Conversely the phenomena of the spectroscope all depend on the fact that the vibrations of atoms and molecules can propagate waves through the ether, as well as absorb ether-waves into their own motions, and thus give spectra distinguished by bright or dark lines peculiar to each substance, by which it can be identified. Whatever ether may be, this much is certain about it: it pervades all space. That it extends to the boundaries of the infinitely great we know from the fact that light reaches us from the remotest stars and nebulæ, and that in this light the spectroscope enables us to detect waves propagated and absorbed by the very same vibrations of the same familiar atoms at these enormous distances as at the earth’s surface. Glowing hydrogen, for instance, is a principal ingredient of the sun’s atmosphere and of those distant suns we call stars, and it affects the ether and is affected by it exactly in the same manner as the hydrogen burning in an ordinary gas-lamp.

In the direction also of the infinitely small, ether permeates the apparently solid structure of crystals, whose molecules perform their limited and rigidly definite movements in an atmosphere of it, as is shown by the fact that in so many cases light and heat penetrate through them. A whole series of remarkable phenomena arise from the manner in which the vibrations of ether which cause light are affected by the structure of the molecules of crystals through which they pass. In certain cases they are what is called polarised, or so affected that while they pass freely if the crystal is held in one direction, they are stopped if it is turned round through an angle of 90° to its former position, so that one and the same crystal may be alternately transparent and non-transparent. It would seem as if its structure were like that of wood, grained, and more easy to penetrate if cut with the grain than against it, so that when a ray of light attempted to penetrate, its vibrations were resolved into two, one with the grain which got through, the other against it which was suppressed; so that the emerging ray, which entered with a circular vibration, got out with only one rectilinear vibration parallel to the diameter which coincided with the grain.

Other crystals of more complicated structure affect transmitted light in a more complex way, developing a double polarity very similar to that induced in the iron filings when brought under the influence of the two poles of the magnet. With this polarised light the most beautiful coloured rings can be produced from the waves of the different colours into which the white light has been analysed in passing through the crystal, which alternately flash out and disappear as the crystal is turned round its axis, and which present a remarkable analogy to the curves into which the iron filings form themselves under the single or double poles of the magnet.

The importance of this will appear afterwards, but for the present it is sufficient to show that the waves of ether which cause light really penetrate through the molecules of crystals, but in doing so may be affected by them.

Rings of Polarised Light, Uniaxial Crystals.Rings of Polarised Light, Biaxial Crystals.

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

The difference between this and xy is xdy + ydx + dx.dy. But dx.dy is, as we have seen, a differential of the second order and may be neglected. Therefore dxy = xdy + ydx. In like manner dx² = (x + dx)²-x² = 2xdx + dx², which last term may be neglected, and dx² = 2xdx. In this way the differentials of all manner of functions and equations of symbols representing dimensions and motions may be found. Conversely the wholes may be considered as made up of an infinite number of these infinitely small parts, and found from them by summing up or integrating the differentials. Thus if we had the equation