Sunday, April 28, 2024

Elements of Newton’s Philosophy, by Voltaire

Elements of the Philosophy of Newton (French: Éléments de la philosophie de Newton) is a book written by the philosopher Voltaire and co-authored by mathematician and physicist Émilie du Châtelet in 1738 that helped to popularize the theories and thought of Isaac Newton. This book, coupled with Letters on the English, written in 1733, demonstrated that Voltaire had moved beyond the simple poetry and plays he had written previously.

Publisher from Kehl’s Preface

[The Publisher from Kehl’s Preface pertains to all of Voltaire’s works on physics. However, we only reproduce here the passage relating to the Elements of Newtonian Philosophy.]

When Voltaire wrote his Elements of Newtonian Philosophy, nearly all French scholars were Cartesian: Maupertuis and Clairaut, both geometers from the Academy of Sciences, but very young at the time, were almost the only Newtonians known to the public.

The bias for Cartesianism was such that Chancellor d’Aguesseau refused a privilege to Voltaire. Forty years earlier, the philosophy of Descartes had been banned in the schools of Paris, and the example of what had happened had not been sufficient to teach that opposing the progress of reason was in vain, and that to judge Newton like Descartes, one must at least be in a position to understand them.

Voltaire’s work was useful: it contributed to making Newton’s philosophy as intelligible as it can be for those who are not geometers.

He took care not to try to enhance these Elements with foreign adornments: only he scattered reflections of a just and moderate philosophy, presented in a piquant manner, a common characteristic of all his works.

He always stood against the misuse of humor in physical discussions. The ingenious Fontenelle had set the example1; Pluche and Castel made its abuse felt2. Some time later, Voltaire was forced to equally rise against another perhaps greater defect, the mania for writing about science in poetic prose. This abuse is more dangerous. The bad jokes of Castel or Pluche can only amuse colleges and perpetuate some prejudices; the abuse of eloquence, on the other hand, can suspend the progress of philosophy.

Three philosophers then shared the honor in Europe of having recalled the enlightenment, Descartes, Newton, and Leibnitz; and those who had not delved deeply into the sciences placed Malebranche almost on the same level.

Descartes was a very great geometer. The idea, so happy and so vast, of applying general quantity analysis to geometric questions changed the face of mathematics; and he shared this glory with none of the geometers of his time, who were very prolific in men endowed with a great genius for mathematics, such as Cavalleri, Pascal, Fermat, and Wallis.

Even if Descartes owed to Snellius the knowledge of the fundamental law of dioptrics, which is far from proven, this discovery had remained absolutely sterile in the hands of Snellius; and Descartes derived from it the theory of spectacles: we owe to him that of mirrors and glasses, whose surfaces would be formed by arcs of conic sections. He discovered, independently of Galileo, the general laws of motion, and developed them better than him; he erred on those of the impact of bodies, but he was the first to imagine seeking them, and he showed what principles should be employed in this research. He is especially owed for having banished from physics everything that could not be reduced to mechanical or calculable causes, and from philosophy the use of authority.

Newton has the unique honor, so far, of having discovered one of the general laws of nature; and, although Galileo’s research on uniformly accelerated motion, those of Huygens on central forces in the circle, and especially the theory of the developed, which allowed considering the elements of curves as arcs of a circle, had opened the way for him, this discovery must place his glory above that of philosophers or geometers who might have had a genius equal to his. Kepler had only found the laws of the motion and celestial bodies; and Newton found the general law of nature on which these rules depend. The discovery of differential calculus places him at the forefront of the geometers of his century; and his discoveries on light, at the head of those who have sought in experience the means to know the laws of phenomena.

Leibnitz disputed with Newton the glory of having found differential calculus; and, examining the documents of this great case, one cannot unjustly deny Leibnitz at least complete equality. Note that these two great men were content with equality, rendered justice to each other, and that the dispute that arose between them was the work of the zeal of their disciples. The calculus of exponential quantities, the method of differentiating under the sign, several other discoveries found in the letters of Leibnitz, and to which he seemed to attach little importance, prove that, as a geometer, he did not yield in genius to Newton himself. His ideas on the geometry of situations, his essays on the game of solitaire, are the first features of a new science that can be very useful, but which has yet made only little progress, although learned geometers have been occupied with it. He did little in physics, although he knew all the known facts of his time, and even all the opinions of physicists, because he did not think of conducting new experiments. He is the first who imagined a general theory of the earth, formed according to observed facts, and not according to theological dogmas; and this essay is far superior to everything that has been done since in this genre.

His genius embraced the whole extent of human knowledge; metaphysics attracted him; he believed he could assign the principles of appropriateness that had presided over the construction of the universe. According to him, God, by his very essence, is necessitated not to act without a sufficient reason, to preserve in nature the law of continuity, not to produce two rigorously similar beings, because there would be no reason for their existence; since he is supremely good, the universe must be the best of possible universes; supremely wise, he regulates this universe by the simplest laws. If all phenomena can be conceived, assuming only simple substances, there is no need to suppose composed ones, nor consequently of extent, susceptible to indefinite division. But simple beings, provided that one supposes them to have an active force, are capable of producing all the phenomena of extent, all those presented by moving bodies.

Some simple beings have ideas: such are human souls. All will therefore be susceptible to having them; but their ideas will be distinct or confused, depending on the order that these beings occupy in the universe. The soul of Newton, the element of a block of marble, are substances of the same nature: one has sublime ideas, the other only confused ones.

This element, placed in another place, in the course of time, may become a reasonable soul. It is not by virtue of its nature that the soul acts on the monads that make up the body, and these on the soul; but, by virtue of the eternal laws, the soul must have certain ideas, the body’s monads certain movements. These two series of phenomena can be independent of each other: they are therefore, since a real dependence is unnecessary to the order of the universe.

These ideas are great and vast; one can only admire the genius who conceived their order and ensemble; but it must be admitted that they are devoid of proofs, that we know nothing in nature, except the succession of facts that it presents to us, and these facts are too few in number for us to be able to guess the general system of the universe. From the moment we step out of our abstract ideas and the truths of definition to examine the tableau presented by the succession of our ideas, what is for us the universe, we can find, with more or less probability, a constant order in each part; but we cannot grasp the whole, and never, no matter how much progress we make, will we know it entirely.

Leibnitz was also a profound publicist, a learned jurist, a scholar of the first order. He embraced everything in the historical sciences, politics, as in metaphysics and natural sciences; everywhere he brings the same spirit, seeking general truths, subjecting to a systematic order the objects most dependent on opinion, and which seem to refuse it the most.

Malebranche was only a disciple of Descartes; superior to his master when he explains the errors of the senses and of the imagination, a more perfect model of a noble style, simple, animated by the sole love of truth, without other ornaments than the greatness or the finesse of ideas. This style, the only eloquence that suits the sciences, in works made to enlighten men, and not to amuse the multitude, was that of Bacon, Descartes, Leibnitz. But Malebranche, writing in his natural language, and when language and taste were perfected, can alone, among the writers of the last century, be regarded as a model; this is almost all his merit today, and France, more enlightened, no longer places him alongside Descartes, Leibnitz, and Newton.

After these great men, Kepler was admired, who discovered the laws of the movement of the planets; Galileo, who calculated the laws of the fall of bodies and their movement in the parabola, perfected the spectacles3, discovered the satellites of Jupiter and the phases of Venus, established the true system of celestial bodies on unshakable foundations, and was persecuted by ignorant theologians, and by the Jesuits, who did not forgive him for being a better astronomer than the professors of the grand Gesù; Huygens finally, to whom we owe the theory of central forces, which led to the method of calculating movement in curves, the discovery of the centers of oscillation, the theory of the art of measuring time, the discovery of the ring of Saturn, and that of the laws of the impact of bodies. He was the man of his century who, by the strength and type of his genius, came closest to Newton, of whom he was the forerunner.

Voltaire here does justice to all these illustrious men; he respects the genius of Descartes and Leibnitz, the good that Descartes has done for men, the service he rendered in freeing the human spirit from the yoke of authority, as Newton and Locke cured it of the mania of systems; but he allowed himself to attack Descartes and Leibnitz, and there was courage in a time when France was Cartesian, where Leibnitz’s ideas reigned in Germany and the North.

This work should be regarded as an exposition of the main discoveries of Newton, very clear and sufficient for those who do not wish to follow demonstrations and details of experience.

When it appeared, it was useful even to scholars; there was still nowhere such a precise tableau of these important discoveries; most physicists fought them without knowing them. Voltaire contributed more than anyone to the fall of Cartesianism in the schools, by popularizing the new truths that had destroyed the errors of Descartes: and when the author of Alzire deigned to write an elementary book on physics, he was entitled to the gratitude of his country, which he enlightened; to that of the scholars, who should see in this work only a tribute to the sciences and their utility by the leading man of literature.

The response to some objections made against the Philosophy of Newton4 proves how little known Newton’s philosophy was then, and consequently how useful Voltaire’s enterprise was. We will note that, in Voltaire’s old age and after his death, the same objections were repeated: so true is it that he then had only enemies who were well below their century.

Preface by Beuchot

The first publication of the Elements of Newtonian Philosophy dates back to April 1738; and it is evident from Voltaire’s correspondence that this was done without the author’s knowledge5. A few months later, the reprint made under the author’s supervision appeared. However, what was published then was only a part of the work as it is today. Other parts did not appear until 1740 and 1741.

Voltaire, having taken refuge in Holland in 1736, handed the first chapters of the Elements of Newtonian Philosophy to the bookseller Lodet. He left Holland without having provided the end of the manuscript. The bookseller had the work completed by a local mathematician and sold the volume containing twenty-five chapters, after adding to the title given by Voltaire the words: Made accessible to everyone, which led to a poor joke. It was said that there was a typographical error in the title, and it should read: Thrown out the door of everyone6.

Voltaire responded with Clarifications7 that he sent to various journals, and decided to have his book printed in France. However, this required a permission known as a privilege. The author wanted to add a first part containing Metaphysics. It was primarily because of this section that Chancellor d’Aguesseau refused the privilege. He nevertheless granted a tacit permission for what had been printed in Holland, that is, to reprint it in France, but under the name of a foreign country: Voltaire prefaced the work with the Clarifications I have mentioned, and added a Chapter XXVI on tides8.

The booksellers in Holland reprinted these additions to add to the copies they had left.

The Dutch mathematician had a piece titled The Truth Discovered published in the Historical Memoirs, July 1738, to which Voltaire’s letter of August 30, 1738, in his correspondence, responds.

Father Regnault, a Jesuit, is the author of the Letter (anonymous) from a physicist on the philosophy of Newton made accessible to everyone, 1738, in-12 of 46 pages.

The author, still unknown, of Reflections on the philosophy of Newton, etc., 1738, in-12 of 82 pages, responds to Father Regnault’s critique, but on other points, contests Voltaire’s opinion.

Jean Banières published Examination and Refutation of the Elements of Newtonian Philosophy, 1739, a large in-8o volume, which led to the writing by Leratz de Lanthenée titled: Examination and Refutation of some opinions on the causes of reflection and refraction of light, spread in the work of Mr. Banières, in-8o of 50 pages.

A few other people skirmished in journals, and Voltaire published a Response to the main objections, etc., which can be found under its date (1739).

The following year, Voltaire had the Metaphysics of Newton printed in Holland, which L.-M. Kahle critiqued in German. See later (year 1744) the pamphlet titled Short Response to the Long Discourses of a German Doctor.

In 1741, Voltaire released in France, but under the address of London, a completely revised edition of the Elements of Newtonian Philosophy. He had divided it into three parts: the first, comprising Metaphysics (published in 1740); the second and third consisting (in 14 and 16 chapters) of what formed the entire work in 1738, that is, physics. The chapters by the Dutch mathematician had been replaced by pieces by Voltaire.

In 1748, Voltaire revisited his work to form volume VI of the edition of his Works that appeared in Dresden, at Conrad Walther’s. In the second part, he omitted chapter XIV; in the third, chapters X and XI. I have preserved these three chapters in notes or variants.

The deletions were no less considerable when, in 1756, Voltaire revisited his book again for the first edition that the Cramer brothers published of the Collection of his Works. This edition was expanded, in the first part, by the chapter titled Doubts on the freedom called of indifference. However, chapters XII, XIII, and XIV of the third part are no longer found, which makes the theory of the planetary system incomplete.

Mr. Lacroix, a member of the Institute, whose insights I sought, believes that the deleted chapters containing some fairly serious errors (I have pointed out two in chapter XII), and many numbers that discoveries by astronomers and geometers had significantly changed, the author preferred to remove these chapters rather than correct or redo them. The works of Euler, d’Alembert, and Clairaut having already perfected Newton’s theory and spread his philosophy, Voltaire no longer felt the same interest in arid details, which had become completely foreign to his habits; and he seems to indicate this quite clearly in the passage that replaced, in 1756, the deleted chapters:

“We will not pursue further research on gravitation here. This doctrine was still very new when the author presented it in 1736. It is no longer so, and one must conform to the times. The more enlightened men have become, the less need there is to write.”

I thought it necessary to include this passage, which I have nonetheless preserved as variants in my note at the end of chapter XI of the third part.

Following the example of some recent editors9, I have nonetheless reproduced the three chapters deleted in 1756. My work differs from theirs mainly in that, instead of amalgamating the chapters from various editions, I have adhered solely to the text from the 1748 edition for the main text. It is in notes or variants that I have included what belongs to the editions of 1738, 1741, and 1756. The chapters from 1741, placed in variants at the end of chapter IX of the third part, had not been collected by any editor.

B.

Paris, January 9, 1830.

Author’s dedicatory letter to Madame du Châtelet

DEDICATORY EPISTLE TO MADAME LA MARQUISE DU CHÂTELET10

Madame,

When I first placed your respectable name at the forefront of these Elements of Philosophy, I was learning alongside you. However, since then you have soared to heights I can no longer follow. I now find myself in the position of a grammarian who would have presented a rhetorical essay to Demosthenes or Cicero. I offer simple Elements to one who has penetrated all the depths of transcendental geometry and who alone among us has translated and commented on the great Newton.

This philosopher gathered all the glory he deserved during his lifetime; he did not incite envy because he could have no rival. The learned world was his disciple, the rest admired him without daring to claim to understand him. But the honor you do him today is undoubtedly the greatest he has ever received. I am unsure whom I should admire more, Newton, the inventor of the calculus of infinity, who discovered new laws of nature, and who dissected light, or you, madame, who amidst the distractions attached to your station, possess so well everything he invented. Those who see you at court would certainly not take you for a philosopher’s commentator; and scholars who are learned enough to read you would be even less likely to guess that you descend to the amusements of this world with the same ease that you rise to the most sublime truths. This naturalness and simplicity, always so estimable, but so rare with talents and with science, will at least ensure that your merit is forgiven. In general, this is all one can hope for from those with whom one spends one’s life; but the small number of superior minds who have applied themselves to the same studies as you will hold you in the highest regard, and posterity will look upon you with astonishment. I am not surprised that persons of your sex have gloriously reigned over great empires: a woman with good counsel can govern like Augustus; but to penetrate tirelessly into truths whose approach intimidates most men, to delve in your leisure hours into what the deepest philosophers study incessantly, is given only to you, madame, and is an example that will be seldom imitated. Etc.

Part One. Metaphysics.

Chap. I. — On God. Reasons not appreciated by all minds. Materialists' reasons.

Newton was deeply convinced of the existence of God, and by this term, he meant not only an infinite, all-powerful, eternal, and creator being but a master who has established a relationship between himself and his creatures: for, without this relationship, the knowledge of a God is but a sterile idea that seems to invite crime through the hope of impunity, a notion any inherently perverse reasoner might entertain.

Thus, this great philosopher makes a remarkable observation at the end of his principles. It is that one does not say my eternal, my infinite, because these attributes have nothing relative to our nature; but one says, and should say my God, and by this, one must understand the master and conservator of our life, and the object of our thoughts. I remember that in several discussions I had in 1726 with Dr. Clarke, this philosopher never pronounced the name of God but with an air of recollection and very remarkable respect. I confessed to him the impression this made on me, and he told me that it was from Newton that he had insensibly taken this custom, which should indeed be that of all men.

All of Newton’s philosophy necessarily leads to the knowledge of a supreme Being who has freely created and arranged everything. For, if according to Newton (and according to reason) the world is finite, if there is a void, then matter does not necessarily exist, it has thus received existence from a free cause. If matter gravitates, as has been demonstrated, it does not gravitate by its nature, just as it is extended by its nature: it has therefore received gravitation from God11. If the planets rotate in one direction rather than another, in a non-resisting space, then the hand of their creator has thus directed their course in that direction with absolute freedom.

Far from it that the so-called physical principles of Descartes lead the mind to the knowledge of its Creator. God forbid that by a horrible slander I accuse this great man of having misunderstood the supreme intelligence to which he owed so much, and which had raised him above almost all the men of his century! I only say that the misuse he sometimes made of his mind led his disciples to precipices far from their master; I say that the Cartesian system produced that of Spinoza; I say that I have known many people whom Cartesianism has led to admit no other God than the immensity of things, and conversely, I have seen no Newtonian who was not a theist in the strictest sense.

Once you are persuaded, with Descartes, that it is impossible for the world to be finite, that motion is always in the same quantity; once you dare say: Give me motion and matter, and I will create a world; then, you must admit, these ideas seem to exclude, by too just consequences, the idea of a being alone infinite, sole author of motion, sole author of the organization of substances.

Several people here may be surprised that among all the proofs of the existence of God, the one from final causes was the strongest in Newton’s eyes. The design, or rather the infinitely varied designs that shine in the vastest and smallest parts of the universe, provide a demonstration that, by being so palpable, is almost despised by some philosophers; but Newton thought that these infinite relationships, which he perceived more than another, were the work of an infinitely skillful artisan12.

He did not much appreciate the great proof derived from the succession of beings. It is commonly said that if humans, animals, plants, everything that composes the world, were eternal, one would be forced to admit a series of generations without cause. These beings, it is said, would have no origin of their existence: they would have no external origin, since they are assumed to trace back generation after generation, without beginning; they would have no internal origin, since none of them would exist by themselves. Thus, everything would be effect, and nothing would be cause.

He found that this argument was based only on the equivocation of generations and beings formed by one another: for atheists, who admit the plenum, respond that, strictly speaking, there are no generations, there are no produced beings, there are not multiple substances. The universe is a whole, existing necessarily, which perpetually unfolds; it is the same being whose nature is to be immutable in its substance, and eternally varied in its modifications; thus the argument drawn solely from successive beings would perhaps prove little against the atheist, who would deny the plurality of beings. The atheist would call to his aid those ancient axioms that nothing comes from nothing, that one substance cannot produce another, that everything is eternal and necessary. It would thus be necessary to fight him with other weapons; one would have to prove to him that matter cannot of itself have any motion; one would have to make him understand that if it had the slightest motion by itself, this motion would be essential to it: it would then be contradictory that there was any rest. But, if the atheist responds that there is nothing at rest, that rest is a fiction, an idea incompatible with the nature of the universe; that an infinitely fine matter eternally circulates in all the pores of bodies; if he maintains that there are always equal motive forces in nature, and that this permanent equality of forces seems to prove a necessary movement; then one must still resort against him to other weapons, and he may prolong the fight: in short, I do not know if there is any metaphysical proof more striking, and which speaks more powerfully to man than this admirable order that reigns in the world; and if there has ever been a more beautiful argument than this verse: The heavens declare the glory of God. Also, you see that Newton brings no other at the end of his Optics and his Principles. He found no reasoning more convincing and more beautiful in favor of the Divinity than that of Plato, who has one of his interlocutors say: You judge that I have an intelligent soul because you perceive order in my words and my actions; judge then, by seeing the order of this world, that there is a supremely intelligent soul.

If it is proven that there exists an eternal, infinite, all-powerful being, it is not similarly proven that this Being is infinitely benevolent in the sense that we give to this term.

This is the great refuge of the atheist: If I admit a God, he says, this God must be goodness itself: whoever gave me being owes me well-being; yet I see in the human race only disorder and calamity; the necessity of eternal matter repels me less than a Creator who treats his creatures so badly. One cannot satisfy, he continues, my just complaints and my cruel doubts by telling me that a first man, composed of a body and a soul, angered the Creator, and that the human race bears the penalty: for first, if our bodies come from this first man, our souls do not come from him, and even if they could come from him, the punishment of the father in all the children appears to be the most horrible of all injustices; secondly, it seems evident that the Americans and the peoples of the old world, the Negroes and the Laplanders, are not descended from the first man. The internal constitution of the organs of Negroes is palpable proof of this; no reason can therefore appease the murmurs that rise in my heart against the evils with which this globe is flooded. I am thus forced to reject the idea of a supreme Being, of a Creator whom I would conceive as infinitely good, and who would have made infinite evils, and I prefer to admit the necessity of matter, and of generations, and of eternal vicissitudes, rather than a God who would have freely made the unhappy.

This atheist is answered: The word good, well-being, is equivocal. What is bad in relation to you is good in the general arrangement. The idea of an infinite, all-powerful, all-intelligent being present everywhere, does not revolt your reason: will you deny a God, because you had a bout of fever? He owed you well-being, you say; what reason do you have to think so? Why did he owe you this well-being? What treaty had he made with you? So you lack only being always happy in life to recognize a God? You, who cannot be perfect in anything, why would you claim to be perfectly happy? But I suppose that, in a continuous happiness of a hundred years, you have a headache: will this moment of pain make you deny a Creator? It seems unlikely. Well, if a quarter-hour of suffering does not stop you, why would two hours, why a day, why a year of torment, make you reject the idea of a supreme and universal artisan?

It is proven that there is more good than evil in this world, since indeed few men wish for death; you are therefore wrong to bring complaints on behalf of humanity, and even more wrong to deny your sovereign on the pretext that some of his subjects are unhappy. When you have examined the relationships found in the springs of an animal, and the designs that shine from all sides in the way this animal receives life, sustains it, and gives it, you readily recognize this sovereign artisan: will you change your sentiment because wolves eat sheep, and spiders catch flies? Do you not see, on the contrary, that these continual generations, always devoured and always reproduced, fit into the plan of the universe? I see skill and power, you reply, and I see no goodness. But what? when in a menagerie you raise animals that you slaughter, you do not want to be called wicked, and you accuse of cruelty the master of all animals, who made them to be eaten in their time? Finally, if you can be happy throughout eternity, are a few pains in this passing instant called life worth speaking about?

You do not find that the Creator is good, because there is evil on earth. But would necessity, which would take the place of a supreme Being, be something better? In the system that admits a God, there are only difficulties to overcome, and in all other systems there are absurdities to swallow.

Philosophy shows us well that there is a God; but it is powerless to teach us what he is, what he does, how and why he does it.

It seems to me that one would have to be him to know.

Chap. II. — On space and duration, as properties of God. — Leibnitz’s view. Sentiment and reasons of Newton. Infinite matter impossible. Epicurus should have admitted a creator and governor God. Properties of space and duration.

Newton considered space and duration as two entities whose existence necessarily follows from God Himself: for the Infinite Being is in all places, therefore all place exists; the Eternal Being endures for all eternity: therefore an eternal duration is real.

Newton had remarked at the end of his Optics queries: Do not these phenomena of nature show that there is a living, intelligent, incorporeal being present everywhere, who in infinite space, as in his sensorium, sees, discerns, and understands everything in the most intimate and most perfect manner?

The famous philosopher Leibnitz, who had previously recognized with Newton the reality of pure space and duration but who had long ceased to hold any of Newton’s views and had established himself in Germany as the leader of an opposing school, attacked these expressions of the English philosopher in a letter he wrote in 1715 to the late Queen of England, wife of George II; this princess, worthy of corresponding with Leibnitz and Newton, engaged a formal dispute by letters between the two parties. But Newton, averse to all disputes and frugal with his time, let Dr. Clarke, his disciple in physics and at least his equal in metaphysics, enter the lists for him. The dispute touched on almost all of Newton’s metaphysical ideas: and it is perhaps the finest monument we have of literary combats.

Clarke began by justifying the comparison taken from the sensorium13, which Newton had used; he established that no being can act, know, see where it is not: thus God, acting, seeing everywhere, acts and sees in all points of space, which in this sense alone can be considered as his sensorium, given the impossibility in any language of expressing oneself when one dares to speak of God.

Leibnitz argued that space is nothing but the relation we conceive between coexisting beings, nothing but the order of bodies, their arrangement, their distances, etc. Clarke, following Newton, argued that if space were not real, it would lead to an absurdity: for if God had placed the earth, the moon, and the sun where the fixed stars are, provided that the earth, the moon, and the sun were among themselves in the same order where they are, it would follow from there that the earth, the moon, and the sun would be in the same place where they are today, which is a contradiction in terms.

According to Newton, one must think of duration as of space, that it is a very real thing: for if duration were only an order of succession among creatures, it would follow that what is done today and what was done thousands of years ago would be in themselves done in the same instant, which is again contradictory.

Finally, space and duration are quantities: therefore, they are something very positive.

It is good to pay attention to this ancient argument, which has never been answered. Let a man at the bounds of the universe extend his arm, this arm must be in pure space: for it is not in nothing; and if one responds that it is still in matter, the world, in this case, is therefore infinite, the world is therefore God.

Pure space, the void exists, as well as matter, and it exists even necessarily, whereas matter exists only by the free will of the Creator.

But, it will be said, you admit an immense infinite space; why not do the same with matter? Here is the difference. Space exists necessarily, because God exists necessarily; it is immense, it is, like duration, a mode, an infinite property of an infinite necessary being. Matter is none of this: it does not exist necessarily; and if this substance were infinite, it would be, or an essential property of God, or God Himself; however, it is neither one nor the other: it is therefore not infinite, and cannot be.

I will insert here a remark that seems to me to deserve some attention.

Descartes admitted a creator God, and cause of everything; but he denied the possibility of the void. Epicurus denied a creator God, and cause of everything, and he admitted the void; now it was Descartes who by his principles should have denied a creator God, and it was Epicurus who should have admitted one. Here is the evident proof.

If the void were impossible, if matter were infinite, if extent and matter were the same thing, it would have to be that matter was necessary; now if matter was necessary, it would exist by itself of an absolute necessity, inherent in its nature, primordial, antecedent to everything: therefore it would be God, therefore he who admits the impossibility of the void must, if he reasons consistently, admit no other God than matter.

On the contrary, if there is a void, then matter is not a necessary being, existing by itself, etc.: for what is not in all places cannot exist necessarily in any place. Therefore matter is a non-necessary being, therefore it has been created, therefore it was for Epicurus to believe, I do not say in useless gods, but in a creator and governor God; and it was for Descartes to deny him. Why, on the contrary, did Descartes always speak of the existence of a creator and conservator being, and Epicurus reject him? It is because men, in their sentiments as in their conduct, rarely follow their principles, and both their systems, as well as their lives, are contradictions.

Space is a necessary consequence of the existence of God; God is, strictly speaking, neither in space nor in a place; but God, being necessarily everywhere, constitutes by that fact the immense space and the place: similarly, duration, eternal permanence is an indispensable consequence of the existence of God. He is neither in infinite duration nor in a time; but, existing eternally, he constitutes by that eternity and time.

The immense extended space, inseparable, can be conceived in several portions: for example, the space where Saturn is is not the space where Jupiter is; but these conceived parts cannot be separated; one cannot put one in place of another, as one can put one body in place of another.

Similarly, infinite duration, inseparable and without parts, can be conceived in several portions, without ever being able to conceive one portion of duration placed in place of another. Beings exist in a certain portion of duration, called time, and can exist in any other time; but a conceived part of duration, any time, cannot be elsewhere than it is; the past cannot be future.

Space and duration are two necessary, immutable attributes of the Eternal and Immense Being.

God alone can know all space. God alone can know all duration. We measure some improperly called parts of space by means of the extended bodies we touch; we measure parts improperly called of duration by means of the movements we perceive.

This chapter does not go into the details of the physical proofs reserved for other chapters; it is enough to note that in everything that regards space, duration, the boundaries of the world, Newton followed the ancient opinions of Democritus, Epicurus, and a host of philosophers rectified by our celebrated Gassendi. Newton told several Frenchmen, who are still alive, that he regarded Gassendi as a very just and very wise mind, and that he was proud to be entirely of his opinion in all the things just discussed.

Chap. III. — On freedom in God, and the great principle of sufficient reason. — Leibnitz’s principles, perhaps pushed too far. His seductive reasoning. Response. New cases against the principle of indiscernibles.

Newton maintained that God, being infinitely free as well as infinitely powerful, has done many things for which the only reason for their existence is His sole will.

For example, that the planets move from west to east rather than otherwise; that there is such a number of animals, stars, worlds, rather than another; that the finite universe is at one point in space rather than another, etc.: the will of the Supreme Being is the only reason for these.

The famous Leibnitz claimed the opposite, based on an ancient axiom once used by Archimedes: Nothing happens without a cause or without sufficient reason, he said, and God did everything in the best way possible because if He had not done it as the best, He would not have had a reason to do it. But there is no best in indifferent things, the Newtonians said; but there are no indifferent things, the Leibnitzians respond. Your idea leads to absolute fate, Clarke said; you make of God a being who acts out of necessity, and consequently a purely passive being: that is no longer God. Your God, Leibnitz retorted, is a capricious workman, who decides without sufficient reason. God’s will is the reason, the Englishman replied. Leibnitz insisted and made very strong attacks in this manner.

We do not know two entirely similar bodies in nature, and there cannot be any: for if they were similar, first, this would denote in God all-powerful and all-fruitful a lack of fecundity and power. Secondly, there would be no reason why one would be in this place rather than the other.

The Newtonians responded:

Firstly, it is false that several similar beings indicate sterility in the power of the Creator: for if the elements of things must be absolutely similar to produce similar effects; if, for example, the elements of the eternally red rays of light must be the same to give these red rays; if the elements of water must be the same to form water; this perfect resemblance, this identity, far from detracting from the greatness of God, is one of the most beautiful testimonies of His power and wisdom.

If I dared here to add something to the arguments of a Clarke and a Newton, and take the liberty to dispute against a Leibnitz, I would say that only an Infinitely Powerful Being can make perfectly similar things. No matter how much effort a man makes to create such works, he can never succeed, because his sight will never be fine enough to discern the inequalities of the two bodies; one must therefore see down to the infinite smallness to make all the parts of one body similar to those of another. This is therefore the unique prerogative of the Infinite Being.

Secondly, the Newtonians might still say, we fight Leibnitz with his own weapons. If the elements of things are all different, if the first parts of a red ray are not entirely similar, there is no longer a sufficient reason why different parts always give a constant color.

Thirdly, the Newtonians might say, if you ask for the sufficient reason why this atom A is in one place, and this atom B, entirely similar, is in another place, the reason lies in the movement that pushes them; and if you ask what is the reason for this movement, you are either forced to say that this movement is necessary, or you must admit that God started it. If you finally ask why God started it, what other sufficient reason can you find, except that God needed to order this movement to carry out the works planned by His wisdom? But why this movement to the right rather than to the left, towards the west rather than towards the east, at this point in duration rather than at another? Must we not then resort to the will of indifference in the Creator? This is what is left to be examined by any impartial reader.

Chap. IV. — On freedom in man. Excellent work against freedom; so good, that Dr. Clarke responded with insults. Freedom of indifference. Freedom of spontaneity. Lack of freedom, a very common thing. Powerful objections against freedom.

According to Newton and Clarke, the infinitely free Being has communicated to man, his creature, a limited portion of this freedom; and here freedom is not simply understood as the power to apply one’s thought to this or that object, and to initiate movement; it is not merely the faculty to will, but the faculty to will very freely with a full and effective will, and even sometimes to will without any reason other than one’s own will. There is no man on earth who does not sometimes feel that he possesses this freedom. Several philosophers think oppositely; they believe that all our actions are necessitated, and that our only freedom is that of sometimes willingly wearing the chains to which fate binds us.

Of all the philosophers who have boldly written against freedom, the one who unquestionably did so with more method, force, and clarity is Collins, a magistrate from London, author of the book On the Freedom of Thought, and several other works as bold as they are philosophical.

Clarke, who was entirely in agreement with Newton on the matter of freedom and who also defended its rights as much as a theologian of a peculiar sect as a philosopher, responded sharply to Collins, mixing so much bitterness with his reasons that he seemed to at least acknowledge the full force of his opponent. He accuses Collins of confusing all ideas because Collins calls man a necessary agent. He says that in this case man is not an agent; but who does not see that this is a mere quibble? Collins calls a necessary agent anything that produces necessary effects. Whether one calls it an agent or a patient, what does it matter? The point is to know if it is necessarily determined.

It seems that if one can find just one case where man is truly free with the freedom of indifference, that alone should suffice to decide the question. Now what case shall we take, if not one where we wish to test our freedom? For example, I am asked to turn to the right or to the left, or to perform some other action, to which no pleasure draws me, and from which no disgust repels me. I then choose, and I do not follow the dictamen of my understanding, which represents the best to me: for there is here neither better nor worse. What then do I do? I exercise the right given to me by the Creator to will and act in certain cases without any reason other than my own will itself. I have the right and the power to initiate movement, and to initiate it in the direction I want. If no other cause of my will can be assigned in this case, why look for it elsewhere than in my own will? It therefore seems probable that we have the freedom of indifference in indifferent matters. For who can say that God has not made, or could not have made this gift to us? And if He could, and if we feel within us this power, how can we assert that we do not have it?

I have often heard this freedom of indifference described as a chimera; it is said that determining oneself without reason would only be the lot of fools; but it is not considered that fools are sick people who have no freedom. They are necessarily determined by the vice of their organs; they are not masters of themselves, they choose nothing. He who determines himself is free. Now why should we not determine ourselves by our own will alone in indifferent things?

We possess the freedom that I call spontaneity in all other cases, that is, when we have motives, our will is determined by them, and these motives are always the final result of the understanding or instinct: thus, when my understanding represents to me that it is better for me to obey the law than to violate it, I obey the law with spontaneous freedom, I voluntarily do what the last dictamen of my understanding obliges me to do.

We never feel this kind of freedom more clearly than when our will combats our desires. I have a violent passion, but my understanding concludes that I must resist this passion; it presents to me a greater good in victory than in yielding to my taste. This last motive prevails over the other, and I combat my desire with my will; I necessarily obey, but willingly, this order of my reason; I do, not what I desire, but what I want, and in this case I am free with all the freedom that such a circumstance can allow me.

Finally, I am free in no sense, when my passion is too strong and my understanding too weak, or when my organs are disturbed; and unfortunately, this is very often the case with people: thus it seems to me that spontaneous freedom is to the soul what health is to the body; some people have it entirely and lastingly; many lose it often, others are sick all their life; I see that all the other faculties of man are subject to the same inequalities. Vision, hearing, taste, strength, the gift of thinking, are sometimes stronger, sometimes weaker; our freedom is like everything else, limited, variable, in a word, very little, because man is very little.

The difficulty of reconciling the freedom of our actions with God’s eternal foreknowledge did not stop Newton, because he did not engage in this labyrinth; once freedom is established, it is not for us to determine how God foresees what we will do freely. We do not know in what manner God currently sees what is happening. We have no idea of His way of seeing, why would we have any of His way of foreseeing? All His attributes must be equally incomprehensible to us.

It must be admitted that there are objections raised against this idea of freedom that are daunting.

First, it is seen that this freedom of indifference would be a very frivolous gift if it only extended to spitting to the right or left, and to choosing odd or even. What matters is that Cartouche and Sha-Nadir have the freedom not to shed human blood. It matters little whether Cartouche and Sha-Nadir are free to advance the left foot or the right foot.

Then, this freedom of indifference is found impossible: for how to determine oneself without reason? You want; but why do you want? You are offered odd or even, you choose odd, and you do not see the reason; but your reason is that odd presents itself to your mind at the moment you need to make a choice.

14Everything has its cause: your will therefore has one. You can only want as a result of the last idea you received.

No one can know what idea he will have in a moment: therefore, no one is the master of his ideas, therefore no one is the master of willing or not willing.

If one were the master, one could do the opposite of what God has arranged in the chain of things in this world. Thus, each man could change, and would indeed change at every moment the eternal order.

That is why the wise Locke does not dare to pronounce the name of freedom; a free will seems to him but a chimera. He knows no other freedom than the power to do what one wants. The gout sufferer does not have the freedom to walk, the prisoner does not have the freedom to leave: one is free when he is cured; the other, when the door is opened for him.

To shed more light on these horrible difficulties, I suppose that Cicero wants to prove to Catilina that he should not conspire against his country. Catilina tells him that he is not the master of it; that his last conversations with Céthégus have imprinted in his head the idea of conspiracy; that this idea pleases him more than another, and that one can only want as a consequence of one’s last judgment. But you could, Cicero would say, take other ideas with me, apply your mind to listen to me and see that one must be a good citizen. I try in vain, replies Catilina; your ideas revolt me, and the desire to assassinate you prevails. I pity your frenzy, Cicero would say; try to take some of my remedies. If I am frenzied, retorts Catilina, I am not the master of trying to cure myself. But, the consul says, men have a foundation of reason that they can consult, and which can remedy this disorder of organs that makes you perverse, especially when this disorder is not too strong. Show me, replies Catilina, the point at which this disorder can yield to the remedy. For my part, I confess that from the first moment I conspired, all my reflections have led me to conspiracy. When did you begin to take this fatal resolution? the consul asks. When I lost my money at gambling. Well! couldn’t you have refrained from gambling? No; for the idea of gambling prevailed in me that day over all other ideas; and if I had not gambled, I would have disrupted the order of the universe, which held that Quarsilla would win four hundred thousand sesterces from me, that she would buy a house and a lover, that a son would be born from this lover, that Céthégus and Lentulus would come to my house, and that we would conspire against the republic. Fate made me a wolf, and it made you a shepherd dog; fate will decide which of the two must strangle the other. To this Cicero would have replied only with a Catiline; indeed, one must admit that one can hardly respond except with vague eloquence to the objections against freedom: a sad subject on which even the wisest fears to dare to think.

One consolation remains: whatever system we embrace, to whatever fatality we believe all our actions are attached, we will always act as if we were free.15

The Kehl editors added the following note:

“Whatever stance one takes on this thorny question, it is impossible not to agree that, in the actions called free, man is aware of the motives that make him act. He can therefore know which actions conform to justice, to the general interest of men, and the motives he might have for performing these actions, and avoiding those that are contrary. These motives act upon him: there is therefore a morality. The hope for rewards, the fear of punishments are among these motives; these feelings can therefore be useful; punishments and rewards can therefore be just. If he yielded to an unjust motive, he will be sorry when this motive ceases to act with the same force: he will repent, he will have remorse. He will believe that, warned by his experience, this motive will not have the power to lead him astray another time; he will therefore promise himself not to fall again. Thus whatever system one takes on freedom, not excluding the most absolute fatalism, the moral consequences will be the same. Indeed, according to fatalism, every man was predestined to do all the actions he has done; but when he determines, he does not know which of the two actions he proposes to himself he must determine: he only knows that it is to the one for which he believes he sees the most powerful motives.”

In editions containing this chapter, the first part has ten chapters. (B.)

Chap. V. — On natural religion. Leibnitz’s unfounded reproach to Newton. Refutation of a sentiment by Locke. The good of society. Natural religion. Humanity.

Leibnitz, in his dispute with Newton, accused him of holding very low ideas of God and of annihilating natural religion.

He claimed that Newton made God corporeal, and this accusation, as we have seen16, was based on the term sensorium. He added that Newton’s God had made this world a very poor machine, which needed to be cleaned (Leibnitz’s term). Newton had said: Manum emendatricem desideraret.

This reproach is based on Newton’s statement that over time the motions will decrease, the irregularities of the planets will increase, and the universe will perish or be reordered by its author.

Experience makes it abundantly clear that God has made machines to be destroyed. We are the work of His wisdom, and we perish; why should it not be the same for the world? Leibnitz wants this world to be perfect; but if God formed it only to last for a certain time, its perfection then consists in lasting only until the appointed moment for its dissolution.

As for natural religion, no man was a greater advocate than Newton, except perhaps Leibnitz himself, his rival in both science and virtue. By natural religion, I mean the principles of morality common to all humanity. Newton indeed did not admit any notion innate in us, neither ideas, sentiments, nor principles. He was persuaded with Locke that all ideas come to us through the senses as they develop; but he believed that God having given the same senses to all men, it results in them having the same needs, the same feelings, hence the same crude notions that are everywhere the foundation of society. It is evident that God has given bees and ants something that makes them live in community, which He has not given to wolves or hawks; it is certain, since all men live in society, that there is in their being a secret bond by which God has wanted to tie them to one another. Thus, if at a certain age, the ideas coming through the same senses to men all organized in the same way did not gradually give them the same principles necessary for any society, it is also very sure that these societies would not survive. That is why, from Siam to Mexico, truth, gratitude, friendship, etc., are honored.

I have always been astonished that the wise Locke, at the beginning of his Essay Concerning Human Understanding, in refuting so well innate ideas, claimed that there is no notion of good and evil common to all men. I believe he fell there into an error. He bases this on the accounts of travelers, who say that in certain countries it is customary to eat children, and to also eat mothers when they can no longer bear children17; that in others, certain enthusiasts who use donkeys instead of women are honored as saints; but should a man like the wise Locke not consider these travelers suspect? Nothing is so common among them as to see poorly, to report poorly what they have seen, to take especially in a nation whose language they do not know, the abuse of a law for the law itself, and finally to judge the customs of a whole people by a particular fact, whose circumstances they still ignore.

