Gauss-Bolyai-Lobachevsky: The Dawn of Non-Euclidean Geometry
“One of Euclid’s postulates — his postulate 5 — had the fortune to be an epoch-making statement — perhaps the most famous single utterance in the history of science.” ~ Cassius Jackson Keyser
Elements — the geometry textbook compiled by Euclid in 300 BC, holds a distinguished place in the history of human thought. It marks nothing less than an epoch in the development of logical thinking. For, it was the first text to demonstrate that any logical system must rest on a few basic facts (axioms/postulates) which are to be taken for granted. Not everything can be proved.
Elements started with laying out a few basic postulates, and then built a marvellously consistent world on top of them. Inside the Elements, anything and everything can be backtracked to the postulates we start with. This was to serve as a blueprint for all scientific thought to come. How seminal a work Elements was can be measured from the fact that it is second only to the Bible in terms of the number of editions published.
The Postulates
So, without further ado, let’s go through the five basic postulates of the Elements (not verbatim).
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
- All Right Angles are equal.
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough.
What’s up with the Last Postulate?
At first glance, perhaps all the postulates seem equally intuitive. However, over time, mathematicians developed a singular obsession around the last one. Why? Key factors were —
- The postulate being overly descriptive, leading to the suspicion that actually it was a theorem, which could be proved by the other four postulates. If this were true, it would become redundant to have it as a postulate.
- The first 26 propositions of Book 1 of Elements (around elementary theory of triangles) can be proven with the help of the first four postulates only, without resorting to the fifth. Hence, we see that a considerable portion of geometry would remain perfectly valid, irrespective of the fifth postulate’s status.
For several centuries, the former point above set mathematicians on course to either simplify the fifth postulate, or prove it using the other postulates (thereby rendering it redundant).
Replacing the Fifth Postulate
Till the beginning of the 19th century, there were attempts galore to rid Euclid’s geometry of the seemingly out of place fifth postulate. A typical example of a substitution/replacement attempt was Playfair’s Axiom —
“Through a given point can be drawn only one parallel to a given line.”
It’s not tough to show that the above is equivalent to the fifth postulate. All such substitutions were attempts to make the postulate less verbose. But, in reality, none of them succeeded in the task. At the best, they were merely superficial improvements over the last postulate.
Proving the Fifth Postulate
Replacements aside, another line of attack for mathematicians was to prove the fifth postulate using the first four. Notable among these attempts was that by Saccheri, an Italian Jesuit priest.
Prior to Saccheri, mathematicians had largely tried to (falsely) prove the fifth postulate by taking an implicit assumption equivalent to the postulate itself. Saccheri’s novelty lied in his approach to tackle the postulate by reductio ad absurdum technique. He was probably the first one bold enough to deny the validity of the postulate (replacing it by differing axioms), and see where logical reasoning led him to.
Saccheri had expected to land up with a few contradictions, which would have validated the fifth postulate. Instead, he was baffled to land up somewhere else entirely! For, he simply couldn’t find any inconsistency in the geometry he constructed by altering the fifth postulate.
Such were the repercussions of this experiment, that most probably even Saccheri himself wasn’t able to comprehend the depth of what he had accidentally trodden upon! It was a pity that Saccheri could not come up with a fitting conclusion to his research. As such, the onus to explore deeper along these lines fell to mathematicians born a century later.
The Eventual Realization
So, why did Saccheri (and others to follow) fail to find any contradictions upon assuming the fifth postulate as false? Was it down to their mathematical shortcomings? No, not at all. Rather, they failed to find one simply because there was none to be arrived at! The search for a contradiction was doomed to end empty handed.
What does that mean exactly? It means that there exist geometries fundamentally different from Euclid’s, geometries where the fifth postulate doesn’t hold! And these geometries are just as logically sound and consistent as Euclid’s, just a lot less intuitive.
What do these geometries look like? It turns out there are two types of Non Euclidean geometries, considering the behaviour of two straight lines indefinitely extended in a two-dimensional plane, both being perpendicular to a third line.
