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. S........34 TABLE I. Shewing the Classification of Subjects. E.G. Euclid. | E.G. Euclid. | E.G.. Euclid. PLANE GEOMETRY. Quad. and Mult. Figs. Book II. Book I. 26.. Rectangles. 27 ....... 46 Cor. Intr” ... 11 Cor. ......... 1, 2, 3 ... 4...IO St. lines, <s and As. ....... 32 Cor. ..... ........ 12 I ............... 4 ........13 .......... VI. 9 2 ............... 5 ........... 14 .. ... ... 46 3 ...............264 ............ 11 4 ... ... ... ...... 6 ..... IV. IO 5 ............... 8 Areas. Sup. to Book II. 31 ......... 35, 36 Q .................. 4 ....... 41, 37, 38 b .................. 7 Right Ls. .......... 39, 40 C.... ............. ............... 15 .....43 ...............IT ....... . ......IO ......47 Inequalities. 13 ......... 16, 17 14 .... 15 ... re..........19 16 ...............20 17 .............. Circles. Chords. ..............., 2 prr......... I, 3 .. ......... 9 ... ... ... ... ... 14 ............... 15 ... ... ... ... ... 25 p................ 2 .......13 ...... 5,32 IO Int. TABLE I. Shewing the Classification of Subjects. E.G. Euclid. ' E.G. Euclid. / E.G. Euclid. Comp. of st. lines. SOLID GEOMETRY, ...............10 St. lines I to planes. ... ... ... ... ... 30 9................ 13a 14 ............... 16 .............. 25 A .......... 10, 15 .............,,I 2 Planes to to planes. 13.,.............13 12 .... XII. 2, &c. Comp. of Ls. en |