If there be three magnitudes, the excess of the first of which above a given magnitude has a given ratio to the second; and the excess of the second 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. A G C F 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 given; and CF is given, wherefore (2. dat.) GH is given: and AG is given, wherefore the whole AH is given: and because as GB to CD, so is GH to CF, and so is (19. 5.) the remainder HB to the remainder FD; the ratio of HB to FD is given: and the ratio of FD to E is given, wherefore (9. dat.) 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. H B Ꭰ E "Otherwise, Let AB, C, D, be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a given ratio to C, and the excess of C above a given magnitude has a given ratio to D: the excess A of AB above a given magnitude has a given ratio to D. E Because EB has a given ratio to C, and the excess of C above a given magnitude has Fa given ratio to D; therefore (24. dat.) the excess of EB above a given magnitude has a given ratio to D: let this given magnitude be B EF; therefore FB, the excess of EB above EF, has a given ratio to D: and AF is given, because AE, EF are given therefore FB, the excess of AB above a given magnitude AF, has a given ratio to D." Ir two lines given in position cut one another, the point or points in which they cut one another are given. * Let two lines AB, CD given in position cut one another in the point E; the point E is given. C Because the lines AB, CD are given in position, they have always the same situation (4. def.), and therefore the point, or points, in which they cut one another, have always the same situation : and because the lines AB, CD can be found (4. def.), the point, or points, in which they cut one another, are likewise found; and therefore are given in position (4. def.). A If 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 (4. def.): let these be the points A, B, between which a straight line AB can be drawn (1. postulate.); this has an invariable po- Asition, because between two given -B 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 straight line AB is given in position and magnitude: 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 B C Because the straight line is given in magnitude, one equal to it can be found (1. def.); let this 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 D 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 AB, that is, than D; therefore the point B is given (4. def.): and it is plain another such point can be found in AC, produced upon the other side of the point A. If 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 posi tion; the straight line drawn through a parallel to BC is given in position. Through A draw (31. 1.) the straight line DAE parallel to BC; the straight D line DAE has always the same position, because no other straight line can be. drawn through A parallel to BC: there- B fore the straight line DAE, which has been found, is given (4. def.) in position. Α E C IF 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 found (1. def.); let this be the angle at D: at the given point C, in the given straight line AB, make (23. 1.) the angle ECB equal to the angle at D: therefore the straight line EC has always the same situation, because any other A D E F 1 B 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. Ir 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. E Through the point A, draw (31. 1.) A F the straight line EAF parallel to BC; and because through the given point A, the straight line EAF is drawn parallel B to BC, which is given in position, EAF is therefore given in position (31. dat.): and because the straight line AD meets the paral lels, BC, EF, the angle EAD is equal (29. 1.) 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, and makes with it a given angle EAD, AD is given (32. dat.) in position. IF 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 posi tion; 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 can be found (1 def.); 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 AE, AE is the straight line given in magnitude, drawn from the given point A to BC: and it is evident that AE is given in position, (33. dat.), because it is drawn from the given point A to BC, which is given in position, and makes with BC the given angle AEC. D. A E C 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 GFH is given in position (6. def.), and the straight line BC is also given in position; therefore their intersection G is given (28. dat.); and the point A is given; wherefore AG is given in position (29. dat."), that is, the straight B line AG given in magnitude, (for it is equal to D) and drawn from the given point A to the straight line BC given in position, is also G E H C F given in position: and in like manner AH is given in position: therefore in this case there are two straight lines AG, AH of * See Note. |