1. It is a commonplace of Biological Science that each individual, in his physical development, reproduces in an abridged form the successive stages through which the race has passed in its remote history. Educational experience suggests that a similar process occurs in mental development, so that, generally speaking, ideas have to be introduced in the order of their historical occurrence, if unnecessary resistance is not to be encountered.

I.The first obstacle may be one of sentiment. It is said that in a certain grassy part of the world a man will walk a mile to catch a horse, whereon to ride a quarter of a mile to pay an afternoon call. Similarly, it is not quite respectable to arrive at a mathematical destination, under the gaze of a learned society, at the mere footpace of arithmetic. Even at the expense of considerable time and effort, one should be mounted on the swift steed of symbolic analysis.

1. In order to understand the meaning to be attached to Non-Euclidean Geometry and the use of the model measuring tape described below, it is advisable to say a few words upon Euclidean Geometry.

Whatever theory is finally adopted for the propagation of light, there will always remain the interesting fundamental problem as to how far an object is exactly reproduced by an optical system. In this paper, the enquiry will be considered in its simplest form, and only the simplest types of objects can be considered. Taking the wave theory, let us consider the reproduction of an infinitely distant self-luminous point source, such as a star, through an optical system, such as a telescope. Sir George Airy investigated this case in 1834 ; he shewed what W. Herschel had recorded in 1782, that the image in the focal plane f of the optical system consisted of a central concentration of light, surrounded by light rings of rapidly diminishing intensity (see Fig 2), and gave formulae to shew the falling off in intensity of the light away from the centre of the image. This disc of light, then, is the approximation to the point-image of geometrical optics. The deformation is due to the distortion of the sphericity of the light waves making the image by diffraction at the aperture of the system. As in the propagation of wireless waves, however, diffraction is not the only cause of distortion.