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. _Lectures on Science, Philosophy and Arts (New York, 1908), p. 2._ =1881.= The ancients, in the early days of the science, made great use of the graphic method, even in the form of construction; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observation of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice; but did not abolish it. The Greeks and Arabians employed it still for a great number of investigations for which we now consider the use of the Calculus indispensable.--COMTE, A. _Positive Philosophy [Martineau], Bk. 1, chap. 3._