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. II. 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 dyɛwμétρntos, 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, March 1, 1850. TRINITY COLlege, R. P. The supplement to the School Euclid (about forty-eight pages) has been incorporated with this impression of the Fifth Edition. CONTENTS. On the Ancient Geometrical Analysis Geometrical Exercises on Book I Note on the Algebraical Symbols & Abbreviations used in Geometry 109 Geometrical Exercises on Book II Geometrical Exercises on Book III Geometrical Exercises on Book IV Solutions, Hints, &c. on Book I. 175 190 196 204 214 EUCLID'S ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS. I. A POINT is that which has no parts, or which has no magnitude. II. A line is length without breadth. The extremities of a line are points. IV. A straight line is that which lies evenly between its extreme points. V. A superficies is that which has only length and breadth. 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 direction. 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. X. 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. An acute angle is that which is less than a right angle. XIII. A term or boundary is the extremity of any thing. XIV. A figure is that which is enclosed by one or more boundaries. |