THE -ELEMENTS O F EUCLID. VIZ. THE FIRST SIX BOOKS, TOGETHER WITH THE ELEVENTH AND TWELFTH. e Errors, by which THEON, or others, have long ago And fome of EUCLID'S Demonftrations are reftored. ALSO THE BOOK OF EUCLID'S DATA, In like manner corrected. BY ROBERT SIMSON, M. D. Emeritus Profeffor of Mathematics in the University of Glasgow. To this SIXTH EDITION are alfo annexed, ELEMENTS of PLAIN and SPHERICAL TRIGONOMETRY. ċ EDIN Printed for J. NOURSE, London, and J. BALFOUR, Edinburgh. PREFAC Ë TH HE Opinions of the Moderns concerning the Author of the Elements of Geometry, which go under Euclid's name, are very different and contrary to one another. Peter Ramus afcribes the Propofitions, as well as their Demonftrations, to Theon; others think the Propofitions to be Euclid's, but that the Demonftrations are Theon's; and others maintain that all the Propofitions and their Demonftrations are Euclid's own. John Buteo and Sir Henry Savile are the Authors of greatest Note who affert this laft, and the greater part of Geometers have ever fince been of this Opinion, ast they thought it the most probable. Sir Henry Savile, after the feveral Arguments he brings to prove it, makes this Conclufion (Page 13. Praelect)" That, excepting a very few "Interpolations, Explications, and Additions, Theon altered "nothing in Euclid." But, by often confidering and comparing together the Definitions and Demonftrations as they are in the Greek Editions we now have, I found that Theon, or whoever was the Editor of the prefent Greek Text, by adding fome things, fuppreffing others, and mixing his own with Euclid's Demonftrations, had changed more things to the worse than is commonly fuppofed, and thofe not of imall moment, especially in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated; for instance, by fubftituting a fhorter, but infufficient Demonftration of the 18th Prop. of the 5th Book, in place of the legitimate one which Euclid had given; and by taking out of this Book, befides other things, the good Definition which Eudoxus or Euclid had given of Compound Ratio, and giving an abfurd one in place of it in the 5th Definition of the 6th Book, which neither Euclid, Archimedes, Appollonius, nor any Geometer before Theon's time, ever made ufe of, and of which there is not to be found the leaft appearance in any of their Writings; and, as this Definition did much embarafs Eeginners, and is quite ufelefs, it is now thrown out of the Elements, and another, which, without doubt, Euclid had given, is put in its proper place among the Definitions of the |