Kepler

"Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God." - Johannes Kepler

The astronomer and mathematician Johannes Kepler was born on this day, Dec. 27, 1571.

Kepler was a committed Christian. Wiki mentions he was perceived to have been a Calvinist. Whether that's true, it's true Kepler considered the Lutheran pastorate prior to astronomy.

I thought it might be worth quickly running through some of Kepler's main achievements since he's one of the many Christians who helped expand the frontiers of scientific knowledge and this is, after all, the anniversary of his birth.

Kepler is, of course, most known for his three laws of planetary motion, particularly the first law which demonstrated planets moved in elliptical orbits rather than circular ones. Although he was originally utterly convinced planets moved in circles. As Richard Feynman describes in The Character of Physical Law:

When all these data [from Tycho Brahe's observations] were collected they came into the hands of Kepler, who then tried to analyze what kind of motion the planets made around the sun. And he did this by a method of trial and error. At one stage he thought he had it; he figured out that they went round the sun in circles with the sun off center. Then Kepler noticed that one planet, I think it was Mars, was eight minutes of arc off, and he decided this was too big for Tycho Brahe to have made an error, and that this was not the right answer. So because of the precision of the experiments he was able to proceed to another trial and ultimately found out three things.

First, he found that the planets went in ellipses around the sun with the sun as a focus. An ellipse is a curve all artists know about because it is a foreshortened circle. Children also know because someone told them that if you put a ring on a piece of cord, anchored at each end, and then put a pencil in the ring, it will draw an ellipse...The two points A and B are the foci. The orbit of a planet around the sun is an ellipse with the sun at one focus.

The next question is: In going around the ellipse, how does the planet go? Does it go faster when it is near the sun? Does it go slower when it is farther from the sun? Kepler found the answer to this too...He found that, if you put down the position of a planet at two times, separated by some definite period, let us say three weeks - then in another place on its orbit two positions of the planet again separated by three weeks, and draw lines (technically called radius vectors) from the sun to the planet, then the area that is enclosed in the orbit of the planet and the two lines that are separated by the planet's position three weeks apart is the same, in any part of the orbit. So that the planet has to go faster when it is closer to the sun, and slower when it is farther away, in order to show precisely the same area.

Some several years later Kepler found a third rule, which was not concerned only with the motion of a single planet around the sun but related various planets to each other. It said that the time the planet took to go all around the sun was related to the size of the orbit, and that the times varied as the square root of the cube of the size of the orbit and for this the size of the orbit is the diameter across the biggest distance on the ellipse.

Kepler then had these three laws which are summarized by saying that the orbit forms an ellipse, and that equal areas are swept in equal times, and that the time to go round varies as a three half power of the size, that is, the square root of the cube of the size. These three laws of Kepler give a complete description of the motion of the planets around the sun.

Feynman also said in his famous Lectures on Physics:

Thus Kepler's three laws are:

1. Each planet moves around the sun in an ellipse, with the sun at one focus.
2. The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time.
3. The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits:T ∝ a^3/2

Kepler's laws had a large impact. His work leading to the discovery of the elliptical orbits of planets made use of conic sections, ellipses, parabolas, hyperbolas, circles, etc., thereby substantiating the ancient Greeks including one named Apollonius who wrote a book titled Conics. Next, although Kepler lived and died before both Newton and Leibniz, Kepler used a crude form of integral calculus to help him in working on the second law as well as when he found a better way to measure the volume of wine in a wine cask. And Kepler's third law laid the foundation for Isaac Newton's law of gravitation, which he had in part modified based on Kepler's law.

Apart from his laws of planetary motion, Kepler also made other significant contributions. Kepler made discoveries in geometry such as the continuity principle. Moreover, while walking across a bridge in a snowstorm, and in part due to his nearsightedness, Kepler was able to closely observe the snowflakes which had fallen on his arm, whereupon he made several more geometric and other observations. Kepler discovered several polyhedra (e.g. antiprism, cuboctahedron, dodecahedron, triacontahedron).

Kepler helped pave much of the way for modern optics. He discovered how our eyes form images by refraction, as well as the fact that two eyes aid in depth perception. He was able to explain how eyeglasses worked, and subsequently fashioned eyeglasses to aid in farsighted and nearsighted vision. Kepler's work on the same as well as understanding magnification and reflection likewise improved the telescope.

Kepler was the first person to hypothesize that the ebb and flow of the tides were caused by the influence of the moon: "The sphere of influence of the attraction which is in the moon extends as far as the Earth, and incites the waters up from the torrid zone." However, Galieo ridiculed the idea as "occultic." Galileo also disbelieved Kepler's idea that planets traveled in an ellipse.

Scientists and writers like Isaac Asimov and Carl Sagan have considered Kepler's Somnium the first work of science fiction.

