The Le Sage Theory of Gravitation
THE LE SAGE THEORY OF GRAVITATION.
(Translation by C. G. ABBOT, with introductory note by S. P. LANGLEY.)
 Le Sage's paper is one much oftener referred to than directly quoted from or read, and this is partly because the original is very little known, although it is in no more obscure a place than the Memoirs of the Berlin Academy, printed in the year 1784.
Le Sage appears to have been one of the academicians who, though in the capital of Prussia, were bound to write French of any sort rather than German, and it is only fair to the present translator to say that certain passages of the original hold the meaning of the author so securely hidden that it is doubtful if anyone could render them into English with entire confidence that their whole meaning had been grasped. Whatever the original obscurity, however, the translation, I believe, means something definite and, I hope, true.
The reader will recall that at the time when Le Sage wrote, the corpuscular theory of light was universally accepted, the laws of the conservation of energy and of matter were as yet unknown, and the kinetic theory of gases was quite beyond the scientific horizon. Hence it is a matter for surprise, not that Le Sage introduces in explanation of the difficulties met with hypotheses now in a form appearing somewhat crude, though doubtless still conceivable, but rather that his statement requires so little modification to fit it to the thought of the present day.
Some of the great objections made to Le Sage's theory, such as the supposed impossibility of this shower of his atoms acting with equal effect in the interior of the densest bodies as on the surface, are made in probable ignorance of how entirely satisfactory the hypothesis of the author is in this respect; I mean so far as the use of the mathematical infinity can render it so; while other difficulties have been, if not deared up, at least rendered less formidable by the advance of modern knowledge, which is on the whole clearly making more for the hypothesis than against, if we put it in the form in which Le Sage would doubtless put it were he living now.
Thus the objection of the hypothesis of countless atoms coming from and going to infinity, to the dissipation of their kinetic energy into heat upon impact with solids-this latter class of objections seems to have been very generally met in recent years. Thus it has been made  evident that the particles in question could vibrate in long closed paths with the same effect as if they came in from outer space and returned to it in straight lines, as the author originally supposed; and as to their infinitesimal smallness, our purely physical conceptions of space and even of time are not only still, as is well known, relative, but have received a curious extension since Le Sage wrote, so that our limit of the physically infinitesimal has been pushed farther back by studies into the nature of the molecule and the atom until we have before us actual things of an order of magnitude incomparably below anything known to the physicists of our author's time.
On the whole, then, the tenor of modern thought goes in the direction in which we are led by this theory, if by that we understand it, not in its first crude enunciations, but with the modifications which can now be legitimately associated with it, and which tend to make it both more suggestive and to maintain a continued interest in it-an interest which seems to justify the present publication of a paper with which so few are familiar at first hand. S. P. LANGLEY.
THE NEWTONIAN LUCRETIUS
By M. LE SAGE. (Read by M. Prevost at a meeting of the Berlin Academy in 1782.)
"In all branches of knowledge the earliest systems are too limited, too narrow, too timid; and it would even seem that the prize of truth is only won by a certain audacity of reason." - Fontenelle, in eulogy of Cassini.
THE AIM OF THIS MEMOIR
 I propose to show that if the earliest Epicureans had possessed as just ideas on cosmography as those of several of their contemporaries, which they neglected, and but a portion of the knowledge of geometry which had then been attained, they would in all probability have easily discovered the laws of universal gravitation and its mechanical cause. Laws, whose discovery and demonstration are the greatest glory of the mightiest genius that has ever lived; and cause, whose comprehension has long been the object of ambition of the greatest physicists and is now the stumbling block of their successors. Such things, for example, as the famous Kepler's laws - discovered scarcely two centuries ago, and founded in part upon gratuitous conjectures and in part upon tedious gropings - would have been nothing but special inevitable corollaries of the general knowledge which the ancient philosophers could easily draw from nature's own mechanism. This conclusion is entirely applicable to Galileos laws on falling bodies, whose discovery has been still slower and more contested. Moreover, the experiments by which this discovery was established were so crude that they left the way open to interpretations which rendered them equally compatible with several other hypotheses, which were in fact urged against him. On the other hand,  the consequences of the theory of atomic collisions would have been unequivocally in favour of the sole right interpretation (equal accelerations in equal times).
The union of the several branches of this conclusion forms not only a philosophic truth of extreme interest, but one from which a very useful consequence may be drawn, which is that in spite of the greater weight due to a posteriori researches a priori ones are not to be wholly neglected, since they may greatly accelerate the success of the former. Already some impartial philosophers are agreed that such conjectures if lucid and capable of evaluation might be useful to the most rigorous physicists, were it only in suggesting to them definite points of view from which to direct experiment, in the place of that indecision in which the mere vague wish for new investigation has often left them.
Let us clearly understand that such speculation is only permissible for the sake of occupation when the skill and patience which new observation and experiment require are lacking. We ought to be thoroughly informed as to all previous observations and experiments on the subject and to keep these steadily in view in forming hypotheses, which are to be tested by them with the aid of every help that mathematics can give in examining as to the exactness of their agreement. Finally, it is such an agreement rather than any elaboration of method which brings conviction to most students of any physical theory, and this whether they are aware of this agreement before their acquaintance with these methods or whether a study of the method led them to the agreement.
If the disciples of Epicurus had been as fully persuaded of the sphericity of the earth  as they were of its flatness, then instead of conceiving their atoms to move in nearly parallel paths, as was suited to a directive force perpendicular to a plane surface, they would undoubtedly have attributed to them motion normal to the surface of a sphere, and consequently directed at all points toward its center. An example of such a condition as I have in mind would be furnished if it hailed simultaneously in all the countries of the earth.
