ARTICLE II.-THE STUDY OF ELEMENTARY
GEOMETRY:

WITH A CRITICISM OF SOME OF THE DEFINITIONS AND METHODS OF TEXT BOOKS.*

use.

IN the following pages the writer has selected for criticism a few of the many books which possess such peculiar merits as to commend them to the public and bring them into extensive He has not pretended to discuss all the points of disagreement between the views of the several authors and his own view of the treatment of the subject. In order not to make an Article of too great length he has not touched upon. the important topic of ratio and proportion. It seemed to him that this subject needed treatment by itself, and might well form the subject of a second Article, to be written when his own mind becomes settled upon the best method of discussing it.

If any excuse were needed for bringing these subjects now discussed to the consideration of the public, it could be found (apart from the intrinsic importance of the study of geometry itself), in the necessity under which an examiner lies, to weigh the merits of the different text books presented by the applicants for admission to college, taken in connection with the fact that the number of such text books is increasing. The movement in England to drive Euclid from the schools, under the auspices of the "Association for the Improvement of Geometrical Teaching," if it has not yet reached this country and if it has not yet resulted here, as it has there, in the introduction of new manuals, will soon be felt in an impulse to establish some standard text book (or order of propositions), other than Euclid's, and will lead, if not properly directed, to the issuing

* The writer is under special obligation to the following authors: Herbert Spencer, J. S. Mill, Whewell, Prof. DeMorgan, J. Todhunter, C. L. Dodgson.

of a greater number of books, thus still further adding to the perplexity of both teachers and examiners.

The question naturally arises why an examiner should be perplexed by the multiplication of text books in geometry more than he should be by the variety of text books on any other elementary mathematical subject, as for instance algebra; or why he should be more embarrassed in weighing the merits of a student's work in geometry, than an examiner in Latin grammar should be by the various Latin grammars used in the schools. In the first place, with regard to grammar, the declensions of nouns and adjectives, the conjugation of verbs and the rules of syntax are essentially the same in all text books. The order of arrangement in presenting these parts of the subject is generally the same, or if not the same, is a matter of indifference. The language has fixed the declension of a noun. Grammars differ so slightly that an examiner has only to make the questions on his paper cover the essential grammat ical points, and he is sure that if the applicant has studied faithfully any one of the various standard grammars he will be able to pass the paper satisfactorily. Again, with regard to algebra, the study is entirely different in its nature from geometry. To the mind of an examiner, it principally differs from geometry in this, that in geometry the sequence of propositions and their logical connection are of main importance, while elementary algebra is nothing more than a general arithmetic, in which numbers are denoted by letters and the processes by symbols. In examining in geometry it is necessary to know the sequence of the propositions belonging to each text book in order to estimate rightly the candidate's proficiency in the study. A few examples will test the applicant's knowledge of algebra, as that knowledge is not dependent upon a certain order of development, but upon his familiarity with the use and application of symbols. For instance, a boy prepared in Euclid ought to prove that "any two sides of a triangle are together greater than the third side" by means of the principle. that the greater angle of every triangle is subtended by the greater side, whereas a candidate offering Davies' Legendre or Loomis' Geometry, proves the second theorem by the first, thus reversing Euclid's order.

Now to throw light upon the question, what are the best methods of teaching geometry to a beginner; and upon a second question, what books most closely follow these methods, it is proposed to discuss, first, what ought the study of geometry to do for a pupil in the way of furnishing him with actual knowledge as a foundation for other acquisitions; and, second, what does it do for the student himself in the way of strengthening or developing his mind and character? Though every study does something for a student in both these ways, it is necessary to consider separately the particular effects of the study of geometry in both these directions in order to come to an understanding of the term geometry, for the word means different things to different persons. As Mr. Todhunter remarks in his Conflict of Studies: "To the admirers of Euclid it means a system of demonstrated propositions valued more for the process of reasoning involved than for the results obtained. With modern reformers, rigor of method is of small account compared with the facts themselves."

