London: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. LEIPZIG: F. A. BROCKHAUS. NEW YORK: MACMILLAN AND CO. EUCLID'S ELEMENTS OF GEOMETRY EDITED FOR THE SYNDICS OF THE PRESS lmy arty w BY H. M. TAYLOR, M.A. FELLOW AND FORMERLY TUTOR OF TRINITY COLLEGE, CAMBRIDGE. BOOKS I-VI, XI, XII. CAMBRIDGE AT THE UNIVERSITY PRESS 1895 [All Rights reserved] 1-15-41 Gift 7-27-26 PREFACE TO BOOKS I. AND II. IT T was with extreme diffidence that I accepted an invitation from the Syndics of the Cambridge University Press to undertake for them a new edition of the Elements of Euclid. Though I was deeply sensible of the honour, which the invitation conferred, I could not but recognise the great responsibility, which the acceptance of it would entail. The invitation of the Syndics was in itself, to my mind, a sign of a widely felt conviction that the editions in common use were capable of improvement. Now improvement necessitates change, and every change made in a work, which has been a text book for centuries, must run the gauntlet of severe criticism, for while some will view every alteration with aversion, others will consider that every change demands an apology for the absence of more and greater changes. I will here give a short account of the chief points, in which this edition differs from the best known editions of the Elements of Euclid at present in use in England. While the texts of the editions of Potts and Todhunter are confessedly little more than reprints of Simson's English version of the Elements published in 1756, the text of the present edition does not profess to be a translation from the Greek. I began by retranslating the First Book: but there proved to be so many points, in which I thought it desirable to depart from the original, that it seemed best to give up all idea of simple translation and to retain merely the substance of the work, following closely Euclid's sequence of Propositions in Books I. and II. at all events. Some of the definitions of Euclid, for instance trapezium, rhomboid, gnomon are omitted altogether as unnecessary. The word trapezium is defined in the Greek to mean 66 any four sided figure other than those already defined," but in many modern works it is defined to be "a quadrilateral, which has one pair of parallel sides." The first of these definitions is obsolete, the second is not universally accepted. On the other hand definitions are added of several words in general use, such as perimeter, parallelogram, diagonal, which do not occur in Euclid's list. The chief alteration in the definitions is in that of the word figure, which is in the Greek text defined to be "that which is enclosed by one or more boundaries." I have preferred to define a figure as "a combination of points, lines and surfaces." That Euclid's definition leads to difficulty is seen from the fact that, though Euclid defines a circle as 66 a figure contained by one line...", he demands in his postulate that "a circle may be described...". Now it is the circumference of a circle which is described and not the surface. Again, when two circles intersect, it is the circumferences which intersect and not the surfaces. I have rejected the ordinarily received definition of a square as "a quadrilateral, whose sides are equal, and whose angles are right angles." There is no doubt that, when we define any geometrical figure, we postulate the possibility of the figure; but it is useless to embrace in the definition more properties than are requisite to determine the figure. The word axiom is used in many modern works as applicable both to simple geometrical propositions, such as "two straight lines cannot enclose a space," and to proposi |