Dec 272020

Today is the birthday (1571) of Johannes Kepler, a German astronomer, mathematician, and astrologer (no doubt named after John the Apostle because this is his saint’s day ). Kepler is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonices Mundi, and Epitome Astronomiae Copernicanae. These works also provided one of the foundations for Newton’s theory of universal gravitation which augmented Kepler’s concepts of planetary motion.

Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting (or Keplerian) telescope, and was mentioned in the telescopic discoveries of his contemporary Galileo Galilei. He was a corresponding member of the Accademia dei Lincei in Rome.

Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of natural philosophy). Kepler pulled together astronomical observations with mathematics and physics to build the foundations of astrophysics. He also incorporated religious arguments and reasoning into his work, motivated by the religious conviction and belief that God had created the world according to an intelligible plan that is accessible through reason. Kepler described his new astronomy as “celestial physics,” as “an excursion into Aristotle’s Metaphysics,” and as “a supplement to Aristotle’s On the Heavens,” transforming the ancient tradition of physical cosmology by treating astronomy as part of a universal mathematical physics. When Newton said that he saw so far because he stood on the shoulders of giants, Kepler was one of those giants.

Delving into all that Kepler accomplished is way too much for a short blog post, so let me focus on his perhaps most famous formulation, his laws of planetary motion.  The three laws state that:

  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the slower its orbital speed, and vice versa. As such, Kepler refined the theories proposed by Copernicus.

If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agreed with Copernicus:

1. The planetary orbit is a circle.
2. The Sun is at the center of the orbit.
3. The speed of the planet in the orbit is constant.

The eccentricities of the orbits of those planets, known to Copernicus and Kepler, are small, so the foregoing rules give fair approximations of planetary motion, but Kepler’s laws fit the observations better than does the model proposed by Copernicus. Kepler’s corrections are:

1. The planetary orbit is not a circle, but an ellipse.
2. The Sun is not at the center but at a focal point of the elliptical orbit.
3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed (closely linked historically with the concept of angular momentum) is constant.

If you are lost at this point – or don’t care about these points – I won’t beat you up, and I certainly won’t add any equations, or the like. It’s enough to be aware that the planetary orbits are elliptical rather than circular. Kepler shows us that physics is in a perpetual state of refinement (that never ceases). Newton added his bits and much later Einstein added his. Understanding is in a constant state of change, leading to a general acknowledgement that change is the only constant. Scientific theories never explain everything: there are always observations that cannot be accounted for by current theories. Hence the periodic revolutions in scientific thinking.

I will admit that my eyes have the habit of glazing over when I see something like r=p/1+ϵcosθ (or more complex) and it’s not because I do not understand the formula.  It’s because my interests do not lie in delving deeper into the implications.  There’s the nub of the matter. I accept the conclusions and move on. But . . . there are people like Kepler who are never satisfied. The work of Copernicus was fairly close to the truth, but the anomalies bothered Kepler and he worked tirelessly to get yet closer to the truth.  This is one of the reasons I get so aggravated with flat earthers and other science deniers.  They think that some vague intuitions based on limited sensory information can challenge the exhaustive work of dedicated professionals.  I despair – sometimes.

Kepler was born in Weil der Stadt, currently in the Stuttgart Region of the German state of Baden-Württemberg. Among other things, it is well-known for its Fasnetsküchle which will be available very soon (after Epiphany). Fasnetsküchle are similar to the so-called “Berliner,” or jelly doughnuts, or “Krapfen,” which are available year round. But the Swabian version is distinctly different: it must be flat and square or rectangular, never round nor as tall as a Berliner, and it’s not supposed to be filled with jam – it should be plain inside and out. True Swabians insist that Fasnetsküchle may not even be sprinkled with cinnamon or powdered sugar. But times change and you now find them with a dusting of sugar (and cinnamon).



2 cups whole milk, warmed to 110F/43C
4½ teaspoons active dry yeast (two packages)
¾ cup and 1 pinch granulated sugar, divided
5 to 6 cups all-purpose flour, divided
2 eggs
1 teaspoon vanilla extract
1¼ teaspoons salt
4 tablespoons unsalted butter, melted
oil for deep- frying (lard is traditional)


Pour the warm milk into bowl. Stir in the yeast and a pinch of granulated sugar. Let stand for 5 to 10 minutes, or until it has become bubbly. Add 2 cups of flour to the mixture and stir with a wooden spoon until a smooth batter forms. Cover with plastic wrap and set in a warm spot for 30 minutes. By now the mixture should have risen and become bubbly.

In a medium bowl, whisk the eggs until pale yellow and frothy (about three minutes). Add the sugar, vanilla extract and salt, and whisk until combined and smooth.

Add the egg mixture to the dough and hand-knead until mostly combined. Add the melted butter and mix. Gradually add three more cups of flour to the mixture and continue to knead until very soft dough comes together (it will be rather slack and a bit sticky.) If necessary, add up to another cup of flour, a spoonful at a time, until the dough firms. Transfer the dough to a lightly greased bowl, cover with plastic wrap, or a kitchen cloth, and let it set in a warm spot until dough has doubled in size (20 to 30 minutes).

Remove the dough from the bowl and turn out on to a floured work surface. With your fingers, or a rolling pin, push down the dough into an even layer. Sprinkle flour on the dough and roll it out to about ½-inch thickness. If the dough doesn’t hold its shape and springs back, cover with a damp towel and let it rest for a few more minutes and try again.

Cut out 3 x 3 inch squares or 3 x 4 rectangles of dough. Transfer the dough pieces to parchment-lined baking sheets. Gather scraps of dough and again roll out and cut until you have used up all of the dough. Cover the baking sheets loosely with a dish-towel, plastic wrap and place in a warm, draft-free spot until they are almost doubled in size, about 30 minutes.

Meanwhile, heat at least 1½ to 2 inches or more of deep frying shortening or oil in a heavy-bottomed pot or deep skillet over medium heat to 350F/176C. Carefully lower about three or four Küchle into the oil one at a time (be sure not to over-crowd the pan) and fry until the bottom is golden brown. Carefully turn them over and continue to fry until the other side is golden brown.

Remove with a slotted spoon, and drain on a wire rack. Repeat for the remaining Küchle. They are best eaten warm.

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