of the parallelism (or in common language equidistance) of straight lines in the same plane which never meet. This is of great consequence, as the leading idea in connection with straight lines called parallel is that suggested by the literal meaning of the word: thus the conception of a parallelogram is not that of a figure whose opposite sides will never meet however far they may be produced in either direction, but rather that of a figure whose opposite sides are equidistant. Again, Euclid's treatment of ratio and proportion is objected to as being only adapted to the grasp of the advanced geometrician; one of the great difficulties to be encountered by a student attempting to master Euclid's Fifth Book being that magnitudes in general are dealt with instead of certain specified magnitudes. Accordingly to supply the want due to the fact of this Book being invariably omitted, the necessary propositions contained in it have been established for those particular species of magnitude to which they are afterwards applied. The method given possesses the combined advantages of being rigorous, geometrical, simple and short. The terms duplicate ratio and compound ratio have been relegated to the Supplement, there placed for the benefit of those likely to be examined thereon, but ejected from the text, as not being generally appreciated by students, few of whom would estimate the relative magnitudes of plane similar rectilineal figures by contemplating their homologous sides and then calculating their duplicate ratio. In order to make the Second Book, which treats of the relations existing between the rectangles under the segments of a line divided in various ways, available for the extensive development of this part of Geometry of which it is capable,. Euclid's second and eight following propositions should be shewn to be deducible from the first one. This has accordingly been done, the method being applicable to all propositions which are the geometrical interpretation of Algebraical identities of two dimensions. At the latter part of the Third Book a section has been added on the equal subdivision of the circumference of a circle, thus preparing the way for a more general consideration of the subject of regular polygons. In the Fourth Book the areas and perimeters of regular polygons inscribed in and circumscribed about a circle have been discussed, and the groundwork laid for a comparison not only of the areas but also of the circumferences of circles. It has been deemed essential that Problems, though kept distinct from Theorems, should nevertheless occupy the attention of the beginner as soon as possible, and also that the necessity of his having to wade through a long book of Theorems only should be avoided. This has been effected by placing the Problems at the end of the various sections, which are never of any great length. Euclid's mode of demonstration, in which the conclusion of each step is preceded by reasoning expressed with all the exactness of the minor premiss of a syllogism, of which some previous proposition is the major premiss, has been adopted as offering a good logical training, and also as being peculiarly adapted for teaching large classes, rendering it possible for the teacher to call first upon one then upon another, and so on, to take up any link in the chain of argument. The principle of superposition has been made use of whenever possible, as affording the most convincing proof of equality of geometrical magnitudes. With a view to facilitating the student's progress it has been arranged that, as a rule, each proposition should have a page to itself, and that those of greater length should occupy pages facing each other: so also that propositions and the corresponding converse ones should be placed on opposite pages. Moreover, Definitions and Axioms, instead of being all at the beginning of the book, have been introduced when required. The great objection entertained by those in authority at the Universities towards modern text-books on Geometry is grounded on the fact that enormous inconvenience would arise in conducting examinations with no recognized sequence of propositions. This objection I believe I have effectually dealt with by giving demonstrations which depend on propositions occupying a prior position not only in this work but also in Euclid. How this and other points of general arrangement have been carried out will to some extent be seen by reference to the accompanying Tables and Scheme for Examinations. It will be readily seen that the scheme referred to offers fair rivalry between Euclid's proofs and others without necessarily displacing Euclid as a text-book. I think that there would thus be found no difficulty in conducting examinations under it, and this opinion is shared by Mr F. C. Wace, M.A., Fellow and Lecturer of St. John's College, Cambridge, whose frequent experience as Moderator and Examiner for the Mathematical Tripos gives him a practical acquaintance with the difficulties which must accompany any such alteration, and to whom I am indebted for several corrections and suggestions in the revision of this work for the press. In conclusion, it remains for me to acknowledge the debt of gratitude due to my former tutor, the Rev. J. G. Mould, B.D.,late Fellow and Tutor of Corpus Christi College, Cambridge: to whom I submitted the work before placing it in the hands of the publishers last September, when he kindly offered many valuable criticisms, especially on my method of treating proportion. Since that time I have been pleased to find that the idea of establishing the various propositions on proportion for particular geometrical magnitudes has been approved of by some of the foremost mathematicians of the day. CITY OF LONDON SCHOOL, March, 1874. |