Auguste Comte: Course of positive philosophy 2 – Celestial geometry

Although one would have assumed that celestial geometry concerns only such properties as the distance, figure and size of celestial objects, Comte defines these as only one class of phenomena in stellar geometry, namely, static phenomena. He spends considerable time explaining how astronomers can determine these static phenomena, that is, measure our distance from various stellar objects, recognise the figure of these objects, investigate their size and even note the density of their atmosphere. Comte also describes the history of attempts to determine the shape of the Earth, noting that even if there has always been room for making details more precise, this does not mean that the advance of science has been just replacing one error with another.

Comte is clearly more interested in the dynamic phenomena involving movement of celestial objects. He recounts in great detail the history of the discovery of Earth’s movement, both around its axis and around the Sun. What Comte finds philosophically interesting in this discovery is that it has forced us to abandon the theological idea of humans as the centre of the whole universe and also the teleological idea of planets moving for some purpose. This does not mean that astronomy has made the world meaningless, he soothes the reader, since through it we have found the lofty idea of humans as intelligences discovering the laws of the universe even from an insignificant vantage point.

A second important consequence of these discoveries is, Comte says, that we must distinguish the notions of world and universe. By world he means our region of universe, consisting of Earth and its nearby celestial objects – effectively, the Solar System. While people of earlier times could have thought that there is nothing beyond this world, modern astronomy must assume that the universe continues beyond our world, even if we cannot say anything certain about what happens beyond the confines of our world.

It is just to be expected that Comte still has much to say about the three laws of Kepler. He is especially keen to point out that Kepler had to overcome former mythological ideas, involving the notion of a circle as the perfect and thus the only suitable orbit for the supposedly divine stars. The great effect of these laws, Comte suggests, is that they allow us to make predictions about the orbits of planets, satellites and comets. Yet, he adds, even these laws are mere approximations of celestial mechanics – the topic of my next post.

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.

Auguste Comte: Course of positive philosophy 1 - Geometry

While many philosophers had considered it an important problem to put geometry on secure foundations, Comte finds such attempts mere unfounded metaphysics. For him, geometry is simply a natural science with an empirical basis. Of course, it is the most abstract natural science, dealing only with static spatial properties of things, in abstraction from any movement. Still, Comte feels no need to prove the basic axioms of geometry, since he can just assume them as bare facts. On the other hand, he also feels no need to consider the possibility of other geometries with other axioms, since experience appears to agree with the ordinary Euclidean geometry.

Comte supposes that this empirical science has a very practical purpose, namely, that of measuring spatial features of things. Of course, he adds, not all measuring is geometry, for instance, if we fill an oddly shaped container with water and then measure the volume of the water, this is still not geometry. Instead, geometry, like all mathematics, is an art of finding out quantities indirectly, through calculations.

When we measure spatial features of things, Comte continues, we can measure all the dimensions of it or then only some of them. When a geometer speaks of planes or lines, he adds, they are considering just such abstractions, that is, they are ignoring some of the dimensions the thing has. Thus, lines in reality always are wide and thick, we just concentrate on their length. We can even ignore all the dimensions of a thing and consider only its position in relation to other things - this is the origin of the notion of point.

While the essence of geometry is indirect measuring of spatial features of things, a geometer must assume some ways to directly measure these spatial features. This implicitly assumed form of measuring, Comte suggests, is the measurement of straight lines or lengths by comparing them with a length of some other thing, like a ruler. Other geometrical figures (curves, areas and volumes) are then to be measured with the help of these straight lines.

Comte admits that geometry is full of other things beyond mere measuring, that is, full of propositions about spatial properties of things. Still, he insists, even these properties are ultimately studied, because they could help with measuring (perhaps in some more concrete science). Because one cannot know beforehand what properties help with measurements, Comte advocates studying as many such properties as possible.

Comte mentions the old distinction between synthetic and analytic methods in geometry, but seems to have no idea what these terms meant originally and what their difference was supposed to be. Instead, Comte suggests that the distinction is just a roundabout way to distinguish ancient from modern geometry. Ancient geometry, Comte suggests, was mainly dealing with concrete and individual figures, taking one type of figure and finding all its characteristics. Because of the uniqueness of the chosen figure, none of these characteristics could be assumed to hold for other entities. Modern geometry, on the other hand, deals with abstract geometrical problems, which can then be applied to many different contexts.

Comte has little patience with ancient geometry, and he especially dislikes the common habit of starting to teach geometry from works like Euclid’s Elements - as he has noticed earlier, history of a discipline is usually not the most convenient way to teach it. Comte is especially critical of the proofs of early propositions in Elements, where Euclid simply places figures on top of each other, to show their similarity - this is just as futile as an attempt to prove the parallel axiom would be.

