of excellence in logic, in drawing, and in arrangement of work, and he fails to acquire the power to read and assimilate mathematical literature, a serious bar to his subsequent progress in more advanced lines.

To meet the first of the above objections, the waste of time in dictation, text-books have been prepared containing merely the definitions, postulates, axioms, enunciations, etc. But while free from the first objection, they are open to the others, and hence have met with only slight favor.

Text-books have also been prepared which, in place of the proofs, submit series of questions, the answers to which lead to the demonstrations. This is the heuristic method put into book form; it substitutes a dead printed page for a live intelligent teacher. The substitution is necessarily a poor one, for the printed questions usually admit of but a single answer each, and hence they merely disguise the usual formal proof. They give the proof, but they give no model of a logical

statement.

The kind of text-book which the world has found most usable, and probably rightly so, is that which possesses these elements: (1) A sequence of propositions which is not only logical, but psychological; not merely one which will work theoretically, but one in which the arrangement is adapted to the mind of the pupil; (2) Exactness of statement, avoiding such slipshod expressions as, "A circle is a polygon of an in

finite number of sides," "Similar figures are those with proportional sides and equal angles," without other explanation; (3) Proofs given in a form which shall be a model of excellence for the pupil to pattern after; (4) Abundant exercises from the beginning, with practical suggestions as to methods of attacking them; (5) Propedeutic work in the form of questions or exercises, inserted long enough before the propositions concerned to demand thought—that is, not immediately preceding the author's proof.

Such a book gives the best opportunity for successful work at the hands of a good instructor. But no book can ever take the place of an enthusiastic, resourceful teacher. In the hands of a dull, mechanical, gradgrind person with a teacher's license, no book can be successful. The teacher who does not anticipate difficulties which would otherwise be discouraging to the pupil, tempering these difficulties (but not wholly removing them) by skilful questions, is not doing the best work. On the other hand, the teacher who overdevelops, who seeks to eliminate all difficulties, who does all of the thinking for the class, is equally at fault. Youth takes little interest in that which offers no opportunity for struggle, whether it be on the playground, in the home games of an evening, or in the classroom.

CHAPTER XI

THE BASES OF GEOMETRY

The bases-Geometry as a science starts from certain definitions, axioms, and postulates. It is hardly

the province of this work to enter into a philosophical discussion of the foundations upon which the science rests, first because such a discussion would require a volume of some size,1 and also because practically the teacher is unable materially to change the definitions, axioms, and postulates of the textbook which happens to be in the hands of his pupils. A brief consideration of these bases of the science may, however, be of service.

The definitions of geometry occupy a position somewhat different from that held by the definitions of algebra and arithmetic. There is not the same necessity for exactness in the definition of monomial

1 The teacher may consult Dixon, E. T., The Foundations of Geometry, Cambridge, 1891; Russell, An Essay on the Foundations of Geometry, Cambridge, 1897; Poincaré, On the Foundations of Geometry, The Monist, October, 1898; Hilbert, D., Grundlagen der Geometrie, in Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Leipzig, 1899; Veronese, G., Fondamenti di Geometria, Padova, 1891; Koenigsberger, L., Fundamental Principles of Mathematics, Smithsonian Report, 1896, p. 93.

as in that of right angle, for the latter is a controlling factor in several logical demonstrations, while the former is not. In the same way more care must be shown in the definition of similar figures than in that of simultaneous equations, of isosceles triangle than of incomplete quadratic, of parallelepiped than of binomial; not that all of these terms must not be well understood and properly used, and not that algebra is less exact than geometry, but that the geometric terms enter into logical proofs in such way as to make their exact statement a matter of greater moment.

Hence the suggestions, already made in Chapter VIII upon accuracy of definition in algebra, apply with even greater force to geometry. Nor should the teacher attend so much to the idea that all the truth cannot be taught at once, as to acquire the dangerous habit of teaching partial truths only, or (as too often happens) of teaching mere words, sometimes unintelligible, sometimes wholly false. A few selections from our elementary text-books will illustrate these points.

We often see, for example, as a definition, "A straight line is the shortest distance between two points." Now in the first place this is absurd, because a line is not distance; distance is measured on a line, and usually on a curved one. Furthermore, the statement merely gives one property of a straight line; it is a theorem, and by no means an easy one to prove. A definition should be stated in

terms more simple than the term defined; but distance is one of the most difficult of the elementary concepts to define.1 Mathematicians have long since

abandoned the statement. "It is a definition almost universally discarded, and it represents one of the most remarkable examples of the persistence with which an absurdity can perpetuate itself through the centuries. In the first place, the idea expressed is incomprehensible to beginners, since it presupposes the idea of the length of a curve; and further, it is a case of reasoning in a circle (c'est un cercle vicieux), for the length of a curve can only be understood as the limit of a sum of rectilinear lengths. And finally, it is not a definition at all, but rather a demonstrable proposition." 2

The fact is, the concept straight line is elementary; it is not capable of satisfactory definition, and hence it should be given merely some brief explanation. From Plato's time to our own, attempts have been made to define such fundamental concepts as straight line and angle, but with no success. As

1 Pascal's rules for definitions are worthy of consideration in this connection: "(1) Do not attempt to define any terms so well known in themselves that you have no clearer terms by which to explain them; (2) Admit no terms which are obscure or doubtful, without definition; (3) Employ in definitions only terms which are perfectly well known or which have already been explained." Rebière, Mathématiques et mathématiciens, p. 23.

2 Laisant, p. 223.