ether proved by light—light-waves—elasticity of ether—its universal diffusion—influences molecules and atoms—is influenced by them—successive orders of the infinitely small—illustrated by the differential and integral calculus—explanation of this calculus—theory of vortex rings.
perhaps the best way to convey some idea of this order of magnitudes to the ordinary reader is to quote sir w. thomson’s illustration, that if we could suppose a cubic inch of water magnified to the size of the earth—i.e. to a sphere 24,000 miles in circumference—the dimensions of its ultimate particles, magnified on the same scale, or, as he expresses it, its degree of coarse-grainedness, would be something between the size of rifle-bullets and cricket-balls.
extraordinary as these dimensions are, they are not more so than those at the opposite extremity of the scale, where the distance of stars and nebul? has to be measured by the number of thousand years their light, travelling at the rate of 192,000 miles per second, takes to reach us. infinitely small, however, as those dimensions appear to our original conceptions derived from our natural senses, they are certain and ascertained facts, if not as to the precise figures, yet beyond all doubt as to the orders of magnitude. in dealing with them also we are to a great extent on familiar ground.[22] molecules are nothing more nor less than small pieces of ordinary matter; and atoms are also matter, for they obey the law of gravity, have definite weights, and build up molecules as surely as molecules build up ordinary matter, and as squared stones build up pyramids.
but to understand the constitution of the material universe we must go a step further, part from the familiar world of sense, and deal with an all-pervading medium, which is at the same time matter and not matter, which lies outside the laws of gravity, and yet obeys other laws intelligible and calculable by us; of which it may be said we know it and we know it not. we call it ether.
ether is a medium assumed as a necessary consequence from the phenomena of light, heat, and electricity—primarily from those of light. respecting light two facts are known to us with absolute certainty.
1st. it traverses space at the rate of 192,000 miles per second.
2nd. it is propagated not by particles actually travelling at this rate, but, like sound through air, by the transmission of waves.
the first fact is known from the difference of time at which eclipses of jupiter’s satellites are seen according as the earth is at the point of its orbit nearest to or farthest from jupiter—i.e. from the time light takes to traverse the diameter of the earth’s orbit, which is about 180 millions of miles; and this velocity of light is confirmed by direct experiments, as by noting the difference of time between seeing the flash and hearing the sound of a gun, which gives the velocity of light compared with the known velocity of sound.
[23]
the second fact is equally certain from the phenomena of what are called interferences, when the crest of one wave just overtakes the hollow of a preceding one, so that, if the two waves are of equal magnitude, the oscillations exactly neutralise one another, and two lights produce darkness. this is shown in a thousand different ways, and for all the different colours depending on different waves into which white light is analysed when passed through a prism. it is a certain result of wave-motion, and of wave-motion only, and therefore we know without a doubt that light is propagated by waves.
but waves imply a medium through which waveforms are transmitted, for waves are nothing but the rhythmic motion of something which rises and falls, or oscillates symmetrically about a mean position of rest, slowly or quickly according to the less or greater elasticity of the medium. the waves which run along a large and slack wire are large and slow, those along a small and tightly stretched wire are small and quick; and from the data we possess as to light, its velocity of transmission, its refraction when its waves pass from one medium into another of different density, and from the distance between the waves as shown by interference, it is easy to calculate the lengths and vibratory periods of the waves, and the elasticity of the medium through which such waves are transmitted.
the figures at which we arrive are truly extraordinary. the dimensions and rates of oscillations of the waves which produce the different colours of visible light have been measured and calculated with the greatest accuracy, and they are as follows:
[24]
dimensions of light-waves.
colours no. of waves
in one inch no. of oscillations
in one second
red 39,000 477,000,000,000,000
orange 42,000 506,000,000,000,000
yellow 44,000 535,000,000,000,000
green 47,000 577,000,000,000,000
blue 51,000 622,000,000,000,000
indigo 54,000 658,000,000,000,000
violet 57,000 699,000,000,000,000
the elasticity of this wonderful medium is even more extraordinary.
the rapidity with which wave-motion is transmitted depends, other things being equal, on the elasticity of the medium, which is proportional to the square of the velocity with which a wave travels through it. as the velocity of the sound-wave in air is about 1,100 feet in a second, and that of the light-wave about 192,000 miles in the same time, it follows that the velocity of the latter is about a million times greater than that of the former, and if the density of ether were the same as that of air, its elasticity must be about a million million times greater. but the elasticity is the same thing as the power of resisting compression, which in the case of air we know to be about 15 pounds to the square inch; so that the ether, if equally dense, would balance a pressure of 15 million million pounds to the square inch—that is, it would require a pressure of about 750 millions of tons to the square inch to condense ether to the density of air. on the other hand, its density, if any, must be so infinitesimally small that the earth moving through it in its orbit with a velocity of 1,100 miles a minute suffers no perceptible retardation.
