cannot make the same angle with a third line, on the same side, for we ask, that lines which have the same direction never meet? De Morgan's criticism was never overcome, and in later editions of Wilson's Geometry, the direction-method was dropped, presumably because it is unsatisfactory and erroneous. The Association for the Improvement of Geometrical Teaching. -At the general meeting of this Association, held January 17th, 1891, Mr. E. T. Dixon gave a short account of his book on the "Foundations of Geometry," which introduced the directionmethod in a form modified to meet some of the objections raised by De Morgan. There was a discussion in which the author failed to meet the just criticism of the chairman (the late R. B. Hayward, F.R.S.). The chairman said: "He nowhere found it defined when the direction from A to B was the same as that from C to D. What was meant in this case by 'in the same direction'?" The method failed, just as Circular 711 fails, to provide any criterion, practical or logical, of the notion of same direction." Not only are we asked to accept "direction" as an indefinable, but also "same direction. same direction" may be used without explicit statement as another indefinable, whenever convenient! United States.-We are indebted to the Bureau of Education, Washington, for a copy of Circular No. 3, 1890, on the Teaching and History of Mathematics in the United States, by Prof. Cajori. In the section "On Parallel Lines and Allied Subjects,' p. 376, we are given a criticism of the many erroneous attempts made to replace Euclid's treatment of Parallels. In particular, on pages 382, 383, we have an able examination of the directionmethod which results in a conclusion quite adverse to its adoption in teaching. From p. 383 we quote the following: "One of the many objections to all attempts to found the elements of geometry on the word direction' is stated by Professor Halstead (in a letter to Prof. Cajori) in the following manner :Direction is a common English word, and in Webster's Dictionary, our standard, the only definition of it in a sense at all mathematical is the fourth the line or course upon which anything is moving .; as, the ship sailed in a south-easterly direction.' Direction, to be understood in any strict sense whatever, posits and presupposes three fundamental geometric ideas, namely, straight line, angle, parallels. After the theory of parallels founded upon an explicit *Now the Mathematical Association. + Association for the Improvement of Geometrical Teaching, Report for 1891, p. 22. assumption has been carefully established, a strict definition of direction may be based upon these more simple concepts, and we may use it as Rowan Hamilton does in his Quaternions." If additional evidence is required, reference may be made to Dodgson's "Euclid and his Modern Rivals"; pages 97-157 contain detailed criticisms of three Geometries by Wilson, Pierce, and Willock respectively, which employ the directionmethod for parallels. No answer to these criticisms can be found. Though the subject of this paper is distinctly narrow and special, the author begs all teachers of geometry to give it their earnest consideration. Nor do we hesitate to appeal to mathematicians of the highest rank to give some consideration to this discussion, for Gauss has by his example led the way for all. The question at issue is not the old one, now settled, of the retention of Euclid's Elements, for it is admitted generally that this work is not suitable for beginners in geometry. But the important principle at issue is whether a fundamental part of geometry should be introduced in such a way as to confuse the later logical development. There is a widespread feeling that it would be wise in elementary teaching to increase the number of explicit assumptions; agreement with this feeling does not require retention of the direction-method, for the explicit assumptions, built upon suitable illustrations and practical work, may surely be arranged otherwise. In this respect we may call attention to the methods advocated by Méray, Poincaré, Hadamard, Borel, and others in France, with the intention of using a method for parallels which in an elementary way is a first introduction to the notion of a group. Borel's Géométrie contains an account prepared for young beginners; while Hadamard's Leçons de Géométrie, in Note B (p. 278), Sur le Postulatum d'Euclide, contains a reasoned discussion of the method in its relation to non-Euclidean geometry. But we mention this method merely for illustrating a point in the argument of the paper-that any method adopted must provide for the whole of Euclid's treatment of parallels, while leaving it possible to refer without contradiction to the work of Lobachewski, Bolyai, Gauss, Beltrami, and others in a wider sphere of work. T. JAMES GARSTANG. THE ALGEBRA SYLLABUS IN THE I. INTRODUCTORY. The object of the present paper is to examine the aims proper to the teaching of algebra at school and to discuss what subject-matter should be comprised in the course in order to attain the aims in view. As a preliminary step we must adopt some rough classification of the pupils who study algebra; that which follows is quoted from a recent report of the Mathematical Association, a report that may be read in connection with the subject of this paper. "It is possible, and in many secondary schools customary, to divide mathematical pupils broadly into a few main classes : "I. Boys reading for mathematical honours† and examinations of similar standard. "II. Boys looking forward to an engineering career, army candidates, and others who need mathematics as one of the main subjects of their education. "III. Boys studying mathematics as part of a general education-an education which University entrance examinations and other examinations of similar standard are designed to test." Classes I. and II. may conveniently be described as specialists, Class III. as non-specialists. It may be assumed that specialists form an important minority among boys, and a minority numerically insignificant among girls. One of the main contentions of the present paper is that the course of algebra teaching usual in this country sacrifices the interests of the non-specialist to those of the specialist; further, certain remedies for this state of affairs are