-as if

editors of Euclid, feeling that there is somethiug wanting in this definition, have (they think) vastly improved it by saying that "a point is that which has position but no magnitude"-as if position is more easily grasped than point. Then again (still at the threshold of the subject) the beginner is taught to believe that he is getting a very definite conception of a right line in the defiuition, "a right line is that which lies evenly between its extreme points the meaning of "evenly" is at once beyond question. But of all the elementary conceptions in Euclid that of an angle is the one which most puzzles a beginner, and remains unrealised for the longest time. "An angle is the inclination of two straight lines to one another." again ws have one obscure term defined by another equally obscure; and we know by experience that, unless the conception is presented in a very different way, the obscurity will be permanent.

Here

Moreover, it is possible to point out a self-contradiction in Euclid. Thus his definition of a circle makes it to be a disc-" a circle is a plain figure bounded by one line called the circumference"-so that, clearly, the whole of the space inside (or, possibly, outside) the circumference is the circle, whose mere boundary is the circumference; and, if so, two circles can, of course, intersect in an infinite number of points-over an extensive area, in fact; but this is contradicted by Euclid in the tenth proposition of Book III., according to which one circle cannot intersect another in more than two points

These, it may be admitted, are comparatively minor considerations, and the defects might be corrected by judicious teaching.

It is chiefly in the way in which the fifth and sixth Books of Euclid are apprehended by boys that the necessity for a change in the system of teaching is to be seen.

Those mediæval technicalities "duplicate ratio," subduplicate ratio," "sesquiplicate ratio," and some others are drummed into the heads of boys as if they were terms of the utmost scientific importance. What mathematician ever uses such terms, or even thinks of them in his investigations?

The simple and extremely important fact that the areas of two similar figures are to each other as the squares of corresponding linear dimensions is presented to the begin

ner in the nineteenth proposition of the sixth Book in the words "similar triangles are to one another in the duplicate ratio of their homologous sides "—a statement which is singularly deficient in accuracy inasmuch as it omits to say precisely what two qualities or quantities connected with the triangles are thus related (colours, shapes, sizes, or what?); and the result is absolute confusion in the minds of a very large number of boys.

Let me illustrate this by a few bona fide examples. In reply to the question, "What are similar triangles, and what is the relation between their areas ?" the following answers were received :

(1) A triangle is similar to another triangle when their sides are proportional, and when the homologous sides of one are in duplicate ratio to the homologous sides of the

other.

(2) If two triangles have the sides about an angle in each proportional and the other angles of the same affection, the triangles are similar. Similar triangles are proportional to the bases on which they stand, and are to one another in the duplicate ratio of their homologous sides.

(3) Similar triangles are those which are equal in area to each other and are in the same proportion to each other as the duplicate ratio of their homologous sides.

(4) When the angles are similar the areas are similar, when the areas are similar the angles are similar, when the sides are similar the areas are similar.

(5) Similar triangles are equal in all respects-sides equal to sides, angles equal to angles, areas equal to areas. Similar triangles are to each other as their bases.

(6) Similar triangles are to one another in the duplicate or subduplicate ratio of their homologous sides. Their areas are as the square or square root of their bases according as it is in the duplicate or subduplicate ratio.

(7) Similar triangles are to one another as their bases They are also to each other in the duplicate ratio of their homologous sides.

(8) Triangles are said to be similar when they have their corresponding sides equal and are equal in area. Similar triangles are to one another in the duplicate ratio of their homologous sides.

Each of these exhibits a pleasing variety and a liberalminded, large-hearted toleration of conflicting views.

Such examples might be multiplied almost indefinitely, and they show clearly the impotence of the dictum "similar triangles are to one another in the duplicate ratio of their homologous sides" to convey any real knowledge to the mind of the ordinary learner. Duplicate ratio" and "homologous" are mere sounds, to the latter of which violence is often done, inasmuch as I have frequently met with "homolicus" and "harmologous" sides.

Now, as regards the amount of time which is spent in the schools by young boys in acquiring the elementary facts and conceptions of geometry from Euclid's book, I know that very many months are occupied in attaining to the twelfth proposition of the first Book. I have before me, in fact, a fair-sized treatise written for the purpose of guiding boys along Euclid's exact path to this proposition.

There is absolutely nothing in the first twelve propositions that could not be taught far more effectively to a boy of ordinary intelligence in a few days, if only a rational style of teaching geometry were adopted; but if the exact language and pedantic professionalism of the school Euclids must be followed, to the weariness of the boy's mind and the quenching of his interest, it becomes a very long process indeed-ending, in the case of a large number, in utter failure.