The following objection would of course have been raised by some to this view: Part of these atoms must necessarily encounter the moon before reaching the earth, and by their pelting would push her toward us; and on the other hand the force exerted upon those terrestrial objects which she shields would be less because of her interposition. Consequently we ought to see the moon descending and a part of the waters of the ocean rising to meet her, as if rendered lighter by the interception of the atoms, and consequently yielding their place to the adjacent waters. In view of these objections the Epicureans would have had to see if some phenomenon of this nature did not really exist. They would have answered their opponents that the moon did not recede from us on a tangent, but really did approach the earth at each instant, and that the alternating motions of the ocean, so accordant with those of the moon, exhibited this very effect in question, due to the inequality introduced in the stream of atoms by the interposition of this great body.
The example of a pebble projected horizontally, which circulates for a few moments about the earth before falling, and longer in proportion as the motion is more rapid, would have made it clear that the moon, which occupies but a month in such a great journey, might' not of necessity actually approach the earth except in the sense of being nearer than if she had gone off on a tangent.
A persistent antagonist, fortified by some theorems of centrifugal force similar to those of Huygens (which are easily demonstrated by elementary geometry for polygonal orbits such as would result from intermittent collisions) might further have objected that the motion of the moon was still 60 times too slow to prevent her actual approach to us, taking into consideration the very considerable force of gravitation found at the surface of the earth. Upon this the Epicureans would not have been slow to reply that since the distance from the  moon to the center of our globe is 60 times as great as our distance from this same center, the spherical surface having the radius of the moon's orbit is 3,600 times as great as that of the earth. So that if the outer surface were traversed by the same number of atoms as the inner, their distribution would be 3,600 times rarer, and they would in consequence cause a gravitation 3,600 times less. This would be exactly that required by the theorems, for this gravitating force would suffice to sustain at a distance 60 times as great a moving body whose absolute velocity was times less than that required by a body revolving at the surface of the earth.
The parallelism of path which Epicurus had introduced in the atomic theory of Leucippus and Democritus was not exact, since had it been so these atoms, all moving with equal velocity, could never have come in collision. But Epicurus required that they should collide in order that he might explain the formation of compound bodies without assuming the intervention of a superior cause. Hence he supposed the paths of the atoms to be slightly inclined to each other, and it is well known that the introduction of the correction subjected him to many pleasantries and objections from philosophers of other sects.
If, however, Epicurus had embraced the doctrine of the convergence of the atoms toward a center, undoubtedly his opponents would have attacked this hypothesis quite as vigorously. The Epicureans in replying would have been able to explain this convergence by returning to the system of Leucippus and Democritus as follows: Imagine the atoms to move fortuitously in every direction, and let us trace the result in  the case of a body near the earth. All the atoms coming toward the body from the direction of the earth would be cut off by it, while from all other directions the body would be subjected to uninterrupted bombardment. Consequently there would be a resultant motion of the earth, that is in the line of diminished resistance, and this resultant motion would be exactly the same as if the bombarding atoms all converged toward the earth's center instead of moving fortuitously.
The Epicureans would have even seized with avidity upon this occasion to give an air of disorder to the primitive movements of the universe. For this would accord the better with their system of the origin of things (otherwise sufficiently absurd and impious) that there was no appearance of parallelism, perfect or imperfect, whereas all tendency to parallelism would appear to be the result of some particular design, and consequently to indicate the operation of some intelligent being.
I speak of disorder in connection with primitive movements only. The resultant motion of bodies having inertia would be directed toward the center of our globe with great exactness, in consequence of the combination of a vast number of impulses in different directions. For it is a well-known result of the doctrine of chances that minor irregularities, when in great number, mutually compensate each other exactly, so that each several inequality becomes imperceptible in its effect upon the resultant.
Still another consideration would have led the atomists to make this same modification of the direction of motion of the gravitational atoms. All will agree with me that they were certain to have met with one or other of these two objections or to have themselves raised them. As the earth revolves without cessation about the sun, the hypothesis that  all the atoms are directed toward the center of the earth would have required that each new shower of atoms must seek it in a different direction from that followed by the shower next preceding, a condition not in accord with the predilection of the sect of the Epicureans for the operations of chance, nor with their antipathy for occult qualities.
In order to extricate themselves from this difficulty the atomists would necessarily have rejoined that there was no place in the heavens, equal in dimension to the earth, toward which there did not advance in a given time quite as many atoms as our planet encounters in the same portion of time, and that these other atoms were in motion exactly like those encountered by the earth. Not that there was any particular relation between places and the streams setting toward them, but, since it was essentially a confused movement, equal areas must naturally intercept, one equally as much as another, the paths of the atoms which blindly traverse space; and in consequence they must be equally exposed to their visits.
When once the Epicureans were thus come to explain the matter so neatly, the most thoughtful and curious among them would certainly have followed out the consequences which could be easily deduced from this hypothesis, and they would necessarily have arrived at the following propositions:
1. The atoms which pass to one side of any central body contribute nothing to the force of gravitation which it exercises toward other bodies, for such atoms are exactly counterbalanced by direct antagonists. Gravitation would be due solely to those atoms which are fortuitously directed toward the central body. As we have seen, the resultant action of these atoms is everywhere directed toward the central body, like the rays of light converging toward a focus when assembled by a convex lens or a concave mirror. Hence, it is proper to apply to them what has been proven in Paragraph IV touching the terrestrial gravitation; that is to say, their gravitational effect is inversely proportional to the square of the distance of the attracted body from the central body. 2. The gravitational atoms are directed not only toward the centers of the greater bodies, but toward each of their particles as well, since they move indiscriminately in all directions in space. The atoms, moreover, act effectively in those directions in which their antagonists are intercepted; that is to say, in all directions in which there are particles of matter. Therefore they tend to move the heavy masses which they encounter not toward the heavenly bodies in gross, but toward each of their particles in detail. Hence the gravitation of masses toward the center of a celestial body is nothing but the resultant  of an imperceptible movement of the masses toward all parts of the great body (as, from certain passages of Cicero and Plutarch, it appears had been before supposed by some of the ancients). Consequently this gravitation would be proportional to the number of the particles; that is to say, to the mass of the central body.