Suppose a Persian arrives in Lisbon, Madrid, or Goa, on the day of an auto-da-fé; he might reasonably believe that Christians sacrifice men to God; if he reads the almanacs distributed throughout Europe to the common folk, he might think that we all believe in the effects of the moon; and yet we laugh at it, far from believing in it. Thus any traveler who tells me, for example, that savages eat their father and mother out of pity will allow me to reply that firstly the fact is highly doubtful; secondly, if it is true, far from destroying the idea of respect owed to one’s parents, it is probably a barbaric way of showing affection, a horrible abuse of natural law: for presumably one only kills one’s father and mother out of duty to deliver them, either from the discomforts of old age or the furies of the enemy; and if then one gives them a tomb in the filial bosom, instead of letting them be eaten by victors, this custom, however horrific it is to the imagination, yet necessarily comes from the goodness of the heart. Natural religion is nothing other than this law known throughout the universe: Do unto others as you would have them do unto you; now the barbarian who kills his father to save him from his enemy, and who buries him in his own body, for fear he might have his enemy as a tomb, wishes that his son would treat him the same in a similar case. This law of treating one’s neighbor as oneself naturally flows from the crudest notions and is eventually heard in the hearts of all men: for, having the same reason, it is inevitable that sooner or later the fruits of this tree resemble each other; and they indeed do resemble each other in that in every society what is believed to be useful to society is called by the name of virtue.

Find me a country, a group of ten people on earth, where what is useful to the common good is not esteemed: and then I will agree that there is no natural rule. This rule varies infinitely, no doubt; but what to conclude, if not that it exists? Matter everywhere receives different forms, but everywhere retains its nature.

It is in vain that we are told, for example, that in Sparta theft was ordered: this is only an abuse of words. What we call theft was not commanded in Sparta; but in a city where everything was communal, the permission given to cleverly take what individuals appropriated against the law was a way to punish the spirit of property, forbidden among these people. Mine and yours was a crime, of which what we call theft was the punishment; and among them and us there was a rule for which God made us, as He made the ants to live together.

Newton thus thought that this disposition we all have to live in society is the foundation of natural law, which Christianity perfects.

Above all, there is in man a disposition to compassion as generally widespread as our other instincts: Newton had cultivated this sentiment of humanity, and he extended it even to animals; he was strongly convinced, along with Locke, that God has given animals (which seem to be only matter) a measure of ideas, and the same feelings as to us. He could not believe that God, who does nothing in vain, had given animals organs of sensation so that they would not feel.

He found it terribly contradictory to believe that animals feel and to make them suffer. His morals agreed with his philosophy in this respect; he reluctantly yielded to the barbarous practice of nourishing ourselves with the blood and flesh of beings similar to us, whom we caress every day; and he never allowed in his house that they be killed by slow and exquisite deaths, to make the food more delicious.

This compassion he had for animals turned into true charity for humans. Indeed, without humanity, a virtue that encompasses all virtues, one would hardly deserve the name of philosopher.

Chap. VI. — On the soul, and how it is united to the body and has its ideas. — Four opinions on the formation of ideas: those of the ancient materialists, that of Malebranche, that of Leibnitz; Leibnitz’s opinion contested.

Newton was convinced, like nearly all good philosophers, that the soul is an incomprehensible substance; and several people who have spent much time with Locke have assured me that Newton admitted to Locke that we do not have enough knowledge of nature to dare declare that it is impossible for God to endow any extended being with the gift of thought. The main difficulty is rather to understand how a being (whatever it is) can think, rather than how matter can become thinking. Thought, indeed, seems to have nothing in common with the attributes we recognize in the extended being we call body; but do we know all the properties of bodies? It seems quite bold to tell God: You could give motion, gravitation, vegetation, life to a being, and yet you cannot endow it with thought!

Those who say that if matter could receive the gift of thought, the soul would not be immortal, are they reasoning consistently? Is it more difficult for God to preserve than to create? Moreover, if an indivisible atom lasts forever, why wouldn’t the gift of thought in it last as well? If I am not mistaken, those who deny God the power to attach ideas to matter must claim that what we call spirit is a being whose essence is to think, excluding any extended being. Now, if it is the nature of the spirit to essentially think, then it necessarily thinks, and it always thinks, just as every triangle necessarily and always has three angles, independently of God. What! As soon as God creates something that is not matter, must that something necessarily think? We are feeble and bold! Do we know if God has not formed millions of beings that have neither the properties of the spirit nor those of the matter known to us? We are like a shepherd who, having seen only oxen, would say: If God wants to make other animals, they must have horns and chew cud. Which is more respectful to the Divinity, or to affirm that there are beings that have the divine attribute of thought without Him, or to suspect that God can grant this attribute to the being He deigns to choose?

It is clear from this how unjust are those who have accused Locke of this sentiment and have combated, with cruel malignity, using the weapons of religion a purely philosophical idea.

Moreover, Newton was far from venturing a definition of the soul, as so many others have dared to do. He believed it possible that there might be millions of other thinking substances, whose nature could be absolutely different from that of our soul. Thus the division that some have made of all nature between body and spirit appears to be the definition of a deaf and blind person who, in defining the senses, would neither suspect sight nor hearing: by what right indeed could one say that God has not filled the immense space with an infinity of substances that have nothing in common with us?

Newton had not devised any system on how the soul is united to the body, and on the formation of ideas. An enemy of systems, he judged nothing except by analysis; and when this torch was lacking, he knew how to stop.

There have so far been four opinions on the formation of ideas in the world. The first is that of almost all the ancient nations who, imagining nothing beyond matter, regarded our ideas in our mind as the impression of a seal on wax. This confused opinion was rather a crude instinct than reasoning; philosophers, who later tried to prove that matter thinks by itself, erred even more: for the common people were mistaken without reasoning, and these erred on principles; none of them could ever find anything in matter that could prove it has intelligence by itself.

Locke appears to be the only one who removed the contradiction between matter and thought, by suddenly resorting to the creator of all thought and all matter, and modestly saying: He who can do everything, cannot He make a material being, an atom, an element of matter think? He limited himself to this possibility as a wise man: to assert that matter actually thinks, because God could have communicated this gift to it, would be the height of recklessness; but to assert the contrary, is it not equally bold?

The second opinion, and the most generally accepted, is the one which, establishing the soul and the body as two beings that have nothing in common, yet affirms that God created them to act on each other. The only proof of this action is the experience that everyone believes to have: we find that our body sometimes obeys our will, sometimes masters it; we imagine that they act on each other really because we feel it, and it is impossible for us to pursue the inquiry further. This system is objected to in a way that seems irrefutable: if an external object, for example, communicates a shock to our nerves, this movement either goes to our soul or it does not: if it goes, it communicates movement, which would suppose the soul to be corporeal; if it does not go, in this case there is no action. All that can be answered to this is that this action is one of those things whose mechanism will always be unknown: a sad way to conclude, but almost the only one suitable for man in more than one aspect of metaphysics.

The third system is that of the occasional causes of Descartes, pushed even further by Malebranche. It begins by supposing that the soul can have no influence on the body, and from there it advances too far: for from the fact that the influence of the soul on the body cannot be conceived, it does not at all follow that it is impossible. It then supposes that matter, as an occasional cause, makes an impression on our body, and then God produces an idea in our soul, and reciprocally man produces an act of will, and God acts immediately on the body in consequence of this will: thus man acts, thinks only in God; which can, it seems, only receive a clear meaning by saying that God alone acts and thinks for us.

We are overwhelmed by the weight of difficulties that arise from this hypothesis: for how, in this system, can man will himself, and not be able to think himself? If God has not given us the faculty to produce movement and ideas, if it is He alone who acts and thinks, it is He alone who wills. Not only are we no longer free, but we are nothing, or rather we are modifications of God Himself. In this case there is no longer a soul, an intelligence in man, and there is no need to explain the union of the body and the soul, since it does not exist, and only God exists.

The fourth opinion is that of the pre-established harmony of Leibnitz. In his hypothesis the soul has no intercourse with its body; they are two clocks that God has made, each with a spring, and which go for a certain time in perfect correspondence: one shows the hours, the other rings. The clock showing the hour does not show it because the other rings; but God has established their movement in such a way that the needle and the chime correspond continuously. Thus Virgil’s soul produced the Aeneid, and his hand wrote the Aeneid without that hand obeying in any way the intention of the author; but God had eternally decreed that Virgil’s soul would write verses, and that a hand attached to the body of Virgil would write them down.

Without discussing the extreme difficulty of reconciling freedom with this pre-established harmony, there is a very strong objection to be made: if, according to Leibnitz, nothing happens without a sufficient reason, taken from the depths of things, what reason did God have to unite together two immeasurable beings, two beings as heterogeneous, as infinitely different as the soul and the body, neither of which influences the other? It would have been just as well to place my soul in Saturn as in my body: the union of the soul and the body here is a very superfluous thing. But the rest of Leibnitz’s system is even more extraordinary: its foundations can be seen in the Supplement to the Acts of Leipzig, volume VII; and one can consult the commentaries that several Germans have made on it extensively with a completely geometric method.

According to Leibnitz, there are four kinds of simple beings, which he calls monads, as will be seen in chapter viii; we speak here only of the species of monad called our soul. The soul, he says, is a concentration, a living mirror of the entire universe, which has within it all the confused ideas of all the modifications of this world, past, present, and future. Newton, Locke, and Clarke, when they heard of such an opinion, showed as much contempt for it as if Leibnitz had not been its author; but since very great German philosophers have made it a point of honor to explain what no Englishman has ever wanted to understand, I am obliged to clearly expose this hypothesis of the famous Leibnitz, which has become more respectable to me since you have made it the object of your research.

Every simple, created being, he says, is subject to change, otherwise it would be God: the soul is a simple, created being; it cannot therefore remain in the same state; but bodies, being composite, cannot make any alteration in a simple being: it must therefore take its source of changes from its own nature. Its changes are therefore successive ideas of things in this universe: it has some of them clear; but all the things in this universe, says Leibnitz, are so dependent on each other, so linked together forever, that if the soul has a clear idea of one of these things, it necessarily has confused and obscure ideas of all the rest.

One might, to clarify this opinion, bring the example of a man who has a clear idea of a game; he has at the same time several confused ideas of several combinations of this game. A man who currently has a clear idea of a triangle has an idea of several properties of the triangle, which may in turn present themselves more clearly to his mind. That is in what sense the monad of man is a living mirror of this universe.

It is easy to reply to such a hypothesis that, if God has made the soul a mirror, He has made it a very dim mirror and that, if there are no other reasons to advance such strange suppositions than this alleged indispensable connection of all the things in this world, this bold edifice is built on foundations that are scarcely visible: for when we have a clear idea of the triangle, it is because we have knowledge of the essential properties of the triangle; and if the ideas of all these properties do not all at once shine brightly in our mind, they are nevertheless there, they are contained in this clear idea, because they have a necessary relation to each other. But is the whole assembly of the universe in this case? If you remove a property from the triangle, you take everything away from it; but if you remove a grain of sand from the universe, is the rest entirely changed? If of a hundred million beings that follow each other two by two, the first two change places among themselves, do the others necessarily change? Do they not maintain the same relations among themselves? Moreover, do a man’s ideas have among themselves the same chain that is supposed in the things of this world? What necessary connection, what necessary medium is there between the idea of night and the unknown objects I see on waking up? What chain is there between the temporary death of the soul in a deep sleep or in a fainting fit, and the ideas that one receives in recovering one’s senses? Even if it were possible that God had done everything that Leibnitz imagines, should it be believed on a simple possibility? What has he proved by all these new efforts? That he had a very great genius; but has he enlightened himself, and has he enlightened others? Strange! We do not know how the earth produces a blade of grass, how a woman makes a child, and we believe we know how we form ideas!

If you want to know what Newton thought about the soul, and how it operates, and which of all these opinions he embraced, I will answer that he followed none18. What did he know then about this matter, he who had subjected infinity to calculation, and who had discovered the laws of gravitation? He knew how to doubt.

Chap. VII. — On the first principles of matter. Examination of prime matter. Newton’s oversight. There are no true transmutations. Newton admits atoms.

This is not about examining which system was more ridiculous, whether the one that made water the principle of everything, the one that attributed everything to fire, or the one that imagines dice placed side by side without interval, turning in some unknown way upon themselves.

The most plausible system has always been that there is a prime matter indifferent to everything, uniform, and capable of all forms, which, differently combined, constitutes this universe. The elements of this matter are the same: it is modified according to the different molds through which it passes, like molten metal becoming now an urn, now a statue. This was the opinion of Descartes, and it agrees very well with the chimera of his three elements. Newton thought about matter in this respect like Descartes; but he had reached this conclusion by a different route. As he almost never formed a judgment that was not founded either on mathematical evidence or on experience, he believed he had experience on his side in this examination. The illustrious Robert Boyle, the founder of physics in England, had long held water in a retort over a constant fire; the chemist working with him believed that the water had finally turned into earth: the fact was false, as later proved by Boerhaave, a physician as accurate as he was skilled; the water had evaporated, and the earth that had appeared in its place came from elsewhere19.

To what extent must one be wary of experience, since it deceived Boyle and Newton? These great philosophers did not hesitate to believe that since the primitive parts of water changed into primitive parts of earth, the elements of things are but the same matter differently arranged.

If a false experience had not led Newton to this conclusion, it is believed he would have reasoned quite differently.

I beg you to read the following carefully.

The only way that belongs to man to reason about objects is analysis. To start directly from first principles belongs only to God; and if one can compare God to an architect, and the universe to a building without blasphemy, what traveler, seeing part of the exterior of a building, would dare to suddenly imagine all the artifice inside? Yet this is what almost all philosophers have dared to do with a thousand times more temerity.

Let us then examine this building as much as we can: what do we find around us? animals, plants, minerals, under which genre I include all salts, sulfurs, etc., mud, sand, water, fire, air, and nothing else, at least so far.

Before even examining whether these bodies are mixtures or not, I ask myself whether it is possible that a so-called uniform matter, which is in itself nothing of all that is, can nevertheless produce everything that is.

1° What is a prime matter that is nothing of the things of this world, and yet produces them all? It is something of which I can have no idea, and therefore should not admit. It is true that I cannot generally form the idea of an extended, impenetrable, and figurable substance without determining my thoughts to sand or mud, or gold, etc.; but either this matter is really one of these things, or it is nothing at all; similarly, I can think of a triangle in general, without stopping at the equilateral, scalene, isosceles, etc.; but it is necessary that an existing triangle be one of these. This idea alone, well considered, is perhaps sufficient to destroy the opinion of a prime matter.

2° If any matter, set in motion, were sufficient to produce what we see on Earth, there would be no reason why dust well shaken in a barrel could not produce men and trees, nor why a field sown with wheat could not produce whales and crayfish instead of wheat.

It would be in vain to reply that the molds and dies that receive the seeds oppose this; for one will always have to come back to this question: Why are these molds, these dies so invariably determined?

Now, if no movement, no art has ever been able to bring forth fish instead of wheat in a field, nor medlars instead of a lamb in the belly of a sheep, nor roses at the top of an oak, nor soles in a beehive of bees, etc.; if all species are invariably the same, should I not first believe, with some reason, that all species have been determined by the Master of the world; that there are as many different designs as there are different species, and that matter and motion would produce only an eternal chaos without these designs?

All experiments confirm me in this sentiment. If I examine on one side a man or a silkworm, and on the other a bird and a fish, I see them all formed from the beginning of things; I see in them only a development. That of man and the insect have some similarities and some differences; that of fish and bird have others: we are a worm before being received into the womb of our mother; we become chrysalids, nymphs in the uterus, when we are in that envelope called the coif20; we emerge with arms and legs, as the worm turned into a fly emerges from its tomb with wings and feet; we live a few days like it, and our body then dissolves like its. Among reptiles, some are oviparous, others viviparous; in fish, the female is fertile without the male’s approach, who only passes over the eggs laid to hatch them. Aphids, oysters, etc., produce their like, alone, and without the mixture of two sexes. Polyps have in them what it takes to regenerate their heads when they are cut off. Legs grow back to crayfish. Vegetables, minerals, form very differently. Each kind of being is a world apart; and far from a blind matter producing everything by simple movement, it is very likely that God formed an infinity of beings with infinite means, because He is infinite Himself.

Here is what I suspect when considering nature. But if I go into detail, if I experiment with each thing, here is what results.

I see mixtures such as plants and animals that I decompose, and from which I extract some crude elements, spirit, phlegm, sulfur, salt, dead head. I see other bodies, such as metals, minerals, from which I can never draw anything other than their own more attenuated parts. Pure gold could only yield gold; never with pure mercury could one get anything but mercury. Simple sand, simple mud, simple water, could never be changed into any other species of beings.

What can I conclude, other than that plants and animals are composed of these other primitive beings that never decompose? These unalterable primitive beings are the elements of bodies: man and the midge are thus a compound of the mineral parts of slime, sand, fire, air, water, sulfur, salt21; and all these primitive, forever indecomposable parts are elements each with its own distinct and unvarying nature.

To dare assert the contrary, one would have to have seen transmutations; but has anyone ever discovered them through the aid of chemistry? Is the philosopher’s stone not regarded as impossible by all wise minds? Is it any more possible, in the current state of this world, for salt to be changed into sulfur, water into earth, air into fire, than to make gold from projection powder?

When men have believed in true transmutations, have they not been deceived by appearances, just like those who believed that the sun moved? For seeing wheat and water converted in human bodies into blood and flesh, who would not have believed in transmutations? Yet is all this anything other than salts, sulfurs, slime, etc., differently arranged in the wheat and in our bodies? The more I reflect on it, the more a rigorous metamorphosis seems to me to be nothing other than a contradiction in terms. For the primitive parts of salt to change into the primitive parts of gold, I believe, two things are necessary: to annihilate these elements of salt, and to create elements of gold. That is fundamentally what these supposed metamorphoses of a homogeneous and uniform matter admitted until now by so many philosophers are, and here is my proof.

It is impossible to conceive of the immutability of species without them being composed of unalterable principles. For these principles, these first constituent parts, not to change, they must be perfectly solid, and therefore always of the same shape: if they are such, they cannot become other elements, for they would have to receive other shapes; therefore, since it is impossible that, in the present constitution of this universe, the element that serves to make salt be changed into the element of mercury, it would be necessary, to make an element of salt in place of an element of mercury, to annihilate one of these elements, and to create another in its place. I do not know how Newton, who admitted atoms, did not draw this natural induction. He recognized true atoms, indivisible bodies like Gassendi; but he had arrived at this assertion through his mathematics; at the same time, he believed that these atoms, these indivisible elements, were continually changing into one another. Newton was human; he could err like us.

One might well ask here how the germs of things, being hard and indivisible, can grow and expand: they probably only grow by assembly, by contiguity; several atoms of water form a drop, and so forth.

It remains to be seen how this contiguity operates, how the parts of bodies are linked together. Perhaps it is one of the Creator’s secrets, which will forever be unknown to men. To know how the constituent parts of gold form a piece of gold, it seems one would have to see these parts.

If it were permissible to say that attraction is probably the cause of this adhesion and continuity of matter, that is what one could advance most likely: for indeed if it is demonstrated, as we will see, that all parts of matter gravitate towards each other, whatever the cause, can one think of anything more natural than that bodies touching more points are more united together by the force of this gravitation? But this is not the place to enter into this physical detail22.

Chap. VIII. — On the nature of the elements of matter, or monads. — Newton’s opinion. Leibnitz’s imagination.

If ever the phrase audax Japeti genus23 was warranted, it was in the inquiries men have dared to make into these first elements, which seem to be placed at an infinite distance from the sphere of our knowledge. Perhaps nothing is more modest than Newton’s opinion, which simply held that the elements of matter are matter itself, that is, an extended and impenetrable being whose intimate nature is unfathomable to the intellect; that God can divide it infinitely as he can annihilate it, but that He does not do so, and maintains these extended and indivisible parts to serve as the foundation for all the productions of the universe.

Perhaps, on the other hand, nothing is bolder than the leap taken by Leibnitz, starting from his principle of sufficient reason, to penetrate, if possible, into the bosom of causes and into the inexplicable nature of these elements. Every body, he says, is composed of extended parts; but what are these extended parts made of? They are currently, he continues, divisible and divided infinitely; thus, you find only extension. However, to say that extension is the sufficient reason for extension is to argue in a circle, to say nothing; thus, one must find the reason, the cause of extended beings in beings that are not extended, in simple beings, in monads; therefore, matter is nothing but an assembly of simple beings. It was seen in the chapter on the Soul that, according to Leibnitz, each simple being is subject to change; but its alterations, its successive determinations that it receives, cannot come from outside, for the reason that this being is simple, intangible, and occupies no space: it thus has the source of all its changes within itself, occasioned by external objects; it thus has ideas. But it has a necessary relationship with all parts of the universe: it thus has ideas relating to the entire universe; the elements of the vilest excrement thus have an infinite number of ideas; their ideas, true, are not very clear, they do not have apperception, as Leibnitz says, they do not have within them the intimate witness of their thoughts; but they have confused perceptions of the present, the past, and the future. He admits four species of monads: 1) the elements of matter, which have no clear thought; 2) the monads of beasts, which have some clear ideas and none distinct; 3) the monads of finite spirits, which have confused, clear, and distinct ideas; 4) finally, the monad of God, which has only adequate ideas.

The English philosophers, I have already said24, who respect no names, have responded to all this with laughter; but I am permitted to refute Leibnitz only by reasoning; it seems I would take the liberty of saying to those who have credited such opinions: Everyone agrees with you on the principle of sufficient reason; but do you draw from it here a just consequence?

  1. You admit matter is actually divisible infinitely; the smallest part is thus not possible to find. There is none that does not have sides, that does not occupy a place, that does not have a shape: how then do you want it to be formed only of beings without shape, without place, and without sides? Do you not clash with the great principle of contradiction in wanting to follow that of sufficient reason?

  2. Is it sufficiently reasonable that a composite should have nothing similar to what composes it? What do I say, nothing similar? there is infinity between a simple being and an extended being; and you want one to be made of the other: whoever would say that several elements of iron form gold, that the constituent parts of sugar make colocynth, would he say anything more revolting?

  3. Can you really claim that a drop of urine is an infinity of monads, and that each of them has the ideas, although obscure, of the entire universe, and that because, according to you, everything is full, because in the full everything is connected, because everything being linked together, and a monad necessarily having ideas, it cannot have a perception that does not relate to everything that is in the world?

25But is it proven that everything is full, despite the multitude of metaphysical and physical arguments in favor of the void? Is it proven that, everything being full, your alleged monad must have the useless ideas of everything that happens in this full? I appeal to your conscience: do you not feel how such a system is purely imaginative? Isn’t the admission of human ignorance about the elements of matter above such vain science? What use of logic and geometry, when one uses this thread to lose oneself in such a labyrinth, and walks methodically towards error with the very torch meant to enlighten us!

Chap. IX. — On the active force, which puts everything in motion in the universe. — Whether there is always the same amount of force in the world. Examination of the force. How to calculate the force. Conclusion of both parties.

First, I assume that we agree that matter cannot move by itself: it must therefore receive motion from elsewhere; but it cannot receive it from another matter, as this would be a contradiction; thus, an immaterial cause must produce motion. God is this immaterial cause, and it must be noted here that the common axiom: “One should not resort to God in philosophy,” is only valid for things that should be explained by physical proximate causes. For example, I want to explain why a weight of four pounds is counterbalanced by a weight of one pound: if I say that God has thus regulated it, I am ignorant; but I satisfy the question if I say that it is because the weight of one pound is four times as far from the pivot as the weight of four pounds. It is not the same for the first principles of things: then not to resort to God is ignorant, for either there is no God, or there are no first principles but in God.

It is He who has imparted to the planets the force with which they move from west to east; it is He who moves these planets, and the sun on their axes.

He has imprinted a law on all bodies, by which they all equally tend towards their center. Finally, He has formed animals to which He has given an active force with which they generate movement.

The big question is whether this force given by God to initiate movement is always the same in nature.

Descartes, without mentioning force, asserted without proof that there is always an equal amount of motion; and his opinion was all the less well-founded as the very laws of motion were completely unknown to him.

Leibnitz, coming at a more enlightened time, was forced to admit, with Newton, that motion is lost; but he claims that, although the same quantity of motion does not persist, the force always remains the same.

Newton, on the contrary, was convinced that it implies a contradiction that motion is not proportional to the force.

Before entering into any mechanical discussion, one must take things in their very nature: for the metaphysician must always lead the geometer. A man has a certain amount of active force; but where was this force before his birth? If it is said that it was in the child’s germ, what is a force that cannot be exerted? But when he has become a man, is he not free? Can he not use more or less of his force? I suppose he exerts a force of three hundred pounds to move a machine; I suppose, as it is possible, that he exerted this force by lowering a lever, and that the machine attached to this lever is in the vacuum receptacle: the machine can easily acquire a force of two thousand pounds.

Once the operation is done, the arm withdrawn, the lever removed, the weight immobile, I ask whether the little matter that was in the receptacle has received from the machine a force of two thousand pounds: do all these considerations not show that active force is continually repaired and lost in nature? Let’s pay a little attention to this argument.

There can be no motion without a vacuum; now let a body A B C D receive an impression in all its parts, I ask if the parts B C D, behind which there will be no body, will not lose motion; and if the parts B C lose their motion, do they not evidently lose their force?

Let us now listen to Newton and experience to conclude this metaphysical dispute. Motion, he says, is produced and lost. But because of the tenacity of fluids and the lack of elasticity in solids, much more motion is lost than is regenerated in nature.

Given this, if you consider this undeniable axiom that the effect is always proportional to the cause, where motion diminishes, force necessarily also diminishes; thus, to always maintain the same amount of forces in the universe, this principle (that the cause is proportional to the effect) would have to cease to be true.

It has been believed that, to always preserve the same force in nature, it suffices to change the usual way of estimating this force: instead of Mersenne, Descartes, Newton, Mariotte, Varignon, etc., who have always, after Archimedes, measured the motion of a body by multiplying its mass by its velocity, Leibnitz, Bernoulli, Herman, Polenis, S’Gravesande, Wolff, etc., have multiplied the mass by the square of the velocity.

This dispute has divided Europe; but finally, it seems to me that it is recognized that it is fundamentally a dispute over words. It is impossible that these great philosophers, although diametrically opposed, are mistaken in their calculations. They are equally just; mechanical effects correspond equally well to one way of counting as to the other. There is undoubtedly a sense in which they are all right. Now the point where they are right is the one that should unite them; and here it is, as Doctor Clarke first indicated, though a bit harshly.

If you consider the time in which a mobile acts, its force at the end of this time is like the square of its velocity by its mass. Why? Because the space traversed by its mass is like the square of the time in which it is traversed. Now time is like velocity: therefore then the body that has traversed this space in this time acts at the end of this time by its mass, multiplied by the square of its velocity: thus, when mass 2 traverses in two times any space with two degrees of velocity, at the end of this time its force is 2, multiplied by the square of its velocity 2; all making 8, and the body makes an impression as 8; in this case the Leibnitzians are not wrong. But also the Cartesians and Newtonians united have great reason when they consider the matter from another angle, for they say: In equal time a body of four pounds weight, with one degree of velocity, acts precisely like a weight of one pound with four degrees of velocity, and elastic bodies that collide always rebound in the reciprocal ratio of their velocity and mass; that is to say, a double ball with a movement like one, and a sub-double ball with a movement like two, launched against each other, arrive in equal time, and rebound to equal heights: therefore, it is not necessary to consider what happens to mobiles in unequal times, but in equal times, and this is the source of the misunderstanding. Thus the new way of considering forces is true in one sense, and false in another; thus it only serves to complicate, to confuse a simple idea; thus it is necessary to stick to the old rule. What to conclude from these two ways of considering things? Everyone must agree that the effect is always proportional to the cause: now, if motion is lost in the universe, then the force that causes it is also lost. This is what Newton thought about most of the questions that pertain to metaphysics: it is up to you to judge between him and Leibnitz.

I will move on to his discoveries in physics26.

Second Part. — Newtonian Physics. — Introduction

My main goal in the research I am about to undertake is to give myself, and perhaps some readers, clear ideas about those primitive laws of nature that Newton discovered. I will examine how far others had gone before him, where he started, where he stopped, and sometimes what has been found after him. I will start with light, which he alone truly understood; I will end by examining gravity, and the general law of gravitation or attraction, the universal spring of nature, whose discovery is solely his.

27We will try to make these Elements accessible to those who only know Newton and philosophy by name. The science of nature is a treasure that belongs to all mankind. Everyone would like to know their heritage, few have the time or the patience to calculate it; Newton has calculated for them. Here, one must sometimes be content with the sum of these calculations. Every day a public man, a minister, forms a correct idea of the results of operations that he himself could not perform; other eyes have seen for him, other hands have worked, and they enable him, through a faithful account, to make his judgment. Any intelligent man will be roughly in the position of this minister.

Chap. I. — Early research on light, and how it comes to us. Errors by Descartes on this matter. — Singular definition by the Peripatetics. The systematic spirit misled Descartes. His false system. On the progressive movement of light. Error of Spectacle of Nature. Demonstration of the motion of light by Roemer. Roemer’s experiment contested and wrongly combated. Proof of Roemer’s discovery by Bradley’s discoveries. History of these discoveries. Explanation and conclusion.

The Greeks, and then all the barbarian peoples who learned from them to reason and to err, have said from century to century: “Light is an accident, and this accident is the act of the transparent as transparent; colors are what move transparent bodies. Luminous and colored bodies have qualities similar to those they excite in us, by the great reason that nothing gives what it does not have. Finally, light and colors are a mixture of hot, cold, dry, and wet: for the moist, the dry, the cold, and the hot, being the principles of everything, it follows that colors must be a compound of them.”

This absurd gibberish was respected by human credulity for so many years, taught by masters of ignorance paid by the public; this is how people reasoned almost about everything until the times of Galileo and Descartes. Even long after them, this jargon, which dishonors human understanding, persisted in several schools. I dare say that man’s reason, thus obscured, is far below those limited but sure knowledge that we call instinct in animals. Thus, we cannot be too happy to be born at a time and among a people where eyes are beginning to open, and to enjoy the most beautiful endowment of humanity, the use of reason.

All the so-called philosophers having thus guessed at random through the veil that covered nature. Descartes came, who lifted a corner of this great veil. He said: Light is a fine and delicate matter, and it strikes our eyes. Colors are the sensations that God excites in us, according to the various movements that carry this matter to our organs. Up to this point Descartes was right: he should have either stopped there or let experience guide him further. But he was possessed by the desire to establish a system. This passion did in this great man what passions do in all men: they lead them beyond their principles.

He had laid down as the fundamental principle of his philosophy that nothing should be believed without evidence: and yet, in contempt of his own rule, he imagines three elements formed from the supposed cubes he assumes were made by the Creator, and having broken in turning on themselves, when they came out of God’s hands. These three imaginary elements are, as is known:

  1. The thickest part of these cubes, and it is this coarse element from which, according to him, the solid bodies of the planets, the seas, even the air were formed;
  2. The impalpable dust, which the breaking of these dice had produced, and which fills infinitely the interstices of the infinite universe in which he assumes no vacuum;
  3. The middles of these supposed broken dice, equally thinned on all sides, and finally rounded into balls, which he is pleased to make light of, and which he spreads freely throughout the universe.

The more ingeniously this system was devised, the more you feel it was unworthy of a philosopher; and since none of this is proven, it was as good to adopt the hot and the cold, the dry and the wet. Error for error, what does it matter which dominates?

According to Descartes, light does not come to our eyes from the sun; but it is a globular matter spread everywhere, that the sun pushes, and which presses our eyes like a stick pushed by one end presses instantly at the other end. He was so convinced of this system that, in his seventeenth letter of the third volume, he positively states and repeats: I confess that I know nothing in philosophy if the light from the sun is not transmitted to our eyes in an instant.

Indeed, it must be admitted that, great genius though he was, he still knew little about true philosophy: he lacked the experience of the century that followed him. This century is as superior to Descartes as Descartes was to antiquity.

1° If light were a fluid always present in the air, we would see clearly at night, since the sun, under the hemisphere, would always push this fluid of light in all directions, and the impression would come to our eyes. Light would circulate like sound. We would see an object beyond a mountain; in fact, we would never have such a beautiful day as during a central solar eclipse, for the moon, passing between us and this star, would press (at least according to Descartes) the globules of light, and only increase their action.

2° The rays that are bent by a prism, and forced to take a new path, demonstrate that light actually moves, and is not merely a cluster of pressed globules; light follows three different paths when entering a prism; its three routes in the air, inside the prism, and upon exiting the prism, are different; moreover, it accelerates its movement within the prism28: is it not a bit strange to say that a body which visibly changes its position three times, and increases its speed, does not move? Yet, there has just been a book published in which it is boldly stated that the progression of light is an absurdity.

3° If light were a cluster of globules, a fluid existing in the air and everywhere, a small hole made in a dark room should illuminate it entirely: for the light, pushed in all directions into this small hole, would act in all directions like ivory balls arranged in a circle or square that would all scatter if one of them were strongly pressed; but the opposite happens: the light received through a small orifice, which only allows a small cone of rays to pass, and reaches twenty-five feet, barely illuminates half a foot of the area it strikes.

4° It is known that light, emanating from the sun to us, traverses in about eight minutes this immense path that a cannonball, maintaining its speed, would not complete in twenty-five years.

The author of The Spectacle of Nature29, a very commendable work, has here made a mistake that could mislead the beginners for whom his book is intended. He states that light takes seven minutes to reach us from the stars, according to Newton: he has mistaken the stars for the sun. Light from the nearest stars takes about six months30, based on a calculation founded on very delicate and highly faulty experiments. It was not Newton, but Huygens and Hartsoeker who made this assumption. He further claims, to prove that God created light before the sun, that light is spread throughout nature, and makes itself felt when the luminous stars push it; but it is demonstrated that it arrives from fixed stars in a very long time. Thus, if it makes this journey, it was not spread beforehand. It is wise to guard against these errors, which are repeated daily in many books that echo one another.

Here, in brief, is the substance of Roemer’s sensible demonstration that light takes seven to eight minutes to travel from the sun to the Earth.

It is known that one hundred and ninety million of our leagues, which the Earth travels at least in a year, are just a point in relation to the distance from the fixed stars to the Earth. The eye cannot perceive whether a star has changed its position relative to us at the end of the diameter of this immense orbit: yet it is certainly true that after six months, there is a difference of about sixty-six million leagues between us and a star located near the pole; and this path, which a cannonball would not complete in fifty years even if it retained its speed, is annihilated by the prodigious distance of our globe from the nearest star: because, when the visual angle becomes so small, it is no longer measurable, it becomes nil.

Finding the secret of measuring this angle, knowing the difference when the Earth is at the Cancer and when it is at the Capricorn, thus obtaining what is called the Earth’s parallax, seemed a problem as difficult as that of the longitudes.

The famous Hooke, well-known for his Micrographia, undertook to solve the problem; he was followed by the astronomer Flamsteed, who had charted the position of three thousand stars; then Sir Molyneux, with the help of the famous mechanic Graham, invented a machine for this operation: he spared neither effort, time, nor expense; finally, Dr. Bradley put the finishing touches to this great work.

The machine used was called the parallactic telescope. Its description can be found in Mr. Smith’s excellent Treatise on Optics. A long telescope suspended perpendicular to the horizon was so arranged that one could easily direct the axis of vision in the plane of the meridian, either a bit north or a bit south, and know, by means of a wheel and an index, with the greatest accuracy, by how much the instrument had been moved south or north. Several stars were observed with this telescope, and among others, a star from Draco was followed for an entire year.

What should have resulted from this diligent search? Certainly, if the Earth, from the beginning of summer to the beginning of winter, had changed its position, if it had traveled those sixty-six million leagues, the ray of light that had been aimed six months before at the axis of this telescope’s vision must have deviated; a new movement had to be imparted to this tube to receive this ray, and one knew, by means of the wheel and the index, how much movement had been given to it, and, as an infallible consequence, by how much the star was more northern or more southern than six months before.

These admirable operations began on December 3, 1725: the Earth was then approaching the winter solstice; it seemed likely that if the star could give some sign of aberration from December, it would appear to cast its light more to the north, since the Earth was then going to the south around the winter solstice. But, from December 17th, the observed star appeared to have advanced in the meridian towards the south. Everyone was very surprised31. They had precisely the opposite of what they had hoped; but through the constant succession of observations, they gained more than they would have ever dared to hope. They distinctly understood the parallax of this fixed star, the Earth’s annual movement, and the progression of light.

If the Earth orbits around the sun, and if light is instantaneous, it is clear that the observed star must always appear to move a little towards the north when the Earth moves to the opposite side; but if light is emitted from this star, and it takes some time to arrive, one must compare this time with the speed at which the Earth moves, there is nothing left but to calculate; by this, they saw that the speed of light from this star was ten thousand two hundred times faster than the average movement of the Earth. Observations on other stars showed that not only does light move at this enormous speed, but it moves uniformly, even though it comes from fixed stars placed at very unequal distances. They saw that the light from each star travels the space determined by Roemer, that is about thirty-three million leagues in almost eight minutes.

They saw, by measuring the annual parallax, that the star observed in Draco is four hundred thousand times further from us than the sun. Now, I implore every attentive reader, who loves the truth, to consider that if light reaches us from the sun uniformly in about eight minutes, it arrives from this star of Draco in six years and more than a month; and that if stars six times smaller are six times further from us, they send us their rays in more than thirty-six and a half years. Moreover, the course of these rays is always uniform. Now judge whether this uniform march is compatible with a supposed matter spread everywhere. Ask yourself if this matter would not slightly disturb this uniform progression of rays; and finally, when you read the chapter on vortices, remember this enormous extent that light traverses in so many years, judge in good faith whether an absolute full would not oppose its passage. See finally how many errors this system must have led Descartes into. He had made no experiments, he was imagining; he did not examine this world, he created one. Newton, on the other hand, Roemer, Bradley, etc., only made experiments, and judged only according to the facts32.

Chap. II. — Malebranche’s system as erroneous as that of Descartes; nature of light; its paths; its speed. — Error of Father Malebranche. Experiment that destroys the chimera of luminous whirlwinds. Definition of the matter of light. Fire and light are the same being. Speed of light. Smallness of its atoms. False idea about how it comes to us. Progression of light. Proof of the impossibility of the plenum. Stubbornness against these truths. Abuse of Holy Scripture against these truths.

Father Malebranche, who, in examining the errors of the senses, was not free from those that the subtlety of genius can cause, adopted without proof Descartes' three elements; but he changed many things in this enchanted castle, and, making even fewer experiments than Descartes, he constructed a system like him.

According to him, vibrations from the luminous body imprint shocks to small soft vortices, capable of compression, all composed of subtle matter. But if one had asked Malebranche how these soft vortices would have transmitted light to our eyes, how the action of the sun could pass instantly through so many small compressed bodies, any very small number of which would suffice to dampen this action; how these soft vortices would not get mixed up while spinning on each other; how these soft vortices would be elastic; finally, why he supposed vortices at all? What would Father Malebranche have answered? On what foundation did he base this imaginary edifice? Must men, who spoke only of truth, have only ever written novels!

An experiment seems to completely destroy all these supposed vortices of luminous matter, which are so gratuitously assumed. Receive sunlight on a concave mirror; oppose a lens to this concave mirror as much as you can, so that the tips of the two light cones join in the air: by this artifice, you create the most intense heat possible on earth. If the tips of these cones were vortices tending to escape in all directions, as claimed, isn’t it true that they would make a tremendous battle at the meeting point? Isn’t it true that the effect would be noticeable some distance from the cone tips? However, an inch from this point, you feel no heat at all: imagine after that, small vortices.

What then is the matter of light? It is fire itself, which burns at a short distance when its parts are less thin, or faster, or more gathered, and which gently lights our eyes when it acts from farther away, when its particles are finer and less rapid, and less united.

Thus, a lit candle would burn the eye only a few lines away from it, and lights the eye that is a few inches away; thus the rays of the sun, scattered in the space of the air, illuminate objects, and, gathered in a burning glass, melt lead and gold.

If asked what fire is, I will answer that it is an element that I know only by its effects, and I will say, here as elsewhere, that man is not made to know the inner nature of things; he can only calculate, measure, weigh, and experiment.

Fire does not always illuminate, and light does not always shine; but only the element of fire can illuminate and burn. Fire that is not developed, whether in a bar of iron or in wood, cannot send rays from the surface of this wood or iron, therefore it cannot be luminous; it only becomes so when this surface is ablaze.

The rays of the full moon give no sensible heat at the focus of a burning glass, although they give a considerable light33. The reason is palpable: the degrees of heat are always in proportion to the density of the rays. It is proven that the sun, at the same height, casts ninety thousand times more rays than the full moon reflects to us on the horizon.

