Johannes Kepler of Mathematics

free essayJohannes Kepler was an important person in the scientific revolution of the 17th century. He leads the list of individuals linked with the emergence of an entirely new era in mathematics, that is, the era of variable quantities. In the early 17th century, his study of laws of governing planetary motion pushed him into the tedious and significant part of integral calculations (Field, 1999). Despite the fact that Kepler did not carry out many integrations, he used a new concept in the area covered by a curve and created procedures that are similar to the contemporary methods for arithmetical integration. He is currently committed to memory as a mathematical astronomer, more so, a pioneer of three laws that explain planetary movement (Berlinghoff & Gouve?a, 2004). His work was remarkable in optics, managed to discover polyhedra in 1619, provided a new mathematical solution for spheres under close proximity and equal measurements, thereby elucidating the shape of honeycomb cells, pioneered a proof of the workings of logarithms in 1624 and created a methodology for determining the volumes of solids; which was a massive contribution to the calculus development in 1616. Additionally, Kepler computed the most accurate astronomical tables used until the present, whose consistent level of accuracy helped in finding the realist of heliocentric astronomy.

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Early Years

Kepler, decorated mathematician, was born on December 27, 1571 in a small town of Weil der Stadt (Field, 1999). Kepler’s father worked as a mercenary soldier while his mother worked as a herbalist and a healer. He was the firstborn among two siblings. Kepler’s father left Free Imperial City when he was only five years. During his childhood, he was raised by his mother and helped in serving food in his grandfather’s inn. His earlier education happened in a local school after which he relocated to a nearby seminary and then registered at the University of T?bingen as it was called then. Maestlin Kepler was Kepler’s mathematics professor from 1580 to 1635 (more than five decades) (Field, 1999). Maestlin was among the first astronomers to adopt the Copernicus theory of heliocentric, even though his lectures only revolved around the Ptolemaic system (Berlinghoff & Gouve?a, 2004). Only during graduate seminars Maestlin got acquainted with his students, one of whom was Kepler. During this time, Kepler was a Copernican for metaphysical reasons. He came out as an excellent mathematician and gained a standing as a competent astrologer, who was able to cast horoscopes on behalf of his fellow colleagues. Under Maestlin instruction, he got acquainted with both Copernican and Ptolemaic systems of planetary motion. During a student disputation, Kepler showed skills and knowledge while defending the heliocentric theory from both theological and theoretical standpoint, insisting that the sun was the primary source of motive power (Gilder & Anne-Lee, 2004). Even though he desired to work as a minister in his future life, towards the end of his education, Kepler was highly recommended for a teaching position at the Protestant school to teach astronomy and mathematics. He took the position in the month of April 1594 when he was just 23 years.

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Applied Mathematics

Across different eras, mathematics featured in various guises and has been applied in several areas of life. Originally, mathematical methods were applied in land surveying, astronomy, and commercial accounting (Berlinghoff & Gouve?a, 2004). In his days, pure mathematics was conventionally divided into arithmetic, the art of using numbers in computation, geometry, the art of formulation and analysis of figures, and algebra, use of symbols in computation. Kepler’s main area of interest was geometry in figures but not in practice and arithmetic which was essential for his astronomical and difficult calculations (Berlinghoff & Gouve?a, 2004). He perceived algebra to be of less value, because he believed that they were not related to the things that actually existed. Consequently, he avoided it and used algebraic terms in expressing equations, which comprised of sentences. He also employed the use of natural language in delineating rules for defining mathematical functions. During Kepler’s days, “applied mathematics” had not been coined as an applicable term (Gilder & Anne-Lee, 2004). Mathematics was known to cover all kinds of problems whose solution need mathematical reasoning (Gilder & Anne-Lee, 2004). By emphasizing the use of geometrical methods, in efforts to have a holistic explanation for things, the use of natural language in the description of methods, and formulation of calculation methods, he anticipated the use of system-oriented and qualitative approaches (Gilder & Anne-Lee, 2004). These approaches are employed today to solve difficult problems by methods of the constraint analysis and requirement engineering which began from conditions that were expressed in natural language. Therefore, Kepler’s use of natural language set a platform for today’s practive as mentioned above.

