Book I. hath been drawn at right angles to the given ftraight line the straight line AB, from any point given in it the point C. A F D C E B For fince the straight line CD is equal to the ftraight line CE (by conft.) and the straight line CF common; certainly the two DC, CF are equal to the two EC, CF each to each; and a base the ftraight line DF is equal (by conft.) to a base the ftraight line EF: wherefore (by prop. 8.) an angle the angle contained by DCF is equal to an angle the angle contained by ECF, and they are adjacent or confequent: but when a ftraight line ftanding upon a straight line makes the adjacent angles equal to one another; each of the equal angles is a right angle: therefore each of the angles contained by DCF, FCE is a right angle. Wherefore a straight line the straight line FC hath been drawn at right angles to the given straight line the ftraight line AB, from any point given in it the point C. Which was to be done. To draw a perpendicular straight line upon any given indefinite straight line, from any given point, which is not in it. Let the one, the given indefinite straight line, be the straight line AB, and the other, the given point, which is not in it; be the point C; and it is required upon the given indefinite straight line, the ftraight line AB, from the given point, the point C, which is not in it, to draw a perpendicular straight line. For on the other fide of the straight line AB take any point which you may accidentally happen upon as the point D and with the point C indeed for a center, but at the distance CD let a circle. be described, the circle FGE; and let the ftraight line FG be cut in halves (by prop. 10.) in the point H, and let the straight lines CF, CH, CG be drawn (by poft. 1.): I fay that upon the given indefinite ftraight line, the straight line AB, from the given point, the point C, which is not in it, a perpendicular hath been drawn, the Straight line CH. For the angle contained by CHF is equal to an angle, the angle contained by GHC, and they are adjacent: but when a straight line standing upon a ftraight line makes the adjacent angles equal to one another; each of the equal angles is a right angle: and the standing straight line is called a perpendicular to that on which it stands. Therefore upon the given indefinite ftraight line the straight line AB, from the given point, the point C, which is not in it, a perpendicular hath been drawn, the ftraight line CH. Which was to be done. Whenever a straight line standing upon a straight line makes angles it will make either two right angles, or angles equal to two right angles. For let any ftraight line, the straight line AB, standing upon a ftraight line the ftraight line CD, make angles, those contained by CBA, ABD: I say that the angles contained by CBA, ABD are either two right angles, or equal to two right angles. If indeed the angle contained by CBA be equal to the angle contained by ABD, they are (by def. 10.) two right angles: but if not, let the ftraight line BE be drawn from the point B (by prop. 11.) at right angles to the ftraight line CD: therefore the angles contained by CBE, D EBD are two right angles: and fince the E A B C angle contained by CBE is equal to two, the angles contained by CBA, ABE; let the angle contained by EBD, a common one be added: Book I. added: therefore (by com. not. 2.) the angles contained by CBE, EBD are equal to three, the angles contained by CBA, ABE, EBD. Again, because the angle contained by DBA is equal to two, the the angles contained by DBE, EBA; let the angle contained by ABC a common one be added: therefore (by com. not. 2.) the angles, the angles contained by DBA, ABC are equal to three, the angles contained by DBE, EBA, ABC: but the angles contained by CBE, EBD have also been proved equal to the fame three: but magnitudes which are equal to the fame magnitude, are equal to one another and the angles contained by CBE, EBD are therefore equal to the angles contained by DBA, ABC: but the angles contained by CBE, EBD are two right angles, and the angles contained by DBA, ABC are therefore equal to two right angles. Whenever therefore a straight line standing upon a straight line makes angles; it will make either two right angles, or angles equal to two right angles. Which was to be demonstrated. PROP. XIV. If, with any straight line, and at the fame point in it, two straight lines, not lying towards the fame parts make the adjacent angles equal to two right angles, the ftraight lines fhall be in a ftraight line with one another. For with any straight line, the ftraight line AB, and at the fame point in it, the point B, let two straight lines, the straight lines BC, BD not lying towards the fame parts, make the adjacent angles, the angles contained by ABC, ABD equal to two right angles: I fay that the straight line BD is in a straight line with the ftraight line CB. For if the straight line BD is not in a straight line with the fraight line CB; let the ftraight line BE be in a straight line with the straight line CB. Because therefore a straight line, A E B D fup.) fup.) equal to two right angles: therefore the angles contained by Book I. CBA, ABE are (by com. not. 1.) equal to the angles contained by CBA, ABD; let what is common, the angle contained by ABC be taken away, therefore what remains the angle contained by ABE is equal (by com. not. 3.) to what remains the angle contained by ABD, the less to the greater, which is impoffible: therefore the straight line BE is not in a straight line with the straight line BC: certainly in the fame manner we shall fhew that neither any other but the ftraight line BD is: therefore the straight line CB is in a ftraight line with the ftraight line BD. Wherefore if, with any straight line, and at the same point in it, two straight lines, not lying towards the fame parts, make the adjacent angles equal to two right angles, the straight lines fhall be in a ftraight line with one another. Which was to be demonstrated. PRO P. XV. If two straight lines cut one another they shall make the angles at the vertex equal to each other. For let two ftraight lines, the ftraight lines AB, CD cut one another at the point E: I say that the one the angle contained by AEC is equal to the angle contained by DEB and the other contained by CEB to the angle contained by AED. C B A E D For fince a straight line, the straight line AE ftands upon a straight line the ftraight line CD, making angles, the angles contained by CEA, AED: therefore the angles contained by CEA, AED are equal to two right angles (by prop. 13.): again, because a straight line, the straight line DE ftands upon a ftraight line, the Straight line AB, making angles the angles contained by AED, DEB: therefore the angles contained by AED, DEB are equal to two right angles: but also the angles contained by CEA, AED have been demonftrated to be equal to two right angles: therefore (by com. not. 1.) the angles contained by CEA, AED are equal to the angles contained by AED, DEB: let the angle contained by AED VOL. I. which C Book I. which is common be taken away, wherefore (by com. not. 3.) the angle contained by CEA which remains is equal to the angle contained by BED which remains: certainly in the fame manner it will be fhewn, that also the angles contained by CEB, DEA are equal. If therefore two straight lines cut one another they shall make the angles, at the vertex equal to each other. Which was to be de-. monftrated. Corollary. Certainly it is manifest from this, that also whatever number of straight lines cut one another, they will make the angles at the fection equal to four right angles. One of the fides of any triangle being produced, the outward angle is greater than either of those two which are within and oppofite. Let there be a triangle, the triangle ABC, and let one fide of it the straight line BC be produced to the point D: I say that the angle without, the angle contained by ACD is greater than either of those two which are within and oppofite, the angles contained under CBA, BAC. Let the straight line AC be cut in halves in the point E (by prop. 10.); and the straight line, BE being drawn (by poft. 1.) let it (by post. 2.) be produced to the point F, and let the straight line EF be placed (by prop. 3.) equal to the ftraight line BE and let the straight line CF be drawn (by poft. 1.), and let the ftraight line. AC be produced to G. |