Now from these two propositions alone there might have been deduced synthetically the entire theory of universal gravitation without further mention of gravitational atoms.
This is the place to insert a certain proposition which is commonly spoken of as if it were distinct from those which teach that gravitation is universal, but which appears to me to be included in that expression. I refer to that which affirms that gravitation is mutual or reciprocal; or, in other words, that it is subject to the ancient law of mechanics, which states that action and reaction are equal.
I say that this is the place to consider this proposition, because it can equally well be proved either through the introduction of the agent of gravitation, as I have done in preceding paragraphs, or by considering gravitation abstractly, as I shall do in those which follow. This proposition therefore forms, as it were, a gradation between those which I have established by the first method and those which I shall establish by the second.
First method: Inasmuch as one body is pushed toward another by the atoms which the second body has deprived of direct antagonists, while the latter body is pushed toward the former by these same antagonists, the two bodies are necessarily pushed toward each other with equal force, whatever be the inequality of their masses or the differences in their forms.
Second method: Since each particle of one of the two bodies tends toward every particle of the other, the first body is urged toward the second with a force proportional to the number of particles which the second contains, or, in other words, with a force proportional to the mass of the second. Furthermore, since the impetus or momentum of the first body is the summation of the impetus of its separate particles, it is proportional to the total mass of the first body. Thus it follows that the impetus of the first body is proportional to the product of the masses of the two bodies.
By a similar train of reasoning the impetus of the second body is also proportional to this product. Therefore the usual bodies are urged together with equal forces.
I am now in a position to examine what other consequences the ancients would probably have drawn from the principle of a mutual gravitation directly proportional to the masses and varying inversely  as the square of the distance. For the sake of brevity the mechanical cause may be left out of consideration in the discussion.
As these philosophers would have foreseen many difficulties in rigorously testing every consequence to see if it coincided exactly with observation, and would therefore have refrained from embarking upon so serious a task before perceiving that the deductions accorded in gross with the results of experience, I presume they would not seriously have applied geometry and computation to this gravitation without having first determined by simple reasonings what, approximately, would be the effects flowing from it, and seeing that these conjectures accorded roughly with the real constitution of the universe I believe I do no violence to probabilities in presuming that the ancient philosophers would have been acquainted with some such reasonings. Having fewer matters than we to distract their attention, they were able to make very exact deductions in subjects requiring nothing but meditation. With reference to the acquired knowledge which would be needed in such reasonings, it will be recalled that the theory of conic sections had been discovered and cultivated before the birth of Epicurus, that Archimedes had made great advance in the doctrine of centers of gravity, and that the ancient geometers, and especially the last named, employed approximations with great ingenuity when they were unable to attain to rigorous precision.
Encouraged by these first successes and animated by the grandeur of the enterprise it is highly improbable that these ardent and subtile geometers would have stopped here. They would doubtless have invented for the purpose some means for passing from the ratio of sensible quantities to that of their imperceptible elements, and conversely from elementary quantities to their summation, at least for the simple case required when one wishes to avoid the numerical computation of the small anomalies of the movements of the celestial bodies.
 Certainly they had sufficient patience and sagacity to succeed in finding such a method, since they had had enough of these qualities to discover and advance in considerable degree the admirable doctrine of incommensurables, and of exhaustions, although these were not ordinarily used except in the consideration of the five regular bodies, and were specially derived, it is said, to examine certain very hazardous and even fantastic conjectures of the Pythagoreans and Platonists.
Practically, if one omits from the theory of central forces those curious propositions and generalizations which can only be regarded as its luxuries, as well as the delicate evaluations which are required only for the perfecting of astronomical tables, all the rest may be demonstrated sufficiently for the uses of the physicist by the aid of lemmas less exact and universal than those of the calculus.
This has indeed been pointed out in some degree by several geometers, but it may be realized still further if the reader will undertake by the same or analogous means of simplification to attack other propositions than those already so treated.
But the probability that the ancients would have been able to accomplish such demonstrations is still less necessary to the plan which I have proposed to myself, as stated at the beginning of this essay, than the probability that they would have discovered the simple relations mentioned in the thirteenth paragraph. Consequently the reader may, if he prefers, ignore the last three paragraphs and give attention only to matters which I have expressly engaged to establish.
I declared that the laws of Kepler were necessary consequences of the doctrine that gravitation results from the impulsion of atoms moving in every direction, since Kepler's laws follow directly from those of Newton. I ought, however, to show, for the benefit of readers less versed in the matter, where it may be found proved that the first-mentioned laws are the natural consequences of the second.
First. That the law of areas proportional to times is a necessary consequence of gravitation, always directed toward a single point, is demonstrated by elementary geometry in the first proposition of Newton's Principia.
Second. That the law of squares of periodic times proportional to the cubes of the distances, for bodies appearing to describe circles, must necessarily follow from a gravitation inversely proportional to the square of the distance constitutes the second part of the sixth corollary to Proposition IV of the same work, and may be demonstrated by elementary methods also for regular polygons, which represent more nearly than exact circles the orbits traversed by bodies diverted slightly from their paths by intermittent collisions.
 Third. That the ellipticity of an orbit is the necessary consequence of gravitation directed toward its focus, and reciprocally proportional to the square of the distance, is the converse of Proposition XI of the same book. This proposition has been more simply demonstrated as a consequence of the fiftieth of Book III of the conics of Appolonius.