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Astronomy

As an astronomer, Kepler applied numerous mathematical principles in solving complex issues in astronomy. In his original works entitled, “Mysterium Cosmographicum,” he attempted to explain how far the planets are from the sun by using the mathematical framework of regular solids inserted in another with a sphere placed between the two solids (Dunham, 1991). Thus, he was able to view the solar system, as God’s masterpiece, to exist as a mathematical model. This perspective not only met physical requirements but also his aesthetic and Christian expectations. In Astronomia Nova, viewed by other mathematicians as his most significant astronomical contribution from a scientific standpoint, Kepler, using Planet Mars as an example, successfully availed a solution to a quantitative and qualitative account of planetary orbits which to date is sufficient in physics (Berlinghoff & Gouve?a, 2004). Applying the extensive data retrieved from Tycho Brahe’s astronomical observations, Kepler managed to prove that planet Mars’ orbit is an exact ellipse, and the moving radius covers equivalent distance. To provide a proof of the validity of all his propositions, Kepler had to engage in a lot of computational work. By the three laws coined by Kepler, a valid mathematical model for explaining the dynamics of planets within the solar system was developed in line with data given by Tycho Brahe’s astronomical measurements (Berlinghoff & Gouve?a, 2004). Actually, Isaac Newton was the first scientist to successfully derive all these laws using one formula: the gravity law and so viewing them as laws that govern mechanical systems (De Gandt, 1995). Therefore, his creativity and original works in astronomy enabled other scientists and mathematicians to understand solar system and laws governing it.

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Stereometry and Gauging

Kepler made massive contributions to stereometry and gauging. This mathematician is among the pioneers of calculus of integration together with Pappus and Archimedes which was finally formulated by Newton and Leibniz, who also invented infinitesimal calculus (De Gandt, 1995). Kepler was first attracted to the subject dealing with integration when he encountered a problem in efforts to establish the exact capacity of a wine-barrel (Gilder & Anne-Lee, 2004). He later did it by the use of a gauging-rod. He noted that the rod was usable no matter the shape of a barrel. Consequently, he solved this problem in his book and further came up with a complete theory related to it. The mathematical theory was about stereometry of Archimedes. He expanded Archimedes stereometry to incorporate other rotational bodies produced using conic sections such as spindle, lemon, and apple (Dunham, 1991). This theory can also be applied in proving that Austrian wine casks are maximal: meaning that the level of accuracy of the method used in measurement is hardly affected by the change in shape of a barrel. In his book, “Nova Stereometria Doliorum Vinariorum,” Kepler launches new mathematical terms in German, the same as Latin used today (Dunham, 1991). Kepler presented a method for determining the content of cask, which is partly filled and further, he provided the practical aspect of stereometry through systematic treatment of various measurement units. His works regarding gauging, published in his two books, was further continued by the Lambert, another German mathematician. Thus, Kepler works acted as a foundation for other mathematicians to formulate their theories.

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Logarithms

Kepler studied John Napier’s introduction of logarithms at the early age. Having struggled with time-consuming calculations for his book, he saw it necessary to change to logarithmic methods while using Rudolphine Tables (Gilder & Anne-Lee, 2004). His contributions on logarithms were featured in his book, Chilias Logarithmorum, which was published in 1625 (Field, 1999). Throughout the content of his book, Kepler played a crucial role, beside Briggs and Napier, in spreading knowledge of this new method and promotes its application in mathematics. In other words, he managed to calculate tables featuring eight-figure logarithms. The tables were not just used in making observations but also in his first two laws. The most remarkable thing concerning Rudolphine Tables was the fact that their accuracy was proven and remained solid for many decades (Dunham, 1991). Therefore, Kepler was the one of the first people to exercise an original theory in constructing logarithms and employ the use of logarithmic calculations in helpful astronomical researches.

In conclusion, the paper discussed mathematical biography of Johannes Kepler, featuring his life story, accomplishments, and contributions to various sub-disciplines of mathematics such as applied mathematics, logarithms, astronomy and gauging. Since contemporary education today stresses on analysis and algebra, his approach to building a sound mathematic reasoning with geometry is often hard to fathom. Kepler endeavored for “mathematics of being” and was not convinced with mere application of mathematical principles. Thus, he offered an original framework of holistic and system-oriented approach, which is presently relevant in solving problems.