This tribute to the great mathematician Lobachevsky reflects much more than a mere craving, on my part, to write a few kind words about an extraordinary Russian genius. Because of my mathematical background, he and I go back together at least half-a-century. Besides, I’ve always had reverence for revolutionary thinking as such, and in this sense I see Lobachevsky not just as a great man of science, but also as a kindred spirit.
Not too many people have ever heard the name of Nikolai Ivanovich Lobachevsky, but to the readers of my blog his must have become a household name by now. “Mr. Non-Euclid” (or, more accurately, “Mr. Super-Euclid,” as we could call him for a good reason explained later) may sound a bit outlandish, but his greatest achievement was indeed proving that the great Euclid was not a General, but only a glorious particular.
The circumstances of his life are very interesting, and I encourage the reader to look up his biographies. His relatively minor accomplishments were the ones for which he was honored during his life by the position of rector (chief administrator) of his alma mater the University of Kazan, where he also taught mathematics, physics, and astronomy, and with several prestigious national decorations, which, however, did not prevent him from ending his life in relative poverty. As for his revolutionary achievement in creating non-Euclidian (or super-Euclidian, as I would again put it) “Lobachevsky” geometry, he was scathingly ridiculed for it, and its submission for publication in 1826 was rejected. Only in 1829 was the first publication of his new theory made possible, but still with no academic support or recognition from his Russian peers. Recognition was to come from abroad, when in 1837 his article Géométrie Imaginaire was published in French in a respectable Berlin scientific journal, followed in 1840 by the small booklet Geometrische Untersuchungen zur Theorie der Parallellinien. Prussia’s great mathematical genius Gauss, happily admitting that he had been working on the same subject for a number of years himself, but had been reluctant to publish anything perhaps out of fear of being ridiculed, commended Lobachevsky for his work quite generously, but without false modesty: “…In the development of the subject, he followed a different path from the one which I had followed; it is done masterfully by Lobachevsky, in the true spirit of geometry…” Gauss promptly recommended to elect Lobachevsky as a foreign correspondent member of the newly established Göttingen Royal Science Society of which he himself was a distinguished member. Lobachevsky’s election did take place in 1842, but, aside from bringing him some token of international recognition, it did not bring him any profit. As is often the case with geniuses, the real fame would come to him posthumously.
Of greatest interest to us in this entry is, just as the title says, the mindwork of a genius. The great Euclidian Bible of Geometry and Mathematics, known as Elements, contains (already in Book I, out of 13) a series of axioms, postulates, and theorems, all of which, with the exception of one are either rigidly provable or made demonstrably self-evident. The one and only exception is the so-called Fifth Postulate on Parallelism. This Postulate is set apart from the others, and it had always been understood as a theorem, needing to be proven. For two thousand years mathematicians had seen this as a magnificent challenge and were going out of their way attempting to prove it, to no avail. Some actually deluded themselves in the belief that they had found the proof, only to be disproved soon thereafter; others had given up on finding the proof, honestly admitting their failure; still others had been loath to confess defeat and insisted that the postulate was unprovable, but that it was true anyway and had to be accepted on faith, because it just had to be true.
Now here comes our Lobachevsky. From an early age he becomes fascinated with the Fifth Postulate, and, like so many others before him, tries to prove it. Having failed to do so, he, however, does not go the ways of those others. Instead, he declares it… wrong! On the basis of this revolutionary declaration, he goes on to develop his non-Euclidian geometry, which is destined to open the door to the geometry of the future (a.k.a. modern geometry).
Simple? Very. Yet for two thousand years before him no one had been able to “find” that simplicity. This is the way genius works!
Curiously, calling Lobachevsky’s geometry “non-Euclidian” is not quite accurate, for which reason I have called it “super-Euclidian.” Proper Euclidian geometry isn’t contrapositive to Lobachevsky’s, it is included in it as a particular instance: when the curvature of the curved surface approaches zero, approximating none other than Euclid’s flat plane.
Now, regarding the primogeniture of Lobachevsky’s discovery among the other sons of Mathematica. There are two more names associated with his discovery. One is the great Gauss who had indeed toyed with this idea in his written drafts, but had never opted to release it into the wild, which disqualifies him from the laurels, obviously without diminishing his genius even by one iota. In this context, as he was happy to admit in the letter that I quoted earlier, his own path to the discovery was different from Lobachevsky’s, and thus the latter’s work is unquestionably retaining its uniqueness as the first of its kind, quod erat demonstrandum.
The third name, representing a tragic case indeed, is that of the Hungarian mathematician Janos Bolyai, who developed a similar theory independently of either Gauss or Lobachevsky, but published his work several years after the latter. There is no doubt that Bolyai was a bona fide genius in his own right, but there is very little sense or profit to argue about winners and losers in this unintended and unaware race. None of the said three geniuses should become a lesser genius if a different adjudication of this case is made. And, for all that we know, it is the Russian genius Lobachevsky who is primarily identified with this grandiose discovery and on the strength of it called “Copernicus of geometry” by posterity. This appellation originates with William Kingdon Clifford (1845-1879), an English mathematical prodigy and an enthusiastic adept of Lobachevsky geometry. His motion was later seconded by the Scottish-American mathematician and author Eric Temple Bell (1883-1960), who wrote this in his renowned, albeit controversial, book Men of Mathematics:
“The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other axioms or accepted truths, for example the law of causality which, for centuries, have seemed as necessary to straight thinking as Euclid’s postulate appeared till Lobachevsky discarded it.
