Kepler before Newton
Two generations before Newton, Johannes Kepler inherited a treasure: decades of astonishingly precise naked-eye planetary observations recorded by Tycho Brahe. By years of stubborn arithmetic Kepler distilled that data into three laws of planetary motion. Crucially, they were empirical — accurate descriptions of what the planets do, with no explanation of why. Newton's law of gravity would later supply the why, and reproduce all three exactly.
First law: orbits are ellipses
Kepler's first law: each planet moves on an ellipse with the Sun at one focus — not at the center. An ellipse is a slightly squashed circle; a circle is just the special case where the two foci merge. Most planetary orbits are only mildly elliptical, but the offset is real and matters.
Because the Sun sits off-center, a planet's distance changes over its year. The closest point is perihelion and the farthest is aphelion. (For a satellite orbiting Earth the matching words are perigee and apogee.) The Earth itself is about 3% closer to the Sun in early January than in early July — a fact independent of, and much smaller in effect than, the seasons caused by axial tilt.
Second law: equal areas in equal times
Kepler's second law: the line joining the Sun to a planet sweeps out equal areas in equal intervals of time. In plain terms, a planet moves fastest at perihelion (close in) and slowest at aphelion (far out). This is not a separate mystery — it is conservation of angular momentum in disguise. Gravity always points straight at the Sun, so it exerts no twist (no torque) about the Sun, and the planet's angular momentum L stays constant.
The rate at which area is swept equals the angular momentum divided by twice the mass. Because L is conserved, the areal rate is constant — which is exactly Kepler's second law.
Third law: the harmony T² ∝ a³
Kepler's third law: the square of a planet's orbital period is proportional to the cube of its orbit's semi-major axis a (the average of its closest and farthest distances). Planets far from the Sun take dramatically longer to go around. We can derive it exactly for a circular orbit from Guide 3: take v = \sqrt{GM/r} and T = 2\pi r / v, then square.
Kepler's third law. The proportionality constant 4π²/GM depends only on the central mass M, not on the orbiting body — so every planet around the Sun obeys the same T²/a³.
One family: orbits as conic sections
Newton proved something beautiful: under an inverse-square force, every trajectory is a conic section — circle, ellipse, parabola, or hyperbola. Which one you get depends only on how much energy the body has. A comet swinging once past the Sun on a hyperbola and a planet looping endlessly on an ellipse obey the very same law. That energy dividing line — bound versus escaping — is exactly where the next guide begins.