contained by FEC (by prop. 15.) for they are vertical wherefore Book I. (by prop. 4.) a base the straight line AB is equal to a base the straight line CF and the triangle ABE is equal to the triangle FEC, and the remaining angles are equal to the remaining angles, under which the equal fides are extended, each to each wherefore the angle contained by BAE is equal to the angle contained by ECF : but the angle contained by ECD is greater (by com. not. 9.) than the angle contained by ECF: wherefore the angle contained by ACD is greater than the angle contained by BAE: in like manner also the straight line BC having been cut in halves, the angle contained by BCG, that is (by prop. 15.) the angle contained by ACD, is also greater than the angle contained by ABC. Wherefore one of the fides of any triangle being produced, the outward angle is greater than either of those two which are within and oppofite. Which was to be demonftrated. The two angles of any triangle are less than two right angles, being interchanged every way. Let there be a triangle the triangle ABC; I say that the two angles of the triangle ABC are less than two right angles, being interchanged every way. For let the straight line BC be produ ced to the point D. C A D And fince the angle contained by ACD is an outward angle of the triangle ABC, it is greater than the inward and oppofite, the angle contained by ABC: B let a common one be added, the angle contained by ACB: therefore the angles contained by ACD, ACB are greater (by com. not. 4.) than those contained by ABC, BCA; but the angles contained by ACD, ACB are equal to two right angles; therefore those contained ABC, BCA are lefs than two right angles. And in like manner we shall fhew that the angles contained by BAC, ACB are lefs than two right angles, as also thofe contained by CAB, ABC. Book I. Wherefore the two angles of any triangle are less than two right angles, being interchanged every way. PROP. XVIII. The greater fide of every triangle fubtends the greater angle. For because the straight line AC is greater than the straight line AB let the ftraight line AD be placed (by prop. 3,) equal to the straight line AB, and let the ftraight line BD be drawn. C D A B And fince the angle contained by ADB is an outward angle of a triangle, the triangle BDC, it is greater (by prop.16.) than the inward and oppofite, the angle contained by DCB. But the angle contained by ADB is equal to the angle contained by ABD (by prop. 5.), because the fide AB is equal to the fide AD: therefore also the angle contained by ABD is greater than the angle contained by ACB; wherefore the angle contained by ABC is much greater than the angle contained by ACB. Therefore the greater fide, of every triangle, fubtends the angle. Which was to be demonftrated. PROP. XIX. greater The greater fide of every triangle is extended under the greater angle. Let there be a triangle, the triangle ABC, having the angle contained by ABC greater than the angle contained by BCA; I fay that also the fide AC is greater than the fide AB. For if not certainly the ftraight line AC is equal to the straight line AB, or less: but indeed the ftraight line AC is not equal to the ftraight line AB, for then A alfo an angle, the angle contained by ABC would be equal (by prop. 5.) to the angle C B line AC is not equal to the ftraight line AB; neither is the ftraight Book L line AC less than the ftraight line AB; for if it were less, then (by prop. 18.) an angle the angle contained by ABC would be lefs than the angle contained by ACB: but (by fupp.) it is not therefore the ftraight line AC is not less than the straight line AB; but it has been demonstrated, that it is not equal to it; wherefore the fide AC is greater than the fide AB. Wherefore the greater fide of every triangle is extended under the greater angle. Which was to be demonftrated. PROP. XX. The two fides of every triangle are greater than the remaining fide, being interchanged every way. For let there be a triangle, the triangle ABC: I say that the two fides of the triangle ABC are greater than the remaining fide, being interchanged every way: viz. BA, AC greater than BC; and AB, BC greater than AC; laftly BC, CA greater than AB. For let the ftraight line BA be produced to the point D, and let the straight line DA be placed (by prop. 3.) equal to CA, and let the ftraight line DC be drawn. Since therefore the ftraight line DA is equal to the ftraight line AC, also B A C D (by prop. 5.) the angle contained by Book I. Therefore of every triangle the two fides are greater than the remaining fide, being interchanged every way. Which was to be demonftrated. PROP. XXI. If two straight lines be joined together within a triangle, upon one of its fides from the extremities of it, the joined lines fhall indeed be less than the two remaining fides of the triangle, but will contain a greater angle. For let two ftraight lines, the straight lines BD, DC be joined together within a triangle, the triangle ABC upon one of the fides the Straight line BC drawn from the extremities the points B, C ; I say that the straight lines BD, DC are indeed less than the two remaining fides of the triangle the Straight lines BA, AC; but contain a greater angle, the angle contained by BDC greater than the angle contained by BAC. For let the ftraight line BD be produced to the point E. A E D C cause the two fides of the triangle CED, the ftraight lines CE, ED are greater than the ftraight line CD, let the ftraight line DB a common one be added; therefore (by com. not. 4.) the ftraight lines CE, EB are greater than the ftraight lines CD, DB: but the Straight lines BA, AC have been fhewn to be greater than the ftraight lines BE, EC; therefore BA, AC are greater by much than BD, DC. Again, because the outward angle of every triangle is greater than the inward and oppofite (by prop. 16.): therefore the outward angle of the triangle CDE, the angle contained by BDC is greater than that contained by CED: For the fame reafon therefore alfo the outward angle of the triangle ABE, the angle con tained by CEB is greater than that contained by BAC: but the Book L angle contained by BDC has been demonstrated to be greater than that contained by CEB: therefore the angle contained by BDC is greater by much than that contained by BAC. If therefore two ftraight lines be joined together within a triangle, upon one of its fides from the extremities of it, the joined lines shall indeed be less than the two remaining fides of the triangle, but will contain a greater angle. Which was to be demonftrated. PROP. XXII. To describe a triangle of three straight lines, which are equal to three given straight lines: but it is neceffary that two of them be greater than the remaining one; being interchanged every way. Let the three given straight lines be A, B, C; of which let the two be greater than the remaining one, being interchanged every way, that is A and B greater than C: A and C greater than B; as alfo B and C greater than A: it required to describe a triangle of straight lines equal to A, B, C. Draw any straight line DE terminated at the point D but indifinite towards the point E; and (by prop. 3.) let DF be made equal to A, and FG equal to B; but GH equal to C and with the point F for a center, and at the distance FD let the circle DKL be describ D K E F G H LA B C ed; and again with the point G for a center, and at the distance GH let the circle KLH be defcribed; and draw KF, KG: I say that the triangle KFG hath been described of three ftraight lines equal to A, B, C. For because the point F is the center of the circle DKL, FD is equal to FK but FD is equal (by conft.) to A, therefore FK is equal to A. Again because the point G is the center of the circle LKH, (by def. 15.) GK is equal to GH; but GH is equal to C, therefore GK is equal to C: but FG is alfo equal to B; therefore the three straight lines KF, FG, GK are equal to the three A, B, C. There |