Comparative Review of some Dynamical Theories of Gravitation

Introduction.
THE modes of accounting for natural phenomena have been very different at different times. The old philosophers had in general scarcely an idea of that which we now call a mechanical explanation; they figured to themselves rather the agencies working in nature as living beings. This applies also to Kepler, who banished from himself any idea of a mechanical explanation of the laws discovered by him. On the basis of the researches of Galileo, Newton was the founder of the Mechanics of to-day; and on his principles the edifice of the action-at-a-distance theory has been founded. Until Newton's time the notion of a direct action at a distance was completely unknown: on the contrary, many experiments exist by the Greek philosophers to account for the seeming action at a distance by the intervention of a medium; therefore Demokritos sought to explain natural phenomena by the motions of very fine bodies. First Boscovich, Mosotti, Wilhelm Weber, and many others developed the aspect of nature on the basis laid down by Newton, in accordance with which the universe consists of a number (if even very great) [146] of material points, which, without anything intervening, act on each other directly at a distance, according to a mathematically exact formulated law. If the initial positions and velocities of all the atoms are given, then their motions can be calculated for any periods of time from the equations formulated by Newton, and so a clearly defined mathematical problem is presented.

It is, however, well to observe that Newton did not believe in such an action at a distance without the intervention of something, as appears from his third letter to Bentley, where he says: —

"That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it" (Newton's third letter to Bentley, February 25, 1692-3).

In the same sense speak many subsequent important scientists. For instance Count Rumford remarks: —

"Nobody surely in his sober senses has ever pretended to understand the mechanism of gravitation, and yet what sublime discoveries has our immortal Newton been enabled to make, merely by the investigation of the laws of its action" ("An Inquiry concerning the Source of the Heat which is excited by Friction" by Count Rumford, Phil. Trans. 1798).

These last scientists are therefore not satisfied with the Boscovich-Mosotti explanation of natural phenomena; they demand rather an explanation (by the intervention of a medium) of the seeming action at a distance. To give such an explanation was never seriously attempted by Newton: the first attempt of that kind is to be found in the mechanical gravitation theory of Le Sage, born at Geneva in 1724. This theory is contained in a memoir published in the Transactions of the Royal Berlin Academy for 1782, under the title Lucrece newtonien. There is also a book, Deux Traités de Physique mécanique, edited by Pierre Prevost, Paris, 1818, which contains a full description of Le Sage's theory.

Le Sage lays emphasis on the probability of the existence of a mechanism of gravitation, and devoted his life to the development of his idea. The introductory paragraph of his memoir (entitled Lucréce Newtonien) is as follows, translated from the French original, viz.: —

"I propose to show that if the first Epicureans had had as healthy ideas of Cosmography as several of their contemporaries [147] (to whom they would not listen), and only a part of the knowledge of Geometry which was then prevalent, they would in all probability have discovered the laws of universal gravitation and its mechanical cause. Laws, the discovery and the demonstration of which constitute the fame of the most powerful genius that has ever existed; and Cause, which after having been the ambition for a long time of the greatest scientists, is at present the despair of their successors. 80 that, for example, the celebrated laws of Kepler, discovered somewhat less than 200 years ago, partly by gratuitous conjectures, and partly by repeated trial and error, would have been no more than inevitable corollaries which could have been arrived at by these ancient philosophers by investigating the mechanism of nature. The same conclusion applies also to the laws of Galileo upon the fall of bodies, the discovery of which took place still later, and which have been more contested, because the experiments upon which this discovery was based permitted a latitude in their results ( necessarily rough), which would make them fit equally well with other laws, so that one did not fail to contest them: whereas the inferred consequences of the shock of atoms would have been unmistakably in favour of the only true principle, viz., equal accelerations in equal times." (Trans. of Royal Berlin Academy, 1782.)

