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His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2.
This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career.
There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager.
He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801.
Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages.
They had an argument over a party Eugene held, which Gauss refused to pay for.
Eugene shared a good measure of Gauss's talent in languages and computation. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.
The son left in anger and, in about 1832, emigrated to the United States.
While working for the American Fur Company in the Midwest, he learned the Sioux language.
Two religious works which Gauss read frequently were Braubach's Seelenlehre (Giessen, 1843) and Siissmilch's Gottliche (Ordnung gerettet A756); he also devoted considerable time to the New Testament in the original Greek.
Gauss's religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods.
Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer.