Wednesday, March 21, 2012

Johannes Kepler (1571-1630): The Secret of the Universe

by G. Jack Urso
 

Early Years
 

Fig. 1: Kepler
Johannes Kepler, the son of a mercenary and an innkeeper's daughter, was born in the Holy Roman Empire, which comprised much of what is now modern Germany. Unlike the noble upbringing of his contemporary Tycho Brahe, Kepler grew up in a working-middle class atmosphere, often helping out at his grandfather's inn (Field). As a Lutheran growing up in the Catholic Holy Roman Empire, Kepler's faith was tolerated under Emperor Rudolf. However, in 1612, Mathias II ascended to the throne of the Holy Roman Empire following the death of his brother, Rudolf. Less tolerant than his brother, this new emperor compelled Kepler to flee to Prague (Burne 572).  


Following his exile, Kepler was employed as an astrologer for Albrecht von Wallenstein (1583-1632), the duke of Friedland, who led his armies to many victories during the Thirty Years War (1618-1648). Kepler was disdainful of conventional astrology and presented interpretations based on the angles between planets and stars. He died while on route to collect money owed him in relation to his work, The Rudolphine Tables, a collection of observations and calculations made during under the rule of Holy Roman Emperor Rudolf (Kepler 235, 250).

 
The Mystery of the Universe 
 
 
Kepler's first major work, Mysterium Cosmographicum (The Mystery of the Universe), was published in 1596. In it, he supports the Copernican heliocentric model with numerous arguments, and in more detail than did Copernicus himself. Kepler applied the full force of mathematics in providing proofs of his positions, moving discussion of the Copernican model to another level. Copernicus, while moving the Earth out of the center of the model of the universe, persisted in attributing special qualities to the Earth regarding its orbit and relationship to other heavenly bodies. Kepler rightly proves in Mysterium Cosmographicum that the Earth should be treated no different than any other planet (Kuhn 209-210).
 

Influences
 

Michael Maestlin introduced Kepler to the Copernican heliocentric model at the University of Tübingen (a theological school). Maestlin was a lecturer who served as an early mentor to Kepler and encouraged him in his studies. A series of letters in 1595 from Kepler to Maestlin outlined Kepler's thoughts on proofs regarding the Copernican system. These letters showed Kepler's thoughts on using three-dimensional polyhedrons as mathematical models to be used to figure the orbit and arraignments of the planets instead of the two-dimensional figures that Copernicus and others had been using (The Secret of the Universe 18, 22).
 

Maestlin helped Kepler by providing methods for calculating distances of the planets to the Sun and serving as a sounding board for his ideas. Maestlin suggested that Kepler present his information supporting the Copernican system as a mathematical hypothesis so as to avoid a confrontation with the Lutheran Church. Although Kepler suspected his mentor of privately supporting the Copernican system, Maestlin never made an overt declaration in support of it. In 1616, the same year the Vatican condemned the Heliocentric model, Maestlin would publicly reject the idea in a letter, but this did not stop Kepler from pursuing his ideas (The Secret of the Universe 17).
 

The Church Intercedes
 

Kepler required the permission of the Rector and Senate of the University of Tübingen in order to publish Mysterium Cosmographicum (The Mystery of the Universe). A chapter that tried to reconcile the Copernican system with the Bible was removed as a result of the University's review (The Secret of the Universe 21). While the work itself was approved to be published, the rector who approved it, Matthias Hafenreffer, would later rail against it in a sermon. This was not the first time Kepler encountered the religious intolerance. As a student at Tübingen University a professor rejected an essay he wrote on the Copernican system. Later, in 1620 he would successfully defend his mother against charges of witchcraft (Somnium xvii, xix).
 

The Noble Dane
 

In December 1597, Kepler sent to Tycho Brahe, “the Noble Dane,” a copy of Mysterium Cosmographicum. In response, Brahe sent a letter to Kepler's mentor Maestlin noting his disagreement with Kepler’s conclusion regarding the heliocentric. However, Brahe did assert that progress could not be made through a priori deductions, but only through accurate observations and calculations. As a consequence, Brahe was sufficiently intrigued by Kepler's work to invite him to Uraniberg; although, frankly, Brahe was more interested in Kepler’s mathematical ability than in his theories (The Secret of the Universe 22-23).
 