On this paragraph the following opinion is emitted by Lord Kelvin, viz.: —

"If Le Sage had but excepted Kepler's third law, it must be admitted that his case, as stated above, would have been thoroughly established by the arguments of his "memoire"; for the Epicurean assumption of parallelism adopted to suit the false idea of the earth being flat, prevented the discovery of the law of the inverse square of the distance, which the mathematicians of that day were quite competent to make, if the hypothesis of atoms moving in all directions through space, and rarely coming into collision with one another, had been set before them, with the problem of determining the force with which the impacts would press together two spherical bodies, such as the earth and moon were held to be by some of the contemporary philosophers to whom the Epicureans "would not listen." But nothing less than direct observation, proving Kepler's third law — Galileo's experiment on bodies falling from the tower of Pisa, Boyle's guinea-and-feather experiment, and Newton's experiment of the vibrations of pendulums composed of different kinds of substance — could either give the idea that gravity is proportional to mass, or prove that it is so to a high degree of [148] accuracy for large bodies and small bodies, and for bodies of different kinds of substance "(Phil Mag. May 1873, p. 323).

Le Sage's Theory.
Le Sage based his theory on perfectly arbitrary assumptions. He assumed (Deux Traits de Physique mécanique, Paris 1818, edition Pierre Prevost): —

1) That a number of streams of atoms, equally distributed in space, exist; of which each stream moves continually in one and the same direction.

2) The length of these streams (at the centre of which the universe known to us is placed) is finite, but very great; therefore gravitation must have a correspondingly limited period for existence.

3) That the streams must be everywhere equally dense.

4) That the mean velocity of the streams is everywhere the same.

The conditions above set forth depend manifestly on perfectly arbitrary assumptions, and it is not easy to see by what mechanism such streams should either originate or be kept up. As regards the behaviour of these streams of atoms towards gross matter, Le Sage assumes the following. Gross matter is chiefly freely penetrated by the streams of atoms, only a small part of their energy is absorbed by gross matter, which implies a continuous annihilation of energy. Whence it arises that every portion of gross matter opposes a certain shelter to every other neighbouring portion from the encounters of the streams of atoms; and from this the apparent attraction of the gross matter according to the Newtonian law of gravity is easily explained.

Lord Kelvin presupposes exactly the same streams of atoms as Le Sage; the mechanism which regulates or maintains these atom-streams therefore remains with him as obscure as with Le Sage. An important progress in Lord Kelvin's case consists, however, in the fact that he regards the atoms as elastic. In order to explain the elasticity, he proposes to regard the atoms as vortex rings in a perfect liquid. The elasticity of these is then explained by the laws which Helmholtz found to apply to the motions of such vortex rings.

The aether atoms then rebound from gross matter in accordance with the laws of elastic collision: instead of the absorption (annihilation) of energy assumed by Le Sage, Lord Kelvin supposes that the aether atoms, in addition to their translatory energy, also possess an energy of internal motion, just as Clausius assumes for the molecules of ordinary gases. [149]

On account of the relatively very large dimensions and superior elastic rigidity of the gross molecule, it is scarcely disturbed by the collision of the very minute atom. On the other hand, the minute atom is thrown into strong vibration and rotation by the blow. This vibration or rotation ("internal motion") cannot evidently be generated out of nothing. The small atom therefore loses at impact a portion of its translatory motion, by converting the same into internal motion (vibration and rotation). The diminution of the translatory motion of the small gravity-atoms at their encounter with gross molecules is therefore rather to be looked upon as a necessary deduction than as an hypothesis. One might, indeed, easily illustrate this fact experimentally.

If an elementary example be excused, we can consider the - case when any small elastic body such as a small polished steel key-ring is thrown against the surface of a polished steel anvil. A key-ring and an anvil (of the same metal) may be equally elastic, but on account of the considerable difference in their dimensions — therefore pliability — only the small ring will be thrown into perceptible vibration by the encounter ( or into rotation, for the anvil cannot rotate on account of its mass). The ring rebounds with a diminution of its translatory motion, by converting the same into vibration and rotation.

