The Principia

by Isaac Newton, Andrew Motte (Translator)

In his monumental 1687 work Philosophiae Naturalis Principia Mathematica, known familiarly as the Principia, Isaac Newton laid out in mathematical terms the principles of time, force, and motion that have guided the development of modern physical science. Even after more than three centuries and the revolutions of Einsteinian relativity and quantum mechanics, Newtonian physics continues to account for many of the phenomena of the observed world, and Newtonian celestial dynamics is used to determine the orbits of our space vehicles.
This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms.
Newton's principles describe acceleration, deceleration, and inertial movement; fluid dynamics; and the motions of the earth, moon, planets, and comets. A great work in itself, the Principia also revolutionized the methods of scientific investigation. It set forth the fundamental three laws of motion and the law of universal gravity, the physical principles that account for the Copernican system of the world as emended by Kepler, thus effectively ending controversy concerning the Copernican planetary system.
The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students.

Genres: Science, Philosophy, Physics, Nonfiction, Mathematics, Classics, History

490 pages, Paperback

First published July 1, 1687

About the author

Sir Isaac Newton, FRS , was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. His Philosophiæ Naturalis Principia Mathematica, published in 1687, is considered to be the most influential book in the history of science. In this work, Newton described universal gravitation and the three laws of motion, laying the groundwork for classical mechanics, which dominated the scientific view of the physical universe for the next three centuries and is the basis for modern engineering. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientific revolution.

In mechanics, Newton enunciated the principles of conservation of momentum and angular momentum. In optics, he invented the reflecting telescope and developed a theory of colour based on the observation that a prism decomposes white light into a visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound.

In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function, and contributed to the study of power series.

Newton was also highly religious (though unorthodox), producing more work on Biblical hermeneutics than the natural science he is remembered for today.

In a 2005 poll of the Royal Society asking who had the greater effect on the history of science, Newton was deemed much more influential than Albert Einstein.

Reviews

[Roy Lotz · Rating 5 out of 5]

It is shown in the Scholium of Prop. 22, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the surface of the earth in the ratio of 30 to 0.0000000000003998, or as 75,000,000,000,000 to 1, nearly.

Marking this book as “read” is as much an act of surrender as an accomplishment. Newton’s reputation for difficulty is well-deserved; this is not a reader-friendly book. Even those with a strong background in science and mathematics will, I suspect, need some aid. The historian of mathematics Colin Pask relied on several secondary sources to work his way through the Principia in order to write his excellent popular guide. (Texts by S. Chandrasekhar, J. Bruce Brackenridge, and Dana Densmore are among the more notable vade mecums for Newton’s proofs.) Gary Rubenstein, a math teacher, takes over an hour to explain a single one of Newton’s proofs in a series of videos (and he had to rely on Brackenridge to do so).

It is not that Newton’s ideas are inherently obscure—though mastering them is not easy—but that Newton’s presentation of his work is terse, dense, incomplete (from omitting steps), and at times cryptic. Part of this was a consequence of his personality: he was a reclusive man and was anxious to avoid public controversies. He says so much himself: In the introduction to Book III, Newton mentions that he had composed a popular version, but discarded it in order to “prevent the disputes” that would arise from a wide readership. Unsurprisingly, when you take material that is intrinsically complex and then render it opaque to the public, the result is not a book that anyone can casually pick up and understand.

The good news is that you do not have to. Newton himself did not advise readers, even mathematically skilled readers, to work their way through every problem. This would be enormously time-consuming. Indeed, Newton recommended his readers to peruse only the first few sections of Book I before moving on directly to Book III, leaving most of the book completely untouched. And this is not bad advice. As Ted said in his review, the average reader could gain much from this book by simply skipping the proofs and calculations, and stopping to read anything that looked interesting. And guides to the Principia are certainly not wanting. Besides the three mentioned above, there is the guide written by Newton scholar I. Bernard Cohen, published as a part of his translation. I initially tried to rely on this guide; but I found that, despite its interest, it is mainly geared towards historians of science; so I switched to Colin Pask’s Magnificent Principia, which does an excellent job in revealing the importance of Newton’s work to modern science.

So much for the book’s difficulty; on to the book itself.

Isaac Newton’s Philosophiæ Naturalis Principia Matematica is one of the most influential scientific works in history, rivaled only by Darwin’s On the Origin of Species. Quite simply, it set the groundwork for physics as we know it. The publication of the Principia, in 1687, completed the revolution in science that began with Copernicus’s publication of De Revolutionibus Orbium Coelestium over one hundred years earlier. Copernicus deliberately modeled his work on Ptolemy’s Almagest, mirroring the structure and style of the Alexandrian Greek’s text. Yet it is Newton’s book that can most properly be compared to Ptolemy’s. For both the Englishman and the Greek used mathematical ingenuity to draw together the work of generations of illustrious predecessors into a single, grand, unified theory of the heavens.

The progression from Copernicus to Newton is a case study in the history of science. Copernicus realized that setting the earth in motion around the sun, rather than the reverse, would solve several puzzling features of the heavens—most conspicuously, why the orbits of the planets seem related to the sun’s movement. Yet Copernicus lacked the physics to explain how a movable earth was possible; in the Aristotelian physics that held sway, there was nothing to explain why people would not fly off of a rotating earth. Furthermore, Copernicus was held back by the mathematical prejudices of the day—namely, the belief in perfect circles.

Johannes Kepler made a great stride forward by replacing circles with ellipses; this led to the discovery of his three laws, whose strength finally made the Copernican system more efficient than its predecessor (which Copernicus’s own version was not). Yet Kepler was able to provide no account of the force that would lead to his elliptical orbits. He hypothesized a sort of magnetic force that would sweep the planets along from a rotating sun, but he could not show why such a force would cause such orbits. Galileo, meanwhile, set to work on the new physics. He showed that objects accelerate downward with a velocity proportional to the square of the distance; and he argued that different objects fall at different speeds due to air resistance, and that acceleration due to gravity would be the same for all objects in a vacuum. But Galileo had no thought of extending his new physics to the heavenly bodies.

By Newton’s day, the evidence against the old Ptolemaic system was overwhelming. Much of this was observational. Galileo observed craters and mountains on the moon; dark spots on the sun; the moons of Jupiter; and the phases of Venus. All of these data, in one way or another, contradicted the old Aristotelian cosmology and Ptolemaic astronomy. Tycho Brahe observed a new star in the sky (caused by a supernova) in 1572, which confuted the idea that the heavens were unchanging; and observations of Haley’s comet in 1682 confirmed that the comet was not somewhere in earth’s atmosphere, but in the supposedly unchanging heavens.

In short, the old system was becoming unsustainable; and yet, nobody could explain the mechanism of the new Copernican picture. The notion that the planets’ orbits were caused by an inverse-square law was suspected by many, including Edmond Haley, Christopher Wren, and Robert Hooke. But it took a mathematician of Newton’s caliber to prove it.

But before Newton published his Principia, another towering intellect put forward a new system of the world: René Descartes. Some thirty years before Newton’s masterpiece saw the light of day, Descartes published his Principia Philosophiæ. Here, Descartes summarized and systemized his skeptical philosophy. He also put forward a new mechanistic system of physics, in which the planets are borne along by cosmic vortexes that swirl around each other. Importantly, however, Descartes’s system was entirely qualitative; he provided no equations of motion.

