Philosophiæ Naturalis Principia Mathematica

The Principia, published in July 1687 by Edmond Halley at the Royal Society, is not merely a book of physics. It is a re-founding of natural philosophy. Newton proposed a new method, to argue from phenomena by induction, to derive forces, then to use those forces to demonstrate all other phenomena. In his Preface he states his purpose plainly, "from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena".

The work is divided into three books. Book I, De Motu Corporum, treats the motion of bodies in free space under centripetal forces, using pure geometry. Book II extends the theory to resistive media, to fluids, to projectiles and to the motion of waves, refuting Cartesian vortices. Book III, De Systemate Mundi, applies the mathematics to the heavens, proves universal gravitation, and derives the motions of planets, moons, comets, and tides from a single law.

This guide follows Newton’s own architecture, but renders his proofs with modern clarity, interactive diagrams, and computational verification.

1. Philosophical Foundations

Newton begins not with hypotheses, but with definitions and axioms. In the Scholium following the definitions he rejects the scholastic philosophy of his education at Cambridge. Where Aristotelians explained motion by substantial forms and occult qualities, Newton demands quantities that can be measured, mass, velocity, force.

Most radical is his concept of absolute space and absolute time. Against Descartes, who held all motion to be relative, Newton argues in the famous bucket experiment that rotation is absolute. The water climbs the sides not relative to the bucket, but relative to space itself. Absolute time "flows equably without relation to anything external".

His method culminates in the General Scholium added to the second edition, 1713, "hypotheses non fingo", I feign no hypotheses. He refuses to invent hidden mechanisms for gravity. He will not, like Descartes with his vortices of subtle matter, or like the scholastics with sympathies, posit causes that cannot be deduced from phenomena. It is enough that gravity exists and acts according to laws, its cause may be left to further inquiry.

Did Newton get his notion from Plato?

The question arises naturally, since Newton revered the ancients and filled his alchemical manuscripts with Platonic language. The answer is nuanced, not directly.

Newton owned and annotated Plato's Timaeus in the Ficino translation. He admired the Pythagorean tradition. Yet his physics is profoundly anti-Platonic in method. Plato's cosmos in the Timaeus is constructed by a Demiurge imposing perfect geometric Forms, triangles composing the elements, final causes and a world soul directing motion toward the Good. Knowledge comes through rational insight into eternal Forms.

Newton's cosmos is mechanical and voluntarist. God creates particles with given masses and imposes laws, then the world runs by efficient causes, forces impressed, not by final causes. Mathematics is not contemplation of Forms, but an instrument to measure forces from phenomena. Newton explicitly bans final causes from natural philosophy.

Newton's true philosophical lineage comes from elsewhere, from Galileo's mathematization of motion, from Kepler's laws as phenomena to explain, from Descartes' mechanical philosophy which he corrected, from Boyle's corpuscularianism, and from the ancient atomists Democritus and Lucretius whom he read through Gassendi. Plato gave Newton reverence for mathematics and belief in a rational Creator, but not his physics.

2. The Mathematics of Principia — Fluxions in Geometric Dress

By 1665 during the plague years at Woolsthorpe, Newton had invented his method of fluxions, our differential and integral calculus. He could differentiate \(x^n\), find tangents, and perform quadratures. Yet when writing the Principia twenty years later, he deliberately concealed the analysis.

Why? Partly to avoid controversy, partly because geometric demonstration carried the authority of the ancients, and partly because fluxions lacked rigorous foundation by contemporary standards. So Newton cast his calculus into synthetic geometry using the "method of first and last ratios".

In Book I, Section I, Lemmas I-XI lay the foundation.

Lemma I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

This is the limit concept. Newton proves it by contradiction, if the ultimate difference were some \(D\), then contrary to hypothesis they could not approach nearer than \(D\). From this follow lemmas on inscribed and circumscribed figures, on vanishing parallelograms, and on curvature.

Lemma II is essentially the Riemann integral, the sum of evanescent rectangles approaches the curvilinear area. Lemma VII proves that the ultimate ratio of arc, chord, and tangent are ratios of equality. This allows Newton to replace curves by their tangents in the limit, precisely what we do with derivatives.

The interactive below shows Newton's technique in action. Archimedes approximated a circle by polygons, Newton makes the polygon sides evanescent.

3. Axioms or Laws of Motion — with Proofs

Following Definitions, Newton states three Axioms, sive Leges Motus.

From these Newton derives Corollaries. Corollary I is the parallelogram of forces. If a body is acted on by two forces simultaneously, it will describe the diagonal of the parallelogram in the same time as it would describe the sides separately.

4. Book I: De Motu Corporum — Central Forces

Proposition I, Theorem I: Equal Areas in Equal Times

Newton proves Kepler's second law not from astronomy, but from dynamics. A body under any centripetal force directed to a fixed point S will sweep equal areas in equal times.

Crucially, the converse holds, if areas are proportional to times, the force is centripetal. This is Proposition II.