Thus, for the moon’s rays at the focus of a burning glass to give only as much heat as the sun’s rays would on a plot of land the same size as this glass, there would need to be ninety thousand times more rays at this focus than there are.

Those who wanted to make two beings out of light and fire were therefore mistaken by basing this on the fact that not all fire illuminates, and not all light warms: it is as if one made two beings out of each thing that can serve two purposes.

This fire is darted in all directions from the radiant point; this is what makes it perceived from all sides: it must always be considered by geometers as lines going from a center to the circumference. Thus any beam, any accumulation, any streak of rays, coming from the sun or any fire, must be considered as a cone whose base is on our pupil, and whose tip is in the fire that darts it.

This fire matter darts from the sun to us and to Saturn, etc., with a rapidity that staggers the imagination.

Calculations teach that if the sun is twenty-four thousand half-diameters of the earth away, it follows that light travels from this star to us (in round numbers) a billion feet per second. Now a cannonball, driven by half a pound of powder, only travels six hundred feet in one second; thus the speed of a ray from the sun is, in round numbers, one million six hundred sixty-six thousand six hundred times greater than that of a cannonball: it is therefore certain that if an atom of light were only about one-sixteen-hundred-thousandth of a pound, it would necessarily result that rays of light would have the effect of a cannon; and even if they were a thousand billion times smaller still, a single moment of light emission would destroy everything that grows on the surface of the earth. How inconceivably small must these rays be to enter our eyes without hurting them?

The sun, which darts this luminous matter to us in seven or eight minutes, and the stars, those other suns, which send it to us in several years, supply it eternally without appearing to be exhausted, much like musk constantly emits fragrant bodies around it without losing any sensible weight.

Finally, the speed with which the sun darts its rays is probably in proportion to its size, which exceeds that of the earth by about a million times, and with the speed with which this immense fiery body rolls upon itself in twenty-five and a half days.

Some people have imagined that I claimed this light was attracted to the earth, from the substance of the sun; but I have never said anything that could have given the slightest pretext for such an idea.

Others have claimed that the sun would lose all its substance in a few days, and that it must send millions of pounds of light every minute; but if attention were paid to the fact that light barely weighs, that the sun perhaps supplies only an ounce per year, and that it receives from all other suns, these hasty criticisms would not be made.

We can conclude in passing from the speed with which the substance of the sun escapes towards us in a straight line, how inadmissible Descartes' plenitude is. For: 1° how could a straight line reach us through so many millions of layers of matter moved in a curved line, and through so many diverse movements? 2° how could such a delicate body travel in seven or eight minutes the space of four hundred thousand times thirty-three million leagues from a star to us, if it had to penetrate a resistant matter in this space? Each ray would have to disrupt thirty-three million leagues of subtle matter four hundred thousand times in an instant.

Note again that this supposed subtle matter would resist in the absolute full as much as the most compact matter. For a pound of gold powder, pressed in a box, resists as much as a lump of gold weighing a pound. Thus a ray from a star would have much more effort to make than if it had to pierce a gold cone, whose axis would be thirteen thousand two hundred billion leagues.

Moreover, experience, this true master of philosophy, teaches us that light, when coming from one element to another, from one medium to another, does not pass through entirely, as we will say: a large part is reflected, air repels more than it transmits; thus it would be impossible for us to receive any starlight, it would all be absorbed, all reflected, before a single ray could even come halfway through our atmosphere. And what would it be if this ray still had so many other atmospheres to cross? But in the chapters where we will explain the principles of gravitation, we will see a host of arguments proving that this supposed fullness was a novel.

Let us pause here for a moment to see how slowly truth establishes itself among men. Nearly fifty years ago Roemer demonstrated, by observations on the eclipses of Jupiter’s satellites, that light emanates from the sun to the earth in about seven and a half minutes; however, not only is the opposite still maintained in several physics books, but here is how they speak in a three-volume collection drawn from observations of all the academies of Europe, printed in 1730, page 35, volume I:

“Some have claimed that from a luminous body like the sun there is a continuous flow of an infinity of tiny insensible parts, which carry light to our eyes; but this opinion, which still smacks a little of old philosophy, is not sustainable.”

This opinion is nevertheless demonstrated in more than one way, and far from smelling of old philosophy, it is directly contrary to it: for what could be more contrary to empty words than so many measurements, calculations, and experiments?

Other contradictors have attacked this truth of the emanation and progression of light with the same weapons with which more respected than enlightened men once so imperiously and vainly attacked Galileo’s view on the movement of the earth.

Those who fight reason with authority employ Holy Scripture, which should teach us to live well, to draw lessons of their philosophy from it; they have actually made Moses a physicist. If it is simplicity, they are to be pitied. If they believe with this artifice to make odious those who do not share their view, they are to be pitied even more; they should remember that those who condemned Galileo on such a pretext have covered their country with a shame that only the name of Galileo can erase. They must believe, they say, that daylight does not come from the sun because, according to Genesis, God created light before the sun.

But these gentlemen do not consider that, according to Genesis, God also separated light from darkness, called light day, and darkness night, and composed a day of evening and morning, etc., and all this before creating the sun.

They would therefore have to, by the account of these physicists, have the sun not make the day, and the absence of the sun not make the night.

They further add that God separated the waters from the waters, and they understand by this separation the sea and the clouds. But, according to them, it would therefore be necessary that the vapors which form the clouds were not, as they are, raised by the sun. For, according to Genesis, the sun was created only after this separation of the lower and upper waters; yet they admit in this case that it is the sun that raises these upper waters. Here they are therefore in contradiction with themselves. Will they deny the movement of the earth because Joshua commanded the sun to stand still34? Will they deny the development of germs in the earth because it is said that the grain must rot before it springs35? They must therefore recognize, with all sensible people, that it is not physical truths that should be sought in the Bible, and that we should learn from it to become better, not to know nature.

Chap. III. — The property that light has to reflect was not truly known; it is not reflected by the solid parts of bodies, as was believed. — No united body. Light not reflected by solid parts. Decisive experiments. How and in what sense light rebounds from the vacuum itself. How to experiment with it. Conclusion of this experiment. The smaller the pores, the more light passes. Poor objections against these truths.

Having learned what light is, where it comes from, how and when it reaches us, let’s examine its properties and effects unknown until our days. The first of these effects is that it seems to rebound from the solid surface of all objects, bringing their images to our eyes.

All men, all philosophers, including Descartes and Malebranche, and those who have strayed furthest from common thoughts, have equally believed that indeed it is the solid surfaces of bodies that reflect the rays back to us. The more uniform and solid a surface, the more it is said to rebound light; the larger and straighter the pores, the more it transmits rays through its substance. Thus, a polished mirror, whose back is covered with a layer of mercury, reflects all rays; likewise, the same mirror without mercury, having straight, large, and numerous pores, lets a significant portion of the rays pass. The more large and straight the pores, the more diaphanous the body: such, it was said, is the diamond; such is water itself; these were the generally accepted ideas, and no one doubted them.

However, all these ideas are entirely false: so much so that what is plausible is often the farthest from the truth. Philosophers have plunged into error in the same way that the common folk are wholly inclined when they think the sun is no larger than it appears to the eyes. Here is where the philosophers erred.

There is no body whose surface we can truly unify. Yet many surfaces appear to us uniform and perfectly polished. Why do we see as uniform and smooth what is not? The most even surface is, relative to the small bodies that compose light, just an accumulation of mountains, cavities, and intervals, just as the tip of the finest needle is actually bristled with prominences and asperities that the microscope reveals.

All the bundles of light rays that would fall on these irregularities would be reflected according to how they had fallen: therefore, having fallen unevenly, they would never reflect regularly, hence one could never see oneself in a mirror. Moreover, glass probably has a thousand times more pores than material; yet every point on the surface returns rays, therefore they are not returned by the glass.

The light that brings us our image from atop a mirror certainly does not come from the solid parts of the mirror’s surface; nor does it come from the solid parts of mercury and tin spread behind this glass. These parts are no flatter, no more unified than the glass itself. The solid parts of the tin and mercury are incomparably larger, wider than the solid constituent parts of light; therefore, if the small particles of light fall on these large parts of mercury, they would scatter in all directions like lead pellets falling on rubble. What unknown power, then, regularly bounces light back to us? It already appears that it is not bodies that return it to us. What seems most known, most indisputable among men, becomes a greater mystery than air’s weight once was. Let’s examine this problem of nature, our astonishment will redouble. We can only learn here with surprise.

Take a piece, a crystal cube for example; here is what happens to the sun’s rays that fall on this solid and transparent body (figure 2).

1° A small part of the rays bounce back to your eyes from its first surface A, without even touching this surface, as will be more amply proven.

2° A very small part of the rays is received in the substance of this body at B; it plays there, gets lost, and extinguishes: this is why there are few perfectly transparent crystals, especially when they are thick.

3° A third part reaches the interior C of the mirror, and from near the surface, it returns to the air, and some rays come to your eyes.

4° A fourth part passes into the air.

5° A fifth part, which is the most considerable, returns from beyond the outer surface D into the crystal, passes back through it, and comes to reflect in your eyes. Let’s examine here only these last rays, which, escaping from the outer surface D, and having found air, rebound from this air towards the eye by re-entering through the crystal. Certainly, they did not encounter solid parts in this air on which they could have bounced: for, if instead of air they encounter water at this surface B, then fewer come back; they enter this water, they penetrate it in large numbers. Now, water is about 800 to 900 times36 heavier, more solid, less rare than air. Yet these rays do not rebound from this water, and they rebound from this air in this glass: therefore, it is not from the solid parts of bodies that light is reflected.

Here is an even more singular and decisive observation: Expose this crystal A B (figure 3) in a dark room to the sun’s rays so that the beams of light reaching its surface B form an angle of more than 40 degrees with the perpendicular P.

Most of these rays then no longer penetrate the air: they all return to this crystal the moment they exit it; they return, as you see, making an imperceptible curvature.

Certainly, it is not the solid surface of the air that has repelled them back into this glass; several of these rays entered the air before when they fell less obliquely; why then at an obliquity of 40 degrees 19 minutes do the majority of these rays no longer pass through? Do they find more resistance, more material in this air, than they find in this crystal that they had penetrated? Do they find more solid parts in the air at 40 degrees and 1/3 than at 40? The air is approximately two thousand four hundred times rarer, less heavy, less solid, than the crystal: therefore, these rays should have passed into the air with two thousand four hundred times more ease than they penetrated the thickness of the crystal. However, despite this prodigious appearance of ease, they are repelled: they are therefore repelled by a force that is here two thousand four hundred times more powerful than air; they are therefore not repelled by air; rays, once again, are therefore not reflected in our eyes by the solid parts of bodies. Light rebounds so little on the solid parts of bodies that it is indeed from the vacuum that it sometimes rebounds: this fact deserves great attention.

You have just seen that light, falling at an angle of 40 degrees 19 minutes on crystal, almost entirely rebounds from the air it encounters at the outer surface of this crystal; that if light falls at an angle one minute smaller, even less passes out of this surface into the air.

Newton asserted that if one could find the secret of removing the air from beneath this piece of crystal, then no more rays would pass through, and all the light would reflect: I experimented; I had an excellent prism encased in the middle of a copper plate; I applied this plate to the top of an open recipient, placed on the pneumatic machine; I brought the machine into my dark room. There, receiving light through a hole onto the prism, and making it fall at the required angle, I pumped air for a very long time; those present saw that as air was pumped, less light passed into the recipient, and finally almost none at all passed. It was a very pleasing spectacle to see this light reflect entirely by the prism onto the floor.

The experiment thus demonstrates that light, in this case, rebounds from the vacuum; but it is well known that this vacuum cannot have any action. What can we then conclude from this experiment? Two very palpable things: first, that the surface of solids does not return light; second, that there is in solid bodies an unknown power that acts on light; and it is this second property that we will examine in its place.

It is only a matter of proving here that light is not reflected to us by solid parts.

Here is another proof of this truth.

Every opaque body, reduced to a thin film, lets rays of a certain kind pass through its substance and reflects other rays; yet if light were returned by bodies, all rays falling equally on these films would be reflected on these films. Finally, we will see that never has such an astonishing paradox been proven in more ways. Let us therefore begin by familiarizing ourselves with these truths.

1° This light, which is believed to be reflected by the solid surface of bodies, actually rebounds without having touched this surface.

2° Light is not sent back from behind a mirror by the solid surface of the mercury; but it is sent back from the depths of the pores of the mirror, and from the pores of the mercury itself.

3° It should not, as has been thought until now, be necessary for the pores of this mercury to be very small to reflect light; on the contrary, they must be large.

It will be a new subject of surprise, for those who have not studied this philosophy, to hear that the secret to making a body opaque is often to enlarge its pores, and that the way to make it transparent is to narrow them. The order of nature will seem entirely changed in appearance: what seemed to should cause opacity is precisely what will create transparency; and what seemed to make bodies transparent will be what makes them opaque. However, nothing is so true, and the most basic experiment proves it.

Dry paper, whose pores are very large, is opaque: no ray of light crosses it; narrow its pores by soaking it, either in water or in oil, it becomes transparent; the same happens with linen, with salt.

It is good to inform the public that a man who recently wrote against these truths, with much more arrogance and contempt than knowledge, wanted to mock Newton over these discoveries. “If the secret,” he said, “of making a body transparent is to narrow its pores, then we will have to make windows smaller to have more light in our room, etc.” I answer that it is quite indecent to be facetious when one pretends to speak philosophically, and that to ridicule Newton is an overly ambitious endeavor; I especially respond that this jester should have considered that it is indeed true that large openings intercepting light would not render light; and that a thin body, pierced by an infinity of small holes exposed to the sun, illuminates us a lot. Oiled paper, wet linen, for example, are thin bodies, whose pores have been shrunk and straightened by oil or water, and light passes through these pores made straighter; but it will not pass through the largest sieves that will cross each other and intercept the rays.

Before taking a mocking tone, one should be very sure that they are right; and when one is finally assured of being right, one should not mock.

Let’s return, and summarize that there are thus unknown principles that perform these wonders, causes that bounce light back before it has touched a surface, that send it back from the pores of the transparent body, that bring it back from the very midst of the vacuum; we are irrevocably obliged to accept these facts, whatever their cause may be.

Let us therefore study the other mysteries of light, and see if from these surprising effects we can reach some undeniable principle, which must be accepted as well as these effects themselves.

Chap. IV. — On the property that light has to break when passing from one substance into another, and to take a new path. How light breaks.

The second property of light rays that must be thoroughly examined is their deviation from their path when passing from the sun into air, from air into glass, from glass into water, etc. This new direction in these different mediums, this bending of light is called refraction; it is by this property that an oar dipped in water appears bent to the sailor handling it; it is what allows us to see an object in a bowl by adding water, which we could not see before from the same position.

Finally, it is through refraction that our eyes enjoy sight. The admirable secrets of refraction were unknown to antiquity, which, though having it before their eyes and using it every day, left no writings that might suggest they had guessed its reason. Thus, even today we are ignorant of the cause of the movements of our bodies and the thoughts of our soul; but this ignorance is different. We will never have instruments refined enough to see the first springs of ourselves; but human ingenuity has made new eyes that have revealed almost everything about the effects of light that men are allowed to know.

Here, one must form a clear idea of a very common experiment (figure 4). A gold coin is in this basin; your eye is placed at the edge of the basin at such a distance that you cannot see the coin.

Pour water into it: you could not see it initially where it was; now you see it where it is not: what has happened?

The object A reflects a ray that strikes the edge of the basin (figure 5), and which will never reach your eye; it also reflects this ray A B, which passes over your eye: now you receive this ray A B; it is not your eye that has changed place, so it must be the ray A B; it has clearly turned at the edge of this basin, passing from water into air; thus it strikes your eye at C.

But you always see objects in a straight line, so you see the object along the straight line C D, so you see the object at point D above the place where it actually is.

If this ray breaks in one direction when it passes from water into air (figure 6), it must break in the opposite direction when it enters from air into water.

I raise a perpendicular over this water, the ray A, which, starting from the luminous point, breaks at point B and approaches the perpendicular in the water following the path B D; and the same ray D B, passing from water into air, breaks going towards A and moving away from this same perpendicular: light thus refracts according to the mediums it traverses. It is on this principle that nature has arranged the different humors that are in our eyes, so that the beams of light passing through these humors break in such a way that they converge afterwards at a point on our retina; it is finally on this principle that we manufacture glasses, whose lenses undergo even greater refractions than those that occur in our eyes, and which, thus bringing together more converged rays, can extend up to two hundred times the strength of our sight; just as the invention of levers has given new strength to our arms, which are natural levers. Before explaining the reason Newton found for this property of light, you want me to explain how this refraction acts in our eyes, and how the sense of sight, the most extensive of all our senses, owes its existence to refraction. Although this matter is well-known, beginners who might read this little work will be glad not to have to look elsewhere for what they would like to know about sight.

Chap. V. — On the conformation of our eyes; how light enters and acts in this organ. — Description of the eye. Presbyopic eye. Myopic eye.

To understand the human eye from a physicist’s perspective, which considers only vision, one must first know that the first white envelope, the rampart and ornament of the eye, transmits no ray. The whiter and more uniform this part of the eye, the more it reflects light; and when some strong emotion sends new spirits to the face, which further tense and shake this tunic, then sparks seem to fly from it.

In the middle of this membrane, the cornea slightly rises, thin, hard, and transparent, exactly like the glass of your watch placed over a ball.

Beneath this cornea is the iris, another membrane which, colored by itself, spreads its colors over the transparent cornea covering it: it is the iris that makes eyes blue or black. It is pierced in its middle, which thus always appears black; and this middle is the pupil of the eye. It is through this opening that light rays are admitted: it enlarges involuntarily in dark places, to receive more rays; it then narrows when bright light offends it.

The rays admitted through this pupil have already undergone quite a strong refraction while passing through the cornea covering it. Imagine this cornea as the glass of your watch; it is convex on the outside and concave on the inside: all oblique rays are broken in the thickness of this glass; but then its concavity approximately restores what its convexity had broken. The same happens in our cornea. The rays, thus broken and refracted, find, after passing through the cornea, a transparent humor through which they pass. This water is called the aqueous humor. Anatomists still do not agree on the shape of this small reservoir; but, whatever its figure, nature seems to have placed there this clear and limpid humor to operate refractions, to transmit light purely, so that the crystalline, which is behind, can move effortlessly and freely change shape, to maintain necessary moisture, etc.

Finally, having exited this water, the rays encounter a kind of liquid diamond, cut into a lens, and set in a thin and diaphanous membrane. This diamond is the crystalline; it breaks all the oblique rays: it is a principal organ of refraction and sight, perfectly similar to a lens in a telescope. Consider this crystalline or lens (figure 7).

The perpendicular ray A penetrates it without deviating; but the oblique rays B C divert within the thickness of the glass approaching the perpendiculars that would be drawn at the points where they land; then, when they exit the glass to pass into the air, they break again moving away from the perpendicular: this new breakage is precisely what causes them to converge at D, the focus of the lens.

Now the retina, this light membrane, this expansion of the optic nerve, which lines the back of our eye, is the focus of the crystalline; it is to this retina that the rays converge; but before reaching there, they encounter another medium they traverse: this new medium is the vitreous humor, less solid than the crystalline, less fluid than the aqueous humor.

It is in this vitreous humor that the rays have time to assemble, before coming to make their final union at points on the back of our eye. So, imagine under this lens of the crystalline, this vitreous humor on which the crystalline rests; this humor holds the crystalline in its concavity, and is rounded towards the retina.

The rays, escaping from this last humor, thus finish converging. Each beam of rays coming from a point of the object strikes a point on our retina.

A diagram, where each part of the eye is seen under its proper name, will explain all this artifice better than could lines, A’s, and B’s (figure 8).

Several ancient philosophers believed37 that, far from the beams of light, reflected from objects, coming to draw their image at the back of our eyes, it was rather from our eyes themselves that beams of light went out to seek the objects, and brought back some kind of intentional species. This idea was worthy of the rest of Greek physics; I don’t say Roman, because the Romans almost never had it.

It was Giovanni Battista della Porta, an Italian, who in 1560 first unfolded the true causes of sight, and by the simple experiment of a white sheet exposed to a ray of sunlight in a dark room38, suspected that the same must happen in the eye. He did not dare to imagine that the rays penetrated as far as the retina; he believed that objects painted themselves on the crystalline, and everyone believed him, until Kepler and Descartes explained all the mechanics of vision, all the refractions that occur in our eyes, and what causes short-sightedness and how it can be aided. Dr. Hooke, precursor to Newton, subsequently showed through experiments that an object, to be perceived, must at least cast an image on the retina that is the eight-thousandth part of an inch.

Having thus developed the structure of the eyes solely for the use of optics, one can easily understand why glasses are so often necessary, and what their use is.

Often an eye will be too flat, either due to the shape of its cornea or because its crystalline, dried by age or disease, then the refractions are weaker and less abundant, the rays no longer converge on the retina. Consider this too flat eye, called the presbyopic eye.

For simplicity, let’s only look at three beams, three cones of rays, that fall on this eye; they will converge at points A A A, beyond the retina: it will see objects blurred (figure 9).

Nature provides a remedy for this inconvenience through the strength it has given to the muscles of the eye to lengthen or flatten the eye, to approach or recede from the retina. Thus, in this elderly or diseased eye, the crystalline has the ability to advance a bit, and go towards D D; then the space between the crystalline and the back of the retina becomes larger, the rays have time to come together on the retina, instead of going beyond; but when this strength is lost, human ingenuity compensates: a lens is placed between the object and the weakened eye. The effect of this lens is to bring the rays it has received closer; the eye thus receives them, more converged and in greater numbers they come to a point on the retina as required then the vision is clear and distinct.

Consider another eye, which suffers from the opposite condition (figure 10); it is too round: the rays converge too soon, as you see at point B; they cross too quickly, separate at B, and create a blur on the retina. This is what is called a myopic eye. This inconvenience diminishes as other age-related issues, which are dryness and weakness, gradually flatten this overly round eye; and that’s why it is said that shortsightedness lasts longer. It’s not that it actually lasts longer than other types; but at a certain age, the dried-out eye flattens: then the person who previously had to hold their book three or four inches from their eye, can sometimes read at a foot’s distance; but then their vision quickly becomes blurry and confused, they cannot see distant objects: such is our condition, that one defect is almost never repaired except by another.

While this eye is too round, it needs a glass that prevents the rays from converging so quickly: this glass does the opposite of the first; instead of being convex on both sides, it will be slightly concave on both sides, and the rays will diverge in this glass, whereas they would converge in the other. Consequently, they will come to unite farther than they did before in the eye; and then this eye will enjoy perfect vision. The convexity and concavity of the glasses are tailored to the defects of our eyes: this is why the same glasses that give clear vision to one elderly person are of no help to another, because there are neither two diseases, nor two men, nor two things in the world that are equal, except for the primary principles of homogeneous bodies.

It is said that antiquity was not aware of these glasses; however, it knew about burning mirrors: a discovered truth is not always a reason for discovering other truths related to it. The attraction of the magnet was known, and its direction escaped observation. The proof of the circulation of the blood was in the very bleeding that all Greek doctors practiced; and yet, no one suspected that the blood circulated. But how could the Greeks and Romans engrave those stones whose details we today can only admire with a magnifying glass? On the other hand, if the art of making glasses was known to the ancients, how did it perish? A secret can be lost, but any useful art persists. It is believed that it was during the time of Roger Bacon, in the early 13th century, that these glasses called spectacles, and the magnifying glasses that give new eyes to the elderly, were discovered: for he is the first to speak of them with some clarity, and they only began to be discussed around that time; for nearly four hundred years these glasses have been used without precisely knowing by what mechanics they aid our eyes, much as we still use the compass without understanding the cause that directs the magnetized needle.

You have just seen the effects that refraction has in our eyes, whether the rays arrive without intermediate aid or have passed through crystals: you understand that without this refraction performed in our eyes, and without this reflection of rays from the surfaces of bodies towards us, the organs of sight would be useless. The means that nature employs to perform this refraction, the laws it follows, are mysteries that we will unfold. We must first finish what we have to say about sight; it is necessary to answer these very natural questions: Why do we see objects beyond a mirror, and not on the mirror itself? Why does a concave mirror make the object larger? Why does a convex mirror make the object smaller? Why do telescopes bring things closer and enlarge them? By what mechanism does nature enable us to recognize sizes, distances, positions? What is the true reason that we see objects as they are, even though they are painted inverted in our eyes? There is nothing here that does not merit the curiosity of every thinking being; but we would not dwell on these subjects, which so many illustrious writers have discussed, and we would refer to them, if we did not have to introduce some quite new truths, curious to a small number of readers.

Chap. VI. — On mirrors, telescopes. Reasons given by mathematics for the mysteries of vision; that these reasons are not sufficient. — Plane mirror. Convex mirror. Concave mirror. Geometric explanations of vision. No immediate correlation between the rules of optics and our sensations. Example in proof.39

The rays, which an unknown power until our days makes reflect back to your eyes from the surface of a mirror without touching it, and from the pores of the mirror without touching the solid parts; these rays, I say, return to your eyes in the same direction they reached the mirror. If it is your face you are looking at, the rays from your face parallel and perpendicular to the mirror bounce back just like a ball that bounces perpendicularly off the floor.

If you look in this mirror M (figure 11), at an object beside you like A, the same thing happens to the rays coming from this object as to a ball that would bounce at B, where your eye is. This is called the angle of incidence being equal to the angle of reflection.

Line A C is the line of incidence, line C B is the line of reflection. It is well understood, and the mere statement of it demonstrates, that these lines form equal angles on the surface of the glass; now why do I not see the object either in A, where it is, or in C, from where the rays come to my eyes, but in D, behind the mirror itself?

Geometry will tell you (figure 12): It’s because the angle of incidence is equal to the angle of reflection; it’s because your eye in B projects the object at D; it’s because objects can only act on you in a straight line, and the straight line continued in your eye B to behind the mirror at D is as long as line A C and line C B taken together.

Furthermore, it will also tell you: You never see objects except from the point where the rays begin to diverge. Consider this mirror M I.

The bundles of rays emanating from each point of object A begin to diverge the moment they leave the object; they arrive on the surface of the mirror: there each of these rays falls, spreads out, and reflects back towards the eye. This eye projects them onto points D D, at the ends of the straight lines where these same rays would meet; but, meeting at points D D, these rays would do the same as at points A A: they would start to diverge; therefore you see object A A at points D D.

These angles and lines certainly help you understand this trick of nature; but they are far from being able to teach you the physical reason why your mind unhesitatingly projects the object beyond the mirror at the same distance as it is in front of it. These lines depict what happens, but they do not teach you why it happens40.

If you want to know how a convex mirror diminishes objects, and how a concave mirror enlarges them, these lines of incidence and reflection will give you the same reason.

You are told: This cone of rays diverging from points A (figure 13), and falling on this convex mirror, forms angles of incidence equal to angles of reflection, whose lines go into our eye. Now these angles are smaller than if they had fallen on a flat surface: therefore, if they are supposed to pass in B, they will converge much sooner, thus the object that would be at B B would be smaller.

Now your eye projects the object at B B to the points from where the rays would begin to diverge: therefore the object must appear smaller, as it indeed does in this figure. For the same reason it appears smaller, it also appears closer, since in fact the points where the rays B B would end up are closer to the mirror than the rays A A.

By the reasoning of opposites, you must see objects larger and farther away in a concave mirror, by placing the object close enough to the mirror (figure 14).

For the cones of rays A A diverging upon the mirror at the points where these rays fall, if they reflected through this mirror, they would only reunite at B B: therefore it is at B B that you see them. Now B B is larger and farther from the mirror than A A is: therefore you see the object larger and farther.

This is generally what happens with the rays reflected to your eyes; and this single principle, that the angle of incidence is always equal to the angle of reflection, is the primary foundation of all the mysteries of catoptrics.

Now it is about understanding how glasses increase these sizes and bring these distances closer; finally, why, with objects painted upside down in your eyes, you still see them as they are.

Regarding sizes and distances, here is what mathematics will teach us. The larger an object makes an angle in your eye, the larger the object will appear to you: nothing is simpler. This line H K, which you see at a hundred steps, makes an angle in the eye A (figure 15); at two hundred steps, it makes an angle half as small in the eye B (figure 16). Now the angle formed in your retina, and of which your retina is the base, is like the angle of which the object is the base. These are angles opposite at the vertex: therefore, by the first notions of geometry, they are equal; therefore if the angle formed in the eye A is double that formed in the eye B, this object must appear twice as large to the eye A as to the eye B.

Now, for the eye being in B to see the object as large as the eye sees it in A, it is necessary to ensure that this eye B receives an angle as large as that of the eye A, which is twice as close. The lenses of a telescope will have this effect (figure 17).

Let’s put here only one lens for simplicity, and disregard the other effects of several lenses. The object H K sends its rays to this lens. They converge at some distance from the lens. Let’s conceive of a lens shaped such that these rays cross to go and form in the eye in C an angle as large as that of the eye in A: then the eye, we are told, judges by this angle. It thus sees the object of the same size as seen by the eye in A. But in A, it is seen at a hundred steps' distance: therefore in C, receiving the same angle, it will see it still at a hundred steps' distance. All the effects of multiplied glasses, of various telescopes, and of microscopes that magnify objects, consist then in making things be seen under a larger angle. The object A B (figure 18) is seen through this glass under the angle D C D, which is much larger than the angle A C B.

You also ask optical rules why you see objects in their situation, although they are painted inverted in our eyes?

The ray that starts from this man’s head A (figure 19) comes to the lower part of your retina at A; his feet B are seen by the rays B B at the upper part of your retina at B. Thus this man is actually painted upside down, head downwards and feet upwards, at the back of your eyes. Why then do you not see this man inverted, but upright, as he is!

To resolve this question, the comparison of the blind man who holds crossed sticks with which he guesses very well the position of objects is used.

For the point that is to the left, being felt by the right hand with the aid of the stick, he immediately judges it to be on the left; and the point that his left hand felt through the other stick, he judges to be on the right without error.

All masters of optics thus tell us that the lower part of the eye immediately relates its sensation to the upper part of the object, and the upper part of the retina naturally relates the sensation to the lower part; thus we see the object in its true position41.

But when you have perfectly understood all these angles, and all these mathematical lines, through which the path of light is followed to the back of the eye, do not believe therefore that you know how you perceive the sizes, distances, positions of things. The geometric proportions of these angles and lines are accurate, true; but they have no more relationship to our sensations than between the sound we hear and the size, distance, position of the heard thing. By sound, my ear is struck; I hear tones, and nothing more. By sight, my eye is affected; I see colors, and nothing more. Not only can the proportions of these angles and lines in no way be the immediate cause of the judgment I form of objects, but in many cases these proportions do not at all agree with the way we see objects.

For example, a man seen at four paces and at eight paces is seen of the same size. However, the image of this man, at four paces, is, with very little approximation, double in your eye than it traces at eight paces. The angles are different, and you always see the object of the same size; therefore, it is evident by this single example, chosen among many, that these angles and lines are not at all the immediate cause of the way we see.

Before then continuing the research we have started on light, and on the mechanical laws of nature, you order me to say here how the ideas of distances, sizes, positions of objects, are received in our soul. This examination will provide something new and true: it is the only excuse for a book.

Chap. VII. — How we know distances, magnitudes, figures, situations. — Neither angles nor optical lines can make us know distances. Example in proof. These optical lines do not make known either magnitudes or figures. Example in proof. Proof by the experience of the blind-born healed by Cheselden. How we know distances and magnitudes. Example. We learn to see as to read. Sight cannot make known extent.42

Let’s start with distance. It is clear that it cannot be perceived immediately by itself, as distance is merely a line from the object to us. This line ends at a point; therefore, we only feel that point, and whether the object is a thousand leagues away, or at a foot, that point is always the same.

Therefore, we have no immediate means to perceive distance all at once, as we do to feel by touch whether a body is hard or soft; by taste, whether it is sweet or bitter; by hearing, whether of two sounds one is low and the other high. For, take careful note, the parts of a body that yield to my finger are the closest cause of my sensation of softness, and the vibrations of the air, excited by the sounding body, are the closest cause of my sensation of sound; hence, if I cannot immediately have an idea of distance, it must be that I know this distance by means of some other intermediate idea. But at least I must perceive this intermediate: for an idea that I do not have will certainly not help me to have another. I say that a certain house is a mile from a certain river; but if I do not know where the river is, I certainly do not know where the house is. A body yields easily to the impression of my hand, I immediately conclude its softness; another resists, I immediately feel its hardness: thus, I would need to feel the angles formed in my eye to immediately conclude the distances of objects. But most people do not even know if these angles exist: thus, it is evident that these angles cannot be the immediate cause of your knowledge of distances.

One who, for the first time in their life, heard the noise of a cannon, or the sound of a concert, could not judge whether this cannon was fired, or this concert performed, at a league, or at thirty paces. Only experience can accustom him to judge the distance between him and the place from where this noise emanates. The vibrations, the undulations of the air, bring a sound to his ears, or rather to his soul; but this noise no more alerts his soul to where the noise begins than it teaches him the shape of the cannon or the musical instruments.

It is precisely the same with respect to the rays of light that come from an object: they do not tell us at all where this object is.

They do not make us more aware of the sizes, or even the shapes.

I see from afar a kind of small tower. I move closer, perceive, and touch a large quadrangular building. Certainly, what I see and what I touch is not what I saw. That small round object, which was in my eyes, is not that large square building.

Thus, the measurable and tangible object is one thing, the visible object another. I hear the noise of a carriage from my room: I open the window, and I see it; I go down, and I get into it. Now, this carriage that I heard, this carriage that I saw, this carriage that I touched, are three completely different objects of three of my senses, which have no immediate relation to one another.

Furthermore, it is demonstrated, as I have said, that an angle twice as large, very nearly, is formed in my eye when I see a man four feet from me, as when I see the same man eight feet from me. Yet I always see this man the same size: how then does my sensation thus contradict the mechanism of my organs? The object is really twice as small in my eyes, and I see it twice as large. It is in vain that one tries to explain this mystery by the path, or by the shape taken by the crystalline in our eyes. Whatever assumption one makes, the angle under which I see a man four feet from me is always twice the angle under which I see him at eight feet; and geometry will never solve this problem, nor will physics: for no matter how much you suppose that the eye takes a new conformation, that the crystalline advances, that the angle enlarges, all this will operate equally for the object that is eight paces and for the object that is four. The proportion will always be the same: if you see the object at eight paces under an angle half as large, you also see the object at four paces under an angle half as large or thereabouts. Thus, neither geometry nor physics can explain this difficulty.

These geometric lines and angles are not really the cause of our seeing objects in their place any more than they are of our seeing them of such a size, and at such a distance.

The soul does not consider whether such a part will be painted at the bottom of the eye; it relates nothing to lines it does not see. The eye merely lowers to see what is near the ground, and raises to see what is above the ground.

All this could not be clarified, and put beyond all contestation, except by some blind-born person who had been given the sense of sight. For if this blind person, the moment he opened his eyes, had judged distances, sizes, and situations, it would have been true that the optical angles, formed all at once in his retina, would have been the immediate causes of his feelings. Thus, Doctor Barclay assured, after Mr. Locke (and even going further than Locke in this), that neither position, size, distance, nor shape would be discerned at all by this blind person whose eyes would suddenly receive light.

But where to find the blind person on whom the indubitable decision of this question depended? Finally, in 1729, Mr. Cheselden, one of those famous surgeons who combine the skill of the hand with the greatest lights of the mind, having imagined that it might be possible to give sight to a blind-born by lowering what are called cataracts, which he suspected had formed in his eyes almost at the time of his birth, he proposed the operation. The blind person was reluctant to consent. He did not quite see how the sense of sight could greatly increase his pleasures. Without the desire that was inspired in him to learn to read and write, he would not have wanted to see. He verified by this indifference that it is impossible to be unhappy from the deprivation of goods of which one has no idea: a very important truth. In any case, the operation was performed, and succeeded. This young man of about fourteen years saw the light for the first time. His experience confirmed everything that Locke and Barclay had so well foreseen. He could not distinguish sizes, positions, or even shapes for a long time. An object an inch in front of his eye, which hid a house from him, seemed as large as the house. Everything he saw at first seemed to be on his eyes, and to touch them as the objects of touch touch the skin. He could not initially distinguish what he had judged round by the help of his hands, from what he had judged angular, nor discern with his eyes whether what his hands had felt to be up or down was indeed up or down. He was so far from knowing the sizes that after he had finally perceived by sight that his house was larger than his room, he could not conceive how sight could give this idea. It was only after two months of experience that he was able to perceive that paintings represented solid bodies; and, when after this long groping of a new sense in him he had felt that bodies, and not just surfaces, were painted in the paintings, he reached out his hand, and was astonished not to find with his hands these solid bodies, of which he was beginning to perceive the representations. He asked which was the deceiver, the sense of touch or the sense of sight.

This was therefore an irrevocable decision that the way we see things is not at all the immediate result of the angles formed in our eyes: for these mathematical angles were in the eyes of this man as in ours, and did not serve him at all without the aid of experience and other senses.

How then do we represent sizes and distances to ourselves? In the same way we imagine the passions of men, by the colors they paint on their faces, and by the alteration they bring to their features. There is no one who does not read at a glance on another’s forehead pain or anger. It is the language that nature speaks to all eyes; but only experience teaches this language. Also, experience alone teaches us that when an object is too far away, we see it indistinctly and weakly. From there we form ideas, which then always accompany the sensation of sight. Thus every man who, at ten paces, has seen his horse five feet high, if he sees, a few minutes later, this horse the size of a sheep, his soul, by an involuntary judgment, concludes instantly that this horse is very far away.

It is indeed true that when I see my horse the size of a sheep, a smaller image forms in my eye, a sharper angle; but this accompanies, not causes, my perception. Similarly, a different kind of stirring happens in my brain when I see a man blush from shame than from anger; but these different impressions wouldn’t teach me anything about what’s going on in that man’s mind without experience, whose voice alone is heard.

Far from this angle being the immediate cause of my judgment that a large horse is far away when I see this horse very small, on the contrary, at all times, I see the same horse equally large at ten, twenty, thirty paces, although the angle at ten paces is double, triple, quadruple.

I look from afar, through a small hole, at a man on a roof; the distance and the few rays initially prevent me from distinguishing if it is a man: the object appears very small, I believe I see a statue of at most two feet; the object moves, I judge that it is a man, and from that very instant, this man appears to be of ordinary size: where do these two different judgments come from?

When I believed I saw a statue, I imagined it to be two feet because I saw it under such an angle: no experience compelled my soul to deny the impressions made on my retina; but as soon as I judged that it was a man, the connection made by experience, in my brain, between the idea of a man and the idea of a height of five to six feet, forces me, without thinking about it, to imagine, by a sudden judgment, that I see a man of such height, and to actually see such a height43.

We must absolutely conclude from all this that distances, sizes, and situations are not, strictly speaking, visible things, that is, they are not the proper and immediate objects of sight. The proper and immediate object of sight is nothing other than colored light: everything else, we only feel gradually and by experience. We learn to see precisely as we learn to speak and to read. The difference is that the art of seeing is easier, and nature is equally our teacher for all.

The sudden judgments, almost uniform, that all our souls, at a certain age, make about distances, sizes, and situations, make us think that it is enough to open our eyes to see in the way we do. We are mistaken; it requires the help of the other senses44. If men had only the sense of sight, they would have no means to know the extent in length, width, and depth; and a pure spirit might not know it either, unless God revealed it to him. It is very difficult to separate in our understanding the extension of an object from the colors of this object. We never see anything but what is extended, and from there we are all inclined to believe that we indeed see extent. We can hardly distinguish in our soul this yellow, which we see in a gold Louis, from that gold Louis of which we see the yellow. It is as when we hear the word gold Louis pronounced, we cannot help but attach the idea of this coin to the sound we hear pronounced45.

If all men spoke the same language, we would always be ready to believe that there is a necessary connection between words and ideas. Now all men have the same language, in terms of imagination. Nature tells them all: When you have seen colors for a certain time, your imagination will represent to you all, in the same way, the bodies to which these colors seem attached. This prompt and involuntary judgment you form will be useful in the course of your life: for if you had to wait, to estimate the distances, the sizes, the situations of everything around you, that you had examined angles and visual rays, you would be dead before knowing if the things you need are ten paces from you, or a hundred million leagues, and if they are the size of a mite or a mountain. It would be much better for you to be born blind.

We are very wrong when we say that our senses deceive us. Each of our senses performs the function to which nature has destined it. They assist each other in sending to our soul, through the hands of experience, the measure of knowledge that our being can contain. We ask our senses for what they are not made to give us. We would like our eyes to make us know solidity, size, distance, etc.; but touch must agree in this with sight, and experience must second them. If Father Malebranche had considered nature from this side, he might have attributed fewer errors to our senses, which are the only sources of all our ideas.

It is not, of course, necessary to extend to all cases this kind of metaphysics that we have just seen: we should only call it to aid when mathematics are insufficient; and this is yet another error that must be recognized in Father Malebranche. He attributes, for example, to the mere imagination of men, effects of which the only rules of optics give reason. He believes that if the stars appear larger on the horizon than at the meridian, it is to the imagination alone that we must blame. We will, in the next chapter, explain this phenomenon, which has exercised so many philosophers for a hundred years.

