Johannes Kepler (1571 - 1630)

The end of the sixteenth century was a time fraught with change, wars and religious strife. Despite this Kepler, an innkeeper’s son, rose to become Imperial Mathematician to the Holy Roman Emperor.

Life was not easy for Kepler though. Born in Protestant Germany he was deeply religious and spent his life moving round the Germanic world occasionally fleeing persecution.

He also had a tragic family life. His beloved first wife and much loved son died within the same year. And in the last years of his life, his mother was arrested for a crime she didn’t commit, dying soon afterwards. He spent the next six years clearing her name.

Against this troubled backdrop, Kepler and others helped to build what we call the Enlightenment. The Enlightenment was the triumph of reason over medieval mysticism.

Despite the part that Kepler played in the Enlightenment, he was a student of both science and astrology. Indeed he would have believed that they were two sides of the same scientific coin.

Astrology, then, was included in the study of astronomy. It was believed that the particular conjunctions of heavenly bodies at the moment of birth absolutely determined the fortunes of the newly born person throughout their subsequent life.

Fame and fortune came to these who understood the motions of the planets and could then cast horoscopes.

But Kepler’s principal interest was scientific. He became mathematical assistant to the great Danish astronomer Tycho Brahe and thereby had access to his detailed and methodical observations.

Bringing his mathematical skills to bear on Brahe’s observations he developed the three laws of planetary motion which still bear his name.

Actually Kepler’s laws are more radical than they first appear. Copernicus had already come to the conclusion that the Earth revolved around the Sun - rather than the other way round.

Galileo had also come to the same conclusion but, on pain of excommunication from the Catholic church, had been forced to deny it. Kepler’s laws put the Sun at the centre of the planetary system and gave precise laws about the planets rotation about it. This Copernican view finally became accepted.

Kepler also helped to publicise and extend Napier’s work on logarithms. The idea of logarithms as a practical means of doing involved calculations was distrusted at the time, much as computers were in the 1960s.

Kepler’s mathematics

Keplers laws are easily stated and understood:

First Law The orbit of a planet/comet about the Sun is an ellipse with the Sun’s centre at one focus

Second Law A line joining a planet (or comet) and the Sun sweeps out equal areas in equal intervals of time.

Third Law The square of the period (T) of a planet (or comet) is proportional to the cube of its semi-major axis (R),

T^2\;\alpha\;{R^3}\ \mbox{or written another way,}\ {T^2\over{R^3}}=\mbox{constant}

The accuracy of this final law is shown in this table:

PlanetYear length  {T^2}\over{R^3}
Mercury  0.241

Kepler published a mathematical account of logarithms and used them in his astronomical work which proved them to be remarkably accurate and very practical.

He also became interested in finding mathematical methods of determining the so-called volumes of revolution.

These are created when a curve is rotated about an axis. For example, if a parabola is rotated about a vertical axis, as shown below, it creates such a volume:

Kepler based his method on the work of Archimedes, who divided up volumes of solid figures in to small slices called ‘indivisibles’: a slice of the paraboloid might be considered as a cylinder.

The volume of the cylinder can easily be calculated; in turn the volume of the paraboloid may also be determined:

Finding the volumes of these and then adding them all together gave him the volume he sought. Kepler’s work on volumes was later developed by Cavalieri and became part of the development of a powerful mathematical tool - calculus.

He perfectly blended the practical man with the man of ideas and theories - he designed two new telescopes and also gave the first correct explanation of the working of the human eye.