* 1 Def. * 12. 1. * 6 Def. between the parallel straight lines * C FK D G Because EF is given in magnitude, a straight line equal to it can be found *; let this be G: in AB take a given point H, and from it draw HK perpendicular to CD: therefore the straight line G, that is, 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 circle * MLN, and the straight line * 28 Dat. CD, are given in position, the points M, N are given: and the point H is given, wherefore the A straight lines HM, HN, * 29 Dat. * A. Def. * 34. 1. * 29. 1. * 1 Def. E.. See N. * H В G : * HN, let HN be that which is not parallel to EF, for EF cannot be parallel to both of them and draw EO parallel to HN: EO therefore is equal to HN, that is, to G; and EF is equal to G; wherefore EO is equal to EF, and the angle EFO to the angle EOF, that is, 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 magnitude*; and consequently the angle EFC. PROP. XXXVII. 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 from the point A, to the straight line BC given in position, E AHF In BC take a given point G, and B DG C draw GH parallel to AD: and be * 29. 1.' 32 Dat. 34. 1. cause HG is drawn to a given point G in the straight line BC given in position, in a given angle HGC, for it is equal to the given angle ADC; HG is given in * position: but it is given also in magnitude, because it is equal to * AD, which is given in magnitude: there- * fore because G, one of the extremities of the straight line GH, given in position and magnitude, is given, the other extremity H is given *; and the straight line * 30 Dat. EAF, which is drawn through the given point H parallel to BC given in position, is therefore given in * 31 Dat. position. PROP. XXXVIII. * If a straight line be drawn from a given point to two parallel straight lines given in position; the ratio of the segments between the given point and the parallels shall be given. Let the straight line EFG be drawn from the given point E to the parallels AB, CD, the ratio of EF to EG is given. 34. From the point E draw EHK perpendicular to CD; and because from a given point E the straight line EK is drawn to CD which is given in position, in a given angle EKC; EK is given in position *; and AB, CD, * 33 Dat. are given in position; therefore the points H, K * 28 Dat. * 29 Dat. * 1 Dat. 35, 36. See N. are given and the point E is given; wherefore* EH, EK are given in magnitude, and the ratio of them is therefore given. But as EH to EK, so is EF to EG, because AB, CD, are parallels; therefore the ratio of EF to EG is given. PROP. XXXIX. If the ratio of the segments of a straight line between a given point in it and two parallel straight lines be given; if one of the parallels be given in position, the other is also given in position. From the given point A, let the straight line AED be drawn to the two parallel straight lines FG, BC, and let the ratio of the segments AE, AD, be given; if one of the parallels BČ be given in position, the other FG is also given in position. From the point A draw AH perpendicular to BC, and let it meet FG in K; and because AH is drawn from the given point A to the straight line BC given in position, and makes a given angle AHD; AH is * 2 Dat. * 30 Dat. B D H C given in position; and BC the point A is also given; where- B fore AH is given in magni tude *, and, because FG, BC, are parallels, as AE to AD, so D HC K G is AK to AH; and the ratio FE parallel to BC which is given in position, therefore* • 31 Dat. FG is given in position. If the ratio of the segments of a straight line into which See N. it is cut by three parallel straight lines be given; if two of the parallels be given in position, the third is also given in position. Let AB, CD, HK, be three parallel straight lines, of which AB, CD are given in position; and let the ratio of the segments GE, GF, into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in position. In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N: because LM is drawn from the given point L to CD which is given in position, and makes a given angle LMD; LM is given in position *; and CD is given in position, where- * 33 Dat. fore the point M is given*; and the point L is given; LM * 28 Dat. is therefore given in magnitude*; and because the ratio 29 Dat. of GE to GF is given, and as GE to GF, so is NL to NM; the ratio of NL to NM is given; and therefore* the ratio of ML to LN is given; but ML is given in magnitude, wherefore LN is given in magnitude: and it is also given in position, and the point L is given, wherefore the point N is given: and because the straight line HK is drawn through the given point N parallel to CD, which is given in position, therefore HK is given in position *. PROP. XLI. If a straight line meet three parallel straight lines which See N. are given in position; the segments into which they cut it have a given ratio. Let the parallel straight lines AB, CD, EF, given 33 Dat. in position, be cut by the straight line GHK; the ratio of GH to HK is given. A GL B H/ M D CH/ In AB take a given point L, and draw LM perpendicular to CD, meeting EF in N; therefore* LM is given in position: and CD, EF, are given in position, wherefore the points M, N, are given and the point L is *29 Dat. given; therefore the straight lines LM, MN, are given in magnitude; and the ratio of LM to MN is therefore given*: but as LM to MN, so is GH to HK; wherefore the ratio of GH to HK is given. 39. 1 Dat. See N. 22.1. *8. 1. *1 Def. 1 Dat. * 3 Def. 40. If each of the sides of a triangle be given in magnitude; the triangle is given in species. Let each of the sides of the triangle ABC be given in magnitude, the triangle ABC is given in species. A D Make a triangle* DEF, the sides of which are equal, each to each, to the given straight lines AB, BC, CA; which can be done, because any two of them must be greater than the third; and let DE be equal to AB, EF to BC, and FD to CA; and because the two sides ED, DF, are equal to the two BA, AC, each to each, and the base EF equal to the base BC; the angle EDF is equal to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given *: in like manner the angles at B, C, are given. And because the sides AB, BC, CA, are given, their ratios to one another are given *; therefore the triangle ABC is given in species. * PROP. XLIII. If each of the angles of a triangle be given in magnitude; the triangle is given in species. Let each of the angles of the triangle ABC be given in magnitude, the triangle ABC is given in species. |