In dealing with these excessively small magnitudes it may assist the reader who has some acquaintance with mathematics in forming some conception of them, to refer to that refinement of calculation, the differential and integral calculus. And even the non-mathematical reader may find it worth while to give a little attention in order to gain some idea of this celebrated calculus which was the key by which Newton and his successors unlocked the mysteries of the heavens. The first rough idea of it is gained by considering what would happen if, in a calculation involving hundreds of miles, we neglected inches. Suppose we had a block of land to measure, 300 miles long and 200 wide; as there are, say, 5,000 feet in a mile, and the error from omitting inches could not exceed a foot, the utmost error in the measurement of length could not exceed 1/1500000th, and in width 1/1000000th part of the correct amount. In the area of 300 × 200 = 60,000 square miles, the limit of error would, by adding or omitting the rectangle formed by multiplying together these two small errors, not exceed 1/1500000 × 1/1000000 = 1/1500000000000th part. It is evident that the first error is an excessively small part of the true figure, and the second error a still more excessively small part of the first error. But, as we are dealing with abstract numbers, we can just as readily conceive our initial error to be the 1/100th or 1/1000th of an inch, as one inch; and, in fact, diminish it until it becomes an infinitesimally small or evanescent quantity. In doing so, however, it is evident that we shall make the second error such a still more infinitesimally small fraction of the first that it may be considered as altogether disappearing. The first error is called a differential of the first order and denoted by _d_, the second a differential of the second order denoted by d₂. Thus if we call the base of our rectangle _x_ and its height _y_, the area will be _xy_. Let us suppose _x_ to receive the addition of a very small increment _dx_, and _y_ the corresponding increment _dy_, what will be the corresponding increment of the area, or _d.xy_? Clearly the difference between the old area _xy_ and the new area (_x_ + _dx_) multiplied by (_y_ + _dy_). This multiplication gives _x_ + _dx_ _y_ + _dy_ ------------ _xy_ + _ydx_ _xdy_ + dx.dy ------------------------------ _xy_ + _xdy_ + _ydx_ + _dx.dy_