Principia Mathematica
in Motion
The mathematical principles of natural philosophy — Newton's 1687 masterwork, with its diagrams set moving. Drag, fire, and tune the very demonstrations he drew by hand.
Auctore Is. Newton · Londini, MDCLXXXVII · rendered anew
De Mundi Systemate · The System of the World
Newton's Cannon — an orbit is only falling, forever
Newton imagined a cannon atop an impossibly high mountain. Fire the ball gently and it arcs to the ground. Fire it harder and it flies farther before landing — until, at one exact speed, the Earth curves away beneath it as fast as it falls, and it never lands at all. Push past that and the ball escapes entirely. Increase the muzzle speed and fire.
Muzzle
vc = the circular-orbit speed. Below it the ball falls back; at 1.00 it circles; above √2 ≈ 1.41 it escapes to infinity.
“…the body… would go on revolving in its orbit as the planets do.” — De Syst. Mundi
Axiomata, sive Leges Motus · Axioms, or the Laws of Motion
The three laws, each a small machine you can run
Everything that follows in the Principia is built on these three axioms. Here each is a live demonstration — the inertia of a drifting body, force building velocity in proportion to mass, and every push answered by an equal push back.
Uniform motion in a right line
“Every body perseveres in its state of rest, or of uniform motion in a right line, unless compelled to change it by forces impressed.”
Change of motion follows the force
“The alteration of motion is proportional to the motive force impressed, and made in the line of that force.”
Equal, and in opposite parts
“To every action there is always opposed an equal reaction; the mutual actions of two bodies are always equal and directed contrary.”
Liber Primus · Prop. I, Theor. I
Equal areas in equal times — the first theorem
The Principia's opening theorem: a body pulled toward a fixed centre sweeps out equal areas in equal times, whatever the shape of its path. Watch the planet race through perihelion and crawl through aphelion — yet each gold wedge, stamped at a fixed tick of the clock, holds the same area. Change the orbit's eccentricity and it stays true.
The orbit
0 is a perfect circle; nearer 1 is a long, comet-like ellipse. The Sun sits at one focus S.
“Areas quas corpora… describunt… temporibus proportionales esse.”