Chap. VIII. — Why the sun and the moon appear larger on the horizon than at the meridian. — System of Malebranche denied by experience. Explanation of the phenomenon.

Wallis was the first to believe that the prolonged interposition of land and even clouds makes the sun and moon appear larger on the horizon than at the meridian. Malebranche reinforced this opinion with all the proofs his ingenuity could muster. Régis had a famous dispute with him over this phenomenon, attributing it to refractions occurring in the vapors of the earth, and he was mistaken, for refractions have exactly the opposite effect to that which Régis attributed to them; but Father Malebranche was no less mistaken in maintaining that the imagination, struck by the long expanse of land and clouds on our horizon, represents the same celestial body as larger at the end of these lands and clouds than when, having reached its highest point, it is seen without any interposition.

The simplest experiments refute Malebranche’s system. A few years ago, curious to examine this phenomenon in sequence, I had cardboard tubes seven to eight feet long, half a foot in diameter made; I had several children whose imaginations were not at all accustomed to judging the size of the celestial body by the expanse that appears between the celestial body and the eyes look at the sun on the horizon. They didn’t even see the ground or the clouds. The tube only allowed them to see the sun, and all saw it as I did much larger than at noon. This experiment and several others led me to imagine another cause; and I already had the misfortune to make a system, when the mathematical solution of this problem by Mr. Smith fell into my hands, sparing me the errors of a hypothesis. Here is the explanation that deserves to be studied.

It must first be established that, according to the rules of optics, the sky should appear to us as a flattened vault. Here is a familiar proof.

Our vision extends distinctly up to the point where objects make an angle in our eye of at least one eight-thousandth of an inch, according to Hooke’s observations. A man O P (figure 20) 5 feet tall looks at the object A B also 5 feet tall and 25,000 feet away: he sees it under the angle A B; but this angle A B not being in the eye one eight-thousandth of an inch, he does not distinguish it. But if he looks at the object C, the angle is even smaller; he sees it as if the object were in A D; thus everything behind C becomes even less distinct; houses, clouds, that will be behind C, must appear to skim the horizon towards G; all the clouds thus lower for us on the horizon at the distance of 25,000 feet, that is about one league of 3,000 steps and two-thirds, and they lower by degrees: therefore all the clouds that rise in G (figure 21), about three-quarters of a league high, must appear to us to skim our horizon; thus, instead of seeing the clouds G as high as the cloud N, we see the clouds G touching the ground, and the cloud N raised about three-quarters of a league above our head; we should therefore not see the sky either as a ceiling or as a circular arch, but as a flattened vault, whose large diameter B B is about six times larger than the small A D.

We thus see the sky in this manner B A B; and when the sun or moon are in B on the horizon, they appear more distant (to us who are in D) by about a third, than when these celestial bodies are in A: now, we must see them under the angles that will come to our eyes from B and from A; it remains then to examine these angles (figure 22). It would seem at first that they should be smaller when the object is more distant; and larger, when it is closer; but here it is quite the contrary.

The real celestial body, the tangible celestial body rolls in B D R E; but the apparent celestial body goes in the curve B A G G. Now the angles are formed by the apparent object; therefore draw angles from the eye which is in P to the real places of the celestial body D, these angles would necessarily graze the apparent celestial bodies: you see, for example, that the angle is considerably large at the horizon in G, and becomes quite small in C; the difference is greater at the meridian. The celestial body at the meridian has its disk as 3, and at the horizon about as 9; for the diameters of the celestial body are as its apparent distances: now, the apparent distance of the celestial body is about 9 at the horizon, and 3 at the meridian; thus is its apparent size.

This truth is confirmed by another experiment of a similar kind: look at two stars really distant from each other by a tenth of a degree; they appear much more distant at the horizon, and much closer towards the meridian.

These two stars always equally distant are seen under the angle F C D towards the horizon (figure 23), which is much larger than the angle F A B at the meridian: you see that this apparent difference comes precisely by the same reason that I have just reported.

Here then, according to this rule and according to the observations that confirm it, are the proportions of the apparent sizes and distances of the sun and moon:

On the horizon, these celestial bodies are seen of size 100; At 15 degrees above, of size 68; At 30 degrees, of size 50; At 90 degrees, of size 30.

Similarly, any two stars that always maintain the same distance between them appear at the horizon as 100 units apart, and at the meridian as 30 units apart; this is always, as you see, the proportion of about 9 to 3.

This theory is further confirmed by another observation. The moon appears considerably larger at certain times of the year than at others; the sun also appears larger in winter than in summer; and the differences in this apparent size being more noticeable towards the horizon than at the meridian, they are more easily noted. The reason for this increase in size is that when the diameter of the moon and sun appear larger, these celestial bodies are in fact closer to us: the sun is closer to the earth in winter than in summer, by about twelve hundred thousand leagues: thus in winter it appears larger; but this width of its disk is slightly reduced by the refractions of the dense air: the moon in summer is at its perigee; thus it appears under a larger diameter, and the width of its disk at the horizon is even less reduced in summer than in winter, because the air, in summer, is more subtle and rarer.

This phenomenon is therefore entirely within the realm of geometry and optics, and Doctor Smith has the glory of having finally found the solution to the problem on which the greatest geniuses had made useless systems46.

Chap. IX. — On the cause that makes light rays break when passing from one substance into another; that this cause is a general law of nature, unknown before Newton; that the inflection of light is also an effect of this cause, etc. — What is refraction. Proportion of refractions found by Snellius. What is the sine of refraction. Newton’s great discovery. Light broken before entering bodies. Examination of attraction. Attraction must be examined before rebelling against this term. Impulse and attraction, both certain and unknown. In what attraction is an occult quality. Proofs of attraction. Inflexion of light near bodies that attract it.

We have already seen the almost incomprehensible artifice of light reflection, which known impulse cannot cause. The refraction we are about to examine again is no less astonishing.

Let’s start by firmly establishing a clear idea of the thing that needs explaining. Remember that when light falls from a rarer, lighter substance, such as air, into a heavier, denser substance, like water, which seems to resist it more, the light then deviates from its path, breaking towards a perpendicular that could be erected on the surface of this water.

Mr. Leclerc, in his Physics, said the complete opposite, due to lack of attention. In his book V, chapter viii: “The greater the resistance of bodies, he says, the more light that falls into them deviates from the perpendicular. Thus the ray deviates from the perpendicular when passing from air into water.”

This is not the only mistake in Leclerc; and a person who had the misfortune to study physics from this author’s writings would have mostly false or confused ideas.

To get a clear idea of this truth, look at this ray that falls from the air into this crystal (figure 24).

You know how it breaks. This ray A E makes an angle with the perpendicular B E when falling on the surface of this crystal. The same ray, refracted in this crystal, makes another angle with this same perpendicular that governs its refraction. Measuring this incidence and the breaking of light was necessary. It seems like a very easy thing; however, the Arab mathematician Alhazen, Vitellio, even Kepler, failed. Snellius Villebrod is the first, according to Huygens, an eyewitness, who found this constant proportion in which light breaks in given mediums. He used secants. Descartes then used sines, which is exactly the same proportion, the same theorem, under other names. This proportion is very easy to understand for those who are the least familiar with geometry.

The larger the line A B you see, the larger the line C D will also be. This line A B is what is called the sine of incidence. This line C D is the sine of refraction47. This is not the place to explain in general what a sine is. Those who have studied geometry know it well enough. Others might be a bit embarrassed by the definition. It is sufficient to know well that these two sines, no matter their size, are always in proportion in a given medium. Now, this proportion is different when refraction occurs in a different medium.

The light that falls obliquely from air into crystal breaks in such a way that the sine of refraction C D is to the sine of incidence A B as 2 to 3: which means nothing else but that this line A B is one third larger in air, in this case, than line C D in this crystal.

In water, this proportion is 3 to 4. Thus, it is palpable that, in all cases, in all possible obliquities of incidence, the refractive force of crystal is to that of water as 9 is to 8; not only must we know the cause of refraction but also of all these different refractions. This is where philosophers have all made hypotheses and been mistaken.

Finally, Newton alone found the true reason that was being sought. His discovery certainly deserves the attention of all ages: for it is not only about a particular property of light, although that would already be significant; we will see that this property belongs to all bodies in nature.

Consider that the rays of light are in motion; that if they turn by changing their course, it must be by some primitive law, and that what happens to light should happen to all bodies of the same smallness as light, all other things being equal.

Let a lead ball A (figure 25) be pushed obliquely from air into water, it initially experiences the opposite of what happened to this ray of light: for this fine ray passes through pores, and this ball, whose surface is broad, meets the surface of the water that supports it.

This ball thus initially moves away from the perpendicular B; but when it has lost all that oblique movement that was imparted to it, it then falls, almost along a perpendicular that would be erected from the point where it begins to descend. It delays, as is known, its fall in the water because the water resists it; but a ray of light increases its speed in water, because the water does not resist those of the rays that penetrate it.

There is therefore a force, whatever it may be, that acts between bodies and light.

That this attraction, this tendency exists, we cannot doubt: for we have seen light, attracted by glass, enter it without touching anything: thus, this force necessarily acts in a perpendicular line, the perpendicular being the shortest path.

Since this force exists, it is present in all parts of the body that exerts it. The parts of the surface of any body therefore experience this power before it penetrates the interior of the substance, before it reaches the point where it is directed (figure 26). Thus, as soon as this ray arrives near the surface of the crystal or water, it already slightly takes the path of the perpendicular.

It already breaks a little at C before entering: the deeper it enters, the more it breaks, because the closer it gets, the more it is attracted. There is another important reason why the ray necessarily bends by a gentle curve before entering straight into the crystal: it is because there are no strict angles in nature; a continuous movement can only change direction by going through all possible degrees of change; it cannot, from a straight line, suddenly move into another straight line without drawing a small curve that joins these two lines together. Thus, the principle of continuity, established by Leibniz, and Newton’s attraction, come together in this phenomenon. Thus, the ray does not fall exactly perpendicularly, and does not follow its initial oblique straight line when crossing this water or glass; but it follows a line that participates from both sides, and descends all the faster as the attraction of this water or crystal is stronger. Therefore, far from water breaking the rays of light by resisting them, as was believed, it actually breaks them because it does not resist, and, on the contrary, because it attracts them. It must therefore be said that the rays break towards the perpendicular, not when they pass from a less resistant medium to a more resistant medium, but when they pass from a less attracting medium to a more attracting medium. Note that by this term ‘attracting’ we only mean the point towards which a recognized force, an undeniable property of matter, which is very sensitive between light and bodies, is directed. Consider that, since 1672 when Newton demonstrated this attraction, no philosopher has been able to imagine a plausible reason for this bending of light.

Some tell you: The crystal refracts light rays because it resists them; but, if it resists them, why do these rays enter it more easily and with greater speed48? Others imagine a matter in the crystal that opens easier paths on all sides; but if these paths are so easy on all sides, why doesn’t the light enter without deviating?

Some invent atmospheres; others, vortices; all their systems collapse in some way: it is necessary, I believe, to stick to Newton’s discoveries, to this visible attraction which neither he nor any philosopher could find the reason for.

You know that many people, as attached to philosophy, or rather to the name of Descartes, as they were previously to the name of Aristotle, have risen up against attraction. Some did not want to study it, others despised it, and insulted it after barely examining it; but I ask the reader to consider the following three reflections:

1° What do we mean by attraction? Nothing other than a force by which one body approaches another, without any visible or known force pushing it.

2° This property of matter is established by the best philosophers in England, Germany, Holland, and even in several universities in Italy, where somewhat strict laws sometimes close the access to truth. Isn’t the consent of so many learned men proof? Certainly; but it’s a strong reason to at least examine whether this force exists or not.

3° One should think that we know no more about the cause of impulsion than of attraction. We have no more idea of one of these forces than of the other: for no one can conceive why a body has the power to move another from its place. We do not understand either, it is true, how a body attracts another, nor how the parts of matter gravitate mutually, as will be proven. Nor does anyone say that Newton boasted of knowing the reason for this attraction. He simply proved that it exists: he saw in matter a constant phenomenon, a universal property. If a man found a new metal in the earth, would this metal exist any less because we do not know the first principles from which it is formed? Let the reader who glances at this work refer to the metaphysical discussion on attraction, made by M. de Maupertuis, in the smallest and perhaps the best book ever written in French on philosophy: through the reserve with which the author has expressed himself, one will see what he thinks and what one should think of this attraction whose name has so frightened many.

It is often said that attraction is an occult quality.

If by this term one means a real principle of which no reason can be given, the whole universe is in this case. We do not know how there is movement, nor how it is communicated, nor how bodies are elastic, nor how we think, nor how we live, nor how or why anything exists: everything is an occult quality.

If by this term one means an expression from the old school, a word without idea, consider only that it is through the most sublime and accurate mathematical demonstrations that Newton has shown men this principle which one tries to treat as a chimera.

We have seen that the rays reflected from a mirror could not come to us from its surface. We have experimented that rays, transmitted through glass at a certain angle, return instead of passing into the air; that if there is a vacuum behind this glass, the rays that were previously transmitted return from this vacuum to us: certainly, there is no known impulsion here. We must necessarily admit another power; we must also admit that there is something in refraction that was not understood until now.

So what will be this power that will break this ray of light in this basin of water? It is demonstrated (as we will say in the next chapter) that what was believed until now to be a simple ray of light is a bundle of several rays that all refract differently. If, from these traits of light contained in this ray, one refracts, for example, at four measures from the perpendicular, another will break at three measures. It is demonstrated that the most refrangible, i.e., for example, those that, in breaking out of glass and taking a new direction in the air, approach less to the perpendicular of this glass, are also those that reflect most easily, most quickly. There is therefore already a good appearance that it will be the same law that will reflect light, and that will refract it.

Finally, if we still find some new property of light that seems to owe its origin to the force of attraction, should we not conclude that so many effects belong to the same cause?

Here is this new property, which was discovered by Father Grimaldi, a Jesuit, around the year 1660, and on which Newton has pushed the examination to the point of measuring the shadow of a hair at different distances. This property is the inflection of light49. Not only do the rays break when passing through the medium whose mass attracts them; but other rays, passing through the air near the edges of this attracting body, noticeably approach this body, and visibly deviate from their path. Put (figure 27) in a dark place this blade of steel, or thinned glass, which ends in a point; expose it near a small hole through which light passes; let this light come to graze the point of this metal: you will see the rays bend near in such a way that the ray that approaches this point the most will bend more, and the one that is further away will bend less proportionally. Is it not highly probable that the same power that breaks these rays when they are in this medium, forces them to deviate when they are near this medium? So here are refraction, transparency, reflection, subject to new laws. Here is an inflection of light that obviously depends on attraction. A new universe presents itself to the eyes of those who want to see.

We will soon show that there is an obvious attraction between the sun and the planets, a mutual tendency of all bodies towards each other. But we also warn in advance that this attraction, which causes the planets to gravitate towards our sun, does not act at all in the same ratios as the attraction of small bodies that touch each other. These are probably attractions of absolutely different kinds. These are new and different properties of light and bodies that Newton has discovered. It is not a matter of their cause, but simply of their effects, unknown until our times. Do not think that light is bent towards crystal and in crystal according to the same ratio, for example, as Mars is attracted by the sun50.

Chap. X. — Continuation of the wonders of the refraction of light. That a single ray of light contains all possible colors within itself. What is refrangibility. New discoveries. — Imagination of Descartes on colors. Error of Malebranche. Experience and demonstration by Newton. Anatomy of light. Colors in the primary rays. Vain objections against these discoveries. Even vainer criticisms. Important experience.

If you ask philosophers what produces colors, Descartes will tell you that “the globules of his elements are determined to spin on themselves, in addition to their tendency to move in a straight line, and that it is the different spins that create different colors.” But his elements, his globules, his spinning, do they even need the touchstone of experience to feel their falsity? A host of demonstrations annihilate these chimeras. Here are the simplest and most sensitive ones.

Line up balls against each other: suppose they are pushed in all directions, and spinning all over themselves in all directions; by the mere statement, it is impossible for these contiguous balls to advance in straight lines regularly. Moreover, how would you see this blue point and this green point on a wall (figure 28)?

There they are marked on this wall; they must intersect in the air at point A before reaching the eyes. Since they intersect, their supposed spinning must change at the intersection. The spins that made the blue and green no longer remain the same: there would then no longer be a green or a blue point. A Flemish Jesuit made this objection to Descartes. He felt all its force; but what do you think he answered? That these balls do not actually spin, but that they have a tendency to spin. This is what Descartes said in his letters. Is the act of the transparent as transparent more intelligible?

You will undoubtedly tell me that this difficulty is equal in all systems. You will say that these rays, which start from this blue point and this green point, necessarily intersect, whatever opinion one embraces concerning colors; that this intersection of the rays should always prevent vision; that in short, it is always incomprehensible that intersecting rays arrive at our eyes in their order; but this scruple will soon be lifted, if you consider that every part of matter has incomparably more pores than substance. A ray of sunlight, which is over thirty million leagues in length, probably does not have one foot of solid matter end to end. It would therefore be very possible for one ray to pass through another in this matter, without disturbing anything (figure 29).

But it is not only in this way that they pass, it is also one over the other51 like two sticks. But, you will say, rays emanating from a center would not precisely, and in mathematical rigor, meet at the same line of circumference. That is true. It will always be a very small amount off. But two men would not see the same points of the same object. That is also true. Of a thousand million people who look at a surface, there will not be two who see the same exact points.

It must be admitted that, in the plenum of Descartes, this intersection of rays is impossible; but everything is equally impossible in the plenum, and there is no movement, whatever it may be, that does not presuppose and prove the void.

Malebranche comes in turn and tells you: “It is true that Descartes was mistaken. His spinning of globules is not tenable; but it is not globules of light, it is small spinning vortices of subtle matter, capable of compression, that are the cause of colors; and colors consist, like sounds, in pressure vibrations.” And he adds: “It seems impossible to discover by any means the exact relationships of these vibrations,” that is, of colors. You will note that he spoke thus in the Academy of Sciences in 1699, and that these proportions had already been discovered in 1675, not proportions of vibration of little vortices, which do not exist, but proportions of the refrangibility of rays, which contain the colors, as we will soon discuss. What he thought impossible had already been demonstrated to the eyes, recognized as true by the senses, which would have greatly displeased Father Malebranche. Other philosophers, sensing the weakness of these assumptions, tell you, at least with more plausibility: “Colors come from more or fewer rays reflected from colored bodies. White is the one that reflects more; black is the one that reflects the least. The brightest colors will be those that bring you the most rays. Red, for example, which tires the eyes a bit, must be composed of more rays than green, which rests them more.” This hypothesis (already suspect, since it is a hypothesis) appears only as a gross error, from the moment one deigns to consider a painting in a weak light, and then in broad daylight. You always see the same colors. White, illuminated only by a candle, is always white; and green, illuminated by a thousand candles, will always be green.

Finally, turn to Newton. He will tell you: Do not believe me; believe only your eyes and mathematics; place yourself in a completely dark room, where daylight enters only through an extremely small hole: the ray of light will come onto paper to give you the color of whiteness.

Expose transversely to a ray of light this glass prism (figure 30); then place at a distance of about sixteen or seventeen feet a sheet of paper P P opposite this prism.

You know that light breaks when entering from the air into this prism; you know that it breaks in the opposite sense, exiting from this prism into the air. If it did not break thus, it would go from this hole to fall on the floor of the room Z. But, since the light must escape away from the line Z, this light will therefore strike the paper. This is where all the secret of light and colors is seen. This ray, which has fallen on this prism, is not, as was believed, a simple ray: it is a bundle of seven main bundles of rays, each carrying its own primitive, primordial color. The mixtures of these seven rays produce all the colors of nature; and the seven together, reflected together from an object, form whiteness.

Explore this admirable artifice. We had already suggested that the rays of light do not refract, do not break all equally; what happens here is an evident demonstration to the eyes. These seven rays of light escaped from the body of this ray, which has been anatomized on leaving the prism, come to place themselves, each in their order, on this white paper, each ray occupying an oval. The ray that has the least force to follow its path, the least stiffness, the least matter, deviates more in the air from the perpendicular of the prism. The one that is the strongest (figure 31), the most dense, the most vigorous, deviates the least. Do you see these seven rays coming to break one above the other?

Each of them paints on this paper the primitive color it carries within itself. The first ray, which deviates the least from this perpendicular of the prism, is the color of fire; the second, orange; the third, yellow; the fourth, green; the fifth, blue; the sixth, indigo; finally, the one that deviates the most from the perpendicular, and rises last above the others, is violet.

A single beam of light, which previously made the color white, is therefore a composite of seven beams, each with its color. The assembly of seven primordial rays thus makes white.

CHAPTER X. Continuation of the wonders of light refraction. That a single ray of light contains within it all possible colors; what is refrangibility. New discoveries. — Descartes' theory on colors. Error of Malebranche. Experiment and demonstration by Newton. Anatomy of light. Colors in the primitive rays. Baseless objections against these discoveries. Even more baseless criticisms. Significant experiment.

If you still have doubts, take one of the lenticular glasses from a telescope, which gather all the rays at their focus; expose this glass to the hole through which the light enters: you will never see anything at this focus but a circle of whiteness.

Expose this same glass to the point where it can gather all seven rays emanating from the prism: it reunites, as you see, these seven rays at its focus (figure 32). The color of these seven combined rays is white: thus, it is demonstrated that the color of all combined rays is whiteness.

Consequently, black will be the body that does not reflect any rays.

For, when you have separated one of these primitive rays using the prism, expose it to a mirror, a burning glass, another prism: it will never change color, it will never separate into other rays. Carrying such a color is its essence; nothing can alter it, and as further proof, take silk threads of different colors; expose a blue silk thread, for example, to the red ray, this thread will become red. Place it in the yellow ray, it will turn yellow; and so on. Finally, neither refraction, reflection, nor any conceivable means can change this primitive ray, similar to gold that has been tested in a crucible, and even more unalterable.

This property of light, this inequality in the refraction of its rays, is called by Newton refrangibility. The fact was initially met with resistance and long denied because Mr. Mariotte in France failed to replicate Newton’s experiments. People preferred to say that Newton boasted of seeing what he had not seen rather than think that Mariotte had not managed well to see, and that he had not been fortunate in the choice of prisms he used. Even later, when these experiments were properly conducted, and the truth was manifest before our eyes, prejudice persisted to the extent that, in several journals and books written since the year 1730, these same experiments are boldly denied, although they are performed throughout Europe. Thus, after the discovery of the circulation of blood, theses were still upheld against this truth, and attempts were made to ridicule those who explained the new discovery by calling them “circulators.”

Finally, when people were forced to yield to the evidence, they still did not surrender: they saw the fact, and quibbled over the expression; they rebelled against the term refrangibility, just as against the terms attraction, gravitation. Ah! what does the term matter, as long as it denotes a truth? When Christopher Columbus discovered the island of Hispaniola, could he not name it whatever he wished? And isn’t it up to the inventors to name what they create or discover? There were outcries, writings against the terms that Newton uses with the utmost caution to prevent errors.

He calls these red, yellow, etc., rays “rubrific, jaunific,” meaning they induce the sensation of red, yellow. He wanted by this to silence anyone who would have the ignorance or bad faith to accuse him of believing, like Aristotle, that colors are in the things themselves, in these yellow and red rays, and not in our souls. He was right to fear this accusation. I have found otherwise respectable men who assured me that Newton was a Peripatetic, that he thought rays were actually colored themselves, as it was once thought that fire was hot; but these same critics also assured me that Newton was an atheist. It is true they had not read his book, but they had heard about it from people who had written against his experiments without having seen them.

The mildest thing first written against Newton is that his system is a hypothesis; but what is a hypothesis? A supposition. Truly, can we call a multitude of times demonstrated facts a supposition? Is it because one is born in France that one blushes to accept the truth from an Englishman’s hands? Such a sentiment would be unworthy of a philosopher. For anyone who thinks, there is neither French nor English; the one who enlightens us is our compatriot.

Refrangibility and reflection evidently stem from the same cause. This refrangibility we’ve just observed, being linked to refraction, must originate from the same principle. The same cause must oversee the operation of all these mechanisms: this is the order of nature. All plants nourish themselves by the same laws; all animals share the same life principles. Whatever happens to bodies in motion, the laws of motion are unchanging. We have already seen that reflection, refraction, and the bending of light are the effects of a power that is not impulsion (at least not as known); this same power is evident in refrangibility; these rays, which scatter at different distances, alert us that the medium through which they pass acts upon them unevenly. A beam of rays is attracted into glass; but this beam of rays is composed of unequal masses. These masses are thus unequally attracted; if so, they must reflect from this prism in the same order as they were refracted: the most reflective must be the most refrangible.

This prism has sent these seven colors onto this paper; turn this prism on itself in the direction A B C, and you will soon find the angle at which all light will reflect from within this prism to the outside, instead of passing onto this paper; as soon as you begin to approach this angle, suddenly the violet ray detaches from this paper, and you see it project onto the ceiling of the room (figure 33), After the violet comes the purple; after the purple, the blue; finally, the red leaves the paper last, where it is painted, to in turn reflect onto the ceiling. Thus, every ray is more reflective as it is more refrangible; thus the same cause operates both reflection and refrangibility.

The solid part of the glass does neither this refrangibility nor this reflection: thus, once again, these properties arise from another cause than known impulsion on Earth. There is nothing to argue against these experiments; one must submit to them, however resistant one might be to the evidence52.

Several authors have admitted only four colors: they eliminated the three intermediate colors, purple, green, and orange, as produced by the mixture of the two neighboring colors; they were confirmed in their opinion by experiments where only four colors are really seen; but this opinion is poorly founded: blue and yellow do indeed make green; but, if you look at a cardboard through a prism, at the green formed by the union of yellow and blue rays, the two colors separate; but if you look at this same cardboard, through a prism, at the image illuminated by the green rays of another prism, you elongate the image, but it remains green.

The prism only gives four colors when the light is weak, or too little spread by the prism; and if it were even weaker, if the image were less extended, you would see only a spectrum of a dirty white or reddish color. This is how the light of a star appears through a prism. If you equip the prism with a powerful lens, then the star’s spectrum will show you distinctly up to four colors, red, yellow, blue, and violet; with a weaker lens, the yellow and white disappear, and you see green instead. These experiments on the light of stars, which prove that this light is of the same nature as that of the sun, that of terrestrial burning bodies, are due to Mr. Abbé Rochon.

Not only is refraction different in different media, but the difference in the refrangibility of different rays is not proportional in these media to the refraction. It follows that one can, by combining different media, make prisms where the rays refract without separating, and eliminate colors in lenses by using lenses composed of several glasses of different nature. This idea, due to Mr. Euler, has produced achromatic lenses that several skilled artists have brought to a very high degree of perfection. Mr. Abbé Rochon has found, by applying lenses to prisms, ways to measure with great precision the ratio of the refractive power of different media with their dispersive power: precision necessary for the theory of lenses and for their construction.

There are substances that have double refraction, so that objects viewed through a prism formed of these substances appear double. Such is rock crystal, Iceland crystal; and these substances likely have this property because they are composed of heterogeneous plates placed one on top of the other; at least the same phenomenon is produced with artificial glasses thus arranged. This double refraction has been employed with great success by

Mr. Abbé Rochon, for the measurement of small angles. The instrument he invented for this purpose is very ingenious and provides these measurements with the greatest precision. It can also be used to measure distances without the need to employ bases of great extent. (K.)

Chap. XI. — On the rainbow; that this meteor is a necessary consequence of the laws of refrangibility. Mechanics of the rainbow unknown to all antiquity, ignorance of Albert the Great. Archbishop Antonio de Dominis is the first to have explained the rainbow. His experience imitated by Descartes. Refrangibility, the sole reason for the rainbow. Explanation of this phenomenon. The two rainbows. This phenomenon always seen in a semicircle.

The rainbow, or iris, is a necessary consequence of the properties of light we have just observed. There is nothing in the writings of the Greeks, Romans, or Arabs that suggests they knew the reasons for this phenomenon. Lucretius says nothing about it; and from all the absurdities he expounds, in the name of Epicurus, on light and vision, it appears that his otherwise refined era was plunged in profound ignorance regarding physics. It was known that a thick cloud resolving into rain needed to be exposed to the sun’s rays, and our eyes had to be between the star and the cloud, to see what was called the iris. Mille trahit varias adverse sole colores53; but that was all that was known: no one imagined why a cloud gives off colors, nor how the nature and order of colors are determined, nor why there are two rainbows one above the other, nor why these phenomena always appear in the shape of a semicircle.

Albert, who was nicknamed the Great because he lived in an era when men were quite small, imagined that the colors of the rainbow came from dew that is between us and the cloud, and that these colors, received on the cloud, were sent to us by it. You will also note that this Albert the Great believed, along with the entire school, that light was an accident.

Finally, the famous Antonio de Dominis, Archbishop of Spalatro in Dalmatia, driven from his bishopric by the Inquisition, wrote around 1590 his treatise De Radiis lucis et de iride, which was not printed in Venice until twenty years later54. He was the first to show that the sun’s rays, reflected from within the raindrops themselves, formed this arc-like painting, which seemed an inexplicable miracle; he made the miracle natural, or rather explained it through new wonders of nature.

His discovery was all the more remarkable as he otherwise had very false notions of how vision occurs. He asserts in his book that the images of objects are in the pupil, and that no refraction occurs in our eyes: quite strange for a good philosopher! He had discovered the then-unknown refractions in the raindrops of the rainbow, yet denied those occurring in the humors of the eye, which were beginning to be demonstrated; but let’s leave his errors to examine the truth he found.

He saw, with uncommon sagacity, that each row, each band of raindrops that forms the rainbow, had to send back rays of light at different angles; he saw that the difference in these angles had to cause the difference in colors. He knew how to measure the size of these angles: he took a very transparent crystal ball filled with water; he suspended it at a certain height, exposed to the sun’s rays.

Descartes, who followed Antonio de Dominis, who corrected and somewhat surpassed him, and perhaps ought to have cited him, also conducted the same experiment. When this ball is suspended at such a height that the ray of light, coming from the sun to the ball, thus makes an angle of 42 degrees 2 or 3 minutes with the ray going from the ball to the eye, this ball always produces a red color.

When this ball is suspended a bit lower, and these angles are smaller, the other colors of the rainbow appear successively so that the largest angle, in this case, makes red, and the smallest angle of 40 degrees 17 minutes forms violet. This is the foundation of the knowledge of the rainbow; but this is still only the foundation.

Refrangibility alone accounts for this phenomenon so ordinary, so little known, and of which very few beginners have a clear idea: let’s try to make the matter understandable to everyone. Let’s suspend a crystal ball full of water, exposed to the sun; let’s place ourselves between the sun and it: why does this ball send me colors? And why certain colors? Masses of light, millions of beams, fall from the sun on this ball: in each of these beams there are primitive streaks, homogeneous rays, several reds, several yellows, several greens, etc.; all break at their incidence in the ball; each of them breaks differently, and according to the type it is, and according to the place it enters.

You already know that the red rays are the least refrangible; the red rays of a certain determined beam will therefore gather at a certain determined point at the bottom of the ball, while the blue and purple rays of the same beam will go elsewhere. These red rays will also exit the ball in one place, and the greens, blues, purples, in another place. That’s not enough: it’s necessary to examine the points where these red rays enter this ball, and exit to come to your eye.

To provide the necessary clarity, let’s consider this sphere as it actually is, an assembly of an infinite number of flat surfaces: since a circle is composed of an infinity of infinitely small lines, the sphere is just an infinity of surfaces at its circumference.

Red rays A B C (figure 34) come from the sun parallel onto these three small surfaces. Isn’t it true that each breaks according to its degree of incidence? Isn’t it evident that the red ray A falls more obliquely on its little surface than the red ray C does on its own? Thus, both come to point R by different paths.

Red ray C, falling on its little surface even less obliquely, breaks much less and also arrives at point R, breaking only very slightly.

So, I already have three red rays, meaning three beams of red rays that converge at the same point R.

At this point R, each makes a reflection angle equal to its angle of incidence, each breaks upon emerging from the sphere, moving away from the perpendicular of the new small surface it encounters, just as each had broken upon incidence by approaching its perpendicular: thus, all return parallel, thus all enter the eye, according to the angle specific to the red rays.

If there are enough of these homogeneous red traits to shake the optic nerve, it is undeniable that you should only perceive red.

These rays A B C are what are called visible rays, effective rays of this drop: because each drop has its visible rays.

There are thousands of other red rays that, coming onto other small surfaces of the sphere, higher and lower, do not converge at R, or which, having fallen on these same surfaces at another obliquity, also do not converge at R: these are lost to you; they will come to another eye, placed higher or lower.

Thousands of orange, green, blue, violet rays have also arrived, indeed, with the visible reds on these surfaces ABC; but you cannot receive them. You know the reason: they are all more refrangible than the reds; as they all enter at the same point, each takes a different path within the sphere: all broken more, they come below the point R; they also break more than the reds when exiting the sphere. The same force, which brought them closer to the perpendicular of each surface inside the sphere, therefore drives them further away as they return to the air. Thus, they all return below your eye; but lower the sphere, you make the angle smaller. Let this angle be about 40 degrees 17 minutes, you only receive violet objects.

No one, based on this principle, should find it difficult to conceive the trickery of the rainbow: imagine several rows, several bands of raindrops; each drop does precisely the same effect as this sphere.

Look at this arch, and to avoid confusion, only consider three rows of raindrops, three colored bands.

It is evident that the angle P O L (figure 35) is smaller than the angle V O L, and the angle R O L is the largest of the three. Therefore, this largest angle of the three is that of the primitive red rays; the middle one is that of the primitive greens; the smallest P O L is that of the primitive purples. Thus, you should see the iris red on its outer edge, green in its middle, purple and violet in its inner band. Just note that the last violet layer is always tinged with the whitish color of the cloud into which it fades.

You can easily understand that you only see these drops under the effective rays that have reached your eyes after a reflection and two refractions, arriving under determined angles. If your eye changes position, from being at O to being at T, you no longer see the same rays: the band that gave you red now gives you orange, or green; and so forth, and with every movement of your head, you see a new iris.

With this first rainbow well understood, you will easily grasp the second, which is usually seen embracing the first, and is called the false rainbow because its colors are less vivid and they are in a reversed order.

To see two rainbows, it is enough that the cloud is sufficiently extensive and thick. This arch, which is painted on the first and embraces it, is formed in the same way by rays that the sun casts into these raindrops, which break there, which are reflected in such a way that each row of drops also sends you primitive rays: this drop, a red ray; another drop, a violet ray.

But everything happens in this large arc in a manner opposite to what happens in the small one. Why is this? It’s because your eye, which receives the effective rays of the small arc coming from the sun in the upper part of the drops, receives on the contrary the rays of the large arc coming through the lower part of the drops.

You see (figure 36) that the water drops of the small arc receive the sun’s rays through the upper part, the top of each drop; the drops of the large rainbow, on the contrary, receive the rays that come through their lower part. It should be easy for you to understand how the rays are reflected twice in the drops of this large rainbow, and how these rays, refracted twice and reflected twice, give you an iris in an order opposite to the first, and more color-faded. You just saw that the rays enter the small lower part of the water drops of this outer iris.

A mass of rays (figure 37) presents itself on the surface of the drop at G; there a part of these rays refracts inward, and another part scatters outward: that’s already a loss of rays for the eye. The part refracted reaches H, half of this part escapes into the air when leaving the drop, and is again lost to you. The little that was preserved in the drop goes to K; there a part escapes again: third reduction. What remained at K goes to M, and at this emergence at M, a part scatters again: fourth reduction; and what remains finally reaches the line M N. Thus, in this drop, there are as many refractions as in the drops of the small arc; but, as you see, there are two reflections instead of one in this large arc. Therefore, double the light is lost in this large arc, where the light is reflected twice; and half less is lost in the small interior arc, where the drops undergo only one reflection. It is therefore proven that the outer rainbow must always be half as weak in color as the small inner arc. It is also proven by this double path made by the rays that they must reach your eyes in a direction opposite to that of the first arc: for your eye is placed at O.

In this position (figure 38), it receives the least refrangible rays of the first outer band of the small arc, and it must receive the most refrangible of the first outer band of this second arc: these most refrangible are the violets. Here are the two rainbows here in their order, putting only three colors to avoid confusion.

All that remains is to see why these colors are always seen in a circular shape. Consider this line Z, which passes through your eye. Imagine these two spheres always moving at an equal distance from your eye: they will describe the bases of cones (figure 39), whose tip will always be in your eye.

Imagine that the ray from this water drop R, coming to your eye O, rotates around this line O Z as around an axis, always making, for example, an angle with your eye of 42 degrees 2 minutes: it is clear that this drop will describe a circle that will appear red to you. Let this other drop V be imagined to rotate in the same way, always making another angle of 40 degrees 17 minutes: it will form a violet circle; all the drops that will be in this plane will therefore form a violet circle, and the drops that are in the plane of the drop R will form a red circle. You will thus see this iris as a circle; but you do not see a full circle, because the earth cuts it; you see only an arc, a portion of a circle.

Most of these truths could not yet be perceived by Antonio de Dominis or by Descartes: they could not know why these different angles gave different colors; but it was a lot to have found the craft. The subtleties of the craft are rarely due to the first inventors. Thus, being unable to guess that the colors depended on the refrangibility of the rays, that each ray contained within itself a primitive color, that the different attraction of these rays made their refrangibility, and effected these deviations that make the different angles, Descartes gave himself over to his inventive spirit to explain the colors of the rainbow55. He employed the imaginary spinning of these globules, and this tendency to spin: proof of genius, but proof of error. Thus, to explain the systole and diastole of the heart, he imagined a movement and a conformation in this organ, the falsity of which all anatomists have recognized. Descartes would have been the greatest philosopher on earth if he had invented less.

Chap. XII. — New discoveries on the cause of colors, which confirm the previous doctrine. Demonstration that colors are occasioned by the thickness of the parts that compose bodies, without the light being reflected from these parts. — Deeper knowledge of the formation of colors. Great truths drawn from a common experience. Experiences by Newton. Colors depend on the thickness of the parts of bodies, without these parts themselves reflecting light. All bodies are transparent. Proof that colors depend on thicknesses, without the solid parts actually reflecting light.

From all that has been said so far, it follows that all colors come to us from the mixture of the seven primary colors that the rainbow and the prism show us distinctly56.

Bodies most suited to reflecting red rays, whose parts absorb or let other rays pass through, will be red, and so on. This does not mean that the parts of these bodies actually reflect the red rays; but that there is a power, a force as yet unknown, which reflects these rays from the surfaces and from the depths of the pores of the bodies.

Colors, therefore, are in the sun’s rays and bounce back to us from the surfaces, and the pores, and the void. Let us now seek to understand what consists the apparent power of bodies to reflect these colors to us, what makes scarlet appear red, meadows green, a clear sky blue: for, to say that this comes from the difference in their parts is to say something vague that teaches us nothing at all.

A child’s amusement, which seems to have nothing in it but contempt, gave Mr. Newton the first idea of these new truths that we are about to explain. Everything should be a subject of meditation for a philosopher, and nothing is small in his eyes. He noticed that in these soap bubbles that children make, the colors change from moment to moment, counting from the top of the bubble as the thickness of this bubble diminishes, until finally the weight of the water and soap that always falls to the bottom breaks the balance of this light sphere and causes it to vanish. He presumed that colors might well depend on the thickness of the parts that make up the surfaces of bodies, and to be sure, he conducted the following experiments.

Let two crystals touch at one point: it doesn’t matter if they are both convex57; it is enough that the first one is, and that it is placed on the other in this manner.

Place water between these two glasses (figure 40) to make the experiment more noticeable, which is also done in air; press these glasses slightly against each other, a small transparent black spot appears at the point of contact of the two glasses: from this point, surrounded by a bit of water, colored rings form in the same order and in the same manner as in the soap bubble; finally, by measuring the diameter of these rings and the convexity of the glass, Newton determined the different thicknesses of the water parts that gave these different colors; he calculated the thickness necessary for water to reflect white rays: this thickness is about four parts of an inch divided into a million, that is to say, four millionths of an inch; the azure blue and the colors leaning towards violet depend on a much smaller thickness. Thus, the smallest vapors rising from the earth, and which color the air without clouds, being of a very thin surface, produce this celestial blue that charms the sight.

Other equally fine experiments have further supported this discovery, that it is the thickness of surfaces to which colors are attached.

The same body that was green when it was a bit thick turned blue when it was made thin enough to only reflect blue rays and let the other rays pass through. These truths, of such delicate research and which seemed to elude human sight, deserve to be closely followed; this part of philosophy is a microscope with which our mind discovers infinitely small magnitudes.

All bodies are transparent, it is just necessary to make them thin enough so that the rays, finding only a blade, only a leaf to cross, pass through this blade. Thus, when gold leaf is exposed to a hole in a dark room, it sends back yellow rays from its surface which cannot transmit through its substance, and it transmits green rays into the dark room, so that gold then produces a green color: new confirmation that colors depend on different thicknesses.