Though Descartes’s hypothesis has no validity, it had a profound effect on Newton, as it provided him with a rival. The very title of Newton’s book seems to allude to Descartes’s: while the French philosopher provides principles, Newton provides mathematical principles—a crucial difference. Almost all of Newton’s Book II (on air resistance) can be seen as a detailed refutation of Descartes’s work; and Newton begins his famous General Scholium with the sentence: “The hypothesis of vortices is pressed with many difficulties.”

In order to secure his everlasting reputation, Newton had to do several things: First, to show that elliptical orbits, obeying Kepler’s law of equal areas in equal times, result from an inverse-square force. Next, to show that this force is proportional to the mass. Finally, to show that it is this very same force that causes terrestrial objects to fall to earth, obeying Galileo’s theorems. The result is Universal Gravity, a force that pervades the universe, causing the planets to rotate and apples to drop with the same mathematical certainty. This universal causation effectively completes the puzzle left by Copernicus: how the earth could rotate around the sun without everything flying off into space.

The Principia is in a league of its own because Newton does not simply do that, but so much more. The book is stuffed with brilliance; and it is exhausting even to list Newton’s accomplishments. Most obviously, there are Newton’s laws of motion, which are still taught to students all over the world. Newton provides the conceptual basis for the calculus; and though he does not explicitly use calculus in the book, a mathematically sophisticated reader could have surmised that Newton was using a new technique. Crucially, Newton derives Kepler’s three laws from his inverse-square law; and he proves that Kepler’s equation has no algebraic solution, and provides computational tools.

Considering the mass of the sun in comparison with the planets, Newton could have left his system as a series of two-body problems, with the sun determining the orbital motions of all the planets, and the planets determining the motions of their moons. This would have been reasonably accurate. But Newton realized that, if gravity is truly universal, all the planets must exert a force on one another; and this leads him to the invention of perturbation theory, which allows him, for example, to calculate the disturbance in Saturn’s orbit caused by proximity to Jupiter. While he is at it, Newton calculates the relative sizes and densities of the planets, as well as calculates where the center of gravity between the gas giants and the sun must lie. Newton also realized that gravitational effects of the sun and moon are what cause terrestrial tides, and calculated their relative effects (though, as Pask notes, Newton fudges some numbers).

Leaving little to posterity, Newton realized that the spinning of a planet would cause a distortion in its sphericity, making it marginally wider than it is tall. Newton then realized that this slight distortion would cause tidal locking in the case of the moon, which is why the same side of the moon always faces the earth. The slight deformity of the earth is also what causes the procession of the equinoxes (the very slow shift in the location of the equinoctial sunrises in relation to the zodiac). This shift was known at least since Ptolemy, who gave an estimate (too slow) of the rate of change, but was unable to provide any explanation for this phenomenon.

The evidence mustered against Descartes's theory is formidable. Newton describes experiments in which he dropped pendulums in troughs of water, to test the effects of drag. He also performed experiments by dropping objects from the top of St. Paul's Cathedral. What is more, Newton used mathematical arguments to show that objects rotating in a vortex obey a periodicity law that is proportional to the square of the distance, and not, as in Kepler’s Third Law, to the 3/2 power. Most convincing of all, Newton analyzes the motion of comets, showing that they would have to travel straight through several different vortices, in the direction contrary to the spinning fluid, in order to describe the orbits that we observe—a manifest absurdity. While he is on the subject of comets, Newton hypothesizes (correctly) that the tail of comets is caused by gas released in proximity to the sun; and he also hypothesizes (intriguingly) that this gas is what brings water to earth.

This is only the roughest of lists. Omitted, for example, are some of the mathematical advances Newton makes in the course of his argument. Even so, I think that the reader can appreciate the scope and depth of Newton’s accomplishment. As Pask notes, between the covers of a single book Newton presents work that, nowadays, would be spread out over hundreds of papers by thousands of authors. The result is a triumph of science. Newton not only solves the longstanding puzzle of the orbits of the planets, but shows how his theory unexpectedly accounts for a range of hitherto separate and inexplicable phenomena: the tides, the procession of the equinoxes, the orbit of the moon, the behavior of pendulums, the appearance of comets. In this Newton demonstrated what was to become the hallmark of modern science: to unify as many different phenomena as possible under a single explanatory scheme.

Besides setting the groundwork for dynamics, which would be developed and refined by Euler, d'Alembert, Lagrange, Laplace, and Hamilton in the coming generations, Newton also provides a model of science that remains inspiring to practitioners in any field. Newton himself attempts to enunciate his principles, in his famous Rules of Reasoning. Yet his emphasis on inductivism—generalizing from the data—does not do justice to the extraordinary amount of imagination required to frame suitable hypotheses. In any case, it is clear that Newton’s success was owed to the application of sophisticated mathematical models, carefully tested against collections of physical measurements, in order to unify the greatest possible number of phenomena. And this was to become a model for other intellectual disciples to aspire to, for good and for ill.

A striking consequence of this model is that its ultimate causal mechanism is a mathematical rule rather than a philosophical principle. The planets orbit the sun because of gravity, whose equations accurately predict their motions; but what gravity is, why it exists, and how it can affect distant objects, is left completely mysterious. This is the origin of Newton’s famous “I frame no hypothesis” comment, in which he explicitly restricts himself to the prediction of observable events rather than speculation on hidden causes (though he was not averse to speculation when the mood struck him). Depending on your point of view, this shift in emphasis either made science more rational or more superficial; but there is little doubt that it made science more effective.

Though this book is too often impenetrable, I still recommend that you give it a try. Few books are so exalting and so humbling. Here is on display the furthest reaches of the power of the human intellect to probe the universe we live in, and to find hidden regularities in the apparent chaos of experience.

[Orhan Pelinkovic · Rating 5 out of 5]

Rather than just reading about the Scientific Revolution (1543-1687), I chose to experience it through some of the books that initiated, shaped, and crystallized the scientific method. My reading journey began with Copernicus' On the Revolutions of the Heavenly Spheres (1543), followed by Kepler's Astronomia Nova (1609), Harmonies of the World (1619), and Epitome of Copernican Astronomy (1621), down south to Galilei's Dialogue Concerning the Two Chief World Systems (1632), then to Descartes' Discourse on the Method (1637), and finally here, the crown jewel, The Principia (1687).

What makes this book one of the greatest scientific works of all time is that Isaac Newton (1643-1727) presents in it the mathematical language that is capable of revealing the marvels of the cosmos. Newton has crafted mathematical expressions that can decipher the behavior of the universe. He has discovered simple formulas and a new branch of mathematics, calculus, that can even predict the time, position, and rate of change of celestial bodies in motion and certain objects on Earth that are set in motion. I can't help but notice that it took a proper interpretation of the behavior of the heavens to initiate a Scientific Revolution here on Earth.

Reading The Principia we see how the fabric of space is governed by a rhythmical pulse; where the invisible, to the naked eye, phenomena such as time, forces, and resistances are in varied proportional (mathematically expressible) relationships to the visible characteristics of space; such as the mass of celestial bodies and other objects, their motions, and the distances between them. We live in a clockwork universe of consistent symmetry.

