The Star-Gazer

The year 1801 was one of the best of Gauss’ life. It is the year that defined the beginning of his remarkable career as he published his monumental textbook Disquisitiones Arithmeticae and rediscovered the planet Ceres, all at just age 24.

On January 1st 1801, Giuseppe Piazzi discovered Ceres, a dwarf planet. However, he could only keep it in his sights as it moved nine degrees across the sky. Then, on February 12th, it disappeared behind the Sun. According to Piazzi’s calculations, it would take a few months to reappear. But it never did. With such little information collected while visible, Piazzi failed to relocate the planet (Tennenbaum & Director, 2019).

A portrait of Giuseppe Piazzi ([Piazzi], 2019).

By October, Gauss had heard about the problem. He decided to try his hand at it and so stepped into a new field - astronomy. Three months later he had his answer and made it public. On December 31st, Franz Xaver von Zach and Heinrich Olbers had both used Gauss’ calculations and spotted the planet again. Gauss had been less than half a degree off. This feat from a non-astronomer seemed fantastical, especially considering Gauss did not let anyone in on his methods until several years later (Biographical, 1991).

Gauss’ brilliance was his recognition that the planet followed an elliptical orbit. Piazzi had failed because he was relying on bad precedent. In 1781, the planet Uranus had faced a similar problem. After its discovery by William Herschel, astronomers looked for a model of its orbit and landed on a circular one. Coincidentally, what we know now to be an elliptical orbit had such a small eccentricity that the circular approximation was satisfactory (Tennenbaum & Director, 2019). When faced with an ellipse with a larger eccentricity (as in the case of Ceres), a circle would not suffice. Even worse, there was much less data from Ceres than Uranus because it was visible for such a short time. Mathematicians such as Euler, Lambert, Lagrange, and Laplace had worked with ellipses before, but none had found a method for deriving an elliptical equation with such little data available. Laplace had gone as far as to state the problem unsolvable (Weiss, 1999).

The orbits of planets in our solar system including Uranus and Ceres ([Dwarf Planet], 2018).

To create such an elliptical orbit, Gauss’ first challenge was to determine the distance between the Earth and Ceres. Over the 42 days Piazzi kept track of the planet, he made 19 separate observations about its position in the sky. Each observation consisted of a timestamp and two angles that determined the direction of the planet in relation to Earth. Gauss’ initial work used three of these - data coming from the day of discovery (January 1st), the last day seen (February 11th), and a day in the middle (January 21st). He chose the middle day as his median with which to calculate the distance from Earth (Tennenbaum & Director, 2019).

In another diversion from the Piazzi, Gauss made use of Kepler’s hypotheses. First, that an orbit is independent of a planet itself; thus, no details are needed about mass. Second, assuming an orbit does not come near another large body, the Sun will be a focal point of the orbit. With these in mind, the orbit is dependent on 5 variables; the relative scale of the orbit, the eccentricity of the orbit, the shortest distance from the orbit to the Sun, the relative tilt of the main axis of the orbit, and the plane of the orbit relative to that of the Earth. Hence, Gauss assumed nothing besides the three observations he had taken from Piazzi. Such a set up led to an equation of degree eight. The execution to find a solution would be complex; it required over 100 hours of manual calculations and approximations.

First, a basic approximation of its solution was found using the three chosen observations. Then, he used the method of least squares, which he had developed at age 18 but failed to publish. He slowly increased precision by adding in the rest of the observations until his solution fit them all. From this, he created an ellipse using the Sun as one focus and keeping his three original data points lying on the ellipse. Then, as Kepler’s Second Law told him how to calculate the length of an arc and he had data telling him how long it took to get from one point to another, the current location could be found (Weiss, 1999).

Gauss’ drawing of the orbits of several planets, including Ceres (Weiss, 1999).

Overwhelmed by the success and recognition this feat won him, Gauss continued work in astronomy. Eight years after the fact, Gauss revealed his methods. In 1809, he had collected and refined his work to meet his satisfaction for publication. His book became Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. He simplified so much of the orbital calculations that existed at the time that his work is still in use today, requiring only a few modifications to be applied to computers. He introduced the Gaussian gravitational constant and successfully proved the method of least squares for normally distributed errors; a theory known as the Gauss-Markov theorem, although there are disputes about Gauss’ priority.

Since Ceres, over 1,500 celestial bodies have been discovered (though Ceres remains the largest). By improving his methods, Gauss calculated orbits for many throughout his life with increasing speed. In 1807, Olbers discovered the planet Vesta and Gauss calculated its orbit in only 10 hours - much faster than the 3 months it took for Ceres. His methods also worked for the parabolic orbits of comets and these came even faster. Gauss could find them in a single hour, while Euler’s previous methods took three days (Weiss, 1999).

