=1870.= Many are acquainted with mathematics, but mathesis few know. For it is one thing to know a number of propositions and to make some obvious deductions from them, by accident rather than by any sure method of procedure, another thing to know clearly the nature and character of the science itself, to penetrate into its inmost recesses, and to be instructed by its universal principles, by which facility in working out countless problems and their proofs is secured. For as the majority of artists, by copying the same model again and again, gain certain technical skill in painting, but no other knowledge of the art of painting than what their eyes suggest, so many, having read the books of Euclid and other geometricians, are wont to devise, in imitation of them and to prove some propositions, but the most profound method of solving more difficult demonstrations and problems they are utterly ignorant of.--LAFAILLE, J. C. _Theoremata de Centro Gravitatis (Anvers, 1632), Praefat._ =1871.= The elements of plane geometry should precede algebra for every reason known to sound educational theory. It is more fundamental, more concrete, and it deals with things and their relations rather than with symbols.--BUTLER, N. M. _The Meaning of Education etc. (New York, 1905), p. 171._ =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.