= Orbital observations behind the area law
Why infer an <area> law from <astronomy>? In the authored account <The great Martian catastrophe and how Kepler fixed it by Owen Gingerich>, a 13 August 1593 <Mars> observation has <ecliptic longitude> $346^\circ7.5'$, against predicted $342^\circ0'$ and $351^\circ26'$. These are Earth-based directions with a date, not measured Sun-based <areas>. The observer's <armillary sphere> used graduated circles and sighting arms to read <declination>; reversing it allowed a second reading. Comparing <stars> at different altitudes exposed <refraction>, and a planet-based check revealed an erroneous solar correction table. The apparent <Mars> distance signal had been contaminated by that correction. <Johannes Kepler> subsequently corrected the observer's <Earth> <orbit>, then tested <Mars> trajectories. The account describes typical angular errors of $2'$ or less ($120$ <arcseconds>): small residuals needed care.<The great Martian catastrophe and how Kepler fixed it by Owen Gingerich>[pp. 51-53: Fig. 2, refraction checks, 13 August log table, Earth-orbit correction and angular errors]{ref}
The Tufts class notes <The Newtonian Revolution – Part One; Philosophy 167: Science Before Newton's Principia/Kepler's Astronomia Nova and the Orbit of Mars> explain the key limitation: an <area> requires a trajectory and distances, so <Kepler's second law> cannot be directly tested from angular observations alone. They describe checking the combined <ellipse> and <area> law against 28 dated observations around seven oppositions during 1583-1595, using an <Earth> model to predict Earth-based directions; agreement improved greatly but some discrepancies remained.<The Newtonian Revolution – Part One; Philosophy 167: Science Before Newton's Principia/Kepler's Astronomia Nova and the Orbit of Mars>[p. 20, items 3-5, and pp. 26-27, section IV.D: predictions, residuals and dependence of the area test on a trajectory]{ref}
The inferential chain is useful today too: recover relative <positions> and <times>, compare swept <areas>, and ask what <forces> are compatible with their evolution. Agreement with equal <areas> supports zero <torque> about the chosen origin under the assumptions below; it alone does not identify an <inverse-square law>. Recovering the radial <force> also needs the trajectory and its timing, as in <inferring an inverse-square force from Kepler laws>. The two diagrams here are calculated examples, not the historical measurements.
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