I may pause here, since in maintaining that the laws of Kepler are an easy consequence of the system of atoms I have not pretended that their application to complex cases readily follows from the slight knowledge of geometry possessed by the ancients. Nevertheless, I may add--
Fourth. That the Proposition XI of the Principia once attained it does not appear to me difficult to establish the fiftieth, which extends our second consequences to ellipses-that is to say, which proves that in ellipses as well, the squares of the periodic times about an attracting body (placed in one of the foci) are proportional to the cubes of the mean distances.
Let us now see how the laws of Galileo may be derived from the hypothesis of the impulsion of the atoms.
The blows of corpuscles, moving with a velocity more rapid than light, upon a body which has fallen three or four seconds, would be sensibly of the same strength as the preceding blows had been upon the same body when it had only fallen one or two seconds. Hence the successive accelerations of the body in equal times must be sensibly equal, and the velocity at any instant must be sensibly proportional to the time elapsed since the beginning of the fall. From this it follows necessarily that the spaces traversed since the beginning are sensibly proportional to the squares of the total times, and will be sensibly proportional to the successive odd numbers.
These synthetic demonstrations of laws of falling bodies by the introduction of mechanism whose existence is only surmised, may perhaps be less philosophical than analytic demonstrations which are based entirely upon observed phenomena. Still it must be recalled that in cases where direct observation has been difficult and inexact, error has frequently attended deductions of this latter kind. At all events the former kind of demonstration is much more philosophical than a gratuitous hypothesis, which is, nevertheless, the means of invention employed by Galileo; and its results are quite as well established as are the laws of Galileo since they are proved by exactly the same means, that is by the sensible accord of their consequences with the phenomena. Nothing else than this is claimed by Galileo himself and his principal successors.
But the atomists would have encountered one very serious objection, to which they were necessarily exposed in common with all physicists who undertake an explanation of gravitation. For by having thickness a roof receives not a whit more of hail, or a shield of arrows; whereas, remaining otherwise unchanged, the weight of all bodies is augmented in direct proportion of their thickness. Conversely when one removes a heavy body from a shop or dwelling, or reduces it to sheets exposed without protection to material influences (the rain, for example) it receives more than when protected or concentrated so as to present a small surface. But it has never been found by merchants and artisans, who are continually in the habit of weighing, that bodies appear heavier in open air than when under cover, and gold-beaters have never perceived that the weight of the metal augments in proportion to the increase of its surface.
In a word, if the collision of atoms is the cause of heaviness, the weight of bodies ought to be proportional to their surface (or rather to their horizontal projection). How, then, does it happen that the weight is proportional to the mass!
Do the gravitational atoms then act across the thickest and most compact envelopes of all substances as fully as through the air? And  does not the very sensible weight which they impart to these envelopes demonstrate the contrary, that is that all substances arrest the passage of a great number of corpuscles?
To this the Epicureans would have been forced to respond that the atoms doubtless traverse very freely all heavy bodies; as freely, for example, as light passes through diamond and magnetic matter through gold, though one of these bodies is the hardest and the other the heaviest of all known bodies (which shows that they are less porous than most substances). Thus the number of atoms which are intercepted by the first layers of a heavy body would be absolutely insensible relatively to the number of those which pass through the last layers. Nevertheless, the relatively small number intercepted would produce a sensible action upon the body, since they have, in virtue of an immense velocity, the force of impact which they would lack by reason of their small mass.
A second difficulty which would have embarrassed the more scrupulous atomists, is that the mutual collision of the atoms would retard their motions repeatedly, and diminish, consequently, the gravitational action. Any such effect, nevertheless, has hitherto been imperceptible.
Now, it would be useless to offer in explanation that the sum of the motions would remain the same, since this is only true when the word sum is used in the sense of geometers, who comprehend by it the difference of contraries. Such a definition is readily seen to offer no assistance to the atomist in the case of equality of contrary movements. For the algebraic sum of the motions of the atoms is zero before as after the collision; but before the collision they were capable of effects of which they are incapable afterwards.
It is apparent that such mutual encounters would be the more rare the smaller the atoms were supposed to be compared with the intervals between them. These intervals can not, however, be assumed very great since gravitation manifests no sensible interruption even in places and times the most adjacent; so that the only conceivable recourse to render the encounter of the gravitational atoms sufficiently rare is to suppose them extremely small. Happily this device is completely sufficient. Conceive two balls whose centers trace given courses in different planes. In order that they may never meet it suffices to diminish the sum of their semi-diameters till it becomes less than the least distance between their paths.
But since, with diminishing size, the atoms would be less efficient to produce gravitation, the intensity of which is fixed by phenomena, it is necessary to see if their effectiveness may be maintained by some other properties. I see no recourse of this nature except in the increase of individual density or of velocity. These two recourses appear very natural, and are at the same time the more satisfactory because they were (very probably) in accord with the spirit of the atomists of whom I speak, and would probably have sufficed to close the mouths of their adversaries.
Third difficulty: Each celestial body perpetually finds atoms in its path which it necessarily displaces in passing onward. This can not occur without the atoms communicating to the body a part of their motion, and in consequence causing its retardation. Exclusive of all other elements except the mass displaced, this retardation is proportional  to the density of the medium made up of these atoms and their interstices. Now, the gravitation of the body (exclusive of all other elements than this atomic mass) is proportional to this same mean density. How, then, can it be that the retardation is imperceptible while the gravitation is so sensible? The objection is rendered the more forcible when we consider that the retardation of a revolving body is brought about by all the atoms which it meets in its orbit, while its gravitation is produced only by those which at any one position in its orbit are directed toward the central body.
