Divine Geometer

Divine Geometer

After three years of calculation Johannes Kepler believed he had found the correct values for a Martian circular orbit, which matched ten of Tycho’s observations within two minutes of arc. Now, there are 60 minutes of arc in an angular degree, and 90 degrees, a right angle, from the horizon to horizon. So a few minutes of arc is a very small quantity to measure – especially without a telescope. It is one-fifteenth the angular diameter of the full Moon as seen from Earth.

But Kepler’s replenishable ecstasy soon crumbled into gloom – because two of Tycho Brahe’s further observations were inconsistent with Kepler’s orbi, by as much as eight minutes of arc:

Divine Providence granted us such a diligent observer in Tycho Brahe that his observations convicted this … calculation of an error of eight minutes; it is only right that we should accept God’s gift with a grateful mind… If I had believed that we could ignore there eight minutes, I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.

Kepler was shaken at being compelled to abandon a circular orbit and to question his faith in the Divine Geometer. Having cleared the stable of astronomy of circles and spirals, he was left, he said, with “only a single cartful of dung”, a stretched-out circle something like an oval.

“The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted… Ah, what a foolish bird I have been” (Johannes Kepler). Image:  © Megan Jorgensen (Elena)

Eventually, Kepler came to feel that his fascination with the circle had been a delusion. The Earth was a planet, as Copernicus had said, and it was entirely obvious to Kepler that the Earth, wracked by wars, famine, pestilence and unhappiness, fell short of perfection. Kepler was one of the first people since antiquity to propose that the planets were material objects made of imperfect stuff like the Earth. And if planets were “imperfect”, why not their orbits as well? He tried various oval-like curves, calculated away, made some arithmetical mistakes (which caused him at first to reject the correct answer) and months later in some desperation tried the formula for an ellipse, first codified in the Alexandrian Library by Apollonius of Perga. He found that it matched Tycho’s observations beautifully: “The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted… Ah, what a foolish bird I have been”.

Kepler had found that Mars moves about the Sun not in a circle, but in an ellipse. The other planets have orbits much less elliptical that that of Mars, and if Tycho had urged him to study the motion of, say, Venus, Kepler might never have discovered the true orbits of the planets.