supply this omission, each proposition of Euclid, Books I. and II., being treated as a geometrical exercise proposed for the investigation of the learner, the reasoning which leads to the solution of the problem, or the demonstration of the theorem, is explained in simple language, with the two-fold object of making the propositions per se more interesting and intelligible, and of affording models for the guidance of the student in his original work. After this analysis a summary is made of the main points on which the solution or demonstration depends. These outlines will, it is hoped, be found especially useful in impressing the argument clearly on the mind, and will also effect a great saving of time in a second reading of the subject for examination. Lastly, each proposition is formally stated succinctly, but avoiding the suppression of any link in the chain of argument. Euclid's demonstration is in all cases given, but a second proof is added where it seemed to present advantages (as in I. 5, &c., and in several propositions of Book II.). The exercises are unusually numerous, and have been principally selected from the last twenty years examination papers of the Universities of London and Cambridge. The method of teaching geometry indicated above has been adopted by many teachers in their oral lessons; it is hoped that this little work may be found a useful accompaniment to such lessons in the hands of the pupils, and that it may induce other teachers to adopt a plan which is usually attended with excellent results. CONTENTS. I. THE RISE AND PROGRESS OF THE STUDY OF GEOMETRY -LIFE OF EUCLID REASONS FOR LEARNING GEO- II. ON THE DEFINITIONS, AXIOMS, AND POSTULATES III. ON THE NATURE AND ARRANGEMENT OF A PROPOSI- TION-PROPOSITIONS 1-3-RESULTS OBTAINED XII. ANALYSIS OF THE FIRST SECTION OF BOOK I.-INTRO- DUCTION TO THE THEORY OF PARALLEL STRAIGHT LINES-PROPOSITIONS 27-29. XIII. PROPOSITIONS 30-31-ANALYSIS OF THE THEORY OF 65 CLASS LESSONS ON EUCLID. LECTURE I. THE RISE AND PROGRESS OF THE STUDY OF GEOMETRY.-LIFE OF EUCLID.-REASONS FOR LEARNING GEOMETRY. THE subject which we are about to begin together may be defined as the science of measurement. It treats of the relations and properties of magnitudes, that is, of those properties of bodies with which we are all familiar under the names of length, breadth or width, and thickness, depth or height. Geometry, the name of this science, is a word of Greek origin, and means literally land-measuring, thus indicating that men's attention was first drawn to it from the necessary measurements of portions of surface of the ground, for divisions of property, building, and so forth. Geometry may be divided into two parts, plane and solid. Plane geometry, which we are now going to study, deals with surfaces, that is with figures having length and breadth only. The Greeks, who may be considered as the educators and thinkers of the old world, just as the Romans were its lawmakers, early paid attention to geometry, and as soon as the science was sufficiently developed made it an important part of their systems of education. From Greece its study spread to other countries, either to the conquered or, as time rolled on, to the conquering races. And from the times of ancient B Greece to our own day, even through the darkness of the early middle ages, geometry has always kept its place as an important part of a liberal education in every civilised country. The text-book of geometry which we are going to study was written by a Greek philosopher more than 2000 years ago, and hence, next to the Old Testament, is the oldest book in the world in everyday use. Eucleides-or, as we call him in English, Euclid-was a mathematician and musician who lived and taught in Alexandria in the third century before the birth of our Lord. This century is a very remarkable era in the history of Greek literature and art. Euclid lived in the same age as Plato and Aristotle, the founders of the two most celebrated systems of philosophy; as the famous orator Demosthenes; as Praxiteles the sculptor, and Apelles the great painter of antiquity. He dwelt, too, in the highly civilised and very learned city of Alexandria, the celebrated library of which was already in existence. Ptolemy Lagus and his son Philadelphus, who then ruled in Egypt, encouraged learned men at their Court, and the second Ptolemy himself attended the lectures of Euclid. Euclid founded the famous mathematical school of Alexandria, to which all mathematicians repaired till the taking of the city by the Saracens in the middle of the seventh century. The work in which he embodied his mathematical teaching is called 'The Elements of Euclid.' It is divided into thirteen books, containing the principles of geometry laid down by Pythagoras, Thales, and other philosophers, collected, systematically arranged, and further developed. To these thirteen books two others are generally added, which are now supposed to have been written by Hypsicles, another mathematician of the Alexandrian school. The first six books of the 'Elements' relate entirely to plane geometry, and have been universally studied from the date of their composition down to the present time; the next four are upon arithmetic and a part of the science of numbers which is now generally included in algebra, and are quite superseded by modern writers; the remaining three books deal with the subject of |