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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
Algebra, 37, 84

Al-Khowarazmi, 37

Allman, G. J., 29

Almagest, 35

Al-Nairizi, 171, 193, 214, 264
Al-Qifti, 49

Analysis, 41, 161

INDEX

Angle, 142, 155; trisection of,
31, 215

Anthonisz, Adriaen, 279
Antiphon, 31, 32, 276

Apollodotus (Apollodorus), 259
Apollonius, 34, 214, 231

Applied problems, 75, 103, 178,
186, 192, 195, 203, 204, 209, 215,
217, 242, 267, 295, 317
Appreciation of geometry, 19
Arab geometry, 37, 51

Archimedes, 34, 42, 48, 139, 141,

215, 276, 278, 314, 327, 328
Aristæus, 310

Aristotle, 33, 42, 134, 135, 137,
145, 154, 177, 209

Aryabhatta, 36, 279

Associations, syllabi of, 58, 60,

64

Assumptions, 116
Astrolabe, 172

Athelhard of Bath, 37, 51

Athenæus, 259
Axioms, 31, 41, 116

Babylon, 26, 272

Bartoli, 10, 44, 238
Belli, 10, 44, 172
Beltinus, 239, 241
Beltrami, 127
Bennett, J., 224
Bernoulli, 280

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Eratosthenes, 48

Euclid, 33, 42, 43, 44, 119, 125,
135, 156, 165, 167 ff., 201 ff.,
et passim; editions of, 47, 52;
efforts at improving, 57; life
of, 47; nature of his "Ele-
ments," 52, 55; opinions of, 8
Eudemus, 33, 168, 171, 185, 216,
309

Eudoxus, 32, 41, 48, 227, 308,
314, 317

Euler, 38, 280, 318
Eutocius, 184

Exercises, nature of, 74, 103;
how to attack, 160
Exhaustions, method of, 31
Extreme and mean ratio, 250

Figures in geometry, 104, 107,
113

Finaeus, 44, 239, 240, 243
Fourier, 142

Fourth dimension, 326

Frankland, 56, 117, 127, 135, 159
Fusion, of algebra and geometry,
84; of geometry and trigonom-
etry, 91

Gargioli, 44
Gauss, 140, 274
Geminus, 126, 128, 149
Geometry, books of, 165, 167,
201, 227, 252, 269, 289, 303,
321; compared with other sub-
jects, 14; introduction to, 93;
modern, 38; of motion, 68, 196;
reasons for teaching, 7, 15, 20 ;
related to algebra, 84; text-
books in, 70

Gerbert, 43

Gherard of Cremona, 37, 51
Gnomon, 212

Golden section, 250

Gothic windows, 75, 221 ff., 274,

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