# The Mathematical Principles of Natural Philosophy (1729)

An English translation by Andrew Motte, based on the 1726 3rd edition of Philosophiae Naturalis Principia Mathematica.
* Volume 1: Scan index
* Volume 2: Scan index
THE
MATHEMATICAL
P R I N C I P L E S
OF
Natural Philosophy.
By Sir ISAAC NEWTON.
Translated into English by Andrew Motte.
To which are added,
The Laws of the MOON's Motion, according to Gravity.
By John Machin Astron. Prof. Gress. and Secr. R. Soc.
In Two Volumes
LONDON:
Printed for Benjamin Motte, at the Middle- Temple-Gate, in Fleetstreet.
MDCCXXIX.
Contents :
| Front Matter ||
| | page |
| Title Page | i |
| Dedication | iii |
| The Author's Preface | vii |
| The Preface of Mr Roger Cotes | xiv |
| Definitions | 1 |
| Scholium | 9 |
| Axioms, or Laws of Motion | 19 |
| Book 1: The Motion of Bodies ||
| Section I | 41 |
| Section II | 57 |
| Section III | 79 |
| Section IV | 94 |
| Section V | 104 |
| Section VI | 143 |
| Section VII | 154 |
| Section VIII | 168 |
| Section IX | 177 |
| Section X | 196 |
| Section XI | 218 |
| Section XII | 263 |
| Section XIII | 292 |
| Section XIV | 311 |
Book 2: The Motion of Bodies (in resisting mediums) :
| Section I | 1 |
| Section II | 12 |
| Section III | 45 |
| Section IV | 55 |
| Section V | 64 |
| Section VI | 80 |
| Section VII | 111 |
| Section VIII | 163 |
| Section IX | 184 |
| Book 3: Of the System of the World ||
| Rules of Reasoning in Philosophy | 202 |
| Phænomena, or Appearances | 206 |
| Propositions I-IX (Force of gravity) | 213 |
| Propositions X-XXIV (Motions of the sea) | 230 |
| Propositions XXV-XXXIII (Motions of the moon) | 262 |
| Propositions XXXVI-XXXVIII (Forces to move the sea) | 305 |
| Lemmas I-III, Proposition XXXIX (Precession of equinoxes) | 315 |
| Lemmas IV-XI, Propositions XL-XLII (Comets) | 323 |
| General Scholium | 387 |
| Index | XX |
| Appendix | i |
| The Laws of the Moon's Motion according to Gravity (John Machin) | (1) |
| Errata | XX |

### Book 1

       THE MOTION OF BODIES
* Of the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow
* Of the invention of centripetal forces
* Of the motion of bodies in eccentric Conic sections
* Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given
* How the orbits are to be found when neither focus is given
* How the motions are to be found in given orbits
* Concerning the rectilinear ascent and descent of bodies
* Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force
* Of the motion of bodies in movable orbits; and of the motion of the apsides
* Of the motion of bodies in given superficies; and of the reciprocal motion of funependulous bodies
* Of the motions of bodies tending to each other with centripetal forces
* Of the attractive forces of sphaerical bodies
* Of the attractive forces of bodies which are not of a sphaerical figure
* Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body

### Book 2

       THE MOTION OF BODIES (In resisting mediums)
* Of the Motion of Bodies that are resisted in the ratio of the Velocity
* Of the Motion of Bodies that are resisted in the duplicate ratio of their Velocities
* Of the Motions of Bodies which are resisted partly in the ratio of the Velocities, and partly in the duplicate of the same ratio
* Of the circular motion of bodies in resisting mediums
* Of the density and compression of fluids; and of Hydrostatics
* Of the motion and resistance of funependulous bodies
* Of the motion of fluids and the resistance made to projected bodies
* Of motion propagated thro' fluids
* Of the circular motion of fluids

### Book 3

       OF THE SYSTEM OF THE WORLD
* Preface to Book 3
* Rules of Reasoning in Philosophy
* The Phaenomena or Appearances
* Propositions
       Motion of the satellites of Jupiter
       Propositions 2: the primary Planets, and 3: the Moon
       Proposition 6: Gravitation towards every Planet
       Proposition 7: Gravity tending to all Bodies
       Proposition 10: Longevity of planetary motions
       Proposition 11: Common centre of gravity of the Earth, the Sun and all the Planets
       Proposition 13: the Planets move in Ellipses
       Proposition 17: the diurnal motions of the Planets are uniform
       Proposition 18: (oblateness of the Planets & the Earth)
       Proposition 21: the equinoctial points go backwards
       Proposition 22: all the motions of the Moon ... follow from the principles ... laid down
       Proposition 24: the flux and reflux of the Sea, arise from the actions of the Sun and Moon
       Propositions 25-35: (disturbances of the motions of the Moon)
       Proposition 36-37: (forces of Sun & Moon to move the Sea)
       Proposition 38: Figure of the Moon's Body
       Proposition 39: precession of the equinoxes
       (Theory of the comets)
* General Scholium
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