Isaac Newton

The Dualistic Celestial-Terrestrial Impasse (1600–1665)

Prior to the publication of the Principia (1687), physical science was split across incompatible paradigms: * Aristotelian/Scholastic Terrestrial Physics: Held that heavy bodies moved toward the center of the Earth due to an intrinsic teleological quality ("gravity" as an internal striving for natural rest), while motion in vacuum was impossible. Horizontal motion was maintained only by physical contact with a surrounding medium (antiperistasis). * Galilean Kinematics (1638): Established uniform gravitational acceleration on Earth ($s = \frac{1}{2}gt^2$) and the parabolic trajectory of projectiles, but treated terrestrial gravity as a local constant rather than a universal attractive force. * Keplerian Planetary Kinematics (1609, 1619): Johannes Kepler formulated three empirical laws of planetary motion from Tycho Brahe’s observational data: 1. Planets trace ellipses with the Sun at one focus. 2. The radius vector sweeps out equal areas in equal times ($\frac{dA}{dt} = \text{const}$). 3. The square of orbital period is proportional to the cube of semi-major axis ($T^2 \propto a^3$). Kepler attributed this to an elusive magnetic "anima motrix" emitted by the rotating Sun, lacking a unified mechanical force model. * Cartesian Ether Vortices (1644): René Descartes proposed that space was a plenum filled with swirling, microscopic ether particles. Celestial bodies were swept along in concentric vortices like flotsam. This model was purely qualitative; it could not predict planetary periods, could not account for comets cutting across orbital planes, and introduced frictional drag incompatible with stable planetary conservation of momentum.

2.1 The Three Axioms of Motion (Axiomata sive Leges Motus)

Codified in the opening of the Principia (1687):

1. First Law (Inertia): Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. $$\mathbf{F}_{\text{net}} = 0 \implies \frac{d\mathbf{p}}{dt} = 0 \implies \mathbf{v} = \text{constant}$$
2. Second Law (Proportional Acceleration): The change of momentum of a body is proportional to the motive force impressed, and takes place along the straight line in which that force is impressed. $$\mathbf{F} = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} = m \frac{d^2\mathbf{r}}{dt^2} \quad (\text{for constant mass } m)$$
3. Third Law (Action and Reaction): To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. $$\mathbf{F}_{12} = -\mathbf{F}_{21}$$

2.2 The Law of Universal Gravitation and Kepler's Third Law

Newton postulated that every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance separating their centers:

$$\mathbf{F}_{12} = -G \frac{m_1 m_2}{r^2} \hat{\mathbf{r}}_{12}$$

Derivation of the Inverse-Square Law from Kepler's Laws

Consider a planet of mass $m$ executing a circular orbit of radius $r$ with velocity $v$ and period $T$ around the Sun of mass $M$: 1. The centripetal acceleration required to maintain the circular path is: $$a_c = \frac{v^2}{r} = \frac{(2\pi r / T)^2}{r} = \frac{4\pi^2 r}{T^2}$$ 2. By Kepler's Third Law: $$\frac{r^3}{T^2} = k \implies T^2 = \frac{r^3}{k}$$ 3. Substituting $T^2$ into the acceleration formula: $$a_c = \frac{4\pi^2 r}{r^3 / k} = \frac{4\pi^2 k}{r^2}$$ 4. Multiplying by planetary mass $m$ gives the gravitational force: $$F = m a_c = \frac{4\pi^2 k \cdot m}{r^2} \propto \frac{m}{r^2}$$ By the Third Law of Motion, the force must be symmetric in $M$ and $m$, requiring $4\pi^2 k = G M$, yielding: $$F = G \frac{M m}{r^2}$$

2.3 The Shell Theorem (Theorema de Conchis)

In Principia, Book I, Propositions 70–76, Newton solved the fundamental integration problem: does a spherical planet act upon an external object as if all its mass were concentrated at its exact center?

