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.
How Principia Mathematica builds its case
Follow how the book develops its argument. Each note is a brief orientation, not a replacement for the chapter.
The Mathematical Worldview
Newton opens with a method for natural philosophy. Forces are inferred from phenomena of motion, and further phenomena are then demonstrated from those forces.
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.
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.
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.
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.
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.
Common Centres, Many Bodies
Earlier chapters treated the force centre as fixed. Newton now asks what changes when attracting bodies move themselves.
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.
Resistance as Geometry
Earlier orbital mechanics idealized motion as resistance-free. Book II asks what changes when a medium continually removes motion.
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.
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.
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.
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.
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.
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.
