Physics — The System of the World
Book 3 applies the mathematics of Books 1–2 to "the system of the world" — planets, moons, comets, and tides.
Definitions
- Quantity of matter (mass): measure from density and bulk conjointly.
- Quantity of motion: $p = m v$
- Vis inertiae: innate force resisting change
- Impressed force: action to change state, subsists only during action
- Centripetal force: directed toward a center, e.g. gravity
The Three Laws of Motion
- Law I: Every body perseveres in its state of rest or uniform motion in a right line, unless compelled to change by forces impressed.
- Law II: Change of motion is proportional to the motive force impressed, and is made in the direction of that force. $\vec F = \dot{\vec p}$. For constant mass, $\vec F = m\vec a$.
- Law III: To every action there is always opposed an equal reaction.
Universal Gravitation
Every particle attracts every other with force $F = G \frac{m_1 m_2}{r^2}$ directed along the line joining them. Newton deduced this from Kepler and proved it explains:
- Elliptical orbits with Sun at focus
- Equal areas in equal times
- $T^2 \propto a^3$ for planets
Mathematics — Fluxions in Geometric Dress
Newton had invented calculus ("method of fluxions") by 1671, but Principia is written in classical geometry to be rigorous by the standards of the time.
Method of First and Last Ratios — The Limit Concept
Book 1, Lemmas 1–11: If quantities approach each other and their difference becomes less than any given assignable quantity, their ultimate ratio is the ratio of equality. This is the limit $\lim_{\Delta t \to 0}$ in geometric language.
Lemma 2: The area under a curve is the ultimate sum of vanishing parallelograms — the definite integral.
Fluxions
- Fluent: flowing quantity $x$
- Fluxion: $\dot x = \frac{dx}{dt}$ — its velocity
- Moment: $o\dot x$ — infinitesimal increment
In Principia, he avoids $o$ and uses ultimate ratios of vanishing triangles.
Key Tools
| Principia | Geometric Proof | Modern Calculus |
|---|---|---|
| Lemma 10 | Spaces under constant force ∝ $t^2$ | $\int_0^t at\,dt = \tfrac12 at^2$ |
| Prop 1 | Centripetal impulse → equal triangles | $\frac{d}{dt}(\vec r \times m\vec v)=0$ |
| Prop 6 | Force from curvature of orbit | $F = \frac{L^2}{mr^2}\left(\frac{1}{r} - \frac{d^2}{d\theta^2}\frac{1}{r}\right)$ |
| Prop 11 | Ellipse implies $1/r^2$ force | Binet equation |
| Prop 71 | Shell theorem | $\oint_{sphere} \frac{\hat r}{r^2} dA = 4\pi$ |
Core Proofs (Translated to Modern Notation)
1. Kepler's Second Law from Central Force (Prop. 1, Book 1)
Newton's construction: A body moves from A to B by inertia. At B, an impulsive centripetal force toward S deflects it to C. Triangles SAB and SBC have equal area (same base SB, equal altitude). Repeating with smaller time steps, the polygonal path approaches a smooth curve. In the limit, equal areas in equal times.
Modern: Central force $\vec F \parallel \vec r$ gives zero torque: $\vec \tau = \vec r \times \vec F = 0$. Hence $\vec L = \vec r \times m\vec v$ is constant. Swept area $dA = \tfrac12 |\vec r \times d\vec r|$, so $\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$.
2. Inverse-Square → Conic Sections (Prop. 11)
For a centripetal force $F \propto 1/SP^2$, Newton proves using ultimate ratios that the trajectory satisfies $l/r = 1 + e\cos\theta$, where $l = L^2/GMm^2$. That's the polar equation of a conic with focus at S. If $0 \le e < 1$, an ellipse — Kepler's First Law.
3. Shell Theorem (Prop. 70–71)
A uniform spherical shell attracts an external particle as if all mass were at its center. Proof: take opposite surface patches; their oblique components cancel, radial components sum to $dF = Gm\,dM\cos\phi / s^2$. Integration gives $F = GMm/R^2$. Inside the shell, the cancellation is perfect: net force zero.