The Göttingen Observatory where Gauss was director ([The Old Observatory], 2014).

This success changed the course of Gauss’ life. As noted in previous posts, Gauss had been receiving a stipend from the Duke of Brunswick, allowing him to leave home and focus on mathematics full-time. However, growing up financially unstable, he always doubted the security the stipend provided. As democratic revolutions were happening around the globe (a cause Gauss did not believe in), he saw the risk they posed to his well-being. After Gauss’ success with Ceres, the Duke increased his stipend, but it was not enough to assuage his insecurities. The obvious path was to become a professor - a path he resisted as he was not one that enjoyed working with students. However, now that he was being celebrated by astronomers, he had another option. The life of a professional astronomer included little teaching and more time for research. He had also seen firsthand the public recognition astronomy could win him, something that was comparatively lacking in mathematics. Gauss had a great ambition for fame and dreamed of rising to the level of renown in the sciences that only comes once in a generation. Thus, he cleared the path for himself and in 1807 was appointed Professor of Astronomy and Director of the observatory at Göttingen. This was certainly a stable position and he held it for the rest of his life (Biographical, 1991).

With two extraordinary accomplishments in a single year, Gauss was made famous in the academic world. While his father had downgraded him as the “star-gazer,” it was a name that ended up becoming true (Eisenhart, 2008, pg. 1).

References

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

Dwarf planet orbit [Digital image]. (2018, January). Retrieved from https://superstarfloraluk.com/9487485-Make-Make-Dwarf-Planet-Orbit.html

Eisenhart, C. (2008). Gauss, Carl Friedrich. Retrieved from https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/carl-friedrich-gauss

Giuseppe Piazzi [Digital image]. (2019, January 1). Retrieved from http://scihi.org/giuseppe-piazzi-ceres/

Tennenbaum, J., & Director, B. (2019). How Gauss determined the orbit of Ceres. Schiller 
Institute.

The old observatory of Carl Friedrich Gauss [Digital image]. (2014, August 13). Retrieved from https://www.flickr.com/photos/unigoettingen/15175546641

Weiss, L. (1999). Gauss and Ceres. Retrieved from http://sites.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html

The Heptadecagon

As discussed in our last blog post, Gauss explored non-Euclidean geometry, but was extremely reluctant to publish his results. Today, we see the opposite; Gauss’ work with Euclidean geometry and his triumph as he makes the first advancement in 2,000 years, earning himself the title of “Prince of Mathematics.” That accomplishment is the proof of the constructibility of the heptadecagon (a 17-sided regular polygon) using only a compass and straightedge. It’s made all the more extraordinary by the fact that Gauss completed it at the age of just 19 (Dunham, 1990).

While at Göttingen University, Gauss read all the journals and mathematical work he could find. This collection was somewhat limited and he often would stumble upon proofs years after he had solved them himself, startled to find they were not original. Work he could find was often done by mathematicians such as A.G. Kastner, who Gauss considered mediocre. Meanwhile, he studied extraordinary classicists like Heyne, leading Gauss down the path of becoming a philologist. However, this all changed with his stunning discovery of the heptadecagon (Biographical, 1991).

While Gauss worked in many areas throughout his life, he considered mathematics the “queen of the sciences” and number theory the “queen of mathematics” (Dunham, 1990, pg. 240). At the time, number theory was a series of scattered results. In 1801, Gauss published Disquisitiones Arithmeticae. This was groundbreaking work as it systematically collected previous work in number theory, solved several outstanding problems, and set the standard of research for the next century and beyond. Gauss’ original work includes a proof of the law of quadratic reciprocity, the inception of modular congruence, the first examples of equivalence relations, further development of quadratic forms, and analysis of the cyclotomic equation. Such a collection instantly brought Gauss to the front of the mathematical world (Biographical, 1991).

Pages from Disquisitiones Arithmeticae (public domain).

Gauss culminated his book with work on the cyclotomic equation and lead it into connections with polygon construction, thus, combining number theory, geometry, and complex numbers. Polygon construction dates back to Book IV of Euclid’s Elements. Euclid showed how to construct regular triangles, squares, pentagons, hexagons, octagons, and pentadecagons using only a compass and straightedge. Although he never explicitly addressed it, it is also clear Euclid knew that doubling the number of sides of a constructible polygon results in another constructible polygon. Unclear, however, is if these were the only constructible polygons. No more had been discovered since Euclid (300 B.C.), so this was the assumption. That is, until out of nowhere, the young and unknown Gauss announced Euclid’s collection was unfinished (Dunham, 1996).

