THE ELEMENTS OF EUCLID; VIZ. THE FIRST SIX BOOKS, TOGETHER WITH THE ELEVENTH AND TWELFTH. The ERRORS by which THEON, or others, have long ago And fome of EUCLID'S DEMONSTRATIONS are Restored. ALSO THE BOOK OF EUCLID'S DATA, In like Manner CORRECTED. By ROBERT SIMSON, M. D.. To this TWELFTH EDITION are also annexed LONDON: Printed for F. WINGRAVE, in the Strand, Successor to Mr. NOURSE. 1804. 1 TO THE KING, THIS EDITION OF THE PRINCIPAL BOOKS OF THE ELEMENTS OF EUCLID, AND OF THE BOOK OF HIS DATA, IS MOST HUMBLY DEDICATED, BY HIS MAJESTY'S MOST DUTIFUL, AND MOST DEVOTED SUBJECT AND SERVANT, ROBERT SIMSON. 1 PREFACE. レ THE 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 ascribes the Propositions, as well as their Demonstrations, to Theon; others think the Propofitions to be Euclid's, but that the Demonstrations are Theon's; and others maintain, that all the Propofitions and their Demonstrations are Euclid's own. John Buteo and Sir Henry Savile are the authors of greatest note who affert this last; and the greater part of geometers have ever since been of this opinion, as they thought it the most probable. Sir Henry Savile, after the several arguments he brings to prove it, makes this conclufion (Page 13. Prælect.) "That, excepting a very few interpolations, explications, and "additions, Theon altered nothing in Euclid." But, by often confidering and comparing together the Definitions and Demonstrations as they are in the Greek editions we now have, I found that Theon, or whoever was the editor of the present Greek text, by adding some things, fuppreffing others, and mixing his own with Euclid's Demonstrations, had changed more things to the worse than is commonly supposed, and those not of small moment, especially in the fifth and eleventh Books of the Elements, which this editor has greatly vitiated; for instance, by substituting a shorter, but insufficient Demonstration 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, besides 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, Apollonius, nor any geometer before Theon's time, ever made use of, and of which there is not to be found the leaft appearance in any of their writings; and, as this Definition did much embarrass beginners, and is quite useless, 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 |