- In our classical Euclidean geometry, the two lines remain equidistant forever, and are known as parallel lines.
- In the first type of Non-Euclidean geometry, called Hyperbolic geometry, the two lines curve away from each other, increasing in distance as one moves further from the point of intersection.
- In the other Non-Euclidean geometry, known as Elliptic geometry, the two lines curve towards each other and intersect eventually. In a way, this geometry is analogous to the geometry on a sphere (spherical geometry).
It was this realization that marked the start of the 19th century. After centuries of toil, finally the fruit (of Non-Euclidean geometry) was ripe for plucking. And to consume it elegantly were present three remarkable mathematicians — Gauss, Bolyai and Lobachevsky. Below we examine some select communications from the trio which provide meaningful insights into the thought process behind Non Euclidean geometry. So, let’s get started!
Gauss
“The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible” ~ Gauss (in a letter written on 8th November, 1824 to F.A. Taurinus)
Carl Friedrich Gauss hardly needs any introduction to mathematics aficionados. Gauss was perhaps the most imposing figure in the 19th century mathematics ecosystem. Hence, it’s but natural that he had a considerable contribution to the early development of Non-Euclidean geometry. In fact, he was the one who christened the new geometry as Non-Euclidean!
Note, however, that even Gauss started off on the same path as Saccheri, in considering the fifth postulate as true and attempting to prove it using reductio ad absurdum. But ultimately (during the second decade of the 19th century), Gauss’ genius was able to recognize the fundamental difficulties with that postulate, and he was bold enough to venture out and discover the alternate postulates of Non-Euclidean geometry!
We let the below two letters, authored by Gauss, speak for themselves —
Letter to F.A. Taurinus, written at Göttingen on November 8, 1824
"I have not read without pleasure your kind letter of October 30th with the enclosed abstract, all the more because until now I have been accustomed to find little trace of real geometrical insight among the majority of people who essay anew to investigate the so-called Theory of Parallels.In regard to your attempt, I have nothing (or not much) to say except that it is incomplete ... I imagine that this problem has not engaged you very long. I have pondered it for over thirty years, and I do not believe that anyone can have given more thought to this second part than I, though I have never published anything on it. The assumption that the sum of the three angles [of a triangle] is less than 180 degrees leads to a curious geometry, quite different from ours (the Euclidean), but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it with the exception of the determination of a constant, which cannot be designated a priori. The greater one takes this constant, the nearer one comes to Euclidean geometry, and when it is chosen infinitely large the two conincide. The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible. For example, the three angles of a triangle become as small as one wishes, if only the sides are taken large enough; yet the area of the triangle can never exceed a definite limit, regardless of how great the sides are taken, nor indeed can it ever reach it. All my efforts to discover a contradiction, an inconsistency, in this Non-Euclidean geometry have been without success ...I do not fear that any man who has shown that he possesses a thoughtful mathematical mind will misunderstand what has been said above ... Perhaps I shall myself, if I have at some future time more leisure than in my present circumstances, make public my investigations."
Letter to Schumacher, written on May 17, 1831
"I have begun to write down during the last few weeks some of my own meditations, a part of which I have never previously put in writing, so that already I have had to think it all through anew three or four times. But I wished this not to perish with me."Gauss did publish some elementary material on Non-Euclidean geometry. However, he could not advance much further. Not because of some limitation on his end, rather because someone else had come up with identical ideas! And that someone had gone on to take the extra pain of making his discoveries public. On February 14, 1832 Gauss received a copy of Johann Bolyai’s iconic work, Appendix (thus named because it was written as an appendix to a mathematics textbook by his father).
Bolyai
“Out of nothing I have created a strange new universe.” ~ Johann Bolyai
Farkas Bolyai (1775–1856) attended Göttingen University from 1796 to 1799. During this period, the Hungarian befriended Gauss, who was also a student there. It is quite probable that the two gave much common thought to problems around the fifth postulate. This is backed by the fact that even after having left the University, the duo’s correspondence included material around the theory of parallels.