An even stronger proof is that, in the experiment of this convex-plane glass, touching at one point this convex glass, water is not the only element that, in various thicknesses, gives various colors: air does the same effect; only the colored rings it produces between the two glasses have a larger diameter than those of the water.

There is thus a secret proportion established by nature between the force of the constituent parts of all bodies and the primitive rays that color the bodies; the thinnest films will give the weakest colors; and to give black, exactly the same thickness, or rather the same thinness, the same thinness, as the upper small part of the soap bubble, in which a small black point was seen, or the same thinness as the point of contact of the convex glass and the flat glass, which contact also produces a black spot.

But, once again, do not believe that bodies send back light through their solid parts, because the colors depend on the thickness of the parts. There is a power attached to this thickness, a power that acts near the surface; but it is not at all the solid surface that repels, that reflects. This truth will be even more visibly demonstrated in the next chapter than it has been proven so far. It seems to me that the reader should have reached the point where nothing should surprise him anymore; but what he has just seen leads even further than one might think, and so many singularities are, so to speak, only the borders of a new world.

CHAPTER XIII. Continuation of these discoveries; mutual action of bodies on light. — Very singular experiment. Consequences of these experiments. Mutual action of bodies on light. This entire theory of light is related to the theory of the universe. Matter has more properties than we think.

With the reflection, bending, refraction, and refrangibility of light known, the origin of colors discovered, and the very thickness of bodies necessary to cause certain colors determined, we still have two properties of light to examine, no less astonishing and no less new. The first of these properties is the very power that acts near surfaces: it is a mutual action of light on bodies, and of bodies on light.

The second is a relationship found between the colors and the tones of music, between the objects of sight and those of hearing. But here we will only discuss the reciprocal action of bodies on light, because it relates to the great principle of nature by which all bodies act upon one another.

Regarding the analogy between the seven primitive colors and the seven tones of music, it is a discovery that is not yet sufficiently deepened, which cannot yet lead to anything.

Thus, we will conclude this small treatise on optics by examining the mutual action of bodies and light.

You have seen that these two crystals, touching at one point, produce rings of different colors, reds, blues, greens, whites, etc. Perform this same test in a dark room, where you have conducted the experiment of the prism exposed to the light coming through a hole. You recall that, in this prism experiment, you saw the decomposition of light and the anatomy of its rays: you placed a sheet of white paper opposite this prism; this paper received the seven primitive colors, each in their order. Now expose your two glasses to any colored ray you wish, reflected from this paper: you will still see colored rings forming between these glasses; but all these rings are then of the color of the rays coming from the paper. Expose your glasses to the light of red rays, you will have only red rings between your glasses (figures 41 and 42); but what is surprising is that between each of these red rings there is a completely black ring. To further establish this fact and the singularities attached to it, present your two glasses, no longer to the paper, but to the prism, so that one of the rays escaping from this prism, a red one for example, comes to fall on these glasses: only red rings form between the black rings; place behind your glasses the sheet of white paper: each black ring produces on this sheet of paper a red ring, and each red ring, being reflected towards you, produces black on the paper.

This experiment demonstrates that the air or water between your glasses reflects light in one place, and in another place lets it pass, transmits it. I admit that I cannot sufficiently admire here the depth of research, the more than human sagacity, with which Newton pursued these imperceptible truths; he recognized by measurements and by calculation these strange proportions.

At the point of contact of the two glasses, no light is reflected to our eyes: immediately after this contact, the first small blade of air or water touching this black point reflects rays to you; the second blade is twice as thick as the first, and reflects nothing; the third blade is triple the thickness of the first, and reflects; the fourth blade is four times thicker, and does not reflect; the fifth is five times thicker, and reflects; and the sixth, six times thicker, transmits, and does not reflect.

Thus, the black rings follow this progression, 0, 2, 4, 6, 8; and the bright and colored rings follow this progression, 1, 3, 5, 7, 958.

What happens in this experiment occurs similarly in all bodies, which all reflect part of the light, and receive another part into their substances. This is therefore another property demonstrated to the mind and to the eyes, that solid surfaces are not what reflect the rays. Because, if solid surfaces did indeed reflect: 1° the point where the two glasses touch would reflect and would not be dark; 2° each solid part that would give you one type of ray should also send you all types of rays; 3° solid parts would not transmit light in one place and reflect it in another, for, being all solid, all would reflect; 4° if solid parts reflected light, it would be impossible to see oneself in a mirror, as we have said, since the mirror, being grooved and rough, could not return light in a regular manner. It is therefore indisputable that there is a power acting on bodies, without touching the bodies, and that this power acts between bodies and light. Indeed, far from the light bouncing off the bodies themselves and returning to us, it must be believed that the majority of the rays that hit solid parts stay there, get lost, are extinguished.

This power, which acts on surfaces, acts from one surface to another: it is mainly from the last outer surface of the transparent body that the rays rebound; we have already proven this. It is, for example, from points B B B (figure 43), more than from point A, that light is reflected.

We must therefore admit a power, which acts on the rays of light from above one of these surfaces to another, a power that transmits and reflects rays alternately. This play of light and bodies was not even suspected before Newton; he counted several thousand of these alternative vibrations, these transmitted and reflected jets. This action of bodies on light, and of light on bodies, still leaves many uncertainties in the way to explain it.

He who discovered this mystery could not, in the course of his long life, conduct enough experiments to assign the certain cause of these effects. But, if his discoveries had taught us only new properties of matter, would it not already be a great service rendered to philosophy59? He did not dwell on it in any way; he was content with the facts, without daring to determine anything about the causes.

We will not go further in this introduction on light, perhaps we have said too much in simple elements; but most of these truths are new for many readers. Before moving to the other part of philosophy, let us remember that the theory of light has something in common with the theory of the universe into which we are about to enter. This theory is that there is a kind of attraction noted between bodies and light, as we are about to observe between all the globes of our universe: these attractions manifest through different effects; but it is always a tendency of bodies towards each other, discovered with the help of experience and geometry.

Among so many properties of matter, such as these accessions of transmission and reflection of light rays60, this repulsion that light experiences in the vacuum, in the pores of bodies, and on the surfaces of bodies; among these properties, I say, it is especially necessary to pay attention to this power by which rays are reflected and broken, to this force by which bodies act on light, and light on them, without even touching them. These discoveries should at least serve to make us extremely cautious in our decisions about the nature and essence of things. Let us think that we know nothing at all but through experience. Without touch, we would have no idea of the extension of bodies; without eyes, we could never have guessed light; if we had never experienced movement, we would never have believed matter to be mobile; a very small number of senses that God has given us serve to discover a very small number of properties of matter. Reasoning complements the senses we lack, and still teaches us that matter has other attributes, like attraction, gravitation; it probably has many more that are related to its nature, and which perhaps one day philosophy will give some ideas to men.

For me, I confess that the more I think about it, the more I am amazed that people fear recognizing a new principle, a new property in matter. It may have an infinite number; nothing is alike in nature. It is very likely that the Creator made water, fire, air, earth, plants, minerals, animals, etc., on different principles and plans. It is strange that we rebel against new riches that are presented to us: for is it not enriching man to discover new qualities of the matter of which he is formed61?

Author’s letter, which may serve as a conclusion to the theory of light

I would have had the honor of responding to you sooner, sir, if not for the continuous illnesses that test my patience more than Newton tests my mind. I believe your doubts, sir, would have given rise to his. You say it is a pity that he did not explain more clearly the reason why the attractive force often becomes repulsive, and about the force by which the rays of light are darted with such prodigious speed; and I would dare to add that it is a pity he could not know the cause of these phenomena. Newton, the foremost of men, was only a man, and the primary mechanisms that nature employs are not within our reach when they are not subject to calculation. It is in vain to calculate the strength of muscles; all the mathematics will be powerless to teach us why these muscles act at the command of our will. All the knowledge we have of the planets will never teach us why they rotate from west to east, rather than the contrary. Newton, having dissected light, has not discovered its intimate nature. He knew well that there are properties in elemental fire that are not in other elements; it travels one hundred and thirty million leagues in a quarter of an hour.

It does not appear to tend towards a center like bodies; but it spreads uniformly and equally in all directions, unlike other elements. Its attraction to the objects it touches, and off the surfaces of which it rebounds, bears no proportion to the universal gravitation of matter.

It is not even proven that the rays of elemental fire do not penetrate each other62. Therefore, Newton, struck by all these peculiarities, always seems to doubt whether light is a body. For me, sir, if I dare venture my doubts, I confess that I do not believe it impossible that elemental fire is a being apart, which animates nature, and which lies between bodies and some other being that we do not know; just as certain organized plants serve as a transition from the vegetable kingdom to the animal kingdom. Everything leads us to believe that there is a chain of beings rising by degrees. We know only imperfectly some links of this immense chain, and we, little men, with our little eyes and our little brains, boldly distinguish all of nature into matter and spirit, including God, and not knowing moreover a word of what spirit and matter truly are at the core. I express my doubts to you, sir, with the same frankness with which you communicated yours to me. I congratulate you on cultivating philosophy, which should teach us to doubt everything that is not within the realm of mathematics and experience, etc.

Third Part

Chap. I. — First ideas concerning the laws of gravity and attraction: that subtle matter, whirlwinds and the plenum must be rejected. — Attraction. Experience that demonstrates the vacuum and the effects of gravitation. Gravity acts in proportion to the masses. Where does the power of gravity come from. It cannot come from a supposed subtle matter. Why one body weighs more than another. The system of Descartes cannot explain this.

A wise reader, having attentively observed these marvels of light and convinced by experience that no known impulse causes them, will undoubtedly be eager to observe this new power we have spoken of under the name of attraction, which acts on all other bodies more noticeably and differently than it does on light. Let us not be alarmed by the names, but simply examine the facts.

I will always use the terms attraction and gravitation interchangeably when speaking of bodies, whether they visibly tend towards each other, rotate in vast orbits around a common center, fall to the earth, unite to form solid bodies, or round into drops to form liquids63. Let’s delve into the matter.

All known bodies have weight, and absolute lightness has long been counted among the recognized errors of Aristotle and his followers. Since the famous air pump was invented, we have been better able to understand the weight of bodies: for when they fall in the air, the air’s particles significantly slow the fall of those with much surface area and little volume; but in this air-free machine, bodies left to whatever force propels them without obstacle, fall according to their full weight.

The air pump, invented by Otto Guerike, was soon perfected by Boyle; then, much longer glass containers were made, which were completely purged of air. In one of these long containers, composed of four tubes and totaling eight feet in height, pieces of gold, paper, and feathers were suspended at the top by a spring. It was to be seen what would happen when the spring was released. The wise philosophers anticipated that all would fall simultaneously; the majority asserted that the most massive bodies would fall much faster than the others: this majority, almost always mistaken, was quite astonished when it saw, in all experiments, the gold, the lead, the paper, and the feather fall at equal speed, reaching the bottom of the container simultaneously.

Those who still held to Descartes' plenum and the supposed effects of subtle matter could give no good reason for this fact: for facts were their stumbling blocks. If everything were full, even granting that there could then be movement (which is absolutely impossible), at least this supposed subtle matter would fill the container exactly as much as water or mercury placed in it; it would at least oppose the bodies' swift descent; it would resist the large piece of paper according to its surface area and let the gold or lead ball fall much faster; but the fall occurs at the same instant: thus, there is nothing in the container that resists; thus, this supposed subtle matter cannot have any sensible effect in this container; thus, there is another force that causes gravity.

It would be vain to say that there might still be subtle matter in this container, since light penetrates it; there is a great difference. The light in this glass vessel certainly does not occupy the hundred thousandth part; but, according to the Cartesians, their imaginary matter must fill the container more exactly than if I were to suppose it filled with gold: for there is much emptiness in gold, and they admit no emptiness in their subtle matter.

Now, by this experiment, the gold piece, which weighs a hundred thousand times more than the piece of paper, fell as quickly as the paper: therefore, the force that caused it to descend acted a hundred thousand times more on it than on the paper, just as it would take a hundred times more force from my arm to move a hundred pounds than to move one pound; therefore, this power that operates gravitation acts in direct proportion to the mass of the bodies. It acts indeed according to the mass of the bodies, not according to the surfaces, as a piece of gold reduced to powder descends in the air pump as quickly as the same quantity of gold spread in a sheet. The shape of the bodies changes nothing here in their gravity: this power of gravitation thus acts on the internal nature of bodies, and not according to their surfaces.

No one has ever been able to counter these pressing truths except by a supposition as fanciful as the vortices. It is supposed that the alleged subtle matter that fills the entire container does not weigh anything: a strange idea that becomes absurd here. For it is not, in the present case, a matter of a matter that does not weigh, but of a matter that does not resist. All matter resists by its force of inertia. Therefore, if the container were full, any matter that filled it would resist infinitely: this seems rigorously demonstrated.

This power does not reside in the alleged subtle matter, which we will discuss in the next chapter; this matter would be a fluid. Every fluid acts on solids according to their surfaces; thus the ship presenting less surface by its prow cuts through the sea which would resist its sides. Now, when the surface area of a body is the square of its diameter, the solidity of this body is the cube of that same diameter: the same power cannot act at the same time in proportion to the cube and to the square; therefore, gravity, gravitation is not the effect of this fluid.

Moreover, it is impossible that this alleged subtle matter could have on one hand enough force to precipitate a body from 54,000 feet high in one minute (for such is the fall of bodies), and on the other hand be so powerless as to be unable to prevent the pendulum of the lightest wood from rising from vibration to vibration in the air pump, which this imaginary matter is supposed to fill exactly all the space.

I will therefore not fear to assert that, if an impulse were ever discovered to be the cause of the gravitation of bodies towards a center, in a word the cause of gravitation, of universal attraction, this impulse would be of a wholly different nature than that which we know.

Thus, here is a first truth already indicated elsewhere, and proved here: there is a power that makes all bodies gravitate in direct proportion to their mass.

If one currently seeks why one body is heavier than another, the only reason will be easily found: it will be judged that this body must have more mass, more matter within the same extent; thus, gold weighs more than wood, because there is in gold much more matter and less void than in wood.

Descartes and his followers (if he can still have any) maintain that one body is heavier than another without having more matter; not satisfied with this idea, they support it with another equally untrue one: they admit a great vortex of subtle matter around our globe, and it is this great vortex, they say, that, by circulating, drives all bodies towards the center of the earth, and makes them experience what we call gravity.

It is true that they have provided no proof of this assertion: there is not the slightest experiment, not the slightest analogy in things that we know a bit, that can found even a slight presumption in favor of this vortex of subtle matter; thus, from the fact alone that this system is a pure hypothesis, it must be rejected. Yet it is by this alone that it has been accredited. The vortex was conceived without effort, a vague explanation of things was given by uttering this word of subtle matter; and when the philosophers felt the contradictions and absurdities attached to this philosophical novel, they thought of correcting it rather than abandoning it.

Huygens and so many others have made a thousand corrections to it, which they themselves admitted were insufficient. But what shall we put in place of the vortices and subtle matter? This reasoning, too ordinary, is what most firmly holds men in error and in the wrong path. We must abandon what we see as false and unsustainable, just as well when we have nothing to substitute for it as when we could have the demonstrations of Euclid to put in its place. An error is neither more nor less an error, whether we replace it or not with truths: should I admit the horror of a vacuum in a pump, because I do not yet know by what mechanism water rises in this pump?

Let us therefore begin, before going further, by proving that the vortices of subtle matter do not exist; that the plenum is no less chimerical; that thus this whole system, founded on these imaginations, is nothing but an ingenious novel without verisimilitude. Let us see what these imaginary vortices are, and then examine whether the plenum is possible.

Chap. II. — That Descartes' whirlwinds and the plenum are impossible, and that consequently there is another cause of gravity. — Proofs of the impossibility of whirlwinds. Proofs against the plenum.

Descartes posits a vast accumulation of imperceptible particles that sweep the Earth in a rapid movement from west to east, moving from pole to pole parallel to the equator; this vortex, which extends beyond the Moon and carries it along in its course, is itself encased in an even larger vortex, touching yet distinct from another vortex, and so on.

1° If this were so, the vortex purported to move around the Earth from west to east should propel bodies across the Earth from west to east; however, all bodies in falling describe a line that, if extended, would pass approximately through the Earth’s center: thus, this vortex does not exist.

2° If the circles of this alleged vortex moved and acted parallel to the equator, all bodies would fall directly beneath the circle64 of this subtle matter corresponding to their location: a body at A near the pole P (figure 45) should, according to Descartes, fall at R.

But it falls approximately along the line A B, which differs by about 1,400 leagues, since one can count 1,400 common leagues of France from point R to the Earth’s equator B: thus, this vortex does not exist.

3° If, to support this novel of vortices, one still likes to suppose that a swirling fluid does not rotate about its axis; if one imagines that it can rotate in circles all having the center of the vortex itself; one only needs to experiment with a drop of oil or a large air bubble enclosed in a crystal ball filled with water: spin the ball on its axis, and you will see the oil or air arrange itself into a cylinder in the middle of the ball, forming an axis from one pole to the other; for every experiment and all reasoning undermine vortices.

4° If this vortex of matter around the Earth, and those other alleged vortices around Jupiter and Saturn, etc., existed, all these immense vortices of subtle matter, rolling so rapidly in different directions, would never allow a ray of light shot from a star to reach us in a straight line. It is proven that these rays arrive in a very short time relative to the immense distance they cover: thus, these vortices do not exist.

5° If these vortices carried the planets from west to east, comets that cross these spaces in every direction from east to west, and from north to south, could never traverse them. And even if one assumed that comets did not actually move from north to south, or from east to west, it would gain nothing by this evasion, for it is known that when a comet is in the region of Mars, Jupiter, Saturn, it moves incomparably faster than Mars, Jupiter, Saturn: therefore, it cannot be carried by the same layer of fluid that is supposed to carry these planets; thus, these vortices do not exist.

6° These alleged vortices would be either as dense, as massive as the planets; or they would be denser, or finally less dense. In the first case, the alleged matter surrounding the moon and the Earth, being supposed dense like an equal volume of Earth, we would experience, for example, the same resistance to lifting a cubic foot of marble as we would to lifting a column of marble with a base of one foot, which would have its length from the Earth to the far end of the alleged Moon’s vortex.

In the other two cases, which are, I believe, impossible, it is rightly disputed what would happen. But here is what resolves all difficulty, and shows that no vortex can press upon the Earth and cause gravity. It is demonstrated, by the theory of motive forces, that a body moving, for example, with ten degrees of speed, receives no force, no movement from a power that also has only ten degrees, and that pursues this moving body.

For this power to add new degrees of movement to this body, it must have more than it; and it only transmits its excess. But the power of gravitational attraction acts equally on bodies at rest and in motion, imparting the same degrees of speed to both: thus, this power cannot come from a fluid that can only act according to the laws of motive forces.

7° If these fluids existed, a minute would suffice to destroy all movement in the stars. Newton has shown that any body moving uniformly in a fluid of the same density loses half its movement after having traveled three times its diameter. This is beyond dispute.

8° Assuming again, which is impossible, that these planets could be moved in these imaginary vortices, they could only move circularly, since these vortices, at equal distances from the center, would be equally dense; but the planets move in ellipses: thus, they cannot be carried by vortices; thus, etc.

9° The Earth has its orbit which it follows between those of Venus and Mars; all these orbits are elliptical and have the sun as their center; now, when Mars, Venus, and the Earth, are closer to each other, then the matter of the alleged torrent carrying the Earth would be much more compressed: this subtle matter should accelerate its course, like a river narrowed on its banks, or flowing under the arches of a bridge; then this fluid should carry the Earth with much greater speed than in any other position; but, on the contrary, it is at this very time that the Earth’s movement is more slowed down.

When Mars appears in the sign of Pisces, Mars, the Earth, and Venus, are approximately in this proximity as you see (figure 46): then the sun appears to delay by a few minutes, that is, it is the Earth that delays. It is therefore demonstrated impossible that there is a torrent of matter carrying the planets: thus, this vortex does not exist.

10° Among the more elaborate demonstrations that annihilate vortices, we choose this one. By one of Kepler’s great laws, every planet describes equal areas in equal times; by another law no less certain, each planet makes its revolution around the sun in such a way that if, for example, its average distance from the sun is 10, take the cube of this number, which will be 1,000, and the time of the revolution of this planet around the sun will be proportionate to the square root of this number 1,000. Now, if there were layers of matter that carried the planets, these layers could not follow these laws: for the speeds of these torrents would have to be at once reciprocally proportional to their distances from the sun, and to the square roots of these distances, which is incompatible.

11° To top it off, everyone sees what would happen to two fluids circulating opposite each other. They would necessarily merge, and form chaos instead of unraveling it. This alone would have cast ridicule on the Cartesian system that would have overwhelmed it if the taste for novelty and the little use then made of examination had not prevailed.

It must now be proven that the plenum, in which these vortices are supposed to move, is as impossible as these vortices.

1° A single ray of light, which does not weigh, by far, the hundred thousandth part of a grain, would have to disrupt the entire universe if it had to make its way to us through an immense space, each point of which would resist by itself, and by the entire line along which it would be pressed.

2° Consider these two hard bodies A B (figure 47); they touch by a surface and are supposed to be surrounded by a fluid that presses them from all sides; now, when they are separated, it is clear that the alleged subtle matter arrives sooner at point A, where they are separated, than at point B: therefore, there is a moment when B is empty; thus, even in the system of subtle matter, there is a vacuum, that is, space.

3° If there were no vacuum and no space, there would be no movement, even in Descartes' system. He supposes that God created the universe full and consisting of small cubes: let there be a given number of cubes representing the universe, without any space between them: it is evident that one of them must leave the place it occupied, for if each remains in its place there is no movement, since movement consists in leaving one’s place, moving from one point of space to another point of space. Now who does not see that one of these cubes cannot leave its place without leaving it empty the moment it exits? for it is clear that this cube, by turning on itself, must present its angle to the cube that touches it, before the angle is broken: therefore, there is space between these two cubes; therefore, even in Descartes' system, there can be no movement without a vacuum.

4° If everything were full, as Descartes wants, we would ourselves experience infinite resistance while walking, instead of only experiencing the resistance of the fluids in which we are: for example, that of water, which resists us 860 times more than that of air; that of mercury, which resists about 14,000 times more than air. Now the resistances of fluids are as the squares of the speeds, that is, if a man travels a foot of mercury space in a third of a second, which resists 14,000 times more than air; if this man, in the second third of a second, has double that speed, this mercury, which is 14,000 times denser than air, will resist as the square of two, the resistance will soon be infinite: thus, if everything were full, it would be absolutely impossible to take a step, to breathe, etc.

5° Attempts have been made to evade the force of this demonstration, but one cannot answer a demonstration except by an error. It is claimed that this infinite torrent of subtle matter, penetrating all the pores of bodies, cannot stop their movement. It is not considered that every mobile moving in a fluid experiences more resistance the more surface it opposes to this fluid; thus, the more holes a body has, the more surface it has: thus the alleged subtle matter, by striking all the interior of a body, would oppose the movement of this body far more than by touching only its outer surface; and this is also rigorously demonstrated.

6° In the plenum, all bodies would be equally heavy; it is impossible to conceive that a body weighs on me, presses me; that by its mass a pound of gold dust weighs as much on my hand as a piece of gold weighing a pound. In vain do the Cartesians reply that the subtle matter penetrating the interstices of bodies does not weigh, and that only what is not subtle matter should be considered heavy: this opinion of Descartes is in him only a pure contradiction, for, according to him, this alleged subtle matter alone causes the weight of bodies, by repelling them towards the earth: thus it weighs itself on these bodies; therefore, if it weighs, there is no more reason why one body should be heavier than another, since everything being full, everything would equally have mass, whether solid or fluid; thus the plenum is a chimera; thus there is a vacuum; thus nothing can be done in nature without a vacuum; thus gravity is not the effect of an alleged vortex imagined in the plenum65.

We have just realized, through experience in the air pump, that there must be a force that causes bodies to descend toward the center of the earth, that is, which gives them gravity, and that this force must act in proportion to the mass of bodies: we must now see what the effects of this force are, for if we discover the effects it is evident that it exists. So let us not first imagine causes and make hypotheses: this is the sure way to go astray; let us follow step by step what actually happens in nature. We are travelers arrived at the mouth of a river: we must ascend it before imagining where its source is.

Chap. III. — Gravitation demonstrated by Newton’s discovery. History of this discovery. That the moon orbits due to the force of this gravitation. — Laws of falling bodies found by Galileo. Whether these laws are the same everywhere. History of the discovery of gravitation. Newton’s process. Theory derived from these discoveries. The same cause that causes bodies to fall to the earth directs the moon around the earth.

Galileo, the restorer of reason in Italy, discovered this important proposition: heavy bodies descending to Earth (disregarding the slight resistance of air) experience an accelerated motion in a proportion I will attempt to explain clearly. A body left to itself from the top of a tower covers, in the first second of time, a space found to be 15 Paris feet, according to the discoveries of Huygens, a mathematician. Before Galileo, it was believed that this body, during two seconds, would have traveled only twice the same space, thus covering 150 feet in ten seconds, and 900 feet in a minute: this was the general, and even quite plausible opinion to those who do not examine closely; however, it is true that in one minute this body would have traveled a distance of 54,000 feet, and 216,000 feet in two minutes.

Here is how this progression, which initially astonishes the imagination, necessarily and simply occurs. A body is precipitated by its own weight: this force, whatever it may be, that propels it to descend 15 feet in the first second acts equally at all moments, for, nothing having changed, it must always be the same: thus, in the second second, the body will have the force it acquired at each instant of the first second, and the force it experiences each instant of the second. Now, by the force animating it in the first second, it covered 15 feet; it thus still has this force when it descends the second second. Besides this, it has the force of another 15 feet which it gained as it descended in that first second: that makes 30; nothing having changed, in the time of this second second, it must still have the force to cover 15 feet: that makes 45; by the same reason, the body will cover 75 feet in the third second, and so forth.

From this it follows: 1° That the moving object acquires infinitely small degrees of speed in infinitely small equal times, which accelerate its movement toward the center of the earth, as long as it encounters no resistance;

2° That the speeds it acquires are proportional to the times it spends descending;

3° That the spaces it covers are proportional to the squares of these times or these speeds;

4° That the progression of the spaces covered by this moving object is like the odd numbers 1, 3, 5, 7. This necessary knowledge of this phenomenon that occurs around us at all times will be made tangible even to those who might initially be a little puzzled by all these relationships: it only requires a bit of attention by looking at this small table, which each reader can expand as they wish.

Time during which the object falls. Spaces it covers in each time. Spaces covered are as the squares of the times. Odd numbers marking the progression of the movement and the spaces covered.
1st Second, one speed. The body descends 15 feet. The square of one is one; the body covers 15 feet. Once 15.
2nd Second, 2 speeds. The body covers 45 feet. The square of 2 seconds or 2 speeds is 4: 4 times 15 makes 60: thus the body has covered 60 feet; that is, 15 in the first second, and 45 in the second. Three times 15; thus the progression is from 1 to 3 in this second.
3rd Second, 3 speeds. The body covers 75 feet. The square of 3 seconds is 9; now, 9 times 15 makes 135: thus the body has covered 135 feet in 3 seconds. Five times 15 feet; thus the progression is clearly according to odd numbers 1, 3, 5, etc.

It is clear that the power that acts equally at every moment, and loses none of its force, must thus increase its effect, until some other force comes to oppose it.

With this small table, a glance will demonstrate that after one minute, the moving object will have covered 54,000 feet, because 3,600 feet make the square of sixty seconds: now, 15 multiplied by the square of 60, which is 3,600, gives 54,000.

From this beautiful discovery by Galileo, a new question arose. It was asked: Will a body always descend about 15 feet in the first second, no matter where in the universe it is placed? We see that the fall of bodies accelerates as they return to our globe: they all evidently tend, in falling, towards the center of this globe; isn’t there some power drawing them towards this center? And does this power not increase its force as the center comes closer? Copernicus had already had some faint glimmer of this idea. Kepler had embraced it, but without method. Chancellor Bacon stated formally that it is likely that there is an attraction of bodies to the center of the Earth, and from this center to the bodies. He proposed, in his excellent book Novum Organum Scientiarum, that experiments be carried out with pendulums on the tallest towers and in the deepest depths: For, he said, if the same pendulums make more rapid vibrations at the bottom of a well than on a tower, it must be concluded that gravity, which is the principle of these vibrations, is much stronger at the center of the Earth, which this well is closer to. He also tried to have moving objects descend from different elevations, and to observe if they would descend less than fifteen feet in the first second; but no variation ever appeared in these experiments, the heights or depths where they were conducted being too small.

Thus, there remained uncertainty, and the idea of this force acting from the center of the Earth remained a vague suspicion.

Descartes was aware of it: he even discussed it in terms of gravity; however, the experiments that would clarify this great question were still lacking. The vortex system carried away this sublime and vast genius: in creating his universe, he wanted to give everything its direction through his subtle matter: he made it the dispenser of all movement and all gravity; gradually Europe adopted his system, despite the protests of Gassendi, who was less followed because he was less daring.

One day, in the year 1666, Newton, having retreated to the countryside, and seeing fruits fall from a tree, as his niece (Mrs. Conduit) told me, gave himself over to deep meditation on the cause that thus draws all bodies along a line which, if extended, would pass near the center of the earth66.

What is, he asked himself, this force that cannot come from all these imaginary vortices shown to be so false? It acts on all bodies in proportion to their masses, not their surfaces; it would act on the fruit that has just fallen from this tree, whether it was raised three thousand toises or ten thousand. If this is so, this force must act from where the moon’s globe is down to the center of the earth; if so, this power, whatever it is, may thus be the same as that which causes the planets to tend towards the sun, and that which causes Jupiter’s satellites to gravitate towards Jupiter. Now, it is demonstrated, from all the inductions drawn from Kepler’s laws, that all these secondary planets weigh towards the center of their orbits, more the closer they are, and less the farther they are, that is, inversely according to the square of their distances.

A body placed where the moon is, circulating around the earth, and a body placed near the earth, must therefore both weigh on the earth precisely according to this law.

Therefore, to be certain if it is the same cause that keeps the planets in their orbits and that causes bodies here to fall, no more than measurements are needed, it is only necessary to examine what space a falling body covers on the earth in a given time, and what space a body placed in the region of the moon would cover in a given time.

The moon itself is that body which can be considered as actually falling from its highest point in the meridian.

But this is not a hypothesis that one adjusts as one can to a system; it is not a calculation where one should be satisfied with approximate results. One must begin by precisely knowing the distance from the moon to the earth, and, to know it, it is necessary to have the measurement of our globe.

This is how Newton reasoned; but he relied, for the measurement of the earth, on the faulty estimate of pilots, who counted sixty miles of England, that is, twenty French leagues, for one degree of latitude, whereas it should have been seventy miles.

There was, indeed, a more accurate measurement of the earth available. Norwood, an English mathematician, had, in 1636, measured a degree of the meridian quite accurately; he had found it, as it should be, to be about seventy miles. But this operation, done thirty years before, was unknown to Newton. The civil wars that had afflicted England, always as detrimental to science as to the state, had buried in oblivion the only accurate measurement of the earth, and people stuck to this vague estimate of the pilots. By this account, the moon was too close to the earth, and the proportions sought by Newton were not found with accuracy. He did not believe he was permitted to supplement anything, or to accommodate nature to his ideas; he wanted to accommodate his ideas to nature: he therefore abandoned this beautiful discovery, which the analogy with other stars made so plausible, and which was so close to being demonstrated; a rare honesty, which alone should lend great weight to his opinions.

Finally, on more accurate measurements taken in France several times, which we will discuss, he found the proof of his theory. The degree of the earth was estimated at twenty-five of our leagues, the moon was found to be at sixty half-diameters from the earth, and Newton thus resumed the thread of his demonstration.

Gravity on our globe is in inverse proportion to the squares of the distances of heavy bodies from the center of the earth; that is to say, a body that weighs a hundred pounds at one diameter from the earth will weigh only one pound if it is ten diameters away.

The force that causes gravity does not depend on vortices of subtle matter, whose existence is proven false.

This force, whatever it may be, acts on all bodies, not according to their surfaces, but according to their masses. If it acts at a distance, it must act at all distances; if it acts in inverse proportion to the square of these distances, it must always act according to this proportion on known bodies, when they are not at the point of contact, I mean as close as possible without being united.

If, according to this proportion, this force causes a body on our globe to travel 54,000 feet in 60 seconds, a body which would be about sixty radii from the center of the earth would, in 60 seconds, fall only 15 feet from Paris or so.

The moon, in its average motion, is distanced from the center of the earth by about sixty radii of the earth’s globe: now, from the measurements taken in France, we know how many feet make up the orbit described by the moon; we know from this that in its average motion it describes 187,001 Parisian feet in one minute.

The moon, in its average motion, has fallen from A to B (figure 48): it has thus obeyed the force of the projectile that pushes it along the tangent AC, and the force that would make it descend along the line AD, equal to BC; remove the force that directs it from A to C, there remains a force that can be estimated by the line CB: this line CB is equal to the line AD; but it is demonstrated that the curve AB, worth 187,961 feet, the line AD or CB is worth only fifteen: thus, whether the moon has fallen in A or in D, it is the same here, it would have traveled 15 feet in one minute from C to B; thus it would have also traveled 15 feet from A to D in one minute. But, in covering this space in one minute, it does precisely 3,600 times less distance than a mobile would here on earth; 3,600 is just the square of its distance: thus the gravitation that thus acts on all bodies also acts between the earth and the moon precisely in this ratio of the inverse square of distances.

But if this power that animates the bodies directs the moon in its orbit, it must also direct the earth in its own, and the effect it has on the moon’s planet, it must have on the earth’s planet, for this power is everywhere the same; all the other planets must be subject to it: the sun must also experience its law, and if there is no movement of the planets with respect to each other that is not the necessary effect of this power, then it must be admitted that all of nature demonstrates it. This is what we will observe more amply67.

Chap. IV. — That gravitation and attraction direct all the planets in their course. — How the theory of gravity should be understood in Descartes. What is centrifugal force and centripetal force. This demonstration proves that the sun is the center of the universe, not the earth. It is for the reasons above that we have more summer than winter.

Almost all of Descartes' theory of gravity is based on this natural law that any body moving in a curved line tends to move away from its center along a straight line that would touch the curve at a point. This is like a sling escaping from the hand, etc.

All bodies, by rotating with the Earth, thus make an effort to move away from the center; but the subtle matter, making a much greater effort, was said to repel all other bodies.

It is easy to see that it was not for the subtle matter to make this greater effort, and to move away from the center of the supposed vortex rather than the other bodies; on the contrary, it was its nature (supposed it existed) to go to the center of its motion, and to let go to the circumference all bodies that would have had more mass. This is indeed what happens on a table that turns around, when, in a tube made in this table, several powders and several liquids of different specific weights are mixed: everything that has more mass moves away from the center; everything that has less mass approaches it. Such is the law of nature, and when Descartes made his supposed subtle matter circulate at the circumference, he began by violating this law of centrifugal forces, which he posited as his first principle. Despite imagining that God had created dice rotating on each other; that the scraping of these dice, which made his subtle matter, escaping on all sides, thereby gained more speed; that the center of a vortex became encrusted, etc.; these imaginations were far from rectifying this error.

Without wasting more time fighting these beings of reason, let us follow the laws of mechanics that operate in nature. A body that moves circularly takes in this manner, at each point of the curve it describes, a direction that would move it away from the circle, making it follow a straight line.

This is true. But it must be taken into account that this body would thus move away from the center only by this other great principle: that every body, being indifferent in itself to rest and to motion, and having this inertia which is an attribute of matter, necessarily follows the line in which it is moved. Now, every body that rotates around a center follows at every instant an infinitely small straight line, which would become an infinitely long straight line if it encountered no obstacles. The result of this principle, reduced to its true value, is therefore nothing else, but that a body that follows a straight line will always follow a straight line: therefore another force is needed to make it describe a curve; thus this other force, by which it describes the curve, would make it fall to the center at every instant, in case this projectile motion in a straight line ceased. Indeed, moment by moment this body would go to A, B, C, if it escaped (figure 49).

But also moment by moment it would fall from A, B, C, back to the center; because its movement is composed of two kinds of movements: the projectile movement in a straight line, and the movement also imparted in a straight line by the centripetal force, the force by which it would go to the center. Thus, from the very fact that the body would describe these tangents A B C, it is demonstrated that there is a power that withdraws it from these tangents at the very instant it begins them. Thus, it must absolutely be considered that any body moving in a curve is moved by two powers, one of which is the one that would make it travel along tangents, and which is called the centrifugal force, or rather the force of inertia, of inactivity, by which a body always follows a straight line if it is not prevented; and the other force that withdraws the body towards the center, which is called the centripetal force, and which is the true force68.

From the establishment of this centripetal force, it first results this demonstration that every mobile that moves in a circle, or in an ellipse, or in any curve, moves around a center to which it tends.

It follows also that this mobile, whatever portions of curve it traverses, will describe, in its largest and smallest arcs, equal areas in equal times. For example, if a mobile in one minute skirts the space A C B (figure 51), which contains one hundred square miles, it must skirt in two minutes another space B C D of two hundred square miles.

This law, inviolably observed by all the planets, and unknown to all of antiquity, was discovered nearly one hundred fifty years ago by Kepler, who deserved the name of legislator in astronomy, despite his philosophical errors. He could not yet know the reason for this rule to which the celestial bodies are subjected. The extreme sagacity of Kepler found the effect of which the genius of Newton found the cause.

I will give the substance of Newton’s demonstration: it will be easily understood by any attentive reader, for humans have a natural geometry in their mind, which makes them grasp relations when they are not too complicated69.

Let the body A (figure 54) be moved to B in a very short space of time: at the end of a like space, a similarly continued movement (for there is here no acceleration) would bring it to G; but at B, it finds a force that pushes it along the line B H S: thus it follows neither this path B H S, nor this path A B C: draw this parallelogram C D B H, then the mobile being moved by the force B C, and by the force B H, goes according to the diagonal B D; and this line B D and this line B A, conceived as infinitely small, are the beginnings of a curve, etc.; thus this body must move in a curve.

It must skirt equal spaces in equal times, for the space of the triangle S B A is equal to the space of the triangle S B D; these triangles are equal: therefore these areas are equal; therefore every body that traverses equal areas in equal times in a curve makes its revolution around the center of forces to which it tends; therefore the planets tend towards the sun, and not around the Earth: for taking the Earth as the center, their areas are unequal in relation to the times; and taking the sun as the center, these areas are always proportional to the times, except for the small disturbances caused by the gravitation of the planets themselves.

To understand even better what these areas proportional to the times are, and to see at a glance the advantage you gain from this knowledge, look at the Earth carried in its ellipse around the sun S, its center (figure 55). When it goes from B to D, it sweeps as large a space as when it travels this large arc H K: the sector H K gains in width what the sector B S D has in length. To make the area of these sectors equal in equal times, the body towards H K must move faster than towards B D. Thus the Earth and every planet moves faster in its perihelion, which is the curve closest to the sun S, than in its aphelion, which is the curve farthest from this same focus S.

One therefore knows what the center of a planet is, and what figure it describes in its orbit, by the areas it traverses; one knows that every planet, when it is farther from the center of its motion, gravitates less towards this center. Thus the Earth being closer to the sun by a thirtieth and more, that is to say, by twelve hundred thousand leagues, during our winter than during our summer, is also more attracted in winter; thus it moves faster then because of its curve; thus we have eight and a half days more summer than winter, and the sun appears in the northern signs eight and a half days more than in the southern ones. Since every planet follows, with respect to the sun the focus of its orbit, this law of gravitation that the moon experiences with respect to the Earth, and to which all bodies are subjected when falling on the Earth, it is demonstrated that this gravitation, this attraction, acts on all bodies that we know.

But another powerful demonstration of this truth is the law followed respectively by all the planets in their courses and in their distances; this must be carefully examined.

Chap. V. — Demonstration of the laws of gravitation drawn from Kepler’s rules: that one of these laws of Kepler demonstrates the movement of the earth. — Great rule of Kepler. False reasons for this admirable law. True reason for this law, found by Newton. Recapitulation of the proofs of gravitation. These discoveries by Kepler and Newton serve to demonstrate that it is the earth that revolves around the sun. Demonstration of the movement of the earth, drawn from the same laws.

Kepler also found this admirable rule, which I will illustrate with an example before giving the definition, to make the matter more tangible and easier to understand.

Jupiter has four satellites revolving around it; the closest is at a distance of 2 diameters of Jupiter and 5 sixths, completing its orbit in 42 hours; the farthest revolves around Jupiter in 402 hours. I want to find out how far this last satellite is from the center of Jupiter. To do this, I apply the following rule: As the square of 42 hours, the orbit of the first satellite, is to the square of 402 hours, the orbit of the last, so the cube of 2 diameters and 5 sixths is to a fourth term. Having found this fourth term, I extract its cube root; this cube root turns out to be 12 and two thirds; thus, I say that the fourth satellite is 12 diameters of Jupiter and two thirds away from the center of Jupiter.

