[1872]. The reason why geometry is not so difficult as algebra, is to be found in the less general nature of the symbols employed. In algebra a general proposition respecting numbers is to be proved. Letters are taken which may represent any of the numbers in question, and the course of the demonstration, far from making use of a particular case, does not even allow that any reasoning, however general in its nature, is conclusive, unless the symbols are as general as the arguments.... In geometry on the contrary, at least in the elementary parts, any proposition may be safely demonstrated on reasonings on any one particular example.... It also affords some facility that the results of elementary geometry are in many cases sufficiently evident of themselves to the eye; for instance, that two sides of a triangle are greater than the third, whereas in algebra many rudimentary propositions derive no evidence from the senses; for example, that a3−b3 is always divisible without a remainder by a−b.—De Morgan, A.
On the Study and Difficulties of Mathematics (Chicago, 1902), chap. 13.
[1873]. The principal characteristics of the ancient geometry are:—
(1) A wonderful clearness and definiteness of its concepts and an almost perfect logical rigour of its conclusions.
(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.