But even though Farkas held a lifelong obsession with the fifth postulate, he wasn’t able to materialize much. Or at least he got nowhere near his son’s contributions to the subject! Johann Bolyai, born in 1802 to Farkas Bolyai, went on to comprehensively eclipse his father’s achievements in Non-Euclidean geometry.
Having been tutored by his father in his formative years, it’s no surprise that Johann also developed a keen interest in Non-Euclidean geometry. So much so that he went on to dedicatedly pursue the subject against the wishes of Farkas, who had advised him to stay away from Non-Euclidean geometry, probably a sensible advice from a father who had returned empty handed from a treasure hunt, and didn’t want his son to wander the same way!
But Johann persevered, and persevered well. And unlike his father, Johann was able to come up with a concrete theory out of his meditations on Non-Euclidean geometry!
After having worked on his ideas for some time, Johann sent the following note to his father.
Letter to Farkas Bolyai, written on November 3, 1823
"It is now my definite plan to publish a work on parallels as soon as I can complete and arrange the material and an opportunity presents itself; at the moment I still do not clearly see my way through, but the path which I have followed gives positive evidence that the goal will be reached, if it is at all possible; I have not quite reached it, but I have discovered such wonderful things that I was amazed and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe. All that I have sent you previously is like a house of cards in comparison with a tower. I am no less convinced that these discoveries will bring me honour, than I would be if they were completed."Farkas replied very positively, urging that Johann publish his work as soon as possible. However, it took some years for the nascent formulations to develop fully. Finally, in 1832, Johann Bolyai was able to stitch his ideas together to come up with his landmark publication, the Appendix.
Farkas, being Gauss’ close friend, sent him an advance copy of the Appendix for his kind perusal. It would be fitting to end this section with Gauss’ response!
Letter from Gauss to Farkas Bolyai, written on March 6, 1832 (Commenting on Johann Bolyai’s work on Non-Euclidean geometry)
"If I begin with the statement that I dare not praise such a work, you will of course be startled for a moment: but I cannot do otherwise; to praise it would amount to praising myself; for the entire content of the work, the path which your son has taken, the results to which he is led, coincide almost exactly with my own meditations which have occupied my mind for from thirty to thirty-five years. On this account I find myself surprised to the extreme.My intention was, in regard to my own work, of which very little up to the present has been published, not to allow it to become known during my lifetime. Most people have not the insight to understand our conclusions and I have encountered only a few who received with any particular interest what I communicated to them. In order to understand these things, one must first have a keen perception of what is needed, and upon this point the majority are quite confused. On the other hand it was my plan to put all down on paper eventually, so that at least it would not finally perish with me.So I am greatly surprised to be spared this effort, and am overjoyed that it happens to be the son of my old friend who outstrips me in such a remarkable way."
Lobachevsky
“It is unlikely that mathematicians have yet exhausted the richness of the world that Lobachevsky discovered in the relative isolation of Kazan.” ~ Jeremy Gray, Historian of Modern Mathematics.
Unlike Gauss and the Bolyais, Nikolai Lobachevsky was unique in the sense that he did not have any active correspondence with other pioneers of Non-Euclidean geometry. Though it’s reasonable to assume that this was a result of Lobachevsky’s geographical positioning, rather than a conscious refrain. Lobachevsky lived his entire life in Russian obscurity, cut off from the European hub of mathematics, a fact which makes his already enviable achievements all the more startling!
Lobachevsky is credited with the first printed material on Non-Euclidean geometry — a memoir on the principles of geometry in the Kasan Bulletin, published in 1829–30 (two years before Johann Bolyai published the Appendix). However, owing to the language being Russian, the text received practically null readership.
But so confident was Lobachevsky of his new theory that he went on to publish material in mainstream European languages as well, in the hope of receiving wider readership. Most important among these publications was a German text — Geometrische Untersuchungen zur Theorie der Parallellinien (Geometric investigations into the theory of parallel lines).
