XV. A circle is a plane 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. XVI. And this point is called the centre of the circle. XVII. A diameter of a circle is a ftraight line drawn through the see N. centre, 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. 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 ftraight lines. XXII. Multilateral figures, or polygons, by more than four ftraight 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 XXVI. Book I. ΔΔΔ XXVI. A scalene triangle, is that which has three unequal fides. A right angled triangle, is that which has a right angle. An obtuse angled triangle, is that which has an obtuse angle. A XXIX. An acute angled triangle, is that which has three acute angles. XXX. Of four fided figures, a fquare is that which has all its fides. equal, and all its angles right angles. XXXI. An oblong, is that which has its angles right angles, but has not all its fides equal as all its fides equal, but its angles XXX I A rhombus, is that which: Sce N. A rhomboid, is that which has its oppofite fides equal to one another, but all its fides are notequal, nor its angles right angles. XXXXIV, 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. LE 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 straight line. III. And that a cirele may be defcribed from any centre, at any diftance from that centre. AXIOM S. I. HINGS which are equal to the fame are equal to one an- TH 11. 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. VIII. Magnitudes which coincide with one another, that is, which exactly fill the fame space, are equal to one another. IX. "If a ftraight line meets two ftraight lines, fo as to make the "two interior angles on the fame fide of it taken together "less than two right angles, these straight lines being con"tinually produced, fhall at length meet upon that fide on "which are the angles which are lefs than two right angles. "See the notes on Prop. 29. of Book I." PROPO. " T PROPOSITION I. PROBLEM. O defcribe an equilateral triangle upon a given fi- Let AB be the given straight line; it is required to defcribe an equilateral triangle upon it. a From the centre A, at the diftance AB, defcribe the circle BCD, and from the centre B, at the diftance BA, defcribe the circle ACE; and from the point C, in which the circles cut one another, draw the ftraight lines b CA, CB to the points A, B; ABC fhall be an equilateral triangle. :15 Book I. a. 3. Pofiulate. finition. Because the point A is the centre of the circle BCD, AC is equal to AB; and because the point B is the centre of the c. 15th Decircle ACE, 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 d; therefore CA is equal to CB; wherefore CA, AB, d. 1ft AxiBC are equal to one another; and the triangle ABC is there-om. fore equilateral, and it is defcribed upon the given ftraight line AB. Which was required to be done. PROP. II. PROB. ROM a given point to draw a ftraight line equal to FR Let A be the given point, and BC the given ftraight line; it is required to draw from the point A a ftraight line equal to BC. From the point A to B draw the ftraight line AB; and upon it defcribeb the equilateral triangle DAB, and produce the ftraight lines DA, DB, to E and F; from the centre B, at the distance BC, defcribed the circle CGH, and from the centre D, at the distance DG, defcribe the circle GKL. AL hall be equal to BC. D K H a. 1. Poft. b. I. I. c. 2. Polt. d. 3. Poft. G E Because |