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 roth, viz. “Equal and " fimilar folid figures are those which are contained by fimilar " planes of the fame 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 afsumed; and therefore, though this were a true Proposition, it ought to have been demonstrated. But, indeed, this Propofition, which makes the 10th Definition of the Irth Book, is not true universally, except in the cafe in which each of the folid angles of the figures is contained by no more than three plane angles; for in other cases, two folid figures may be contained by fimilar planes of the fame 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 plain angles of the fame num ber and magnitude, placed in the saine order; but neither is this universally true, except in the cafe in which the solid angles are contained by no more than three plain angles; nor of this cafe is there any Demonstration in the Elements we now have, though it be quite necessary there should be one. Now, upon the roth Definition of this Book depend the 25th and 28th Propofitions of it; and, upon the 25th and 26th depend other eight, viz. the 27th, 31st, 32d, 33d, 34th, 36th, 37th, and 40th of the fame Book; and the 12th of the 12th Book depends upon the eighth of the fame; and this eighth, and the Corollary of Propofition 17th and Propofition 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 fimilar plane figures, which are not fimilar to one another, in the true sense of fimilarity received by geometers; and all these Propofitions have, for these reafons, been infufficiently demonftrated fince Theon's time hitherto. Besides, there are several other things, which have nothing of Euclid's accuracy, and which plainly shew, that his Elements have been much corrupted by unskilful geometers; and, though. these are not fo 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 fince these Elements are the foundation of a science by which the investigation and difcovery of useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewife is of the greatest use in the arts both of peace and war, to many of which geometry is absolutely neceffary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of fome 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 fuppressed, and which have these many ages been buried in oblivion. In this twelfth edition, Ptolemy's Propofition concerning a property of quadrilateral figures in a circle, is added at the end of the fixth Book. Also the Note on the 29th Proposition, Book Ift, is altered, and made more explicit, and a more general Demonftration is given, instead of that which was in the Note on the 10th Definition of Book 11th; befides, the Tranflation 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 Ι. DEFINITIONS. I. A Point is that which hathno parts, or which hathno magnitude. A line is length without breadth. II. III. The extremities of a line are points. See Notes. IV. A straight line is that which lies evenly between its extreme points. V. A fuperficies is that which hath only length and breadth. VI. The extremities of a superficies are lines. VII. A plane superficies is that in which any two points being taken, See N. 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 See N. " plane, which meet together, but are not in the fame direction." IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the fame straight linę. A N.B. When several angles are at one point B, any one of them ' is expreffed 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, it may be expressed by a • letter placed at that point; as the angle at E.' An obtufe angle is that which is greater than a right angle. XII. An acute angle is that which is less than a right angle. XIII. " A term or boundary, is the extremity of any thing." XIV. A figure is that which is inclosed by one or more boundaries |