The Principia is composed of 3 books. In the 1st and 2nd book, Newton presents his mathematical principles of nature through his theories, countless calculations, and experimental results which felt like reading a textbook, but the 3rd book reads like an astronomy science book and was more of an enjoyable read with the last few pages of the book being a real treat.

In my view, Newton's, mathematical approach to codifying the celestial mechanics of the solar system is considerably Keplerian. It's clear that Johannes Kepler's three laws of planetary motion inspired Newton to devise his three laws of motion and possibly laid the foundation for Newton's law of universal gravitation. Although to my surprise, Newton, does not give Kepler any recognition for it, other than briefly mentioning Kepler's laws of planetary motions.

Even though, Newtonian mechanics were considered, for over 200 years, to reflect an accurate interpretation of the behavior of the physical space around us, but with the introduction of the theory of relativity and quantum mechanics, at the turn of the 20th century, Newton's laws became great approximations for everyday life purposes. All in all, this was a short and insightful journey, and I look forward to revisiting this period to fulfill other reading aspirations.

(4.5/5.0)

[India M. Clamp · Rating 4 out of 5]

Naturally when delving into a book initially published in 1687, investigation into the life of the author is customary. Warmth, kindness and a loving home were not what we could hope to describe for this physicist/mathematician whose literary progeny became the authority on physics called “Philosophiae Naturalis Principia Mathematica” (aka The Principia).

Sir Issac Newton—as a toddler—was in for a surprise when his mother remarried a man of the cloth and he was packed up and sent away to live with his grandmother (maternal). “The Principia: Mathematical Principles of Natural Philosophy” had its beginning when awkward Newton undertook a work-study program to serve affluent students.

“This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being...

This “Being” governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκρατωρ or Universal Ruler.”
---Isaac Newton

During a hiatus from Trinity College (closed due to Great Plague) Newton orchestrated the methodology for infinitesimal calculus thus becoming the founding father to how the planets moved/behaviors of light and color. The first law (as is true in a large scale/small scale). A stationary body will stay stationary unless an external force is applied to it.

Though he had a clearly rough start, never married and was on the brink of madness (not an uncommon affliction to genius) his discovery of gravity and his work in optics and the pin hole camera was remarkable. A nervous breakdown (1678) brought on from his critics Hooke/Huygens steamrolled Isaac. The brilliance the ellipse was telling. Read.

[Matt]

I tried. But this is Newton using geometry to explain the calculus behind his theory of gravity. Every few pages, between the charts and equations, he writes a one or two sentence introduction to the proposition about to be proved. I understood those. Mostly. And I could see this is where Newton’s Laws of Motions come from. His proofs are beyond me though.

Interestingly, one of the few other things I could understand, beyond his Preface, was the General Scholium at the end. After describing the heliocentric solar system, he launches into the modern equivalent of an Intelligent Design argument:

All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. Pg. 442.

Newton’s fascination with Biblical history, alchemy and the occult has been credited with helping him believe in a gravitational force that pervades all matter and affects things unseen at distance. A fascinating mix of science and faith. He was probably as enigmatic as his equations seem to me.

I’m sure this book is worth ten stars but, in the interest of intellectual honestly, I’m personally not qualified to rate it.

[Beauregard Bottomley · Rating 5 out of 5]

I only read the guide to the principia. Newtown's actual principia is monstrously difficult and I gave up.

[Conrad · Rating 5 out of 5]

First, A Clarification: The publication I have is the hardcover revision by Florian Cajori of Andrew Motte's 1729 English translation, copyrighted in 1934 by the Regents of the University of California, and published by UC Berkeley and UCLA Press.

I should also note that, although I have read Newton's Principia several times over several years and for various reasons, I doubt I have ever completed the whole book. To do so would be advisable only under limited circumstances.

For whatever reason, Newton did not meticulously document his propositions. Hence, the Principia requires its reading audience to do a fairly significant amount of sleuthing to reach a workable grasp of just one proposition. Once completed, congratulate yourself. You have extracted the ten or twenty steps needed to prove a proposition. Now you can confidently advance to the next proposition--on page two.

To describe Newton's Principia as dense is clichéd, fuzzy, and simplistic, but for 98% of us, dense is most appropriate. If previous generations truly had less trouble with reading Principia, then... WOW... our reading skills have certainly plummeted.

Yes, it's true that Newton's Principia changed the world, and is undoubtedly near or at the top of the greatest work ever. Unfortunately, few will directly experience its unvarnished power. Regardless, the endeavor to undertake the challenge is highly recommended and greatly rewarding. Good luck!!

[Laura · Rating 4 out of 5]

Free download available at Project Gutenberg

I made the post-processing of this book for DistributedProofreaders and Project Gutenberg will publish it pretty soon.

The work is a foundational text in the fields of physics and mathematics, addressing the laws of motion and universal gravitation. It explores the mathematical principles governing the motion of celestial bodies and the forces that act upon them. The opening of this monumental work introduces the context and purpose behind Newton's explorations into mathematics and natural philosophy. It discusses the importance of mechanics in understanding the natural world, emphasizing the relationship between mathematics and philosophical inquiry. Newton outlines his method while addressing prior beliefs and theories, affirming his aim to derive natural phenomena from mathematical laws. This introduction sets the stage for a detailed examination of physical principles, beginning with definitions and laws that illustrate his revolutionary ideas in motion and gravitation.

[Aleksandar Janjic · Rating 5 out of 5]

Одма да кажем да немам шта паметно да кажем о овој књизи јер је Њутн ипак можда мало пренапредан за мене. На почетку сам покушавао да пратим и донекле успијевао, али то није дуго потрајало јер су ствари врло брзо постале прекомпликоване. Наравно, овакве научне класике данас је практично немогуће читати без разноразних додатних објашњења и коментара, којих у овом издању нажалост нема, мада морам да напоменем да имам (у папирном облику) једну апсолутну звијер од издања, тешку једно сто кила, са ооооооооооогромним уводом и гомилетином коментара, што такође планирам некад да прочитам, е не бих ли много боље разумио. Ово издање које сам читао је за Киндл и није препоручљиво ако планирате да се удубљујете у материју, зато што су неке формуле тешко читљиве, неким табелама фале комади, а и неке слике се не виде добро. Препоручљиво је ако желите, као ја сад, само да прелетите да бисте видјели о чему се ту отприлике ради.

Мало конкретнији ривју слиједи... ахем... за неколико година :-)

[Abe · Rating 5 out of 5]

One of the most intelligent and influential books of all time. Period. This is an older read I remember fondly enough to rate the full 5 stars even though it has been a while.

[Leftbanker · Rating 5 out of 5]

Newton unleashed one of the most startling scientific undertakings in history with his seemingly simple question posed in this hallowed treatise: what would happen if seven people representing various socio-economic strata of American life were stranded together on a desert island following a mishap during a three-island tour?

In the centuries since the publication of this philosophical juggernaut, men and women have agonized over the fundamental question of whether to sleep with Ginger or Mary Ann. Newton posed a question most poeple had never stopped to consider: what about the old broad? Why doesn’t anyone go that route? Newton himself was obviously enthralled with the Skipper and his ample buttocks. He liked a big ass, that’s just how they rolled back then. That's what British public school did to a tender, thoughtful lad like Newton.