Proposition XI, Problem VI: Inverse-square implies Ellipse

Given a centripetal force varying as \(1/r^2\), prove the orbit is a conic section with focus at center of force. Newton’s proof occupies three pages of dense geometry. The modern essence is simple.

For central force, specific angular momentum \(h = r^2\dot{\theta}\) is constant by Prop I. Write acceleration in polar coordinates, radial component \(a_r = \ddot{r} - r\dot{\theta}^2 = -k/r^2\). Substitute \(u = 1/r\), and using \(d/dt = h u^2 d/d\theta\), the equation becomes $$ \frac{d^2u}{d\theta^2} + u = \frac{k}{h^2} $$ whose solution is \(u(\theta) = \frac{k}{h^2}(1 + e\cos(\theta-\theta_0))\), the polar equation of a conic with eccentricity \(e\). Newton proves the same result with Lemma XI on curvature and without explicit calculus.

Proposition LXXI: The Shell Theorem

No result impressed Newton’s contemporaries more. Prove that a uniform spherical shell attracts an external particle as if all its mass were concentrated at its centre.

Modern integral proof: for shell radius \(R\), surface density \(\sigma\), distance \(d>R\): $$ F = \int_0^\pi \frac{G m (2\pi R^2 \sigma \sin\phi d\phi) \cos\alpha}{s^2} = \frac{GmM}{d^2} $$ where \(s^2 = R^2 + d^2 -2Rd\cos\phi\). Newton performed this geometrically, avoiding explicit integration.

5. Book III: System of the World

Having established mathematics, Newton turns to phenomena. He lists Kepler's laws as Phenomena I-III, then derives universal gravitation.

Derivation from Kepler III: For circular orbit, centripetal force \(F = m v^2 / r = m (2\pi r / T)^2 / r = 4\pi^2 m r / T^2\). Kepler's third law states \(T^2 \propto r^3\), thus \(T^2 = k r^3\). Substituting, $$ F = \frac{4\pi^2 m}{k} \frac{1}{r^2} $$ Hence force is inverse-square. By Prop LXXI, this holds for spherical bodies like Sun and planets.

By Law III, if Earth pulls Moon, Moon pulls Earth. Generalizing, $$ F = G \frac{m_1 m_2}{r^2} $$ Newton computed \(G\) implicitly from terrestrial gravity and lunar motion, finding agreement within 15 percent, later refined.

From this one law Newton explained:

• Tides: differential lunar and solar attraction across Earth, principal lunar tide twice solar, spring and neap tides derived in Prop XXIV.
• Precession of equinoxes: torque of Sun and Moon on Earth's equatorial bulge, computed as 50 arcseconds per year.
• Comets: show they move in highly eccentric conics obeying same law, Halley's comet predicted to return.
• Irregularities of Moon: from solar perturbation, founding perturbation theory.

6. Newton vs Leibniz — Why Calculus Appeared Twice

The calculus controversy has obscured a deeper truth, calculus was inevitable by 1670. The problems demanding it were everywhere, tangents to curves for optics, quadratures for areas under curves, maxima and minima for projectile motion, and inverse tangent problems for central forces.

Both Newton and Leibniz built on a common foundation: Cavalieri's indivisibles, 1635, Fermat's adequality for extrema, 1637, Wallis's Arithmetica Infinitorum, 1656, Barrow's geometric lectures, 1669, and Huygens's work on evolutes.

As Whiteside, Hall, and Guicciardini demonstrate, the inventions were independent and differently motivated. Newton sought to describe motion, fluxions are velocities. Leibniz sought a universal characteristic, differentials are infinitesimal differences amenable to algebraic manipulation. Newton's notation \(\dot{x}\) is physical but opaque for partial derivatives. Leibniz's \(dx\) generalizes, becomes the language of Euler, Lagrange, and modern analysis.

The priority dispute after 1710 damaged British mathematics for a century, as continentals advanced with Leibnizian notation while British stuck to dots.

7. GNU Octave Examples — Principia in Code

Newton performed calculations by hand. We can verify his propositions numerically in Octave, free software compatible with MATLAB.

1. kepler_second_law.m — Equal areas

Copy% Verify Proposition I: constant areal velocity
GM = 1.0; dt = 0.005; N = 5000;
x = 1; y = 0; vx = 0; vy = 0.9; % elliptical initial conditions
areas = zeros(1,N);
for i=1:N
  r = sqrt(x^2 + y^2);
  ax = -GM*x/r^3; ay = -GM*y/r^3;
  vx = vx + ax*dt; vy = vy + ay*dt;
  xnew = x + vx*dt; ynew = y + vy*dt;
  areas(i) = 0.5*abs(x*ynew - xnew*y); % cross product
  x = xnew; y = ynew;
end
mean_area = mean(areas); std_area = std(areas);
printf('Mean dA/dt = %.6f, relative std = %.2e\n', mean_area, std_area/mean_area);
plot(areas); title('Areal velocity is constant');

