(2) A complete want of general principles and methods.... In the demonstration of a theorem, there were, for the ancient geometers, as many different cases requiring separate proof as there were different positions of the lines. The greatest geometers considered it necessary to treat all possible cases independently of each other, and to prove each with equal fulness. To devise methods by which all the various cases could all be disposed of with one stroke, was beyond the power of the ancients.--CAJORI, F. _History of Mathematics (New York, 1897), p. 62._ =1874.= It has been observed that the ancient geometers made use of a kind of analysis, which they employed in the solution of problems, although they begrudged to posterity the knowledge of it.--DESCARTES. _Rules for the Direction of the Mind; The Philosophy of Descartes [Torrey] (New York, 1892), p. 68._ =1875.= The ancients studied geometry with reference to the _bodies_ under notice, or specially: the moderns study it with reference to the _phenomena_ to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They abstract, to treat by itself, every question relating to the same geometrical phenomenon, in whatever bodies it may be considered. Geometers can thus rise to the study of new geometrical conceptions, which, applied to the curves investigated by the ancients, have brought out new properties never suspected by them.--COMTE. _Positive Philosophy [Martineau], Bk. 1, chap. 3._ =1876.= It is astonishing that this subject [projective geometry] should be so generally ignored, for mathematics offers nothing more attractive. It possesses the concreteness of the ancient geometry without the tedious particularity, and the power of the analytical geometry without the reckoning, and by the beauty of its ideas and methods illustrates the esthetic generality which is the charm of higher mathematics, but which the elementary mathematics generally lacks. _Report of the Committee of Ten on Secondary School Studies (Chicago, 1894), p. 116._ =1877.= There exist a small number of very simple fundamental relations which contain the scheme, according to which the remaining mass of theorems [in projective geometry] permit of orderly and easy development.