External Potential of a Spherically Symmetric Shell

Let a thin spherical shell have radius $R$, uniform surface mass density $\sigma$, and total mass $M = 4\pi R^2 \sigma$. Let point mass $m$ be situated at distance $r > R$ from the shell's center.

The gravitational potential $V(r)$ is the surface integral over the shell: $$V(r) = -G \sigma \int_0^{2\pi} d\phi \int_0^\pi \frac{R^2 \sin \theta \, d\theta}{\sqrt{r^2 + R^2 - 2rR \cos \theta}}$$

Let $s = \sqrt{r^2 + R^2 - 2rR \cos \theta}$, so $s \, ds = r R \sin \theta \, d\theta$: $$V(r) = -G \frac{2\pi \sigma R^2}{r R} \int_{r-R}^{r+R} ds = -\frac{2\pi G \sigma R}{r} [(r+R) - (r-R)] = -\frac{4\pi G \sigma R^2}{r} = -\frac{G M}{r}$$

The gravitational field outside the shell is: $$\mathbf{g}(r) = -\nabla V(r) = -\frac{G M}{r^2} \hat{\mathbf{r}}$$ This proved rigorously that planets and moons could be treated mathematically as dimensionless point masses centered at their geometric centers.

2.4 The Method of Fluxions (Differential Calculus)

Newton formulated the calculus of fluents (quantities $x, y$ flowing with time) and fluxions (instantaneous rates of flow $\dot{x}, \dot{y}$).

Let a curve be defined by polynomial $f(x, y) = 0$. Replace $x \to x + \dot{x}o$ and $y \to y + \dot{y}o$, where $o$ is an infinitely small interval of time: $$f(x + \dot{x}o, y + \dot{y}o) = 0$$ Expanding by binomial theorem, canceling zero-order terms $f(x, y) = 0$, dividing through by $o$, and dropping remaining terms containing factor $o$: $$\frac{\partial f}{\partial x} \dot{x} + \frac{\partial f}{\partial y} \dot{y} = 0 \implies \frac{\dot{y}}{\dot{x}} = -\frac{\partial f / \partial x}{\partial f / \partial y} = \frac{dy}{dx}$$

3.1 Newton's Cannon: Unification of Terrestrial and Orbital Dynamics

                         (Orbital Insertion: v = v_orbit)
                                    . - ~ ~ - .
                                . '             ' .
                            . '                     ' .
                           /                           \
                          /    * Mountain Top (Cannon)  \
                         |    / |                        |
                         |   /  |                        |
                         |  /   | (Earth: Mass M, R)     |
                          \     |                       /
                           \    |                      /
                            . ' |                   . '
                                ' - . _ _ _ . - '
                        Sub-orbital Impact (v < v_orbit)

3.2 Prism Dispersion and White Light Decomposition (1672)

flowchart LR
    Sun[Sunlight Beam] --> Aperture[Pinhole Aperture in Shutter]
    Aperture --> Prism1[Glass Prism 1: Differential Refraction]
    Prism1 --> Red[Red Ray: Minimal Refraction]
    Prism1 --> Violet[Violet Ray: Maximum Refraction]
    Red --> Screen[Chromatically Dispersed Spectrum: ROYGBIV]
    Violet --> Screen
    Violet --> Aperture2[Slit Isolating Monochromatic Violet]
    Aperture2 --> Prism2[Glass Prism 2]
    Prism2 --> UnalteredViolet[Unaltered Violet: Proves Color is Invariant]