Corollary: Earth acts as point mass for Moon, and as shells for internal gravity.
4. Rules of Reasoning in Philosophy (Book 3)
- Admit no more causes than are true and sufficient.
- Therefore to the same natural effects we must assign the same causes.
- Qualities of bodies found universally by experiments are universal.
- Propositions inferred by induction are true until new phenomena contradict them.
GNU Octave — Run Newton's Mathematics
Copy these scripts into Octave to reproduce Principia results numerically.
1. Law II — Projectile Parabola
% projectile.m
g = 9.81; v0 = 50; theta = 45*pi/180;
t = linspace(0, 2*v0*sin(theta)/g, 300);
x = v0*cos(theta)*t;
y = v0*sin(theta)*t - 0.5*g*t.^2;
plot(x,y,'LineWidth',2); axis equal; grid on;
xlabel('x (m)'); ylabel('y (m)');
title('Projectile: F = ma integrates to parabola');2. Universal Gravitation — Elliptical Orbit
% orbit.m
GM = 1.0;
y0 = [1; 0; 0; 0.8]; % [x y vx vy], e ~ 0.6
f = @(t,y) [y(3); y(4); -GM*y(1)/norm(y(1:2))^3; -GM*y(2)/norm(y(1:2))^3];
[t,y] = ode45(f, [0 20], y0);
plot(y(:,1), y(:,2)); axis equal; grid on;
title('Orbit from F = -GMm/r^2'); xlabel('x'); ylabel('y');3. Equal Areas in Equal Times
% kepler2.m
[t,y] = ode45(f, linspace(0,10,2000), y0);
r1 = y(1:end-1,1:2); r2 = y(2:end,1:2);
dA = 0.5*abs(r1(:,1).*r2(:,2) - r1(:,2).*r2(:,1));
A = cumsum(dA);
plot(t(2:end), A, 'LineWidth',1.5); grid on;
xlabel('time'); ylabel('cumulative area');
title('Prop 1: dA/dt constant for central force');4. Shell Theorem — Numerical Integration
% shell.m
G=1; M=1; R=1; rp=2.5; n=3e5;
phi = acos(2*rand(n,1)-1); lam = 2*pi*rand(n,1);
xs = R*sin(phi).*cos(lam); ys = R*sin(phi).*sin(lam); zs = R*cos(phi);
dx = rp - xs; dy = -ys; dz = -zs; r = sqrt(dx.^2+dy.^2+dz.^2);
Fx = sum(G*(M/n) .* dx ./ r.^3);
printf('Numerical Fx = %.6f, Theory GM/rp^2 = %.6f\n', Fx, G*M/rp^2);5. Book 2 — Resisted Motion $F = -k v^2$
% resistance.m
k=0.05; m=1; v0=20; t=linspace(0,10,400);
v = v0 ./ (1 + k*v0*t/m); % analytic solution
x = (m/k)*log(1 + k*v0*t/m);
subplot(2,1,1); plot(t,v); grid on; ylabel('v'); title('Quadratic drag');
subplot(2,1,2); plot(t,x); grid on; ylabel('x'); xlabel('t');6. Tides — Sun and Moon
% tides.m
days = 0:0.05:60;
moon = cos(2*pi*days/12.42/2); % semi-diurnal lunar
sun = 0.46*cos(2*pi*days/12/2); % solar
tide = moon + sun;
plot(days, tide); grid on;
xlabel('hours'); ylabel('relative height');
title('Spring-neap beats: Principia Book 3, Prop 24');Interactive Demonstrations
1. Newton's Cannon
From Principia, Book 1, inspired by "a stone projected" from a mountaintop. Too slow: falls. Just right (~7.1): circular orbit. Faster: ellipse. Much faster: escape.
2. Equal Areas in Equal Times
Reading the Principia Today
Start with the Definitions and Axioms (Laws). Then Book 1, Propositions 1–13: these prove that central forces give area law, and inverse-square gives Kepler ellipses. Props 70–75 are the shell theorem. Book 3 translates this to real data — the Moon's motion, the tides, the precession of the equinoxes, and the orbits of comets.