The following isn't an accomplishment, but it's noteworthy Kepler also asked as well as sought to answer an ancient riddle known as Olbers' paradox. That is, why is the night sky dark?

Kepler wasn't the first to do so, nor was he the last, but he did recognize it. Edward Harrison writes in Cosmology: The Science of the Universe:

There is a simple and important experiment in cosmology that almost everybody can perform. It consists of gazing at the night sky and noting its state of darkness. When we ask, why is the sky dark at night?...the natural response is the Sun is shining on the other side of the Earth and starlight is weaker than sunlight. It takes an unusual mind to realize that the relative weakness of starlight is of cosmological significance, and such a person was the astronomer Johannes Kepler, imperial mathematician to the emperor of the Holy Roman Empire.

In a forest...a line of sight in any horizontal direction must eventually intercept a tree trunk, and the distant view consists of a background of trees. Similarly, on looking away from Earth at night, we see a "forest" of stars...If the stars stretch away endlessly, a line of sight must eventually intercept the surface of a distant star. The distant view of the universe should consist of a continuous background of bright stars with no separating dark gaps.

We know that the universe consists of clusters of stars, galaxies, and clusters of galaxies, and has astronomical structure more complex than the uniform distribution of stars often supposed in previous centuries. The universe expands and may consist of non-Euclidean geometry. Even so, we shall continue to think of the riddle of darkness in its original form, and later show that developments in cosmology have made little difference to the riddle.

In a universe of infinite extent, populated everywhere with bright stars, the entire sky should be covered by stars with no separating dark gaps. Hence, when all stars are bright like the Sun, the entire sky at every point should blaze with a brilliance equal to the Sun's disk. The sky is 180,000 times larger than the Sun's disk, and starlight falling on Earth should be 180,000 times more intense than sunlight. In the midst of this inferno of intense light, life would cease in seconds, the atmosphere and oceans boil away in minutes, and the Earth turn to vapor in days. Fortunately, the sky at night is dark. What then is wrong with the forest analogy?

Kepler's proposed "solution" to the paradox was to deny an infinite or boundless universe in favor of a finite one with boundaries. Harrison continues:

According to Kepler, the universe is not like an endless forest; instead, it is like a finite clump of trees in which we looked out between the tree trunks to a dark enclosing wall. He did not use the forest analogy. Nonetheless, he realized that in an endless universe the stars would collectively outshine the Sun and flood the heavens with light far more intense than we actually observe.

The infinite stellar universe in Kepler's Conversation its most devastating criticism. The choice was clear: either a cosmic edge and a dark night sky, or no cosmic edge and a blazing sky. Astronomers who followed disliked the idea of a cosmic edge, and over the centuries have sought for the solution of the dark night-sky riddle.

There are two alternative interpretations of the darkness of the night sky...Either the sky is covered with overlapping stars or is not covered with stars.

According to the first interpretation (interpretation A), the sky is actually covered by stars with no gaps in between, and the riddle in effect asks: what has happened to the missing starlight that never reaches us on Earth? Most studies of the riddle have assumed that this is the correct interpretation...All presently known scientific solutions are given in Darkness at Night: A Riddle of the Universe by the author...

In the second interpretation (interpretation B), the sky is not covered with stars and the dark gaps are real. The riddle in effect asks: what has happened to the missing stars? Kepler chose this interpretation when he argued that the dark gaps between stars was evidence of a finite universe bounded by a dark enclosing wall.

For all Kepler's accomplishments, however, he lived a hard life. Mathematician Howard Eves writes in his book In Mathematical Circles:

It is sad that Kepler's personal life was made almost unendurable by a multiplication of worldly misfortunes. An infection with smallpox when he was but four years old left his eyesight much impaired.

In addition to his general lifelong weakness, he spent a joyless youth, his marriage was a constant source of unhappiness, his favorite child died of smallpox, his wife went mad and died, he was expelled from his lectureship at the University of Gratz when that city fell to the Catholics, his mother was charged and imprisoned for witchcraft and for almost a year he desperately tried to save her from the torture chamber, he himself very narrowly escaped condemnation of heterodoxy, and his stipend was always in arrears.

One report says that his second marriage was even less fortunate than his first although he took the precaution to analyze carefully the merits and demerits of eleven girls before choosing the wrong one.

He was forced to augment his income by casting horoscopes, and he died of a fever while on a journey to obtain some of his long overdue salary.

Notwithstanding, Kepler apparently voiced in his later years:

I had the intention of becoming a theologian...but now I see how God is, by my endeavors, also glorified in astronomy, for "the heavens declare the glory of God."

His epitaph reads:

Mensus eram coelos, nunc terrae metior umbras.
Mens coelestis erat, corporis umbra jacet.

I used to measure the heavens, now I shall measure the shadows of the earth.
Although my soul was from heaven, the shadow of my body lies here.