I apply the same rule to all the planets revolving around the sun. I say: Venus rotates in 224 days, and the Earth in 365; the Earth is 30 million leagues from the sun; how many leagues is Venus? I say: As the square of the Earth’s year is to the square of Venus’s year, so the cube of the Earth’s average distance is to a fourth term whose cube root will be about 21 million 700,000 leagues, which is the average distance of Venus from the sun; I say the same for the Earth and Saturn, etc.

This law is therefore that the square of a planet’s orbit is always to the square of the orbits of other planets as the cube of its distance is to the cubes of the distances of the others from the common center.

Kepler, who discovered this proportion, was far from finding the reason for it. A less good philosopher than a remarkable astronomer, he said (in the fourth book of his Epitome) that the sun has a soul, not an intelligent soul, animum, but a vegetative, acting soul, animam; that by rotating on itself it attracts the planets; but that the planets do not fall into the sun, because they also revolve on their axis. By making this revolution, he said, they present to the sun sometimes a friendly side, sometimes an enemy side: the friendly side is attracted, and the enemy side is repelled; this produces the annual course of the planets in ellipses.

It must be admitted, to the humiliation of philosophy, that it is from this reasoning, so unphilosophical, that he concluded that the sun must rotate on its axis: error led him by chance to the truth; he guessed the rotation of the sun on itself more than fifteen years before Galileo’s eyes recognized it with the help of telescopes.

Kepler adds, in the same Epitome, page 495, that the mass of the sun, the mass of all the ether, and the mass of the spheres of the fixed stars, are perfectly equal, and that these are the three symbols of the most holy Trinity.

The reader who, in reading these elements, has seen such great fantasies alongside such sublime truths, in such a great man as Kepler, in such a profound mathematician as Kircher, should not be surprised; one can be a genius in terms of calculation and observations, and sometimes use one’s reason poorly for the rest; there are such minds that need to rely on geometry, and fall when they want to walk alone. It is therefore not surprising that Kepler, in discovering these laws of astronomy, did not know the reason for these laws70.

This reason is that the centripetal force is precisely in inverse proportion to the square of the distance from the center of motion, towards which these forces are directed: this must be followed carefully. It must be well understood that in a word this law of gravitation is such that any body that approaches three times closer to the center of its motion gravitates nine times more; that, if it moves away three times more, it will gravitate nine times less; and that if it moves away a hundred times more, it will gravitate ten thousand times less.

A body moving circularly around a center thus weighs in inverse proportion to the square of its current distance to the center, as also in direct proportion to its mass; and it is demonstrated that it is gravitation that makes it revolve around this center, since, without this gravitation, it would move away by describing a tangent. This gravitation will thus act more strongly on a mobile that rotates faster around this center; and the more distant this mobile is, the more slowly it will rotate, for then it will weigh much less.

Thus, this law of gravitation, based on the square of the distances, is demonstrated:

1° By the orbit described by the moon, and by its distance from the Earth, its center;

2° By the path of each planet around the sun in an ellipse;

3° By comparing the distances and revolutions of all the planets around their common center.

It is not useless to note that this same rule of Kepler, which serves to confirm Newton’s discovery regarding gravitation, also confirms the Copernican system on the motion of the Earth. It can be said that Kepler, by this single rule, demonstrated what had been discovered before him, and opened the way to truths that were to be discovered one day. For, on one hand, it is demonstrated that if the law of centripetal forces did not exist, Kepler’s rule would be impossible; on the other, it is demonstrated that, according to this same rule, if the sun revolved around the Earth, it would be necessary to say: As the revolution of the moon around the Earth in a month is to the supposed revolution of the sun around the Earth in a year, so the square root of the cube of the distance of the moon to the Earth is to the square root of the cube of the distance of the sun to the Earth. By this calculation, one finds that the sun is only 510,000 leagues from us; but it is proven that it is at least about 30 million leagues away; thus, the motion of the Earth has been rigorously demonstrated by Kepler. Here is another very simple demonstration, derived from the same theorems.

If the Earth were the center of the sun’s motion, as it is of the moon’s motion, the sun’s revolution would be 475 years, instead of one year: for the average distance at which the sun is from the Earth is to the average distance at which the moon is from the Earth, as 337 is to 1. Now, the cube of the distance of the moon is 1; the cube of the distance of the sun 38,272,753: complete the rule, and say: As the cube 1 is to this cube number 38,272,753, so the square of 28, which is the periodic revolution of the moon, is to a fourth number; you will find that the sun would take 475 years, instead of a year, to revolve around the Earth: it is therefore demonstrated that it is the Earth that revolves.

It seems all the more appropriate to place these demonstrations here as there are still men destined to instruct others in Italy, in Spain, and even in France, who doubt, or who pretend to doubt the motion of the Earth.

It is therefore proven, by Kepler’s law and by that of Newton, that each planet gravitates towards the sun, the center of the orbit they describe: these laws are fulfilled in the satellites of Jupiter with respect to Jupiter, their center; in the moons of Saturn, with respect to Saturn; in ours, with respect to us: all these secondary planets, which roll around their central planet, also gravitate with their central planet towards the sun; thus the moon, drawn around the Earth by the centripetal force, is at the same time attracted by the sun, around which it also makes its revolution. There is no variety in the course of the moon, in its distances from the Earth, in the shape of its orbit, sometimes approaching an ellipse, sometimes a circle, etc., that is not a consequence of gravitation due to changes in its distance to the Earth, and its distance to the sun.

If it does not exactly traverse equal areas in equal times in its orbit. Mr. Newton has calculated all the cases where this inequality occurs: all depend on the attraction of the sun; it attracts these two globes in direct proportion to their masses, and in inverse proportion to the square of their distances. We will see that the slightest variation of the moon is a necessary effect of these combined powers.

Chap. VI. — New proofs of attraction: that the inequalities in the movement and orbit of the moon are necessarily the effects of attraction. — Example in proof. Inequalities in the course of the moon, all caused by attraction. Deduction of these truths. Gravitation is not the effect of the course of the stars, but their course is the effect of gravitation. This gravitation, this attraction may be a first principle established in nature.

The moon has only one uniform motion, which is its rotation around itself on its axis, and this is the only one we do not perceive: it is this motion that always presents us with nearly the same lunar disk; so that while actually rotating on itself, it appears not to rotate at all, and to have only a slight swaying motion, a libration, which it does not have71, and which all of antiquity attributed to it72.

All its other motions around the earth are uneven, and must be so if the rule of gravitation is true. The moon, in its monthly course, is necessarily closer to the sun at a certain point and at a certain time in its course: now, at this point and at this time, its mass remains the same; its distance only being changed, the sun’s attraction must change in inverse proportion to the square of this distance: the moon’s course must therefore change, it must go faster at certain times than the earth’s attraction alone would make it go; for, by the earth’s attraction, it must cover equal areas in equal times, as you have already observed in chapter iv.

One cannot help but admire with what sagacity Newton unraveled all these inequalities, regulated the course of this planet, which had eluded all the researches of astronomers; here especially one can say:

Nec propius fas est mortali attingere divos73.

Among the examples one might choose, take this one: Let A be the moon (figure 56); A B N Q, the moon’s orbit; S, the sun; B, the location where the moon is in its last quarter. It is then evidently at the same distance from the sun as the earth is. The difference in the obliquity of the moon’s line of direction to the sun being counted as nothing, the earth’s and moon’s gravitation towards the sun is therefore the same. However, the earth advances on its annual path from T to V, and the moon, in its monthly course, advances to Z: now, at Z, it is clear that it is more attracted by the sun S, being closer to it than the earth is; its motion will therefore be accelerated from Z to N; the orbit it describes will therefore be changed. But how will it be changed? By flattening a bit, becoming more straightened from Z to N: thus, from moment to moment, gravitation changes the course and shape of the ellipse in which this planet moves.

For the same reason, the moon must delay its course, and change again the shape of the orbit it describes, when it passes back from the conjunction N to its first quarter Q: for, since in its last quarter it accelerated its course by flattening its curve towards its conjunction N, it must delay this same course in rising from the conjunction towards its first quarter.

But, when the moon rises from this first quarter towards its full phase A, it is then farther from the sun, which attracts it less; it gravitates more towards the earth. Then, the moon accelerating its motion, the curve it describes flattens a bit again as in the conjunction, and this is the sole reason why the moon is farther from us in its quarters than in its conjunction and opposition. The curve it describes is a kind of oval approaching a circle.

Thus, the sun, which it approaches or moves away from every moment, must at every moment vary the course of this planet. It has its apogee and perigee, its greatest and smallest distance from the earth; but the points, the places of this apogee and perigee must change.

It has its nodes, that is, the points where the orbit it traverses precisely meets the orbit of the earth; but these nodes, these points of intersection, must also always change.

It has its equator inclined to the earth’s equator; but this equator, sometimes more, sometimes less attracted, must change its inclination.

It follows the earth despite all these variations: it accompanies it in its annual course; but the earth, in this course, is found to be a million leagues closer to the sun in winter than in summer. What happens then, independently of all these other variations? The earth’s attraction acts more fully on the moon in summer: then the moon completes its monthly course a bit faster; but in winter, on the contrary, the earth itself, more attracted by the sun and moving faster than in summer, lets the moon’s course slow down, and the lunar winter months are a bit longer than the summer months. This little we say will suffice to give a general idea of these changes.

If someone were to make here the difficulty that I have sometimes heard proposed: how, being more attracted by the sun, does the moon not then fall into this star? one has only to consider that the force of gravitation that directs the moon around the earth is only reduced here by the action of the sun; we will see further, in the article on comets, why a body that moves in an ellipse, and approaches its focus, does not fall into this focus.

From these inequalities in the course of the moon, caused by attraction, you will rightly conclude that any two planets, close enough and large enough to act on each other sensibly, can never rotate in circles around the sun, nor even in absolutely regular ellipses. Thus, the curves described by Jupiter and Saturn experience, for example, noticeable variations when these stars are in conjunction; when, being as close to each other as possible, and farthest from the sun, their mutual action increases, and the sun’s action on them decreases.

This gravitation, increased and weakened according to the distances, thus necessarily assigned an irregular elliptical shape to the path of most planets: thus the law of gravitation is not the effect of the course of the celestial bodies; but the orbit they describe is the effect of gravitation. If this gravitation were not, as it is, in inverse proportion to the squares of the distances, the universe could not subsist in the order in which it is.

If the satellites of Jupiter and Saturn make their revolution in curves that are closer to the circle, it is because, being very close to the large planets, which are their center, and very far from the sun, the sun’s action cannot change the course of these satellites, as it changes the course of our moon; it is therefore proven that gravitation, whose name alone seemed such a strange paradox, is a necessary law in the constitution of the world: so much that is improbable is true sometimes!

At present there is no good physicist who does not recognize both Kepler’s rule and the necessity of admitting a gravitation such as Newton proved it; but there are still philosophers attached to their vortices of subtle matter, who would like to reconcile these imaginary vortices with these demonstrated truths.

We have already seen how inadmissible these vortices are; but does not this very gravitation provide a new demonstration against them? For, assuming these vortices existed, they could only rotate around a center by the laws of this very gravitation; one would therefore have to resort to this gravitation as the cause of these vortices, and not to the supposed vortices as the cause of gravitation.

If, finally forced to abandon these imaginary vortices, one is reduced to saying that this gravitation, this attraction depends on some other known cause, some other secret property of matter, this might indeed be the case; but this other property would itself be the effect of another property, or else be a primary cause, a principle established by the Author of nature: but why should the attraction of matter not itself be this primary principle?

Newton, at the end of his Optics, said that perhaps this attraction is the effect of an extremely elastic and rare spirit spread throughout nature; but then where would this elasticity come from? would it not be as difficult to understand as gravitation, attraction, the centripetal force? This force is demonstrated to me; this elastic spirit is barely suspected; I stick with that, and I cannot admit a principle of which I have not the slightest proof, to explain a true and incomprehensible thing of which all nature demonstrates the existence74.

It is good to observe here that great geometers of the Academy of Sciences in Paris believe they find other relationships of gravitation, between the moon and the earth, than those assigned by Newton. I do not enter into this dispute: it serves only to show that gravitation is a quality of nature as recognized as its extent, and to make blush the ignorant who, believing themselves learned, dared to combat this demonstrated quality.

Chap. VII. — New proofs and new effects of gravitation: that this power is in every part of matter: discoveries dependent on this principle. General and important remark on the principle of attraction. Gravitation, attraction is in all parts of matter equally. Bold and admirable calculation by Newton.

Let us gather from all these notions that the centripetal force, attraction, gravitation is the indubitable principle of the motion of the planets, the fall of all bodies, and the gravity we experience in bodies. This centripetal force causes the sun to gravitate towards the center of the planets, as the planets gravitate towards the sun, and draws the earth towards the moon as the moon towards the earth.

One of the primitive laws of motion is yet another demonstration of this truth: this law is that reaction is equal to action. Thus, if the sun gravitates on the planets, the planets gravitate on it; and we will see, at the beginning of the next chapter, in what manner this great law operates.

Now, this gravitation acting necessarily in direct proportion to mass, and the sun being about 464 times larger than all the planets combined (not counting Jupiter’s satellites, and Saturn’s ring and moons), it must be that the sun is their center of gravitation: thus they must all revolve around the sun.

We must always carefully note that, when we say that the power of gravitation acts in direct proportion to the masses, we always mean that this power of gravitation acts more on a body the more parts it has; and we have demonstrated this by showing that a straw falls as fast in an air-purged machine as a pound of gold. We have stated (ignoring the slight resistance of the air) that a lead ball, for example, falls 15 feet to the earth in one second; we have shown that the same ball would fall 15 feet in a minute, if it were at 60 radii from the earth, as is the moon: therefore the power of the earth on the moon is to the power it would have on a lead ball transported to the elevation of the moon: as the solid body of the moon would be with the solid body of that small ball. It is in this proportion that the sun acts on all the planets; it attracts Jupiter and Saturn, and the satellites of Jupiter and of Saturn, in direct proportion to the solid matter75 that is in the satellites of Jupiter and of Saturn, and in that which is in Saturn and Jupiter.

From this flows an indisputable truth: that this gravitation is not only in the total mass of each planet but in every part of this mass; and thus there is not an atom of matter in the universe that is not endowed with this property.

We choose here the simplest manner in which Newton demonstrated that this gravitation is equally in each atom. If all parts of a globe did not equally have this property, if there were weaker and stronger ones, the planet, in turning on itself, would necessarily present weaker sides, and then stronger sides at equal distance: thus the same bodies, on all possible occasions, experiencing sometimes a degree of gravitation, sometimes another at equal distance, the law of the inverse square of the distances and Kepler’s law would always be inverted; but they are not, therefore there is in all the planets no part less gravitating than another.

Here is yet another demonstration. If there were bodies in which this property was different, there would be bodies that would fall more slowly, and others faster, in the vacuum machine; but, all bodies fall in the same time, all pendulums even make in the air similar vibrations at equal length; pendulums of gold, silver, iron, maple wood, glass, make their vibrations in equal times: therefore all bodies have this property of gravitation precisely to the same degree, that is to say, precisely as their masses; so that gravitation acts as 100 on 100 atoms, and as 10 on 10 atoms.

From truth to truth one rises insensibly to knowledge that seemed to be beyond the scope of the human mind.

Newton dared to calculate, with the help of the sole laws of gravitation, what should be the weight of bodies on other globes than ours: what should weigh in Saturn, in the sun, the same body that we call here a pound; and as these different weights depend directly on the mass of the globes, it was necessary to calculate what should be the mass of these stars. After that, say that gravitation, attraction is an occult quality! dare to call such a name a universal law, which leads to such astonishing discoveries76!

We cannot know the mass of all the planets, for those that do not have moons, no satellites, lacking planets of comparison, cannot be subjected to our research; thus we do not know the ratio of gravitation that is between Mercury, Mars, Venus, and us, but we do know that of the other planets77.

I am going to give a small theory of our entire planetary world, as Newton’s discoveries help to make it known; those who want to give themselves a more thorough reason for these calculations will read Newton himself, or Gregory, or Mr. de S’Gravesande. It is only necessary to note that by following the proportions discovered by Newton we have adhered to the astronomical calculation of the Paris Observatory. Whatever the calculation, the proportions and the proofs are the same.

Chap. VIII. — Theory of our planetary world. — Demonstration of the movement of the earth around the sun, drawn from gravitation. Size of the sun. It rotates on itself around the common center of the planetary world. It always changes place. Its density. In what proportion bodies fall on the sun. Newton’s idea on the density of Mercury’s body. Copernicus' prediction on the phases of Venus.

THE SUN.

The sun is at the center of our planetary world, and must necessarily be there. It is not that the middle point of the sun is precisely the center of the universe; but this central point, towards which our universe gravitates, is necessarily within the body of this star, and all the planets, having once received projectile motion, must all revolve around this point, which is in the sun. Here is the proof.

Consider these two globes A and B (figure 57), the larger representing the sun, the smaller representing any planet. If they are both left to the law of gravitation, and free from any other movement, they will be attracted in direct proportion to their masses; they will be directed in a perpendicular line towards each other; and A, being a million times larger than B, will move towards it a million times faster than globe A will move towards B.

But if they each have a projectile motion proportional to their masses, the planet in B C, the sun in A D: then the planet obeys two movements: it follows the line B C, and gravitates towards the sun along the line B A; it will therefore travel the curved line B F; the sun itself will follow the line A E; and, gravitating towards each other, they will revolve around a common center. But the sun, surpassing the earth a million times in size, and the curve A E, which it describes, being a million times smaller than that described by the earth, this common center is necessarily almost at the middle of the sun.

It is also demonstrated that the earth and the planets revolve around this star; and this demonstration is all the more beautiful and powerful as it is independent of all observation, and based on the primordial mechanics of the world.

If the sun’s diameter is made equal to a hundred diameters of the earth, and thus surpasses the earth a million times in size, it is 464 times larger than all the planets combined, not counting the satellites of Jupiter or Saturn’s ring. It gravitates towards the planets, and makes them all gravitate towards it; this gravitation makes them orbit by pulling them from the tangent, and the attraction that the sun exerts on them surpasses that which they exert on it, as much as it surpasses them in quantity of matter. Never lose sight of the fact that this mutual attraction is nothing else than the law of bodies all gravitating, and all turning towards a common center.

The sun, therefore, rotates on this common center, that is, on itself, in 25 and a half days; its middle point is always a little away from this common center of gravity, and the body of the sun moves away from it proportionally as several planets in conjunction attract it towards them; but, when all the planets are on one side and the sun on the other, the common center of gravity of the planetary world barely exits the sun, and their combined forces could barely disturb and move the sun by one entire diameter.

It actually changes position at every moment, as it is more or less attracted by the planets; and this slight approach of the sun restores the disturbance that the planets cause on each other; thus the continual disturbance of this star maintains the order of nature.

Although it surpasses the earth a million times in size, it does not have a million more material, as has already been mentioned.

If it were indeed a million times more solid, more full than the earth, the order of the world would not be as it is: for the revolutions of the planets and their distances from their center depend on their gravitation, and their gravitation depends directly on the quantity of the material of the globe where their center is; thus, if the sun surpassed our earth and our moon to such an excess in solid material, these planets would be much more attracted, and their ellipses very disturbed.

Secondly, the material of the sun cannot be like its size; for, being entirely on fire, rarification is necessarily very great, and the material is all the less as the rarification is stronger.

By the laws of gravitation, it appears that the sun has only 250,000 times more material than the earth; thus, the sun, being a million times larger, is only a quarter of a million times more material, the earth, being a million times smaller, will therefore proportionally have four times more material than the sun, and be four times denser.

The same body, in this case, which weighs a pound on the surface of the earth would weigh 35 pounds on the surface of the sun; but this proportion is 24 to one, because the earth is not in fact four times denser, and because the diameter of the sun is here supposed to be a hundred times that of the earth.

The same body that falls here from 15 feet in the first second, would fall about 415 feet on the surface of the sun, all else being equal78.

The sun always loses, according to Newton, a little of its substance, and would eventually be reduced to nothing, if the comets that occasionally fall into its sphere did not serve to repair its losses: for everything alters and everything repairs itself in the universe.

MERCURY.

From the sun to about eleven or twelve million leagues, or thereabouts, no globe appears79.

Eleven or twelve million leagues from the sun is Mercury at its average distance. It is the most eccentric of all the planets: it orbits in an ellipse that brings it about a third closer at perihelion than at aphelion; such is, approximately, the curve it describes (figure 58).

Mercury is about twenty-seven times smaller than the earth; it orbits the sun in 88 days, which constitutes its year. Its rotation on itself, which constitutes its day, is unknown; neither its weight nor its density can be assigned. It is only known that if Mercury is precisely an earth like ours, the material of this globe must be about eight times denser than ours, to prevent everything on it from being in a degree of effervescence that would instantly kill animals of our kind, and would make all matter of the consistency of our world’s waters evaporate.

Here is the proof of this assertion. Mercury receives about 7 times more light than us, due to the square of the distances, because it is about 2 times 2/3 closer to the center of light and heat; thus it is 7 times more heated, all else being equal. Now, on our earth, the great heat of summer being increased about 7 to 8 times immediately makes water boil vigorously; therefore, everything must be about 7 times denser than it is, to resist 7 or 8 times more heat than the hottest summer gives in our climates; therefore, Mercury must be at least 7 times denser than our earth, for the same things that exist on our earth to subsist on the globe of Mercury, all else being equal. Moreover, if Mercury receives about 7 times more rays than our globe, because it is about 2 times 2/3 closer to the sun, by the same reason the sun appears from Mercury about 7 times larger than from our earth.

VENUS.

After Mercury comes Venus, twenty-one or twenty-two million leagues from the sun at its average distance; it is as large as the earth; its year is 224 days. What constitutes its day, that is, its rotation on itself80, is not yet known. Very great astronomers believe this day to be 25 hours, others believe it to be 25 of our days. Sufficiently certain observations have not yet been made to know which side the error is on; but this error, in any case, can only be a mistake of the eyes, an error of observation, and not of reasoning.

The ellipse that Venus travels in its year is less eccentric than that of Mercury; one can form some idea of the path of these two planets around the sun by this figure (figure 58).

It is not out of place to note here that Venus and Mercury, in relation to us, have different phases just like the moon. Copernicus was once reproached that, in his system, these phases should appear; and it was concluded that his system was false, because they were not perceived. If Venus and Mercury, they said to him, revolve around the sun, and we revolve in a larger circle, we must see Mercury and Venus, sometimes full, sometimes crescent, etc.; but this is what we never see. Yet this is what happens, Copernicus told them, and this is what you will see, if you ever find a way to perfect your sight. The invention of telescopes, and Galileo’s observations, soon served to fulfill Copernicus’s prediction. In any case, nothing can be assigned about the mass of Venus, and about the gravity of bodies81 on this planet82.

Chap. IX. — Theory of the earth: examination of its shape. — History of opinions on the shape of the earth. Discovery by Richer, and its consequences. Theory by Huygens. That by Newton. Disputes in France on the shape of the earth.

I will elaborate more on the theory of the earth.

First, I will examine its shape, which necessarily results from the laws of attraction and the rotation of this globe on its axis.

I will demonstrate the movements it has, and I will conclude this theory of our globe with the most evident proofs of the cause of the tides, a phenomenon inexplicable until Newton, and which became the most beautiful testimony to the truths he taught.

I begin with the shape of our globe.

The earliest astronomers, in Asia and Egypt, soon noticed from the projection of the earth’s shadow in lunar eclipses that the earth is round; the Hebrews, who were very poor physicists, imagined it flat; they pictured the sky as a half-arch covering the earth, of which they knew neither the shape nor the size, but of which they hoped to eventually be the masters. This notion of a narrow and flat earth long prevailed among Christians. Among many scholars, in the fifteenth century, it was quite accepted that the earth was flat and elongated from east to west, and very narrow from north to south. A bishop of Avila, who wrote at that time, deemed the opposing view heretical and absurd; finally, reason and Christopher Columbus’s voyage restored the earth to its ancient spherical shape. Then one extreme was switched for another; the earth was believed to be a perfect sphere, as it was then believed that the planets made their revolutions in a true circle.

However, as soon as it was well known that our globe rotates on itself in twenty-four hours, it could have been judged from this alone that a truly round shape could not belong to it. Not only does centrifugal force significantly raise the waters in the equatorial region due to the twenty-four-hour rotation, but they are also raised about twenty-five feet twice per day by the tides; it would thus be impossible for the lands around the equator to be perpetually flooded; yet, they are not: thus, the equatorial region is much more elevated in proportion than the rest of the earth; therefore, the earth is a spheroid elevated at the equator, and cannot be a perfect sphere. This simple proof escaped the greatest geniuses, because a universal prejudice seldom allows for scrutiny.

It is known that in 1672, Richer, on a voyage to Cayenne, near the equator, undertaken by order of Louis XIV under the auspices of Colbert, the father of all arts; Richer, I say, among many observations, found that the pendulum of his clock no longer made its oscillations, its vibrations as frequent as in the latitude of Paris, and that it was absolutely necessary to shorten the pendulum by a line and more than a quarter.

Physics and geometry were not then, by far, as cultivated as they are today: which man could believe that from such a small observation, and from a line more or less, could come the greatest physical truths? It was first found that gravity must necessarily be lesser at the equator than in our latitude, since gravity alone causes a pendulum’s oscillation.

Consequently, since the gravity of bodies is less strong the further these bodies are from the center of the earth, it was absolutely necessary that the region of the equator be much more elevated than ours, further from the center; thus the earth could not be a true sphere.

Many philosophers made, regarding these discoveries, what all men do when it is necessary to change their opinion: they disputed Richer’s experiment; they claimed that our pendulums made their vibrations less prompt towards the equator because the heat elongated the metal; but it was seen that the heat of the hottest summer elongates it by a line over thirty feet in length, and here it was a matter of one and a quarter lines, one and a half lines, or even two lines on an iron rod three feet eight lines long.

A few years later, Messrs. Varin, Deshayes, Feuillée, Couplet, repeated near the equator the same pendulum experiment; it always had to be shortened, although the heat was very often less intense under the line itself than at fifteen or twenty degrees from the equator. This experiment has just been confirmed anew by the academicians whom Mr. the count de Maurepas sent to Peru, and it is learned at the moment that near Quito, on mountains where it was freezing, the second pendulum had to be shortened by about two lines83.

Around the same time, the academicians who went to measure an arc of the meridian in the north found that at Pello, beyond the Arctic Circle, the pendulum had to be lengthened to have the same oscillations as in Paris: consequently, gravity is greater at the Arctic Circle than in the climates of France, as it is greater in our climates than towards the equator. If gravity is greater in the north, the north is therefore closer to the center of the earth than the equator; the earth is therefore flattened towards the poles.

Never did experience and reasoning concur with such agreement to prove a truth. The famous Huygens, by the calculation of centrifugal forces, had proven that gravity must be greater at the equator than in the polar regions, and that consequently the earth must be a spheroid flattened at the poles. Newton, by the principles of attraction, had found the same ratios to a close degree; it is only necessary to observe that Huygens believed that this inherent force in bodies that determines them towards the center of the globe, this primitive gravity, is the same everywhere. He had not yet seen Newton’s discoveries; he therefore considered the decrease in gravity only by the theory of centrifugal forces. The effect of centrifugal forces reduces the primitive gravity under the equator. The smaller the circles in which this centrifugal force is exerted, the more this force yields to that of gravity: thus, under the pole itself, the centrifugal force, which is null, must leave all its action to the primitive gravity.

But this principle of an always equal gravity falls into ruin by the discovery that Newton made, and which we have discussed so much in this work, that a body transported, for example, to ten diameters from the center of the earth, weighs a hundred times less than at one diameter.

Thus, it is through the laws of gravitation, combined with those of centrifugal force, that the true shape of the earth is shown. Newton and Gregory were so sure of this theory that they did not hesitate to state that experiments on gravity were more reliable for determining the shape of the earth than any geographical measurement84.

Louis XIV had marked his reign with this meridian that crosses France; the illustrious Dominique Cassini had begun it with his son; he had, in 1701, drawn from the foot of the Pyrenees, to the Observatory, a line as straight as possible, through the almost insurmountable obstacles that the heights of the mountains, the changes in refraction in the air, and the alterations of the instruments, constantly opposed to this vast and delicate undertaking; he had thus, in 1701, measured 6 degrees 18 minutes of this meridian. But from whatever source the error came, he had found the degrees towards Paris, that is to say, towards the north, smaller than those that went to the Pyrenees towards the south; this measurement contradicted both that of Norwood and the new theory of the earth flattened at the poles.

However, this new theory began to be so widely accepted that the secretary of the Academy did not hesitate, in his history of 1701, to say that the new measurements taken in France proved that the earth is a spheroid whose poles are flattened. The measurements of Dominique Cassini indeed led to the opposite conclusion; but, as the shape of the earth was not yet a question in France, no one at the time contested this false conclusion. The degrees of the meridian from Collioure to Paris were considered to be accurately measured, and the pole, which, by these measurements, should necessarily have been elongated, was considered flattened.

An engineer named Mr. des Roubais, surprised by the conclusion, demonstrated that, by the measurements taken in France, the earth had to be an oblong spheroid, whose meridian, which goes from one pole to the other, is longer than the equator, and whose poles are elongated85. But of all the physicists to whom he addressed his dissertation, none wanted to publish it, because it seemed that the Academy had spoken, and it appeared too bold for an individual to contest.

Sometime later, the error of 1701 was recognized, there was a retraction, and the earth was elongated by a just conclusion drawn from a false principle. The meridian was continued on this principle from Paris to Dunkirk; the degrees of the meridian were always found smaller when going towards the north.

Around this time, mathematicians, who were conducting the same operations in China, were surprised to see a difference between their degrees, which they thought should be equal, and to find them, after several verifications, smaller towards the north than towards the south. This was another powerful reason to believe in the oblong spheroid, that this agreement of the mathematicians of France and those of China.

More was done in France, parallels to the equator were measured. It is easy to understand that, on an oblong spheroid, our degrees of longitude must be smaller than on a sphere. Mr. de Cassini found the parallel that passes through Saint-Malo shorter by one thousand thirty-seven toises than it should have been in the hypothesis of a spherical earth. This degree was therefore incomparably shorter than it would have been on a spheroid with elongated poles.

So many measurements overturned for a time, in France, the demonstration of Newton and Huygens, and there was no doubt that the poles were of a shape entirely opposite to that which they had been thought to be at first.

Finally, the new academicians who went to the Arctic Circle in 1736, having found, by the measurements taken with the most scrupulous accuracy, that the degree was in these climates much longer than in France, there was doubt between them and the Cassinis. But soon after there was no more doubt: for the same astronomers who returned from the pole examined again this degree, measured in 1677 by Picard, to the north of Paris; they verified that this degree is 123 toises longer than Picard had determined it. If therefore Picard, with his precautions, had made his degree 123 toises too short, it was very likely that subsequently the degrees towards the south had been found longer than they should be. Thus the first error of Picard, which served as the basis for the measurements of the meridian, also served as an excuse for the almost inevitable errors that very good astronomers could have made in this great work.

The academicians returned from the pole had for them in this dispute both theory and practice. Both were confirmed by an admission made in 1740 to the Academy by the grandson of the illustrious Cassini, heir to the merit of his father and grandfather. He had just completed the measurement of a parallel to the equator; he confessed that finally this measurement, taken with all the care required by the dispute, gave the earth as flattened. This courageous admission should end the dispute honorably for all parties.

Furthermore, the difference between a sphere and a spheroid does not result in a larger or smaller circumference: for a circle changed into an oval neither increases nor decreases in area. As for the difference from one axis to another, it is not seven leagues: an immense difference for those who take sides, but negligible for those who consider the measurements of the earth only for their practical applications; there is no geographer who could show this difference on a map, nor any pilot who could ever know whether he is sailing on a spheroid or on a sphere. However, the measurements that made the spheroid oblong, and those that made it flattened, differed by about a hundred leagues, which then became relevant to navigation.

Here are the numbers accepted today, resulting from the discussion of the measurements by Bessel:

Equatorial radius = R = 6,377,398 meters. — polar. . . = r = 6,356,680 — Difference. . . . . . = R — r = 21,318— Flattening

{displaystyle scriptstyle { rac {1}{299}}}
of R. (D.)

Chap. X. — Of the 25,920 year period, caused by attraction. — General misunderstanding in the language of astronomy. History of the discovery of this period, not favorable to Newton’s chronology. Explanation given by the Greeks. Research on the cause of this period.

If the shape of the earth is an effect of gravitation, of attraction, this powerful principle of nature is also the cause of all the earth’s movements in its annual course. It has, in this course, a movement whose period is completed in nearly twenty-six thousand years: this is the period called the precession of the equinoxes; but, to explain this movement and its cause, one must go back a bit further.

Common language, in terms of astronomy, is perpetually incorrect. It is said that the stars make their revolution around the equator; that the sun each day turns with them around the earth from east to west; yet the stars, by another motion opposite to the sun, turn slowly from west to east; that the planets are stationary and retrograde. None of this is true; it is known that all these appearances are caused by the motion of the earth.

Yet we always express ourselves as if the earth were immobile, and we retain common language because the language of truth would contradict our eyes too much, and the prejudices received are even more deceiving than sight.

But never do astronomers express themselves less in accordance with the truth than when they say in all almanacs: The sun enters spring at such degree of Aries. Summer begins with the sign of Cancer; autumn, with Libra. It has been a long time since all these signs have new places in the sky, relative to our seasons, and it would be time to change the way of speaking, which will indeed have to be changed one day: because, in fact, our spring begins when the sun rises with the fishes; our summer, with the twins; our autumn, with the virgin; our winter, with the archer; or, to speak more precisely, our seasons begin when the earth, in its annual route, is in the signs opposite to the signs that rise with the sun.

Hipparchus was the first among the Greeks to notice that the sun no longer rose in spring in the signs where it had formerly risen. This astronomer lived about sixty years before our common era; such a discovery made so late, which should have been made much earlier, proves that the Greeks had not made great progress in astronomy.

It is said (but it is only one author who says it, in the second century) that at the time of the voyage of the Argonauts, the astronomer Chiron established the beginning of spring, that is, the point where the earth’s ecliptic crossed the equator, at the fifteenth degree of Aries. It is certain that, more than five hundred years later, Meton and Euctemon observed that the sun, at the beginning of summer, entered the eighth degree of Cancer; and consequently, the spring equinox was no longer at the fifteenth degree of Aries, and the sun had advanced seven degrees toward the east since the expedition of the Argonauts. It is on these observations, made five hundred years later by Meton and Euctemon, one year before the Peloponnesian War, that Newton partly based his system of reforming all chronology; and it is on this that I cannot help but submit here my scruples to the enlightenment of knowledgeable people.

It seems to me that if Meton and Euctemon had found such a noticeable difference as that of seven degrees between the position of the sun in the time of Chiron and the time when they lived, they could not have helped discovering this precession of the equinoxes, and the resulting period. It was only a simple rule of three, and to say: If the sun advances about 7 degrees, in 500 some years, in how many years will it complete the entire circle? The period was all found.

Yet nothing was known of it until the time of Hipparchus. This silence makes me believe that Chiron did not know as much as they say, and that it was only afterward that it was believed he had fixed the spring equinox at the fifteenth degree of Aries. It was imagined he had done it because he had to do it. Ptolemy says nothing about it in his Almagest, and this consideration could, in my opinion, somewhat shake Newton’s chronology.

It was not by the observations of Chiron, but by those of Aristillus and Meton, compared with his own, that Hipparchus began to suspect a new vicissitude in the course of the sun. Ptolemy, more than two hundred and fifty years after Hipparchus, confirmed the fact, but confusedly. It was believed that this revolution was of one degree in one hundred years; and it is according to this false calculation that the great year of the world was composed of thirty-six thousand years.

But this movement is really only of one degree or thereabouts in seventy-two years, and the period is only twenty-five thousand nine hundred and twenty years, according to the most accepted calculations. The Greeks, who had no notion of the ancient system once known in Asia, and renewed by Copernicus, were far from suspecting that this period belonged to the earth. They imagined I know not what prime mover, which carried all the stars, the planets, and the sun, in twenty-four hours around the earth; then a crystal heaven, which turned slowly in thirty-six thousand years from west to east, and which made, I know not how, the stars retrograde despite this prime mover; all the other planets, and the sun itself, made their annual revolution, each in its crystal heaven; and this was called philosophy!

At last it was recognized in the last century that this precession of the equinoxes, this long period comes only from a movement of the earth whose equator, year by year, crosses the ecliptic at different points, as will be explained.

Before explaining this movement and showing its cause, may I still be permitted to research what could be the reason for this period.

However bold it may be to determine the reasons of the Creator, it seems at least excusable to dare to say that one guesses the utility of the other movements of our globe. If it traverses year by year, in its great orbit, at least one hundred ninety-eight million leagues around the sun, this course brings us the seasons. If it rotates in twenty-four hours on itself, the distribution of day and night is probably one of the objectives of this rotation ordered by the Master of nature.

It seems to me that there is still another necessary reason for this daily movement: that if the earth did not rotate on itself, it would have no centrifugal force; all its parts, pressed toward the center by the centripetal force, would acquire an adhesion, an invincible hardness, which would make our globe sterile.

In short, one easily understands the utility of all the earth’s movements; but, for this movement of the pole in 25,920 years, I discover no sensible use: it results from this movement that our polar star will one day no longer be our polar star, and it is proven that it has not always been so; the equinox and the solstices change; the sun is no longer in relation to us in Aries at the spring equinox, whatever all the almanacs say: it is in the fishes, and with time it will be in the water-bearer. But what does it matter? This change produces neither new seasons, nor new distribution of heat and light: everything remains in nature sensibly equal,

What, then, is the cause of this twenty-five thousand nine hundred year period, so long and at the same time apparently so useless?

In all the composite machines that we see, there is always some effect that, by itself, does not produce the utility derived from the machine, but which is a necessary consequence of its composition: for example, in a water mill, a great part of the water that falls on the paddles is lost; this water, which the movement of the wheel scatters in all directions, serves no purpose for the machine; but it is an indispensable effect of the movement of the wheel.

The noise that a hammer makes has nothing in common with the bodies that the hammer shapes on the anvil; but it is impossible that the shaking of the anvil does not accompany this action. The steam that escapes from a liquid that we boil necessarily comes out without contributing at all to the use we make of this liquid; and he who judges that all these effects are necessary, although they are often of no sensible utility, judges well.

If we are permitted to compare for a moment the works of God to our feeble efforts, it can be said that, in this immense machine, he has arranged things so that several effects necessarily follow without being of any utility to us. This period of twenty-five thousand nine hundred and twenty years appears to be exactly in this case: it is a necessary effect of the attraction of the sun and the moon.

To gain a clear understanding of this periodic movement of 25,920 years, let’s first consider the Earth (figure 60) carried annually along its major axis A B, parallel to itself around the sun, the North Star.

This axis, moving from west to east, always appears to point towards this North Star; the Earth, in half of its annual journey, that is, from spring to autumn, travels about ninety-five million leagues; but this distance is negligible compared to the extreme remoteness of this star which it would always equally face if this axis of the Earth were always in the same direction A B as you see it.

However, this axis does not persist in this position, and after a very long number of years, this conceived axis along this line of the ecliptic is no longer in the A B position; it no longer looks at its movement of parallelism, it is no longer directed towards the North Star. This different direction is almost nothing compared to the immense expanse of the heavens; but it is significant in relation to the movement of our pole.

Imagine then this small globe of the Earth making its very small revolution of about one hundred ninety-eight million leagues, which is but a point in the vast space filled with fixed stars (figure 61). Its pole, which corresponds to this North Star at P, after seventy-two years will be one degree away.

In six thousand five hundred years this pole will face the star T, and after about thirteen thousand years it will correspond to the star at Z; successively our axis from Z will go to F and return to P, so that at the end of 25,920 years, or thereabouts, we will have the same North Star as today.

Having presented the figure of this revolution of our axis, it will be easy to understand the physical reason. Remember, when discussing the inequalities in the moon’s course, Newton demonstrated that they all depend on the combined attraction of the sun and the Earth. It is this attraction, this gravitation that continually changes the position of the moon, as we have already seen in chapter vi; reciprocally, the attraction of the sun and that of the moon, acting on the Earth, continually changes the position of our globe; do not forget that the Earth is much higher at the equator than towards the poles. Imagine the Earth at T, the moon at L, the sun at S (figure 62).

If the Earth and the moon were always to rotate in the plane of the equator, it is certain that this elevation of the lands D E would always be equally attracted; but when the Earth is not at the equinoxes, this elevated part E, for example, is attracted by the sun and by the moon, which I suppose in that situation: then what must happen to a ball which, loaded unevenly, would roll on a plane; it would wobble, it would tilt. Imagine this part D fallen towards E, by the attraction of the sun, it can only go from D to E while at the same time the terrestrial pole P changes its situation, and goes from P to Z; but this pole can only fall from P to Z as the equator of the Earth responds to another part of the sky than to the one it corresponded to before: thus the points of the equinox and the solstice successively correspond, after seventy-two years, to a different degree in the sky; thus the equinox formerly arrived, in the time of Hipparchus, when the sun appeared to be in the first point of Aries, that is, when the Earth actually entered Libra, the sign opposite Aries; and this same equinox now arrives when the sun appears to be in Pisces, that is, when the Earth is in Virgo, the sign opposite Pisces. By this, all constellations have changed places; Taurus is where Aries was, Gemini where Taurus was.

