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.
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