Reply: Other things being equal, the force of gravitation, being produced by the single stream of atoms deprived of antagonists, is proportional to the square of the velocity of the atoms (by a proposition demonstrated generally), while the retardation above spoken of, being caused by the stream opposing the planet in its motion, is proportional to the product of this velocity of the atoms by that of the revolving body (as we shall prove directly). Consequently (things being equal) the gravitation is to the retardation as the velocity of the atoms is to that of the revolving body.
Now, it is not hard to believe that the velocity of the atoms is greater than that of the revolving body; and, indeed, all that we have heretofore said would lead to the presumption that it is incomparably greater. Hence the system of thin-sown atoms moving in every direction agrees very well with a condition of gravitation incomparably greater than the retardation, and it agrees still, despite the consideration which fortifies the difficulty which we are considering, since a velocity has always been assigned to the atoms greater than would have been necessary to obviate this latter difficulty alone.
Remark: I have said that the retardation of a great body caused by the opposing, stream of atoms moving much more rapidly than the body itself would be proportional to the product of the velocity of the atoms by that of the great body. I shall first demonstrate this proposition with respect to the couple of opposed streams parallel to the direction of the great body, and in so doing I shall have proved it for the case of opposing streams oblique to this direction, since their motions may be decomposed in two directions, the one parallel and the other perpendicular to the direction of the body, of which the first is nearly always much greater than the motion of the body, and of which the second produces no effect.
Demonstration: The total retardation of the body is the excess of the simple retardation it experiences from the stream which it encounters over the simple acceleration which it experiences on the part of the stream which pursues it. Now, these simple factors are proportional to the squares of velocities, which are respectively the sum and difference of the absolute velocity of the atoms and the absolute velocity  of the body. Consequently, the resultant retardation is proportional to the excess of the square of the sum over the square of the difference, which (by the eighth proposition of the second book of the Elements of Euclid) is four times the product of the absolute velocities in question.
To the three difficulties above mentioned may be reduced all those which are plausible, since there can be no other changes in the motions of a heavy body, or in the motions of the gravitational fluid, or in their constitution, except those which proceed from some opposition or interposition, either on the part of the particles of the heavy body, which hinder the atoms composing the fluid from reaching their destination, or from particles of the fluid itself, the one opposing the other, or, finally, from the effect of the latter on the path of the heavy body. The solutions of all these difficulties depend either on the permeability of the heavy body or the subtlety and rapidity of the gravitational atoms - properties to none of which we are obliged to assign two opposing limits.
This last expression signifies that while several considerations may unite to augment the intensity of such or such property, yet no consideration requires a diminution in the intensity of the same property, and that reciprocally no considerations tend to limit the diminutiveness of properties of which certain other considerations limit more and more the magnitude. There are no conditions which give opposing indications, and which therefore obstruct the choice of remedies. This assertion would be tedious to establish, but very few readers will contest its correctness.
While we speak of alterations and remedies it is for me to conform to the irregularity of our ordinary progress in research. Truth never permits us to discover her at first seeking, with all her following train of verities, but we proceed gradually in discovery by tedious gropings and corrections. To this procedure a writer ought also in some measure to conform, in the exposition of truths which he has finally discovered, when the greatness or smallness of the objects discussed transcends that of the majority of those objects with which we are familiar, and when he believes that his reader will not at first be disposed to countenance suppositions so excessive, but only in a measure as he shall have shown him their necessity. For the reader will have had no perspective to apply to this immensity or that diminutiveness if it has been assumed at the start in sufficient measure to satisfy all phenomena.
The author might with equal justice assume at the start a magnitude or diminutiveness sufficient for the purpose, since in explaining the phenomena the physicist takes the place (so far as he may) of the Creator - a being who, having determined precisely in advance all the  consequences of the different intensities which might be given to such or such properties, has chosen in each case that intensity most proper to attain the desired result and has precisely determined the consequences without any preliminary trial.
All other conceivable objections are founded on certain regularities or irregularities of detail which have not been minutely set forth, but gratuitously assumed, and which, in consequence, ought not seriously to be taken into account. Or, in the second place, such objections may be founded on the tenets of some metaphysical sect. Before responding to such objections I pray these metaphysicists to first agree among themselves. Or, finally, they address themselves to the imagination rather than to the understanding. Thus some may be shocked at what in this system is extreme, strange, or extraordinary-as if it was after our gross and limited measures that the subtlety and grandeur of Nature must be evaluated! As if a confused repugnance sufficed to condemn a theory which depends neither on taste nor sentiment! Or as if one ought to follow servilely the beaten track, even in researches where no success has ever come to those who have followed it!
If one is satisfied with the exact agreement of this system with physical astronomy and with terrestrial phenomena, he ought not to distrust it, as if this apparent conformity were the effect of the artfulness with which I have adjusted matters or as if other systems also might be rectified so as to agree throughout with the phenomena should a hand more skilful take the same pains to accommodate them to each other.
I have not added to the atoms sung by Lucretius any feature directed solely toward the explanation of the great laws discovered by the Moderns. But, on the contrary, I have merely divested the motion of these atoms of an arbitrary feature (the nearly perfect parallelism) by which Epicurus had disfigured the unrestricted motion assumed by Democritus. That was a motion so simple that it would appear as if its inventor had proposed it with no other end but the most absolute simplicity, unconcerned that it might in no way explain real phenomena, but rather, perhaps, contradict them; so that it is impossible that any system can equal this in simplicity.
I would even have had no need to advise myself of this correction, in reading the poem of Lucretius, if I had been instructed beforehand in the system of Leucippus and Democritus as I was long after this reading.
