19. IF PROP. XXVII. there be three magnitudes the excess 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 excefs of the first of which AB above the given magnitude AG, viz. GB has a given ratio to CD; and FD the excefs 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. Because the ratio of GB to CD is given, as GB to CD, so make GH to CF; therefore the ratio of GH to CF is 2. 2. Dat. given; and CF is given, wherefore a GH is gi b. 19. 5. A C ven; and AG is given, wherefore the whole G c. 9. Dat. and the ratio of FD to E is given, wherefore c the ratio of HB to E is given. and AH is given; B'D'E therefore HB the excefs of AB above the given magnitude AH has a given ratio to E. "Otherwife. 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 gi-A ven magnitude has a given ratio to D. Because EB has a given ratio to C, and the E excess of C above a given magnitude has a given 4. 24. Dat. ratio to D; therefore the excess of EB above a F given magnitude has a given ratio to D. let this therefore FB the excess of AB above the given magnitude AF has a given ratio to D." IF F 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 giver in position cut one another in the point E; the point E is given. A PROP. XXIX. C F the extremities of a ftraight line be given in pofition; the straight line is given in pofition and magnitude. Because the extremities of the ftraight line are given, they can 26. be found ; let these be the points A, B, between which a straight a. 4. De. line AB can be drawn b; this has an invariable position, because between A two given points there can be drawn b. 1. Poftu -B late. but one ftraight line. and when the ftraight line AB is drawn, its magnitude is at the fame time exhibited, or given. therefore the ftraight line AB is given in pofition and magnitude. 27. IF PROP. XXX. 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 straight line given in magnitude, and which lies in the straight line AC given in pofition; the other extremity is also given. Because the straight line is given in magnitude, one equal to it 2. 1. Def. 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 other extremity B ୯ upon the fame fide of A, cuts off between it and the point A a greater or lefs ftraight line than AB, that is than D. therefore the b. 4. Def. 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. a. 31. I. 28. IF PROP. XXXI. a ftraight line be drawn through a given point parallel to a straight line given in pofition; that ftraight line is given in pofition. Let A be a given point, and BC a straight line given in position; the straight line drawn thro' A parallel to BC is given in position. Thro' A draw a the straight line DAE has always the fame pofition, because B b A C b. 4. Def. straight line DAE which has been found is given ↳ in position. IF PROP. XXXII. Fa ftraight line be drawn to a given point in a given straight line, and makes a given 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 straight line drawn to C 29. C B b. 23. I. D b C in the given ftraight line AB make the angle ECB equal to the angle at D. therefore the straight line EC has always the fame fituation, because 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 pofition. It is to be obferved 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 side. IF PROP. XXXIII. a 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 pofition. From the given point A let the ftraight line AD be drawn to the straight line BC given in position, and make with it a given angle ADC; AD is given in pofition. Thro' the point A draw line EAF parallel to BC; and because 30. the ftraight E A Ta. 31. I. thro' the given point A the ftraight line EAF is drawn parallel to BC which is gi- B D ven in pofition, EAF is therefore given in position b. and because the ftraight line AD meets the parallels BC, b. 31. Dat. EF, the angle EAD is equal to the angle ADC; and ADC is c. 29. 1. C 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, and makes with it a given angle EAD; d. 32. Dat. AD is given din pofition. See N. 31. IF PROP. 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 also given in pofition. Let A be a given point, and BC a straight line given in pofition ; a straight line given in magnitude drawn from the point A to BC is given in position. A. Because the straight line is given in magnitude, one equal to it a. 1. Def. can be found; 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 B less than AE. If therefore D be equal to AE, D E C AE is the straight line given in magnitude drawn from the given b. 33. Dat. point A to BC. and it is evident that AE is given in pofition b be cause 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 ftraight line D be not equal to AE, it must be greater than it. produce AE, and make AF equal to D; and from the center A, at the distance AF defcribe the circle GFH, and join AG, c. 6. Def. AH. because the circle GFH is given in pofition, and the straight line BC is alfo given in pofition; therefore their interfection G is gi d. 28. Dat. ven ́d; and the point A is given; B G/E A H C that is, the straight line AG given in F D magnitude (for it is equal to D) and e. 29. Dat. wherefore AG is given in position *, drawn from the given point A to the straight 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 ftraight lines AG, |