Isaac Newton : Principia

translated and annotated by
Ian Bruce

            This is a new translation of the Principia. The Principia in the 3rd edition was translated in 1729 by Andrew Motte, and his is the translation usually referred to; however, most helpful as it is as a contemporary source, the Latin sentences have been largely paraphrased, which means that you are not reading what Newton said, but rather what Motte thought he said, and the two things are not always exactly the same; for example, it is due to Motte that we have Newton's first law in the form we know it, which is not exactly what Newton intended. Note especially in Motte's translation that sentences in italics have been added by Motte himself trying to make clear what Newton  is saying. Occasional sentences in addition have just been mistranslated; this error in translation can be held against the two modern translators, Cajori and Cohen, and rather than getting closer to what Newton actually said, or more precisely what Cotes and Bentley made out that he wanted to say, they have on numerous occasions strayed away from the original text even further; these comments apply mostly to the introductory material, which is mainly non mathematical in nature. The Principia is just as much a book of problems as it is a book of solutions to problems; Newton has solved a great many problems that he posts as Corollaries, but leaves for his readers ; lemmas are often new results; a thorough knowledge of the geometry of conic sections is required to tackle this work. Various notes have been added to help you do this.  One gradually becomes aware that differential and integral calculus form the backbone of this work, but the use of these tools has been carefully concealed, or at least not being given pride of place; there are however, occasions when a proposition can only be demonstrated by calculus, such as Prop. 39 of Section 7 in Book I ; in fact, Newton kept scurrying backwards and forwards from the brave new world of differential and integral calculus to the world of Euclidian geometry, with which he was far more familiar, and this may account for some of the apparent lack of detail in proofs, which are often given in the form of conclusions, with no mathematics. One should not expect everything written down by Newton, in the original Latin, or in some translation, to be understood without coming across items that require some or considerable effort ; thus, even matters that were mainly incorrect, such as the flow of water from an aperture in a tank, need to be looked at more carefully; some enlightenment can be derived from Leseur & Jacquier, but of course they themselves did not know the full answers to some of the questions posed by Newton. What one can say with some definiteness is, that Newton's work was the common starting point for numerous subsequent investigations by others, and it is this aspect perhaps that makes Newton's Principia still a useful book to read, at least superficially, at least as a starting point for some subsequent discussion, or for an in depth historical investigation. The gravitational theory cannot really be faulted, as it represents the classical view of gravitation, still a valid approximation to what we now believe to be reality ; as such it is an enduring monument to Newton's genius.

Some books and websites of value  are :

1. Whiteside's Mathematical Papers of Isaac Newton, especially Volume VI. CUP.

2 Routh and Broughams's Analytical View of Sir Isaac Newton's Principia (1855); available as a pdf file from the Open Library website, and also as a books on demand book, but be careful with these, as pages may be missing or not scanned correctly.

3. Le Seur and Jacquier edition of the Principia, which expands greatly on the original text, in Latin of course. Available from the e-rara website.

4. Cohen's Principia. California ; and his : Introduction to Newton's Principia CUP. I have not looked at his translation in depth, apart from seeing that we agree on difficult points; not wishing to be influenced in any way; however, before reading any translation, it will not be a bad thing to read Cohen's chapter on the Structures of Books 1, 2, and 3. A lot of the relevant history of the Principia is given here, and any of the various references quoted can be pursued as desired.

5. Densmore and Donahue. Newton's Principia. The Central Argument. Green Lion. Deals with some aspects in depth.

6. Isaac Newton 1642 – 1727 . A memorial volume published for the Mathematical Association, edited by W. J. Greenstreet. Bell & Sons.

7. S. Chandrasekhar. Newton's Principia for the Common Reader. Oxford. I think this is a must read book for anyone trying to work out Newton's Principia. What Newton did can be recast in a modern light, which Chandrasekhar has done ; I know there are uncorrected errors in the latter parts of the text, and that the author ignored the labour of others over the intervening centuries. Yet it is not enough to present a translation of the Principia, it needs to be explained in modern terms for the full wonder to become apparent, and Chandrasekhar accomplishes this task, at least in the early sections.

