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The Teaching of Geometry

by David Eugene Smith

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L’Envoi

In the Valley of Youth, through which all wayfarers must pass on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time.

The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, “My mountain may not be as lofty as yours, but it is easier to ascend”; or “There are quarries on my peak, and points of view from which a building may be seen in process of erection, or a mill in operation, or a canal, while your mountain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit.” Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and these say to the weaker pilgrims, “Why tire yourselves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice?” The result is not wholly disadvantageous, for many who pass through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes.

Then, too, there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is assured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself.

In the midst of all the advocacy of sand-dune climbing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn in the Valley of Youth. Among the mountains that have for ages attracted the pilgrims is Mons Latinus, usually called in the valley by the more pleasing name Latina. 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 rubber-tired 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.

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