The atom gains its full translatory motion gradually again by collisions against atoms of its own kind, — from the fact that the proportionality existing between the amount of translatory motion and the amount of internal motion of the atom continually strives to maintain itself constant; which is a known consequence of the kinetic theory of gases, demonstrated by Clausius.

So is explained how the aether atoms, in being sifted through gross matter, on the average lose a certain velocity of translatory motion, and that therefore a portion of gross matter "shelters" any other neighbouring portion from the impacts of the aether atoms.

The penetration of the two masses by the flying aether atoms brings about the fact that on the adjacent sides of the two masses the pressure of the medium is smaller than on the remote sides of the molecular surfaces of the two masses. The remote sides encounter the full or undiminished translatory velocity of the atoms. Therefore the two masses are naturally driven together, and with a force which obviously, from the nature of the case, must be proportional to the square of the distance of the masses. The further explanation of the gravitation effect is then exactly as by Le Sage's theory.

The present writer attempted in some papers, of which the [150] first appeared in the Philosophical Magazine, Sept. 1877, to replace the arbitrarily assumed atom streams of Le Sage and Lord Kelvin by a motion which is exactly analogous to that which belongs to the kinetic theory of gases.

In that way the most obscure assumption of Le Sage's theory finds an unforced explanation — namely, how the symmetrical motion of the atoms under the continual changes of their direction produced by their collisions against gross matter, is kept up.

Now it has already been mathematically demonstrated in the case of ordinary gases, that an automatic correction goes on in a system of bodies or particles in free collision, and such a one that the particles are forced to move equally in all directions: and this is the absolutely necessary condition for equal pressure in all directions. The rate of establishment of this automatic correction, which is chiefly brought about by the oblique encounters, has, in fact, been calculated mathematically by Prof. Ludwig Boltzmann for ordinary gases. This adjustment (or correction) is in fact of such a stable character, that if the motion of the gas particles were artificially disturbed, the particles would of themselves equalize the motion again, so that an equal number of particles are moving in any two opposite directions. The motion can also be described so, that if we think of any small point situated anywhere in space, the atoms are at every instant flying towards and from this point, exactly as if it were a luminous point.

Hence it follows that when a system of atoms is left to itself, it will, by the principles of dynamics, automatically adjust the character of its motion in such a way that this motion is adapted to produce the gravitation effects. The motion of streams of atoms equally at all angles, which Le Sage gave forth as an arbitrary postulate, is attainable in a gas without any postulate. Instead of streams, each of which for itself maintains a constant direction of motion, and which cease to flow after a long epoch of time, we have a permanent motion of atoms correcting itself in a self-acting manner; and which fulfils the wished-for object.

So, therefore, we have succeeded, by starting from a very simple and thoroughly natural foundation, in establishing ail those conditions which Le Sage needs for his theory.

Nevertheless there are certain assumptions concerning quantitative relations to be added. In the first place, the mean length of path of an aether atom must be assumed to be exceedingly great. If, namely, the same were small in proportion to the distance between two influencing masses, [151] then in the intervening space between these masses, by the collisions of the aether atoms among themselves, the normal proportionality between translatory and internal motion of the atoms would be nearly restored (by encounters), and therefore the mutual shelter of the two masses would nearly be nullified.

The range of gravitation (its sphere of action) is therefore conditioned by the mean length of path of the atoms, and this may be regarded as an interesting deduction from the theory. Accordingly, on the assumption that the mean distances of the stars (excepting, of course, the relatively approximated double stars) are large in proportion to the mean length of path of the atoms, the inference would follow that the stars do not gravitate towards each other — and apparently in that way the universe would rather gain than lose in stability. One sees then that the mean length of path of the aether atoms must be great in comparison with those distances across which Newton's law has been demonstrated to apply with exactness.

