If one of the extremities of a straight line given in position and magnitude be given; the other extremity shall also be given. Let the point A be given, to wit, one of the extremities of a straight line given in magnitude, and which lies in the straight line AC given in position; the other extremity is also given. A D B C Because the straight line is given in magnitude, one * 1 Def. equal to it can be found a: let ti.is be the straight line D: From the greater straight line AC cut off AB equal to the lesser D: Therefore the other extremity B of the straight line AB is found: And the point B has always the same situation; because any other point in AC, upon the same side of A, cuts off between it and the point A a greater or less straight line than 4 Def. AB, that is, than D: Therefore the point B is given b: And it is plain another such point can be found in AC, produced upon the other side of the point A. b Ir a straight line be drawn through a given point parallel to a straight line given in position; that straight line is given in position. Let A be a given point, and BC a straight line given in position; the straight line drawn through A parallel to BC is given in position. Through A drawa the straight D a 31. 1. line DAE parallel to BC; the straight line DAE has always the same position, because no other B A E C straight line can be drawn through A parallel to BC: Therefore the straight line DAE which has been found is b4 Def. givenb in position. PROP. XXXII. Ir a straight line be drawn to a given point in a straight line given in position, and makes a given angle with it; that straight line is given in position. G F Let AB be a straight line given in position, and C a given point in it, the straight line drawn to C, which makes a given angle with CB, is given in position. Because the angle is given, one equal to it can be founda; A let this be the angle at D. At the given point C, in the given straight line AB, make the angle ECB equal to the angle at 29. E F B1 Def. D: Therefore the straight line EC has always the same situation, because any other straight line FC, drawn to the point C, makes with CB a greater or less angle than the angle ECB, or the angle at D: Therefore the straight line EC, which has been found, is given in position. It is to be observed, that there are two straight lines EC, GC, upon one side of AB that make equal angles with it, and which make equal angles with it when produced to the other side. b 23. 1. PROP. XXXIII. Ir a straight line be drawn from a given point to a straight line given in position, and make a given angle with it, that straight line is given in position. From the given point A let the straight line AD be drawn to the straight line BC given in position, and make with it a given angle ADC: ADE is given in position. Through the point A draw a the straight line EAF parallel to BC; and because through the B F C 30. a 31. 1. given point A the straight line EAF is drawn parallel to BC, which is given in position, EAF is therefore given in position. And because the straight line ADmeets the paral- 31 Dat. b e 29. 1. lels BC, EF, the angle EAD is equal to the angle ADC; and ADC is given, wherefore also the angle EAD is given: Therefore, because the straight line DA is drawn to the given point A in the straight line EF given in position, d 32 Dat. and makes with it a given angle EAD, AD is given in position. PROP. XXXIV. See N. Ir from a given point to a straight line given in position, a straight line be drawn which is given in magnitude; the same is also given in position. Let A be a given point, and BC a straight line given in position, a straight line given in magnitude, drawn from the point A to BC, is given in position. Because the straight line is given in magnitude, one * 1 Def. equal to it can be found a; let this be the straight line D: From the point A draw AE perpendicular to BC: and because AE is the shortest of all the straight lines which can be drawn from the point A to BC, the straight line D, to which one equal is to be drawn from the point A to BC, cannot be less than AE. If therefore D be equal to B D E AE, AE is the straight line given in magnitude drawn from the given point A to BC: And it is evident that AE b33 Dat. is given in position, because it is drawn from the given point A to BC, which is given in position, and makes with BC the given angle AEC. But if the straight line D be not equal to AE, it must be greater than it: Produce AE, and make AF equal to D; and from the centre A, at the distance AF, describe the circle GFH, and join AG, AH: Because the circle 6. Def. GFH is given in position, and the straight line BC is also d28 Dat. given in position; therefore their intersection G is givend; and the point A is given; wherefore AG is given in с 29 Dat. positione; that is, the straight drawn from the given point A A to the straight line BC given in position, is also given in position; And in like manner AH is given in position: Therefore in this case there are two straight lines AG, AH, of the same given magnitude which can be drawn from a given point A to a straight line BC given in posi tion. PROP. XXXV. Ir a straight line be drawn between two parallel straight lines given in position, and makes given angles with them, the straight line is given in magnitude. Let the straight line EF be drawn between the parallels AB, CD, which are given in position, and make the given angles BEF, EFD: EF is given in magnitude. EHB 32. a 31. 1. In CD take the given point G, and through G drawa GH parallel to EF: And because CD meets the parallels GH, EF, the angle EFD is equal to the angle HGD; ↳ 29. 1. And EFD is a given angle; wherefore the angle HGD is given: and because HG is drawn to the given point G, in the straight line CD, given in position, and makes a given angle HGD; the straight line HGC FG D C d is given in position: And AB is given in position: there- 32 Dat. fore the point H is givend: and the point G is also given, a 28 Dat. wherefore GH is given in magnitude: And EF is equal 29 Dat to it, therefore EF is given in magnitude. e If a straight line given in magnitude be drawn be- See N. tween two parallel straight lines given in position, it shall make given angles with the parallels. EHB Let the straight line EF given in magnitude be drawn between the parallel straight lines AB, A CD, which are given in position: The angles AEF, EFC, shall be given. Because EF is given in magnitude, C FK D'1 Def. G a straight line equal to it can be found"; EF cannot be less than HK: And if G be equal to HK, EF also is equal to it; wherefore EF is at right angles to CD; for if it be not, EF would be greater than HK, which is absurd. Therefore the angle EFD is a right, and consequently a given angle. But if the straight line G be not equal to HK, it must be greater than it: Produce HK, and take HL equal to G; and from the centre H, at the distance HL, describe the circle MLN, and join HM, HN: And because the 6 Def. circle MLN, and the straight line CD, are given in posi428 Dat. tion, the points M, N ared given: And the point H is given, wherefore the straight lines HM, HN, are given in * 29 Dat. position: And CD is given in position; therefore the angles HMN, HNM, are given 'A Def. in positionf: Of the straight lines HM, HN, let HN be that which is not parallel to H Ᏼ K OM ND G EF, for EF cannot be parallel to both of them: and draw 34. 1. EO parallel to HN: EO therefore is equals to HN, that is, to G; and EF is equal to G; wherefore EO is equal to 29. 1. EF, and the angle EFO to the angle EOF, that ish, to the given angle HNM, and because the angle HNM, which is equal to the angle EFO, or EFD, has been found; therefore the angle EFD, that is, the angle AEF, is given in * 1 Def. magnitudek; and consequently the angle EFC. See N. IF a straight line given in magnitude be drawn from a point to a straight line given in position, in a given angle; the straight line drawn through that point parallel to the straight line given in position, is given in position. Let the straight line AD given in magnitude be drawn In BC take a given point G, and draw is drawn to a given point G in the straight line BC |