In connection with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra.
For example, suppose a cube is inscribed in a hemisphere of radius r and we wish to find its edge, and thereby its surface and its volume.
If x = the edge of the cube, the diagonal of the base must be x√2, and the projection of r (drawn from the center of the base to one of the vertices) on the base is half of this diagonal, or (x√2)/2.
Hence, by the Pythagorean Theorem,
r2 = x2 + ((x√2)/2)2 = (3/2)x2
∴ x = r√(2/3),
and the total surface is 6x2 = 4r2,
and the volume is x3 = (2/3)r3√(2/3).
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.