After this important digression, we will pursue the philosophical examination of special geometry, always considered as reduced to its least possible development, as an indispensable introduction to general geometry. We have now sufficiently considered the graphical solution of the fundamental problem relating to the right line—that is, the determination of the different elements of any right-lined figure by means of one another—and have now to examine in a special manner the algebraic solution.
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.
TRIGONOMETRY.
The difficulty in resolving algebraically such a question