XV. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another: Book I. XVI. And this point is called the centre of the circle. XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 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. Rectilineal figures are those which are contained by straight lines. XXI. Trilateral figures, or triangles, by three straight lines. XXII. Quadrilateral, by four straight lines. XXIII. Multilateral figures, or polygons, by more than four straight lines. XXIV. Of three sided figures, an equilateral triangle is that which has three equal sides. An isosceles triangle, is that which has only two sides equal. Book I. Δ Δ Δ XXVI. A scalene triangle, is that which has three unequal sides. XXVII. A right-angled triangle, is that which has a right angle. An obtuse-angled triangle, is that which has an obtuse angle. 44 XXIX. An acute-angled triangle, is that which has three acute angles. Of four sided figures, a square is that which has all its sides XXXI. An oblong, is that which has all its angles right angles, but has not all its sides equal. XXXII. A rhombus, is that which has all its sides equal, but its angles are not right angles. ن XXXIII. A rhomboid, is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. XXXIV. All other four sided figures besides these, are called trapéziums. XXXV. Parallel straight lines, are such as are in the same plane, and which being produced ever so far both ways, do not meet. Book I. POSTULATES. I. LET it be granted that a straight line may be drawn from any one point to any other point. II. That a terminated straight line may be produced to any length in a straight line. III. And that a circle may be described from any centre, at any distance from that centre. AXIOMS. I. THINGS which are equal to the same are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, are equal to one another. VII. Things which are halves of the same, are equal to one another. Magnitudes which coincide with one another, that is, which ex- 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 pbable. 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, |