Of course, I didn't read this. Do I look like a guy who's read The Principia? No, I look like a guy who grew up watching Gilligan's Island (Definitely Mary Ann, but why not a threesome with Ginger? I'm just thinking out loud here, no harm done I hope).

[Joshua · Rating 5 out of 5]

I learned that there are some problems which simply cannot be solved with a particular framework; that Bezier curves are a fantastic introduction to the philosophical principles of the calculus; that I can, in fact, do math.

[Ehsan Ebrahimian · Rating 4 out of 5]

دیدن کتاب بنیادین رشته ای که بیش از ده ساله در حال خوندنش هستم اصولا هیجان انگیزه، فاصله روشهای نیوتون با روشهای استاندارد فیزیک نسبتا زیاده و کمابیش خبر زیادی از جبر در این کتاب نیست به جاش با زبان هندسه اقلیدسی نوشته شده، نگاه نیوتون به قوانین، فلسفه، شیطنت هایی که وسط کتاب میکنه و مسائلی که با قصدی مشخص حلشون میکنه واقعا جالبه و حقیقتا انتظار نداشتم نیوتون این همه گسترده راجع به فیزیک حرف زده باشه.

دیدگاه های تاثیر گذار نیوتون بر عصر روشنگری هم آشکارا در این کتاب بیان شده.

نمی تونم بگم ترجمه اش بهتر از این نمیشد، با توجه به این که زمینه تاریخی کتاب فوق العاده با ما متفاوته انتظار داشتم مترجم پاورقی های زیادی داشته باشه تا بگه که مثلا فلان کلمه که نیوتون استفاده می کنه در فیزیک امروز معادل این مفهومه اما خب چنین اشاراتی در کتاب غایب هستند و من فیزیک خونده وسطش گیج می شدم و مطمئن نبودم مثلا منظور نیوتون از کشسانی همون کشسانی هست که ما تو فیزیک امروز میگیم یا نه.

[Thomas Preusser · Rating 5 out of 5]

This book, written by Isaac Newton in 1588, served as the foundation of physics for more than 300 years, or up to the time Einstein developed relativity theory. The fact that it is still in print more than 400 years after being written puts it in nearly the same class as the bible. One does not actually read this book so much as marvel at it. The book is chock full of hundreds of geometric diagrams which essentially deal with systematic measurement and calculation. The thing that strikes one most is the lack of elaborate equations, even though Newton was a major impetus in the development of equation-centric calculus. Contrast this with the typical hard-core science works of today which can be full of elaborate equations of arcane notation and interest. This is a slow contemplative read, and deserves to be on your science book shelf.

[Jonathan Morrow · Rating 1 out of 5]

I stopped reading it after the first couple dozen pages. It's a brilliant book, but boy, he did not try at all to make it accessible. He gives a few hints as to the importance of his subject matter at the very beginning, but then he just launches into some very dry geometric proofs and continues that way for what looks like the vast majority of the book. He doesn't really tell you what the destination is, so it's hard to follow him on a journey that is such a slog. The ideas, of course, are world-changing, but it's too bad he wasn't a storyteller.

[Richard Marney · Rating 5 out of 5]

For arguably one of the greatest achievements of mankind, my only comment is I hope I can someday appreciate, truly appreciate the unfathomable genius and beauty of this work.

[Xander · Rating 3 out of 5]

The Principia (1687) was Isaac Newton's grand synthesis of (1) Copernicus' heliocentric theory, (2) Kepler's three planetary laws, (3) Galilei's study's of motion and forces and (4) Netwon's own mathematical analysis. It was more than this though; it was the first philosophical system of the world since Aristotle's philosophy (which had been used by christian theologians since the 12th century as the system of the world).

Newton writes this book in the style of Euclidean geometry: starting with axioms and then deducing, step by step, new truths. This, in combination with the complexity and Newton's notation of the mathematics used, makes the Principia almost impossible to read for modern day readers. Not that it was easier for contemporaries - it was only in the 18th century that this raw material was digested enough for third parties to write more accessible accounts of the new mechanics.

In essence, Newtons explains the motion of all the matter in the universe; he does this with three laws of motion and the (infamous) universal law of gravitation.

Netwon's three laws of motion:
1. All bodies remain at rest or move in uniform, rectilinear motion, unless acted upon by a net force.
2. The net force acting upon a body is proportional to the product of its mass and acceleration.
3. When a body is acted upon by another body, the net force of the one body on the second body is reciprocal to the net force of the second body on the one body (i.e. action = reaction).

Newton's law of universal gravitation:
1. The gravitational force between two bodies is proportionate to the product of both masses and inversely proportional to the square of the distance between the centres of both bodies.

With these four propositions in his hands, Newton is able to explain why apples fall from trees, why planets move in their orbits, why the oceans on Earth have tides and why comets have the strange orbits they have (and why they return after some amount of time). The universality, consistency and totality of this system was amazing; I think we moderns cannot truly understand the shift in thinking this has brought about.

I think it is good to mention that Newton clearly describes the assumptions (or axioms) that underlie his system. These axioms have become the cornerstones of modern day science:

1. No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena.
2. Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same.
3. Those qualities of bodies that cannot be intended and remitted [that is, qualities that cannot be increased or diminished] and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally.
4. In experimental philosophy, propositions gathered from phenomena by induction should be considered exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions.

In other words, we should give up Aristotle's vain attempt to discover truth by applying axiomatic-deductive systems - gone is the philosopher who can get to know Nature from his armchair. What we should do, according to Newton (and he bases this on his ancestors Bacon and Galilei) is make observations, use the method of induction to discover explanations and by synthesizing these explanations into complete theories, assume that all similar effects have similar causes, throughout the whole universe and all of time. This is (almost exactly) the modern method of doing science.

There are three important remarks to make on Newton's mechanics, as outlined in his principia.

The first is that with Newton, the notions of absolute, infinite time and space become necessary. This is because, if everything attracts everything else in a circumscribed universe, the universe would collapse in on itself. Infinite space leaves open the possiblity that every piece of matter is counterbalanced by (infinite) other pieces; therefore no 'Big Crunch'. Infinite time, also means no possiblity of a definite event of Creation; it is not strange that many theologians weren't happy with Newton's system - their conception of the Universe would demand a beginning of time when God created the world, as mentioned in Genesis.

The second remark is that Newton says in his Principia "hypotheses non fingo" - I don't feign hypotheses. He posits gravity as a force to explain the planetary orbits and the movements of matter on Earth. He doesn't know the mechanism by which gravity works or 'what gravity is', but that's not necessary for his theory. (This, by the way, is one of the reasons why later physicists would postulate ethers, because if gravity works instantaneously between two bodies, what is the medium through which it works?). This not using 'occult qualities' to explain natural phenomena, in effect, cuts religion from science - this would become as ground breaking as Newton's mechanics itself.

