Vectors in two dimensions are generally studied in schools; I do not suppose that the study is often pushed into three dimensions. But it is difficult to understand a modern book on electricity without some knowledge of three-dimensional vector notation.

The particular point that I wish to bring before you is that the two-dimensional case is often presented in such a way that its relation to the more general case is obscured.

To cut up an isosceles triangle so that the parts may be put together to form two triangles similar to the original triangle.

Let ABC be the triangle, having the angles at B and C equal, and let D be the middle point of the base. Draw a circle, passing through A, having its centre on AD, and cutting AB and AC in E and F. Join DE, cutting the circle in G.

The Dalton Plan, which has been discussed so much recently, is a change of organisation within school which affects a girl’s whole school life and her general development more than her work in individual subjects. I am obliged in describing it to devote a few minutes to speaking of general school life, and when I turn to the teaching of mathematics there seems so little to say, that could not be said as appropriately of teaching in ordinary circumstances that I do not know if it will justify the consideration of this meeting.

The process known as the differentiation of functions in the Calculus does not present the insurmountable difficulties one experiences in attacking the converse problem associated with the Integral Calculus. As is well known, there are many functions which we cannot integrate, and as the process of integration is really a summation, this means there are many areas which cannot be evaluated by exact methods. There are many cases in various branches of science in which it is required to determine the area of a closed curve obtained by plotting a series of observations. The engineer, for example, has to determine the areas of indicator and similar diagrams, the naval architect has to calculate volumes, and centres of gravity or buoyancy by approximate methods. The mathematician may be called upon to evaluate definite integrals which cannot be integrated directly, such as the elliptic integrals. The problem reduces simply to the consideration of the determination of the area between the curve y=f(x), the ordinates x=x0 and x =x0+nh, and the axis of x; and the first part of the paper will be devoted to the methods of obtaining empirical rules for approximation to the evaluation of such an area when f(x) cannot be directly integrated. In the second part, the mechanical methods of integration will be considered, including demonstrations with the different types of instruments illustrating these methods.