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.

The early textbooks in geometry contained only the propositions, with the proofs in full, preceded by lists of definitions and assumptions (axioms and postulates). There were no exercises, and the proofs were given in essay form. Then came treatises with exercises, these exercises being grouped at the end of the work or at the close of the respective books. The next step was to the unit page, arranged in steps to aid the eye, one proposition to a page whenever this was possible. Some effort was made in this direction in France about two hundred years ago, but with no success. The arrangement has so much to commend it, however, the proof being so much more easily followed by the eye than was the case in the old-style works, that it has of late been revived. In this respect the Wentworth geometry was a pioneer in America, and so successful was the effort that this type of page has been adopted, as far as the various writers were able to adopt it, in all successful geometries that have appeared of late years in this country. As a result, the American textbooks on this subject are more helpful and pleasing to the eye than those found elsewhere.

The latest improvements in textbook-making have removed most of the blemishes of arrangement that remained, scattering the exercises through the book, grading them with greater care, and making them more modern in character. But the best of the latest works do more than this. They reduce the number of proved theorems and increase the number of exercises, and they simplify the proofs whenever possible and eliminate the most difficult of the exercises of twenty-five years ago. It would be possible to carry this change too far by putting in only half as many, or a quarter as many, regular propositions, but it should not be the object to see how the work can be cut down, but to see how it can be improved.

What should be the basis of selection of propositions and exercises? Evidently the selection must include the great basal propositions that are needed in mensuration and in later mathematics, together with others that are necessary to prove them. Euclid's one hundred seventy-three propositions of plane geometry were really upwards of one hundred eighty, because he several times combined two or more in one. These we may reduce to about one hundred thirty with perfect safety, or less than one a day for a school year, but to reduce still further is undesirable as well as unnecessary. It would not be difficult to dispense with a few more; indeed, we might dispense with thirty more if we should set about it, although we must never forget that a goodly number in addition to those needed for the logical sequence are necessary for the wide range of exercises that are offered. But let it be clear that if we teach 100 instead of 130, our results are liable to be about 100/130 as satisfactory. We may theorize on pedagogy as we please, but geometry will pay us about in proportion to what we give.

And as to the exercises, what is the basis of selection? In general, let it be said that any exercise that pretends to be real should be so, and that words taken from science or measurements do not necessarily make the problem genuine. To take a proposition and apply it in a manner that the world never sanctions is to indulge in deceit. On the other hand, wholly to neglect the common applications of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil, to arouse new interest, and to give a meaning to it that is otherwise wanting. It should always be remembered that mental discipline, whatever the phrase may mean, can as readily be obtained from a genuine application of a theorem as from a mere geometric puzzle. On the other hand, it is evident that not more than 25 per cent of propositions have any genuine applications outside of geometry, and that if we are to attempt any applications at all, these must be sought mainly in the field of pure geometry. In the exercises, therefore, we seek to-day a sane and a balanced book, giving equal weight to theory and to practice, to the demands of the artisan and to those of the mathematician, to the applications of concrete science and to those of pure geometry, thus making a fusion of pure and applied mathematics, with the latter as prominent as the supply of genuine problems permits. The old is not all bad and the new is not all good, and a textbook is a success in so far as it selects boldly the good that is in the old and rejects with equal boldness the bad that is in the new.

Lest the nature of the exercises of geometry may be misunderstood, it is well that we consider for a moment what constitutes a genuine application of the subject. It is the ephemeral fashion just at present in America to call these genuine applications by the name of "real problems." The name is an unfortunate importation, but that is not a matter of serious moment. The important thing is that we should know what makes a problem "real" to the pupil of geometry, especially as the whole thing is coming rapidly into disrepute through the mistaken zeal of some of its supporters.

A real problem is a problem that the average citizen may sometime be called upon to solve; that, if so called upon, he will solve in the manner indicated; and that is expressed in terms that are familiar to the pupil.

This definition, which seems fairly to state the conditions under which a problem can be called "real" in the schoolroom, involves three points: (1) people must be liable to meet such a problem; (2) in that case they will solve it in the way suggested by the book; (3) it must be clothed in language familiar to the pupil. For example, let the problem be to find the dimensions of a rectangular field, the data being the area of the field and the area of a road four rods wide that is cut from three sides of the field. As a real problem this is ridiculous, since no one would ever meet such a case outside the puzzle department of a schoolroom. Again, if by any stretch of a vigorous imagination any human being should care to find the area of a piece of glass, bounded by the arcs of circles, in a Gothic window in York Minster, it is fairly certain that he would not go about it in the way suggested in some of the earnest attempts that have been made by several successful teachers to add interest to geometry. And for the third point, a problem is not real to a pupil simply because it relates to moments of inertia or the tensile strength of a steel bar. Indeed, it is unreal precisely because it does talk of these things at a time when they are unfamiliar, and properly so, to the pupil.

