MATHEMATICS It is beyond doubt that in the present century a revolution in the teaching of school mathematics has taken place. In part it con sists in breaking away from the teaching of geometry by means of Euclid's elements and in part that break is typical of the change that has also taken place in the other branches of school mathe matics, algebra, trigonometry, Cartesian geometry, mechanics, and arithmetic.
Up to about the year 190o school mathematics centred around "Euclid" as the subject of geometry was then universally called. The teaching was based on the belief that the subject had a sure foundation in the fundamental assumptions, that the super structure was raised on the foundations by a process of irre fragable logic, and that the best training consisted in the repro duction of Euclid's reasoning. The impulse towards reform came from the realization by the teachers that the teaching did not always furnish a training in reasoning to certain of their pupils, who committed the book to memory and reproduced it mechani cally and without understanding.
Reformed Methods.—The reform was due to the Mathemati cal Association. The reformers set about the designing of a course suited to the average child. Text books were written from which the more subtle of Euclid's propositions were omitted and only those propositions retained that had a substantial meaning. To make it possible to carry out the reform it was necessary to secure the sympathy of examining bodies. The Civil Service Com mission was the first to support the movement. The old examina tion paper consisted of propositions, the writing out of which might not indicate understanding, and of problems which certain pupils could not touch. The task of the examining bodies was to find questions on the subject matter that would test under standing and would at the same time be within reach of the aver age child.
On the old system each branch of mathematics was a separate subject fashioned as far as possible on Euclidian lines with a series of set propositions based upon definitions and axioms. The reform movement led to the disappearance of the lines of de marcation between the branches and their fusion into the single subject of mathematics. At the same time things of little practical value like the theory of numbers were dropped. The time saved
by this fusion and pruning was utilized to carry the more im portant subjects farther and to introduce new subjects. The infinitesimal calculus thus entered the school curriculum and is now sometimes taught from the age of 54. Before the reform numerical work was almost confined to the teaching of arithmetic, being rare even in the supposedly practical subject of trigonom etry. It has since played a great part in all varieties of mathe matics.
Mechanics.—Mechanics gained greatly by the reform. The pupil now verifies the laws by laboratory experiments. Formerly much energy was expended in deducing the parallelogram of velocities and the parallelogram of forces from fundamental as sumptions. The conception of "simultaneous velocities" which is the last remnant of the old system must soon disappear. The sub ject has been extended to include moments of inertia and simple motions of rigid bodies.
Geometry.—In geometry after the reform the methods of Euclid and Descartes enjoyed equal status and each problem was treated by the most appropriate method. The set propositions of conic sections disappeared. In the time saved geometry of three dimensions was taught, chiefly on the methods of Euclid and Descartes, but also to a small extent on the method of Monge, the graphical method by means of which every point and line of a three-dimensional figure can be represented on a two-dimensional sheet of paper. This method which has great educational value and is in constant use in the engineering world will no doubt be given its proper place in the school ; geometry will then be treated by the three co-equal methods, the synthetic method of Euclid, the analytical method of Descartes and the graphical method of Monge.