The third remark is that Newton's switch from deduction to induction would bring back the problem of induction into science, as already mentioned by Sextus Empiricus in the second/third century AD. If you use particular events (e.g. planetary orbits) to discover, by means of induction, universal truths (e.g. law of gravity), you will encounter a problem: there's no way to garantuee that the next observation will not falsify your theory. To prove your theory, you need to have access to all observations - past, present and future - and this is simply not possible. So absolutely proving inductive reasoning is impossible; this leaves room for doubt - therefore skepticism about scientific theories. It is a problem that has never been solved satisfactorily (Popper got close, but failed in the end; while Bayesian probability theory is just a logical rule - trash in leads to trash out).

Safe to say, this is a book that has been influential for centuries; in science, philosophy, religion, culture, literature, and what else. Newton's mechanics are still used by astronomers who work in the range of everyday motions and masses (anything approaching the enormous needs general relativity and anything approaching the sub-atomic world needs quantum mechanics). But this book is un-readable for contemporary people, it is too complex and too obscure for that. One can read the first part of the work to get a good insight, but additional information (i.e. interpretation) on the Principia is necessary.

[Chris Duval · Rating 5 out of 5]

The original book is one of the foundational books for modernity, expounding both mechanics and the calculus while explaining astronomy. (The little digression at the end into theology can be ignored.)
One can imagine an e-edition of this book where, as one reads the description of the ratio of this or that, the relevant lines on the diagram were highlighted. Even better, when areas are described by line segments belonging to the same line, the e-edition could add a side diagram with links to the original diagram.
That lacking, a print edition could make sure the diagrams are always available without flipping pages--not true for this one, and make sure the labels are always clear and mutually distinguishable.
Any edition would be improved with some supplemental materials. It is perhaps reasonable to expect familiarity with Euclid's rules of triangles, but a glossary entry for items like the 'latus rectum' of a conic section would be most helpful.
It would also be nice to explain verbal terms like 'sub-duplicate ratio' that moderns bury in algebraic notation. The help could be in an introduction, in a footnote upon first use or in a glossary. Their pretty clear from context, so this isn't a necessity.

[Dipesh · Rating 5 out of 5]

- an ingenious and energetic builder who's astonishingly brilliant at composing gorgeous monuments of the most intensely clever design. Sometimes these appear as great books like the Principia itself. Sometimes they appear in experiments. But we would be wrong to look for a single key which unlocks the whole mystery of Isaac Newton.

The Mathematical Principles of Natural Philosophy (1729) ... An English translation by Andrew Motte, based on the 1726 3rd edition of Philosophiae Naturalis Principia Mathematica.

Download Link: https://archive.org/download/newtonsp...
copyright status: NOT_IN_COPYRIGHT

[Scott · Rating 5 out of 5]

This book stands as one of the great monuments of science. If you can peer through the ponderous geometric proofs of Newton's physical principles, there is an elegance to his theories that transcends mere science and mathematics and touches the sublime! He actually formulated his theories using his newly-invented methods of Calculus, but few educated readers of his day understood the Calculus, so he proved his ideas using the methods of geometry (which all educated persons knew). We owe much of modern civilization to this book.

[Jeff Gabriel · Rating 3 out of 5]

I don't want to create a whole new shelf for this, but I didn't read it - I gave up after reading as far as I could. My giving up has nothing to do of course with this historical book of the highest importance. However, given that the subject is complex and the language arcane I am afraid I would need an interpreter for both concept and language.

I'll stick to learning my physics from more modern sources. I love reading original sources, and for the things I could grasp this book was very intriguing. I just wish my auto-didactic capabilities could reach this far.

I yield.

[David Darson · Rating 5 out of 5]

Do we dare review the greatest scientific work of the smartest man that ever lived? Can we as mere mortals even comprehend his genius? Touched by divinity, Newton writes, privately, the sum of all human knowledge that came before him and more. For the sum of all human knowledge before him, becomes smaller than what he would create, discover and pen on his own.

Newton shows that words like 'autodidact' and 'polymath' should be defined simply as 'Isaac Newton' in the dictionary. They were created to represent him.

No other man, before or after him, has achieved more for humanity.

[John · Rating 5 out of 5]

Hard going since Newton was so shy about using easy calculus when hard analytic geometry could do the job. Still, this is one of the most important books ever written and anyone with an interest in the history of science (or in seeing Newton draw up an epistemology at the start of book three to keep his critics from savaging him like they did with his Optics) should carve out a few months, get a bunch of paper, and go to.

[Nwe Mon · Rating 5 out of 5]

This book helps me a lot.

[Pranav Jeevan · Rating 5 out of 5]

To see how the great man thought...

[Laura L. Van Dam · Rating 5 out of 5]

Leído para una tesina que escribí para la facultad, está genial poder recurrir al material original, pero está escrito de una forma difícil de seguir, mediante argumentos geométricos.

[Stasia · Rating 5 out of 5]

Another one for future purchase. Ah, Isaac Newton, how you woo me with your writings on gravity...

[Ahmed Sobhy · Rating 5 out of 5]

Therefore, He was a real genius.
Sir Isaac Newton,
Chapeau!

[William Bies · Rating 5 out of 5]

Newton’s Principia, famed for its difficulty, must be oftener cited than actually studied. Indeed, anyone approaching the masterpiece through the old Andrew Motte translation of 1729, as revised by Florian Cajori in 1934 – long the sole edition available in English – as did this reviewer over a decade ago, would be apt to find it off-putting. The bulk of the Principia consists of solutions to abstruse problems, such as to find the ellipse which is tangent to five given lines. All this can strike one as almost perversely opaque. Newton does not define his terms and launches into complicated geometrical constructions without explaining what he is doing. He asserts without proof properties of the ellipse that can take a page of algebra to derive. The modern reader will frequently find it easier to rederive his results for himself rather than to try to puzzle out Newton’s arguments.

Clearly, there ought to be a better way. The great works are learned most profitably from the original, where one has the context to help one to appreciate what the author does and why – and this comment applies no less to natural science than it does to literature. Therefore, the authoritative new translation of Newton’s Principia undertaken by the renowned historians of science, I. Bernard Cohen and Anne Whitman, comes as long overdue. Compare with the Cajori translation? It is hard for one insufficiently versed in Latin to judge which might be truer to the original, but Cohen and Whitman describe in their introduction the great lengths they went to, as professional historians of science, to ensure that the terminology they use is precise, consistent and scientifically accurate. Thus, their version is probably more reliable, however much its literary quality may suffer in consequence. To a fluent speaker of the language, their English suppresses archaisms and strikes one as erring rather to the side of being too simple. But who cares about Newton as a prose stylist? His scientific ideas are what ought to matter.

Cohen also outfits this edition of the Principia with an extended 370-page guide that comments on all aspects of the work, from its origins and the circumstances leading up to its publication to a detailed commentary on individual hard-to-understand propositions. Perhaps most useful are the sections covering general aspects of the Principia (chapter three) and some fundamental concepts in it (chapter four). Here, Cohen’s unsurpassed erudition is put on display, deployed not so as to overawe the reader but to help him find his way from an exposure to the modern curriculum back into Newton’s mindset. Chapter five on axioms, or the laws of motion, is excellent as well. As for the rest of the commentary, it can be invaluable for rendering old-fashioned notation into terms a modern reader can recognize.