suggested. But let it be granted at the outset that no remedy is satisfactory which reverses the alleged injustice, and sacrifices the interests of the specialist to those of the non-specialist. This consideration must be and has been kept steadily in view. * Report of Mathematical Association Committee on the Teaching of Elementary Algebra and Numerical Trigonometry, 1911. (London: G. Bell & Sons. 3d.) †i.e. With a view to the study of Higher Mathematics at a University. When the interests of two groups of students appear to diverge, the first remedy to hand is to separate the groups and teach them in separate classes. This course is more feasible in an English school than in the schools of many foreign countries, inasmuch as the practice of providing different courses for different classes of students in a single school is very familiar in this country. But this is anything but an ideal method. In the first place it is not easy to distinguish future specialists from future non-specialists at an early age: bifurcation cannot well be resorted to before the age of 16 or so. Secondly, it complicates organisation, and tends to impair the solidarity of a school. A better way is to devise some curriculum which will carry all students together up to a certain stage. This elementary course must be designed to occupy the attention of the average non-specialist up to the close of his mathematical education at school. Further, it must be a suitable introduction to a subsequent specialist course, in order that the specialist student may pass normally through the elementary classes studying side by side with the other students. The specialist will, as a rule, be endowed with some degree of mathematical ability, and will pass more or less rapidly through the elementary classes, emerging finally from the highest elementary class with a year or more of his school life in hand to devote to special study of mathematics. This scheme is in accordance with the system of English schools, and corresponds closely to the arrangement very generally adopted; the only modification. needed is to frame an elementary course that shall be for the non-specialist a fairly self-contained and harmonious whole, while serving as a suitable basis for a specialist superstructure. Mathematics appears to hold in some respects a favoured position in the classical schools of this country; the time given to the subject in English classical schools is in excess of that given in the corresponding schools in France or Germany. The reason of the continued existence of mathematical teaching in the classical departments of the great schools is partly to be found in the fact that the Universities of Oxford and Cambridge' demand elementary mathematics as a necessary preliminary to a degree; so long as they continue to make this demand, public schools must comply with it. It is not altogether certain, however, that the demand will continue to be made. When the two ancient Universities cease to require Greek of students in mathematics and science (and * In English schools classical specialists often drop mathematics for the last two years of their school-life. It is hoped that this tendency may be arrested by the adoption of such a syllabus as that now proposed. + Foreign readers will understand that no attempt is made in most English schools to carry all boys of the same age through the same course; the principle is rather that each boy progresses at the rate determined by his ability and industry. the time cannot be far distant), it is by no means impossible that they will cease to require mathematics of classical students. It is doubtful if compulsory mathematics rests, at present, on a consensus of educated opinion that mathematics is a necessary part of a liberal education. Among the headmasters of public schools there are probably many who would deny that mathematics, as taught in their schooldays, was of perceptible value educationally, though no doubt there are many who hold the contrary opinion very strongly. It appears to be a matter of pressing importance that this question of the non-specialist mathematical curriculum shall be taken in hand at once. What we need is a curriculum of admitted educational value, a value that shall be admitted not merely by mathematicians, but by the great body of educated men. This is not a question that can be settled by mathematicians alone, in a high-handed way. We must carry public opinion with us; we have to convince people that the curriculum can, and will, be recast in such a form as to provide an indispensable element in education. If we can do this, mathematics will no longer be dependent upon the protection afforded by a precarious University regulation. There can be little question that, if the compulsion applied by the Universities were removed to-morrow, the time given to mathematics at many public schools would tend gradually to be reduced to a vanishing point. Public schools are governed, in the main, by classical men, and many classical men regard the time they spent over mathematics as time wasted. "Many "of us still smart with indignation at the hours which we were compelled to spend in learning by heart the first two "books of Euclid, and though he has since been deposed from "the eminence which we always grudged him, the tyranny is by no means overpast. We are repeatedly told that there are young scientists whose early life was blighted by having to "read the Apology in a crib, but why is no sympathy ever expended on those upon whose early years there fell the "transient but blighting shadow of x+y?" This is the opinion of many able men of literary tastes. That such an opinion can be held widely points to the conclusion that something must be done." 46 46 46 We who believe in the value of mathematical education have not the slightest doubt that something can be done, and we have clear ideas as to what this something should be. The average boy is perfectly ready to take an interest in mathematics if it is taught with some appeal to the imagination, with free use of intuition, and above all, if he is allowed to see that it is leading him somewhere. In the system of teaching that awakens such bitter memories for the headmaster of Shrewsbury, the imagina * Letter to the "Times," December 29, 1910, by the Headmaster of Shrewsbury School. |