Mathematical Principles, Newton - du Châtelet

A translated excerpt of Principes mathématiques de la philosophie naturelle itself a translation of Isaac Newton’s celebrated work on geometry and physics.

Notes on the translation

My own notes on the text or the translation will be made in footnotes, the below below text consists of the original editor’s note, an introduction to the text and its translator by Voltaire, and three notes by Newton himself — on each version of Principia published.

I will follow the pattern established in previous work in naming various translations:

• Principia will refer to Newton’s original latin text;
• Principes to the Marquise du Chatellet’s French translation; and

• Principles to my own translation of the latter.

  Other references, where relevant, will be given in the usual manner and noted as bibliographic entries in footnotes.

Mathematical Principals of Natural Philosophy

Introductions and Prefaces

Editor’s Note

This work is composed in two parts. The first is a translation of the literal text of the Mathematics Principles of Natural Philosophy. It is almost superfluous to mention that it was made from the recent 1726 edition, an edition which takes precedence over all preceding editions and incorporates corrections suggested by later ideas and by the remarks of some well-known Mathematicians. The illustrious Translator, more zealous to capture the spirit of the Author than his words, has not feared in some places to add or transpose some ideas to give the meaning more clarity. In consequence, one finds often Newton more intelligible in this translation than in the original; and even than in the English translation. In fact, such an attachment was had in that version to the literal words of the Author that if there is some ambiguity in the Latin, it is found also in the English. Such timidity gives place to suspect the Author of having weakly understood the original, and of having used ordinary resources in such a case : that is, of rendering¹ words when unable to render the meaning. We, however, prefer to thing that this scrupulous fidelity comes from some other motive, and to attribute it to a certain respect so rightly acquired of this immortal Work, a respect which inspires its Translator to render it letter for letter².

In regards to the confidence the Public³ should have in this translation, it will suffice to say that it was made by the late Madame la Marquise du Chatellet, and that it was revised by M. Claurault.

The second part of the Work is a Commentary on the place of these principles, relative to the system of the world.⁴ This Commentary is itself divided into two parts, in the first of which are exposed in a more sensitive manner the principal phenomena dependant on attraction; those discoveries until now so entirely prickled with problems, will become accessible to all Readers capable of some measure of attention and having some light notions of Mathematics.

Following this part of the Commentary is a more learned one. Here is given, by analysis, the solution to the most beautiful problems of the system of the world⁵: Will be examined

• the shape that orbits of planets really have or would have under different hypotheses of gravitation,
• the attraction exerted on different shaped bodies,
• the refraction of light,
• the attractive effects of insensitive parts of bodies,
• the theory of the shape of the Earth and of its tides.

All of this research is taken for the most part from either the Works of M. Clairault or the notebooks he gave as lessons to M. le Compte du Chatellet Lomont, son of the illustrious Marquise. The last but one section is an excellent résumé⁶ of his Treatise on the shape of the Earth. The dissertation of the Learned Daniel Bernoulli, which won the prize proposed for answering the question of the tides, forms the basis of the final section; it is augmented by a variety of notes and clarifications communicated by the Author.

It is no doubt surprising that these Commentaries go no farther; but as I have already said, its author was sure of staying limited to what concerns more specifically the world system. In this light, it was not judged necessary to comment on the part of the Principles which contains the theory of fluids. Besides, that theory had already been taken up by so many hands and in particular with much success by MM. Daniel Bernoulli and d’Alembert⁷, whose writings are in everybody’s hands and so touching on it is superfluous. In regard to the theory of Comets, found in the first part of the Commentaries is an entire article about them and which ought to suffice. The geometric determination of the shape of their orbits is contained in the general problem of trajectories and it is in the treatises on Astronomy that one should look for how to determine their shape and position based on observations. M. le Monnier has sufficiently handled that subject in his Elements of Astronomy⁸, and those who do not find sufficient clarity in the text of the third book of Newton’s Principles, can find recourse in those Elements as an excellent Commentary.

There remains but the theory of secondary planets whose absence in those Work seems hard to justify, but at the time M. Clairault worked with Madame du Chatellet he was still too little content with about what Netwton had made of the subject and with his own ideas to communicate anything about it. This so interesting part of the system of the World hadn’t received until recently the perfection it was missing. To make up for this fault, one should look to M. Clairault’s piece — which took the Peterbourg Academy prize — on the theory of the Moon and the first part of the Work that M. d’Alembert has just published under the title Research on several important points of the system of the World⁹.

This is all that we in our capacity as Editors can say about this work. M. de Voltaire has gone to the of trouble tracing the character of the learned Lady who is our author. The Historical Preface which follows this Note is by this celebrated man.

