But the appeal to geometrical notions helps intuition and so is extremely useful. For this reason we will explain briefly a system of co-ordinates and show what geometrical significance the derivative has
We take in a plane two straight lines X'X and Y'Y, intersecting at 0 at right angles, so that X'OX is horizontal extending to the left and right of 0, and YOY', is vertical, extending above and below 0, as a frame of reference for the locations of point, lines, and other geometrical figures in the plane. We call this a two-dimensional rectangular system of co-ordinates. This method may be extended to three dimensions, and our points, lines, and other geometrical figures referred to a three-dimensional rectangular system of co-ordinates consisting of three mutually perpendicular and intersecting planes.
As we see in Fig. 2, we have four quadrants I, II, III, IV, formed by the intersecting axes X'X and Y'Y. The co-ordinates of a point P, by which we