Excerpt from A Treatise on Conic Sections: Containing an Account of Some of the Most Important Modern Algebraic and Geometric Methods, 2d Rev. And Enl Art. 1. Geometrical theorems may be divided into two classes: theorems concerning the magnitude of lines, and concerning their position ; for example, that "the square of the hypothenuse is equal to the sum of the squares of the sides," is a theorem concerning magnitude; that "the three perpendiculars of a triangle meet in a point," is a theorem concerning position. 2. Theorems of the former class can easily be expressed algebraically. To take the example already given, if the lengths of the sides of a right-angled triangle be a, b, c, the proposition alluded to is written c2 = a2 + b2. The learner is probably already .familiar with this application of algebra to geometry, as the propositions of the Second Book of Euclid all relate merely to the magnitude oflines, and the demonstration of them is much simplified by the use of algebraical symbols. 3. But it is by no means so easy to see how to express algebraically theorems involving the position of lines. Accordingly, although algebra was, soon after its introduction into Europe, applied to the solution of the first class of questions, its use was not extended to this latter class until the year 1637, when Des Cartes, by the publication of his "Geometrie,' laid the foundation of the science on which wc arc about to enter. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.