It has been the fashion in some branches of mathematics to use a rather high sounding and mystifying terminology for comparatively simple ideas and I fear that Geometry has fallen to the lure. It is, of course, essential that new conceptions should have new titles. There is, however, the very serious danger that, through familiarity and, perhaps, moments of carelessness more may be read into such titles than was originally intended. I propose to discuss Infinity in Euclidean Geometry, as I find continually, especially with young students, that the phrases “point at infinity” and “line at infinity” have led to concepts of concrete existence which, whatever may be the case in other philosophies and branches of science, are in this one quite unjustified and which would be exceedingly awkward if they were correct.