Book I. Rectilineal figures are those which are contained by straight lines. XVI. Trilateral figures, or triangles, by three straight lines. XVII. Quadrilateral, by four straight lines. XVIII. Multilateral figures, or polygons, by more than four straight lines. XIX. Of three-sided figures, an equilateral triangle is that which has three equal sides. XX. An isosceles triangle is that which has only two sides equal. A scalene triangle, is that which has three unequal sides. XXII. A right angled triangle, is that which has a right angle. XXIII. An obtuse angled triangle, is that which has an obtuse angle. ོ XXIV. An acute angled triangle, is that which has three acute Book I. angles. XXV. Of four sided figures, a square is that which has all its sides equal, and all its angles right angles. XXVI. An oblong, is that which has all its angles right angles, but has not all its sides equal. XXVII. A rhombus, is that which has all its sides equal, but its angles are not right angles. XXVIII. 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. XXIX. All other four-sided figures besides these, are called Trapeziums. XXX. Book I. Straight lines, which are in the same plane, and being produced ever so far both ways, do not meet, are called Parallel Lines. POSTULATES. I. 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. N. THINGS which are equal to the same thing are equal to one another. 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 doubles of the same thing, are equal to Book I. one another. VII. Things which are halves of the same thing, are equal to one another. VIII. Magnitudes which coincide with one another; that is, which exactly fill the same space, are equal to one another. IX. The whole is greater than its part. X. All right angles are equal to one another. XI. "Two straight lines which intersect one another, cannot "be both parallel to the same straight line.” Book I. late. PROPOSITION I. PROBLEM. To describe an equilateral triangle upon a given finite straight line. Let AB be the given straight line; it is required to describe an equilateral triangle upon it. From the centre A, at the distance AB,de a 3. Postu- scribe the circle BCD, b. 1. Post. straight lines CA, nition. CB to the points A, B; ABC is an equilateral triangle. Because the point A is the centre of the circle BCD, c. 11. Defi- AC is equal to AB; and because the point B is the centre of the circle ACE, BC is equal to AB: But it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB; now, things which are d1. Axiom. equal to the same are equal to one another; therefore CA is equal to CB; wherefore CA, AB, CB are equal to one another; and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB. Which was required to be done. a 1. Post. b 1. 1. PROP. II. PROB. From a given point to draw a straight line equal to a given straight line. 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. From the point A to B draw the straight line AB; and upon it describe the equilateral triangle DAB, b |