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Historical Preface - Voltaire

This translation which the most learned men of France should read¹⁰ and that the others should study, a woman undertook and achieved to the astonishment and glory of her country. Gabrielle-Emilie de Breteuil, Marquise du Châtelet, is the Author of this Translation, which has become necessary to all those who would acquire its profound knowledge for which the world is in debt to the great Newton.

It was something already for a woman to understand ordinary Geometry, which itself is not even an introduction to the sublime truths contained in this immortal Work. It is clear that Madame la Marquise du Chaſtelet¹¹ had entered long before this carrier which Newton had opened and that she possessed what that great man had taught. We have seen two miracles¹²: the firt, that Newton had made this Work; the other, that a Lady had translated and brought clarification to it.

This was not her first essay¹³; she had already given the Public an explanation of Leibniz’s Philosophy under the title of Institutions of Physics, addressed to her son, to whom she herself had taught Geometry.

The preliminary Discours which is the beginning of her Institutions is a chef d’œuvre of reason and eloquence: she shows in the rest of the Book a method and a clarity which Leibniz never had and which his ideas had need of whether one’s intention is to hear them or refute them.

Having rendered Leibniz’s imagination intelligible, her mind which had grown still more in power and in maturity through this self-same work understood that this Metaphysics — so bold, but so unfounded — little merited her study. Her soul was made for the sublime, still more for the true. She felt that the preëstablished monads and harmony ought to be let with the three elements of Descartes and that systems which were simply ingenious were not worthy to occupy her. And so, having had the courage to embellish¹⁴ Leibniz she had the courage to abandon him: courage rare in anyone who has taken up an opinion, but which costs little effort to the soul impassioned by the truth.

Unlinked from all thought of system, she took for guide that rule of the Royal Society of London, Nullius in verba; and it is because the bounty of her wit had made her an enemy of parts and of systems that she gave herself completely to Newton. Indeed Newton never used a system¹⁵, supposed nothing, taught no truths which was not based in the sublime Geometry or incontestable experimentation.¹⁶ The conjectures which he hasards at the end of his Book under the name Research are no more than doubts, and he gives them only as such ; it would be almost impossible for someone who never affirms but evident truths to not doubt all the rest.

All that is given here as principle is indeed worthy of that name, these are the first principles¹⁷ of nature, unknown before him: and now it is no longer possible to try at being a Physicist without knowing them.

One must therefore be on guard from envisaging this Book as a system — that is, as a grouping of probabilities which can serve to better or worse explain some aspects of Nature.

If there was still someone absurd enough to support subtle matter (ether) and fluted matter¹⁸, to say that the Earth is a crusted sun and that the Moon was caught up in the whirlwind of the Earth, that ether is the cause of weight¹⁹, and all those other Romanesque opinions substituted for the ignorance of the Ancients, we would say: That man is Cartesian. If he believed in monads, we would say: He is Leibnizian; but we do not say that someone who knows Euclid’s elements that he is Euclidean. Nor of one who knows from Galileo in what proportion bodies fall that he is Galileist²⁰. So in England those who have learned infinitesimal calculus, who have done the experiments with light, who have learned the laws of gravitation, they are not remotely called Newtonians: it is the privilege of error to give its name to a Sect.

If Plato had found some truths, there would never have been Platonists ant all the men would have learned little by little what Plato had taught; but because of the ignorance that covers the Earth, some get attached to an error, others a different one — they fought under different standards : we got Peripatetics, Platonists, Epicureans, Zenonists²¹, when we expected Sages.²²

If we in France still call Newtonians the Philosophers who have joined there knowledge with those Newton gratified the human race with, it is by a leftover ignorance and prejudice. Those who know little and those who know wrong, which make up a multitude, imagined that Newton had done nothing more than combat Descartes, a little like Gassendi had done²³; they heard of his discoveries and took them for a new system. This is how, when Harvée brought forward²⁴ the circulation of blood, we rose against him in France: we called those who dared to embrace this new truth Harveyists and Circulators and took it or an opinion. It must be admitted, all the discoveries come to us from elsewhere and all have been fought against. It were none up until Newton’s experiments with light which have not suffered violent contradiction. It is not surprising after all that the universal gravity of matter — having been demonstrated — has been fought against, too.

It has been necessary, to establish in France all the sublime truths the we owe to Newton, to let pass the generation of those who, having grown old in the errors of Descartes, turpè putaverunt parere minoribus, & quæ imberbes didicêre, ſenes perdneda fateri²⁵.

Madame du Châtelet has rendered a double service to posterity in translating the Book of Principles and in enriching it with a Commentary. It is true that the Latin language it is written in is understood by all the learned²⁶; but it costs them some fatigue to read abstract topics in a foreign language. What more, Latin does not have terms to express the mathematical and Physical truths that the ancients did not know.

