Book I. XV. the circumference, and is such that all straight lines drawn a XVI. XVII. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by that diameter. XIX. “ A segment of a circle is the figure contained by a straight line, " and the circumference it cuts off.” XX. XXI. XXII. XXIII. XXIV. three equal sides. An isosceles triangle, is that which has only two sides equal Book I. XXVI. XXVII. XXVIII. XXIX. XXX. equal, and all its angles right angles. XXXI. XXXII. not right angles. On XXXIII. another, but all its sides are not equal, nor its angles right Book I. XXXIV. XXXV. which being produced ever so far both ways, do not meet. POSTULATES. I. LET it be granted that a straight line may be drawn from any one point to any other point. II. III. tance from that centre. AXIOMS. I. II. III. IV. V If equals be taken from unequals, the remainders are unequal. VI. VII. VIII. actly fill the same space, are equal to one another. 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 ascr bes 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 p bable. Sir Henry Savile, after the several arguments he brings to prove it, makes this conclusion (page 13 Praelect.), “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, |