As a first result of the labours of the International Commission on the Teaching of Mathematics, appointed in 1908 on the initiative of Professor D. E. Smith, the distinguished author of Rara Arithmetica, some 9000 printed pages were presented to the International Congress of Mathematicians at Cambridge. It is obviously impossible to go into details: I can only hope to mention one or two facts, and to give a general impression of the problems raised.

1. It seems most convenient, in virtue of the well-established pedagogic maxim that “algebraic subjects should, as far as possible, be introduced as generalisations of known principles in Arithmetic,” to base our teaching of limits and convergence upon the arithmetic theory of “Approximations,” with which all scholarship candidates are familiar as a practical instrument.

The formal definition of a limit by means of a sequence is usually found by the beginner very hard to understand, but if presented from the point of view of approximations it is less difficult.

1. Let AP, BQ, CR be three lines through the vertices of the ΔABC which rotate with equal angular velocity ω, and which make equal angles with the opposite sides, the initial position being that of the three altitudes of ABC. Then the three circles AQR, BRP, CPQ, which meet in the well-known point O, will be shewn to form three coaxal systems; and among other things we shall shew that the locus of O is the circle on GH as diameter, G being the centroid and H the orthocentre.

I propose in this paper to deal with the development during the last twenty or thirty years of the dominating ideas of the reform of mathematical teaching in Germany, and must beg for your indulgence if the theme is one which is already familiar.

In the Mathematical Gazette of December last there appeared a translation of the so-called Meraner Lehrplan of 1905, which, with the appended notes, formed a kind of Declaration of Eights of the Reform Party. It is my intention to give some account of the movement which led to its publication, to indicate briefly the substance of these appended notes, and to give some details of the way in which the general ideas have been introduced into the curricula.