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| 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.
