nature of reasoning, in the earliest speculations on this subject, the Dialogues of Plato; we find geometrical proof one of the main subjects of discussion in some of the most recent of such speculations, as those of Dugald Stewart and his contemporaries. The recollection of the truths of Elementary Geometry has, in all ages, given a meaning and a reality to the best attempts to explain man's power of arriving at truth. Other branches of Mathematics have, in like manner, become recognized examples, among educated men, of man's powers of attaining truth." Dr. Pemberton, in the preface to his view of Sir Isaac Newton's Discoveries, makes mention of the circumstance, “that Newton used to speak with regret of his mistake, at the beginning of his Mathematical Studies, in having applied himself to the works of Descartes and other Algebraical writers, before he had considered the Elements of Euclid with the attention they deserve.” To these we may subjoin the opinion of Mr. John Stuart Mill, which he has recorded in his invaluable System of Logic, (Vol. 11. p. 180) in the following terms. “The value of Mathematical instruction as a preparation for those more difficult investigations (physiology, society, government, &c.) consists in the applicability not of its doctrines, but of its method. Mathematics will ever remain the most perfect type of the Deductive Method in general; and the applications of Mathematics to the simpler branches of physics, furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education, and regarding, with Plato, one who is áyewuétontos, as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy.” In addition to these authorities it may be remarked, that the new Regulations which were confirmed by a Grace of the Senate on the 11th of May, 1846, assign to Geometry and to Geometrical methods a more important place in the Examinations both for Honors and for the Ordinary Degree in this University, TRINITY COLLEGE, R. P. R. P. • CONTENTS. ............ .. ........ PREFACB to the Third Edition ...... First Book of the Elements ...... Questions on the First Book ...... On the Ancient Geometrical Analysis ... Geometrical Exercises on Book I Notes to the Second Book ..... Note on the Algebraical Symbols & Abbreviations used in Geometry Questions on the Second Book ....... Geometrical Exercises on Book II ..... Questions on the Third Book .. Geometrical Exercises on Book III ........ Fourth Book of the Elements ... Geometrical Exercises on Book IV ..... Notes to the Fifth Book ............ Sixth Book of the Elements .... Questions on the Sixth Book ........ Geometrical Exercises on Book VI Solutions, Hints, &c. on Book I 259 LIBER CANTABRIGIENSIS. An Account of the Aids afforded to poor Students, the encouragements offered to diligent Students, and the rewards conferred on successful Students, in the University of Cambridge ; to which is prefixed a Collection of Maxims, Aphorisms, &c. Designed for the Use of Learners. 'By ROBERT Potts, M.A., Trinity College. Fcap. 8vo., pp. 570, price 5s. 6d.' "It was not a bad idea to prefix to the many encouragements afforded to students in the University of Cambridge, a selection of maxims drawn men, who have shown that learning is to be judged by its fruits in social and individual life.”—The Literary Churchman. "A work like this was much wanted.”- Clerical Journal. “ The book altogether is one of merit and value.”—Guardian. -“The several parts of this book are most interesting and instructive."--Educational Times. “No doubt many will thank Mr. Potts for the very valuable information he has afforded in this laborious compilation.”-Critic. "A vast amount of information is compressed into a small compass, at the cost evidently of great labour and pains. The Aphorisms which form a prefix of 174 pages, may suggest useful reflections to earnest students.”—The Patriot. John W. PARKER & Son, West Strand, London. EUCLID'S BOOK I. A POINT is that which has no parts, or which has no magnitude. II. III. IV. V. VI. The extremities of a superficies are lines. VII. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. A plane angle is the inclination of two lines to each other in a plane, which meet together, but are not in the same straight line. IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. N.B. If there be only one angle at a point, it may be expressed by a letter placed at that point, as the angle at E: but when several angles are at one point B, either of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of these straight lines, and the other upon the other line. Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA ; that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD. When a straight line standing on another straight line, makes the adjacent angles equal to one another, each of these angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. XI. An obtuse angle is that which is greater than a right angle. XII. XIII. XIV. |