Finally, the explanations which I have offered ought not to be regarded as in any respect modifications of this system of atoms, for it would be impossible not to fall upon these explanations in seeking to follow out the necessary consequences of this system.
I did not take undue credit to myself when as a child I rectified the system taught by Lucretius and drew from it immediately its most important consequences, for this was extremely easy or rather entirely natural. Besides, I knew but little more the value and solidity of my little views than the child ordinarily knows the wit or sense which we find in its repartees and sallies. Indeed, the extremely simple idea of trying to explain the principal natural phenomena by the aid of a subtle fluid vigorously agitated in every direction has come to many writers who have before presented it in a vague and ill-assured fashion, not to mention that there has been without doubt a still greater number who have not even deigned to communicate at all. I am well convinced that since the law governing the intensity of universal gravitation is similar to that for light, the thought will have occurred to many physicists that an ethereal substance moving in rectilinear paths may be the cause of gravitation, and that they may have applied to it whatever of skill in the mathematics they have possessed.
But we may say, How is it that none of these physicists have pushed these consequences to their conclusion and communicated the research? Doubtless because the most of them having no clear view of this chaos (of which the first glance is, I admit, frightful) they have not known how to disentangle it and subject it to their calculations. Or not having firmly grasped the principles of the theory, they have allowed themselves to be seduced by specious sophisms, by which men have pretended to refute in advance all imaginable explanations of gravitation. Or they will have had the foible of bowing to the authority of great names, when it is alleged (whether justly or falsely) that they have pronounced upon the impossibility of this or upon the uselessness of that branch of knowledge. Or they have lacked sufficient love of truth or courage of their convictions to abandon easy pleasures and exterior advantages in order to devote themselves simply to researches at the time difficult and little welcome. Or, finally, they have failed to become impressed with the strength and fecundity of this beautiful system so distinctly as to lead them, in their enthusiasm, to sacrifice to it their other views and projects.
Constitutions which I assign to heavy bodies and to the gravitational fluid; followed by a mathematical conception and some remarks to fix the ideas of geometers who desire to follow out for themselves the consequences of this mechanism, and who may desire first to know precisely what are the hypotheses from which I claim all the phenomena to follow necessarily. 
CONSTITUTION OF HEAVY BODIES.
First. Their indivisible particles are cages; for example, hollow cubes or octohedra. (They are, in other words, skeletons of solids of which there is nothing material except the edges.)
Second. The diameters of the bars of these cages, even if supposed increased by the diameter of the gravitational corpuscles (as they must be in order to conveniently evaluate the portion of the atoms intercepted), are so small, relative to the distances between the parallel bars of the same cage, that all the particles included in the terrestrial globe intercept not the ten-thousandth part of the corpuscles which present themselves to traverse it.
Third. These diameters are all equal, or if they are unequal their inequalities sensibly compensate each other. If, for instance, in the smallest portions of matter separately ponderable (which, it has been stated, may weigh one thirty-second part of a grain) the mean diameter of the bars of the one portion does not differ a tenth part from the mean diameter of the bars of the other, then it would follow that in the greatest ponderable masses the mean diameters do not differ by a ten-thousandth part, for every such great ponderable mass is composed of so large a number of indivisible particles that simple chance suffices to almost perfectly effect a compensation of diameters.
CONSTITUTION OF GRAVITATIONAL CORPUSCLES.
First. Conformably to the second of the preceding suppositions, the diameter of the gravitational corpuscle added even to that of the bars of the indivisible particles is so small relatively to the mutual distance of the parallel bars of a single cage that the weight of celestial bodies does not sensibly vary from the ratio between their masses.
Second. The gravitational corpuscles are isolated, so that their progressive movements are necessarily rectilinear.
Third. They are so thinly scattered-that is to say, their diameters are so small relative to their mutual mean distance-that there are no more than a few hundreds which encounter one another in the course of a thousand years. Hence the uniformity of their movements is never sensibly disturbed.
Fourth. They move in several thousand of thousands of different directions, even counting as one all those which are parallel to the same line. The distribution of these directions may be conceived as follows: First, imagine all the points conceived to lie in different directions strewn upon a sphere as uniformly as is possible, and consequently separated from one another by less than a second of arc; then imagine a corpuscular path radiating from each of these points.
Fifth. Parallel to each of these directions there moves a stream or torrent of corpuscles. Now, in order to give it no more than the necessary size, the transverse section of this current has the same contour as  the orthogonal projection of the visible universe upon the plane of this section.
Sixth. The different parts of a single current are sensibly of equal density, either where contemporary portions of sensible magnitude or successive portions occupying sensible times in traversing a given surface are compared. The densities of different currents are also equal.
Seventh. The mean velocity determined in the same manner as the mean density is also sensibly constant.
Eighth. This velocity is several thousand times as great, relative to the velocities of the planets, as is the gravitation of the planets toward the sun relative to the greatest resistances which secular observations permit us to suppose they experience. For example, several hundred times greater relative to the velocity of the earth than the gravitation of the earth toward the sun multiplied by the number of times the firmament would contain the disc of the sun is greater than the greatest resistance which the secular differences in the length of the year permit us to suppose the earth experiences from celestial matter.
CONCEPT, WHICH FACILITATES THE APPLICATION OF MATHEMATICS TO DETERMINE THE MUTUAL INFLUENCE OF THE HEAVY BODIES AND THE CORPUSCLES.
First. Decompose all heavy bodies into equal masses so small as to allow them to be treated without sensible error as attractive particles are treated in those theories of gravitation in which no hypothesis is made as to its cause. In such a small mass the effects of unequal distance and position of its particles relative to those of the mass which is conceived to attract it, and to be attracted by it, may be neglected. Such masses will have a diameter no more than one one-hundred-thousandth as great as the mutual distance of the two masses under examination. Thus the apparent semi-diameter of one as viewed from the other does not exceed one second.