It must not be thought that puzzle problems, and unreal problems generally, have no value. All that is insisted upon is that such problems as the above are not "real," and that about 90 per cent of problems that go by this name are equally lacking in the elements that make for reality in this sense of the word. For the other 10 per cent of such problems we should be thankful, and we should endeavor to add to the number. As for the great mass, however, they are no better than those that have stood the test of generations, and by their pretense they are distinctly worse.

It is proper, however, to consider whether a teacher is not justified in relating his work to those geometric forms that are found in art, let us say in floor patterns, in domes of buildings, in oilcloth designs, and the like, for the purpose of arousing interest, if for no other reason. The answer is apparent to any teacher: It is certainly justifiable to arouse the pupil's interest in his subject, and to call his attention to the fact that geometric design plays an important part in art; but we must see to it that our efforts accomplish this purpose. To make a course in geometry one on oilcloth design would be absurd, and nothing more unprofitable or depressing could be imagined in connection with this subject. Of course no one would advocate such an extreme, but it sometimes seems as if we are getting painfully near it in certain schools.

A pupil has a passing interest in geometric design. He should learn to use the instruments of geometry, and he learns this most easily by drawing a few such patterns. But to keep him week after week on questions relating to such designs of however great variety, and especially to keep him upon designs relating to only one or two types, is neither sound educational policy nor even common sense. That this enthusiastic teacher or that one succeeds by such a plan is of no significance; it is the enthusiasm that succeeds, not the plan.

The experience of the world is that pupils of geometry like to use the subject practically, but that they are more interested in the pure theory than in any fictitious applications, and this is why pure geometry has endured, while the great mass of applied geometry that was brought forward some three hundred years ago has long since been forgotten. The question of the real applications of the subject is considered in subsequent chapters.

In Chapter VI we considered the question of the number of regular propositions to be expected in the text, and we have just considered the nature of the exercises which should follow those propositions. It is well to turn our attention next to the nature of the proofs of the basal theorems. Shall they appear in full? Shall they be merely suggested demonstrations? Shall they be only a series of questions that lead to the proof? Shall the proofs be omitted entirely? Or shall there be some combination of these plans?

The natural temptation in the nervous atmosphere of America is to listen to the voice of the mob and to proceed at once to lynch Euclid and every one who stands for that for which the "Elements" has stood these two thousand years. This is what some who wish to be considered as educators tend to do; in the language of the mob, to "smash things"; to call reactionary that which does not conform to their ephemeral views. It is so easy to be an iconoclast, to think that _cui bono_ is a conclusive argument, to say so glibly that Raphael was not a great painter,--to do anything but construct. A few years ago every one must take up with the heuristic method developed in Germany half a century back and containing much that was commendable. A little later one who did not believe that the Culture Epoch Theory was vital in education was looked upon with pity by a considerable number of serious educators. A little later the man who did not think that the principle of Concentration in education was a _regula aurea_ was thought to be hopeless. A little later it may have been that Correlation was the saving factor, to be looked upon in geometry teaching as a guiding beacon, even as the fusion of all mathematics is the temporary view of a few enthusiasts to-day.[35]

And just now it is vocational training that is the catch phrase, and to many this phrase seems to sound the funeral knell of the standard textbook in geometry. But does it do so? Does this present cry of the pedagogical circle really mean that we are no longer to have geometry for geometry's sake? Does it mean that a panacea has been found for the ills of memorizing without understanding a proof in the class of a teacher who is so inefficient as to allow this kind of work to go on? Does it mean that a teacher who does not see the human side of geometry, who does not know the real uses of geometry, and who has no faculty of making pupils enthusiastic over geometry,--that this teacher is to succeed with some scrappy, weak, pretending apology for a real work on the subject?

No one believes in stupid teaching, in memorizing a textbook, in having a book that does all the work for a pupil, or in any of the other ills of inefficient instruction. On the other hand, no fair-minded person can condemn a type of book that has stood for generations until something besides the mere transient experiments of the moment has been suggested to replace it. Let us, for example, consider the question of having the basal propositions proved in full, a feature that is so easy to condemn as leading to memorizing.

