hesitation, and to an extent somewhat unprecedented. The usual expedients for avoiding this, result in tedious methods, involving the same principle, only under a more covert form. The idea of the infinite is certainly a simple idea, as natural to the mind as any other, and even an antecedent condition of the idea of the finite. A peculiar feature of the work will be observed in the "Exercises," which occur at intervals, commencing immediately after the Axioms. These are intended to develop the original powers of the learner, and to bring into play his inventive faculties, the ordinary text tasking the powers of perception alone. The Exercises are so arranged as to make the progress from the easy to the more difficult so gradual that they will be found to excite a lively interest even in students of moderate capacity. They will be especially convenient in the instruction of large classes, the members of which may all pursue the text, while the exercises upon it will afford scope for the students of greatest ability. The appendices will be found to contain some recent and elegant improvements. They leave much to be done by the learner, it being supposed that none will be likely to attempt them except such as have some taste and talent for geometry. The previous exercises will have furnished the skill requisite to master this part of the work with facility. The work might have been arranged in more ele gant form by a rigid classification of subjects, all the theorems relating to a particular class of magnitudes being given together, after the manner of some of the latest and best French and German treatises. This arrangement, though in the main preserved, has been occasionally departed from, for the sake of rendering a knowledge of the whole subject most easy of acquisition by the student. By omitting the fine print in the volume, the student will obtain a very short course of geometry, but one fully adequate as a preparation for the study of all the higher branches of mathematics, while the whole work contains, probably, the most complete system of purely elementary geometry to be found in any single treatise in any language. A 2 Exercises upon the foregoing theorems Theorems upon intersecting lines Relative magnitudes of angles of triangles Theorems upon secant lines and parallels Theorems relating to parallels Relations of outward and inward angles of triangles of the squares of the sides of triangles 59 |