János Bolyai: The science absolute of space: independent of the truth or falsity of Euclid's axiom XI (which can never be decided a priori) (1832)

 

(1802–1860)

A good example of a remarkable coincidence of two persons having almost the same idea roughly simultaneously is the discovery of non-Euclidean geometries, and more precisely, the so-called hyperbolic geometry. We have already seen how Russian mathematician Lobachevsky approached the idea of not assuming Euclid’s parallel axiom, and we are now about to see the Hungarian János Bolyai do it in his own manner.

One might say that the interest in the parallel axiom ran in the Bolyai family, since János’s father, Farkas Bolyai had for a long time tried to deduce the axiom. Indeed, he had warned his son that the parallel axiom was something younger Bolyai should keep away from, since one could waste a lifetime thinking about it. To older Bolyai’s surprise, younger Bolyai sent his father a short paper dealing with the issue, which older Bolyai published as an appendix to his own textbook on mathematics.

The topic of Bolyai junior’s article is primarily the absolute geometry, that is, a geometry where neither Euclid’s parallel axiom nor its denial is assumed. Thus, just like Lobachevsky, Bolyai is interested firstly in the similarities between the Euclidean and the hyperbolic geometry. Again like Lobachevsky, Bolyai defines as a parallel line to a given line X as that precise line, which is a sort of limit of all the lines drawn through the same point y and not cutting the given line X, in the sense that any line falling from that point y more toward the given line X will cut this given line.

Bolyai also shows, like Lobachevsky before him, that parallelism, defined in this manner, is a transitive relation. Bolyai goes even further and shows that with the parallel lines, the congruence of line segments is also a transitive relation. He then notes that given a line segment AM, one can consider a collection of all such points B that if a line segment BN is parallel to AM, it is also congruent to it. Bolyai doesn’t really give any other name to this collection, but F. In addition to F, Bolyai considers the intersection of F with any plane containing AM, which he calls L, while AM he calls the axis of L. He also notes that F can be described by revolving L around AM. Because parallelism and congruence of line segments is transitive, every line segment BN starting from a point B in L and parallel and congruent to AM is also an axis of L.

In Euclidean geometry, Bolyai notes, this L is simply a line perpendicular to AM - and F, similarly, a plane perpendicular to AM. In hyperbolic geometry, on the other hand, L is not a straight line, but a curved line, and similarly F is a curved surface. Indeed, they are what Lobachevsky called respectively oricycle and orisphere. Like Lobachevsky, Bolyai notes that oricycles in an orisphere work like straight lines in a Euclidean plane.

Now, Bolyai pictures an axis AM move through its oricycle L, always staying at same angle to L. He saw that the other points of the axis AM described further oricycles, that is, taken C from AM would describe an oricycle, for which AC would be the axis. Furthermore, taking corresponding parts of two such oricycles, the relation X of their lengths is always a constant, which depends not on the length of the parts, but only of the distance x of the points on AM. Bolyai also notes that these oricycles are always congruent, although part of one appears to be multiple of the corresponding part of the other.

Bolyai further notes that whether one supposes Euclidean or hyperbolic geometry to hold, spherical trigonometry – that is, study of triangles, as it were, on the surface of a sphere – always follows the same rules. Then again, he adds, with the ordinary geometry the case is quite reversed. In Euclidean geometry, given the length of at least one side of a triangle and at least two other elements of the same triangle (whether angles or sides) are known, the other elements can be solved. In hyperbolic geometry, on the other hand, one has to also refer to some length x, of which the corresponding relation X is known, to make similar calculations. Bolyai suggests using length i, defined by having as the corresponding relation e - the basis of natural logarithms. This length i would then work as a sort of natural unit of length in the hyperbolic geometry.

The intriguing question is then, which of the two, Euclidean or hyperbolic geometry, is the one describing the world we live in. Bolyai notes that we cannot really determine this without any empirical facts to guide us, since neither of the two geometries has any intrinsic flaw in it. This statement goes against the common idea of Bolyai’s contemporaries that Euclidean geometry is somehow inherently intrinsic. As if to just spite such thinkers, Bolyai ends his short article by showing how one can construct in hyperbolic geometry a square equal in area to a circle – something that is impossible in Euclidean geometry.

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.