A guide to what one can get out of the text by skipping the pointless exercises, which appear to be designed mainly to exhibit the author’s ingenuity but which are of little relevance to the impact of Newton’s work. For they have been superseded by the calculus anyway, whereas the physics itself continues to have lasting import. Focus on the logical flow. Book I starts with definitions and axioms, or the laws of motion, and proceeds to demonstrate several elementary propositions, such as the parallelogram of forces, parabolic free fall and motion under centripetal forces. The later sections of Book I descend into a welter of particular but not intrinsically very interesting problems that illustrate the general theory.

Book II continues in the same vein by taking up a series of problems concerning motion in a resistant medium. Thus, what everyone rightly deems the supreme accomplishment of the Principia is postponed to Book III, entitled the System of the World. Here, Newton begins by stating the phenomena and reproduces tables of astronomical data. Then he proceeds to their theoretical interpretation in a series of propositions that apply the general theory to the case of celestial mechanics. Here is where the celebrated law of universal gravitation is inferred from the phenomena. Later sections in Book III become more problematic and demonstrate the sweeping scope of Newton’s classical mechanics by solving hitherto-perplexing riddles concerning the geographical variation of the gravitational force at the earth’s surface, the tides and the lunar theory, and comets.

What may we say about Newton with a view towards Kant? Newton appears to resemble more Kepler or Descartes, viz., to be primarily interested in explaining a variegated assemblage of puzzling phenomena in the physics of his day and not so much in system-building for its own sake. For instance, he declines to define space and time, seeing as, in his view, they are sufficiently well known. Kant tends to the opposite extreme, the system-builder par excellence.

The six-page general scholium tacked on to the second edition of 1713, however, deserves close scrutiny. For here, Newton (after discarding Descartes’ vortex theory) unveils the theological motivation behind his researches. At the time, unlike today, theology and natural science were allied subjects and the great figures of the early modern scientific revolution, all of whom were deeply religious (each in his own way), felt no need to conceal their theological views in print. Kepler is a born mystic who manages to overcome imprudent enthusiasm to attain to a mature contemplative attitude late in life (let us refer to our reviews of two of Kepler’s minor works, here and here). Descartes institutes his revolutionary turn to the subject and to rigorous method because he genuinely wants to put religious conviction on a sounder basis; as anyone who has read his meditations on first philosophy knows, what drives him is a vision of the divine perfection and it is here – only here – that he alights upon something that is objective and that must originate from outside his own mind, on which to anchor his certainty. One could go on and analyze the religious attitudes of other major seventeenth-century figures, though here is not the place. But what about Newton himself? It will be well to stick to the text of the general scholium and not to bring in what else one knows about his lifelong preoccupation with alchemy and biblical prophecy, for thus we stay with what Newton wants the public to understand about his religious views. To quote his own words:

This most elegant system of the sun, planets and comets could not have arisen without the design and dominion of an intelligent and powerful being….He rules all things, not as the world soul but as lord of all. And because of his dominion he is called Lord God Pantokrator [παντοκράτωρ]. For ‘god’ is a relative word and has reference to servants, and godhood is the lordship of God, not over his own body as is supposed by those for whom God is the world soul, but over servants. The supreme God is an eternal, infinite and absolutely perfect being; but a being, however perfect, without dominion is not the Lord God. For we do say my God, your God, the God of Israel, the God of Gods and Lord of Lords, but we do not say my eternal one, your eternal one, the eternal one of Israel, the eternal one of the gods; we do not say my infinite one, or my perfect one. These designations do not have reference to servants. The word ‘god’ is used far and wide to mean ‘lord’, but every lord is not a god. The lordship of a spiritual being constitutes a god, a true lordship constitutes a true god, a supreme lordship a supreme god, an imaginary lordship an imaginary god. And from true lordship it follows that the true God is loving, intelligent and powerful; from the other perfections, that he is supreme, or supremely perfect. He is eternal and infinite, omnipotent and omniscient, that is, he endures from eternity to eternity, and he is present from infinity to infinity; he rules all things, and he knows all things that happen or can happen. He is not eternity and infinity, but eternal and infinite; he is not duration and space, but he endures and is present. He endures always and is present everywhere, and by existing always and everywhere he constitutes duration and space. Since each and every particle of space is always, and each and every indivisible moment of duration is everywhere, certainly the maker and lord of all things will not be never or nowhere. [pp. 940-941]

We know him only by his properties and attributes and by the wisest and best construciton of things and their final causes, and we admire him because of his perfections; but we venerate and worship him because of his dominion….For all discourse about God is derived through a certain similitude from things human, which while not perfect is nevertheless a similitude of some kind. This concludes the discussion of God, and to treat of God from phenomena is certainly a part of natural philosophy. [pp. 942-943]

There is something impressive, if not exactly resplendent about Newton’s austere piety in which obedient submission to God’s dominion figures far more prominently than love, the very word Newton studiously avoids – perhaps it reflects the ideology of absolute state power popular in England after the Reformation and Henry VIII’s Act of Supremacy of 1534. For Newton does not, after all, believe in the God of scriptural revelation but inclines rather to the Arian heresy, then undergoing a revival among intellectuals – what is consistent with these passages from the general scholium. As the second passage quoted above indicates, for Newton dominion is not a mere theoretical term; he views it as providentially active in ensuring the continuance of the world, which repeatedly threatens to veer off course due to the accretion of unavoidable perturbations. Thus, in his view, dominion can be a proper subject of natural philosophy, in as much as God’s providential ordering of the world can be discerned from observation (at a high level). Who imposes the restriction that physics can be about only low-level instrumental readings and not proceed to higher-level inferences therefrom? In everyday life, one interacts all the time with other entities and persons who, strictly from sense perception represent high-order constructs by which we render our experienced world intelligible. Nobody is a solipsistic Machian instrumentalist in everyday life! Newton, therefore, regards it as being unproblematical to go one step further and to entertain the possibility of observable action by angelic spirits or divine providential intervention into the course of worldly affairs.

One wonders what Newton would make of the argument to design from fine-tuning of the parameters in the standard model and concordance cosmology – one could construe this as a modern-day enterprise very much in a Newtonian spirit! To close, let us commend Cohen and Whitman for their untiring effort in making the Principia much more accessible to the educated reader of today than it ever was before. For he is now enabled to approach Newton’s masterpiece as a living document that speaks to perennial concerns in the physics of our natural world rather than regard it as but a museum collection of curious Mathematical Tripos exercises, as do Motte and Cajori (who, tellingly, omit the vital general scholium).

[Kenneth Andreassen · Rating 5 out of 5]

What an incredible scientific genius Isaac Newton was. Even though most of what is written in this book, at least from a mathematical point of view, is hard for me to fully comprehend, the big picture was not. Newton writes about his conceptions and experiments and what can be concluded on the basis of them in an interesting way. In this book he formulates the laws of motion and universal gravitation. Newton uses his mathematical description of gravity to derive Kepler's laws of planetary motion, account for tides, the trajectories of comets, the precession of the equinoxes and other phenomena. Challenging to read due to the mathematical knowledge it demands to fully appreciate it, but Newton gets his message across in the best way possible (also for us non-geniuses). Mostly, I read the parts in between the mathematical calculations, and still was able to understand the essence of it and appreciate what Newton writes about. This book provided the foundation that triggered hundreds of years of development within physichs and astronomy. There is hardly any other piece of work within this field that can be said to be as important. Hence, this book is a cornerstone in the history of science, and deserves all the praise it can get.