consider what this means. air blowing at the rate[25] of 100 miles an hour is a hurricane uprooting trees and levelling houses. if ether were as dense as air the resistance to the earth in passing through it would be 600 times that of going dead to windward in a tropical hurricane. but in point of fact there is no sensible resistance, for the earth and heavenly bodies move in their calculated paths according to the law of gravity exactly as they would do if they were moving in a vacuum. even the comets, which consist of such excessively rare matter that when one of them got entangled among the satellites of jupiter it did not affect their movements, are not retarded by the ether, or so slightly, that any retardation in the case of one or two of them is suspected rather than proved. but, if the ether has no weight, how can we call it material, weight being, as we have seen, the invariable test and measure of all matter down to the minutest atom? and yet how can we deny its existence when it is demonstrably necessary to account for undoubted facts revealed to us every day by the prism, the spectroscope, electricity, and chemical action, and deductions from these facts based on the strict laws of mathematical calculation? for the existence of the ether is not based only on the phenomena of light: it is an equally necessary postulate to explain those of heat, electricity, and chemical action. we must conceive of our atoms and molecules as forming systems and performing their movements, not in vacuo, but in an all-pervading medium of this ether, to which they impart, and from which they receive, impulses.
these impulses are excessively minute, and when they occur in irregular order they produce no appreciable effect; but when the vibrations of the ether keep[26] time with those of the atoms, the multitude of small effects becomes summed up into one considerable enough to produce great changes. just so a rhythmic succession of tiny ripples may set a heavy buoy oscillating, and the footfalls of a regiment of soldiers marching over a suspension-bridge may make it swing until it breaks down, while a confused mob could traverse it in safety. the latter affords a good illustration of the way in which molecular structures may be broken down, and their atoms set free to enter into other combinations, by the action of heat, light, or chemical rays beyond the visible end of the spectrum.
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 nebul?, 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[27] 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[28] 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[29] 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 d?. 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[30] 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
xdy + ydx = 2zdz
we know that the left-hand side is the differential of xy, and therefore that by integrating it we shall get xy; while the right side is the differential of z2 which we shall get by integrating it. the relation expressed therefore is that xy = z2, or, in other words, that a rectangle whose sides are x and y exactly equals a square whose side is z.
fig. 1. fig. 2. fig. 3.
the use of this device in assisting calculation will be apparent if we take the case of an area bounded by a curved line. we cannot directly calculate this area, but we can easily tell that of a rectangle. now it is evident that if we inscribe rectangles in this area abc, the more rectangles we inscribe the less will be the error in taking their sum as equal to the curved area. this is apparent if we compare fig. 2 with fig. 3. suppose we take a point p on the curve, call bn = x and pn = y, and suppose nn to be dx, the differentially small increment of x, and pq = dy the corresponding small increment of y. the area of the rectangle pqnn = pn ×[31] nn = ydx, and differs from the true curvilinear area ppnn by less than the little rectangle of pq × pq or of dx.dy. but, as we have seen, if we push our division to the first infinitesimal order, or make nn and pq differentials of x and y, dx.dy may be neglected—i.e. multiply the number of rectangles indefinitely, and the sum of their areas will differ from the true area inclosed by the curve by an error which is evanescent.
if then x and y are connected by some fixed law, as must be the case if the extremity of y traces out some regular curve, the relation between them may be expressed by an equation, which will remain one however often it may be differentiated or again integrated, and whatever modifications or transformations it may receive by mathematical processes which do not alter the essential equality of the two sides connected by the symbol of equality =. thus by differentiating and casting off as evanescent all differentials of a lower order than that which we are working with, we may arrive at forms of which we know the integrals, and by integrating get back to the results in ordinary numbers, which we were in search of but could not attain directly.
the same thing will apply if our symbols are more numerous, and if they express relations of motion as well as of space, or, in fact, any relations which are governed by fixed laws expressible by equations. if i have succeeded in conveying to the readers any idea of this celebrated calculus, they will perceive what an analogy it presents to the idea of modern physical and chemical science, that of molecules, atoms, and ether, forming differentials of successive orders of the infinitely small. it is certainly most remarkable that while the former was a purely intellectual idea based[32] on mathematical abstractions, and which was invented and worked as an instrument for solving the most intricate astronomical problems for nearly two centuries, without a suspicion that it represented any objective reality: the latter idea, based on actual experiment, seems to show that differentials and integrals have their real counterpart in nature and represent fundamental facts in the constitution of the universe.
