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Physics and Natural Philosophy · Treatise
The Mathematical Principles of Natural Philosophy
by Newton · The Analyst of Light · Scientific Revolution, 1642-1727
Newton's full Principia reader through the Author's Preface, definitions, laws of motion, Books I-III, figures, formulas, tables, and the General Scholium.
- Brutal
- 1687
- ~15 hr 45 min read
- 22 chapter guides
- Public domain
Start reading · Read the Scholium (continued) guide free
Before the text
Editor's Prelude
When The Heavens Became A Problem In Motion
A falling body and the Moon’s turning path present Newton with a shared question: what force makes motion change? The Principia pursues an answer that can reach from a laboratory pendulum to the sea and the planets. Newton credits Edmond Halley with pressing him to publish his demonstrations and helping with the printing and diagrams. The growing work also made room for problems that resisted a simple answer, especially the Moon’s unequal motions.
Newton does not write like a modern textbook. Definitions fix the terms; propositions state results; problems ask for constructions. A corollary draws a consequence from a proof, while a scholium adds discussion or explanation. Book I builds the mathematics of motion under force. Book II brings in resistance and fluids. Book III sets calculations beside observations of the heavens and seas. Diagrams, formulas and tables carry parts of the argument itself.
The repeated movement is from visible motion to invisible cause, then back to motion that can be measured. A path becomes evidence. So does a swing that grows shorter. The ambition lies in bringing unlike events under common laws while judging how much the evidence can bear.
The opening pages
Definitions I–VIII
BOOK I.
DEFINITIONS.
DEFINITION I.
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.
DEFINITION II.
The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.
The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.
DEFINITION III.
The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.
This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertiæ, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.
Chapter guides · 22
Scholium (continued)
Motion Needs More Than A Nearby Landmark
Newton argues that changes in motion reveal more than a body’s position relative to its neighbours.
Main idea
Relative motion compares one body with another. Newton distinguishes this from true motion by looking at the forces that cause motion and the effects it produces.
Recap
The rotating bucket’s water becomes concave as the water begins to turn, even when it moves little relative to the bucket. Two globes joined by a cord offer another test: the cord’s tension reveals their rotation. Newton closes by making the discovery of true motions from their causes and effects the aim of the work.
Corollaries V–VI
The Laws Become A Method
Bodies keep the same motions among themselves when their whole system receives the same motion or acceleration.
Main idea
A uniformly moving ship illustrates why a shared motion leaves internal motions unchanged. Equal accelerations in parallel directions also preserve the bodies’ motions relative to one another.
Recap
Newton states these two corollaries, then discusses earlier work on falling bodies and collisions. His pendulum trials allow for air resistance and imperfect rebound as he tests equal action and reaction. He ends with machines, where a gain in force is balanced by the speeds of the parts transmitting it.
Book I · Scholium
Vanishing Quantities Can Still Be Compared
Newton’s first and last ratios describe limits approached during continuous change.
Book I · Proposition X
An Ellipse With Force At Its Centre
A force proportional to distance can keep a body moving in an ellipse around its geometric centre.
Book I · Lemma XXVI to Proposition XXIX
Orbits Can Be Reconstructed
A few known positions or directions can determine an orbit of a given kind.
Book I · Proposition XLV (continued)
The Orbit Itself Can Turn
The turning of an orbit can help identify how its central force changes with distance.
Book I · Proposition LXIX
Attraction Reveals The Amount Of Matter
Newton links a body’s power to attract with the amount of matter it contains.
Book I, Section XIV
Forces Can Bend A Path Through A Surface
Newton uses forces near a surface to model the bending and reflection of a small body’s path.
Book II, Section III
Two Kinds Of Resistance Act Together
Resistance proportional to speed and resistance proportional to speed squared can act on the same body.
Book II · Propositions XXI–XXIII
Compression Gives A Model Of An Elastic Fluid
Newton calculates how a fluid’s density changes under pressure and works out what a possible particle explanation would require.
Book II · General Scholium (continued)
Experiment Enters Through The Pendulum
The shrinking swing of a pendulum lets Newton compare resistance in different fluids.
Book II · Scholium (continued)
Falling Bodies Test The Resistance Model
Newton compares predicted falls with experiments, then uses resistance to argue for nearly empty celestial space.
Book II · Proposition LIII
The Vortex Model Has Failed Its Test
Book II ends by arguing that a fluid vortex cannot preserve the observed planetary motions.
Book III: Phenomena
Observation Now Sets The Terms
Measured paths and periods supply the patterns a theory of celestial motion must explain.
Book III · Propositions VI–IX
Universal Gravity Is Now The Claim
Newton extends attraction to all matter and adds up the effects of a sphere’s particles.
Book III · Proposition XXIV (continued)
Channels Can Change The Pattern Of A Tide
Water’s continuing motion and the routes it takes can delay, combine or cancel tides.
Book III · Proposition XXXV
The Moon’s Tilt Changes With Its Position
Newton calculates the changing tilt of the Moon’s orbit and gathers corrections to its motion.
Book III: The Sea and the Moon's Body
The Sea Becomes A Gravitational Instrument
Tide measurements let Newton estimate the Moon’s pull, mass and shape.
Book III: The Precession of the Equinoxes
The Equator’s Bulge Helps Turn Earth’s Axis
Unequal pulls on Earth’s equatorial bulge account for the slow shift of the equinoxes.
Book III · Proposition XL to Lemma XI
A Comet’s Path Can Be Calculated
Newton places comet motion under the same central-force mathematics used for planets.
Book III · Proposition XLII (continued)
The Comet Argument Ends In Measurement
Comparing observed comet paths can identify a returning comet and estimate its period.
General Scholium
The System Ends At The Edge Of Its Evidence
Newton defends gravity’s observed laws while leaving its physical cause unexplained.
Questions readers ask
- How long does it take to read The Mathematical Principles of Natural Philosophy?
- About 15 hr 45 min in total.
- How difficult is The Mathematical Principles of Natural Philosophy?
- Brutal. A long treatise; about 16 hours of reading.
- Which translation is this?
- The Andrew Motte translation from the Latin, first published in 1687. The text follows Newton's Principia: The Mathematical Principles of Natural Philosophy, first American edition, carefully revised and corrected (1846).
- Can I read The Mathematical Principles of Natural Philosophy for free?
- The Mathematical Principles of Natural Philosophy is part of Nexus Premium, so reading it needs an account and a Premium subscription. The opening above and two chapter guides on this page are free to read.
- How many chapter guides are there?
- 22 chapter guides, written by the Nexus editors. Two open in full here; the rest show their opening line.
About Newton
Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher. In Opticks he investigated light, color, reflection, refraction, and the experimental method that helped define modern science.
“What can experiment reveal when ordinary sight is not enough?”
Newton gives Nexus a scientific reader path where experiment, mathematical discipline, and open questions work together rather than merely reporting discovery.
Source text: public domain. Chapter guides: original Nexus writing. The Mathematical Principles of Natural Philosophy is part of The Nexus Archive, guided Quests through the great works.