This gravitation, which is the sole cause of the revolution of twenty-five thousand nine hundred and twenty years in our globe, is also the cause of the lunar revolution of nineteen years, called the lunar cycle, and of the revolution of the moon’s apsides in nine years. It happens to the moon, rotating around the Earth, precisely the same thing as to this elevation of our globe towards the equator; so that one can consider the moon as if it were an elevation, a ring attached to the Earth; and one can similarly consider this prominence of the equator as a ring of several moons.

It is clear that the sun must have more influence than the moon on this movement of the Earth that causes the precession of the equinoxes. The action of the sun is to that of the moon in this case precisely as that of the moon is to that of the sun in the tides86.

The reader no doubt suspects that since the seas rise at the equator, the sun and the moon, which act on this equator, act more noticeably on the tides. The sun contributes about three to this movement of the precession of the equinoxes, and the moon about one. In the tides, on the other hand, the sun acts only as one and the moon as three: an astonishing calculation, reserved for our age, and a perfect agreement of the laws of gravitation that all of nature conspires to demonstrate.

Chap. XI. — Of the ebb and flow: that this phenomenon is a necessary consequence of gravitation. — The alleged whirlwinds cannot be the cause of the tides: proof. Gravitation is the only evident cause of the tides. Refutation of those who claim to assign the final cause of the tides.

If the whirlpools of subtle matter ever seemed plausible, it is in the ebb and flow of the Ocean: that the waters sink under the tropics when they rise towards the poles, it is said that the air presses them under the tropics. But why does the air press there more than elsewhere? It is because it is itself more pressed; it is because the path of the subtle matter is narrowed by the passage of the moon. The ultimate plausibility was that the tides are higher at the new and full moon than at the quadratures, and finally, the return of the tides to each meridian roughly follows the return of the moon to each meridian. What seems so plausible is, however, in fact, very impossible. It has already been shown that this whirlpool of subtle matter cannot subsist; but, even if it did exist, despite all the contradictions that annihilate it, it could in no way cause the tides.

1° In the assumption of this alleged whirlpool of subtle matter, all lines would press equally towards the center of our globe; thus the moon (figure 63) should press equally in its quarters at R and in its full at P, assuming that it pressed. Thus there would be no tide.

2° By an equally strong reason, no body carried by any fluid can certainly press this fluid more than would a similar volume of this fluid; a body in equilibrium in water replaces a similar volume of water. Whether one puts in a fishpond a hundred cubic feet of water more, or a hundred fish swimming in mid-water, each of a cubic foot; or whether one puts a single fish with ninety-nine feet of water more in the fishpond, it is absolutely the same; the bottom of the fishpond will be neither more nor less loaded in any of these cases. Thus, whether there were a moon above our seas or a hundred moons, it is absolutely the same in the imaginary system of whirlpools and fullness: none of these moons should be considered as a similar quantity of fluid matter.

3° The flow occurs at the circumference of the Ocean under the same meridian at the same time at opposite points; the sea sinks at once in A and in B (figure 64). Now, assuming that the moon could press the alleged torrent of subtle matter on the Ocean A, the waters would then rise in B, instead of sinking: for gravity towards the center, in this system, is the effect of the alleged subtle matter. Now this imaginary fluid, pressing in A the waters on the earth, must raise the waters on which it presses less: now on which waters will it press less than on B? What is meant, when it is claimed that B also sinks by the counter-shock? Since when, when one strikes on one side of a body, whatever it may be, does one sink inwards the opposite side? Press a bladder sufficiently filled with air, will it also sink at one end when you push it at the other? Will it not rise on the contrary at the end opposite to the struck side?

"If this chimerical pressure had a place, would not the air pressed under the tropics then raise the mercury in the barometer? But, on the contrary, the mercury is always a little lower in the torrid zone than towards the poles. What appeared so plausible therefore becomes impossible upon examination.

Gravitation, this principle so recognized, so demonstrated, this force so inherent in all bodies, is here displayed in a very tangible manner: it is the obvious cause of all the tides; this will be easy to understand. The earth rotates on itself; the waters that surround it rotate with it; the great circle of any spheroid rotating on its axis is the one that has the most movement; centrifugal force increases as this circle is large.

This circle A (figure 65) experiences more centrifugal force than the circles B; the waters of the sea therefore rise towards the equator by this sole centrifugal force; and not only the waters, but the lands that are towards the equator are also necessarily raised.

This centrifugal force would carry away all parts of the earth and the sea, if the centripetal force, its antagonist, did not hold them back by attracting them towards the center of the earth; now, any sea that is beyond the tropics towards the poles having less centrifugal force, because it rotates in a much smaller circle, obeys more to the centripetal force; it gravitates more towards the earth; it presses the same oceanic sea that extends towards the equator, and contributes a little more, by this pressure, to the elevation of the sea under the line. This is the state of the Ocean by the mere combination of central forces. Now, what should happen by the attraction of the moon and the sun? This constant elevation of the waters between the tropics must still increase, if this elevation is opposite some globe that attracts it. Now, the region of the tropics of our earth is always under the sun and under the moon: therefore the elevation of the sun and the moon must have some effect on these tropics.

1° If the sun and the moon exert an action on these waters that are in these regions, this action must be greater at the time when the moon is more opposite the sun, that is to say in opposition and in conjunction, in full and new moon, than in the quarters: for in the quarters, being more oblique to the sun, it must act on one side when the sun acts on the other: their actions must harm each other, and one must diminish the other; also the tides are higher in the syzygies than in the quadratures.

2° The moon being new, being on the same side as the sun, must act all the more on the earth as it attracts it almost in the same direction as the sun attracts. The tides must therefore be a little stronger, all things being equal, in the conjunction than in the opposition; and this is what is experienced.

3° The highest tides of the year must occur at the equinoxes, and be higher in the new moon than in the full. Draw a line from the sun passing near the moon L (figure 66), and arriving on the equator of the earth. The equator A Q is attracted almost in the same line by these globes; the waters must rise more than at any other time; and as they can only rise by degrees, their greatest elevation is not precisely at the moment of the equinox, but a day or two after in D Z.

4° If by these laws the tides of the new moon at the equinox are the highest of the year, the tides, in the quadratures after the equinox, must be the lowest of the year: for the sun is still almost on the equator, but the moon is then very far from it, as you see.

For the moon L (figure 67), in eight days, will be towards R. Then the same thing happens to the Ocean as to a weight pulled by two powers acting perpendicularly at once on it, and which no longer act obliquely: these two powers no longer have the same force; the sun no longer adds to the moon the power it added, when the moon, the earth, and the sun, were almost in the same perpendicular.

5° By the same laws we must have stronger tides immediately before the spring equinox than after, and conversely stronger immediately after the autumn equinox than before. For, if the action of the sun at the equinoxes adds to the action of the moon, the sun must add all the more action the closer we are to it; now we are closer to the sun before March 21 at the equinox than after, and we are on the contrary closer to the sun after September 21 than before that time: therefore the highest tides, common year, must occur before the spring equinox, and after the autumn equinox, as experience confirms.

Having proven that the sun conspires with the moon to the elevations of the sea, it is necessary to know what amount of concurrence it brings. Newton and others have calculated that the average elevation in the middle of the Ocean is twelve feet; the sun raises it two and a quarter, and the moon eight and three quarters.

Many clever people, to whom Newton’s discoveries are not familiar, make a specious objection against this action that raises the waters.

If the sun and the moon, they say, raise the waters in G on the earth by attraction (figure 68), the waters in D, under the same meridian, must therefore lower.

You have, they will say, the same difficulty to solve as the Cartesians; and, if they cannot explain how the alleged pressure of the moon sinks the waters at the same time at two opposite points, you will not be able to explain either how your gravitation raises the waters at the same time in G and in D, and the phenomenon of the tides will always remain a problem. Such an objection can only come from a straightforward mind; there is merit in being mistaken in this way, and in objecting by reason what enlightened reason then resolves: here is the solution to this difficulty. What makes it impossible, in Descartes' hypothesis, for the waters to sink at the same time at the opposite points of the same meridian, is that gravity is supposed by him to be only the result of a whirlpool, and that, in this case, the moon supposed to press this alleged whirlpool (if it were possible that it pressed) could not press at the same time two opposite places.

But here there is no hypothesis, only the laws of gravity, of gravitation are considered; all waters gravitate towards the center of the earth, every fluid must be in equilibrium: here are the waters raised in C (figure 69), thus the balance is broken; the waters in V then have more gravitation towards the center of the earth: thus they press more than they pressed; thus the waters in F must approach closer, flatten, sink towards the earth.

The waters in F cannot press, flatten in proportion to the elevation of the waters in C unless they force the waters in D to stretch, to rise in proportion to the pressure in F: thus the waters in D must be as high as in C; and when this pressure is made at the equinoxes, the oval of the earth is increased. Thus, not only the sun is one of the causes of the flow of the sea (which was far from being suspected), but the moon, which was thought to crush the waters by its pressure, raises them instead by the force of attraction. We thought that when the Ocean recedes from our coasts, it was because nothing was acting on it anymore; on the contrary, it withdraws like this, and only piles up under the equator by a very great force that constrains it; and the time of the flow, which we call tide, is the time when the sea descends by its own weight, when this force of attraction diminishes.

You clearly see that when the moon raises the waters at L (figure 70), six hours later, as the earth has made a quarter of its turn around itself, the waters that were at L find themselves at S, and must consequently lower, since nothing is raising them anymore. When will these same waters begin to rise again by the immediate action of the moon? When they find themselves under this planet; it will not be after twenty-four hours, but after twenty-four and three-quarters hours, because the moon advances about three-quarters of an hour each day in its course around the earth: thus, the lunar day, that is, the return of the moon to our meridian, is three-quarters of an hour longer than our day.

Moreover, these tides of the oceanic sea seem to be, just as the precession of the equinoxes and the period of the earth in 25,900 years, a necessary effect of the laws of gravitation, without the final cause being assignable: for to say, with so many authors, that God gives us tides for the convenience of our commerce, is to forget that men have only been trading across the Ocean for two hundred years. It is also much to hazard to say that the ebb and flow make ports more advantageous; and even if it were true that the ocean tides were useful for trade, should we say that God sends them with this in view? How many centuries have the earth and the seas existed before we used navigation for our new needs? What! said a clever philosopher, because after a prodigious number of years spectacles were finally invented, should we say that God made our noses to wear glasses?

The same authors also assert that the ebb and flow are ordained by God lest the sea stagnate and become corrupt: they forget again that the Mediterranean does not stagnate, although it has no tide. When one dares to assign the reasons for everything that God has done, one falls into strange errors. Those who limit themselves to calculating, weighing, and measuring often deceive themselves: what then of those who only want to guess87?

In the 1756 edition and its reprints, immediately after chapter xi was, under the title of Chapter xii, conclusion, the end of the work, starting from the word Conclude, etc. (hereinafter, page 581); i.e., the editions of 1756 and others do not contain chapters xii, xiii, and xiv (in large part), which are in the editions of 1741, 1748, 1750, 1732, and in some editions since 1819; see the Warning from Beuchot.

Chap. XII. — Theory of the moon and the rest of the planets. — Why the moon orbits faster around the earth than the earth around the sun. It never shows us but the same side. Why the moon’s year is only 354 days. Its various movements; movements of the apsides in 9 years, that of the nodes in 19 years; the moon moves faster than it used to. It exerts on the sun 40 times less force than the earth. Gravity of bodies at the surface of the moon. Distance from Mars to the sun. Its size. Size and mass of Jupiter. Gravity and fall of bodies on Jupiter. Plane raised at the equator, flattened at the poles. Satellites of Jupiter. How Saturn is seen from the sun. Its density. Remark on the density of the planets. Gravity of bodies on Saturn, and of this globe on the sun. Disturbance between the orbits of Saturn and Jupiter, quite noticeable, and caused by attraction.

The moon, which is the satellite of the earth, is only about ninety thousand leagues away at its average distance.

It gravitates towards the earth as the earth does towards it; hence they have a common center of gravity. This common center of gravity is located near the surface of the earth; it is this common center of gravity that carries the earth and the moon around the sun, the universal focus of all planets and satellites.

The moon, being much closer to the earth than the earth is to the sun, must, according to the laws of attraction, orbit the earth much faster than the earth orbits the sun in its large orbit. Indeed, the moon completes its orbit around our globe in about 27 and a half days, while it takes the earth 365 days to travel around the sun.

The moon rotates on its axis in exactly the same time it takes to complete its 27.5-day orbit around us; thus, the earth always sees the same side of the moon, with minor differences. If the moon rotated on its axis in half the time it takes to travel its monthly orbit, we would successively see its entire surface. If, in its current state, it rotated precisely in a circle around the earth, we would always see precisely the same half of its surface; but it travels an ellipse with the earth at one focus: thus, it sometimes moves slower, sometimes faster, and shows us a little more, sometimes a little less, of this half facing us.

The earth, being carried around the sun in a year by its gravitation, also carries the moon, which must follow it in its large orbit.

But this annual revolution of the moon cannot be the same as that of the earth. For, completing its periodic month of 27 and a half days, it completes its synodic month, its lunation, in 29 and a half days, meaning it takes 29 and a half days to go from one conjunction with the sun to the next. Twelve times 29 and a half makes 354. Thus, the common year of the moon can only be about three hundred fifty-four days, while that of the earth is about three hundred sixty-five.

It completes a revolution in nine years: this is the revolution of its apsides. The apsides are the points of greatest distance of a planet from the center of its revolution: in the moon, these are the apogee and perigee. The apogee is the point furthest from the earth, the perigee the closest. The line crossing these points is the line of the moon’s apsides, which moves from west to east over about nine years, so that at the end of nine years the distance of the moon from the earth is the same.

Its greatest revolution is another movement of nineteen years. This nineteen-year period is what is called the lunar cycle. It occurs from east to west across the poles of the moon, so that the moon’s nodes constantly change, and are the same again after nineteen years. These nodes of the moon are the points at which the orbit it describes around the earth cuts the earth’s ecliptic; the movement of the nodes of these orbits occurs from east to west, just like the precession of the equinoxes.

We can thus consider five revolutions in the moon: 1° its nodes in nineteen years; 2° its apsides in nine years; 3° its year around the sun in 354 days; 4° its motion around the earth in 27 and a half days, which should be regarded as the same as the synodic month of 29 and a half days, since one only differs from the other by the time; 5° the rotation on its axis, which is completed in the same time as it orbits the earth.

The moon has gradually accelerated its average motion around the earth, if we believe the philosopher Halley, who, having compared the oldest observations we have of lunar eclipses with the latest, found that the moon, since the time of these first observations, has increased the speed of its course.

The moon is about fifty times smaller than our earth, and fifty million times smaller than the sun; the matter of the moon is about one fifth denser, more compact than that of the earth, and about five times more than that of the sun; and thus the sun, which surpasses it 50 million times in size, only surpasses it 10 million times in amount of matter.

The earth weighs on the sun more than the moon does, and this in direct proportion to the mass of the earth and the mass of the moon. Now the size of the earth being to that of the moon as 50 to 1, and the mass, the amount of matter being only as 40, the weight of the earth is forty times greater than the weight of the moon, that is to say, gravitation, making the earth and the moon tend towards the sun in direct ratios of their masses, acts on the earth as 40, and on the moon as 1.

It attracts bodies towards its center about 30 times less than the earth does, and not 40 times less: for if its attraction is 40 times smaller by reason of the amount of matter, this attraction is on the other hand 10 times greater than on the earth, due to the smallness of its diameter: take 10 from 40, leaving 30.

Thus, for example, the same bodies that weigh 400 pounds on the sun, weigh about 15 pounds on the earth, and about half a pound on the globe of the moon.

MARS.

Mars is more than 50 million leagues from the sun at an average distance; it encompasses within its vast orbit the Earth, the Moon, Venus, and Mercury; it revolves around its ellipse in almost two years, and rotates on its axis in twenty-four hours and forty-five minutes. It is five times smaller than our globe. It should be noted here that, as we also rotate in an ellipse around the same center, sometimes we are much closer and at other times much farther from each other. At our closest, we are 12 million leagues apart, and at our furthest, we are 60 million leagues apart; thus, we are about five times farther apart in this way (figure 71).

The amount of illumination, as we have mentioned, is inversely proportional to the square of the distances: 25 is the square of 5; thus, by this rule, we should see Mars sometimes 25 times larger, sometimes 25 times smaller; but, as it also receives less sunlight when it is farther away, this loss of light prevents it from appearing 25 times larger; likewise, when it is farther from Earth, it does not appear 25 times smaller, since it is then more brightly illuminated: what it loses by its distance from our globe, it somewhat regains through its illumination; and conversely, the same can be said of the other planets.

Nothing can be definitively stated about the effects of gravity on the planets of Mars.

JUPITER.

At approximately 150 million leagues is Jupiter, at the average distance from the sun. Here there is a great disproportion: from Mercury to Mars, there are planets approximately 10 million to 11 million leagues apart, or close to it. Mercury, Venus, Earth, Mars, are at not too disproportionate distances; but from Mars to Jupiter, there is a gap of more than 100 million leagues, without any apparent reason for this inequality. It might be said that perhaps there were once planets in this space; but what can be made of a perhaps?

All the other stars we have spoken of are each smaller than Earth; but Jupiter is 1,170 times larger than it.

It revolves around the sun in its ellipse in almost twelve years, due to its distance, following Kepler’s rule; and yet it rotates on its axis in 9 hours 56 minutes: clear proof that the rotation of the planets on their axes is the result of a law of which we have no knowledge.

Jupiter sees the sun 25 times smaller than we see it, and receives 25 times less light, since it is 5 times farther from our globe: therefore, in the hottest times on Jupiter, it is 25 times colder than our summer, all else being equal; but also its material is more than 5 times less dense, and thus it heats up about 5 times more easily.

Although it is 1,170 times larger than Earth, it only has 220 times more matter.

Jupiter, given its distance and its orbital period, exerts 30 times less gravitational pull on the sun than Earth does, despite its enormous size.

Bodies that weigh one pound here weigh about two pounds on the surface of Jupiter; bodies that fall 15 feet in the first second on Earth fall 30 feet on Jupiter.

Astronomers have recognized that the axis of Jupiter’s equator is significantly larger than the axis of the poles, meaning that Jupiter is a spheroid flattened at the poles, like Earth, and probably like all the other planets.

Of the four moons that orbit around Jupiter, the first is only about 35,000 leagues away from it.

Our moon is nearly three times farther from our Earth than the first of Jupiter’s satellites is from its planet, and the last of its satellites is 360,000 leagues away, providing little assistance.

SATURN.

Saturn, at the average distance, is 286 million leagues from the sun. It orbits this star in nearly thirty years, encompassing in an orbit of almost 1,800 million leagues all the planets we have just seen. Its rotation on its axis is unknown; but it is believed likely that it rotates in ten hours like Jupiter, because the distance of its moons is approximately the same. It is as large as 980 of our Earths, and therefore much smaller than Jupiter, although much further from the sun.

As it is about ten times farther from the sun than we are, it is one hundred times less illuminated, and, all else being equal, less heated; and it does not see the sun as large as we see Venus.

The material it is made of is probably less dense than ours in the proportion of 15 to 100, that is, the material of the Earth is 6 and two-thirds times more massive than that of Saturn.

Thus, it is seen that the farther a planet is from the sun, the less dense and hard its material; consequently, it heats up more easily: the material of which Mercury is composed is all the more dense as Mercury is closer to the fire it must resist; and the material of Saturn all the more rare and loose as it is farther from the fire that must animate it. Bodies weigh on its surface a little more than on that of the Earth: what weighs 4 pounds on Earth weighs about 5 pounds on Saturn.

Saturn itself weighs nearly a hundred times less than the Earth on the sun; the same body that falls here from 15 feet in the first second will fall from 12 feet on Saturn.

It has around it five moons; the nearest is 30,000 leagues away, and the fifth about 160,000 leagues, about the same as the first and last of Jupiter’s satellites are distant from Jupiter. We do not go into any detail here about its ring, for which a separate volume would be needed.

Between Jupiter and Saturn there is a noticeable attraction that is not marked among the other main planets: for example, when Venus, Earth, and Mars approach, are in conjunction, their gravitation disturbs their movement in their orbits very little, because their orbits are quite close to the sun; and the mass of this star so greatly surpasses the combined mass of these planets that their centripetal forces are not capable of offering a noticeable resistance against the resultant centripetal force of the mass of the sun that attracts them.

This is not the case with Jupiter and Saturn. These two globes, enormous compared to ours, are at an immense distance from the center that attracts them.

Jupiter is twenty-five times less attracted than we are, and Saturn is nearly a hundred times less attracted than we are, due to the square of the distances; when these two stars are in conjunction, they are much closer to each other than Jupiter is from the sun: thus they gravitate more towards each other, and they noticeably deviate from their usual orbit. Their course is disturbed; this is the greatest triumph of attraction: these two globes, which so rarely find themselves in conjunction, were so at the time of Newton; he calculated, by the laws of attraction, how much their course should be altered. The illustrious Halley observed these stars, and his observations demonstrated what Newton had guessed, as the measurements taken at the pole have since confirmed what Newton had said about the shape of the Earth.

Thus what happens on Earth and what happens at one hundred and fifty, nearly three hundred million leagues from Earth, equally proves this admirable property of matter that Newton discovered.

Chap. XIII. — Of comets: of the power of attraction on them. — Old ideas on comets rectified by Tycho Brahe. Truth and error in Descartes. Comets must necessarily describe a conic section around the sun. Path of comets. Why a comet passing near the sun does not fall on this star. Comets are opaque bodies. They are planets. Difficulty of knowing their return. What is the tail of comets. Descartes' mistake on the tail of comets. Newton measured the line that a comet’s tail must describe over several years. Probable use of comets.

Since attraction acts thus on all celestial bodies, it is easily seen that its power must extend over the comets that come to traverse a sky at the center of which is the sun. To see the progress of human reason, it is not useless to recall here the thought of Aristotle and all the peripatetics about comets: they believed that they were exhalations. These globes, whose orbit extends so far above Saturn, appeared to them as will-o'-the-wisps placed very below the moon, which was, according to them, the sphere of fire.

It is true that, long before Aristotle, in Egypt and Babylon, much sounder notions of astronomy had been held. Pythagoras, who had traveled in the East, brought back not only the knowledge of the true system of the world, renewed later by Copernicus, but he also derived the idea that comets are planets that orbit around the sun.

It is to be believed that the Easterners had guessed these truths through a series of consequences that apparently did not reach the Greeks when Alexander sent the Babylonian observations to Aristotle. One must give the Greeks the credit for believing that they would not have corrupted well-proven systems, to substitute them with such false and unphilosophical ones.

Tycho Brahe was the first modern who dared to say that comets were not below the moon, and that they went as far as the apogee of Venus. He was too cautious.

Descartes, who had not observed any, nevertheless judged that they could in their courses rise very above Saturn; but where he was mistaken was in asserting without any proof, and even without likelihood, that comets never came closer to us than around the orbit of Saturn: what led him into this error was this hypothesis of vortices of subtle matter, which always leads to falsity.

He felt the difficulty there would have been in his system to circulate, against the order of the signs, these foreign globes among our planets, and in this full of subtle matter.

He therefore regarded them as celestial globes; but, using only his imagination in this examination, he said that they were encrusted suns that, having left the center of their vortex, went eternally and as much as they could in a straight line from the confines of one vortex to the confines of another vortex, without being interrupted in this infinite fullness, and in the course of these immense torrents differently carried away. How the greatest geniuses are susceptible to wandering when the spirit of system and hypothesis leads them!

Comets do not go in a straight line, and could not go: for, since they cross the orbits of the planets, they are in the sphere of activity of the sun’s gravitation, as are the planets. Therefore, one of two things must happen, either the sun attracts them to its center by a perpendicular line, or they describe some conic section around the sun. Now Newton, aided by the famous astronomer Halley, the Cassini of England, having followed in its course this comet of 1680, which made so much noise, invented a new theory by which he determined the shape of the orbit that this comet should describe. Cassini the father had already fixed the route that the comet of 1664 should describe; he had dared to be the first to predict the course of a comet: astronomy had yet produced nothing so bold. Newton embraced a general theory; he proves that every comet must appear to describe a parabole around the sun, and assigns the kind of parabole it must appear to describe in all cases.

Then, by this same theory, he determines how this apparent parabole indeed changes into an ellipse; and he shows that the comet of 1680 completes its course in an ellipse so close to the parabole, and so eccentric to the sun, that it must make its way in 500 and so many years: which proves the extreme length of its orbit, since Saturn, so distant from the sun, yet completes its course in thirty years.

Here is the path of comet A (figure 72), in an ellipse around the sun; this comet would follow its course in G, and would not return if it followed a parabole.

But, since it is in the sphere of activity of the sun, it must have it for the center of its movement; thus, as it describes the parabole A G, it is brought back by gravitation towards the sun, in this other curve A E D: those who ask why the planets, being in their perihelion, do not fall into the sun, can, even more so, wonder why a comet that passes so close to this star is not engulfed by the force of the attraction, which increases according to the square of proximity, that is to say, the comet being a hundred times closer, is ten thousand times more attracted towards the center of the sun.

The comet of 1680, for example, descended so close to the sun that it was only a sixth part of this star away from it.

Remember here the great rule of Galileo: a falling body always acquires new degrees of velocity; now, this planet falling almost in a parabolic line towards the body of the sun, keeps at each moment the sum of the forces acquired in the preceding moments: thus this force increases so much that it has as much to climb up as it had to descend; and it passes again through the same degrees of speed, like a pendulum that makes its vibrations.

If one now asks what proof there is that comets are opaque bodies like planets, and not exhalations of fire, this proof is as easy as it is indisputable.

1° The comet of the year 1680 was not, in its perigee, distant from the edge of the sun by a sixth part of the disk of this star. It is easy to calculate by how much this comet had to be more heated than the earth: therefore it had to be a very solid body for that burning not to destroy it.

2° The brightness of comets increases to our eyes when they are near the sun, and decreases when they move away from it: therefore they reflect the sunlight like the other planets.

Thus, our world is indeed much enlarged from what it was formerly. Before Galileo, seven planets were counted, including very improperly the sun; today there are sixteen, in which the earth is found, not counting the ring of Saturn; and there is some likelihood that one day a certain number of these other planets, which, under the name of comets, orbit like us around the sun; but it is not to be hoped that we will know them all.

It is true that very fine observations, and exact measurements to the utmost scruple, are needed to determine the orbit of these globes; the slightest error can make a difference of several hundreds of years.

It is perhaps one of these small errors that deceived the famous mathematician Jacques Bernoulli. He assured that the comet of 1680 would reappear in the month of May 1719; he only gave it a period of about forty years, which was only ten years more than that of Saturn; yet its orbit was incomparably more eccentric to the sun than that of Saturn. Newton finds that the orbit of this comet is to that which describes Saturn, about as 16 is to 1, and thus its course had to be more than five hundred years.

To ensure the course and return of comets, it would first be necessary to have a long series of well-preserved exact observations; then, if a comet at the same time makes the same path at the same distance, with the same hair and the same tail as a comet observed previously, one will not yet be absolutely certain that this comet is the same: for it may very well be that a comet whose return was expected has been diverted from its path by the attraction of some celestial bodies, which has changed its curve. This curve, which previously passed at some distance from the sun, will have passed since into this star, and the comet will have been engulfed there; another will have taken its place by the attraction of the same celestial body, and it will be this other comet that will be seen again in place of the one that was expected. Thus, after observations of several thousands of centuries, one could not flatter oneself of having a well-demonstrated theory of comets.

As for what is called the tail, the hair, and the beard of the comet, it is a long trail of light fairly weak that accompanies it, as long as it is exposed to our view: it is called a beard, when the comet appears to the east of the sun, and that light seems to precede it; it is called a tail, when it is to the west, and that light seems to follow it. It is called hair when, being in opposition to the sun, its light seems more spread around it.

The situation of this light, which varies with respect to us, is always the same with respect to the sun; it is always opposite to this star; and this truth was known as early as the sixteenth century; it had been discovered by Pierre Appien.

The tails of comets are always less brilliant as they move away from the sun.

Descartes was mistaken in the explanation of this tail of comets; he claimed that it was a refraction of the light of these stars. A single reflection overturns this system. The planets have much more light than the comets: they should therefore have longer tails, hairs, beards; they have none at all. This explanation of Descartes is therefore clearly false.

Newton adds to this argument against Descartes another objection no less decisive: it is that if the refraction of the reflected light from the bodies of comets caused these trails of light, one would see different colors, given the great inequality of refractions in the length of these tails.

These trails of light are nothing other than inflamed parts of the comet itself, which the sun detaches from these globes that approach it. The proof of this is that these vapors are very weak and barely visible when the comet begins to come into its perihelion; but, as it approaches it, the trail of fire increases in size and brilliance; its greatest extent and greatest clarity appear when it comes out of the vicinity of the sun, like coals that come smoking out of a fiery hearth.

What is most surprising is that Newton measured the line that this smoke of the comet describes, and by how much it is less curved when the comet climbs back in its elliptical line; and he has shown that this trail of light is continually renewed.

If in a philosophy entirely mathematical, entirely based on experience and calculation, it is permitted to advance probabilities, I would say that Newton suspected in comets an end and a use very contrary to what was established by the superstition of all times.

Far from comets being dangerous, far from them needing to arouse fear, they are, according to him, new blessings of the Creator. People, who, for some unknown fate, always represent the Divinity as malevolent, regarded them as signs of anger and as omens of destruction. Newton, on the contrary, regards them with reason as effects of divine goodness, and physically necessary to the worlds in the vicinity of which they travel; he suspects that the vapors that come out of them are drawn into the orbits of the planets, and serve to renew the humidity of these terrestrial globes, which is always diminishing. He also thinks that the most elastic and subtle part of the air we breathe comes from comets. He especially, it seems to me, has great reason to believe that they sometimes renew the substance of the sun. The curve that they describe, the proximity in which they are often to this star, make this opinion more than probable. It seems to me that this is guessing wisely, and that if it is to be mistaken, it is to be mistaken as a great man.

But what is, it seems to me, neither guessing nor being mistaken, is to conclude from the path of comets that the full and the vortices are impossible: for several comets have crossed from east to west, and from south to north, and from north to south the orbits of the planets; and every comet that is found in the region of Mars, of Jupiter, or of Saturn, goes incomparably faster than Mars, Jupiter, and Saturn, as I have already said. Therefore, finally, the planets, subject to the laws of gravitation like all other bodies, conclusively annihilate the hypothesis of the full and the vortices.

Chap. XIV. — That attraction acts in all operations of nature, and that it is the cause of the hardness of bodies. — Attraction cause of adhesion and continuity. How two coarse parts of matter do not attract each other. How smaller parts attract each other. Attraction of fluids. Experiments that prove attraction. Attraction in chemistry. Conclusion and recapitulation.

You see that all the phenomena of nature, experiments, and geometry converge from all sides to establish attraction. You see that this principle acts from one end of our planetary world to the other, on Saturn and on the smallest atom of Saturn, on the sun and on the thinnest ray of the sun.

Does this power, so active and so universal, not seem to dominate throughout nature? Is it not the sole cause of many effects? Does it not mix with all the other mechanisms with which nature operates?

For example, it is quite likely that it alone causes the continuity and adhesion of bodies: for attraction acts in direct proportion to mass; it acts on each corpuscle of matter; thus it makes each corpuscle gravitate in this sense, as Saturn gravitates towards Jupiter.

Let’s see what happens to bodies on the surface of the Earth.

1° If I place these two ivory balls A B, C D, against each other (‘figure’ 73), they attract; but their reciprocal tendency is destroyed by their gravitation towards the Earth.

2° Let the diameter of each ball be two lines, that is 120 seconds of line for each diameter; let there be a space of one second between these two bodies.

Point D is 120 seconds away from C. Bodies at the point of contact attract each other in inverse proportion to the cube of distances, and in an even greater proportion. Let us take only the cube here; then point D attracts less, and is attracted less than point C by one million seven hundred twenty-eight thousand times; and as points A and D are four lines away from each other, these points A and D will attract each other ten million nine hundred forty-four thousand times less than points B and C.

Now the mass of the Earth is to the mass of each of these two balls as the cube of fifteen hundred small French leagues, worth three billion three hundred twenty-five million leagues, is to the cube of two lines which is worth eight lines. The gravity of each ball towards the center of the Earth is therefore incomparably greater than their mutual attraction.

3° But if the two balls are of the utmost smallness, then their diameter is considered infinitely small; almost all their substance touches at the point of contact; the force of attraction can become immense compared to other opposing forces: then the two small bodies, joined together, compose a massive and continuous body.

4° The smallest bodies are those that have the most surface area, and therefore those that will have the most points of contact. The masses of solid bodies will thus be composed of smaller molecules, attracted to each other.

5° Attraction acts in fluids as in solids. Two drops of water, two globules of mercury, join, and, at the very moment, they form a single globule. Air cannot be the cause, since the same effect occurs in an air-purged machine. No ether, no subtle matter assumed to press these drops, can cause this union: for the supposed subtle matter could only press these drops on the plane where they are; it would prevent their contact by pressing in between; it would divide them, scatter them, far from uniting them by pressing on them.

Thus it is by attracting each other that they join, it is by equally attracting each other that they compose a round body.

6° Every solid and every fluid, being thus subject to attraction, the hardness of palpable bodies is nothing but an attraction of parts. The more a metal contains matter under a small volume, the harder it is; but the more it contains matter, the more each part has immediate contact with its neighboring part, that is when the greatest attraction occurs; think about it well. It is in the enlightened times we live in that no philosopher can find anything satisfactory about the cause of continuity, adhesion, coherence, hardness of bodies. I am not surprised: they find none, and will never find any, because there is none. Whatever fluid, whatever linkage one imagines, it always remains to be understood why the parts of this fluid, why these linked parts are contiguous. There must be a force given by God to matter that thus binds its parts together, and it is this force that I call attraction; I have already said it, there is no philosophy that places man more under the hand of God.

7° If you place one on top of the other two bodies as polished as they can be, whether steel, tin, or crystal, you can no longer separate them easily; and if you place between them some matter that fills the inequalities of their surfaces, such as pitch, then you cannot separate them at all. Why? Because the parts of the pitch touch immediately the parts of these glasses, which did not touch in this way before. Then the attraction increases in proportion to the fullness of the contact.

8° Why do the tubes called capillaries attract in their capacity all the liquids into which they are dipped? It is not, once again, the air that is the cause: for the weight of the air, which raises the mercury to nearly 28 inches in the barometer, cannot do so at all in the capillary tube; moreover, this experiment of liquids, rising in this extremely small capacity, is done in the pneumatic machine as well as in the air. Ether, the subtle matter would do no more. On the contrary, it would press the cavity of this tube, it would prevent water from rising.

It is therefore the sole attraction from the top of the glass that is the cause of this phenomenon. The evidence is palpable.

1° Water always rises the more in these capillary tubes the longer they are; and air, on the other hand, never lets the mercury rise to a greater height than its weight determines, however long the barometer may be.

2° The alteration of the weight of the air, its elasticity, varies the height of the mercury in the same barometer, and never does the height of the water vary in the same capillary tube, because the attraction is always the same.

Now, if this force dominates over all bodies, it must play a significant role in a multitude of physical and chemical experiments whose causes have never been understood.

The actions of acids on alkalis could well be philosophical chimeras, as well as vortices. It has never been possible to define what an acid and an alkali are; when one has well assigned the properties of one, one finds at the first experiment that these properties also belong to the other; thus all we know so far is that there are bodies that ferment with other bodies, and nothing more. But if one considers that there is a real force in nature, which operates the gravitation of all bodies towards each other, one might believe that this force is the cause of all the dissolutions of bodies and their greatest effervescences.

Let’s examine here the simplest of dissolutions, that of salt in water.

Throw a piece of salt into the middle of a basin full of water, the water at the edges will be long without becoming salty; it can only become salty through movement. It can only be in motion by central forces; the water parts nearest to the mass of the salt must gravitate towards this body of salt; the more they gravitate, the more they divide it, and this in proportion compounded from the square of their speed and their mass; the parts divided by this necessary effort are set in motion; their movement carries them throughout the extent of the basin: this explanation is not only simple, but based on all the laws of nature.

88 Let’s conclude, by summarizing here the substance of everything we have said in this work:

1° That there is an active power that imparts to all bodies a tendency towards each other;

2° That, in relation to celestial globes, this power acts in inverse proportion to the squares of the distances from the center of motion, and in direct proportion to the masses; and this power is called attraction with respect to the center, and gravitation with respect to the bodies that gravitate towards this center;

3° That the same power causes these mobiles to descend on our earth, in the progressions that we have seen;

4° That a similar power is the cause of adhesion, of its continuity, and of hardness, but in a proportion entirely different from that in which the celestial globes attract each other;

5° That a similar power acts between light and bodies, as we have seen, without knowing in what proportion89.

Regarding the cause of this power, so uselessly sought by Newton and all those who followed him, what could be better than to translate here what Newton says on the last page of his Principles?

Here is how he explains himself as a physicist as sublime as he is a profound geometer.

“I have so far demonstrated the force of gravitation by the celestial phenomena and by those of the sea; but I have nowhere assigned its cause. This force comes from a power that penetrates to the center of the sun and the planets without losing any of its activity, and which acts, not according to the quantity of the surfaces of the particles of matter, as do mechanical causes, but according to the quantity of solid matter; and its action extends to immense distances, always diminishing exactly according to the square of the distances, etc.”

This clearly and expressly states that attraction is a principle that is not mechanical.

And a few lines later, he says: “I frame no hypotheses, hypotheses non fingo. For what is not deduced from the phenomena is a hypothesis; and hypotheses, whether metaphysical or physical, or suppositions of occult qualities, or mechanical suppositions, have no place in experimental philosophy.”

I am not saying that this principle of gravitation is the only spring of physics; there are probably many other secrets that we have not wrested from nature, and which conspire with gravitation to maintain the order of the universe.

Gravitation, for example, does not account for the rotation of the planets on their own centers, nor for the determination of their orbits in one direction rather than another, nor for the surprising effects of elasticity, electricity, magnetism. There may come a time, perhaps, when we will have a large enough collection of experiments to recognize some other hidden principles. Everything warns us that matter has many more properties than we know. We are still only at the edge of an immense ocean: how many things remain to be discovered! But also, how many things are forever out of the sphere of our knowledge!

END OF THE ELEMENTS OF THE PHILOSOPHY OF NEWTON.

Plates for understanding the text of the Elements of Newton’s Philosophy

Plate II. — Fig. 7 to 14.

  1. In his Plurality of Worlds.↩

  2. One, in his Spectacle of Nature; the other, in his Ocular Harpsichord.↩

  3. The theater telescope is his invention. He found it while trying to reproduce the astronomical telescope, of which he had learned the recent discovery and without further indication. (D.)↩

  4. See later, year 1739.↩

  5. See also, page 267, the Clarifications; and page 389, the Memo sent to the Journal des Savants.↩

  6. It was Madame du Châtelet who, in her letter to Maupertuis on May 9, 1738, accused the Dutch bookseller of making additions to the title. However, it is important to note that in his letter to d’Argens, dated November 19, 1736, Voltaire said, regarding the Philosophy of Newton, that he had made it accessible to the public.↩

  7. See page 267.↩

  8. The volume lists the address as: In London.↩

  9. The edition by Mr. A.-A. Renouard is the first which, in 1819, republished these chapters.↩

  10. This Dedicatory Epistle, undated in the 1748 edition, is, in the 1756 edition and in all those that followed to this day, given as coming from the 1745 edition. In the 1738 editions, there was: 1° an epistle in verse to Madame du Châtelet, which has long been placed among the Poems (see volume X, page 299); 2° a prose piece, or letter of presentation to the same lady, which is as follows:

    TO MADAME LA MARQUISE DU CH******.
    Preface.

    "Madame,

    This is not about a marchioness or an imaginary philosophy. The solid study you have made of several truths, and the fruit of a respectable effort, are what I offer to the public for your glory, for that of your sex, and for the utility of anyone who wishes to cultivate their reason and enjoy your research effortlessly. Not all hands know how to cover the thorns of science with flowers: I must confine myself to trying to understand some truths well, and to show them with order and clarity; it would be up to you to lend them adornments.

    "This name of New Philosophy would be merely the title of a new novel if it only announced the conjectures of a modern opposed to the fantasies of the ancients. A philosophy that would be established only on risky explanations would not deserve, strictly speaking, the slightest examination: for there are innumerable ways to arrive at error, and there is only one path towards truth; there is therefore an infinity against one to bet that a philosophy which relies only on hypotheses will say nothing but chimeras. That is why all the ancients who have reasoned about physics, without having the torch of experience, have been but blind men explaining the nature of colors to other blind men.

    "This writing will not be a complete course of physics. If it were, it would be immense; a single part of physics occupies the life of several men, and often leaves them dying in uncertainty.