2. shell_theorem.m — Numerical integration

Copy% Proposition LXXI numerical test
R=1; M=1; G=1; d=2.5; % distance of particle
nphi=2000; ntheta=2000; Fnum=0;
sigma = M/(4*pi*R^2);
for i=1:nphi
  phi = pi*(i-0.5)/nphi;
  ring_area = 2*pi*R^2*sin(phi)*pi/nphi;
  s = sqrt(R^2 + d^2 - 2*R*d*cos(phi));
  cos_alpha = (d - R*cos(phi))/s;
  dF = G*sigma*ring_area / s^2 * cos_alpha;
  Fnum += dF;
end
Fpoint = G*M/d^2;
printf('Shell force = %.8f, point mass = %.8f, error = %.2e\n', Fnum, Fpoint, abs(Fnum-Fpoint)/Fpoint);

3. inverse_square_orbits.m — Conic sections

Copy% Prop XI: orbits under 1/r^2
GM=1; dt=0.01; steps=10000;
for e = [0.3, 0.9, 1.0, 1.3] % ellipse, parabola, hyperbola
  x=1; y=0; v0=sqrt(GM*(1+e)); vx=0; vy=v0;
  xs=[]; ys=[];
  for i=1:steps
    r=sqrt(x^2+y^2); ax=-GM*x/r^3; ay=-GM*y/r^3;
    vx+=ax*dt; vy+=ay*dt; x+=vx*dt; y+=vy*dt;
    xs(end+1)=x; ys(end+1)=y; if r>5, break, end
  endfor
  plot(xs,ys); hold on;
endfor
axis equal; legend('e=0.3','e=0.9','e=1.0','e=1.3'); title('Inverse-square orbits');

4. three_body_problem.m — Newton Prop LXVI

Copy% Restricted three-body, Lagrange points
mu=0.01215; % Earth-Moon mass ratio
omega=1; % rotating frame
% Equations of motion in rotating frame
f = @(t,z) [z(3); z(4); 2*omega*z(4)+omega^2*z(1) - (1-mu)*(z(1)+mu)/r1^3 - mu*(z(1)-1+mu)/r2^3; ...
           -2*omega*z(3)+omega^2*z(2) - (1-mu)*z(2)/r1^3 - mu*z(2)/r2^3];
% with r1=norm([z1+mu,z2]), r2=norm([z1-1+mu,z2])
% Integrate to show tadpole orbit around L4

Full script available in source. Numerically integrates Prop LXVI corollaries on lunar perturbations.

5. tides.m — Lunar vs solar tides

Copy% Book III, Prop XXIV: tidal forces
G=6.674e-11; M_moon=7.35e22; M_sun=1.99e30;
R_earth=6.37e6; D_moon=3.84e8; D_sun=1.496e11;
% tidal acceleration difference across Earth: ~2GMR/D^3
a_moon = 2*G*M_moon*R_earth / D_moon^3;
a_sun  = 2*G*M_sun*R_earth / D_sun^3;
ratio = a_moon / a_sun;
printf('Lunar tidal acceleration: %.2e m/s^2\n', a_moon);
printf('Solar tidal acceleration: %.2e m/s^2\n', a_sun);
printf('Moon/Sun tide ratio = %.2f (Newton predicted ~2.2)\n', ratio);

8. Interactive Principia Lab

Lab A: Projectile in Resisting Medium, Book II

Newton proves in Book II Prop IV that in a medium with resistance proportional to velocity, the trajectory is not a parabola but a logarithmic curve. Compare.

Lab B: Planetary Orbit and Equal Areas

Lab C: Two-Body Mutual Attraction

Newton's third law implies both bodies orbit their common centre of mass. No body is truly fixed.

9. Philosophy Legacy

Newton's Principia transformed philosophy as much as science. By asserting absolute space and time as the sensorium of God, Newton provoked Leibniz to develop a relational metaphysics. In the Leibniz-Clarke correspondence, 1715-16, Leibniz argued space is merely the order of coexistences, time the order of successions, a debate that prefigures Mach and Einstein.

Ernst Mach in 1883 criticized Newton's bucket, arguing inertia arises from relation to distant masses. Einstein's general relativity, 1915, realized a version of Mach's principle, spacetime is dynamical, not absolute, yet retains Newtonian limits.

"Hypotheses non fingo" became a motto for positivism. Newton refused to feign a mechanism for gravity, yet later generations, from Euler to field theorists, supplied them. The tension between mathematical description and causal explanation remains central to philosophy of science.

Finally, Newton the natural philosopher cannot be separated from Newton the theologian and alchemist. He believed the unity of the world reflected a reasonable Creator, and that the same God who gave absolute laws gave revealed religion. The Principia's General Scholium ends not with a formula but with a confession, "This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being." For Newton, mathematical philosophy and natural theology were two volumes of the same book.

The Principia remains unique, a work where philosophy, mathematics, and experiment are interwoven without seam. Its proofs, though cast in ancient geometry, contain the limit, the derivative, the integral, and the differential equation. To read it is to watch modern science being born.