In an article in the Encyc. Brit. 1875 (or Scientific Papers, vol. ii. p. 476) Maxwell raises the objection that by the atomic encounters gross matter would be raised to a white heat; he grounds this inference on the theorem that for thermal equilibrium between atoms or molecules the mean energy of translatory motion must be equal. Now the pressure (to take some symbol) is equal to the product of the mean energy of translatory motion L of an atom into the number N of atoms contained in the unit of volume. If, therefore, the mean energy of translatory motion of an aether atom be equal to that of a molecule of gross matter, which we can calculate in the case of ordinary gases, then N for the aether must have an enormous value, in order to be able to account for the gravitation pressure. Now Maxwell says: we are tolerably certain that N for the aether is small compared with the value of N for gross matter. From this he concludes that in order to explain the gravitation pressure, it is necessary to assume L enormously great. And according to the theorem that for thermal equilibrium L must be the same for all atoms (or molecules), it would follow that L also for the molecules of gross matter must finally assume a value which is much greater than that which we find in the case of gases. In other words, that all gross matter must be raised to a white heat by the collisions of the aether atoms. But, independently of the fact that the above-named theorem, relating to thermal equilibrium, for molecules or atoms of very different size is still contested, it seems to me that no cogent reason exists for the assumption that N is smaller for the aether than for gross matter. One can, in fact, imagine the aether atoms as small as one pleases; [152] then an enormous number of them can exist in the unit of volume combined with an enormously great length of path.

In general, in putting forward a theory of this kind, we should see no improbability in the assumption of either a very great or a very small number. Our objection to uncommonly great or uncommonly small numbers rests in fact upon custom, and regularly disappears as soon as the theory in question has further introduced itself.

There exists in space field enough, when necessary, for finer material, as our conceptions are not limited in the direction of smallness, and the smaller the particles, the quicker is their natural speed of motion, and the more intense the enclosed store of concealed energy: also the whole arrangement becomes all the less appreciable by our senses. The effects — called gravitational effects — on the other hand, do not escape detection by our senses; and reasoning from these effects, we trace and infer the invisible causes which lie at the basis of these effects.

Evidently there exists just as little an obstacle in space to smallness of size as to any given velocity of motion, and there are reasons for supposing that gravity must propagate itself with great velocity. Precisely because the normal velocity of the atoms is great, the material concerned in producing gravity can be very limited in quantity, and notwithstanding that exert a very considerable pressure. The atoms are therefore to be assumed very small, almost points, the condition adapted for a great length of path. The analogy of this gravitation mechanism (at least in principle) with the generally assumed structure of our atmosphere, may be regarded as a recommendation to the theory.

A further objection of Maxwell's, that according to this theory the action of gravity could only be kept up by an enormous expenditure of external work little short of ruinous, applies in fact to the theory of Le Sage in the form presented by Lord Kelvin; also to the theories of Isenkrahe and Bock considered further on; not, however, to the theory set forth by the present writer, because, according to this latter theory, the maintenance of the motion of the aether atoms demands just as little an expenditure of energy as the maintenance of the motion of the molecules of a gas in the ordinary gas theory. Moreover, the "shelter" of one mass by another is explained without any absorption of energy.

The large store of energy contained in the aether atoms is moreover of use for the explanation of the most varied natural phenomena; and it may be observed that the intervention of [153] a medium is wanted in other respects, for instance for the elucidation of magnetic and electric phenomena. It may be mentioned, further, that the explanation of gravitation carries with it the great advantage of rendering superfluous the idea of the existence of two (inherently different) kinds of matter, "ponderable" and "imponderable." The smaller atoms in space do not gravitate, only because the mechanism of gravitation cannot itself be subject to the conditions for producing gravitation. So, therefore, disappear the almost contradictory properties, "ponderable" and "imponderable," which have been arbitrarily attributed to matter: and we have therefore no reason for believing that the atoms diffused in space differ essentially from gross molecules, excepting in their dimensions. To the abandonment of the idea of two inherently different kinds of matter, the abandonment of two supposed different kinds of energy is analogous — viz., energy with motion, and energy without motion. Accordingly there would remain only one kind of energy, namely, that which a moving body possesses.

