Nikolai Lobachevsky: Geometrical researches on the theory of parallels (1840)

 

(1792-1856)

Euclid’s book on geometry has for ages been seen as an ideal of an axiomatic theory, in which everything is based on a solid basis of definitions and evidently certain axioms and proven through strict demonstrations, making the results presented appear indubitable. No wonder many works of philosophy tried to imitate Euclid’s style, to make their theories seem as indubitable and necessary, usually failing miserably to be as convincing as Euclid.

If you know your Euclid by heart, you know that he had not really achieved the ideal many want to see in his book. There are sometimes slight hidden assumptions in his proofs - and isn’t it a bit too empirical to carry around triangles and put them on top of one another (Euclid, I.4)?

The most glaring fault in Euclid’s work is, of course, the infamous parallel postulate. When compared with other postulates of Euclid, it appears complex and far from self-evident: “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles”. The postulate can be made a bit clearer with some rewording and use of more modern phraseology - if line A cuts two other lines on the same plane, B and C, and the sum of interior angles on one side equals pi (or 180, if you are more into degrees), B and C eventually cut one another on that side and are therefore not parallel. Even with this rewording, it seems like a theorem we should prove, not a postulate to be just assumed.

Many professional geometers and even more geometry dilettantes were equally unimpressed by this postulate and tried to demonstrate it from other postulates. Their efforts led at most to finding other postulates that could replace Euclid’s. Most famous of them is the so-called Playfair’s axiom: given a line and a point, we can draw through the point, at the same plane as the line and the point, only one line that does not cut the first one. This does sound simpler, but still lacks the self-evidency of the other postulates.

Dissatisfaction with Euclid’s postulates and its alternatives continued, but no solution was forthcoming. All of this was changed by Lobachevsky’s seminal paper, Geometrische Untersuchungen zur Theorie der Parallellinien. Well, to be truthful, he had already written papers on the topic in his native language, Russian, in 1820s, but these did not circulate very widely (and in addition, I cannot read Russian).

Lobachevsky’s starting point is the Playfair’s axiom, but instead of just assuming it, he asks what would happen, if there were more than one line we could draw through the point - lines which would not cut the given line. He notes that even then we could find a single particularly interesting one among those non-cutting lines, namely, the limit between lines that do and those that do not, and suggests calling this the parallel line. Then, by tying Playfair’s axiom back to the original framing of Euclid’s postulate (C being the original line, B being parallel to it, A cutting them both and A and B meeting at the given point) and by making the assumption that A cuts C perpendicularly, he notes that in this peculiar setting B and A form an angle less than half the pi (or 90 degrees), thus contradicting Euclid’s postulate. The angle formed by A and B (and dependent on the distance of the given point from the line C), Lobachevsky calls the angle of parallelism.

Although Lobachevsky’s new geometry - later dubbed hyperbolic geometry, although he himself called it imaginary - has clearly different properties from Euclidean geometry, a significant portion of Lobachevsky’s paper is committed to show similarities to Euclidean geometry. The simplest similarity is that the new definition of parallel lines works similarly enough to the Euclidean notion, for instance, parallelism is a symmetrical and transitive relation.

A more intricate similarity Lobachevsky finds through notions of oricycle and orisphere. By oricycle Lobachevsky means such a curve in hyperbolic geometry, all perpendiculars or axes of which are parallel to each other. Furthermore, oricycle is also a sort of limit for circles - by enlarging the ray of the circle indefinitely, in hyperbolic geometry, the curve tends toward the oricycle. In a figurative way, we could say that an oricycle is an infinite circle. Interestingly, the same concept in Euclidean geometry means ordinary straight line.

The notion of oricycle taken into three dimensions forms, then, an orisphere. Technically, an orisphere can be formed from an oricycle by turning it around one of its axes. What is interesting is that oricycles on orisphere work like straight lines on a plane in Euclidean geometry, for instance, in a “triangle” formed of segments of three different oricycles, the sum of the angles equals pi.In effect, two-dimensional Euclidean geometry can be ingrained within three-dimensional hyperbolic geometry.

Although the aforementioned pseudotriangles in hyperbolic geometry do follow same rules as regular triangles in Euclidean geometry, regular triangles in hyperbolic geometry do not. Yet, Lobachevsky points out, when sides of the triangles in hyperbolic geometry decrease indefinitely, the more they start to resemble the triangles in Euclidean geometry, for instance, the sum of their angles approaches pi.An interesting consequence of this is that the bigger the triangles in question are, the more apparent the difference of the two geometries becomes. It becomes then an empirical problem to decide whether we live in a space with a Euclidean or a hyperbolic geometry - just make astronomical measurements of distances between stars and you might notice signs of non-Euclidean properties.