The moderns have had to create new words to render²⁷ these new ideas. It is a great inconvenience of Science Books, and it must be admitted that it is hardly worth the trouble of writing in a Dead Language, one to which must continually be added expressions unknown to antiquity and which could cause embarrassment. The French which is the current Language of Europe and which is enriched with all the new and necessary expressions is much more proper than Latin to spread this new knowledge throughout the world.

In regards to Algebraic Commentary, this is a work above translation. Madame du Châtelet worked on it using M. Clairaut’s ideas: she did all the calculations herself and when she had achieved a chapter, M. Clairaut examined and corrected it.²⁸ That’s not all: in such an arduous project some errors might easily slip through; it is quite easy to substitute one character for another when writing. M. Clairaut had the calculations looked over by a third party once they had been cleaned up, such that it would be morally impossible for any error of inattention could find its way into the Work. And what would give it already at least that much, is that a Work in which M. Clairaut has had a hand has in every case been excellent in its genre.²⁹

As much as we should be stunned that a woman were capable of an enterprise which required such great faculties³⁰ and such obstinate work, we should as much deplore her premature passing. She had not completely finished the Commentary when she saw that death could take her; she was jealous for glory and had none of that pride of false modesty, which consists of seeming to hate what one wants and to want to seem superior to that true glory — the sole recompense of those who serve the Public, the only one worthy of the great souls, which is good to look for, and which we do not scorn but when we ourselves cannot reach it.

She joined with that taste for glory a simplicity which does not always accompany it, bu which is often the fruit of serious study. Never a woman has been as learned as she, and never has anyone merited less that one should say of her: She is a learned woman. She never spoke of science but with those she believed she could learn from and she never spoke to make herself noticed. She was never seen to assemble those Circles where one makes battle of the mind; where one establishes a sort of tribunal; where one judges her century; by which, in recompense, one is judged and severely. She had lived long in those societies where one know what what she was and she took no stock of that ignorance.

Born with a singular eloquence, that eloquence unfurled only when she had an objective worthy of her. Those Letters which serve only as manner to show one’s wit, one’s finesse, those delicate turns of phrase that are given to ordinary things, these did not rise to the immensity of her talents. The right word, precision, correctness, and force were the character of her eloquence. She would rather write like Pascal and Nicole, than Madame de Sevigné. But the severe closedness and that vigorous moral fibre of her mind did not leave to her inaccessible the beauty of sentiments; the charms of Poetry and of Eloquence flowed through her and never was ear more sensitive to harmony. She new by hard the best verses and could not suffer mediocre ones. It was an advantage that she had over Newton, to unite the profoundness of Philosophie with the most lively and delicate taste for the Belles Lettres.

One could only pity a Philosopher reduced to the dryness of truths and for whom the beauty of imagination and sentiment were lost.

From her tender youth she had nourished her mind with the reading of good Authors, in more than one Language: she had started a translation of the Aeneid of which I have seen several samples filled with the spirit of its Author. She learned from Italian and English: Le Taffe and Milton were as familiar to her as Virgile. She made less progress in Spanish, because she was told that there were hardly, in that Language, but one well-known Book and that this Book was frivolous.

The study of Language was one of her principle occupations: there are written remarques of hers in which we discover, in the midst of the incertitude of Grammar, that philosophical spirit that must dominate by all means and which is the strong of all labyrinths.³¹

Among so many works that this most laborious of the learned would but just have undertaken, who would have believed that she found the time, not just to fill all the duties of society but also to look avidly for amusements? She gave herself to the people as to her studies: all things that occupy society was her interest, aside from gossip. Never had anyone heard her respond to a ridicule, she had neither the time nor the interest to notice them; and when she was told that some people had not done her justice, she responded that she would prefer not to hear about it. She was told once that I do not know which miserable paper in which an author who was not of distance to know her, had dared to speak ill of her. She said that if the author had wasted his time on writing useless things, she would not waste hers on reading them; and the morrow, having learned that the author had been locked up for libel, she wrote in his favour without his ever knowing.

Her passing was felt in the French Court, as much as one can be in a country where personal interest so easily allow the rest to be forgotten. Her memory was precious to those who knew her well, and who were close enough to have seen the breadth of her mind and the greatness of her soul.

It would have been happier for her friends had she not undertaken this enterprise from which the learned will benefit. One could say of her, while deploring her fate, periit arte ſua.

She believed herself beat to death long before the blow that took her from us: and from then she thought of little more than using the little time that she saw remaining to finish what she had undertaken, and to rob death of what she she thought to be the best part of herself. Her ardour and opinionatedness of the work, the continued long nights, in a time when repose would have saved her, brought her finally to that death she had foreseen. She felt the end approaching, and by a singular mixture of sentiments that seemed to fight amongst themselves, we saw her missing life and watching death with intrepidity, the paint of an eternal separation afflicted her soul deeply, and the Philosophy with which that soul was filled left her her courage. A man who tears himself in tears from a family who mourns him, and who hands tranquilly the preparations for a long voyage is but a poor portrait of her pain and her steadfastness³²: so strong that those who were witness to her last moments felt doubly the loss by their own affliction and by her regrets and admired at once³³ the force of her sprit, which held along with such touching regrets a constance so immovable.

