The Beautiful Ellipse

To many who read this, geometry is a distant memory, and the mere mention of it may cause some to recollect nightmares of protractors, compasses and constructing proofs. That certainly is not my intent. For the willing soul who cares to read further, I would ask you to consider a geometric figure which has transformed history, science and theology like no other. Most would refer to this shape as an oval, but it is more particular than that. This shape conforms to a particular mathematical equation, and its shape can vary from a line segment to a circle: the ellipse.

In an ellipse, the sum of (a + b) is the same for every point on the ellipse. The two points on the longer axis running through the ellipse are referred to as foci. The ellipse in the figure has an eccentricity of 0.7.

Menaechmus (ca. 380-320 BC) was a Greek mathematician who is credited as the first to investigate the ellipse. His discovery of the ellipse, along with the parabola and hyperbola, came as a result of his attempt to solve the problem of duplicating the cube (i.e. deriving the cube root of 2 with the use of a ruler and compass). These geometric figures are obtained by slicing a cone with a plane which is not parallel to its base (i.e. conic sections). If you are beginning to break into a sweat after reading the last two sentences, don’t panic. The point to be made here is that the ellipse was discovered as an exercise of the mind. Unlike geometric figures such as circles, squares, triangles and hexagons whose analogs can be found in nature, the ellipse was not described or analyzed based on its existence as a feature found in nature.

The ellipse remained as a geometric concept for the next 2,000 years until it met the mind of Johannes Kepler (1571-1630). In his study of astronomy, he was taught both the Ptolemaic (geocentric) and Copernican (heliocentric) systems, but Kepler argued for the Copernican system on both scientific and theological grounds. Ptolemy’s model of the universe, which developed from Aristotle, located the divine in the eternally existing planets and stars with the earth situated at its center – the place to which all the filth of the universe settled – not a place of distinction (another post about this later). These religious beliefs were all counter to his Christian beliefs. On the side of science, the Aristotelian model required an elaborate scheme of epicycles, eccentrics and equants devised by Ptolemy to account for some of the oddities of planetary motion. Even with all these adjustments, it failed to account for the vast variance in distances of planets from earth during their orbits, as well as the tremendous speed required of the distant stars in their course around the earth in 24 hours.

The Copernican model was not problem-free, however. While it addressed most of the problems inherent in the Ptolemaic system, it failed to predict the future position of planets any better than Ptolemy’s model. Kepler discovered the problem lay in Copernicus’ conception of planetary orbits as perfect circles (a remnant of Aristotelian thinking). With the help of precise data obtained from Tycho Brahe’s observatory, Kepler could discern the orbit of Mars did not map out into a circle. But if not a circle, what shape was it? Kepler’s search was an arduous one as he tested out various ovoid shapes doing all his calculations by hand. After many years of trial and error, Kepler realized the shape of planetary orbits conformed to the shape of an ellipse (a shape he dismissed at one point). The sun is not located in the center of the ellipse, but at one of the two foci of the ellipse. We should not be surprised this fact went undiscovered for so long because the oval-ness of planetary orbits (i.e. eccentricity) is most nearly a circle. A circle has an eccentricity of zero while the orbit of Mars has an eccentricity of 0.09.

This realization subsequently allowed Kepler to discover other patterns in planetary motion. While the elliptical shapes of the planets’ orbits are different from one another, their movement through their orbits are constrained by physical laws described by mathematical equations – one harmonious dance around the sun. Kepler’s revelation was a turning point on many levels, but what I wish to focus on here is the curious effectiveness of geometry in which the ellipse, which was conceived of first as a product of the mind, ended up corresponding to something in nature. This interesting connection provides us inferential evidence the planetary orbits are not a product of random happenstance, but are themselves the product of a mind.  As Wiker and Witt note,

If the order of nature preexists our attempts to grasp it and, consequently, if the strange effectiveness of mathematics depends on the preexistent order of nature to be effective, then nature is intelligibly and ingeniously ordered. Exemplifying both surprising depth and a stunning harmony and elegance, such ingenious design necessarily implies a designing genius.

– A Meaningful World (pg 109)

Sources

Hummel, Charles E. The Galileo Connection: Resolving Conflicts between Science and the Bible. Downers Grove, IL: IVP, 1986.

School of Mathematics and Statistics, University of St Andrews, Scotland. “Maneaechumus.” Accessed 7/25/19. https://www-history.mcs.st-and.ac.uk/Biographies/Menaechmus.html.

Wiker, Benjamin and Jonathan Witt. A Meaningful World: How the Arts and Sciences Reveal the Genius of Nature. Downers Grove, IL: IVP, 2006.

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