    “You confine yourself in this study, which I report, to just getting a clear idea of those so delicate and so powerful springs, those primitive laws of nature that Newton discovered; to examining how far others went before him, where he started, and where he stopped. We will begin, like him, with light: it is, of all the bodies that make themselves felt to us, the most delicate, the closest to the infinite in miniature; yet it is the one we know the most about. It has been followed in its movements, in its effects; we have managed to dissect it, to separate it into all its possible parts. It is the one of all bodies whose intimate nature is most developed; it is the one that brings us closest to the first springs of nature.”

    Here, in 1738, were the last two of the three paragraphs that, since 1741, make up the Introduction to the second part.

    "Here will be found all those that lead to establishing the new property of matter discovered by Newton. We will have to talk about some singularities that were found on the path in this field; but we will not stray from the goal.

    “Those who want to learn more will read the excellent Physics of S’Gravesande, Keill, Musschenbroek, Pemberton, and will approach Newton by degrees.”

    This is to the first sentence of this Preface of 1738 that Madame du Châtelet refers, in a letter a passage of which has been transcribed, page 277.

    In the 1741 edition, the Preface was conceived in these terms:

    "Madame,

    Philosophy belongs to every state and every sex: it is compatible with the cultivation of the fine arts, and even with what the imagination has of the brightest, provided that one has not allowed this imagination to get used to adorning falsehoods, nor to flutter too much on the surface of objects.

    It also agrees very well with the spirit of business, provided that, in the tasks of civil life, one has accustomed oneself to bring things back to principles, and that one has not burdened one’s mind too much with details.

    It is certainly within the realm of women, when they have managed to mix with the amusements of their sex that constant application which is perhaps the rarest gift of the spirit.

    Who ever proved this truth better than you, madame? Who has made more use of their mind and brought more honor to the sciences, without neglecting any of the duties of civil life? Your example should encourage or make blush those who give as an excuse for their lazy ignorance these vain occupations that are called pleasures or duties of society, and which are almost never either.

    Before I give under your eyes an idea of Newton’s discoveries in physics, as I had already attempted in previous editions, allow me first to make known what he thought in metaphysics; not that I want to just teach the public the vain anecdotes with which it likes to feed its curiosity about extraordinary men, but because his thoughts on what is least accessible to men can still be very useful to them; indeed, it is to be believed that he who discovered so many admirable truths in the sensible world has not strayed far in the intellectual world. I want to make known both the opinions that you accept and those that you combat. Sure of finding myself in the path of truth when I follow Newton and after you, uncertain when you do not agree with him, I will faithfully report either what I gathered in England from the mouth of his disciples, particularly from the philosopher Clarke, or what I drew from Newton’s own writings, and from the famous dispute between Clarke and Leibnitz. I submit the account that I am going to give, and especially my own ideas, to your judgment and to that of the small number of enlightened minds who are, like you, judges of these matters."↩

  11. This reasoning is not rigorous: it is possible that gravitation is essential to matter like impenetrability, although this general property strikes us less and was observed later. The equation that exists between the ordinate of a parabola and its area is as essential to this curve as the relation with the sub-tangent, although the parabola and this second property were known long before the first was discovered. (K.)↩

  12. This proof is regarded by all enlightened theists as the only one not above human intelligence; and the difficulty between them and the atheists is reduced to determining how far the proof can be taken that there exists in the universe an order indicating it has an intelligent author. Mr. de Voltaire believed, with Fénelon and Nicole, that this probability was equivalent to certainty; others find it so weak that they believe they must remain in doubt; others finally have thought that this probability favored a blind cause. What should console those who are afflicted by these contradictions is that all these philosophers agree on the same morality, and prove equally well that there can be no happiness for man except in the rigorous practice of his duties. (K.)↩

  13. In 1771, in the Questions on the Encyclopedia (see the Philosophical Dictionary, under the word Space, volume XIX, page 2), Voltaire said: “I thought I understood this great word formerly, for I was young; now I understand it no more than his explanations of the Apocalypse.”↩

  14. Here is what is read in a 1751 edition:

    "Everything has its cause: your will therefore has one. You can only want as a result of the last idea you received. This idea depends on our organs.

    "If your blood is inflamed, if your nerves and muscles are steeped in an acrid liquid, your thoughts are violent; they are gentle in the opposite disposition. Your organs are out of your power; you receive everything, you form nothing; you cannot give yourself an idea any more than you can add a hair to your head: therefore you are no more the master of your will than of being blond when you are born brunette.

    “If one were the master, etc.”↩

  15. The 1756 edition and its reprints, which are missing several chapters from 1741 and 1748, also contained an additional chapter V, as follows:

    CHAPTER V.

    Doubts about the freedom called indifference.

    "1. Plants are organized beings in which everything happens necessarily. Some plants are part of the animal kingdom and are in fact animals attached to the earth.

    "2. These plant-animals, which have roots, leaves, and sensation, would they have freedom? It seems unlikely.

    "3. Do animals not have sensation, instinct, a beginning of reason, a measure of ideas and memory? What fundamentally is this instinct? Isn’t it one of those secret springs that we will never know? Nothing can be known except through analysis, or by a sequence of what are called first principles: but what analysis or synthesis can let us understand the nature of instinct? We only see that this instinct is always necessarily accompanied by ideas. A silkworm perceives the leaf that nourishes it; the partridge, the worm it seeks and swallows; the fox, the partridge it eats; the wolf, the fox it devours. It is unlikely that these beings possess what is called freedom. One can thus have ideas without being free.

    "4. Humans receive and combine ideas in their sleep. One cannot say that they are free then. Isn’t this another proof that one can have ideas without being free?

    "5. Man has, above animals, the gift of a more extensive memory. This memory is the sole source of all thoughts. Could this source, common to animals and humans, produce freedom? Would reflected ideas in one brain be absolutely of a different nature than unreflected ideas in another brain?

    "6. Are not all men determined by their instinct? And isn’t that why they never change character? Isn’t this instinct what is called nature?

    "7. If one were free, who would not change their nature? But has anyone ever seen a person give themselves even a taste for something? Has anyone ever seen a person, born with an aversion to dancing, acquire a taste for dancing? A sedentary and lazy person, seek movement? And do not age and food diminish the passions that reason believes it has subdued?

    "8. Is not the will always the result of the last ideas one has received? These ideas being necessary, isn’t the will also necessary?

    "9. Is freedom anything other than the power to act, or not to act? And was Locke not right to call freedom power?

    "10. A wolf perceives some sheep grazing in a field; its instinct drives it to devour them; dogs prevent it. A conqueror perceives a province that his instinct drives him to invade; he finds fortresses and armies that block his way. Is there a big difference between this wolf and this prince?

    "11. Doesn’t this universe appear to be subject in all its parts to immutable laws? If a man could direct his will at will, is it not clear that he could then disturb these immutable laws?

    "12. By what privilege would man not be subject to the same necessity as the stars, animals, plants, and all the rest of nature?

    "13. Is it reasonable to say that in the system of this universal fatality, punishments and rewards would be useless and absurd? Isn’t it rather obviously in the system of freedom that the uselessness and absurdity of punishments and rewards appear? Indeed, if a highway robber possesses a free will, solely determining itself, the fear of punishment might very well not determine him to give up banditry; but if physical causes act solely, if the sight of the gallows and the wheel makes a necessary and violent impression, it necessarily corrects the criminal, a witness to the punishment of another criminal.

    "14. To know if the soul is free, wouldn’t one need to know what the soul is? Is there a man who can boast that his reason alone demonstrates the spirituality, the immortality of this soul? Almost all physicists agree that the principle of sensation is at the place where the nerves meet in the brain. But this place is not a mathematical point. The origin of each nerve is extended. There is a bell on which the five organs of our senses strike. Who can conceive that this bell does not take up space? Are we not automata born to always want, to sometimes do what we want, and sometimes the contrary? From the stars to the center of the earth, outside of us and within us, all substance is unknown to us. We see only appearances: we are in a dream.

    “15. Whether in this dream one believes the will to be free or enslaved, the organized mire from which we are molded, endowed with an immortal or perishable faculty; whether one thinks like Epicurus or like Socrates, the wheels that move the machinery of the universe will always be the same.”↩

  16. Chapter ii, page 408.↩

  17. The original edition of Newton’s Metaphysics, Amsterdam, 1740, carried engender; and the authors of the French Library, volume XXXII, page 130, said about this: “The Academy has decided that the word engender is properly said only of the male. This decision is not without appeal, since here is Mr. Voltaire making the female engender.” (B.)↩

  18. This passage is cited in Voltaire’s letter to the Marquis d’Argenson, dated April 15, 1744.↩

  19. This conversion of water into earth is still a question, although Boerhaave’s opinion is the most plausible. Moreover, it would not be a true transmutation: water is a kind of earth fusible at a very low degree of heat, and this earth could lose this property by digestion in closed vessels, either by combining with the free fire that passes through the vessels or by a new combination of its own elements. (K.) — Except for rainwater carefully collected and certain waters from granite mountains, all natural waters contain dissolved solid materials that can deposit. This earth of Boyle is just a deposit. (D.)↩

  20. Voltaire here follows the system of spermatic worms. See the notes on the article Generation, in the Philosophical Dictionary. (K.) — The editions of Kehl have made no note on the Generation article: see volume XIX, page 223; but they did make one on chapter vii of The Man of Forty Crowns: see volume XXI, page 339.↩

  21. Voltaire here uses the language of chemists from the time he wrote. (K.)↩

  22. If the question of a prime matter is not insoluble for the human species, it is certainly so for philosophers of our century. Chemists are forced to recognize a very large number of elements in bodies, some simple and unalterable in our experiments, others composite and destructible, but whose principles are still little known. It is especially in theoretical chemistry, since this science has, like others, submitted to the analytical approach, to recognize simple principles well, to analyze composite principles, to try to reduce the former to a smaller number, to try to guess the secret of the combination of the others, which nature has so far reserved the means for; but there is a long way from what we know to the knowledge of a prime matter, or even a small number of simple, invariable, primitive principles. (K.)↩

  23. Horace, Book I, Ode iii, verse 27.↩

  24. Chapter vi, page 425.↩

  25. In the edition of 1756 and its reprints, instead of this last paragraph there was:

    “Yet these are the things that have been explained by lemmas, theorems, and corollaries. What has been proven by that? What Cicero said: There is nothing so strange that it is not supported by philosophers. Oh metaphysics! we are as advanced as in the days of the first druids.”

    It is in his work De Divinatione, II, 58, that Cicero said: Nescio quomodo nihil tam absurde dici potest quod non dicatur ab aliquo philosophorum. (B.)↩

  26. The principle of the conservation of living forces generally applies in nature, as long as it is assumed that changes are made by imperceptible degrees; that is, as long as the law of continuity is observed. The same is true of the principle of conservation of action. The principle of least action is also generally true, in the sense that the movement is determined by the same general equations that would have been found by assuming that the action is a minimum. But this is not enough for the action to actually be a minimum; it can be a maximum, or neither, although these equations apply. The agreement of these equations with nature proves only that, in the infinitely small changes that occur in an infinitely short time, the quantity of action remains the same.

    Moreover, it would be vain to believe that final causes are seen in these different laws: they are, as Mr. d’Alembert has demonstrated, only the necessary consequence of the essential and mathematical principles of motion. The discovery of these principles, which he extended to solid, flexible, and fluid bodies, finding at the same time the new calculus necessary to apply mathematical analysis to them, must be regarded as the greatest effort that the human mind has made in this century. (K.)↩

  27. This paragraph and the following were part of the Preface to Madame du Châtelet in 1738; see the note on page 400.↩

  28. This acceleration is a consequence of Newton’s theory. The wave theory leads logically to the opposite result. The famous experiments by Mr. L. Foucault have shown that the speed is smaller in more refracting bodies. This fact decides between the two theories. (D.)↩

  29. Voltaire frequently combats, and sometimes ridicules, the opinions of the Abbé Pluche, author of The Spectacle of Nature and The History of Heaven. See volumes XVII, page 27; XVIII, pages 20, 50, 189, 190, 329 and following, 533; XIX, 65, 137, 558, 559; in the Miscellanies, year 1750, the Sincere Thanks; and paragraph vii of the Instructions for the Guardian of the Capuchins of Ragusa.↩

  30. What is known for certain on this subject only dates from 1838. It is due to Bessel, of Königsberg. The sixty-first of Cygnus, which this illustrious astronomer believes to be one of the nearest, is still at such a distance that it takes more than nine years for its light to reach us. At the time Voltaire was writing, one could only have assumptions. (D.)↩

  31. Picard, long before, in seeking the parallax of the great orbit, also found in the polar star an apparent movement contrary to what the parallax should have caused. Roemer, who was also observing these star movements in his search for the same parallax, did not think to explain them by the progressive movement of light, which he had discovered. Yet it only required this very simple remark. If the time it takes light to traverse the Earth’s orbit delays the appearance of a phenomenon, it must also influence the apparent position of the stars. (K.)↩

  32. In the 1756 edition and its reprints, this chapter ends with: “All these truths are now acknowledged: they were all contested in 1738, when the author published these ‘Elements of Newton’ in France. This is how the truth is always received by those raised in error.”↩

  33. A little bit of radiant heat has been found. In Voltaire’s time, thermoscopic apparatuses were too insensitive to indicate it. (D.)↩

  34. Joshua, chapter x, verse 12.↩

  35. Saint Paul, I, Cor., xv, 36; Saint John, xii, 24.↩

  36. Only 772 times according to Mr. Regnault. (D.)↩

  37. It was the opinion of the Pythagoreans. Empedocles approaches the truth by considering light as emanating from bodies, and the eye as a mirror. Had he known about the camera obscura, he might have discovered the true theory of the eye. (D.)↩

  38. This is the camera obscura. (D)↩

  39. Chapters IV and V preceding, not being in the edition of 1756 or its reprints, what forms chapters VI-XI here were chapters IV-IX. (B.)↩

  40. This explanation shows that we see the object A A exactly as we would see a similar object placed at D D, if there were no mirror. We therefore project it to that point, because the impression is the same as if we actually saw it there. This secret judgment of the soul, which leads us to conclude the location of objects from the impression they make on our senses, was formed based on direct vision; and consequently, it is as if it always were so that we must judge. (K.)↩

  41. M. l’abbé Rochon has rigorously proven by experiment, according to the ingenious conjecture of M. d’Alembert, that we see objects in the direction of the perpendicular drawn from the object to the back of the eye: hence it follows that we must relate upwards the object whose image is traced at the bottom of the eye, and downwards the one whose image is traced at the top of the eye. The judgment of the soul is therefore not necessary to straighten the images of the objects, although it may be so to teach us to generally refer them to a place in space (K.)↩

  42. Voltaire provided, in 1771, in the fourth part of his Questions on the Encyclopedia (see volume XVIII, page 402), an article on Distance, which was almost textually extracted from this chapter.↩

  43. If you examine an object with an instrument that gives two images that are very nearly equal, and you place them in the same horizontal line, you will see both equally distant; if you place them in the same vertical line, the upper object will appear more distant than the other, exactly as two objects placed on an inclined plane, one lower and closer to us, the other higher and further away. We thus place these two images in space as two real objects, which would make the same impression on our eyes, would be placed. This ingenious observation is due to Mr. l’abbé Rochon. (K.)↩

  44. All this is verified by the illusion given by the stereoscope. (D.)↩

  45. It is very likely that a being limited to the sense of sight would at first come to see objects as placed on the same plane, but with the extent and contours that they have on this plane, since that is the only way to order among themselves the successive sensations it would experience: this painting would not seem difficult to it at first, but it would learn by habit to distinguish objects and to place them. For the same reason, from the moment it has an idea of space and movement related to this plane, why, by ordering its successive sensations, by seeing the same object become more visible, occupy more space on this plane, and successively cover other objects, or occupy less space, make a less strong impression, and gradually reveal new objects, could it not form an idea of space in all senses, and order all the objects that strike its gaze? No doubt its ideas of extent, distance would not be rigorously the same as ours, since the sense of touch would not have contributed to form them; no doubt its judgments on place, shape, distance, would be more often wrong than ours, because it could not have corrected them by touch; but it is very likely that this would be the whole difference between it and us. (K.)↩

  46. This solution by Smith exactly matches that of Father Malebranche, since in both opinions we see the celestial bodies larger at the horizon because we judge them to be more distant. These two philosophers only differ in the way they explain why we judge celestial bodies placed at the horizon to be more distant; but they are still very close. Malebranche appears to regard as the immediate cause of this judgment the objects interposed in the plane of the horizon. According to Smith, these interposed objects have accustomed us to judge the vault of the sky as if it were lowered, and this appearance is the immediate cause of the judgment we form on the size of the celestial bodies. (K.)↩

  47. Provided that A E and E D are equal. (D)↩

  48. See note 1 on page 441.↩

  49. This is diffraction. The laws of this phenomenon and its complete theory are due to Fresnel. (D.)↩

  50. To date, nothing has been discovered about the laws of attraction at very small distances. It is in the examination of the phenomena of crystallization that one may one day find these laws; but so far these phenomena have not even been sufficiently observed to know the manner in which this operation is executed. M. l’abbé Haüy has recently given several papers on the formation of crystals that have shed great light on this important matter. However, we may still be far from knowing enough to be able to apply calculation and understand the laws of the attractive force that presides over crystallization. (K.)↩

  51. See, volume XIX, page 119, what Voltaire said in 1771 about the solution he gives here.↩

  52. A light beam, however small, is composed of an infinity of differently refrangible rays. Without this, using a prism with a larger angle would result in seven separated circles, not a continuous image with sides that are noticeably straight lines.

    It is true that this continuous spectrum seems to offer only seven distinct colors; the transition from one color to another is shaded only over a very small space, while the color appears pure over a larger extent of the spectrum. One might therefore suspect that the sensation of color depends on a property of the rays, different from their degree of refrangibility. Newton appears to have believed that there are really only seven rays; he often seems to reason on this assumption; however, as he sensed insurmountable difficulties in this opinion, he never explained himself precisely on this subject.↩

  53. Virgil, Æn., IV, 701.↩

  54. Antonio de Dominis was one of the most illustrious victims of the Roman Inquisition. He renounced his archbishopric and retired around 1603 to England, where he published the history of the Council of Trent by Fra-Paolo, his friend. He worked on the project of reconciling Christian communions: a project that was that of a great number of wise and peace-loving minds in an era where the principles of tolerance were unknown. He was somehow persuaded in 1612 to return to Italy, being promised that it would suffice to retract a few supposedly heretical propositions he was accused of having upheld. But, shortly after this retraction, other crimes were imputed to him. He was placed in the Castel Sant’Angelo, where he died in 1625, at the age of sixty-four. The inquisitors had the barbarity to exhume and burn his corpse. Besides his work on optics, he had written a book titled De Rebublica christiana, which was burned with him. This book was condemned by the Sorbonne because it contained principles of tolerance and maxims favorable to the independence of secular princes. Fra-Paolo, wiser than the Archbishop of Spalatro, stayed all his life in Venice, where he only had to fear assassins. Shortly after, the illustrious Galileo, the honor of Italy, was forced to apologize for having discovered new evidence of the Earth’s motion, and was dragged to prison at the age of over seventy by the same inquisitors. Thus, let us not be surprised if not a single Roman is found among the illustrious men of all kinds, who, in recent centuries, have honored Italy. (K.)↩

  55. This chapter is a very remarkable exposition of the theory of the rainbow, as given by Descartes. Voltaire could have insisted on the part that belongs to Newton following the discovery of the unequal refrangibility of the various rays. (D.)↩

  56. See page 485.↩

  57. Voltaire means two lenses.↩

  58. This refers to the average thickness of the air blades. The diameters are proportional to the square roots of the thicknesses. (D.)↩

  59. In the editions of 1738, after the word philosophy, it read here: "He conjectured that light emanates from the sun and luminous bodies by access, by vibrations; that from these vibrations of the luminous body the first operates a reflection, the second a transmission, and so on indefinitely. He had also prepared experiments that led to show how this play of nature is related to the grand principle of attraction; but he did not have time to complete his experiments. He had also conjectured that there is in nature a very elastic and very rare matter, which becomes less rare the farther it is from opaque bodies; that the rays of light excite vibrations in this elastic matter; and it must be admitted that this hypothesis would explain almost all the mysteries of light, especially attraction and the gravitation of bodies; but a hypothesis, even if it explained everything, should not be admitted. It is not enough that a system is possible to deserve to be believed, it must be proven. If Descartes' vortices could be maintained against all the difficulties with which they are assailed, they would still have to be rejected, because they would be only possible: thus we will not build any real foundation on even Newton’s conjectures.

    “If I speak of it, it is rather to make known the history of his thoughts than to draw the slightest induction from his ideas, which I regard as the dreams of a great man: he did not dwell on it in any way, he was content with the facts, without daring to determine anything about the causes. Let us move on to the other discovery about the relationship that exists between the rays of light and the tones of music.”

    From the edition of 1741, almost all this passage was suppressed. The author had retained only the last four lines, starting from the words: He did not dwell, etc. (B.)↩

  60. Voltaire only cites the name of this famous theory. Newton admits that the luminous molecules in which he constitutes luminous matter have ends of different shapes, and that they acquire a rotational motion on themselves, in addition to the motion of translation; depending on the end that presents itself, there is easy reflection or easy transmission. This ingenious theory has fallen with the system. (D.)↩

  61. In the editions of 1738, and even in that of 1741, chapter thirteen ended with the variant that was read on page 501. After which came a chapter fourteen, which the author removed after 1741, and here it is:

    CHAPTER XIV.

    On the relationship of the seven primitive colors with the seven tones of music. — Very remarkable in Kircher. Way to know the proportions of the primitive colors of light. Analogy of the tones of music and the colors. Idea of an ocular harpsichord.

    «You know that, long before Descartes, it was noticed that a prism exposed to the sun gives the colors of the rainbow; these colors had often been seen painted on a cloth or on white paper, in an order that is always the same; soon they went, from experiment to experiment, to measuring the space that each of these colors occupies; finally, it was noticed that these spaces are between them the same as those of the lengths of a string that gives the seven tones of music.

    «I had always heard that it was in Kircher that Newton had drawn this discovery of the analogy of light and sound. Kircher, indeed, in his Ars magna lucis et umbrae, and in other books as well, calls sound the monkey of light. Some people inferred that Kircher had known these relationships; but it is good, for fear of mistake, to put here under the eyes what Kircher says, on pages 146 and following. ‘Those,’ he says, ‘who have a loud and strong voice are of the nature of the donkey: they are indiscreet and petulant, as we know donkeys are; and this voice resembles the color black. Those whose voice is grave at first, and then sharp, are like the ox: they are, like him, sad and choleric, and their voice corresponds to the celestial blue.’

    «He takes great care to strengthen these beautiful discoveries with the testimony of Aristotle. That is all that Father Kircher teaches us, otherwise one of the greatest mathematicians and most learned men of his time; and this is roughly how all those who were not learned reasoned then. Let’s see how Newton reasoned.

    «As you know, in a single ray of light, there are seven principal rays each with its own refrangibility: each of these rays has its sine; each of these sines has its proportion with the common sine of incidence; observe what happens in these seven primordial features, which escape by diverging in the air.

    «It is not a question here of considering that in this very glass all these features are diverged, and that each of these features there takes a different sine: one must regard this assembly of rays in the glass as a single ray, which has only this common sine A B; but at the emergence of this crystal, each of these features diverging perceptibly, takes each its different sine; that of the red (the least refrangible ray) is this line C B, that of the violet (the most refrangible ray) is this line C B D (figure 44).

    «These proportions established, let’s see what is this relationship, as exact as it is peculiar, between the colors and the music. That the sine of incidence of the white beam of rays be to the sine of emergence of the red ray, as this line A B is to the line A B C.

    Given sine in the glass A B. Given sine in the air A B C.

    «That this same sine A B of common incidence be to the sine of refraction of the violet ratio as the line A B is to the line A B C D.

    A B

    A B C D

    «You see that point C is the term of the smallest refrangibility, and D the term of the greatest: the small line C D thus contains all the degrees of refrangibility of the seven rays. Now double C D above, so that I becomes the middle, as below:

    A I C H G F E B D

    «Then the length from A to C makes red: the length from A to H makes orange; from A to G, yellow; from A to F, green; from A to E, blue; from A to B, purple; from A to D, violet. Now, these spaces are such that each ray may well be refracted, a little more or less, in each of these spaces, but it will never leave this space prescribed to it; the violet ray will always play between B and D; the red ray, between C and I; and so on, all in such proportion that if you divided this length from I to D, into three hundred sixty parts, each ray will have for itself the dimensions you see in the large figure attached See at the end of the note.

    «These proportions are precisely the same as those of the tones of music: the length of the string which being plucked will make re is to the string that will give the octave of re, as the line A I, which will give the red in I, is to the line A D, which gives the violet in D; thus the spaces that mark the colors, in this figure, also mark the tones of music.

    «The greatest refrangibility of the violet corresponds to re; the greatest refrangibility of the purple corresponds to mi; that of the blue corresponds to fa; that of the green, to sol; that of the yellow, to la; that of the orange, to si; that of the red, to do; and finally the smallest refrangibility of the red relates to re, which is the upper octave. The lowest tone thus corresponds to the violet, and the highest tone corresponds to the red. One can form a complete idea of all these properties by looking at the table that I have prepared, and that you must find next to it.

    «There is still another relationship between sounds and colors: it is that the most distant rays (the violets and the reds) come to our eyes at the same time, and that the most distant sounds (the lowest and the highest) come also to our ears at the same time. This does not mean that we see and that we hear at the same distance: for light is felt at least six hundred thousand times faster than sound; but it means that the blue rays, for example, do not come from the sun to our eyes sooner than the red rays, just as the sound of the note si does not come to our ears sooner than the sound of the note re.

    «This secret analogy between light and sound gives rise to the suspicion that all things in nature have hidden relationships, which perhaps will be discovered someday. It is already certain that there is a relationship between touch and sight, since colors depend on the configuration of parts; it is even claimed that there have been born blind people who could distinguish by touch the difference between black, white, and some other colors.

    «An ingenious philosopher wanted to push this relationship of the senses and light perhaps further than it seems allowed for men to go. He imagined an ocular harpsichord, which should successively display harmonious colors, as our harpsichords make us hear sounds: he worked on it with his hands; he finally claims that one could play airs to the eyes. We can only thank a man who seeks to give others new arts and new pleasures. There have been countries where the public would have rewarded him. It is certainly to be hoped that this invention will not be, like so many others, an ingenious and useless effort: this rapid passage of several colors before the eyes may perhaps amaze, dazzle, and tire the view: our eyes may want some rest to enjoy the pleasure of colors. It is not enough to propose a pleasure, nature must have made us capable of receiving this pleasure; it is only experience that can justify this invention. In the meantime, it seems to me that any fair-minded person can only praise the effort and the genius of the one who seeks to enlarge the career of arts and nature.»

    In the 1741 edition, the end of this last paragraph was shortened. After the words new arts and new pleasures, it was only read:

    «Moreover, this idea has not yet been executed, and the author did not follow Newton’s discoveries. In the meantime, it seems to me that any fair-minded person can only praise the effort and the genius of anyone who seeks to enlarge the career of arts and nature.»

    Table of colors and tones of music.

    Table of colors and tones of music

    Table of colors and tones of music

    In the editions of 1738, as in that of 1741, after these last words, were the last three paragraphs of chapter thirteen. This arrangement is in the edition of 1748.

    Father Castel is the one Voltaire refers to here by the words ingenious philosopher, and whom he calls Euclid-Castel in his letter to Thieriot, from November 18, 1736. But in the letter of March 22, 1738, it is Zoilus-Castel; in the one to Rameau, from March 1738, it is the Don Quixote of mathematics; finally, in the letter to Maupertuis, from June 15, 1738, he disavows the praise he had given to Father Castel, which he nevertheless let stand still in 1741. (B.)↩

  62. If Newton means here that calorific rays can cross each other without destruction, the fact is constant today, whether it is considered as proven by experience or when one thinks about the identity of heat and light. (D.)↩

  63. Today, the term “molecular attraction” is specifically used for the forces that push molecules or atoms together to form solids or liquids. (D.)↩

  64. Or rather, perpendicular to the axis of rotation, or to the center of the circle described. (D.)↩

  65. One cannot regard as absolutely rigorous the demonstration of the impossibility of the plenum, because movement would be very possible in an indefinitely expansible fluid, whose density would vary according to a certain law, since the weight, action, resistance of an infinite column of such a fluid could be expressed by a finite amount. It is therefore impossible to know anything precise about this question, as long as we do not know the nature of expansible fluids and the cause of expansibility. We can only say that it is impossible for us to conceive how the same substance can occupy a space double that which it occupied, without forming a vacuum between its parts. (K.)↩

  66. A foreigner once asked Newton how he had discovered the laws of the world system: By thinking about it incessantly, he replied. This is the secret of all great discoveries: genius in sciences depends only on the intensity and duration of the attention a man’s mind can sustain. (K.)↩

  67. This chapter is as one would write it today. Nothing to add. (D.)↩

  68. The 1738 editions also contained the passage here:

    “Thus a body moving along the horizontal line G E (figure 50), and along the perpendicular line G F, obeys at every instant these two powers by traversing the diagonal G H.”

    This paragraph was removed by Voltaire as early as 1741. (B.)↩

  69. In the editions of 1738 and 1741, it was also read here: “You will find the demonstration more extensively in notes.”

    And indeed in notes were read the two demonstrations that follow:

    Demonstration. That every mobile attracted by a centripetal force describes in a curved line equal areas in equal times (figure 52).

    "Every body moves with uniform motion when there is no accelerating force: thus the body A, moving in a straight line in the first time from A to B, will go in a like time from B to C, from C to Z. These spaces conceived as equal, the centripetal force, in the second time, gives this body at B a certain motion, and the body, instead of going to C, goes to H: what direction has it had different from B C? Draw the four lines C H, G B, C B, G H, the mobile has followed the diagonal B H of this parallelogram.

    "Now, the two sides B C, B H of the parallelogram are in the same plane as the triangle A B S: therefore the forces are directed towards G S and towards the straight line A B C Z.

    “The triangles S H B, S C B, are equal, since they are on the same base S B, and between the parallels H C, G B; but S B, A S, C B, are equal, having the same base and the same height: therefore S B, A S, H B, are also equal. The same must be said of the triangles S T H, S D H: therefore all these triangles are equal. Reduce the height to infinity, the body, at each infinitely small moment, will describe the curve, from which all lines tend to the point S: therefore in all cases the areas of these triangles are proportional to the times.”

    Demonstration. That every body, in a curve describing equal triangles around a point, is moved by the centripetal force around that point (figure 53).

    "Let this curve be divided into equal parts A B, B H, H F, infinitely small, described in equal times; let the force act at the points B H F; let A B be prolonged in C, let B H be prolonged in T, the triangle S A B will be equal to the triangle S B H; for A B is equal to B C: therefore S B H is equal to S B C: therefore the force at B G is parallel to C H; but this line B G, parallel to C H, is the line B G S, tending towards the center. The body at H is directed by the centripetal force along a line parallel to F T, just as at the point B, it was directed by the same force along a line parallel to C H; now the line parallel to C H tends towards S: therefore the line parallel to F T will also tend towards S; therefore all the lines thus drawn will tend towards the point S.

    “Conceive now in S triangles similar to those above; the smaller these above triangles are, the more the triangles in S will approach a physical point, which point S will be the center of the forces.”

    These notes or demonstrations were not preserved in the editions of 1748 or 1756. (B.)↩

  70. In Kepler’s time, there was no idea of the methods of calculating movement in curved lines. He supposed that the planets described ellipses around the sun, because being attracted by this star, they had a progressive movement. He called it animal movement, because he did not know that a body that encounters no obstacle continues to move indefinitely in a straight line; he believed that, in this case, a new force was occasionally needed, and he supposed this force to reside in the planets themselves. This second hypothesis is not ridiculous like that of the friendly and enemy sides. (K.)↩

  71. This is an appearance due to the elliptical motion of translation, while the rotational motion is uniform. (D.)↩

  72. See page 559, chapter x on the cause of the moon’s libration.↩

  73. Halley.↩

  74. Perturbations of a planet are changes caused by the attraction of celestial bodies in the orbit that this planet would have described, if it had been attracted only by the sun or the main planet. Newton was unable to provide a sufficiently accurate method to calculate these perturbations. This method was found only about sixty years after the publication of the Principia, by three great geometers of the continent, MM. d’Alembert, Euler, and Clairaut. (K.)↩

  75. By this expression of solid matter, we must understand the condensed matter that constitutes the planet, whether it be gaseous and liquid as well as solid. (D.)↩

  76. In the editions of 1738 (where this chapter was the twenty-second), and in the edition of 1741, we read further here:

    « It is nothing easier than to know the size of any star, as soon as we know its diameter: for the product of the circumference of the great circle by the diameter gives the surface of the star, and the third of the product of this surface by the radius makes the size.

    « But, knowing this size, we do not know at all the mass, that is to say, the amount of matter that the star contains; it can only be known by this admirable discovery of the laws of gravitation.

    « 1° When we say density, quantity of matter, in any globe, we mean that the matter of this globe is homogeneous; for example, that every cubic foot of this matter is equally heavy.

    « 2° Every globe attracts in direct proportion to its mass; thus, all things being equal, a globe that has ten times more mass will attract ten times more than a body ten times less massive would attract at the same distance.

    « 3° It is absolutely necessary to consider the size, the circumference of this globe; for, the larger the circumference, the greater the distance to the center increases, and it attracts in the inverse ratio of the square of this distance. Example: if the diameter of planet A is four times greater than that of planet B, both having equally of matter, planet A will attract bodies at its surface sixteen times less than planet B; and what weighs a pound on planet A will weigh sixteen pounds on planet B.

    « 4° It is necessary to know especially in how much time the mobiles attracted by this globe, which we seek the density of, make their revolution around this globe; for, as we saw in chapter xix, page 201, any body circulating around another gravitates all the more as it turns faster; and it only gravitates more by one of these two reasons, either because it approaches more the center which attracts it, or because this center attracting contains more matter. If, therefore, I want to know the density of the sun relative to the density of our earth, I must compare the time of the revolution of a planet like Venus around the sun with the course of the moon around our earth, and the distance of Venus to the sun with the distance of the moon to the earth.

    « 5° Here is how I proceed: the amount of matter of the sun, in relation to « that of the earth, is as the cube of the distance of Venus to the center of the sun is to the cube of the moon to the center of the earth (taking the distance of Venus to the sun two hundred fifty-seven times greater than that of the moon to the earth), and also in the reciprocal ratio of the square of the periodic time of Venus around the sun, to the square of the periodic time of the moon around the earth.

    « This operation done, always assuming that the sun is to the earth in size as one million to the unit, and counting roundly, you will find that the sun, larger than the earth a million times, has only about two hundred fifty thousand times or so more matter.

    « That being assumed, I want to know what proportion is found between the force of gravitation at the surface of the sun, and this same force at the surface of the earth; I want to know, in a word, how much weighs on the sun what weighs here a pound.

    « To achieve this, I say: The force of this gravitation depends directly on the density of the attracting globes and the distance from the center of these globes to the heavy bodies on these globes: now, the heavy bodies being at the surface of the globe, their distance is precisely the radius of the globe; but the radius of the globe of the earth is to that of the sun as 1 is to 100, and the respective density of the earth is to that of the sun as 4 is to 1. Say therefore: As 100, radius of the sun, multiplied by 1, is to 4, density of the earth multiplied by 1, thus is the weight of bodies on the surface of the sun to the weight of the same bodies on the surface of the earth; this ratio of 100 to 4, reduced to the smallest terms, is as 25 to 1: thus a pound weighs twenty-five pounds on the surface of the sun; what I was looking for. »

    « We cannot have the same notions of all the planets, for those that have no moons, no satellites, etc. »

    A note, which is only in the edition of 1741, is thus conceived: « All this is put in italic letters to warn the less practiced readers that one can pass the calculations and go straight to chapter viii. »

    None of what has just been transcribed is found in the editions of 1748 and 1756. What Voltaire had put in italic letters, in 1741, is here in quoted lines. (B.)↩

  77. The masses of these planets are deduced from the perturbations they produce in the motion of other elements of the solar system. (D.)↩

  78. These determinations are those found in the Mathematical Principles. More exact observations have since taught that some changes had to be made in the elements adopted by Newton, and consequently in these different results. (K.)↩

  79. In 1859, a new planet was announced to be closer to the sun than Mercury. (D.)↩

  80. Schrœter found it to be 23 hours 21 minutes. (D.)↩

  81. It is only through the calculation of perturbations, or through the movement of the axes of the planets (see chapter V), that one can know the masses of the planets. For example, to know that of Venus, it would be necessary, after concluding the proportion of the mass of the moon to that of the sun, from their action on the motion of the earth, to seek the alteration produced by Venus in the Earth’s orbit; and, knowing the one given by the phenomena, the mass of Venus would be known, assuming it to be such as to produce this alteration.

    This mass once found, by comparing observation to theory for a given moment, the theory would provide the tables of perturbations caused by Venus, and the agreement of these tables with the observations would prove the truth of the general law of the world system. (K.)↩

  82. The 1738 editions contained the following passage here:

    THE EARTH.

    "After Venus is our earth, placed about 30 million leagues from the sun or thereabouts, at least at its average distance.

    "It is about 1 million times smaller than the sun: it gravitates towards him, and orbits around him in an ellipse in 365 days 5 hours 48 minutes, and travels at least 180 million leagues per year. The ellipse it travels is greatly disturbed by the action of the moon on it; and while the common center of the earth and the moon describes a true ellipse, the earth actually describes this curve at each lunation (figure 59).

    "Its motion of rotation on its axis, from west to east, constitutes its day of 23 hours 56 minutes. This motion is not that of gravitation. It seems especially impossible to resort here to this sufficient reason spoken of by the great philosopher Leibnitz. One must absolutely admit that the planets and the sun could rotate from east to west: thus, it must be conceded that this rotation from west to east is the effect of the free will of the Creator, and that this will is the sole reason for this rotation.

    "The earth has another movement which its poles complete in 25,920 years: it is the gravitation towards the sun and towards the moon that evidently causes this movement, for the same reasons that the sun and the earth evidently act on the moon.

    “The earth may also experience a much stranger revolution, whose cause is more hidden, whose length astonishes the imagination, and which seems to promise humanity a duration that one would not dare to conceive. This period could be 1,944,000 years. This is the place to insert what is known about this astonishing discovery, before finishing the chapter on the earth.”

    Here the 1738 editions contain a lengthy piece titled Digression on the period of 1,944,000 years newly discovered, which Voltaire reproduced almost entirely in 1741, and which can be found (forming chapter xi) in the lengthy note following chapter ix, hereinafter. (B.)↩

  83. This was written in 1736. (Note by Voltaire.)↩

  84. This can only be said in the hypothesis of the earth being homogeneous, having a regular shape, and only for large measurements, the variations in gravity being imperceptible at small distances. (K.)↩

  85. His paper is in the Literary Journal. (Note by Voltaire.)↩

  86. It was Mr. d’Alembert who first solved, by a certain method, the problem of the precession of the equinoxes, that is, who determined the movements that the attraction of the sun and the moon cause in the axis of the Earth.

    But besides this great revolution, which causes the precession of the equinoxes, the axis of the Earth has another movement, called nutation; this movement, whose revolution is the same, in terms of duration, as that of the moon’s nodes, depends mainly on the attraction of this planet. Mr. d’Alembert employed this phenomenon observed by Bradley, and whose cause he was the first to elucidate, to determine more precisely than had been done before the mass of the moon, that is, the ratio of its attractive force with that of the sun. The attraction of the sun and the Earth produces a movement in the axis of the moon, and this movement is the cause of the phenomenon called libration of the moon.

    This phenomenon is calculated by the same principles, so that Mr. d’Alembert is owed the discovery of the laws of celestial phenomena caused by the shape of the stars, as Newton is owed the discovery of the phenomena caused by their attractive forces, assumed to be united at their center. (K.)↩

  87. In the 1756 edition and in its reprints, the chapter ended thus:

    “We will not pursue further inquiries into gravitation here. This doctrine was still quite new when the author presented it in 1736. It is no longer new; one must conform to the times. The more enlightened men have become, the less there is to write.”

    On the first sentence, the Kehl editors noted:

    “Let us observe here that we also owe to Newton the proof that comets are planets that describe around the sun ellipses elongated enough to be confused with parabolas throughout the extent where comets are visible. Thus a single appearance is not sufficient to determine the entire orbit and predict the return of a comet, which has been seen only once. Halley, a disciple of Newton, calculated the orbit of some comets whose period was roughly known because they had been seen twice, and tried to determine their return taking into account the disturbances caused by the planets near which the comets pass. One of these planets was expected to reappear in 1759; it actually reappeared very close to the time it was supposed to appear according to the calculations of its disturbances made by Mr. Clairaut, using a method much more certain than that which Halley could have used. Another is expected around 1789. The period of the first comet is about seventy-six years, and that of the second about one hundred thirty.”↩

  88. In the 1756 edition and its reprints, which do not contain the beginning of this chapter, nor the two preceding it, what follows had been formed into a Chapter XII, titled Conclusion. (B.)↩

  89. Always the attraction or repulsion exerted on the light rays when they refract or reflect. (D.) — This paragraph was also removed in 1756. (B.)↩

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