EUCLID AND HIS MODERN RIVALS by CHARLES L. DODGSON. Originally published in 1879. PROLOGUE: THE object of this little book is to furnish evidence, first, that it is essential, for the purpose of teaching or examining in elementary Geometry, to employ one text book only secondly, that there are strong a priori reasons for retaining, in all its main features, and specially in its sequence and numbering of propositions and in its treat ment of parallels, the Manual of Euclid and thirdly, that no sufficient reasons have yet been shown for aban doning it in favour of any one of the modern Manuals which have been offered as substitutes. It is presented in a dramatic form, partly because it seemed a better way of exhibiting in alternation the argu ments on the two sides of the question partly that I might feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and a little more acceptable to unscientific readers. In one respect this book is an experiment, and may chance to prove a failure 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 f 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 Geometry 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 moments 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, 1 shall alienate the sympathies of all true scientific readers, who will regard the book as a m iejeu 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, than that, with the deepest conviction of its seriousness of purpose, he should leave it unopened on the shelf. To all the authors, who are here reviewed, I beg to tender my sincerest apologies, if I shall be found to have transgressed, in any instance, the limits of fair criticism. To Mr. Wilson especially such apology is due partly because I have criticised his book at great length and with no sparing hand partly because it may well be deemed an impertinence in one, whose line of study has been chiefly in the lower branches of Mathematics, to dare to pronounce any opinion at all on the work of a Senior Wrangler. Nor should I thus dare, if it entailed my following him np f yonder mountain height which Jie has scaled, but which I can only gaze at from a distance it is only when he ceases c to move so near the heavens and comes down into the lower regions of Elementary Geometry, which I have been teaching for nearly five and-twenty years, that I feel sufficiently familiar with the matter in hand to venture to speak...