8. W.W. Rouse Ball. An Essay on Newton's Principia. C.U.P. (1892). Interesting in that a number of details regarding the origins of Newton' s Principia are given here in this small book.

9. The Correspondence of Isaac Newton, in seven volumes.  This is a very useful set of books, as some of the more obscure points in the Principia are dealt with both by Newton in answering queries, and by the several editors asking for his advice, and in turn trying to give him advice. This is the best place in my opinion, to read about Newton the man, as here he emerges as a whole, rather than having just one aspect of his work and nature investigated by a biographer, who invariably present a narrow view, depending on their own interests and expertise. Newton was really the equivalent of a number of people : he is remembered mostly as a mathematician and natural philosopher of extraordinary abilities ; but there were other aspects to his being – his work in chemistry, his experimental work, his investigations into the accuracy of the books of the Bible, his later life at the Mint, etc. One should note however, that only the first three volumes were edited by a mathematician (Turnbull), while the last three, depending on your point of view, are rather disappointing, as there are hardly any mathematical notes, and goodness knows, there could have been many, while Cotes and later Pemberton took on Newton to try to get explanations of some of his more obscure proofs from him; as it was, the editors of the Letters contented themselves with the old nit-picking letters of Johan Bernoulli & Leibniz on who invented the calculus....

10. The Principes Mathematiques de la Philosophie Naturelle, the Principia translated into French by Madame La Marquise du Chastellet. This is a delightful translation, and more down to earth than Motte's version, with less extra waffle and more common sense extra words to help it along. It was produced in the 1740's by this lady, the amoureuse of Voltaire, when there was still disbelief that the earth was squashed a little at the poles, rather than lemon shaped.....

11. Sir Isaac Newton's Mathematische Principien der Naturlehre : the German translation by Prof. Wolfers.


Prefaces to the three editions by Newton and Cotes's editorial material :

Link to Prefaces by clicking here.

Definitions and Axioms :

Link to Definitions and Axioms by clicking here.


Section 1 : : Link to Section 1 by clicking here.  

Concerning the method of first and last ratios, with the aid of which the following are demonstrated.

     The mathematical foundations of the work are set out here in a series of Scholia, which are of considerable interest, as the are given in terms of what Newton calls the first and last ratios of sums and ratios, being a geometric approach to the limiting processes involved in integration and differentiation. A short summary by the translator is given at the end which may be helpful.

Section 2 : : Link to Section 2 by clicking here.   

On the finding of centripetal forces.

     A series of propositions leads Newton to a geometrical construction for the centripetal force acting on a body in orbit about some immoveable center of force ; this derivation considers two nearby points on the orbit, and a tangent is drawn from the first; from the similar triangles drawn it is shown that the versed sine of the arc subtended and vanishing is proportional to the deviation of the body from rectilinear motion along the tangent, and the area under the arc is proportional to the time taken to travel between the two points; this area is found as that of a triangle, which on squaring and dividing provides the centripetal acceleration.  A note inserted by the translator shows how this procedure is related to the modern formula for centripetal acceleration. Newton justifies his formula by examining several examples, ending with motion in an ellipse about the centre, for which the force is proportional to the distance. Some additions to the notes have been made.

Section 3 : Link to Section 3 by clicking here.   

Concerning the motion of bodies in eccentric conic sections.

     The serious business of applying dynamics to bodies orbiting on conic sections gets under way in this section. A number of results concerning conic sections are referred to without proof, which I have tried to remedy; for example, the Rectangle Theorem is of great assistance. Motion in elliptical, hyperbolic, and parabolic orbits are attended to here. Notes are added to the main theorems by this translator, but much is left for the reader to explore. Some additions to the notes have been made.

Section 4 : Link to Section 4 by clicking here.   


Concerning the finding of elliptical, parabolic and hyperbolic orbits from a given focus.


This section is geometrical in content, and sets out a number of ways of describing ellipses, parabolas, and hyperbolas according to various given boundary conditions where a focus is given.

Section 5 : Link to Section 5 by clicking here.   

