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.

George Peacock: Treatise on Algebra (1830)

 

(1791–1858)

On a surface level, Peacock’s Treatise on Algebra seems a rather humdrum textbook on algebra, with no particularly original innovations to be found in it. Yet, what is most important in the treatise are not any purely mathematical results, but the more philosophical considerations on the nature of algebra and its relation to other parts of mathematics and especially arithmetics.

What Peacock is particularly interested in is the question whether algebra should be somehow based on arithmetics, which is based on concrete groups of objects, like seven apples. Peacock’s answer is a resounding no: arithmetics is far more constrained, allowing e.g. no negative numbers, since you cannot produce a group of, say, minus seven apples, although negative numbers are a common occurrence in algebra.

Not restricted by characteristics of arithmetical numbers, what is to determine what rules are to be assumed in algebra? Peacock makes the bold suggestion that these rules are just assumed: algebra concerns only symbols, and we could in principle choose any rules to govern our calculations in it.

Still, Peacock does not yet reach the modern notion of algebra, where we can have many different systems of calculation with different rules. Instead, he thinks that algebra should be especially the most general system of calculation, where all sorts of calculation are possible. Thus, he accepts not just calculations leading to negative numbers, but also roots of negative numbers, which we nowadays call imaginary numbers. Peacock goes even so far as to suggest that such seemingly nonsensical symbols like 0/0 could be used in a meaningful manner (although he mentions this possibility only in passing, he is referring to certain readings of the infinitesimal calculus).

Peacock still wants that algebra would be of practical use. Here, he suggests that the rules of algebra should at least correspond to the rules of arithmetics at least in those cases where the calculations and their results make arithmetical sense. Then again, he continues, results of algebra could also be used in other mathematical sciences, where some of them might be of relevance. Thus, negative numbers do make sense, for instance, when speaking of debt or of movement to an opposite direction. Even imaginary numbers can have concrete meaning, as describing movement perpendicular to a line.

Algebra becomes then, for Peacock, like a general tool for mathematical problem solving. When a concrete problem is given, it is turned into algebraic symbols: what quantities are known and what are unknown that are supposed to be determined in terms of the known quantities? After doing the algebraic manipulation of the symbols, one must still interpret the results. Often the context of the problem restricts this interpretation, and although algebra would lead to a number of possible results, only some of them might make sense in the context. Sometimes no result suggested by algebra would make sense, and then the problem itself will be impossible.