EDITORIAL PREFACE THE volume on Euclid is one of those which have been added to the series of the WORLD'S EPOCH-MAKERS since the prospectus was issued. Although I had long cherished the desire to include the founder of what may be called the "Science of Geometry" in the series, I did not see my way at the outset to do so, first, owing to the difficulty of getting any one to undertake the subject under the limitations of space necessarily imposed; and, second, owing to the doubt I entertained whether the subject could be treated in a manner sufficiently popular to warrant its inclusion in a Series such as this. After several conversations with my friend, Emeritus Professor Thomas Smith, D.D., LL.D., and after hearing him read some parts of the work, I felt assured that the plan of treatment proposed by Dr. Smith was not only exhaustive from a scientific point of view, but was sufficiently popular in style to win the interest and attention of the non-scientific reader. Dr. Smith has had to encounter and overcome difficulties neither few nor small in the accomplishment of his task, and it is not the least interesting feature of this volume that it has been produced after its venerated author had reached his eighty-fifth year. In his case, however, the intellectual bow has abode in strength long after the time when it might reasonably have been expected to become relaxed. O. S. EUCLID I In order to justify the inclusion of the name of Euclid in the list of epoch-making men, we must first of all intimate who Euclid was; then, what influence he exerted in his own day and in subsequent times; thirdly, we must inquire in what sense, or to what extent, the introduction and the general study of geometry formed an epoch in the history of Europe, and consequently of the world; lastly, it may not be out of place to hazard a forecast as to the future cultivation of the science, and to consider how far its development is to be effected by adherence to Euclid's methods, how far by a modification of them, or whether by a virtual abandonment of them. Such, then, briefly, is a summary of our present undertaking, a condensed table of contents of the present volume. In all these branches of our task we shall have serious difficulties to encounter. In the earlier ones we shall have to regret the paucity of authentic information, and the inconsistency of such as might have been expected to be authentic. In two of our branches we shall have not easy argumentative work, in the course of which we shall have to deal with living opponents far more than "worthy of our steel," and with some of the mighty men of the past whose authority it seems almost profanity to question. But our main difficulty throughout will be to determine the character which our work is to assume with reference to the class of readers who may be expected to take interest in its subject. Certainly the book is not designed for mathematicians; and if any such deign to peruse it, they will find in it much that will be to them unedifying, and will even seem trivial. Yet we see not, on the other hand, how it will be possible to treat some parts of our subject without introducing technicalities which will be repulsive to such as are altogether ignorant of even elementary mathematics. In these circumstances we must endeavour to steer a middle course. Our aim shall be to write for the reader who has just the amount of knowledge of, and interest in, mathematical subjects which may be reasonably expected to pertain to intelligent, though not necessarily intellectual, men and women of the twentieth century; while it is evident that, for the last section, as indicated above, we shall have to bespeak especially the attention of such as are interested, professionally or otherwise, in education. Be it frankly said that we have no expectation of producing a great or classical work. But we are not without hope that our little volume may be suggestive of thoughts which may be conducive to the augmentation and diffusion of intelligence, and even, in some cases, to the quickening of mathematical tastes, and so ultimately to the advancement of mathematical science. At the least, we |