Second. For the surfaces of this mass, accessible but impermeable to the gravitational fluid, substitute a single spherical surface equal to their sum.
Third. Decompose these first surfaces into facets sufficiently small to be treated as planes without sensible error.
Fourth. Transport all these facets to the spherical surface above mentioned. Each one of the facets should in this transformation occupy that point of the spherical surface at which the tangent plane is parallel to the original position of the facet.
First. It is not necessary to be very expert to deduce upon these suppositions all the laws of gravitation, both terrestrial and universal (and consequently those of Kepler and some others), with as much of  precision and more as the phenomena themselves furnish, for these laws are the inevitable consequences of the constitutions I have supposed.
Second. Although I here present these constitutions crudely and without proof, as if they were gratuitous hypotheses and adventurous fictions, the fair-minded reader will perfectly comprehend that I have at hand some presumptions, at least, in their favour (independent of the perfect accord with all the phenomena), but which I withhold as too extended for development in this place. These suppositions may then be regarded as theorems published without demonstration.
Third. Their number is likely to inspire some opposition at first glance; but the attentive mind will not fail to see that they are but details into which I have wished to enter because of the novelty of this doctrine, and that they will be readily understood when it shall have become sufficiently well known that its students may attend under favourable circumstances to the details. If the authors who have written upon hydrodynamics, aeronautics, or optics had had readers who doubted the existence of water, air, and light, and who consequently indulged no tacit supposition upon equalities or compensations of which no express mention was made, they, too, would be obliged to add a great number of explanations to their definitions which instructed or indulgent readers might well dispense with. We do not accept of hints, and sano sensu, except for propositions which are familiar and in whose favour there is a predisposition.
- ↑ Translated by C. G. Abbot from Nouveaux Memoires de L' Academie Royale des Sciences et Belles-Lettres. Annee, MDCLXXXII. A Berlin, MDCLXXXIV, pp. 404-427.
- ↑ I say only the earliest; for after a system has survived several centuries it leads men to the one or the other of two extremes. Some reject everything pertaining to the system disdainfully, while others, on the contrary, embrace reverently all its traditions, without offering to make the least correction. It is this latter faction who have adopted the atoms of Epicurus, Lucretius, Gassendi, and all the intervening Epicureans.
- ↑ One of these hypotheses was that the total time being as the arc of a certain circle, the total distance fallen through was as the versed sine of this arc. Now if the magnitude of this circle had been better chosen, I do not see how one would be able to refute this hypothesis, starting from the simple phenomena.
- ↑ Plato and Aristotle had discoursed at great length upon the sphericity of the earth; Archimedes and Aristarchus had assumed it; Thales and Zeno had taught it, and all the astronomers believed it. (See the Timams of Plato, the close of the second book of Aristotle upon the Heavens, the Hour-Glass of Archimedes, and the tenth chapter of the third book of Plutarch upon the Opinions of the Philosophers.)
- ↑ Neither Epicurus nor Lucretius discovered the figure of the earth. But it seems probable that they conformed to the opinions of Democritus upon all questions where they did not expressly oppose him. Moreover, Gassendi (in his Commentaries on Epicurus, p. 213 of the edition of 1649) alleges strong reasons for believing that they supposed the earth's surface to be flat.
- ↑ Instead of which they entirely rejected this centripetal tendency.
- ↑ This is not precisely the actual state of affairs, but it is thus that the case would present itself at first view. As an exact recognition of the laws of this phenomenon would be more slowly acquired than an exact knowledge of the laws of atomism, there would never be a time when that theory would have been found at fault in this respect.
- ↑ If the force of gravitation were the same at all distances, the period would be reciprocally proportional to the square root of the distance (Hugenii Theor. IV.) instead of to the three halves power as follows from the Newtonian law (Phil. nat. Prine. Math. Prop. IV. Cor. 6). Then the period of the moon, as compared with that of a body revolving at the surface of the earth, would be expressed by instead of , the value derived from the Newtonian law of gravitation.
- ↑ Combine the second and third theorems of Huygens published in 1673 following his Horologium oscillatorium.
- ↑ It was natural enough to greatly diversify this motion which tended to deflect the atoms.
Lucretius, even, despite his devotion to Epicurus, expressed himself several times conformably to the system of Democritus. His first book with the first 216 lines of the second ignored the imperfect parallelism that he lent to the paths of the atoms, for instead of speaking of this parallelism he seems to say three times that they come from all directions (undique, lines 986, 1041, and 1050), that they waver (volitare, 951), trying several kinds of collisions (multi modis plagis, 1023 and 1024), essaying all kinds of movements (omne genus motûs, 1025), finding room to advance in whatever direction they move (motfls quacumque feruntur, 1075). He adds, in the second book, that they wander in space (per Inane vagantur, line 82), that they are agitated by various movements (varioque exercita motu, 96), and that all those which have not been able to associate themselves together to form great masses are always agitated in the great void (in mllgno jactari semper Inani, 121) in the same, way as the dust that one sees in a dark chamber into which the sun's rays penetrate is moved about in all directions (nunc huc nunc illilc, in cunctas denique parties, 130). Finally, several of his commentators convey the same idea.
