What you'll learn
Key ideas from Principia Mathematica
These ideas compress the book's argument without treating the author's view as settled fact. Use them as an orientation before reading the full work or listening in Wiseley.
Natural philosophy infers forces from observed motion, then derives further phenomena from those forces.
Inertia preserves rest or uniform straight-line motion until an impressed force changes it.
Inverse-square attraction makes conic sections with the force centre at a focus, and the converse identifies the same law.
Swept areas provide a geometric measure of elapsed time, allowing positions on elliptic and hyperbolic paths to be constructed.
Spherical shells and concentric layers reduce symmetric inverse-square particle forces to centre-directed laws, including linear attraction inside a sphere.
The Moon’s measured curvature agrees with terrestrial falling after inverse-square scaling, identifying orbital centripetal force with ordinary gravity.
The System of the World unifies inverse-square celestial order, while the General Scholium separates established gravity from its undeduced cause.
Inside Principia Mathematica
Read the first chapter in full here. The other 14 continue in the Wiseley app.
Chapter 1 of 15 · 4 min · Audio & text
The Mathematical Worldview
Principia Mathematica, by Isaac Newton.
Newton opens with a method for natural philosophy. Forces are inferred from phenomena of motion, and further phenomena are then demonstrated from those forces. The work reflects this order: the first two books develop general mathematical propositions, while the third applies them in the System of the World. Celestial phenomena can disclose gravitational forces toward the Sun and planets, after which mathematics addresses planetary, cometary, lunar, and sea motions. The method begins with observable effects, even when the physical mechanism remains unknown.
The Author’s Preface places this method within mechanics. Rational mechanics seeks accurate demonstrations; practical mechanics includes manual arts. Imperfect workmanship does not make the art itself imperfect. Mechanics supplies accurately constructed lines and circles, while geometry uses them to solve problems and measure magnitudes. Geometry concerns magnitudes; rational mechanics concerns motion and the forces producing it.
His definitions begin with matter. Quantity of matter is density multiplied by bulk, or mass; pendulum experiments connect weight with it. Quantity of motion combines mass and velocity. Inertia is matter’s resistance to leaving rest or uniform straight-line motion, proportional to mass. An impressed force changes that state and ends when its action ends; inertia preserves the acquired motion.
Centripetal force draws or impels a body toward a centre. Gravity, magnetism, and the force bending planetary paths are examples. Without such inward action, curved motion would become uniform motion in a straight line. Newton distinguishes absolute force, accelerative force, and motive force: they concern, respectively, the cause’s efficacy, velocity generated in time, and motion generated in the body. Accelerative force varies with place, while motive force depends on mass. These are mathematical descriptions, not explanations of the force’s physical cause.
Time and space receive the same careful separation. Absolute time is uniform duration independent of external things; relative time is a sensible measure based on motion. Since natural days are unequal, astronomy corrects common time. Absolute space is immovable and independent of bodies; relative space is a movable measure fixed by bodily positions. Place is the portion of space occupied by a body. Absolute motion carries a body between absolute places, while relative motion carries it between relative places. A body can be at rest within a moving ship yet move absolutely with the ship. Relative measures are useful, but they are not the quantities themselves.
The rotating-bucket experiment tests whether relations alone define circular motion. At first, the vessel moves relative to the water, but the surface stays flat. Later, the water rotates, rises at the sides, and becomes concave; the greatest rise occurs when the water is relatively still in the vessel. Its recession from the axis therefore reflects real circular motion, not merely relations to surrounding bodies. Newton’s boundary is clear: mathematics may describe effects exactly while leaving attraction’s underlying cause unidentified. This vocabulary prepares the later laws, proofs, and applications.
Chapter 2 of 15 · 4 min · Audio & textIn the app
Laws of Motion
With the earlier vocabulary in place, Newton turns definitions into rules for bodies in interaction. True motion remains indirectly known: apparent changes show differences, while forces reveal causes and effects.
Chapter 3 of 15 · 4 min · Audio & textIn the app
Limits and Central Force
Newton now addresses the problem continuous motion creates: how can finite mechanical actions yield a curved path? He begins with action and reaction.
Chapter 4 of 15 · 5 min · Audio & textIn the app
Conics as Dynamics
Newton’s central-force theory runs in two directions. It asks what force could produce an observed orbit, and what orbit a known force must produce.
Chapter 5 of 15 · 4 min · Audio & textIn the app
Constructing the Orbit
Once the orbit has been identified as a conic, Newton turns to a practical question: how can its curve be recovered from incomplete geometric information? His answer is a chain of determinate constructions.
Chapter 6 of 15 · 5 min · Audio & textIn the app
Time Written in Areas
Once orbit shape is known, Newton asks how geometry can tell us when a body arrives, how fast it moves, and what force law produces its timing. In central-force motion, the radius from the centre sweeps area at a rate tied to elapsed time.
Chapter 7 of 15 · 3 min · Audio & textIn the app
Common Centres, Many Bodies
Earlier chapters treated the force centre as fixed. Newton now asks what changes when attracting bodies move themselves.
Chapter 8 of 15 · 4 min · Audio & textIn the app
Perturbations and Distributed Gravity
After the ideal two-body orbit, Newton examines what an additional attraction leaves behind. The key is its inequality: if an outside body pulls the orbiting body and its primary nearly equally and in parallel, their relative orbit changes little.
Chapter 9 of 15 · 5 min · Audio & textIn the app
Resistance as Geometry
Earlier orbital mechanics idealized motion as resistance-free. Book II asks what changes when a medium continually removes motion.
Chapter 10 of 15 · 4 min · Audio & textIn the app
Fluids, Waves, and Measurements
After treating resistance geometrically, Newton turns to the medium. A fluid is a body whose parts yield to force and move among one another.
Chapter 11 of 15 · 4 min · Audio & textIn the app
Particles, Efflux, and Pulses
Newton now moves from resistance in general to the behavior of media. In a rare medium of separate particles, resistance comes from impacts.
Chapter 12 of 15 · 5 min · Audio & textIn the app
Against Vortices, Toward Gravity
Book III changes the task. Newton now uses the mathematical laws of motion and force to infer the world system from phenomena.
Chapter 13 of 15 · 5 min · Audio & textIn the app
Earth, Moon, and Rotation
Having established universal gravity, Newton turns from ideal orbits to worlds that move, rotate, and have measurable shapes. The system’s centre is not an immovable Sun: Earth, Sun, and the planets attract one another, so their centres continually move.
Chapter 14 of 15 · 4 min · Audio & textIn the app
Lunar Inequalities and Tides
Newton’s lunar theory begins where an ideal orbit ceases to be enough. Solar attraction changes the Moon’s speed, distance, inclination, nodes, and apogee.
Chapter 15 of 15 · 5 min · Audio & textIn the app
Comets and Cosmic Order
After the detailed lunar and tidal arguments, Newton asks how far the gravitational framework reaches. Precession tests Earth’s response to celestial forces.
Chapter 1 of 15 · 4 min · Audio & text: The Mathematical Worldview
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Continue in WiseleyAbout Isaac Newton
Isaac Newton was an English polymath. “Principia Mathematica” explores the mathematical principles of motion and universal gravitation.
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