Ramble, Part 43: Non-Euclidean Prehistory
Jan. 9th, 2008 10:38 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe standard history tells us that non-Euclidean geometry was discovered, independently, by Bolyai and Lobachevsky around 1830. The standard history is, as often, somewhat incomplete; several mathematicians had worked in or near the area prior to those two. I'd like to talk about those, first, before going on to the actual achievements of Bolyai and Lobachevsky.
The first actor in this story was a Jesuit by the name of Girolamo Saccheri, who set out to prove Euclid's parallel postulate by reductio ad absurdum - by assuming it was false, and deriving a contradiction. He began by considering the following construction: take a line segment AB. Raise perpendiculars to the segment at A and B, and mark off congruent segments AD and BC on those perpendiculars, on the same side of AB. Then, complete the quadrilateral ABCD. (It is now known as a "Saccheri quadrilateral".) It can be shown, without using the parallel postulate, that the angles ADC and BCD are congruent. Saccheri then identified three possibilities: that these angles were acute, right, or obtuse. The hypothesis of the right angle, he showed, implied the parallel postulate. He then showed that, on the hypothesis of the obtuse angle, there is an upper limit to the lengths of line segments. This contradicts Euclid's second axiom, that line segments are indefinitely extensible; thus, he was able to discard this possibility. His assault on the hypothesis of the acute angle led him to a number of theorems, but no visible contradiction; he eventually threw up his hands, declared the results obviously absurd, and exited the field. (His book on the subject, published in 1733, attracted little attention and was soon forgotten.)
A generation later, Johann Lambert tackled the problem similarly. He considered quadrilaterals ("Lambert quadrilaterals") three of whose angles were right; like Saccheri, he identified three cases, depending on whether the fourth angle was acute, right, or obtuse. Lambert went considerably further than Saccheri did, but his researches ended on a similarly unsatisfactory note. (Lambert had other, more successful work to his credit; among other things, he was the first to prove that pi is irrational.) He did notice the (not quite perfect) resemblance between the hypothesis of the obtuse angle and the geometry on the sphere; the latter had been much studied for astronomical purposes. He speculated that the hypothesis of the acute angle might be similarly related to geometry on "a sphere with imaginary radius", whatever that might mean.
Adrien-Marie Legendre, author of the most successful textbook on geometry since Euclid, continued the struggle; his three hypotheses had to do with whether the angle-sum of a triangle is less than, equal to, or greater than 180 degrees. As with his two predecessors, he was able to dispose of the third case, but could not find a contradiction in the first.
It was Carl Friedrich Gauss who finally took the plunge. There were two psychological hurdles to be overcome: first, to recognize the possibility that the hypothesis of the acute angle did not, in fact, lead to a contradiction, and second, to conclude that this was enough to make the subject worth studying, whether it described the real world or not. Gauss took both of these steps. Unfortunately, he did not take a third: to publish his conclusions, and for this reason he forfeits his claim to mathematical priority.
That Gauss did not publish this work may not require special explanation; he was notorious for selecting only the choicest of his results ("few, but ripe") for dissemination. However, this is easily the largest and most significant body of work that he left unpublished, and one might wonder at his judgment. There is, however, some evidence of his reasons for not publishing, and it rests with the prominence of Immanuel Kant's philosophy in Germany. Kant, you may recall, had concluded that the results of Euclidean geometry were necessary truths, embedded in the very structure of the human mind. To challenge the Kantian orthodoxy on this would have exposed Gauss to the carping of Kant's followers, and apparently he held back for this reason.
This is rather odd. It is not that Gauss lacked courage; when he found that the M. Leblanc with whom he had been corresponding on various mathematical topics was a woman - Sophie Germain - he pushed hard to have her awarded an honorary doctorate. (Unfortunately, she died before the honor could be bestowed.) That his decision not to publish may have rankled him is suggested by his later interaction with Janos Bolyai - but that is a story for the next post.