Another important quantitative relation is so conditioned that the "shelter" is evidently proportional to the surface exposed to the moving atoms; the gravitational effect, on the other hand, is proportional to the mass, as experiment shows. This result can only be achieved by supposing gross matter to possess a very porous structure. In that way, the gross molecules inside a body are reached or affected by the penetrating aether atoms almost with the same facility as the external molecules of the body. If we assume that the quantity of material contained in the substance of any molecule is very small compared with the vacant space contained in that same molecule, and if one does not suppose any superfluous material in the structure of the molecule; the proportionality existing between gravitation and mass cm be satisfied as closely as observation requires.

Some Remarks on Crystal Structure.
Even Le Sage recognized that for the elucidation of the gravitational effects the assumption of a porous or open structure in matter is necessary. Lord Kelvin draws a curious inference from this. In the Philosophical Magazine, May 1873, postscript p. 331, Lord Kelvin supposes that it might be probable that bi-axial crystals would not be penetrable with equal facility in all directions by the aether atoms. If that were so, such crystals would possess a (even if very small) difference of weight, according as the one or the other axis is [154] vertical. Have, however, sufficiently delicate experiments been made on this point?

A contribution published by me in the Philosophical Magazine for April 1880 on crystalline structure might be mentioned here.

I have tried to define further this open structure, so that it- appears to be well adapted for the explanation of cohesion, adhesion, and chemical affinity.

One knows how the cells of bees are formed by pressure, and how by pressure elastic spheres may be converted into angular, such as hexagonal-shaped, bodies.

As remarked, the gravitation theory (and many independent facts) demand that the molecules of bodies shall possess an open structure; which also satisfies the conditions of lightness and economy of material. As crystals exist, it is sometimes supposed that the molecules of bodies (whose open structure is often illustrated by cubes and other figures formed of wire) themselves represent the shapes of the crystals. We do not, however, need to assume that the molecules possess exactly such shapes, because if the separate molecules themselves possessed even a rounded structure, they must be pressed into angular forms as soon as two or more of them were pressed together by impacts of the aether atoms. Let us take for illustration the simplest open structure, viz. rings; although it is not thereby implied that this is the sole ground- form of the molecules. Elastic molecules of any very open structure of three dimensions would probably give a greater stability to the crystal mass formed out of them.

Simple elastic rings can then by pressure of their boundaries against each other (as caused by the flying of very minute Fig. 1. aether atoms through the structure) conceivably be changed into hexagonal, square figures, &c. Fig. 1 may serve to [155] show such a pressure-effect produced by atomic motion, where elastic rings are converted into the hexagonal forms of crystals. Cohesion (as now is generally supposed) is only gravitation by contact. From the above considerations, it becomes easy to understand that elastic crystal forms can sometimes change into non-crystalline forms; so some crystallizable metals, such as iron and zinc, lose all crystalline structure by rolling and hammering, and become ductile. Crystalline sulphur can, by mere warming, pass over into a sort of indiarubber sulphur. It is evident in fact that the irregular arrangement of the elastic molecules of a substance favours the gliding of the molecules over each other; while, on the other hand, the regularly arranged molecules which are in contact at their boundaries (corresponding to crystalline structure) cannot be displaced at all without separating entirely: take for instance crystallized cast-iron and some other metals.

The freedom allowed to the molecules to arrange themselves in the case of solutions may be favourable for the production of crystalline structure, while rolling, hammering, &c., manifestly forces the molecules to aggregate in an irregular manner.

When elastic molecules of very open structure cross each other irregularly in all directions, or are arranged in parallel layers (as produced by the rolling of a metal), then it becomes obvious that a subsequent displacement of the molecules, as by a tensile-stress for example, does not necessarily produce actual severance; but the atomic streams — from the very nature of this cause — can easily produce contact in fresh places, and so a gliding of molecules over each other is possible, without separation. So a bar of malleable iron gradually lengthens itself under a tensile-stress. On the other hand, because crystalline structure prevents all gliding of the molecules, it becomes impossible in this case for the atomic streams to cause contact in fresh places. The attempt to stretch a cast-iron bar, then, means separation of the molecules. I will not pursue these considerations more at length here: they may well be thought out into greater detail.