those who are of a mystic or metaphysical turn of mind may discern in this, arguments for matter and laws of matter being after all only manifestations of one universal, all-pervading mind; but in following such speculations we should be deserting the solid earth for cloudland, and passing the limit of positive knowledge into the region where reflections of our own hopes, fears, religious feelings, and poetical sentiments form and dissolve themselves against the background of the great unknown. for the present, therefore, i confine myself to pointing out how these undoubted truths of mathematical science, which have verified themselves in the practical form of enabling us to predict eclipses and construct nautical almanacs, correspond with and throw light upon the equally certain facts of this succession of infinitely small quantities of successive orders in the constitution of matter.
an attempt has recently been made, based on abstruse mathematical calculations, to carry our knowledge of the constitution of matter one step further back, and identify atoms with ether. this is attempted by the vortex theory of helmholz, sir w. thomson, and professor tait. it is singular how some of the ultimate facts discovered by the refinements of science correspond with some of the most trivial amusements. thus[33] the blowing of soap-bubbles gives the best clue to the movement of waves of light, and through them to the dimensions of molecules and atoms; and the collision of billiard-balls, knocked about at random, to the movements of those minute bodies, and the kinetic theory of gases. in the case of the vortex theory the idea is given by the rings of smoke which certain adroit smokers amuse themselves by puffing into the air. these rings float for a considerable time, retaining their circular form, and showing their elasticity by oscillating about it and returning to it if their form is altered, and by rebounding and vibrating energetically, just as two solid elastic bodies would do, if two rings come into collision. if we try to cut them in two, they recede before the knife, or bend round it, returning, when the external force is removed, to their original form without the loss of a single particle, and preserving their own individuality through every change of form and of velocity. this persistence of form they owe to the fact that their particles are revolving in small circles at right angles to the axis or circumference of the larger circle which forms the ring; motion thus giving them stability, very much as in the familiar instance of the bicycle. they burst at last because they are formed and rotate in the air, which is a resisting medium; but mathematical calculation shows that in a perfect fluid free from all friction these vortex rings would be indivisible and indestructible: in other words, they would be atoms.
the vortex theory assumes, therefore, that the universe consists of one uniform primary substance, a fluid which fills all space, and that what we call matter consists of portions of this fluid which have become[34] animated with vortex motion. the innumerable atoms which form molecules, and through molecules all the diversified forms of matter of the material universe, are therefore simply so many vortex rings, each perfectly limited, distinct, and indestructible, both as to its form, mass, and mode of motion. they cannot change or disappear, nor can they be formed spontaneously. those of the same kind are constituted after the same fashion, and therefore are endowed with the same properties.
the theory is a plausible one, and the reputation of its authors must command for it respectful consideration; but it is as yet a long way from being an established theory which can be accepted as a true representation of facts. in the first place it is based solely on mathematical theory, and not, as in the case of atoms and light-waves, upon actual facts of weight and measurement tested by experiment, and to which mathematical reasoning affords only an aid and supplement. no one has proved the existence of such a medium or of such vortex rings, much less weighed or measured them.
moreover the theory is open to some very obvious objections. how can aggregations of imponderable matter acquire weight, and become subject to the law of gravity, which, as we have seen, is one of the essential and permanent qualities of atoms? if a cubic millionth of a millimetre of ether formed into a big vortex ring of, say, an atom of mercury, has a weight equal to 200 times that of an atom of hydrogen, which itself has a definite weight, why has it no weight in its original form? and if it had weight, however small, how could the enormous mass of ether filling all space produce no perceptible effect on bodies, even of attenuated cometic[35] vapour, revolving through it with immense velocities? again, how could these innumerable vortex rings be formed out of the ether without disturbing the uniformity and continuity of the medium, which are essential for the propagation of the light-waves through it? and how could the motions requisite to form the vortex rings be impressed on them de novo consistently with the principle of the conservation of energy? energy can no more be created out of nothing than matter, by any process known in nature or conceivable by the human intellect; and to assume it is simply a more refined manner of falling back on the supernatural, which is itself only a more refined manner of saying that we know nothing.
for the present, therefore, we must be content with atoms and ether as the ultimate terms of our knowledge of the material or quasi-material components of the universe.
