Maintaining the Status Quo

Carl Friedrich Gauss was constantly in pursuit of the next big idea; he is credited with more than 400 original ideas throughout his long life. However eager he was to uncover new truths though, he was hostile towards anything that disrupted established tradition (Biographical, 1991).

Gauss lived in a time of instability. A stalwart conservative, he was unsettled by the turmoil caused by the French Revolution and the Napoleonic period and was distrustful of the democratic revolutions happening in his homeland of Germany (McElroy, 2005). To Gauss, Napoleon epitomized all that was wrong with revolution. He remained a firm nationalist and royalist throughout his life (Biographical, 1991).

To understand why he took this position, it can help to look back towards his youth. Gauss grew up in a household that was financially uncertain. The child prodigy wished for a higher education, but was denied by his father. His talents were clear and at age 14, the Duke of Brunswick supplied Gauss with a stipend, making him financially independent of his parents. With this, he left home and began studies and research in the subject he loved; mathematics. The Duke provided Gauss with his “golden years of freedom” and became, in Gauss’ mind, the personification of an enlightened monarchy (Biographical, 1991, pg. 864). He would be honored with 75 official awards throughout his life, but often brushed them off. That is, unless a royal was present (Young, 1998).

Portrait of Charles William Ferdinand (1735-1806), the Duke of Brunswick during Gauss’ youth (public domain).

Then, tragedy struck. In 1806, the Duke lead the Prussian armies against Napoleon. In a humiliating defeat, the Duke was killed and Gauss’ anti-revolutionary beliefs were confirmed (Biographical, 1991).

This reluctance for radical change continued into Gauss’ academic life, as demonstrated in his dislike for the most revolutionary mathematical idea of the 19th century: non-Euclidean geometry (Biographical, 1991). His journey began at the University of Göttingen in 1795. As a student, he bore witness to the popular attempts at proving Euclid’s parallel postulate, but was quick to spot the inevitable misconceptions in each of them. None of them satisfied his infamously rigorous standards of proof. Gauss was so selective, much of his own work failed to meet his standards and went unpublished. Only his most polished work (work beyond criticism) was released. It wasn’t that he didn’t want the satisfaction that came with being first to discover, he just didn’t care if others knew of his preeminence. He had a great desire to be the first discoverer (going so far as to keep a dated record of all his results and those of others), but was satisfied for this to be known only to himself (Biographical, 1991).

As will be discussed in a later post, Gauss held jobs in astronomy and surveying. Thus, when working with geometry, he used the globe as his model. Here it was possible to create a series of vertical lines (longitudes) that each meet the equator at a right angle. However, every longitude intersects at the North and South Poles - a contradiction in Euclidean geometry. Gauss now had a concrete example showing the parallel postulate did not hold on curved surfaces (Development, 2002). Slowly, he ventured into the consequences of excluding it and cautiously moved towards the “new” geometry. However, he never dared make his thoughts known publicly. Mathematicians at the time believed Euclidean geometry was the “inevitable necessity of thought” (O’Connor & Robertson, 1996). Gauss, infamous in his dislike of controversy, was not going to be the one to disrupt this.

A model of the Earth lead Gauss to non-Euclidean geometry ([Latitude and Longitude], 2018).

As the years went on, his correspondence shows him clearly, but warily, growing in his belief that Euclid’s fifth postulate could not be proved. In 1816, he wrote a review of a book that contained explorations in non-Euclidean geometry and insinuated his agreement. His worst fears were confirmed when the idea was “besmirched with mud” by critics (Biographical, 1991, pg. 865). He never made the mistake again. He encouraged others who shared his thoughts, but only in private. He was committed to not publishing his own results (although he considered having them published after his death), but showed more willingness for someone else to take the fall.

In 1831, word came that János Bolyai was up to the challenge. Gauss wrote to his father, Wolfgang Bolyai, endorsing the work but (in Gauss’ off-putting style), declaring his own superiority. He wrote, “I regard this young geometer Bolyai as a genius of the first order” but, “to praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years” (O’Connor & Robertson, 2004, pg. 1). After reading Gauss’ letter, János thought it a conspiracy to take credit for his work and quickly cut all ties with Gauss. It seems Gauss learned his lesson, because a decade later when Lobachevsky came on the scene, Gauss’ letter was full of praise and offered membership to the Göttingen Academy where he was working as an astronomer.

