Regarded as one of the greatest scientific geniuses of all time, Carl Friedrich Gauss (1777-1855) made influential contributions to the study of electricity and magnetism, number theory, astronomy, differential geometry, and many other fields.
Born into a humble family in Brunswick, Germany, Gauss caught the attention of his elementary school teachers by being able to instantly add up the integers from 1 to 100 by noticing that the sum was 50 pairs of numbers, each summing to 101. His mathematical talent earned him the attention of the local Duke, who sponsored his education from secondary school through post-graduate study. Despite his evident ability, Gauss did not decide to devote himself mathematics until age 18, as a result of his discovery of the "constructability of the 17-gon" (see below). He displayed a keen interest in foreign languages in secondary school and at Göttingen University.
Gauss thought his discovery of the "17-gon" was so significant that he asked to have one carved on his tombstone.
It was generally believed that polygons with a prime number of sides greater than five could not be constructed with a ruler and compass, since the number of sides could not be factored. (A "prime number" has no other factors besides 1 and itself).
Constructing a polygon in the manner of ancient Greek geometers entailed essentially "factoring" the number of sides.
Yet Gauss found that while the number 17 was prime in the domain of the integers, it was not prime and could be factored within the domain of complex numbers. He noted that his discovery constituted "a corollary to a theory of greater scope," the theory of complex magnitudes.
An article by R. C. Archibald in The American Mathematical Monthly, "Gauss and the Regular Polygon of Seventeen Sides" discusses Gauss' method.
Gauss' book Disquisitiones Arithmeticae, completed at age 23, represented the most significant contribution to the field of number theory up to that time.
Gauss took an increasing interest in astronomy during his years as a student at Göttingen University and at age 24 solved the foremost problem of mathematical astronomy, the computation of planetary orbits from inaccurate and scarce observations.
Investigations of Curved Surfaces
In 1818, Gauss shocked his colleagues by accepting a commission to carry out a geodetic survey of the Kingdom of Hanover, a job which many regarded as beneath his intellectual stature. For more than a decade he traveled across the country on horseback during the summer with a surveying crew.
The problem of mapping an irregularly curved surface to a plane enabled Gauss to develop his (long-held) ideas on non-Euclidean geometry, published in his General Investigations of Curved Surfaces.
Einstein later wrote, "if [Gauss] had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it." He continued, "The importance of Gauss for the development of modern physical theory and especially for the mathematical fundamentals of the theory of relativity is overwhelming indeed."
We briefly review Gauss' theory of curved surfaces here.
Gauss' and his colleague Wilhelm Weber are said to have built the world's first telegraph line, stretching 5000 feet across the campus of Göttingen University from Gauss' observatory to Weber's physics lab. He also built a "magnetic observatory" to study the earth's magnetic field and developed a theory of the causes and properties of the field.
We encourage interested readers to pick up one of the biographies listed at the top left of the page. The biography Gauss: Prince of Mathematics provides a suitable introduction to his wide-ranging career for elementary and middle school students. We also recommend Peter Pesic's introduction to the new Dover edition of General Investigations of Curved Surfaces.