Did Gauss ever make a mistake?

Question

I have read a bit about Gauss, who was well known for being careful in only publishing work he had perfected (or in his own words "few, but ripe"). What is interesting to me about Gauss though is that all accounts of his students and contemporaries essentially make him appear flawless. Dedekind's recollections of how whenever Gauss wrote out an argument he used exactly the amount of space he had available, etc. In Disquistiones Arithmeticae, Gauss for example points out the flaws and shortcomings of other famous mathematicians such as Euler, who some might rank close to Gauss.

So I am curious if there is a single recorded instance of Gauss ever having made a mathematical error.

Thanks, Matt

Answer 1

Euler ranks higher than Gauss in my opinion. Having said that, I do not know if there is a recorded instance of Gauss making a mathematical error, but it's worth pointing out that the work of the 18th and early 19th century mathematicians was plagued by lack of rigor. Only later in the 18th century did the strides in analysis give mathematics a solid grounding.

Answer 2

You are seeing the down side of the "few but ripe" idea. He comes off badly in the episode with Bolyai. He did not publish his findings n a new, non-Euclidean geometry, over fears of shrill criticism from followers of Immanuel Kant. For the moment, I am going to assume that the word he actually used referred to Boethius; perhaps someone else knows. EDIT: no, it is an ancient insult, the citizens of Athens thought of their neighbors as dull and stupid.

The part I dislike is when Janos Bolyai's father, Farkas Bolyai, wrote to Gauss, asking his opinion of this new world. Gauss replied that he could not praise Janos, as he would then be praising himself, Gauss having done much the same thing decades earlier. I've always thought this a mixture of cowardice and lack of generosity. There really is a difference between doing a bunch of calculations and saying "If so and so happens, here are some rules that apply" compared with saying "Here is the actual thing."

Well, the full story is intricate, but Gauss's reply hit Janos very hard, and appears to have convinced him to drop mathematics; at least, he published no more.

Answer 3

how about errors of omission? Gauss was famous for not publishing monumental discoveries until after they were rediscovered.

Answer 4

There is at least one record that I'm aware of in which Gauss effectively concedes to having made a mathematical error, although it appears in casual correspondence rather than in publication. This correspondence involved some recreational mathematics involving the eight-queens problem that was posed when Gauss was in his 70s. In that correspondence, Gauss asks that his value of 76 confirmed solutions be reduced to 72 (although he correctly asserts that there may be others, as the full answer is 92). Details of this episode can be found in the article "Gauss and the eight queens problem: A study in miniature of the propagation of historical error" by Paul J. Campbell (1977).