Preface to the 1^st Edition

to the first edition of Principes in 1686.

The Ancients, as Pappus shows us, made many examples of Mechanics by interpretation of nature, and the moderns have finally, for some time now, rejected the substantial shapes and occult qualities to bring Natural Phenomena back to mathematical laws. We propose in this Treatise to contribute to this goal, in cultivating in Mathematics its rapport with Natural Philosophy.³⁴

The Ancients taught Mechanics through³⁵ two classes: one theoretical, which procedes by exact demonstration; the other practical. In this way are born all the Arts called Mechanical, from which its name is taken; but as Artisans have a habit of operating with little exactness, from this fact we have come to distinguish so clearly Mechanics from Geometry that all things exact are connected to the latter, and those which are less so the former. This being so, the errors that someone exercising an art makes come from the artist and not the art. The one who who operates less exactly is a less perfect Mechanician, and consequently one who operates perfectly will be the best.

Geometry belongs in some way to Mechanics, for it is on that other that it depends for its definitions of straight lines and circles on which it is founded. It is truly necessary that one who wishes to instruct themself in Geometry know³⁶ how to describe these lines before taking the first lessons in that science: after which they are taught how problems are resolved using those operations. We borrow the solution from Mechanics: Geometry teaches their usage and glorifies itself on the magnificent edifice that is raised by taking so little from elsewhere. Geometry is therefore founded on a mechanical practice and is nothing more than a branch of that Universal Mechanics which deals with and demonstrates the art of measuring. But as the usual Arts are principally occupied with moving of bodies, from this has come the assignment of Geometry to the size of the object and Mechanics to the movement: so the theoretical Mechanics will be the demonstrative science of movements which result from given forces, whatever forces are necessary to engender a given movement.

The Ancients who considered hardly anything more than gravity of the body to move, cultivated that part of Mechanics, in their five powers, which deal with the manual arts; but we have for subject, not the Arts, but the advancement of Philosophy; not content to deal with only the manual powers, but also those that nature employs in its operations. We deal principally with heaviness³⁷, lightness³⁸, the electric force, the resistance of liquids, and other forces of this kind — whether attractive or repulsive. This is why we propose what we give here as Mathematical Principles of Natural Philosophy. Indeed all the difficulty of Philosophy seems to consist of finding the forces that nature employs, through the Phenomena that we know and demonstrating then, by them, the other Phenomena. This is the objective we have had in view of the general propositions of Book I and II and we give yet another example in Book III in explaining the system of the Universe; for we there determine by the Mathematical propositions demonstrated in the first two Books the forces that cause bodies to tend towards the Sun and the Planets, after which — by the same Mathematical propositions — the movements of the Planets, the Comets, the Moon and the Seas³⁹. It would be desired that the other Phenomena which nature presents us with could so happily be derived from these mechanical principles, as several reasons lead me to suspect that they all depend on some forces for which the causes are unknown and by which the particles of a body are pushed against one another, are unified in regular shapes or are repulsed and flee one another. It is this ignorance of those forces where we have been until now which has prevented the Philosophers from trying to explain nature successfully. I hope that the Principles which I have suggested in this Work might be of some use in this manner of philosophising or to some more true, if I have not hit the mark.

The ingenious M. Halley, whose knowledge touches all genres of literature, has not only given his aid to this Edition, in correcting several printing errors and in engraving the figures, he is also the one who has caused me to give it. For after having heard from me what I had demonstrated on the shape of planetary orbits, he prayed me unendingly to show it also to the Royal Society whose pleas and gracious exhortations gave me determination to dream of publishing something on the subject. I worked on it; but after having taken on the question of the Moon’s irregularities and diverse others concerning the laws and measures of gravity and other forces (the shapes that bodies attracted by a force would describe, the movements of several bodies among themselves, those which occur in resistant mediums, the densities and movements of these mediums, finally the orbits of Comets) I thought it would be better to propose postponing the edition to another time, in order to have the luxury of meditating on what remained to find and to give a more complete Work to the public: which I am doing at present. In regards to the movements of the Moon, what I say on the subject being still imperfect, I have placed it in the corollaries to proposition LXVI⁴⁰ of Book I for fear of being obliged to expose and demonstrate each point in particular, which would have engaged me in a superfluous prolixity and would have perturbed the flow of the propositions.

I have preferred to place in several places, however little convenient, the things that I have found later, rather than change the numbers of the oppositions⁴¹ and citations that refer to them.

I pray that the Learned read this Work with indulgence and take the faults as they find them, less as worthy of blame than as points which merit more profound research and renewed efforts.

Cambridge, Trinity College, 8. May 1686.

IS. NEWTON.

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Préface to the 2^nd Edition

at the head of the second Edition.

This Second Edition appears corrected in several Articles and with several additions. In the second Section of the First Book we have more simply rendered the manner of finding the forces necessary to make a body move in a given orbit, and in Section VII of the Second Book, we have researched still further the theory of the resistance of fluids, which is confirmed by new experiences. In Book III, we deduce in a more complete fashion the theory of the Moon and the precession of the Equinoxes, and we have added a large number of calculated orbits to the theory of Comets — done with more care — which gives yet another confirmation.

London. 28. March 1713.

IS. NEWTON.

Préface to the 3^rd Edition

to the third edition.

In this Third Edition — which has benefited from⁴² care of M. Camberton, Doctor of medicine and very capable in his fields — we explain in more length several points concerning the resistance of mediums and we add several new experiments on the falling of weights in air. We explain also in the Third Book, with more detail, the demonstration which proves that the Moon is held in its orbit by gravity. The same Book is augmented with new Observations done my M. Pound on the proportion of the several axes of Jupiter with each other, as well as several concerning the Comet of 1680 made by M. Kirsch and which have only recently come to us. They show again how parabolic orbits approach those of Comets. we determine with more exactitude the orbit of the famous Comet, following the calculations of M. Halley and this on an ellipse. From this we show that that Comet, moving un an orbit of that shape, has had during nine signs⁴³, a course that has been no less regular than those of the Planets in their own orbits. Finally, we have added the determination of the Comet of 1723’s orbit, calculated by M. Bradley, Astronomy Professor at Oxford.

London, 12. January 1725-6

IS. NEWTON.

“On Newton’s Physics”, a poem by Voltaire.

[Trans. note:
My translation is especially rough towards the beginning, and needs a lot of work. The original is written in rhyming alexandrine couplets; here I try simply to retain the rough meaning of each line at its place in the poem.
]

You call me to you, vaste and powerful genius,
Minerva of France, immortal Emilie.
I wake at your voice, I walk at your clarity,
In the prints of your virtues and of your truth.
I leave behind Melpomène⁴⁵ and the player’s⁴⁶ games,
Those combats, those laurels, which I idolised.
By those vain triumphs my heart is no more affected.
Let jealous Rufus, attached to the Earth,
Walk the long of the tomb in senseless fury
To bury in a verse a false thought;
Let him arm against me his languid hands,
Those marks he destined for the remaining men⁴⁷;
Four times a month, let an ignorant Zoïle
Raise, in trembling, an imbecile voice⁴⁸;
I do not hear their cries, which hatred formed.
I do not see their tracks in the printed mire.
The all-powerful charm of Philosophy,
Raises a sage spirit above all desire.
Tranquil in highest heaven, to which Newton submits,
He does not know if he has ennemies.
I know them no more. Already leaving carrier⁴⁹
The august truth comes to open the barrier for me;
already these whirlwinds, each pressure by the others,
Moving around in space, and untethered by rules,
These learned phantoms disappear to my eyes.
A more pure day brightens me; movements spring forth;
Space, which contains the immensity of God,
Sees rolling in its bosom the bound Universe,
This Universe so vaste to our feeble sight,
And which itself is but an atom, a point in the expanse.
— God speaks, and the chaos dissipates at his voice.
Towards a common centre all gravity at once.
This great powerful resource, nature’s soul,
Was enfolded by obscure night.
Newton’s compass, measuring the Universe,
Hoist at last the great veil, and the Heavens are opened.
— He makes my eyes discover, my learned hand,
Of the season’s star a sparkling gown;
The emerald, the azure, the purple, the ruby,
Are the immortal tissue that make its clothing shine.
Each of the rays of its pure substance,
Holds in itself the colours which nature paints,
And all taken together they light up our eyes,
They animate the world, they fill the Heavens.
— Confidentes of the Most-High, eternal substances,
Who burn of his fires, who cover with your wings
The Throne where your Master sits among you,
Speak; of the great Newton were you not at all jealous?
— The ocean hears his voice. I see the humid empire
Raie, advance on the Heavens which attract it:
But a central power stops its efforts;
The sea falls, collapses, and rolls over its edges.
Comets, which are feared same as thunder,
Quit frightening the peoples of the Earth;
In an immense ellipse make your course;
Arise, descend near the day star;
Throw your fires, fly; and come again without end,
Reanimate the aged of the weary worlds.
— And you—heart of the sun—star which in Heaven
Tricked the weak eyes of dazzles sages,
Newton has marked the limits of your career:
Work, illuminate the night, your boundaries are prescribed.
— Earth, change your form, and may gravity,
In lowering the Pole, raise the Equator.
Pole, immobile to the eyes, so slow in your course,
Flee the icy chariot of Ursa’s seven Stars:
Embrace in the course long of your long movements
Two hundred centuries and again six thousand years.⁵⁰
— How beautiful these objects are! May our purified soul
Fly to this truths by which it is clarified.
Yes, in God’s bosom, far from mortal body,
The spirit seems to hear the voix of the Eternal.
— You, to whom this voice makes itself so clear,
How were you able, in yet so tender an age,
Despite the vain pleasures, the pitfalls of good weather,
take such bold flight, follow so vaste a course,
Walk after Newton in that unknown path
Of the immense labyrinth where nature gets lost?
Might I for you, in this isolated Temple,
To French eyes show the Truth,
While Algaroti⁵¹, sure to teach and to please,
To the stunned Tiber bear that Stranger⁵².
— If with new flowers he adorns his appeal,
With compass in hand I will trace my marks;
With my thick pencils I will paint the immortal;
Searching to embellish, I will render it less beautiful.
She was, as are you, noble, simple and without embellishment,
Above praise, above my art.

