THE ELEMENTS OF EUC L LID. BOOK I. DEFINITION S. I. A Point is that which hath no parts, or which hath no magnitude. See Not: 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. VI. The extremities of a fuperficies are lines. VII. A plane fuperficies is that in which any two points being taken, See N. the straight line between them lies wholly in that fuperficies. VIII. "A plane angle is the inclination of two lines to one another in a See N. 66 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 line. A Book I. A ته E B '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 ver "tex 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 fomewhere 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.' X. When a ftraight line standing on another is called a perpendicular to it. XI. An obtufe angle is that which is greater than a right angle. XII. An acute angle is that which is lefs than a right angle. XIII. "A term or boundary, is the extremity of any thing." XIV. A figure is that which is inclofed by one or more boundaries. C e 15 XV. A circle is a plane figure contained by one line, which is called the circumference, and is fuch that all ftraight 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 center of the circle. XVII. A diameter of a circle is a ftraight line drawn thro' the center, and see N, terminated both ways by the circumference. XVIII. A femicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A fegment of a circle is the figure contained by a straight line " and the circumference it cuts off." XX. Rectilineal figures are thofe 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 fided figures, an equilateral triangle is that which has three equal fides. XXV. An ifofceles triangle, is that which has only two fides equal Book I. See N. AAA XXVI. A fcalene triangle, is that which has three unequal fides. XXVII. A right angled triangle, is that which has a right angle. An obtufe angled triangle, is that which has an obtuse angle. XXIX. An acute angled triangle, is that which has three acute angles. Of four fided figures, a fquare is that which has all its fides XXXI. An oblong is that which has all its angles right angles, but has not all its fides equal. XXXII. A rhombus is that which has all its fides equal, but its angles, are not right angles. 00 XXXIII. A rhomboid is that which has its oppofite fides equal to one another, but all its fides are not equal, nor its angles right angles. XXXIV. All other four fided figures befides thefe, are called Trapeziums. XXXV. Parallel ftraight lines, are fuch as are in the fame plane, and which being produced ever fo far both ways, do not meet. Book I. POSTULATES. I.. ET it be granted that a straight line may be drawn from II. That a terminated ftraight line may be produced to any length in a ftraight line. III. And that a circle may be defcribed from any center, at any diftance from that center. AXIOM S. I. HINGS which are equal to the fame are equal to one 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 fame, are equal to one another, VII. Things which are halves of the fame, are equal to one another. Magnitudes which coincide with one another, that is which exact |