It may just be remarked, in passing, how elastic rings, fig. 2, can at first repel each other, merely on account of their elasticity of form; and how in fig. 3, if the molecules are made to approach closer by force, this has as a consequence [156] that the pressure of the atomic streams over the enlarged surface of contact overcomes the elasticity of form, and the molecules cohere (which one calls "attraction"). As the converse of this, when the molecules are by a tensile- stress pulled nearly apart: then their elasticity of form can make the molecules suddenly spring apart of themselves, as, for instance, "unbreakable" glass flies into dust, when the molecular equilibrium is upset by a very sharp blow. Also, in general, if something is broken, the pieces will not readily unite of themselves, when placed in contact. The natural elasticity of shape of the elastic molecules causes an initial repulsion. By suitable assumption regarding thickness, stiffness, &c., of these ring-like molecular forms, the differences between the "chemical affinity" of different molecules might doubtless be accounted for.

Some may think that the above considerations are too simple to contain truth. Nevertheless one may rightly ask whether it is not precisely simplicity that one in general seeks in mechanism? The elucidation may serve as an initial explanation of certain obscure facts, which may develop itself further.

Respecting the elasticity of molecules (or atoms) Lord Kelvin makes the following observation: —

"We are forbidden by the modern physical theory of the conservation of energy to assume inelasticity, or anything short of perfect elasticity, in the ultimate molecules" (Phil. Mag. May 1873, p. 329).

The conception of elastic molecules (also illustrated in a striking manner by spectroscopic observations) appears, as said, to be a very practical conception for Physics, which is much needed. By this assumption the almost inconceivable idea of the sharp blows of "infinitely hard" molecules is avoided. On account of the perfect elasticity, all motions take place with "elegance" and smoothness, which permits a mobile equilibrium in nature, and (without due precautions) may well deceive the senses into the idea that in so-called "space" all is in repose.

It is a known consequence of the Newtonian law of gravitation, that the increase of attraction by diminution of distance is so small that two massive bodies, when they touch [157] each other, attract each other so much the less the smaller they are; which one at once sees in the case of two spheres in contact, and can also demonstrate for bodies of other shapes. It, therefore, one attributes a massive structure to molecules, then, for the explanation of cohesion and chemical affinity, forces must be assumed which, by diminution of distance, increase quicker in intensity than according to the Newtonian gravitation law. On the other hand, the attraction of two cylinders of finite length and infinitely small section becomes infinitely great so soon as they touch each other. It is possible, therefore, on the basis of the here assumed open structure of matter, also to account for cohesion, adhesion, and chemical affinity, without necessarily having recourse to forces which, with diminished distance, augment quicker in intensity than the Newtonian law of gravity demands.

The Theory of Isenkrahe.
Two years after the present writer's first published paper, appeared the gravitation theory of Dr. Isenkrahe, which attempts an explanation of gravity based on the kinetic theory of gases, and which seems in Germany to have become very well known. The author of this theory makes no mention of my theory, and it doubtless escaped his attention at that time. The gravitation theory of Dr. Isenkrahe is founded on inelastic collision, which obviously involves the annihilation of energy, whereby the gas producing gravity would, after a certain (even very long) epoch, come to rest, and so gravitation cease to exist.

In a plausible enough way (at first sight at least) the author despises elasticity as a qualitas occulta, which, as he thinks, needs an explanation just as much as gravity itself. It seems, however, to have been overlooked that elasticity (at the encounters of molecules) is already demonstrated to exist by the principle of the conservation of energy, and of the centre of mass. The explanation of elasticity is a deeper one than the explanation of gravitation: therefore let us advance from step to step forward in the elucidation, without pushing on in too great a hurry.

