The philosophy of mathematics · Auguste Comte
IMPERFECTIONS OF ANALYTICAL GEOMETRY.
Having now considered the fundamental conception of analytical geometry under its principal elementary aspects, it is proper, in order to make the sketch complete, to notice here the general imperfections yet presented by this conception with respect to both geometry and to analysis. _Relatively to geometry_, we must remark that the equations are as yet adapted to represent only entire geometrical loci, and not at all determinate portions of those loci. It would, however, be necessary, in some circumstances, to be able to express analytically a part of a line or of a surface, or even a _discontinuous_ line or surface, composed of a series of sections belonging to distinct geometrical figures, such as the contour of a polygon, or the surface of a polyhedron. Thermology, especially, often gives rise to such considerations, to which our present analytical geometry is necessarily inapplicable. The labours of M. Fourier on discontinuous functions have, however, begun to fill up this great gap, and have thereby introduced a new and essential improvement into the fundamental conception of Descartes. But this manner of representing heterogeneous or partial figures, being founded on the employment of trigonometrical series proceeding according to the sines of an infinite series of multiple arcs, or on the use of certain definite integrals equivalent to those series, and the general integral of which is unknown, presents as yet too much complication to admit of being immediately introduced into the system of analytical geometry.
Ideas discussed in this chapter
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