Mathematical Principles
Natural Philosophy

Sir Isaac Newton

Translated into English by Andrew Motte

Text and images from Newton's Principia:
The Mathematical Principles of Natural Philosophy;
to which is added,
Newton's System of the World / translated by Andrew Motte
[first American edition; New York: Daniel Adee, c1846]
and based on the transcription done by Wikisource,
with images for mathematical expressions replaced by text.

This web edition published by
Originally published by eBooks@Adelaide.


Table of Contents

  1. Title Page
  2. The Author's Preface
  3. Chapter 1: Definitions
  4. Chapter 2: Axioms, or Laws of Motion
  5. Book 1: Of the Motion of Bodies
    1. Book 1.1: Of the method of first and last ratios of quantities.
    2. Book 1.2: Of the Invention of Centripetal Forces.
    3. Book 1.3: Of the motion of bodies in eccentric conic sections.
    4. Book 1.4: Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given.
    5. Book 1.5: How the orbits are to be found when neither focus is given.
    6. Book 1.6: How the motions are to be found in given orbits.
    7. Book 1.7: Concerning the rectilinear ascent and descent of bodies.
    8. Book 1.8: Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force.
    9. Book 1.9: Of the motion of bodies in moveable orbits; and of the motion of the apsides.
    10. Book 1.10: Of the motion of bodies in given superficies, and of the reciprocal motion of funependulous bodies.
    11. Book 1.11 Of the motions of bodies tending to each other with centripetal forces.
    12. Book 1.12: Of the attractive forces of sphaerical bodies.
    13. Book 1.13: Of the attractive forces of bodies which are not of a sphaerical figure.
    14. Book 1.14: Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body.
  6. Book 2: Of the Motion of Bodies
    1. Book 2.1: Of the motion of bodies that are resisted in the ratio of the velocity.
    2. Book 2.2: Of the motion of bodies that are resisted in the duplicate ratio of their velocities.
    3. Book 2.3: Of the motions of bodies which are resisted partly in the ratio of the velocities, and partly in the duplicate of the same ratio.
    4. Book 2.4: Of the circular motion of bodies in resisting mediums.
    5. Book 2.5: Of the density and compression of fluids; and of hydrostatics.
    6. Book 2.6: Of the motion and resistance of funependulous bodies.
    7. Book 2.7: Of the motion of fluids, and the resistance made to projected bodies.
    8. Book 2.8: Of motion propagated through fluids.
    9. Book 2.9: Of the circular motion of fluids.
  7. Book 3
    1. Book 3.1: Rules of Reasoning in Philosophy.
    2. Book 3.2: Phaenomena, or Appearances.
    3. Book 3.3: Propositions
    4. Book 3.4: General Scholium.
  8. Index


The Author's Preface

SINCE the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things : and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description if right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics was cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), ho Otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive ; and therefore we offer this work as the mathematical principles of philosophy ; for all the difficulty of philosophy seems to consist in this from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World : for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto at tempted the search of nature in vain ; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing ; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity, and other forces ; and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of mediums ; the orbits of the comets, and such like ; deferred that publication till I had made a search into those matters, and could put forth the whole together. What relates to the lunar motions (being imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and interrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers.

Isaac Newton.

In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon's theory and the praecession of the equinoxes were more fully deduced from their principles ; and the theory of the comets was confirmed by more examples of the calculation of their orbits, done also with greater accuracy.

In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other : there are, besides, added Mr. Kirk's observations of the comet in 1680 ; the orbit of that comet computed in an ellipsis by Dr. Halley ; and the orbit of the comet in 1723 computed by Mr. Bradley.

> Chapter 1

Table of Contents