The Geometrical Root of the Area-Measure of Time (From Kepler’s Astronomia nova)

Abstract

In Astronomia nova (1609), Kepler quantified a geometrically exact measure of time. His innovative approach was to consider intervals (discrete but) so small that in his terms the results became exact. He divided the orbit (initially circular) into 360 equal arcs from its centre to establish uniformity of time, and combined these arcs with 360 distances drawn from the Sun, to represent variable increments of time in orbit. Those increments did not accurately measure areas; but subsequently Kepler discovered a sound geometrical reformulation of the increment which produced an area-representation that he was able straightforwardly to sum. (He wrongly described his method as ‘distance-summation’). Then the time in orbit was measured by a sector of the exact ellipse he had concurrently constructed, founded on the work of Archimedes. Essential diagrams, and the notable justification, from Euclid’s Elements, of the reformulation of the increment, have been added on Kepler’s behalf.