Finding the orbits where neither focus is given.

This section is an extended version of a previously unpublished work of Newton on finding the locus of conic sections for which the focus is unknown; clearly of great use in the determination of the orbits of planets. As is customary with Newton's work, the establishment of the Lemmas and Propositions is often sketched, leaving much for the readers to uncover for themselves. A knowledge of the Conics of Apollonius and the work of Pappus is of help. I have found Whiteside's Volume VI of his Mathematical Papers of Isaac Newton to be of great assistance, and some of the more involved ideas I have left as references to this work.

Section 6 : Link to Section 6 by clicking here.   

Concerning the finding of motions in given orbits.

     This section sets out to establish the point in the orbit that the body will reach in a given time, be it parabolic, elliptic, or hyperbolic. I have included some of notes from Le Seur and Jacquier here; Whiteside, as always, is very helpful. The case of the ellipse includes a proposition on the non-rectifiable nature of the ellipse : Kepler's problem. Thus a mechanical method and iterative schemes are investigated. Some additions to the notes have been made.

Section 7 : Link to Section 7 by clicking here.   

Concerning the rectilinear ascent and descent of bodies.

     There is a great deal of material in this chapter, which sets out to prove what is now the most trivial of tasks, the acceleration of a body under the inverse square law in a straight line. Newton, however, tackles the problem as the limiting case of an orbit, and so brings to bear a number of results from previous chapters ; essentially he solves the Kepler problem in a straight line. The last proposition shows the grasp Newton had about these matters, and uses the calculus to solve in principle the general case.

Section 8 :  Link to Section 8 by clicking here.  

Concerning the finding of orbits in which rotating bodies are acted on by any centripetal forces.

     This chapter is a short continuation of the matter contained at the end of the last section. Here the orbital trajectory is completely solved in principle, essentially by two integrations, so that the velocity, and position of the body in the orbit is demonstrated in principle at some time. All the sections to date have dealt with fixed centres of forces.

Section 9 :  Link to Section 9 by clicking here.  

Concerning the motion of bodies in moving orbits, and from that the motion of the apses.

     This is a topic which is still of some considerable interest to astronomers and historians of astronomy. Here Newton introduces what has come to be called Newton's rotating ellipse proposition ; in which a rotating ellipse is described by a body, for which the central force is found to include a term in proportion to the cube of the distance from the focus ; this rotating ellipse is compared with a similar stationary ellipse in which another similar body rotates about the same centre of force. Careful analysis leads to expressions for the extra inverse cube force, depending on the number of upper and lower apses found in the moving ellipse.

Section 10 :  Link to Section 10 by clicking here.  

Concerning the motion of bodies given surfaces, and from that the repeating motions of string pendulums.

     Here Newton sets to work in proving and extending the work of Huygens on cycloids, extending the isochronous pendulum to include epicycloids. Newton investigates motions derived from such pendulums where the acceleration along the arc is proportional to the decrease in the arc length and vice versa; an interesting idea appears in the energy integral as such curves store potential energy in the string wrapped on the curve

Section 11 :  Link to Section 11 by clicking here.  

Concerning the motion of bodies with centripetal forces mutually attracting each other.

     This is the major introduction to Newton's Lunar Theory. Two body inverse square law problems are discussed, followed by Prop. 66, which contains 22 corollaries and which presents here in a qualitative manner mainly the perturbations of a remote body on a two body system. An introduction to the theory of the tides falls into place; This is not a chapter for the faint hearted; I have added some of the extensive notes prepared by Leseur and Jacquier in their edition.  However, the more difficult material cannot be understood by reading this section alone, and perhaps the best thing to do in that case is to read what Chandrasekhar has explained in great depth; clearly there are a number of issues on which Newton was not entirely happy, and material was withheld until such times as he understood matters clearly. This of course is one of the most difficult problems to be found in dynamics.

Section 12 :  Link to Section 12 by clicking here.


Concerning the attractive forces of spherical bodies.