The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the “Copernicus of Geometry,” for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.”
These words of Clifford and Bell seem like a fitting conclusion to this entry.
Not too many people have ever heard the name of Nikolai Ivanovich Lobachevsky, but to the readers of my blog his must have become a household name by now. “Mr. Non-Euclid” (or, more accurately, “Mr. Super-Euclid,” as we could call him for a good reason explained later) may sound a bit outlandish, but his greatest achievement was indeed proving that the great Euclid was not a General, but only a glorious particular.
The circumstances of his life are very interesting, and I encourage the reader to look up his biographies. His relatively minor accomplishments were the ones for which he was honored during his life by the position of rector (chief administrator) of his alma mater the University of Kazan, where he also taught mathematics, physics, and astronomy, and with several prestigious national decorations, which, however, did not prevent him from ending his life in relative poverty. As for his revolutionary achievement in creating non-Euclidian (or super-Euclidian, as I would again put it) “Lobachevsky” geometry, he was scathingly ridiculed for it, and its submission for publication in 1826 was rejected. Only in 1829 was the first publication of his new theory made possible, but still with no academic support or recognition from his Russian peers. Recognition was to come from abroad, when in 1837 his article Géométrie Imaginaire was published in French in a respectable Berlin scientific journal, followed in 1840 by the small booklet Geometrische Untersuchungen zur Theorie der Parallellinien. Prussia’s great mathematical genius Gauss, happily admitting that he had been working on the same subject for a number of years himself, but had been reluctant to publish anything perhaps out of fear of being ridiculed, commended Lobachevsky for his work quite generously, but without false modesty: “…In the development of the subject, he followed a different path from the one which I had followed; it is done masterfully by Lobachevsky, in the true spirit of geometry…” Gauss promptly recommended to elect Lobachevsky as a foreign correspondent member of the newly established Göttingen Royal Science Society of which he himself was a distinguished member. Lobachevsky’s election did take place in 1842, but, aside from bringing him some token of international recognition, it did not bring him any profit. As is often the case with geniuses, the real fame would come to him posthumously.
Of greatest interest to us in this entry is, just as the title says, the mindwork of a genius. The great Euclidian Bible of Geometry and Mathematics, known as Elements, contains (already in Book I, out of 13) a series of axioms, postulates, and theorems, all of which, with the exception of one are either rigidly provable or made demonstrably self-evident. The one and only exception is the so-called Fifth Postulate on Parallelism. This Postulate is set apart from the others, and it had always been understood as a theorem, needing to be proven. For two thousand years mathematicians had seen this as a magnificent challenge and were going out of their way attempting to prove it, to no avail. Some actually deluded themselves in the belief that they had found the proof, only to be disproved soon thereafter; others had given up on finding the proof, honestly admitting their failure; still others had been loath to confess defeat and insisted that the postulate was unprovable, but that it was true anyway and had to be accepted on faith, because it just had to be true.
Now here comes our Lobachevsky. From an early age he becomes fascinated with the Fifth Postulate, and, like so many others before him, tries to prove it. Having failed to do so, he, however, does not go the ways of those others. Instead, he declares it… wrong! On the basis of this revolutionary declaration, he goes on to develop his non-Euclidian geometry, which is destined to open the door to the geometry of the future (a.k.a. modern geometry).
Simple? Very. Yet for two thousand years before him no one had been able to “find” that simplicity. This is the way genius works!
Curiously, calling Lobachevsky’s geometry “non-Euclidian” is not quite accurate, for which reason I have called it “super-Euclidian.” Proper Euclidian geometry isn’t contrapositive to Lobachevsky’s, it is included in it as a particular instance: when the curvature of the curved surface approaches zero, approximating none other than Euclid’s flat plane.
Now, regarding the primogeniture of Lobachevsky’s discovery among the other sons of Mathematica. There are two more names associated with his discovery. One is the great Gauss who had indeed toyed with this idea in his written drafts, but had never opted to release it into the wild, which disqualifies him from the laurels, obviously without diminishing his genius even by one iota. In this context, as he was happy to admit in the letter that I quoted earlier, his own path to the discovery was different from Lobachevsky’s, and thus the latter’s work is unquestionably retaining its uniqueness as the first of its kind, quod erat demonstrandum.
The third name, representing a tragic case indeed, is that of the Hungarian mathematician Janos Bolyai, who developed a similar theory independently of either Gauss or Lobachevsky, but published his work several years after the latter. There is no doubt that Bolyai was a bona fide genius in his own right, but there is very little sense or profit to argue about winners and losers in this unintended and unaware race. None of the said three geniuses should become a lesser genius if a different adjudication of this case is made. And, for all that we know, it is the Russian genius Lobachevsky who is primarily identified with this grandiose discovery and on the strength of it called “Copernicus of geometry” by posterity. This appellation originates with William Kingdon Clifford (1845-1879), an English mathematical prodigy and an enthusiastic adept of Lobachevsky geometry. His motion was later seconded by the Scottish-American mathematician and author Eric Temple Bell (1883-1960), who wrote this in his renowned, albeit controversial, book Men of Mathematics:
“The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other axioms or accepted truths, for example the law of causality which, for centuries, have seemed as necessary to straight thinking as Euclid’s postulate appeared till Lobachevsky discarded it.
The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the “Copernicus of Geometry,” for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.”
These words of Clifford and Bell seem like a fitting conclusion to this entry.
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