[Hangci Du · Rating 5 out of 5]

I wrote in Chinese, very long, too lazy to translate~

总评：

当初想到读这本经典的缘起是什么呢？
是因为我读广义相对论的时候，意识到爱因斯坦所破的是很多传统观念的冗余，于是我尝试去读类似于《费恩曼物理学讲义》，然而却没有感觉，并且意识到——这种冗余已经很沉重，需要追根溯源，同时我还思考为什么物理学没有进行公理化，很多地方是漫漶不清的。于是开始阅读《自然哲学的数学原理》，追根溯源进行研究，并且我坚持认为运动学的本质应该是变分法，因此计划读完之后去读朗道的力学。同时发现，牛顿当初已经进行了公理化尝试，然而并没有归结到本质性问题（数学工具的不足）。在阅读到一半时，为了追求更本源，去读了亚里士多德的《物理学》，大失所望，才明白牛顿的正经祖宗并不是亚里士多德，而是阿基米德。
牛��的写书意识逻辑是非常优美清晰的，比伽利略的作品要好很多了。然而其很多观点是建立在伽利略的观点基础上（如惯性的观念等），是伽利略观点的体系化、规范化。虽然都伽利略的书会少趣味，但还是看一看伽利略的力学观点《关于两门新科学的谈话》吧，动力学概念的源头，还是要寻找到伽利略头上啊。牛顿是一个伟大的演绎者，而非概念的原创者~
在第一和第二编里，牛顿的“力”的概念，均是从作用的效果而非原因来看待的，包括向心力和阻滞力，这是牛顿提出为了解释运动的工具，而并不关注其如何产生以及机理本质是什么。（在第11章最后附注中有很好的说明）
因此，第一编牛顿关于“效果力 ”的概念为“向心力”，就是指指向点中心的吸引力，探讨不同轨迹下向心力的不同形式最终回到主体对圆锥曲线轨道的探讨上来；第二编，则引入新的“效果力”阻滞力，探讨引入不同形式阻滞力之后物体的运动状态。综上所述，提出这两个力的概念无不是为了研究运动的状态。注意：牛顿的“向心力”与我高中教育时候所灌输的“向心力”有很大的区别。牛顿的向心力的概念与“吸引力”是同一个层次，而我们的“向心力”的概念则更冗杂，所有指向中心的力皆源于此。


导读：

牛顿认为宇宙的优美来自于上帝。
哥白尼的天体运行论是用神学的语言书写的，伽利略认为自然的语言是数学，并且写了两部名著，伽利略死后十天，牛顿出生。
牛顿出世前面对的最大的宇宙体系是笛卡尔体系，其创造了一些描述性的模型，并且笛卡尔是首次将运动引入几何的人（这很牛逼啊，因为之前的力学实际上就是静力学；当然之后的牛顿把运动的几何学发展到巅峰了），然而其自然哲学多有舛误不合验证（星体旋转的涡旋假说）

牛顿的作品非常注重格式规范，模仿欧几里得。分为“定义”、“运动的公理或定律”、“引理”（数学工具）、“命题”（本书正题）。
牛顿的命题分为“定理”和“问题”，定理是得出的基础性的结论，而问题则是通过基础性的结论来解决实在的问题。或者说，“定理”是建立在自己假说之上的演绎，是纯数学的，而“问题”往往和物理世界联系。

牛顿通过圆周运动的规律，提出平方反比定律，这是牛顿宇宙论最重要的基石。进而提出万有引力定律。（从由天体运行的椭圆轨道，得到平方反比力的规律，并比不上什么（牛顿在第一编做完的事）；然而从平方反比定律推广到万有引力定律，这实际上是非常疯狂的想象力！非常非常宏大、优美的推广！而且最终发现其在数学上的完美符合现实）并能通过万有引力定律预言星体的质量。然而牛顿无法解释万有引力的成因。（实际上这就是他“猜”出来的理论规律，是一个通过不可观察的“力”概念建立的理论模型从而实现对于观察结果的解释。）

全书分为三编：
第一编几乎已经囊括了所有的内容。
第二编是第一编的应用（物体在阻滞介质中的运动、流体）
第三编是牛顿的宇宙论，解释天体运行。
牛顿认为以太是不存在的（颠覆从亚里士多德到笛卡尔，然而直到十九世纪末人们还拿着以太模型，可见革新之困难）

定义：
牛顿三定律，以及定义的去歧义的工作，是牛顿自己的思想吗？力的加速度定律应该不是其原创？还是伽利略早就有此观念，牛顿将其标准化以及进行演绎？需要阅读伽利略。真的是做笔记开始思考的时候才能明白自己要做啥呀

运动的公理或定律：
以上两部分提纲挈领，等读完第一编再写读后感。需要读完伽利略再来写读后感。这两部分是牛顿自然哲学很重要的核心，也是后来爱因斯坦等人着力修正的地方。


第一编 物体的运动

第一章 （全为引理，共11条引理，创造出数学工具：微积分）

第二章 向心力的确定 （命题1~10，引理12）
命题1~4均为“定理”，面积与时间成正比（开普勒第二定律），是与“存在指向点的向心力”等价的（并不依赖于力的形式）。将轨道问题转化为用力描述的问题。
命题5~10为“问题”，根据轨道的不同形态，导出向心力的不同解。在命题10里提出指向椭圆中心对应的力的形式（不是焦点而是中心，结论并非平方反比定律，而是与距离成正比）（此时牛顿仅是根据轨道的不同形式提出力的形式，都是数学，处于假说演绎阶段，没有验证）
最后的附注指出，将椭圆的中心移动到无穷远处，就得到抛物线！
这一章明确了利用向心力的理论工具入手。

第三章 物体在偏心的椭圆轨道上的运动 （命题11~17，引理13、14）
命题11~13均为“问题”，依旧延续第二章的解决思路，但是指出：轨迹为椭圆、双曲线、抛物线并以焦点为力心时候，力的形式为平方反比定律。（以开普勒第一定律为条件，得出力的形式）
命题14~16为“定理”，得出了环绕中心按照平方反比向心力运行的天体体系之间的关系，进而得出开普勒第三定律的形式。
命题17为“问题”，解决了轨道的预言问题，已知平方反比力和速度，预言其未来轨道。
到第三章为止，牛顿已经可以将开普勒的规律完全按照伽利略的思路来解释，纳入一个优美的体系。

第四、五、六章 （命题18~31，引理15~28）
已知焦点求椭圆、抛物线和双曲线轨道；
焦点未知时怎样求轨道；
焦点未知时怎样求轨道。
这三章所有命题均为“问题”，都是探讨的圆锥曲线的几何问题。

第七章 物体的直线上升或下降 （命题32~39）
（很令人惊奇，平方反比吸引力下求给定时间内的下落距离，牛顿是用面积法则来做的！）
命题32是“问题”
命题33~35、38是“定理”
命题36~39是“问题”
这一章里的探讨离开静态的“轨道”的范畴，进而开始研究时间、运动、距离的问题。圆锥曲线上的运动问题。

