19. PROP. XXVII. F there be three magnitudes: The excefs of the first of which above a given magnitude has a given ratio to the fecond; and the excefs of the fecond above a given magnitude has also a given ratio to the third: The excess of the first above a given magnitude shall have a given ratio to the third. Let AB, CD, E be three magnitudes, the excess of the first of which AB above the given magnitude AG, viz. GB, has a given ratio to CD; and FD the excess of CD above the given magnitude CF, has a given ratio to E: The excess of AB above a given magnitude has a given ratio to E. ·H Because the ratio of GB to CD is given, as GB to CD, fo make GH to CF; therefore the ratio of GHAI 2. dat. to CF is given; and CF is given, wherefore a GH is given; and AG is given, wherefore G the whole AH is given: And because as GB b 19. 5. to CD, fo is GH to CF, and fo is b the remainder HB to the remainder FD; the ratio of HB to FD is given: And the ratio of FD c9. dat. to E is given, wherefore the ratio of HB to E is given And AH is given; therefore HB the excess of AB above a given magnitude AH has a given ratio to E. "Otherwife, B DE Let AB, C, D be three magnitudes, the excefs EB of the firft of which AB above the given magnitude AE has a given ratio to C, and the excefs of C above a given magnitude has a given ratio to D: The excess of AB above a given magnitude has a given ratio to D. E+ Because EB has a given ratio to C, and the excefs of C'above a given magnitude has a gi- F dat, ven ratio to D; therefore d the excefs of EB above a given magnitude has a given ratio to D: Let this given magnitude be EF; therefore FB the excels of EB above EF has a given ra- B C D tio to D: And AF is given, Because AE, EF are i are given: Therefore FB the excess of AB above a given magnitude AF has a given ratio to D.” two lines given in pofition cut one another, the See N. point or points in which they cut one another are given. Let two lines AB, CD given in pofition cut one another in the point E; the point E is gi they cut one another, are like wife found; and therefore are given in pofition a. F the extremities of a straight line be given in po Ifition, the straight line is given in polition and magnitude. b 1. Poftu Because the extremities of the ftraight line are given, they can be found a: Let these be the points A, B, between which a 4 def. a ftraight line AB can be drawn b; this has an invariable pofition, because between two given points there A B late, can be drawn but one straight line: And when the straight line AB is drawn, its magnitude is at the fame time exhibited, or given Therefore the ftraight line AB is given in pofition and magn ude. 27. IF PROP. XXX. F one of the extremities of a straight line given in pofition and magnitude be given; the other extremity fhall also be given. Let the point A be given, to wit, one of the extremities of a ftraight line given in magnitude, and which lies in the straight line AC given in pofition; the other extremity is alfo given. Because the straight line is given in magnitude, one equal a 1. def. to it can be found a; let this be the ftraight line D: From the greater ftraight line AC cut off AB equal to the leffer D: Therefore the A other extremity B of the straight line AB is found: And the point B has al-D ways the fame fituation; because any. B C other point in AC, upon the fame fide of A, cuts off between it and the point A a greater or less straight line than AB, that b 4. def. is, than D; Therefore the point B is given b: And it is plain another fuch point can be found in AC produced upon the other fide of the point A. @ 31. 1. 28. PROP. XXXI. Fa ftraight line be drawn through a given point I parallel to a straight line given in pofition; that ftraight line is given in pofition. Let A be a given point, and BC a ftraight line given in pofition; the ftraight line drawn through A parallel to BC is given in pofition. D Through A draw a the ftraight line DAE parallel to BC; the ftraight line DAE has always the fame pofition, because no other ftraight line B can be drawn through A parallel to A E BC Therefore the ftraight line DAE which has been found b 4. def. is given b in pofition. PROP. PROP. XXXII. F a ftraight line be drawn to a given point in a given in angle with it; that straight line is given in pofition. Let AB be a straight line given in pofition, and C a given point in it, the ftraight 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 found a; let this be the angle at D, at the given point C, in the given ftraight A line AB, make the angle ECB equal to the angle at D: There G 29. F E F a I. def. B b 23. I. fore the straight line EC has al ways the fame fituation, because D any other straight line FC, drawn to the point C, makes with CB a greater or lefs 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 ftraight lines EC, GC upon one fide of AB that make equal angles with it, and which make equal angles with it when produced to the other fide. PROB. XXXIII. Fa ftraight line be drawn from a given point to a ftraight line given in pofition, and makes 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 ftraight line BC given in pofition, and make with it a given angle ADC : AD is given in po- E fition. Thro' the point A,draw a the straight line EAF parallel to BC; and because thro' the given point A, the ftraight line EAF is drawn parallel to BC, B A which is given in pofition, EAF is therefore given in pofitionb: b 31. dat. And because the ftraight line AD meets the parallels BC, C 29, I. EF, the angle EAD is equal c to the angle ADC; and ADC is given, wherefore alfo the angle EAD is given: Therefore, because the straight line DA is drawn to the given point A in the ftraight line EF given in pofition, and makes with it a d 32. dat. given angle EAD, AD is given din pofition. 31. See N. a 1. def. PRO P. XXXIV. F from a given point to a straight line given in pofition; a ftraight line be drawn which is given in magnitude; the fame is alfo given in pofition. Let A be a given point, and BC a straight line given in pofition, a ftraight line given in magnitude drawn from the point A to BC is given in pofition. A Because the straight line is given in magnitude, one 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 ftraight lines which can be drawn from the point A to BC, the ftraight line D, to which one equal is to be drawn from the B point A to BC, cannot be less than AE. D E C If therefore D be equal to AE, AE is the ftraight line given in magnitude drawn from the given point A to BC: And it 33. dat. is evident that AE is given in pofition b, because it is drawn from the given point A to BC, which is given in pofition, 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 diftance AF, defcribe the circle GFH, and join AG, AH: Because the circle GFH is given in pofitionc, and the straight line BC is alfo given in pofition; therefore their interfection c 6. def. A d 28. dat. c 29. dat. G is given d; and the point A is (for it is equal to D) and drawn D from the given point A to the ftraight line BC given in pofition, is alfo given in pofition: And in like manner AH is given in pofition: Therefore in this cafe there are two straight lines |