1. It is now nearly half a century since H. Lebesgue, whose obituary the reader may have seen in Nature not so long ago, created his theory of the integral which since then has superseded in modern analysis the classical conception due to B. Riemann. It is, I think, regrettable that knowledge of the Lebesgue integral seems to be still largely confined to the research worker. There is nothing unduly abstract or unnatural in this theory, nor anything in the proofs which would be too difficult for a good honours student to grasp. If the aim of university education be the teaching of general ideas and methods rather than that of technicalities, then the modern notion of the integral should not be omitted from the mathematical syllabus. It is the purpose of this purely expository note to sketch the build up of both the Riemann and the Lebesgue integral on the common geometrical basis of “measure” and thus to make evident to the uninitiated reader the striking advantages of the new integral.

1. The following description of the projective geometry of a finite number of points in 2-space is almost certainly known to those acquainted with projective geometry or with modern algebra. The object of this brief account is to show how certain finite systems can be presented in a form easily understood by students, and how they provide simple but instructive examples of fundamental ideas and “constructions.” The fact that these examples belong to a geometry which is essentially non-Euclidean has great teaching value to those students who are apt to confuse projective geometry with the “method of projection” in Euclidean geometry. The underlying algebra is described briefly in § 4, but an understanding of this is not essential to the geometry. This algebraic work may, however, be of interest to those to whom Galois fields are fairly new.

1. Imagine that a number of straight lines, coplanar and concurrent, are united so as to form as it were a rigid frame. Imagine that this frame is moved (continuously) in the plane in such a way that two selected lines always touch two cycloids traced in the plane. Then it will be found that

Every line of the frame will move so as to envelope a cycloid.