angles of the same number and magnitude, placed in the same order; but neither is this universally true, except in the case in which the solid angles are contained by no more than three plane angles; nor of this case is there any Demonstration in the Elements we now have, though it is quite necessary there should be one. Now, upon the 10th Definition of this Book depend the 25th and 28th Propositions of it; and upon the 25th and 26th depend other eight, viz., the 27th, 31st, 32d, 33d, 34th, 36th, 37th, and 40th, of the same Book; and the 12th of the 12th Book depends upon the 8th of the same; and this 8th, and the Corollary of Proposition 17th and Proposition 18th of the 12th Book, depend upon the 9th Definition of the 11th Book, which is not a right definition, because there may be solids contained by the same number of similar plane figures, which are not similar to one another, in the true sense of similarity received by all geometers; and all these Propositions have, for these reasons, been insufficiently demonstrated since Theon's time hitherto. Besides, there are several other things, which have nothing of Euclid's accuracy, and which plainly show, that his Elements have been much corrupted by unskilful geometers; and, though these are not so gross as the others now mentioned, they ought by no means to remain uncorrected. Upon these accounts it appeared necessary, and I hope will prove acceptable, to all lovers of accurate reasoning, and of mathematical learning, to remove such blemishes, and restore the principal Books of the Elements to their original accuracy, as far as I was able; especially since these Elements are the foundation of a science by which the investigation and discovery of useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewise is of the greatest use in the arts both of peace and war, to many of which geometry is absolutely necessary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of geometers, and by restoring to him those things which Theon or others have suppressed, and which have, these many ages, been buried in oblivion. In this edition, Ptolemy's Proposition concerning a property of quadrilateral figures in a circle, is added at the end of the sixth Book. Also the Note on the 29th Proposition, Book 1st, is altered, and made more explicit, and a more general Demonstration is given, instead of that which was in the Note on the 10th Definition of Book 11th; besides, the Translation is much amended by the friendly assistance of a learned gentleman. To which are also added, the Elements of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid. THE ELEMENTS OF EUCLID. . BOOK I. DEFINITIONS. I. A point is that which hath no parts, or which hath no magnitude. II. A line is length without breadth. III. The extremities of a line are points. a IV. * A straight line is that which lies evenly between its extreme points. V. VI. VII. A plane superficies is that in which any two points being taken,* the straight line between them lies wholly in that superficies. VIII. “ A plane angle is the inclination of two lines to one another* in a plane, which meet together, but are not in the same direction.” IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. A D V B * See Notes. 6 N. B. “When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the 'vertex of the angle, that is, at the point in which the straight lines • that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus the angle • which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named • the angle ABD, or DBA; and that which is contained by DB, CB is called the angle DBC, or CBD; but, if there be only one angle at a point, may be expressed by the letter placed at that point; * as the angle at E.' X. straight line makes the adjacent angles XI. XII. XIII. XIV. XV. 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: XVI. 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. XXI. XXII. 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. XXV. A 1 XXVI. XXVII. XXVIII. A A A XXIX. XXX. and all its angles right angles. 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 its sides equal, but its angles are not right angles. 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. XXXV. being produced ever so far both ways, do not meet. POSTULATES. 1. 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. II. |