XV. A circle is a plain figure contained by one line, which is called the circumference, and is fuch that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. Book I. N XIX 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, See N. and 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. "A fegment 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 ftraight lines. XXII. Quadrilateral, by four ftraight 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. ΑΔΑ XXIV XXVI. A scalene 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. AAA 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 See N. 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. Parallel ftraight lines are fuch as are in the fame plane, and Book I. L POSTULATE S. I. ET it be granted that a straight line may be drawn from That a terminated ftraight line may be produced to any length in a straight line. III. And that a circle may be described from any center, at any distance from that center. THINGS 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 same, 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 "If a straight line meets two straight lines, fo as to make the "two interior angles on the fame fide of it taken together less "than two right angles, thefe ftraight lines being continually "produced fhall at length meet upon that fide on which are "the angles which are less than two right angles. See the notes on Prop. 29. of Book I." T PROPOSITION I. PROBLEM. O defcribe an equilateral triangle upon a given Let AB be the given straight line, it is required to describe an equilateral triangle upon it. From the center A, at the di stance AB describe the circle BCD. and from the center B, at Book I. a. 3d Poftu late. the distance BA describe the circle D A ACE; and from the point C in BE which the circles cut one another draw the straight lines b CA, CB to the points A, B. ABC fhall be an equilateral triangle. Because the point A is the center of the circle BCD, AC is equal b. 2d Poft. finition. to AB. and because the point B is the center of the circle ACE, c. 15th De BC is equal to BA. but it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB. but things which are equal to the fame are equal to one another; therefore d. 1ft AxiCA is equal to CB. wherefore CA, AB, BC are equal to one another. and the triangle ABC is therefore equilateral, and it is defcribed upon the given straight line AB. Which was required to be done. ROM a given point to draw a ftraight line equal FRO Let A be the given point, and BC the given straight line; it is required to draw from the point A a straight line equal to BC. om. n a. 1. Poft. K H b. I. I. c. 2. Poft. d. 3. Poft. E |