     This is a most satisfactory chapter, in which Newton sets out various  schemes of increasingly mathematical sophistication for calculating the force on a corpuscle, initially outside a spherical gravitating mass.  This tour de force eventually is applied to general forces acting between particles. Newton also introduces the inversion formula, later considered in solving electrostatics problems by William Thompson some two centuries later, here applied to gravitating masses. Unfortunately, Chandrasekhar did not unraveled Newton's Latin instructing on how to dissect a sphere into constant force shells for an external corpuscle, which is now presented in this translation, so that his book is of less value here than it would have been otherwise.

Section 13 :  Link to Section 13 by clicking here.

Concerning the attractive forces of non–spherical bodies.

     As the title suggests, Newton turns his attention to the attraction of non-spherical bodies under various laws of force. This includes surfaces of revolution such as discs, cylinders, infinite solids with a plane face, etc. He also shows how the attraction of a spheroid can be resolved to a certain extent, ending with the area of a segment of an ellipse; I have included the extra notes supplied by Leseur and Jacquier to show just how much work has to be done to produce this particular result quoted by Newton. Much of the present chapter finds a similar place in works on electrostatics.

Section 14 :  Link to Section 14 by clicking here.  


Concerning the motion of the smallest bodies, which may be set in motion by attracting centripetal forces towards the individual parts of some great body.

     Finally in Book I, Newton considers essentially optical problems, from the point of view of small particles attracted by some unknown constant force in a medium. There is a close relation to Fermat's principle in the final proposition, which shows how to construct an aspheric lens along the lines of Descartes ovals.



Section 1  : Link to Section 1 by clicking here.  

Concerning the motion of bodies being resisted in the ratio of the velocities.

     This section sets out Newton's method for determining the trajectory of bodies under various circumstances with the resistance of the medium depending on the velocity; he considers simple retarded motion without gravity, rising and falling with gravity separately, as well as the motion of a projectile. A large amount of extra material that has subsequently become forgotten has been inserted in another file from the Leseur and Jacquier edition, in in effort to understand the unusual approach adopted by Newton – in which he effectively performs the integration of the second order differential equation for the unbalanced acceleration with resistance and gravity present, in two stages, and makes use of the area under the rectangular hyperbola divided up into the equal time intervals of an arithmetic progression, with the velocities of the body forming a geometric progression along an asymptote.

Section 1 : Link to Section 1 extra notes by clicking here.

I have removed some of the L & J clutter in Section I into a separate file, which may make reading Section I a little easier; there is some important information in the extra notes that should be looked at, certainly the understanding of the exponential or logarithmic function at the time. Section II is too convolved to allow this separation to be done without rewriting the whole section.

Section 2  : Link to Section 2 by clicking here.

Concerning the motion of bodies resisted in the square ratio of the velocities.

     There is a wealth of material here, as Newton extends his analysis to motion with resistance proportional to the square of the distance; I have included much of the material presented by Leseur & Jacquier, which gives a contemporary understanding of the work, rather than just quoting the modern way of doing things. There is a link here to the fall of spheres with resistance proportional to the velocity squared, discussed in Section 7. Newton did not consider what would have been the main result : the trajectory of a projectile under this kind of resistance. Instead, he found the kind of resistance necessary for the body to traverse several well-known curves, and employed himself fitting hyperbolas to various trajectories. One might concede that the analytical method of Leibniz was better suited for such investigations. However, Lemma II shows the origins of Newton's differential calculus, and that should be of interest.

Section 3  : Link to Section 3 by clicking here.  

Concerning the motion of bodies in which it is resisted partially in the ratio of the velocity and partially as the square ratio of the same.

     This is a rather short section in which Newton considers resistance with contributions both from the velocity and the velocity squared, as well as due to the constant acceleration of gravity, that he finally allows to be changed by a constant 'sticky' force present in the fluid. A schematic manner of integration is introduced, relating the constant increase of the area of a circular sector or rectangular hyperbolic sector with time to the length of the side of a triangle representing the velocity and the square of the velocity, the first of which is multiplied by a constant, and by this geometric means, for various relative sized of the acceleration of gravity to the resistive forces, the underlying differential equation for the time in terms of the velocity can be solved, which is not of course presented in any shape or form; the further differential equation relating the forces to the distance gone is then found schematically from the same setup, now using the customary rectangular hyperbola to get an arithmetic progression for uniform increases in the distance. Newton presents some calculus in arriving at this final result. I have not included the material of Leseur & Jacquier in this section, but rather gone for the more modern explanation afforded by Routh and Brougham. I have a feeling that the former authors at least in this case, though they fill in the mechanical details of calculations, do not always get to the heart of the matter.