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Definitions

Definition I

Air becoming of a double density is quadruple in quantity when the space is doubled, and sextuple when the space is tripled. We can say the same of snow and of powders condensed by liquefaction or compression, as well as of all condensed bodies whatever the cause may be.

I pay no attention here to the medium that passes freely between the pars of the bodies, supposing that such a medium exists. I designate the quantity of matter by the words body or mass. This quantity is known by the weight of the body: for I have found through very precise experimentation on pendulums that the weight of a body is proportional to its mass. I will describe these experiments further on.

Details

[Trans. note:
The same given in equation form:

\begin{equation*} \text{mass} = \text{density} \cdot \text{volume} \end{equation*}

]

Definition II

The total movement is the sum of the movement of each part; so that quantity of movement is doubled in a body which⁵⁴ mass is double, if the speed remains the same; but if the speed is doubled, the quantity of movement would quadruple.

Details

[Trans. note:
The same given in equation form:

\begin{equation*} \text{mass} \cdot \text{speed} = \text{momentum} \end{equation*}

]

Definition III

This force is proportional to the quantity of matter of the body, and is not different than what we call the inertia of matter except in how we think of it: since inertia is what makes it such that one cannot change without effort the current state of a body, whether it moves or is at rest. So one can give name to this force which resides in bodies, the very expressive name force of inertia.

The body exercises this force every time it changes its current state, and one can consider it therefore through two separate aspects: as resisting or as imparting⁵⁶; as resisting, when the body is opposing the force that attempts to change its state; as imparting, when the same body exerts some effort to change the state of an obstacle that resists it.

We commonly attribute resistance to bodies in rest and imparting force to those which move; but the movement and the rest, as we commonly conceive of them, are but respective: those bodies we believe to be at rest are not always completely at rest.

Definition IV

This force is had only when the force is in motion, and it does not subsist in the body when the motion has stopped. Rather the body perseveres solely by the force of inertia in the new state it finds itself in. The imparted force can have diverse origins; it can be produced by shock, by pressure, and by centripetal force.

Definition V

The gravity which attracts bodies towards the centre of the Earth, the magnetic force which attracts iron towards a magnet, and the force — whatever it may be — that continually pulls planets from rectilinear motion and which causes them to circulate in curves, are forces of this type.

The stone that is swung from a sling, acts on the hand holding the sling, by an effort which is greater the faster we swing it and escapes as soon as it is let go. The force exerted by the hand to keep the stone, which is equal to and opposite to the force by which the stone pulls on the sling, being always directed towards the hand — which is the center of the described circle — is that which I call centripetal force. It is the same for all bodies which move in circles; they put all effort into distancing themselves from the centre of their revolution, and without recours to some force to oppose that effort and hold them in their orbit, that is some centripetal force, the would go off in a straight line with in a uniform movement.

A projectile would not fall back to the ground if it was not animated by the gravitational force, instead it would go on in a straight line into the heavens in a uniforme movement, if the air resistance was null. It is therefore by its gravity⁵⁸ that it is pulled from the straight line, and it inclines without fail towards the ground⁵⁹; and it is pulled more or less according to its gravity and the speed of its movement.⁶⁰ The smaller the gravity of the projectile in relation to the quantity of its matter [mass], the more speed it will have; the less it will be pulled from its straight line, and will go further before falling again to the ground.

So, if a cannonball is fired horizontally from the top of a mountain at a speed capable of sending it a distance of two leagues before falling back to the ground, with twice the speed the cannonball would not fall back to the ground until it had reached nearly four leagues. And with a decuple⁶¹ speed it would go ten times as far — that is, ignoring all resistance of the air. By augmenting the speed, the path it would describe before falling could be controlled at will and it would lessen the curvature of the line described such that it would not fall until it reached a distance of 10, 30, or 90 degrees. Indeed it could circulate around the Earth without ever falling back down, or go off in a straight, infinite line in the sky.

