CONTAINING THE FIRST SIX BOOKS OF EUCLID: WITH A SUPPLEMENT ON THE QUADRATURE OF THE CIRCLE AND TO WHICH ARE ADDED ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY. BY JOHN PLAYFAIR, F.R.S. LOND. & EDIN. LATE PROFESSOR OF NATURAL PHILOSOPHY (FORMERLY OF MATHEMATICS) IN THE UNIVERSITY WITH ADDITIONS, BY WILLIAM WALLACE, A.M., F.R.S.E., &c. LATE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF EDINBURGH. THIRTEENTH EDITION, IN WHICH THE NOTES ARE ADAPTED TO STUDENTS, AND THE TREATISES BY THE REV. P. KELLAND, A.M., F.R.S. LOND. & EDIN. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF EDINBURGH. EDINBURGH: BELL & BRADFUTE, AND OLIVER & BOYD. SIMPKIN & CO., AND WHITTAKER & CO., LONDON. MDCCCLXXV. PREFACE TO THE ELEVENTH EDITION. THE first six books of the present Treatise are precisely the first six books of Euclid's Elements. No alterations whatever have been made in the arrangement of the propositions, nor any of importance in the demonstration of those of the first four and sixth books. The same does not apply to the fifth book. The doctrine of Proportion laid down by Euclid in that book is an admirable specimen of reasoning based on an abstract definition. In simplicity of treatment, and in rigour of demonstration, this book leaves nothing to be desired. But the geometrical representation of what is essentially an arithmetical multiplication, renders the doctrine, as Euclid delivered it, somewhat difficult to be mas tered. In the present treatise this difficulty has been obviated by the introduction of the concise language of algebra, whereby the reasoning is condensed and simplified, whilst the character of the demonstration remains unchanged. By this means the steps of the argument are brought near to one another, and the force of the whole is so clearly and distinctly perceived, that no more difficulty should be experienced in understanding the propositions of the fifth book than those of any other book of the Elements. The Supplement consists of three books. The First Book treats of the rectification and quadrature of the circle. In the present edition, this book has been condensed and simplified. The Second Book treats of the intersections of planes, and contains the most important propositions of the Eleventh Book of Euclid. The Third Book treats of Solids, and exhibits, in a simple form, the most important propositions of the Twelfth Book of Euclid. The treatise on Plane Trigonometry has, in the present edition, been increased by an additional section, containing some numerical examples, with a popular account of the nature and application of logarithms. The treatise on Spherical Trigonometry, and the Notes, are reprinted with little alteration from the last edition. The whole work is now so well known and appreciated, that a detailed explanatory preface is altogether superfluous. PHILIP KELLAND. COLLEGE OF EDINBURGH, June 1, 1859. ELEMENTS OF GEOMETRY. BOOK FIRST. DEFINITIONS. I. A point is that which has position, but not magnitude.* COROLLARY. The extremities of a line are points; and the intersections of one line with another are also points. III. If two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line. COR. Hence, two straight lines cannot enclose a space. Neither can two straight lines have a common segment; for they cannot coincide in part, without coinciding altogether. IV. A superficies is that which has only length and breadth. COR. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines. V. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. ע VI. A plane rectilineal angle is the inclination of two straight lines to one another, which A meet together, but are not in the same straight line. N.B.-When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is B E 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 See Notes. A |