第八章 受任意向心力作用的物体环绕轨道的确定 （命题40~42）
这一章3个命题，分为一个定理两个问题。内容如题目所示，任意向心力的轨道，可以用计算机模拟，不要太舒服（但是是非分析解）。

第九章 沿运动轨道的物体运动；回归点运动 （命题43~45）
这一章的意义在于，探讨限定轨道的运动，也就是一维的运动。

第十章 物体在给定表面上的运动；物体的摆动运动 （命题46~56）
探讨限定曲面的，也就是二维的运动。同时探讨了物体摆动的规律（伽利略时钟）
这里提到，（命题47）如果中心力的形式是正比于到中心的距离，那么轨道也会是椭圆；如果是直线运动，那么在相同时间里完成各自的周期（这不就是谐振运动吗？）我在这里惊愕了一下，想不应该是平方反比吗？然后意识到这是中心而非焦点。高中物理以圆周运动而非椭圆运动切入是非常不良的，因为中心与焦点重合。而牛顿所探讨的概念，实际是泛函层面的了，按照圆周运动研究，很容易混淆两个函数。

第十一章 受向心力作用物体的相互吸引运动 （命题57~69）
这一章开始，牛顿开始研究二体运动，而非固定吸引中心，这离现实世界的情形更加接近了（在命题66推广到三体）。另外牛顿也指出：他目前的研究都是纯数学的，将物理放到一边。

第十二章 球体的吸引力 （命题70~84，引理29）
如在牛顿第十一章最后附注指出，从这一章开始，牛顿不再以“力的效果”（运动产生的可观察效果）来进行考虑，转而开始考虑“力的成因”（平方反比力所具有的性质）。这里，牛顿将平方反比力的概念替换掉观察到的椭圆运动的概念，用“力”来刻画运动给了其强大的研究武器。这一章，牛顿不再提“向心力”而是“吸引力”，其将力的概念从天体间的吸引推广到每一个无限小质量部分了。
命题70~80为定理，将平方反比力模型应用到球体的每个部分，并分别考察球体内、球体外的作用效果（得出的结论惊人的优美，球壳内不受力球壳外等效于受质点力等等发）

第十三章 非球体的吸引力 （命题85~93）

第十四章 受指向击打物体各部分的向心力推动的极小物体的运动 （命题94~98）
这里通过力的观念的除了类似于光的折射规律。这一章的理念不是很明白，题目的命名也令人费解。

第二编 物体（在阻滞介质中）的运动
引入了另一种“效果力”——阻滞力，并研究不同形式阻滞力对运动的影响。

第一章 受与速度成正比的阻力作用的物体运动

第二章 受正比于速度平方的阻力作用的物体运动

第三章 物体受部分正比于速度部分正比于速度平方的阻力的运动

第四章 物体在阻滞介质中的圆运动

第五章 流体密度和压力；流体静力学 （命题19~23，流体定义）
这一章给出了流体的定义，并研究了流体静力的观念（没有阻滞力的），给之后进一步的讨论奠定理论基础。（这一章内容是站在哪位巨人的肩膀上呢？静力学的基础是阿基米德，流体静力学也是吗？）

第六章 摆体的运动与阻力
在阻滞空力作用下单摆的运动规律。
牛顿对于摆体的研究似乎非常在意，疑惑：这是很重要的模型吗？

第七章 流体的运动，及其对抛体的阻力
这一章开始研究流体内的阻力作用，并有实验验证

第八章 通过流体传播的运动
牛顿将流体看作粒子组成。看来牛顿已经有默认的“微粒组成万物”的观念，或者说原子的观念了。
这一章的主体，是研究流体上的波。

第九章 流体的圆运动
主要是研究涡旋，同时给笛卡尔学说知名的打击

第三编 宇宙体系（使用数学的论述）
按照这一编的前言，其数学基础是：定义、公理部分加上第一编的前三章。牛顿开始用其创造出的理论工具，按照最简单的原则，有序地解释这个宇宙了！

哲学中的推理规则 （牛顿四规则）：
这是牛顿的科学哲学：
规则一，简单性（自然倾向简单，这就是奥卡姆剃刀！）
规则二，普适规律（寻找能普遍解决的规律）
规律三，实验规律视作普适规律（默认实验的可重复性）
规律四，对经验规律可靠性的坚持（我想，其实牛顿对于其平方反比的万有引力定律也是不自信的，因为他并不明白其机制是什么）

现象：
牛顿以开普勒定律为底本，加上木星土星卫星以及月球的运行规律，描述观察到的天文现象

命题 （共33条命题）
感觉牛顿虽然是按照标准的体例写的，但其实有点乱。定理、问题分得不太清，层次递进的关系也不明确。而其万有引力定律也是在一个定理中（命题7、8）提出，这实际上是极其重大的推广。而牛顿当初却似乎把这重要的原理，隐藏在了定理与实际问题之中，从这个程度而言，牛顿的作品很伟大，但没有达到“艺术品”高度，当然，这是指他写的书（和欧几里得相比）。但其思想经过归纳整理，是当之无愧的艺术平。

月球交会点的运动
这是处理具体的问题了，实际上上一部分已经处理了好多了


总释
总释真的非常精彩！
总释里牛顿先是打击了用涡旋解释行星运动的模型。
然后提出了波义耳的真空实验，验证其阻滞观念（波义耳也是牛顿站立其上的一个“巨人”，有必要去读一读这位化学家，我发现，无论物理、生物、化学，都是自然哲学的不同剖面！）。
然后牛顿赞美了伟大的上帝，牛顿相信全知全能的上帝！并且牛顿表示，要想认识上帝，没有比自然哲学更好的手段了！（是的，我们学习物理，实际上才是一群真正的朝圣者）
在倒数第二段，牛顿很明确理解自己学说的局限性：引力理论是一个对运动的解释，而牛顿对于其原理一无所知。（牛顿说的“我不构造假说”，意思是我不会提出假设的东西来探讨引力的原因是什么，我以前的理解都断章取义了！可见读原典可以消除断章取义！）在这里牛顿不得不化为实用主义者，他表示“对于我们来说，能知道引力确实存在着，并按我们所揭示的规律起作用，并能有效地说明天体和海洋的一切运动，即已足够了”。
因此，从这个层面而言，牛顿的“引力理论”，他自己也意识到不过是“经验定律”而算不上是引力理论。爱因斯坦的广义相对论的引力场方程 ，并不是颠覆牛顿的学说，实际上是补上了一块拼图。爱因斯坦用时空弯曲的概念，通过对运动更加深入的探讨，将引力的本质解释提出来。爱因斯坦不是牛顿的颠覆，而是牛顿的补完。由于是本质解释的优越性，其精度定然是会超过牛顿定律这是毋庸置疑的（牛顿四规则里实际上已经预言到了），并且即便是广义相对论没有实现任何对牛顿定律的修正，它也是伟大的，因为它实现了更本质的统一，将这个“力”的概念消除了，划归为了自然运动。
在最后一段里，牛顿意识到自己虽然解释了天空和大海，但自己依旧面对着真理的海洋。要解释世界，还是很远的路途。牛顿是人类历史的一个里程碑，功在万代千秋。