First, the study of geometry gives the student clear ideas of those elementary space relations involved in the properties of lines, surfaces, and solids, which are the indispensable basis of all studies, either practical or theoretical, which have to do with position, form, or measurement. No one expects geometry to teach all these properties, but only the essential ones, so that a student knowing these is prepared to pursue his investigations still further in the same direction, if he desires to do so. It is especially necessary for the right understanding of the analyti cal mathematics that these elementary space relations should be thoroughly mastered. The first applications of the symbols and processes of algebra to lines, surfaces, and solids, are so novel of themselves, that all the learner's mental energy is required to grasp the idea for which the symbols stand. If along with this difficulty he is required to meet also that of learning for the first time the elementary space relations, he is given a task to which few beginners are equal. It is for this reason that geometry is best taught to beginners as a pure science, not mixed with arithmetic or algebra.

Next, what does the study of geometry do for the mind of the student in the way of discipline? What traces does the

study leave on his mind, even though its principles in their sequence have almost faded out of memory? It leaves, among other residua, what Mr. Herbert Spencer calls an "unshakable belief in necessity of relation "-that is, that some facts or truths. are necessarily related to other facts or truths-that given certain data, certain conclusions must follow and in such a way that they can not be conceived as not following. Whether this necessity of relation arises from the peculiar nature of the ideas of space already in the mind, as the idealists maintain, or from the hypothetical nature of the subject, as Mill asserts, may be an interesting question, but does not alter the fact, which is admitted by both Mill and the idealists.

This belief in necessity of relation is peculiarly well inculcated and most easily fostered by the study of geometry. It is peculiarly well inculcated because premises and conclusions are repeatedly brought close together in the demonstration of propositions, so that the mind sees the relation without the intervention of symbols. The continual juxtaposition of antecedent and consequent facts is peculiar to geometry. If it is said that the contemplation of the relations of numbers as presented in arithmetic or algebra teaches the same lesson, the statement is admitted to be true, but in these studies the necessity of the relation is not so evident in thought, because the mind does not always pass beyond the numbers or the symbols to the things represented by them. The relation between the data and the conclusions following from the data is often lost in the effort to understand and manage the symbols. Thus it comes to pass that algebra and arithmetic become to most students a matter of rules rather than of methods. The same might be said of the discipline of logic, which is akin to the discipline of geometry. The mind is distracted by the symbols and loses the relation of the thought and comes to depend upon rules again. Logic in such a case, like algebra, becomes mechanical. Again, belief in necessity of relation is most easily fostered by the study of geometry. In this study, out of a few elementary conceptions capable of a clear representation, the whole science can be built up by simple methods, without the use of puzzling symbols. Algebraic processes and the use of the symbols of logic often seem to the learner like jugglery.

The conclusions are frequently surprises, admitted it may be, but with a feeling of distrust. The student rests in the conclusions of geometry, after he has mastered the demonstrations, for he has been able to follow every transition of thought, and has seen that every fact or thought necessarily follows from the preceding fact or thought.

In attaining this belief in necessity of relation the student acquires a habit of exact thought, and strengthens his reasoning power, for of course the necessity of relation could not have been seen unless the thoughts were exact and the reasoning rigorous. Moreover this habit of exact thought includes the habit of being satisfied with nothing less than clear ideas, for exact thought is only possible to one who holds clear ideas.

In consequence of the necessity of representing the concep tions of geometry by diagrams, in which lines, in order to be perceived are given breadth, the visual perceptions are trained by the study, and exactness of sight is cultivated as well as exactness of thought. The effort to picture in the mind these lines in their relative positions, and again the effort to reproduce them in a diagram gives exercise to the imagination of form. In this process of construction of figures by geometric laws we have two powers of mind brought into play; first the power of abstraction, and second the power of reproduction. Both powers must be exercised while the mind keeps clearly in view the laws of space relation. Geometry is thus the first elementary study by which a rigorous discipline in abstract thought is given to a student, for these geometric lines, surfaces and solids, though capable of concrete representation, are abstract things, hypothetical objects, not existing in fact. The study teaches, therefore, the two important elements of abstract thought, conceiving and judging. The things to be conceived by the mind are few, and in their nature simple, and these simple things are combined in judgments by methods as easy to be understood as the things themselves.

The study of geometry cultivates a habit of examination into a subject without a servile reliance upon authority. For in the beginning of the study, the definitions, postulates, and axioms call clearly to the mind of the learner the principles upon which the science is founded and the methods by which