Since, by the same reason a projectile might turn around the Earth by the gravitational force, it might be that, by its gravitational force (supposing that it gravitates) or by some other force which pulls it towards the ground, the moon is detoured at every moment from a straight line and pulled closer to the Earth, and that it is constrained to circulate along a curve and without such a force the moon could not be held to its orbit.

If that force was less, it would not suffice. It would not pull the moon enough from its straight line; and if it was stronger, it would pull too much and pull it from its orbit towards the Earth. The quantity of this force must therefore be given and it is for the Mathematicians to find the centripetal force necessary to cause a circular orbit in a given body and reciprocally to determine the curve in which a body with a given centripetal force circulates, starting at whatever given point and with a given speed.

The quantity of centripetal force can be considered: absolute, accelerative, or motivating.

Definition VI

This is how magnetic force is stronger in one magnet than in another, according to the size of the stone and the intensity of its virtues⁶².

Definition VII

The magnetic force of a single magnet is stronger at a smaller distance than at a greater one. The gravitational force is stronger in the plains than in the mountains, and must probably by even less (as well will prove later) at even greater distances from the ground. At equal distances, it is the same on all sides; this is why it accelerates equally all falling bodies, whether lighter or heaver, larger or smaller — abstraction made of air resistance.⁶³

Details

[Trans. note:

\begin{equation*} {\vec a}_c=-{\frac {v^2}{r}}=-\omega^{2} \end{equation*}

Where, in modern terms, \(\omega={\frac{v}{r}}\) is the angular momentum and the unit vector, \({\hat {{\mathbf {r}}}}\), is one.
]

Definition VIII

The weight of a body is greater the more mass it has, and the same boy weighs more closer to the surface of the Earth than if it were transported into the sky. The motive quantity of centripetal force is the total force with which the body is attracted towards the centre: simply put, its weight. We can always find it by measuring the opposite and equal force that can prevent that body from descending.

I have called these different quantities of centripetal force motive, accelerative, and absolute in order to be shorter.

We can, to distinguish them, relate them to the bodies which are attracted to a centre, the positions of these bodies, and the centre of the forces.

We can relate motive centripetal force to the body, by considering it as an effort which the whole body exerts to approach the centre, which effort is composed of that of all its parts.

The accelerative centripetal force can be related to the position of the body, by considering that force as it emanates from the centre out to all the positions around it, to move the bodies which encounter it.

Finally, we can relate the absolute centripetal force to the centre, as a certain cause without which the motive and forces would not propagate into all the places surrounding the center, whether it be that this cause be whatever kind of central body (like a magnet at the centre of a magnetic force and the Earth at the center of the gravitational force) or that it some other cause that we do not perceive. This way of thinking about centripetal force is purely mathematical and I do not intend at all to give a physical cause.

The accelerative centripetal force is therefore to the motive centripetal force what speed is to movement; since as how the quantity of movement is the product of the mass by its speed, the quantity of motive centripetal force is the product of the accelerative centripetal force by its mass. Since the sum of all the accelerative centripetal force acting on each particle of the body is the motive centripetal force of the entire body. So at the surface of the Earth where the accelerative force of gravity is the same on all bodies, motive gravity or the weight of the bodies is proportional to their mass; and if we were placed in the regions were the accelerative force was lesser, the weight of the bodies would be, to the same degree, less. Therefore it is always equivalent to the product of the mass by the accelerative centripetal force. In those regions where the accelerative centripetal force is half as much, the weight of the body a half or a third the size would be four or six times less.⁶⁴

Details

[Trans. note:
\[ {\vec p}_c = {\vec a}_c m \] ]

To the rest, I take here in the same meaning the attractions and accelerative and motive impulsions and I use indifferently the words impulsion, attraction, and propensity of whatever sort towards a centre: because I consider all these forces mathematically and not physically. So the Reader must take care to not to think that I have tried to designate by these words some sort of action, cause, or physical reason, for the forces. They should not think that I have wanted to attribute whatever real force to these centres which I consider as mathematical points.

