The philosophy of mathematics · Auguste Comte
ALGEBRAIC SOLUTIONS.
This kind of solution, the evident superiority of which need not here be dwelt upon, belongs necessarily, by the very nature of the question, to the system of the ancient geometry, although the logical method which is employed causes it to be generally, but very improperly, separated from it. We have thus an opportunity of verifying, in a very important respect, what was established generally in the preceding chapter, that it is not by the employment of the calculus that the modern geometry is essentially to be distinguished from the ancient. The ancients are in fact the true inventors of the present trigonometry, spherical as well as rectilinear; it being only much less perfect in their hands, on account of the extreme inferiority of their algebraical knowledge. It is, then, really in this chapter, and not, as it might at first be thought, in those which we shall afterwards devote to the philosophical examination of _general_ geometry, that it is proper to consider the character of this important preliminary theory, which is usually, though improperly, included in what is called _analytical geometry_, but which is really only a complement of _elementary geometry_ properly so called. Since all right-lined figures can be decomposed into triangles, it is evidently sufficient to know how to determine the different elements of a triangle by means of one another, which reduces _polygonometry_ to simple _trigonometry_.
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