entering on Geometry as evinced by sameness of relative magnitude, and not invent a new notion for the occasions of Geometry." The reasoning is exquisite and profound; it is too exquisite; it leaves on most men's minds the half-defined impression that all profound reasoning is something far-fetched and artificial, and differing altogether from good clear common sense. In common then with some of our ablest mathematicians, and with many who are engaged in teaching mathematics, I am of opinion that the time is come for making an effort to supplant Euclid in our schools and universities. Already the fifth book has practically gone; and in consequence the study of the sixth book has become somewhat irrational. For the improvement of our Geometrical teaching in England two things seem to be wanted. First, that Cambridge, and the Government Examiners, should follow the example of Oxford and London, and examine not in Euclid only, but in the Geometry of specified subjects, according to a programme for each examination. Secondly, that textbooks should be written to illustrate what is required. This book is the first part of such a text-book of Elementary Geometry. Probably its method of treating propositions may be considered as sufficiently different from that common in England to justify its publication. At the same time it may be remarked, that the forthcoming second part, which is to embrace the Geometry of the Circle and the applications of proportion to Plane Geometry, ought to bring out in a stronger light than is here possible the superiority of modern to ancient methods of Geometry. The present Part may be used either as introductory to Euclid, or as replacing the first two books. In a few years I hope that our leading mathematicians will have published, perhaps in concert, one or more text-books of Geometry, not inferior, to say the least, to those of France, and that they will supersede Euclid by the sheer force of superior merit. In the compilation of this little book, I lay claim to no originality. I have read several French Geometries, and am under some obligations to them. I owe more to the valuable suggestions of my colleague the Rev. C. E. Moberly, who has the spirit without the prejudices of a geometrician. But much of what is most characteristic in the book is due to Dr Temple. It was at his wish that I undertook the work, as he is strongly impressed with the need of it; and his criticisms and his contributions to it have enabled me to rearrange it and improve it in some important respects. And this gives me confidence in publishing it. At the same time we feel that the experience of a few years in teaching with this book will enable me to make improvements in it, without departing widely from the lines here laid down. To Professor Hirst also I have the pleasure to express my thanks for some corrections and remarks. The distinctive features of this work are intended to be the following. The classification of Theorems according W. G. b to their subjects; the separation of Theorems and Problems; the use of hypothetical constructions; the adoption of independent proofs where they are possible and simple; the introduction of the terms locus, projection, &c; the importance given to the notion of direction as the property of a straight line; the intermixing of exercises, classified according to the methods adopted for their solution; the diminution of the number of theorems; the compression of proofs, especially in the latter part of the book; the tacit, instead of explicit, reference to axioms; and the treatment of parallels. RUGBY, April 28, 1868. J. M. WILSON. CONTENTS. THEOREM I. When a straight line stands upon another straight line, it will make the adjacent angles together equal to two right angles and conversely, if two adjacent angles are equal to two right angles, the exterior arms of these angles will be THEOREM 2. The sum of all the angles made by any number of lines taken consecutively which meet at a point will be four PAGE THEOREM 4. If two lines are respectively parallel to two other lines, the angles. made by the first pair will be equal or sup- plementary to the angles made by the second pair; equal, if both are taken in the same or both in the opposite direction; THEOREM 5. If two straight lines meet in a point they will make THEOREM 6. The exterior angles of any convex polygon, made by producing the sides in succession, will be together equal to THEOREM 7. If one side of a triangle is produced, the exterior angle will be equal to the two interior and opposite angles, and the three interior angles of any triangle will be together THEOREM 9. Oppositely. A triangle which is not isosceles will have the angles at its base unequal, the greater angle being |