So how did he do it? Descartes proved early in the seventeenth century that the set of constructible numbers (i.e. numbers that can be constructed using only a compass and straightedge) includes positive integers, as well as any number resulting from a finite number of constructible numbers combined through addition, subtraction, multiplication, division, and square roots (Dunham, 1996). Gauss sought to prove cos(2π/17) was constructible. If he could do this, then the 17-gon was constructible. To see why, consider the following figure. We start with a unit circle with center O and construct segment OC with length cos(2π/17). Construct a line perpendicular to OC at C and label the intersection of the perpendicular and the circle as point B. Then, cos(<BOC)=OC/OB=cos(2π/17), implying <BOC=2π/17. Next, extend segment OC and label the intersection of OC and the circle as point A. If we then copy the chord AB seventeen times around the circle, we land exactly back at point A and have successfully constructed the 17-gon (Dunham, 1996).

The construction of a 17-gon (Dunham, 1996).

So, Gauss’ proof hinges on his ability to prove cos(2π/17) is indeed constructible. To do so, he ventures into complex numbers - a surprising move. If the geometric construction is, by definition, a concrete real-world representation, why make use of imaginary numbers? Perhaps this is why this proof took so long to come to light. Complex numbers had certainly been explored long before Gauss by some of the greatest mathematicians of their age, but imaginary numbers continued to live in the shadows. Gauss was the first to use them in a “really confident and scientific way” (Hardy & Wright, 2000, pg. 244). According to Gauss, this was because of their branding. He writes,

“If this subject has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1, and √-1, instead of being called positive, negative, and imaginary (or worse still, impossible) unity, been given the names say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity” (Merino, 2006, pg. 5).

While Gauss never got his wish for direct, inverse, and lateral unity, he did coin the term “complex numbers.” His continued work with them resulted in four proofs of the Fundamental Theorem of Algebra which says the field of complex numbers is algebraically closed. The first of these earned Gauss his doctorate from the University of Helmstedt (McElroy, 2005). Uncharacteristically, he published it before satisfying his standards of rigor and thus it contained a gap that was not patched until 1920 (Jackson, 2017; Cain, 2005).

Back in Disquisitiones Arithmeticae, Gauss recognized that the number cos(2π/17) is a component of the complex number z = cos(2π/17) + i sin(2π/17) and thus one of the seventeenth roots of unity. Continuing down the path of complex numbers, he is able to prove the following:

As this expression is composed of only positive integers added, subtracted, multiplied, divided, and square-rooted, it is constructible. Thus, by our previous logic, the 17-gon must be constructible (Dunham, 1996).

A regular 17-gon (Kuh, 2013).

Gauss actually went beyond proving the 17-gon was possible; he proved it was possible to construct a polygon with a number of sides equivalent to a Fermat prime. That is, a prime of the form 22^n+1 for some non-negative integer n. Known to Fermat and Gauss were the first five Fermat primes, 3, 5, 17, 257, and 65,537, corresponding to n values of 0, 1, 2, 3, and 4. In fact, these remain the only five Fermat primes known today (Kuh, 2013). Hence, Gauss proved the constructibility of the regular 17-gon, the 257-gon, and the 65,537-gon.

Gauss did not actually construct each of these polygon; he simply proved it was possible. The constructions would not come until 1800, 1832, and 1894 by Erchinger, Richelot, and Hermes, respectively. They hold no practical use, but their possibility uncovered a mystery lurking in Euclidean geometry. The field was so familiar, Gauss made the first true innovation in Euclidean geometry in 2,000 years. He revelled in his discovery. So proud was he, that above all the original work he would create over his 77 year life, he asked for the 17-gon to be inscribed on his tombstone. Unformately, this was never done (Dunham, 1990). The comparison to Gauss’ work in non-Euclidean geometry is startling. While the 17-gon earned Gauss his name, his insight that Euclidean geometry was not the only geometry was arguably more revolutionary and impactful. However, one discovery he despised and the other he revered. His work with complex numbers though, brought him to the forefront of a field that would grow dramatically in the next century.

References

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

Cain, H. (2005). C.F. Gauss's proofs of the fundamental theorem of algebra. Einstein Institute of Mathematics.

Dunham, W. (1990). Journey through genius. New York, NY: Penguin Books.

Dunham, W. (1996). 1996—A triple anniversary. Math Horizons,4(1), 8-13. Retrieved from http://www.jstor.org/stable/25678076

Hardy, G. H., & Wright, E. M. (2000). An introduction to the theory of numbers (4th ed.). Oxford: Oxford University Press.

Jackson, T. (Ed.). (2017). Mathematics: An illustrated history of numbers. New York, NY: Shelter Harbor Press.

Kuh, D. (2013). Constructible regular n-gons. Whitman College.

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

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.