Endnotesby Le Sage
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.
I am not here speaking of the actual Epicureans who were really lazy and consequently ignorant of astronomy and physics.
I am speaking simply of philosophers.
Of them, the Epicureans respected 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.
Chapter 8 of Book 1 of Aristotle’s Generation and Corruption wrote 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
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 2, Part 3, 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.