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The first actor in this story was a Jesuit by the name of Girolamo Saccheri, who set out to prove Euclid's parallel postulate by reductio ad absurdum - by assuming it was false, and deriving a contradiction. He began by considering the following construction: take a line segment AB. Raise perpendiculars to the segment at A and B, and mark off congruent segments AD and BC on those perpendiculars, on the same side of AB. Then, complete the quadrilateral ABCD. (It is now known as a "Saccheri quadrilateral".) It can be shown, without using the parallel postulate, that the angles ADC and BCD are congruent. Saccheri then identified three possibilities: that these angles were acute, right, or obtuse. The hypothesis of the right angle, he showed, implied the parallel postulate. He then showed that, on the hypothesis of the obtuse angle, there is an upper limit to the lengths of line segments. This contradicts Euclid's second axiom, that line segments are indefinitely extensible; thus, he was able to discard this possibility. His assault on the hypothesis of the acute angle led him to a number of theorems, but no visible contradiction; he eventually threw up his hands, declared the results obviously absurd, and exited the field. (His book on the subject, published in 1733, attracted little attention and was soon forgotten.)
A generation later, Johann Lambert tackled the problem similarly. He considered quadrilaterals ("Lambert quadrilaterals") three of whose angles were right; like Saccheri, he identified three cases, depending on whether the fourth angle was acute, right, or obtuse. Lambert went considerably further than Saccheri did, but his researches ended on a similarly unsatisfactory note. (Lambert had other, more successful work to his credit; among other things, he was the first to prove that pi is irrational.) He did notice the (not quite perfect) resemblance between the hypothesis of the obtuse angle and the geometry on the sphere; the latter had been much studied for astronomical purposes. He speculated that the hypothesis of the acute angle might be similarly related to geometry on "a sphere with imaginary radius", whatever that might mean.
Adrien-Marie Legendre, author of the most successful textbook on geometry since Euclid, continued the struggle; his three hypotheses had to do with whether the angle-sum of a triangle is less than, equal to, or greater than 180 degrees. As with his two predecessors, he was able to dispose of the third case, but could not find a contradiction in the first.
It was Carl Friedrich Gauss who finally took the plunge. There were two psychological hurdles to be overcome: first, to recognize the possibility that the hypothesis of the acute angle did not, in fact, lead to a contradiction, and second, to conclude that this was enough to make the subject worth studying, whether it described the real world or not. Gauss took both of these steps. Unfortunately, he did not take a third: to publish his conclusions, and for this reason he forfeits his claim to mathematical priority.
That Gauss did not publish this work may not require special explanation; he was notorious for selecting only the choicest of his results ("few, but ripe") for dissemination. However, this is easily the largest and most significant body of work that he left unpublished, and one might wonder at his judgment. There is, however, some evidence of his reasons for not publishing, and it rests with the prominence of Immanuel Kant's philosophy in Germany. Kant, you may recall, had concluded that the results of Euclidean geometry were necessary truths, embedded in the very structure of the human mind. To challenge the Kantian orthodoxy on this would have exposed Gauss to the carping of Kant's followers, and apparently he held back for this reason.
This is rather odd. It is not that Gauss lacked courage; when he found that the M. Leblanc with whom he had been corresponding on various mathematical topics was a woman - Sophie Germain - he pushed hard to have her awarded an honorary doctorate. (Unfortunately, she died before the honor could be bestowed.) That his decision not to publish may have rankled him is suggested by his later interaction with Janos Bolyai - but that is a story for the next post.
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Date: 2008-01-10 07:48 pm (UTC)There is a lot of confusion yet today confusing reality and math. The article I recently referred to in the NYTimes listed a couple such - not that it is a peer-reviewed journal (well, maybe it is, but different peers and a different review system), but the writer presumably got those quotes from people who ought to know better.
Since this is, after all, a Historical Ramble, I trust that the section on Lobachevsky will properly weigh the contributions of noted mathematical (pop)historian T. Lehrer?
;<)
no subject
Date: 2008-01-10 11:53 pm (UTC)no subject
Date: 2008-01-11 05:19 am (UTC)