All the consequences of the kinetic theory of gases are already built on the assumption of elasticity: the application of this principle to the smaller particles is therefore to be viewed as a perfectly natural and logical consequence. In fact the gravitation pressure by the encounters of elastic [158] particles is explained quite as completely as the air-pressure is explained by the encounters of such particles (molecules). If one only accepts as valid the two principles of the conservation of energy and of the centre of mass, then one must attribute elasticity as well to the aether as to gas molecules, without being troubled about the further explanation of its nature.

Therefore I have without hesitation regarded molecules of open structure as elastic, which implies that by the impact of such molecules no energy is annihilated. Dr. Isenkrahe regards the molecules of bodies as absolutely hard solid spheres which, in order that gravity by atomic encounters (i. e. its proportionality to mass) may be explained, must be far apart from each other. How can one imagine to oneself a structure composed of perfectly hard molecules situated far apart which shall have only tolerable stiffness and stability? Such a body made up of widely separated spherical molecules, if no other forces but gravity acted, could at the most behave like a gas, but never as a solid or liquid body.

On the other hand, elastic molecules of open structure may be made to cohere at their boundaries by the pressure of the smaller atoms, which at the .same time easily fly through the open parts of the structure. Have we not here at least a groundwork for the conceptions upon which we may hope to build further?

Dr. Isenkrahe gives no limits for the value of the mean length of path, whereas it seems to me to be a very important point of my theory, that the mean length of path must be assumed great in comparison with the planetary distances.

Concerning the calculations which Dr. Isenkrahe attaches to his theory, Dr. A. M. Bock, who made the theory of Isenkrahe the basis of his ' Inaugural Dissertation ', expresses himself as follows: —

"The aim and the purpose of the atomic aether theory, namely to construct universal gravitation, is, as mentioned in the introduction to the Räthsel von der Sehwerkraft, not fully attained. There is no formula deduced from which, as a starting-point, one could follow out the theory further. One sees oneself forced therefore, in the sense of the theory, to deduce an (if only in some measure rigorous) expression for the attraction" (p. 18, under the paragraph-title Die Anziehung zweier Körper).

On the developments and modifications which Dr. Isenkrahe's calculations received through Dr. Bock I allow myself [159] no opinion. The bases of them are nevertheless quite unaltered, and therefore open to the same objections; namely, he sets himself in contradiction with the principle of the conservation of energy, which, moreover, Dr. Bock himself admits.

Hamburg, 1894.

APPENDIX (added Jan. 1895). — All who have thought on the subject know that, in the case of a fulling body, the motion generated comes from the aether, according to any dynamical explanation of gravitation: and when the body strikes the earth's surface, shaking its molecules into vibration by the concussion, these ("heat") vibrations develop waves in the aether, or are "radiated" away. So we have a cyclical process here, where motion passes from a material agent and back again to that agent, in a circle.

In accordance with the above we see, then, that stars or stellar suns do not "pour their heat unrequited into space," but return their stores of motion to the source whence they were obtained. For if gravity be caused by a material agent, and if solar energy be derived from gravity, then manifestly solar energy is returning only to its original source, to be again available for generating heat (through gravitation) in some other regions of the universe.

Evidently, if chemical action be caused by a material medium, then an animal or a steam-engine lifting a weight is an instance (again) of motion coming from a material substance, and going back to it in a circle at the same time. A locomotive, as we know, converts all its energy into heat (which is radiated into the aether) as it progresses with its train: so clearly we have the cyclical process of exchange of motion again here: the same being true of work ierived from falling water (cataracts) or from winds. If, finally, one pure speculation be permitted, we might suggest that overgrown stars may, towards their centres, become from excessive compression inadequately penetrable by the atoms of the aethereal gas, and so the overgrown masses be broken up by conversion of the aethereal motion into heat. Thus cyclical change would apply to the Universe generally: the stellar bodies constituting in sum a gigantic grained gas inside an excessively fine atomic one. For the tentative development of this idea, a paper in the Philosophical Magazine, August 1879, also Sitzungsberichte, April 1883. Vienna, may be mentioned. So it appears that the Universe may at present (in the same sense as a gas is) be in equilibrium of temperature.