5th Book, by which the doctrine of compound Ratios is ren Upon thefe Accounts it appeared neceffary, and I hope will the 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 Discovery 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 abfolutely neceffary. This I have endeavoured to do, by taking away the inaccurate and falfe Reasonings which unskilful Editors have put into the place of fome of the genuine Demonftrations of Euclid, who has ever been juftly celebrated as the most accurate of Geometers, and by restoring to him thofe Things which Theon or others have fuppreffed, and which have these many ages been buried in Oblivion. In this Ninth Edition, Ptolemy's Propofition concerning a Property of quadrilateral Figures in a Circle is added at the End of the fixth Book. Alfo the Note on the 29th Prop. Book ift, 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; befides, the Tranflation is much amended by the friendly affiftance 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 EUCLI D. воок ь. DEFINITIONS. I. A Point is that which hath no parts, or which hath no See Notes. magnitude. II. A line is length without breadth. III. The extremities of a line are points. IV. A ftraight line is that which lies evenly between its extreme points. V. A fuperficies is that which hath only length and breadth. The extremities of a fuperficies are lines. VII. A plane fuperficies is that in which any two points being taken, the straight line between them lies wholly in that fuperficies. VIII. See N. "A plane angle is the inclination of two lines to one another "in a plane, which meet together, but are not in the fame See N. "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 ftraight line. N B. Book I. (y) C N. B. When feveral 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 ftraight lines that contain the angle meet one another, is put between the other two letters, and one of thefe two is fomewhere upon one of those straight lines, and the other upon the other line: Thus the angle which is contained by the 'ftraight lines AB, CB, is named the angle ABC, or CBA; that ' which is contained by AB, BD is named the angle ABD, or " DBA; and that which is contained by BD, CB is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expreffed by a letter placed at that point; as the Angle at E.' X. it. XI. 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. "A term or boundary is the extremity of any thing." |