It was this German text that reached Gauss’s hands, and as was befitting, Gauss accorded the highest merit to the Russian’s splendid work! This is clearly visible from the below communication.
Letter from Gauss to Schumacher, written in 1846 (Commenting on Lobachevsky’s Untersuchungen)
"I have recently had the occasion to look through again that little volume by Lobachevsky (Geometrische Untersuchungen zur Theorie der Parallellinien). It contains the elements of that geometry which must hold, and can with strict consistency hold, if the Euclidean is not true ... You know that for fifty-four years now (since 1792) I have held the same conviction (with a certain later extension, which I will not mention here). I have found in Lobachevsky's work nothing that is new to me, but the development is made in a way different from that which I have followed, and certainly by Lobachevsky in a skilful way and in truly geometrical spirit. I feel that I must call your attention to the book, which will quite certainly afford you the keenest pleasure."Interestingly (and somewhat expectedly), Gauss recommended the Untersuchungen to Farkas Bolyai (Johann’s father) in 1848. And it was thus that Johann Bolyai also came to know about Lobachevsky’s ideas. The positive mood with which Johann received them is evident from the below remarks found in his unpublished notes — Bemerkungen über Nicolaus Lobatchefskij’s Geometrische Untersuchungen (Remarks on Nikolai Lobachevsky’s Geometric Investigations).
Johann Bolyai on Lobachevsky’s work on Non-Euclidean Geometry
"Even if in this remarkable work different methods are followed at times, nevertheless, the spirit and result are so much like those of the Appendix ..., that one cannot recognize it without wonder. If Gauss was, as he says, surprised to the extreme, first by the Appendix and later by the striking agreement of the Hungarian and Russian mathematicians: truly, none the less so am I.The nature of real truth of course cannot but be one and the same in Maros-Vásárhely as in Kamschatka and on the Moon, or, to be brief, anywhere in the world; and what one finite, sensible being discovers, can also not impossibly be discovered by another."
That’s some high praise indeed. Note, however, that based on available evidence, it seems Lobachevsky never even heard of Johann Bolyai! Missed correspondences such as this were typical of Lobachevsky’s life, and ultimately resulted in the genius getting scant recognition during his lifetime. It’s safe to say that the true depth and novelty of his work was recognized and appreciated only posthumously.
Concluding Remarks
The study of Non-Euclidean geometry, initiated by the likes of Gauss, Bolyai and Lobachevsky, now forms an indispensable part of higher mathematics. Following the above three, many others went on to refine and advance the area conceptually. Most notable among them was the legendary Bernhard Riemann, who approached the subject from the viewpoint of differential geometry, rather than the synthetic methods used previously.
And then came Albert Einstein and The General Theory of Relativity, which transposed Non-Euclidean geometry from the realms of the purely academic to something having grave practical consequences. For, the spatial framework which embeds The General Theory of Relativity is Non-Euclidean in nature, thereby suggesting that that the physical space we live in may not be Euclidean after all!
Acknowledgements
Almost all the information contained in this article is derived from Introduction to Non-Euclidean Geometry, by Harold E. Wolfe (in particular Chapter 3: The Discovery of Non-Euclidean Geometry). The book, in turn, draws most of its material from the works of Paul Stäckel (1862–1919), a German mathematician who made significant contributions to the history of mathematics as well (most important, for this article, is Stäckel’s study of the letters exchanged between Gauss and Farkas Bolyai).
References
- Harold E. Wolfe, Introduction to Non-Euclidean Geometry
- Benjamin Wardhaugh, The Book of Wonders: The Many Lives of Euclid’s Elements
- Paul Stäckel, Wikipedia
- Euclid’s Elements, Wikipedia
- Euclid’s Postulates — https://people.math.harvard.edu/~ctm/home/text/class/harvard/113/97/html/euclid.html
- Non-Euclidean Geometry, Wikipedia
