The most broad-minded of the great mathematicians who have recently given attention to secondary problems is Professor Klein of Göttingen. He has had the good sense to look at something besides the mere question of good mathematics.[34] Thus he insists upon the psychologic point of view, to the end that the geometry shall be adapted to the mental development of the pupil,—a thing that is apparently ignored by Méray (at least for the average pupil), and, it is to be feared, by the other recent French writers. He then demands a careful selection of the subject matter, which in our American schools would mean the elimination of propositions that are not basal, that is, that are not used for most of the
exercises that one naturally meets in elementary geometry and in applied work. He further insists upon a reasonable correlation with practical work to which every teacher will agree so long as the work is really or even potentially practical. And finally he asks that we look with favor upon the union of plane and solid geometry, and of algebra and geometry. He does not make any plea for extreme fusion, but presumably he asks that to which every one of open mind would agree, namely, that whenever the opportunity offers in teaching plane geometry, to open the vision to a generalization in space, or to the measurement of well-known solids, or to the use of the algebra that the pupil has learned, the opportunity should be seized.
CHAPTER VII
THE TEXTBOOK IN GEOMETRY
In considering the nature of the textbook in geometry we need to bear in mind the fact that the subject is being taught to-day in America to a class of pupils that is not composed like the classes found in other countries or in earlier generations. In general, in other countries, geometry is not taught to mixed classes of boys and girls. Furthermore, it is generally taught to a more select group of pupils than in a country where the high school and college are so popular with people in all the walks of life. In America it is not alone the boy who is interested in education in general, or in mathematics in particular, who studies geometry, and who joins with others of like tastes in this pursuit, but it is often the boy and the girl who are not compelled to go out and work, and who fill the years of youth with a not over-strenuous school life. It is therefore clear that we cannot hold the interest of such pupils by the study of Euclid alone. Geometry must, for them, be less formal than it was half a century ago. We cannot expect to make our classes enthusiastic merely over a logical sequence of proved propositions. It becomes necessary to make the work more concrete, and to give a much larger number of simple exercises in order to create the interest that comes from independent work, from a feeling of conquest, and from a desire to do something original. If we would "cast a glamor over the multiplication table," as an admirer of Macaulay has said that the latter could do, we must have the facilities for so doing.
It therefore becomes necessary in weighing the merits of a textbook to consider: (1) if the number of proved propositions is reduced to a safe minimum; (2) if there is reasonable opportunity to apply the theory, the actual applications coming best, however, from the teacher as an outside interest; (3) if there is an abundance of material in the way of simple exercises, since such material is not so readily given by the teacher as the seemingly local applications of the propositions to outdoor measurements; (4) if the book gives a reasonable amount of introductory work in the use of simple and inexpensive instruments, not at that time emphasizing the formal side of the subject; (5) if there is afforded some opportunity to see the recreative side of the subject, and to know a little of the story of geometry as it has developed from ancient to modern times.
But this does not mean that there is to be a geometric cataclysm. It means that we must have the same safe, conservative evolution in geometry that we have in other subjects. Geometry is not going to degenerate into mere measuring, nor is the ancient sequence going to become a mere hodge-podge without system and with no incentive to strenuous effort. It is now about fifteen hundred years since Proclus laid down what he considered the essential features of a good textbook, and in all of our efforts at reform we cannot improve very much upon his statement. "It is essential," he says, "that such a treatise should be rid of everything superfluous, for the superfluous is an obstacle to the acquisition of knowledge; it should select everything that embraces the subject and brings it to a focus, for this is of the highest service to science; it must have great regard both to clearness and to conciseness, for their opposites trouble our understanding; it must aim to generalize its theorems, for the division of knowledge into small elements renders it difficult of comprehension."
It being prefaced that we must make the book more concrete in its applications, either directly or by suggesting seemingly practical outdoor work; that we must increase the number of simple exercises calling for original work; that we must reasonably reduce the number of proved propositions; and that we must not allow the good of the ancient geometry to depart, let us consider in detail some of the features of a good, practical, common-sense textbook.