Section 4  : Link to Section 4 by clicking here.  

            Concerning the circular motion of bodies in resisting media.

     Now Newton returns to the Archimedes' spiral with resistance added, and is able to produce the main results for a body attracted towards a centre in a resisting medium, following a curve of this form, for various kinds of force laws and laws governing the density of the medium. Brougham & Routh provide us with an analytical demonstration that complements Newton's original writing and the L. & S. treatment.

Section 5  : Link to Section 5 by clicking here.

Concerning the hydrostatic density and compression of fluids.

     Newton moves on to his consideration of fluids, and relates the pressure, density and force of gravity under various conditions; these are mainly theoretical models, as the equation of state of fluids was of course unknown at the time; various notes by L. & J. have been added to aid in one's understanding of Newton's calculations, which rely on geometric and arithmetic progressions of the variables making use of the area under a rectangular hyperbola.

Section 6  : Link to Section 6 by clicking here.

Concerning the motion and resistance of simple pendulums.

     A lot of work, both analytical and experimental, have gone into producing this section by Newton, in which he demonstrates his immense skills in taking ideas from humble beginnings to respectable enduring physics. In this case he examines the frictional motion on a body moving in a fluid by using the decaying motion of a pendulum ; his results come as no surprise to us:  the resistance on the pendulum bob depends on the velocity squared and on the density of the medium. However, excellent physics; a joy to read; here we have added some material from Brougham & Routh to make the analysis more transparent to the modern reader.

Section 7  : Link to Section 7 by clicking here.

Concerning the motion of fluids & the resistance of projectiles.

     This is one of the longest expositions so far in the book, in which Newton tackles the motion of fluids with resistance, especially the celebrated problem of water flowing out from a hole in the bottom of a cylinder, for which Newton's  analysis is now considered inadequate, or just wrong, as he made unphysical assumptions about the water in the vessel ; this however leads on to an analysis of the frictional forces on projectiles, and especially on pendulums and falling bodies. I have finally got round to sorting out most of the mathematics here in the form of added notes; you need to understand the work covered in Section 2 to appreciate the experiments involving falling spheres in air, with resistance proportional to the speed squared.

Section 8  : Link to Section 8 by clicking here.

Concerning motion propagated by fluids.


This is a work of pure genius or quintessential Newton, as he carves out what amounts to a second order partial differential equation for the differential pressure force exerted on a displaced element of air executing simple harmonic motion, according to Boyle's Law.  As usual, Newton hides his analytical techniques, and presents the work in terms of Euclidean geometry in a sparse manner. Initially,  he considers water waves in a canal as a way of introducing wave motion. The theory of sound wave transmission has of course been refined and elaborated on to give the present day text book treatment with which the reader may well be familiar, but this work is more or less where it all started.

The original text of 'Newton's Theory of Sound. ' can be found here: if you wish

Link to Newton on Sound by clicking here.

Section 9  : Link to Section 9 by clicking here.

Concerning the circular motion of fluids.

     In this short section Newton considers vortex motion in fluids according to mechanics; he then shows that the vortex nature of planetary motion is completely at odds with the observed phenomena. This is the final section in Book II.






Section 1  : Link to Section 1 by clicking here.

      This book is not divided into sections like the previous two books, but it has been useful to retain this scheme here . Section I thus contains the first 21 propositions of the third book; however, there is considerable difficulty in providing adequate notes for some of the material discussed, such as the figure of the earth, as it has not been performed by later commentators in the manner that Newton had originally performed the calculations himself; at times he gives few clues as to what he actually did, just quotes results. Nevertheless, the material is of considerable interest, and ties in closely with the work of later investigators. Chandrasekhar provides a very useful commentary for most of the material in this Book.

