[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.
By a proper appropriation of a few fundamental relations one becomes master of the whole subject; order takes the place of chaos, one beholds how all parts fit naturally into each other, and arrange themselves serially in the most beautiful order, and how related parts combine into well-defined groups. In this manner one arrives, as it were, at the elements, which nature herself employs in order to endow figures with numberless properties with the utmost economy and simplicity.—Steiner, J.
Werke, Bd. 1 (1881), p. 233.
[1878]. Euclid once said to his king Ptolemy, who, as is easily understood, found the painstaking study of the “Elements” repellant, “There exists no royal road to mathematics.” But we may add: Modern geometry is a royal road. It has disclosed “the organism, by means of which the most heterogeneous phenomena in the world of space are united one with another ” (Steiner), and has, as we may say without exaggeration, almost attained to the scientific ideal.—Hankel, H.
Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1869).
[1879]. The two mathematically fundamental things in projective geometry are anharmonic ratio, and the quadrilateral construction. Everything else follows mathematically from these two.—Russell, Bertrand.
Foundations of Geometry (Cambridge, 1897), p. 122.
[1880]. ... Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint,—an enchanted realm where thought is double and flows throughout in parallel streams.—Keyser, C. J.