4.1 The "Moon Test" of 1666 (Woolsthorpe Verification)

Newton tested the inverse-square hypothesis by comparing the acceleration of the Moon in orbit against the acceleration of falling bodies on Earth: * Distance to Moon: $r_{\text{Moon}} \approx 60 \, R_{\text{Earth}}$. * Expected Moon Acceleration: By inverse-square law: $$a_{\text{Moon}} = g \cdot \left(\frac{R_{\text{Earth}}}{r_{\text{Moon}}}\right)^2 = \frac{9.81\,\text{m/s}^2}{60^2} = \frac{9.81}{3600} \approx 0.00272\,\text{m/s}^2$$ * Calculated Orbital Acceleration: From the Moon's orbital period $T \approx 27.32\text{ days} \approx 2.36 \times 10^6\text{ s}$ and radius $r \approx 3.84 \times 10^8\text{ m}$: $$a_c = \frac{4\pi^2 r_{\text{Moon}}}{T^2} \approx \frac{4\pi^2 (3.84 \times 10^8)}{(2.36 \times 10^6)^2} \approx 0.00272\,\text{m/s}^2$$ The empirical agreement ("answered pretty nearly") confirmed that terrestrial gravity and celestial orbital forces were identical.

4.2 Equivalence of Inertial and Gravitational Mass (The Pendulum Trials)

To test whether different materials experience identical gravitational acceleration in vacuum, Newton constructed two identical box pendulums, filling them with identical weights of diverse substances (gold, silver, lead, glass, sand, salt, water, wheat): * Period Formula: $T = 2\pi \sqrt{\frac{m_i}{m_g} \frac{L}{g}}$. * Result: In all trials, the oscillation frequencies matched to within one part in a thousand ($\frac{m_i}{m_g} = 1.000 \pm 0.001$), establishing the weak Equivalence Principle that Einstein would elevate to general relativity.

5.1 The Perihelion Precession of Mercury

Newtonian mechanics predicts that in an isolated two-body system, orbits are closed ellipses. Planetary perturbations (Venus, Jupiter, Earth) induce an advance of Mercury's perihelion of $531.6''$ per century: * Failure Mode: Le Verrier (1859) discovered an anomalous residual precession of $43''$ per century that could not be accounted for by any known Newtonian mass. * Resolution: Solved in 1915 by Einstein's General Relativity via spacetime metric curvature around solar mass.

5.2 Action-at-a-Distance & Instantaneous Propagation

Newtonian gravity assumes gravitational potential propagates instantaneously ($\Delta t = 0$): $$\nabla^2 \Phi = 4\pi G \rho$$ * Failure Mode: Violates Special Relativity: no causal influence can exceed the speed of light $c$. * Resolution: Gravitational waves propagate at $c$, governed by the linearized Einstein field equations.

(Badge Key: V Verified Verified Primary Source; C Corroborated Corroborated Multi-Source; S Single-Source Single Source; I Interpretation Critical Interpretation).

1. Newton, Isaac (1669, pub. 1711). De analysi per aequationes numero terminorum infinitas [On Analysis by Infinite Equations]. London: William Jones. * Archival Role: The earliest manuscript circulating among Cambridge scholars documenting fluxions, fractional exponents, and infinite series.
2. Newton, Isaac (1672). "A Letter of Mr. Isaac Newton... Containing His New Theory about Light and Colors." Philosophical Transactions of the Royal Society of London, Vol. 6, No. 80, pp. 3075–3087. * Archival Role: Newton’s first published scientific paper. Details prism experiments, spectral decomposition, and the physical nature of color.
3. Newton, Isaac (1687). Philosophiae Naturalis Principia Mathematica [Mathematical Principles of Natural Philosophy]. London: Printed by Joseph Streater for the Royal Society (financed by Edmond Halley). * Archival Role: The foundational treatise of modern physical science. Axiomatizes the laws of motion, universal gravitation, and celestial mechanics.
4. Newton, Isaac (1704). Opticks: or, a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light. London: Printed for Sam. Smith, and Benj. Walford. * Archival Role: Comprehensive experimental treatise on geometric and physical optics, thin-film interference (Newton's rings), and particle corpuscular light theory.
5. Newton, Isaac (1713). "General Scholium." Added to the Second Edition of the Principia (edited by Roger Cotes). Cambridge University Press. * Archival Role: Famous methodological defense containing "Hypotheses non fingo", defining empirical scientific deduction and natural theology.