IF which above a given magnitude has a given ratio 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. 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 A; ven; and AG is given, wherefore the whole G AH is given. and because as GB to CD, so is C b. 19. 5. GH to CF, and fo is b the remainder HB to the HF remainder FD; the ratio of HB to FD is given. 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 « Otherwise. Becaufe EB has a given ratio to C, and the E excess of C above a given magnitude has a given given magnitude has a given ratio to D. let this B C D therefore FB the excess of AB above the given magnitude AF has a given ratio to D.” IF point or points in which they cut one another are given. Let two lines AB, CD giver in position çut one another in the point E; the point E is given. Because the lines AB, CD are CH given in position, they have always the fame Gituation, and therefore A B a. 4. Def. the point, or points, in which they cut one another have always the D fame situation, and because the lines AB, CD can be found a, the point, A B or points, in which they cut one another, are likewise found; and C D therefore are given in position a. F the extremities of a straight line be given in position; the straight line is given in position and magnitude. Because the extremities of the straight line are given, they can be found a; let these be the points A, B, between which a straight a. 4. Delo line AB can be drawn b; this has an b. 1. Poftu. late. invariable position, because between A -B two given points there can be drawn but one straight line. and when the straight line AB is drawn, its magnitude is at the same time exhibited, or given. therefore the Itraight line AB is given in position and magnitude. IF F 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. Because the straight line is given in magnitude, one equal to it a. I. Def. can be found a ; let this be the straight line D. from the greater straight line AC cut off AB equal to the A. B c leffer D. therefore the other extremity B of the straight line AB is found, and. the point B has always the same fitua-D tion, because any other point in AC, upon the same fide of A, cuts off between it and the point A a greater or less straight line than AB, that is than D. therefore the b. 4. Def. 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. JF a straight line be drawn through a given point parallel to a straight line given in position; that straight line is given in position. &. 31. I, 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 parallel to BC; the straight line DAED A E has always the same position, because. B no other straight line can be drawn C through A parallel to BC. therefore the D. 4. Def. straight line DAE which has been found is given b in position. F E a. I. Def. PROP. XXXII. 29. IF F a straight 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 position. 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 F equal to it can be found a; let this be the angle at D. at the given point Α. C B C in the given straight line AB make the angle ECB equal to the b. 23. I. angle at D. therefore the straight line EC has always the same situa D tion, 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. PROP. XXXIII. . 30. IF a straight line be drawn from a given point, to a straight line given in position, 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 straight line BC given in position, and make with it a given angle ADC; AD is given in position. Thro’ the point A draw a the straight E A F a. 31. 1. line E AF parallel to BC; and because thro' the given point A the straight line EAF is drawn parallel to BC which is gi-B D ven in position, 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. 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 d in position. See N. IF tion, a straight 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. Because the straight line is given in magnitude, one equal to it 2. 1. Def. can be found * ; let this be the straight line D. from the point A draw AE perpendicular to BC; and because A, 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 E C less than AE. If therefore D be equal to AE, 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 position o be cause 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 its produce AE, and make AF equal to D; and from the cen ter A, at the distance AF describe the circle GFH, and join AG, c. 6. Def. AH. because the circle GFH is given in position, and the straight line BC is also given in position ; Α, therefore their interfection G is gid. 28. Dat. ven d; and the point A is given; B G E H C e. 29. Dat. wherefore AG is given in position , that is, the straight line AG given in Dmagnitude (for it is equal to D) and drawn from the given point 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, |