GEOMETRICAL Solution of a problem, is when the problem is directly resolved according to the strict rules and principles of geometry, and by lines that are truly geometrical. This expression is used in contradistinction to an arithmetical, or a mechanical, or instrumental solution, the problem being resolved only by a ruler and compasses.
The same term is likewise used in opposition to all indirect and inadequate kinds of solutions, as by approximation, infinite series, &c. So we have no geometrical way of finding the quadrature of the circle, the duplication of the cube, or two mean proportionals, though there are mechanical ways, and others, by infinite series, &c.