Mathematica, and Rhetorica, and Grammatica are also among the best known. A group known as Montes Naturales comprises Physica, Biologica, and Chemica, and one great peak with minor peaks about it is called. by the people Philosophia. There are those who claim that these great masses of rock are too old to be climbed, as if that affected the view; while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary. But this affects only a few who belong to the real mountains, and the others labor diligently to improve the paths and to lessen unnecessary toil, but they seek not to tear off the summits nor do they attend to the amusing attempts of those who sit by the hillocks and throw pebbles at the rocky sides of the mountains upon which they work. Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating. A guide who points out the beauties, the grandeur, and the special places of interest commands the admiration of his group of pilgrims. One who fails to do this, who does not know the paths, who puts unnecessary burdens upon the pilgrim, or who blindfolds him in his progress, is unworthy of his position. The pretended guide who says that the painted panorama, seen from the rubbertired car, is as good as the view from the summit is simply a fakir and is generally recognized as such. The mountain will stand; it will not be used as a mere commercial quarry for building stone; it will not be affected by pellets thrown from the little hillocks about; but its paths will be freed from unnecessary flints, they will be straightened where this can advantageously be done, and new paths on entirely novel plans will be made as time goes on, but these paths will be hewed out of rock, not made out of the dreams of a day. Every worthy guide will assist in all these efforts at betterment, and will urge the pilgrim at least to ascend a little way because of the fact that the same view cannot be obtained from other peaks; but he will not take seriously the efforts of the fakir, nor will he listen with more than passing interest to him who proclaims the sand heap to be a Matterhorn. Ahmes, 27, 254, 278, 306 Alexandroff, 164 Al-Khowarazmi, 37 Allman, G. J., 29 Almagest, 35 Al-Nairizi, 171, 193, 214, 264 Analysis, 41, 161 INDEX Angle, 142, 155; trisection of, Anthonisz, Adriaen, 279 Apollodotus (Apollodorus), 259 Applied problems, 75, 103, 178, Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328 Aristotle, 33, 42, 134, 135, 137, Aryabhatta, 36, 279 Associations, syllabi of, 58, 60, 64 Assumptions, 116 Athelhard of Bath, 37, 51 Athenæus, 259 Babylon, 26, 272 Bartoli, 10, 44, 238 Eratosthenes, 48 Euclid, 33, 42, 43, 44, 119, 125, Eudoxus, 32, 41, 48, 227, 308, Euler, 38, 280, 318 Exercises, nature of, 74, 103; Figures in geometry, 104, 107, Finaeus, 44, 239, 240, 243 Fourth dimension, 326 Frankland, 56, 117, 127, 135, 159 Gargioli, 44 Gerbert, 43 Gherard of Cremona, 37, 51 Golden section, 250 Gothic windows, 75, 221 ff., 274, |