Moreover, the current practice which insists on compelling boys to study geometry in an order and language characteristic of medieval times, when no physical sciences existed, is a hindrance to the study of such sciences now, inasmuch as geometry is one of the foundations of all exact science; and it is obvious that if an intelligent knowledge of geometry is postponed, the physical sciences must be kept back also.

The plea that Euclid's book is unrivalled as an exposition of clear logical method and arrangement, and, as such, must be the foundation on which to build geometry, is vain-for the simple reason that it is not in England (where Euclid is worshipped), but in France and Germany (where Euclid is unknown as a text-book), that the great discoverers in geometry have been produced.

The late M. Paul Bert, Minister of Public Instruction in France, published a little book on the proper method of teaching geometry to beginners, in which he severely satirised the faults of the existing procedure; and, again,

the late Rev. W. A. Willock (father of Dr. Sophie Bryant), in his "Elementary Geometry of the Right Line and Circle," has similar excellent remarks on this subject. "It is almost certain," says Dr. Willock, "that Euclid wrote his 'Elements' not for boys, but for grown-up, hard-headed thinking men."

Certain concessions have been made to the advocates of reform, led chiefly by Mr. Hayward—notably by the University of Oxford and the Civil Service Commissioners; and, in the existing state of affairs, it is not reasonable to expect more.

It will be clear from the foregoing that, in my opinion, a more rapid progress in the study of science generally would ensue from any system which would facilitate and accelerate the understanding of geometry by boys in the very elementary stage; and to this end I would suggest that the initiative should be taken by the Universities of Oxford and Cambridge. Our vast system of competitive examinations renders it necessary that a fixed source of authority on the order of deduction in geometry should exist. Such a source is Euclid at present; but a better one might, without serious difficulty, be drawn up by a University Committee, and its adoption by the schools and colleges throughout the country would follow as a matter of course. The chief difficulty is to avoid "fads"; but I learn, from conversation with a distinguished master in the largest of our public schools, that sympathy would not be wanting in an attempt to improve existing methods.

GEORGE M. MINCHIN.

Practical Hints and Examination Papers.

-THE BREADTH OF EDUCATION.-Dr. William R. Harper, in stating in the Methodist Magazine the views on Education of Dr. Vincent, the promoter of the Chautauqua movement, says: "Education is not to be confined to formal study. It includes this, but it includes much more. Books alone are insufficient. One must come in contact with people, and especially with the ablest men and women specialists, scientists, littérateurs, great teachers who know how to inspire and quicken minds, and from whom a special inspiration may be gained for the doing of special service. One must travel at home and abroad, and bring

himself into contact with the locality in which the great lives of the world have been lived and its great events enacted. Perhaps more may be gained than in any other way from personal thought and meditation, in hours during which one is able to examine himself and hold before his soul a mirror in which shall be reflected his inner life and thought." It has always been a source of grief to Dr. Vincent that he did not avail himself of a college training.

-PRACTICAL GEOGRAPHY.—This is the month to review the summer trips of teacher and pupils. No excursion, if it be the only one the child has taken during the holiday season, no matter how short it may have been, should be slighted. The short journey is the connecting link in the child's mind between his home and the great world beyond. Let the child find on the map the first large place that he could reach by the road on which he was travelling. Encourage children to gather specimens of the natural products of the places they visit or of their own locality if they have not been away-grasses, flowers, minerals, the products that make the country's wealth. Every Canadian school-house might have hung upon its walls, as the work of the pupils, pieces of cardboard with the various natural products neatly mounted ou them and correctly labelled. One large card might have specimens of the most important productions of the forests, lakes, plains, rivers and mountains of the Dominion, a smaller chart specimens from the province, and a still smaller card those of the locality. The process of selection would be an admirable exercise for teacher and children. The work should be well done. It is not work for the teacher only but for the children. It will be "education by doing."

-GOOD FOOD FOR THE CHILDREN.—“In order to do good work in this world," says Huxley, "one must be a good animal." We want our children to be good animals, sound of body and strong of muscle. In several respects children brought up on the farm have an advantage over city children. The foundation stone of success at college and in after life, in the various fields of activity to which college graduates find an entrance, was laid on the farm in the plentiful exercise of ploughing, hoeing and general work which called forth a healthful appetite for bread, butter, milk, cream and salt pork, for which the farm is famous. There the physical strength was built up, without