Scholus (Annotation)

I have just gone over the meaning I give in this Work to the terms that are not commonly used. As for those like time, space, place, and movement, they are known to everyone; but we should note that by considering these quantities by their relations to sensitive things, we have fallen into several errors.⁶⁵

To avoid them, we must distinguish time, space, place, and movement into absolute and relative, true and apparent, mathematical and common.⁶⁶

1. Absolute, true and mathematical time, without relation to the exterior, passes uniformly and is called duration. Relative, apparent and common time is that sensitive and external measure of a part of any duration (equal or unequal) taken of movement: its measures are hours, days, months, &c which is ordinarily used in the place of true time.
2. Absolute space, without relation to external things, is always similar and fixed. Relative space is that measure or unfixed dimension in absolute space, which falls under the nose by relation to the body and which the common confound with fixed space. It is thus, for example, that a space taken on Earth or in the sky is determined by its position⁶⁷ in relation to the Earth. Absolute space and relative space are the same in kind and size, but not always the same in number; as, for example, when the Earth has changed place in space, the space that contains our air is the same in relation to the Earth even as the air occupies necessarily different parts of the space through which it passes and it changes this continually.⁶⁸
3. Place⁶⁹ is the part of space occupied by a body and in relation to the space is either relative or absolute. I say that place is a part of space and not simply the location of the body or the surface that covers it because equal solids always have equal places though their surface area is often unequal because of their different shapes; their placement, to be precise, doesn’t have a quantity, it is an allocation of space rather than a location, properly-speaking. Just as movement or translation of all out of its location is the sum of all the movements or translations of the parts out of their own, so place of the whole is the sum of the places of its parts, and it must be internal, and take up the whole of the body (et propterea internus et in copore toto)⁷⁰.
4. Absolute movement is the translation of bodies form an absolute place to another absolute place; the relative movement is the translation from one relative place into another. In this way, on a vessel pushed by the wind, the relative place of a body is the part of the vessel in which the body is found or the space it occupies in the cavity of the vessel. But the true rest of the body is its permanence in part of fixed space, in which we suppose the ship and all it contains to move. So, if the Earth is at rest, the body the is at rest relative to the Vessel will have a true and absolute movement, which speed will be equal to that at which the vessel moves across the surface of the Earth. But should the Earth move through space, the true and absolute movement of the body is composed of the true movement of the Earth in fixed spance and the relative movement of the vessel on the surface of the earth. And if the body made some movement in its vessel, its true and absolute movement would be made up of its movement relative to the vessel, the relative movement of the vessel on the Earth and the of the real movement of the Earth in absolute space. As for movement relative to this body on the Earth, it would be formed of its movement relative to the vessel and the relative movement of the vessel on the Earth. In such a way that if the part of the Earth where the vessel is found moved towards the east at a speed measured at 10 100 units⁷¹; the vessel moved towards the west at a speed of 10 units; and the pilot walked along the vessel towards the east at 1 unit; this pilot would have a real and absolute movement in fixed space of 10001 units towards the east, and a relative movement on the Earth towards the west of 9 units.

Details

Trans. note: I’m not sure how this math works out. I’ve shown my work here, but it’s not coming out right:

Each starts out with a speed (in the given direction) of:

\begin{align*} &\vec{\text{E}}_{Earth} =&\vec{E}_E& = &10100 \text{ units} \\
&\vec{\text{W}}_{vessel} =&-\vec{E}_v& = &10 \text{ units} \\
&\vec{\text{E}}_{pilot} =&\vec{E}_p& = &1 \text{ units} \\
\end{align*}

Since east and west are opposite directions along a single axis⁷², we express \(\vec{W}\) as a negative \(\vec{E}\). To solve for the absolute (\(M_{abs}\)), we use the following equations:

\begin{align*} \vec{E}_{E} - \vec{E}_v + \vec{E}_p &= M_{abs} \\
10100 - 10 + 1 &= 10091\\
\end{align*}

To solve for the relative movement (\(M_{rel}\)) we can set \(\vec{E}_E\) to \(0\):

\begin{align*} 0 - \vec{E}_v + \vec{E}_p &= M_{rel} \\
0 - 10 + 1 &= -9 \vec{E}_E = 9 \vec{W}_E\\
\end{align*}

This results in the correct value of \(9\) units for \(M_{rel}\) but an incorrect value (\(91\) units) for \(M_{abs}\).

In astronomy, we can distinguish absolute time from relative time by the time equation. Since days are not equally long, even if they are commonly used as an equal measure of time, Astronomers correct that inequality in order to measure the celestial movements in a more exact time.

It is quite possible that there is no perfectly steady movement that might serve as an exact measure of time, since all movements are accelerated or slowed, whereas absolute time must flow always in the same way.

The time passed or perseverance of things is therefore the same, whether their movements be prompt or show; and they will be the same where there is no movement.⁷³ Therefore, it is important to distinguish time from its common⁷⁴ measures, and this is what the astronomical equation does. The necessity of this equation in the determination of Phenomena is proved sufficiently by the pendulum clock experiments and by the observations of the Eclipses of Jupiter’s sattelites.

The order of parts of space is as immutable as that of time’s parts, since if parts of space left their places, that would be — so to speak — to leave themselves. Time and space have no other place than themselves and are the place of everything else.

IT GETS MATHS-Y HERE AND I’M TIRED, I’LL PICK IT UP AGAIN LATER If you want to see more of this translation let me know.