A portrait of János Bolyai (1802-1860), credited as one of the founders of non-Euclidean geometry (Márkos, 2012).

While Gauss was privately supportive (or at least, tried to be), in public he remained stubborn. Such a radical discovery was bound to be met with cynicism and a well-known mathematician like Gauss could have gone a long way towards bringing the idea into the mainstream if he had thrown support behind a like-minded mathematician. Instead, he refused to make his ideas known. Despite Gauss’ correspondence, analysis of Bolyai’s and Lobachevsky’s writings shows all three made their discoveries independently. It seems Gauss held a neutral (and sometimes negative) role in the development of non-Euclidean geometry. His public silence was taken as complicity in the general ridicule that prevented progress in the field. It would be decades before this was surpassed and came in part when it was revealed that the prince of mathematics had secretly been a non-Euclidean (Biographical, 1991).

While Gauss never published his work with non-Euclidean geometry, it does show up somewhat indirectly. His work as a surveyor led to work in differential geometry and in 1827 he published Theorema Egregium (Latin for “Remarkable Theorem”) which described what is now known as Gaussian curvature. In this, each surface is assigned a curvature number, the sign of which tells you the type of surface you are working with. A constant curvature of zero indicates the surface is in Euclidean geometry. If there is a constant positive curvature, the surface is a sphere and thus is in spherical geometry - a subset of elliptical geometry. If instead there is a constant negative curvature, the surface is “pseudospherical” and is in hyperbolic geometry (Mastin, 2010; Henderson & Taimina, 2017).

There are three types of constant Gaussian curvature (Mastin, 2010).

The main theorem of the paper states the curvature of a surface is not changed if it is bent without being stretched. Hence, it depends on how distances are measured, but not how they are embedded in space. For example, a cylinder has a Gaussian curvature of zero; equal to that of a flat surface. Thus, by “unrolling” a cylinder, it can be seen that it is isometric to a flat surface. Meanwhile, since a sphere has constant positive curvature and a plane has zero curvature, the two are not isometric. Therefore, any attempt to transform a sphere into a flat representation will distort measurements. This has clear connections to Gauss’ work as a surveyor, as this proves there is no perfect model of the Earth on a flat map (Mastin, 2010).

Gauss had a life full of uncertainty, both financially and politically. He linked Euclidean geometry with the past tradition he revered and feared the ridicule that would come with publishing his thoughts. Despite his reluctance to put his name on his work, he still left his mark. His work is seen in Gaussian curvature, a field mathematicians of the time did not connect with non-Euclidean geometry. More directly, the term “non-Euclidean geometry” was coined by Gauss himself.

References

Biographical dictionary of mathematicians : Reference biographies from the dictionary of
scientific biography (Vol. 2). (1991). New York, NY: Charles Scribner’s Sons.

Henderson, D. W., & Taimina, D. (2017, January 15). Non-Euclidean geometry. Retrieved from https://www.britannica.com/science/non-Euclidean-geometry

[Latitude and Longitude]. (2018, October 16). Retrieved from https://www.cemc.uwaterloo.ca/events/mathcircles/2018-19/Fall/Junior6_Oct16.pdf

Márkos, F. (2012). János Bolyai [Painting].

Mastin, L. (2010). 19th century mathematics - Gauss. Retrieved from https://www.storyofmathematics.com/19th_gauss.html

McElroy, T. (2005). A to Z of mathematicians (Notable Scientists). New York City, NY: Facts 
on File.

O'Connor, J. J., & Robertson, E. F. (1996, February). Non-Euclidean geometry. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html

O'Connor, J. J., & Robertson, E. F. (2004, March). János Bolyai. Retrieved from http://www-history.mcs.st-and.ac.uk/Biographies/Bolyai.html

The development of non-Euclidean geometry. (2002, July 16). Retrieved from http://www.math.brown.edu/~banchoff/Beyond3d/chapter9/section03.html

Young, R. V. (Ed.). (1998). Notable mathematicians: From ancient times to the present. Detroit, MI: Gale Research.