Section 2  : Link to Section 2 by clicking here.  

      This section starts off considering the production of the tides in a non-technical manner, and then quickly gets immersed in the complex details of Newton's Lunar Theory; this he tells us, was done to establish beyond doubt his theory of gravitation. The work of Chandrasekhar is indispensable here, and in fact, it seems to be the only valid modern large scale connection with Newton's work on the moon. Thus, anyone interested in pursuing this part of the work needs to read a translation together with this modern version worked out by Chandrasekhar, or else delve into Laplace, etc. ; thus, I gave up writing extra notes, as this is the place to go. Newton's work read here rather like the articles that used to appear in the Scientific American; you are told a great deal, but you must take it on trust as being true, as only a fraction is made intelligible mathematically: in Newton's case the mathematics  just is not there, and with the long passage of time, there is no way to find out exactly how Newton came upon his results. He seems to have left little trace of his calculations for the Principia for posterity; and it appears that Halley, Cotes and Pemberton, his editors for the three editions, were left in the dark, more or less.

Section 3  : Link to Section 3 by clicking here.  

      This section concludes the production of the tides in a technical manner; there are a number things in this part of the work that have been criticized, and Westfall even went as far as writing a paper on Newton's Fudge Factors; though that seems excessive, as the lunar mass could not be evaluated adequately at the time, a source of a lot of the difficulties. This was also Newton's excursion into the mechanics of bodies rather than particles, and one should be more impressed than critical about what was achieved, considering the elementary nature of the mathematical tools available at the time. A. P. French has given an elementary derivation of the precession of the equinoxes, and Chandrasekhar has caused me some trouble, as he retranslated Lemma I to suit his own purposes.

Section 4  : Link to Section 4 by clicking here.  

      This section is concerned with the experimental evidence available at the time for comets, which is quite extensive, and Newton's masterful analysis of the orbits so carefully observed. It is a tour de force of some magnitude, and one which Newton was clearly very proud. The relevant chapters in Chandrasekhar's book are essential for an understanding of what is going on theoretically. Newton adopts a sort of 'dog in the manger' approach : he knows what he has done, and he is most willing to give you the results, but he is not going to tell you exactly how he worked out all the mathematics. The work ends with the Grand Scholium, rather religious in parts, but interesting. Finally, Newton points to the future of electro-magnetism.

      What more can I say ?  You need to read the work for yourself to even begin to understand the magnitude of Newton's undertaking, and the huge leap it provided in man's understanding of the natural world, and of man's place in the universe.

      Modern readers of the work may wonder that they have not come across the famous formula " F= ma" for Newton's Second Law, and of course it just is not present there, at least not in this form. The first historical appearance of a similar formula is due to Jacob Hermann in his Phoronomia of 1716 :

It comes from p.50 of Book I, On the Forces and Motions of Bodies, which we render from Latin as follows :


§130. Every solicitation [i.e. force or agent] acting uniformly is equivalent to the motion generated, applicable to the time [i.e. in proportion to], in which the same motion is produced.

     The mass of the body to be moved may be called M, the speed to be acquired V, and the motion to be generated M.V, the time in which it must be produced shall be T ; the force to generate the motion shall be G acting uniformly in this time, and it is required to show that G = M.V:T. Since the force (§9), in as much as no motion may produced from a dead force [1], unless it were continued for a certain time in the body, or repeated, now may be considered to be acting uniformly, thus so that in equal times equal quantities of motion may be produced in the body, and thus the motions produced shall be as the times in which they are generated, that which will produce a motion M.V in the time T shall be equivalent to the force G, from which since G = M.V:T multiplied by T will produce M.V, it follows that G = M.V:T.


1. A reference to the vis viva (§8.) or living force notion of the time, said to be the kind of force required for motion to be increased ; other forces which do not result in a change in motion, or state of rest, thus could be called dead forces, or vis mortua (§9), such as action reaction pairs.



Ian Bruce. 26th Feb., 2012 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational use.