- ↑ Democritus was a century and a half later than Pythagoras, who had secretly taught the revolution of the earth. He might even have seen Philolaus who more openly proclaimed it, and Timaeus who appears to have had the same belief. He ought also to have been informed of the opinion of the Pythagoreans upon the subject, for Heraclides had been of this sect before he listened to Plato and Aristotle, and he maintained at least that the earth rotated about its center. According to the report of Diogenes, Laertius, and of Porphyry, Democritus had attended the teaching of the Pythagoreans; and besides, the Eleatic sect (if one may credit Strabo) was nothing bnt an offshoot of the Italic. Finally, the atomists, following Democritus, would have had opportunity to be even better instructed than he in regard to the earth's motion. For this doctrine was supported by a multitude of philosophers of all countries, among whom the principal names, in addition to those already cited, are Archimedes and Nicetas, of Syracnse; Aristarchus and Cleanthus, of Samos; Architas, of Tarento; Seleucus, Eophantus, and even (according to Theophrastus) Plato in his later years.
- ↑ I had intended to insert here some preliminary observations which the atomists would probably have made. I had collect eel them in part from various researches (or incidental points) made by good geometers who have undertaken to illustrate to readers but little advanced in mathematics some of the truths of physical astronomy. The remainder were from notes of lectures which I have myself given upon these matters. But I have omitted this digression on account of its length. Perhaps I may be permitted to remark that these elementary tests may be rendered very convincing, although some of them presuppose so little knowledge of geometry that they may even be stated without reference to figures.
- ↑ It should be borne in mind that we are not here speaking of the Epicureans as some have really been-that is to say of a nature decidedly lazy and consequently ignorant of astronomy and physics-but of philosophers simply, Epicureans as respecting the fundamental propositions of physics only, but resembling rather their contemporaries of other sects in general enlightenment and taste for research. Such a supposed character for these philosophers is by no means forced, since the physical and speculative dogmas of Epicurus did not necessarily entail his moral precepts and practices.
- ↑ To assign to these corpuscles the velocity of sound even would be sufficient. For the velocity of sound is more than thirty-four times as rapid as that of a body which has fallen one second, or more than seventeen times as great as that of one that has fallen two seconds, etc. Hence with the increasing velocity of the falling body the accelerating impulses impressed by the corpuscles would be more feeble than at the beginning of the fall by one thirty-fourth at the end of one second, by two thirty-fourths at the end of two seconds, etc. This gradual decrease of acceleration would not be perceived in the longest times of fall which are ordinarily measured. How much less therefore would they be perceived if we assume for the corpuscles the velocity of light, which is nine hundred thousand times as great as that of sound.
- ↑ Demonstration: I divide the two times which are to be compared into an equal number of parts, so small that the body may be conceived as falling with equal rapidity during the whole duration of one of these parts. And I observe that the two bodies which are compared will have, at the beginning of each of the corresponding parts of the two times, velocities proportional to the times then elapsed, and consequently to the entire times. Hence the small spaces traversed at these corresponding instants will be traversed with a velocity proportional to the times compared.
But the elementary spaces fallen through will be proportional not only to the velocities with which they are traversed, but also to the portions of time occupied in traversing them, and consequently to the whole times. Therefore the small corresponding spaces will be proportional to the squares of the whole times, and the sums of the (equally numerous) small spaces-that is to say, the whole distance traversed-will also be proportional to the squares of the whole times.
Remark: The assumption with which I started, and which is tacitly made in the other demonstrations of this law, is a sort of license equivalent to supposing that the parts of the times and spaces are infinitely small, and is less conceivable than one is accustomed to suppose. It is an inevitable inconvenience of the collision hypothesis of the continuity of the action of gravitation. But this inconvenience is not encountered when we substitute the hypothesis of discontinuity. I mean to say that there arises no contradiction when the time increments are taken equal to the Intervals between the blows of the gravitational agency.
- ↑ Several ancient physicists recognized the pores in bodies. It may be seen, for example, in the eighth chapter of the first book of Aristotle on Generation and Corruption, that Empedocles, Leucippus, and Democritus had made a great deal of use of them to explain sensations and mixtures. Galen reports in his works on the Natural Properties, that Erasistratus (the grandson, it is believed, of Aristotle), a celebrated corpuscular physician who denied attraction, believed in the existence of a vacuum and attempted to reduce all natural properties from the size of the pores. Coelius Aureliauus speaks of them also in connection with Asclepiades, of Bithynia, a physician of the time of Pompey. And Sextus Empiricus assures us that not only Asclepiades but also other physicians and physicists of the sect of Epicureans made many applications of the pores. Finally, in the first book of Lucretius there are ten or twelve lines upon the great permeability of bodies, concluding as follows: Usque adeo, in rebus, solidi nil esse videtur.
- ↑ However considerable we assume the number n of horizontal layers going to compose a body of uniform density, the number (and consequently the effectiveness) of the gravitational atoms is diminished in passing each one of them, because some atoms are intercepted by the solid material composing the layer. The number of atoms transmitted by a layer, and remaining effective to produce weight in the next lower one, will bear the same ratio to the number reaching the first that the volume of the spaces or pores in the layer bears to its total volume. Assuming the body to be of uniform density, this ratio will be constant, and since the weight of each layer is proportional to the number of atoms available to collide with its substance, this ratio represents the relative weight of any layer to that next above It. However nearly equal we may suppose the numbers a and b, which express the ratio which is assumed between the weight of the highest layer of the body and that of the lowest (the two layers being supposed equal in volume and density), it is possible to express in numbers the ratio of the entire volume to that occupied by the pores as to . Such a ratio may be obtained by experiments with several sorts of tissues, as, for example, by means which Newton indicates in his Optics (Book II, Part III, Prop. 8), the number of the orders of pores being the excess of the logarithm of over that of divided by the logarithm of two.
- ↑ The movement of the atoms is so rapid, according to Epicurus (in his letter to Herodotus), that they traverse the greatest imaginable spaces in a time inconceivably short.
- ↑ A little metaphysical consideration suffices to dispose of this instance; but, as will be seen in a moment, I am able to supplement it by two separate physical conceptions.
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