The argument in favor of a book with every basal proposition proved in full, or with most of them so proved, the rest having only suggestions for the proof, is that the pupil has before him standard forms exhibiting the best, most succinct, most clearly stated demonstrations that geometry contains. The demonstrations stand for the same thing that the type problems stand for in algebra, and are generally given in full in the same way. The argument against the plan is that it takes away the pupil's originality by doing all the work for him, allowing him to merely memorize the work. Now if all there is to geometry were in the basal propositions, this argument might hold, just as it would hold in algebra in case there were only those exercises that are solved in full. But just as this is not the case in algebra, the solved exercises standing as types or as bases for the pupil's real work, so the demonstrated proposition forms a relatively small part of geometry, standing as a type, a basis for the more important part of the work. Moreover, a pupil who uses a syllabus is exposed to a danger that should be considered, namely, that of dishonesty. Any textbook in geometry will furnish the proofs of most of the propositions in a syllabus, whatever changes there may be in the sequence, and it is not a healthy condition of mind that is induced by getting the proofs surreptitiously. Unless a teacher has more time for the course than is usually allowed, he cannot develop the new work as much as is necessary with only a syllabus, and the result is that a pupil gets more of his work from other books and has less time for exercises. The question therefore comes to this: Is it better to use a book containing standard forms of proof for the basal propositions, and have time for solving a large number of original exercises and for seeking the applications of geometry? Or is it better to use a book that requires more time on the basal propositions, with the danger of dishonesty, and allows less time for solving originals? To these questions the great majority of teachers answer in favor of the textbook with most of the basal propositions fully demonstrated. In general, therefore, it is a good rule to use the proofs of the basal propositions as models, and to get the original work from the exercises. Unless we preserve these model proofs, or unless we supply them with a syllabus, the habit of correct, succinct self-expression, which is one of the chief assets of geometry, will tend to become atrophied. So important is this habit that "no system of education in which its performance is neglected can hope or profess to evolve men and women who are competent in the full sense of the word. So long as teachers of geometry neglect the possibilities of the subject in this respect, so long will the time devoted to it be in large part wasted, and so long will their pupils continue to imbibe the vicious idea that it is much more important to be able to do a thing than to say how it can be done."[36]

It is here that the chief danger of syllabus-teaching lies, and it is because of this patent fact that a syllabus without a carefully selected set of model proofs, or without the unnecessary expenditure of time by the class, is a dangerous kind of textbook.

What shall then be said of those books that merely suggest the proofs, or that give a series of questions that lead to the demonstrations? There is a certain plausibility about such a plan at first sight. But it is easily seen to have only a fictitious claim to educational value. In the first place, it is merely an attempt on the part of the book to take the place of the teacher and to "develop" every lesson by the heuristic method. The questions are so framed as to admit, in most cases, of only a single answer, so that this answer might just as well be given instead of the question. The pupil has therefore a proof requiring no more effort than is the case in the standard form of textbook, but not given in the clear language of a careful writer. Furthermore, the pupil is losing here, as when he uses only a syllabus, one of the very things that he should be acquiring, namely, the habit of reading mathematics. If he met only syllabi without proofs, or "suggestive" geometries, or books that endeavored to question every proof out of him, he would be in a sorry plight when he tried to read higher mathematics, or even other elementary treatises. It is for reasons such as these that the heuristic textbook has never succeeded for any great length of time or in any wide territory.

And finally, upon this point, shall the demonstrations be omitted entirely, leaving only the list of propositions,--in other words, a pure syllabus? This has been sufficiently answered above. But there is a modification of the pure syllabus that has much to commend itself to teachers of exceptional strength and with more confidence in themselves than is usually found. This is an arrangement that begins like the ordinary textbook and, after the pupil has acquired the form of proof, gradually merges into a syllabus, so that there is no temptation to go surreptitiously to other books for help. Such a book, if worked out with skill, would appeal to an enthusiastic teacher, and would accomplish the results claimed for the cruder forms of manual already described. It would not be in general as safe a book as the standard form, but with the right teacher it would bring good results.

In conclusion, there are two types of textbook that have any hope of success. The first is the one with all or a large part of the basal propositions demonstrated in full, and with these propositions not unduly reduced in number. Such a book should give a large number of simple exercises scattered through the work, with a relatively small number of difficult ones. It should be modern in its spirit, with figures systematically lettered, with each page a unit as far as possible, and with every proof a model of clearness of statement and neatness of form. Above all, it should not yield to the demand of a few who are always looking merely for something to change, nor should it in a reactionary spirit return to the old essay form of proof, which hinders the pupil at this stage.

The second type is the semisyllabus, otherwise with all the spirit of the first type. In both there should be an honest fusion of pure and applied geometry, with no exercises that pretend to be practical without being so, with no forced applications that lead the pupil to measure things in a way that would appeal to no practical man, with no merely narrow range of applications, and with no array of difficult terms from physics and engineering that submerge all thought of mathematics in the slough of despond of an unknown technical vocabulary. Outdoor exercises, even if somewhat primitive, may be introduced, but it should be perfectly understood that such exercises are given for the purpose of increasing the interest in geometry, and they should be abandoned if they fail of this purpose.

=Bibliography.= For a list of standard textbooks issued prior to the present generation, consult the bibliography in Stamper, History of the Teaching of Geometry, New York, 1908.

FOOTNOTES:

[35] For some classes of schools and under certain circumstances courses in combined mathematics are very desirable. All that is here insisted upon is that any general fusion all along the line would result in weak, insipid, and uninteresting mathematics. A beginning, inspirational course in combined mathematics has a good reason for being in many high schools in spite of its manifest disadvantages, and such a course may be developed to cover all of the required mathematics given in certain schools.

[36] Carson, loc. cit., p. 15.