Kepler refined the art of observation and data collection under Brahe. Equipped with the Noble Dane's extensive data from a lifetime of obsessive observation, Kepler applied the research towards his next published work, Astronomie Nova (The New Astronomy) (Stephenson 21). Despite religious convictions that prevented Brahe from seeing the logic of a heliocentric model, his dedication to accurate scientific data and mathematical calculations is a lasting tribute to the Danish astronomer and a major contribution to the Copernican Revolution.
 

The New Astronomy
 

Astronomie Nova, published in 1609, introduces Kepler's first two laws of motion. Ptolemy, Copernicus, and Brahe all explained the apparent retrograde orbit of the planets as the result of a series of epicycles that take place within a planet's orbit, both of which were assumed to be perfect circles (Kuhn 212). This was in accordance with the belief that the heavens were divine in origin. Since anything divine was assumed to be perfect, the orbits must therefore be perfect 360-degree circles. Kepler's observations proved this doubtful and he used Brahe's data to establish the first two of three laws that would serve as the foundation of Isaac Newton's Principia Mathematica (Stephenson 1).
 

Kepler's Laws of Motion
 

Kepler's First Law: Planets move in simple elliptical orbits around a focal point, the Sun (Kuhn 212).

This is the equation for an ellipse:
X2
A2
+
Y2
B2
= 1
 
Kepler's Second Law: The speed of the orbit of each planet varies in proportion to its distance. The closer to the focal point, the Sun, the faster the speed (Kuhn 212).

Fig. 2: The space between the lines represents a unit of time.
This means that a planet will cover the same distance in one hour, for example, whether moving quickly close to the Sun, or slowly further away from it. Ironically, even though Kepler's Second Law was proven true, Isaac Newton showed Kepler's mathematical methodology and premise regarding velocity were fallacious (Westfall 8).
 

The Harmony of the World


Ten years after Astronomie Nova, Kepler publishes Harmonices Mundi (The Harmony of the World). In it, he establishes a third law regarding the motion of the planets.
 

Kepler's Third Law: The squares of the periods of the planets are proportional to the cubes of their semi-major axes. In other words, as a planet moves away from the Sun its orbital speed decreases (Westfall 216).
 
T= The orbits of two planets

R= Mean distance between planets and sun

Ta2 / Tb2 = Ra3 / Rb3

Based on the second law it may be logically assumed that the further away from the Sun a planet got the slower it would go, but Kepler had insufficient data to support a third law to be published in Astronomie Nova. By 1619, however, he had sufficient data to back a third law and presented it in Harmonices Mundi. It is this third law, and the data that supports it, that inspired Newton's law of gravitation; an idea that Kepler himself touched upon via his "theory of heaviness" that he mentions briefly in the introduction to Astronomie Nova (Stephenson 4).
 

With the publication of Astronomie Nova by Kepler in 1609, yet another figure enters the stage whose personality and assiduous dedication to the scientific method will bring Copernicus’ theories into full direct conflict with religious authority…Galileo Galilei.
 


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 Works Cited
 
Burne, Jerome, ed. Chronicle of the World. Paris: Jacques
                LeGrand S.A. International Publishing, 1989. Print.
 
Field, J.F. The MacTutor History of Mathematics Archive:
                Johannes Kepler The University of St. Andrews, Scotland,
                School of Mathematics and Statistics, 1997. Web. 7 Dec.
                2000. <http://www-groups.dcs.st-and.ac.uk/~history/
                Mathematicians/Kepler.html>.

Kepler, Johannes. The Secret of the Universe. Trans. A.M.
                Duncan. New York: Abaris Books, 1981. Print. 

---. Somnium. Trans. Edward Rosen. Madison,
                Milwaukee and London: The University of Wisconsin
                Press, 1967. Print.

Kuhn, Thomas, S. The Copernican Revolution: Planetary
                Astronomy in the Development of Western Thought.
                Cambridge: Harvard University Press, 1957. Print.

Stephenson, Bruce. Kepler's Physical Astronomy. New York,
                London, Paris, Berlin and Tokyo: Springer-Verlag, 1987.
                Print.

Westfall, Richard, S. The Construction of Modern Science.
                Cambridge, New York and Melbourne: Cambridge
                University Press, 1971. Print.
 

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