5. Shall a textbook be used in which the basal propositions are proved in full, the exercises furnishing the opportunity for original work and being looked upon as the most important feature, or shall one be employed in which the pupil is expected to invent the proofs for the basal propositions as well as for the exercises?
6. Shall the terminology and the spirit of a modified Euclid and Legendre prevail in the future as they have
in the past, or shall there be a revolution in the use of terms and in the general statements of the propositions?
7. Shall geometry be made a strong elective subject, to be taken only by those whose minds are capable of serious work? Shall it be a required subject, diluted to the comprehension of the weakest minds? Or is it now, by proper teaching, as suitable for all pupils as is any other required subject in the school curriculum? And in any case, will the various distinct types of high schools now arising call for distinct types of geometry?
This brief list might easily be amplified, but it is sufficiently extended to set forth the trend of thought at the present time, and to show that the questions before the teachers of geometry are neither particularly novel nor particularly serious. These questions and others of similar nature are really side issues of two larger questions of far greater significance: (1) Are the reasons for teaching demonstrative geometry such that it should be a required subject, or at least a subject that is strongly recommended to all, whatever the type of high school? (2) If so, how can it be made interesting?
The present work is written with these two larger questions in mind, although it considers from time to time the minor ones already mentioned, together with others of a similar nature. It recognizes that the recent growth in popular education has brought into the high school a less carefully selected type of mind than was formerly the case, and that for this type a different kind of mathematical training will naturally be developed. It proceeds upon the theory, however, that for the normal mind,—for the boy or girl who is preparing to win out in the long run,—geometry will continue to be taught as demonstrative geometry, as a vigorous thought-compelling subject, and along the general lines that the experience of the world has shown to be the best. Soft mathematics is not interesting to this normal mind, and a sham treatment will never appeal to the pupil; and this book is written for teachers who believe in this principle, who believe in geometry for the sake of geometry, and who earnestly seek to make the subject so interesting that pupils will wish to study it whether it is required or elective. The work stands for the great basal propositions that have come down to us, as logically arranged and as scientifically proved as the powers of the pupils in the American high school will permit; and it seeks to tell the story of these propositions and to show their possible and their probable applications in such a way as to furnish teachers with a fund of interesting material with which to supplement the book work of their classes.
After all, the problem of teaching any subject comes down to this: Get a subject worth teaching and then make every minute of it interesting. Pupils do not object to work if they like a subject, but they do object to aimless and uninteresting tasks. Geometry is particularly fortunate in that the feeling of accomplishment comes with every proposition proved; and, given a class of fair intelligence, a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that brings the most pleasure, and that seems the most profitable of all that are studied in the first years of the high school.
Continually to advance, continually to attempt to make mathematics fascinating, always to conserve the best of the old and to sift out and use the best of the new, to believe that "mankind is better served by nature's quiet and progressive changes than by earthquakes,"[3] to believe that geometry as geometry is so valuable and so interesting that the normal mind may rightly demand it,—this is to ally ourselves with progress. Continually to destroy, continually to follow strange gods, always to decry the best of the old, and to have no well-considered aim in the teaching of a subject,—this is to join the forces of reaction, to waste our time, to be recreant to our trust, to blind ourselves to the failures of the past, and to confess our weakness as teachers. It is with the desire to aid in the progressive movement, to assist those who believe that real geometry should be recommended to all, and to show that geometry is both attractive and valuable that this book is written.
CHAPTER II
WHY GEOMETRY IS STUDIED
With geometry, as with other subjects, it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages. Although such a negative course is not satisfying to the mind as a finality, it possesses definite advantages in the beginning of such a discussion as this. Whenever false prophets arise, and with an attitude of pained superiority proclaim unworthy aims in human life, it is well to show the fallacy of their position before proceeding to a constructive philosophy. Taking for a moment this negative course, let us inquire as to what are not the reasons for studying geometry, or, to be more emphatic, as to what are not the worthy reasons.
In view of a periodic activity in favor of the utilities of geometry, it is well to understand, in the first place, that geometry is not studied, and never has been studied, because of its positive utility in commercial life or even in the workshop. In America we commonly allow at least a year to plane geometry and a half year to solid geometry; but all of the facts that a skilled mechanic or an engineer would ever need could be taught in a few lessons. All the rest is either obvious or is commercially and technically useless. We prove, for example, that the angles opposite the equal sides of a triangle are equal, a fact that is probably quite as obvious as the postulate that but one line can be drawn through a given point parallel to a given line. We then prove, sometimes by the unsatisfactory process of reductio ad absurdum, the converse of this proposition,—a fact that is as obvious as most other facts that come to our consciousness, at least after the preceding proposition has been proved. And these two theorems are perfectly fair types of upwards of one hundred sixty or seventy propositions comprising Euclid's books on plane geometry. They are generally not useful in daily life, and they were never intended to be so. There is an oft-repeated but not well-authenticated story of Euclid that illustrates the feeling of the founders of geometry as well as of its most worthy teachers. A Greek writer, Stobæus, relates the story in these words:
Some one who had begun to read geometry with Euclid, when he had learned the first theorem, asked, "But what shall I get by learning these things?" Euclid called his slave and said, "Give him three obols, since he must make gain out of what he learns."
Whether true or not, the story expresses the sentiment that runs through Euclid's work, and not improbably we have here a bit of real biography,—practically all of the personal Euclid that has come down to us from the world's first great textbook maker. It is well that we read the story occasionally, and also such words as the following, recently uttered[4] by Sir Conan Doyle,—words bearing the same lesson, although upon a different theme:
In the present utilitarian age one frequently hears the question asked, "What is the use of it all?" as if every noble deed was not its own justification. As if every action which makes for self-denial, for hardihood, and for endurance was not in itself a most precious lesson to mankind. That people can be found to ask such a question shows how far materialism has gone, and how needful it is that we insist upon the value of all that is nobler and higher in life.
An American statesman and jurist, speaking upon a similar occasion[5], gave utterance to the same sentiments in these words:
When the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence.
There have not been wanting, however, in every age, those whose zeal is in inverse proportion to their experience, who were possessed with the idea that it is the duty of the schools to make geometry practical. We have