Although Comte is very critical of ancient geometry, he does admit that Greek mathematicians did make some progress. Particularly, they perfected the study of the simplest kind of figures, namely straight lines and polygons and polyhedras. Furthermore, while Comte ridicules the Greek use of diagrams as proofs of proposition, he does suggest it to be a sort of precursor for modern projective geometry. Another field developed by ancient geometers to perfection, Comte concludes, is trigonometry, where they used certain lines (sines,tangents etc.) to represent angles and their relations and so simplified calculations.

Comte associates the birth of modern geometry with Descartes. While it had been long known that geometric forms could be described in terms of spatial situations of their limiting points etc., it was the invention of Descartes, Comte says, to reduce the talk of situations to talk of lengths and other magnitudes through the notion of a coordinate system. In effect, Comte concludes, Descartes was able to transform at first sight qualitative properties like geometric figures into quantitative properties.

The outcome of the Cartesian transformation of geometry, Comte explains, was that now in two-dimensional geometry lines could be expressed by equations and equations by lines (in three-dimensional case, Comte adds, equations express surfaces, while lines are expressed by pairs of equations). These equations do not just characterise some random properties of the lines, Comte notes, but explicate how they could be generated. He adds that when a geometer is looking for an equation to describe a line, there is no need to choose any specific coordinate system - sometimes the searched for equation might be easier to describe e.g. in terms of polar coordinates, which are especially convenient when describing rotations. Then again, he admits, when finding lines to describe equations, it is best to pick the rectilinear coordinate system, which is the most natural for us to decipher.

Although Comte at first appears to say that all lines correspond to an equation and all equations correspond to a line, he is well aware that analytic geometry of his time has imperfections in this department. Firstly, he notes, discontinuous lines cannot be expressed so well in terms of a single equation. Furthermore, he continues, equations with more than three variables or equations with imaginary solutions have no proper geometric model.

Like in the case of abstract mathematics, Comte is mostly interested in the uses geometry could be put to and the problems that could be solved by its help. Solving some of these problems relies on simple algebraic means, such as when we try to find the number of points that are required for determining the course of a curve. Still, most of these problems, Comte notes, rely on the help of differential and integral calculus. Differentiation is useful not just for finding tangents, but also for describing e.g. the curvature of curves. Then again, Comte concludes, integration is the most useful tool, because it helps us to fulfill the true task of geometry, that of measuring lengths, areas and volumes.

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.

August Ferdinand Möbius - Barycentric calculus: new aid for analytical consideration of geometry (1827)

1790-1868

Concrete sciences have often been a spur on the development of new mathematical methods. Take as an example Archimedes, who used the idea of balancing a mass of parabola with a mass of a triangle and thus finding out the size of an area limited by the parabola. The exact method Archimedes used in determining this area was a rudimentary form of integration, with Archimedes, as it were, adding infinitely small pieces of the area together.


The idea of mass and weight was, coincidentally, a spur for Möbius’ text called Der barycentrische Calcül: ein neues Hilfsmittel zur analytischen Behandlung der Geometrie. Möbius starts with the notion of the center of mass, determined by mass of three objects situated in a triangle. He then quickly finds out that in fact any point within the triangle can be expressed as such a center, by just changing the masses assigned to the apex. Indeed, one need not be confined to points in this triangle, but allowing that in this case some of the masses could have a negative mass.

The end result is then the expression of points - and ultimately lines and figures - through three points of an arbitrary triangle. One can expand this meth to three-dimensional case by switching triangle to a pyramid. One can even express various simple functions in these terms - e.g.express curves like parabola, hyperbole and ellipse in terms of a three points of a triangle - and translate some concepts of infinitesimal calculus in this “barycentric” form.

A more interesting question is why Möbius uses such a peculiar form of expression instead of the more familiar method of expressing geometrical figures through two coordinate axes - that one can do seems a poor reason for actually doing so. The answer appears to be simply that some important relations are easier to express in this manner. Möbius is especially interested of what we nowadays might call equivalence relations between different figures. Two examples of such relations have been known since the beginning of mathematics - first is the equality of the sizes of the areas, no matter what their shape is, and second is the similarity of shapes, no matter what their size.

Beyond these two equivalence relations Möbius considers two others, first of which, affinity, had been introduced already by Euler. One might say that this affinity is a relaxed version of similarity, that is, a relation that could hold even between non-similar figures, although similar figures are always affine. Möbius introduced this relation by asking his reader to change the triangle used to determine figures with some completely different (not even similarly shaped) triangle. The figures defined by such change need not be similar - an example of such a pair is formed by a square and a rhombus. Still, they do have some resemblances, for instance, quantitative relations between respective parts (more precisely, parallel line segments) of these figures remain same.

A further form of equivalence relation Möbius discovers is what he calls collinear resemblance. Simply put, Möbius starts from affinity, takes away even further properties of the two figures and leaves only one point of resemblance - that all straight lines in one figure correspond to a straight line in the other figure. This definition loses even the identical ratios of lines, resulting in even more abstract resemblance. Some common quantitative properties still remain, namely, the so-called cross-ratios (a specific type of ratio for four points in the same line). Still, this is a very loose sort of relation - even circle and parabola have this sort of collinear resemblance.