neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the Mathematics: of which sort are perspective, music, astronomy, cosmography, architecture, enginery, and divers others. "In the Mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the pure Mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For, if the wit be dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect that it maketh a quick eye, and a body ready to put itself into all postures; so in the Mathematics, that use which is collateral and intervenient, is no less worthy than that which is principal and intended. And as for the mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed.". How truly has this prediction been fulfilled in the subsequent advancement of the Mixed Sciences, and in the applications of the pure Mathematics to Natural Philosophy! Dr. Whewell, in his "Thoughts on the Study of Mathematics," has maintained, that mathematical studies judiciously pursued, form one of the most effective means of developing and cultivating the reason: and that "the object of a liberal education is to develope the whole mental system of man;-to make his speculative inferences coincide with his practical convictions;—to enable him to render a reason for the belief that is in him, and not to leave him in the condition of Solomon's sluggard, who is wiser in his own conceit than seven men that can render a reason." And in his more recent work entitled, "Of a Liberal Education, &c.” he has more fully shewn the importance of Geometry as one of the most effectual instruments of intellectual education. In page 55 he thus proceeds :-" But besides the value of Mathematical Studies in Education, as a perfect example and complete exercise of demonstrative reasoning; Mathematical Truths have this additional recommendation, that they have always been referred to, by each successive generation of thoughtful and cultivated men, as examples of truth and of demonstration; and have thus become standard points of reference, among cultivated men, whenever they speak of truth, knowledge, or proof. Thus Mathematics has not only a disciplinal but an historical interest. This is peculiarly the case with those portions of Mathematics which we have mentioned. We find geometrical proof adduced in illustration of the 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. PREFACE to the Third Edition 64 69 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 VI Eleventh Book of the Elements, Props. 1-21 Notes to the Eleventh Book..... Questions on the Eleventh Book Geometrical Exercises on Book XI. Twelfth Book of the Elements, Props. 1, 2 Geometrical Exercises on Book XII. Index to the Geometrical Exercises .... LIBER CANTABRIGIENSIS. PART I. An Account of the Aids, the encouragements, and rewards offered to 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. Svo., pp. 570, bds. 4s. 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 from the writings of 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. PART II. An Account of the Changes made by recent Legislation in the Colleges, and the University of Cambridge, with an Appendix, containing the Examination Papers for the Open Minor Scholarships in 1861-1862. Feap. 8vo., pp. 462, bds. 4s. 6d. "We regard Liber Cantabrigiensis, Part II., as an invaluable addition to an invaluable work."-English Journal of Education. "Mr. Potts has very meritoriously presented, in a manageable compass, not only the present code of the University and its Professorships, Scholarships, and Prizes, but also a full abstract of the Statutes of all the Colleges, with particulars of their Fellowships, Scholarships, and Exhibitions. It is a permanent Companion to the Calendar, the importance of which latter will henceforth depend chiefly on the Class lists and other lists of names."- The Reader. LONGMAN & Co., London. A View of the Evidences of Christianity, and the Hora Pauline; by William Paley, D.D., formerly Fellow and Tutor of Christ's College, Cambridge. A new Edition, with Notes, an Analysis, and a selection of Questions from the Senate-House and College Examination Papers; designed for the use of Students. By ROBERT POTTS, M.A., Trinity College. 8vo. pp. 568, price 10s. 6d. in cloth. "The theological student will find this an invaluable volume. In addition to the text there are copious notes, indicative of laborious and useful rescarch; an analysis of great ability and correctness; and a selection from the Senate-House and College Exa mination Papers, by which great help is given as to what to study and how to study it. There is nothing wanting to make this book perfect."-Church and State Gazette. "Mr. Potts' is the most complete and useful edition yet published."--Electic Review. "We feel that this ought to be henceforth the standard edition of the Evidences' and Horæ.'"-Biblical Review. "Without this volume the library of any Christian Man is incomplete.”—Church of England Quarterly Review. LONGMAN & Co., London. 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. D / A К B B |