In one respect this book is an experiment, and may chance to prove a failure 3 I mean that I have not thought it necessary to maintain throughout the gravity of style which scientific writers usually affect, and which has some how come to be regarded as an 'inseparable accident' of scientific teaching. I never could quite see the reason ableness of this immemorial law: subjects there are, no doubt, which are in their essence too serious to admit of any lightness of treatment — but I cannot recognise Geo metry as one of them. Nevertheless it will, I trust, be found that I have permitted myself a glimpse of the comic side of things only at fitting seasons, when the tired reader might well crave a moment's breathing-space, and not on any occasion where it could endanger the continuity of a line of argument. Pitying friends have warned me of the fate upon which I am rushing: they have predicted that, in thus abandon ing the dignity of a scientific writer, I shall alienate the sympathies of all true scientific readers, who will regard the book as a mere jozo d'esprit, and will not trouble them selves to look for any serious argument in it. But it must be borne in mind that, if there is a Scylla before me, there is also a Charybdis — and that, in my fear of being read as a jest, I may incur the darker destiny of not being read at all. In furtherance of the great cause which I have at heart -the vindication of Euclid' s masterpiece — I am content to run some risk; thinking it far better that the purchaser of this little book should read it, though it be with a smile.