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 VITIATED THESE BOOKS ARE CORRECTED, AND SOME OF EUCLID'S DEMONSTRATIONS ARE RESTORED. ALSO, THE BOOK OF EUCLID'S DATA, IN LIKE MANNER CORRECTED. BY ROBERT SIMSON, M. D. EMERITUS PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW. TO THIS EDITION ARE ALSO ANNEXED, ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY. PHILADELPHIA: DESILVER, THOMAS & CO. 1838. PREFACE. 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 Propositions to be Euclid's, but that the Demonstrations are Theon's; and others maintain, that all the Propositions and their Demonstrations are Euclid's own. John Buteo and Sir Henry Savile are the authors of greatest note who assert 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 conclusion (page 13, Prælect.) "That, excepting a very few interpolations, explications, and additions, Theon altered nothing in Euclid." But, by often considering 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, suppressing 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 absurd 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 use of, and of which there is not to be found the least 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 5th Book, by which the doctrine of compound ratios is rendered plain and easy. Besides, among the Definitions of the 11th Book, there is this, which is the 10th, viz. "Equal and similar solid figures are those which are contained by similar planes of the same number and magnitude." Now this Proposition is a Theorem, not a Definition; because the equality of figures of any kind must be demonstrated, and not assumed; and therefore, though this were a true Proposition, it ought to have been demonstrated. But, indeed, this Proposition, which makes the 10th Definition of the 11th Book, is not true universally, except in the case in which each of the solid angles of the figures is contained by no more than three plane angles; for in other cases, two solid figures may be contained by similar planes of the same number and magnitude, and yet be unequal to one another, as shall be made evident in the Notes subjoined to these Elements. In like manner, in the Demonstration of the 26th Prop. of the 11th Book, it is taken for granted, that those solid angles are equal to one another, which are contained by plane |