easy to show that pairs or sets of lines may be found, in endless variety, which have no common measur. , so that there are no finite numbers whatsoever that will express thein exactly. But suppose we have two incommensurable lines, A and B. By taking a linear unit sufficiently small, definite numbers a and b can always be found such that the lengths which they denote differ from A and B by quantities X and Y smaller than any that we choose to name; and the area of the rectangle which contains ab square units differs from that contained by A and B by a quantity less than any that we choose to name. Consequently, although all definite arithmetical calculations respecting such lines and the rectangles formed by them are incapable of absolute exactness, the error involved in treating the calculations as exact may always be made smaller than any quantity that The full discussion of this question, however, would take us too far for our present purpose. From the relation between the sides of a rectangle and its area certain obvious inferences niay be drawn. Thus if a represent the area of a rectangle and b the length of one side, the length of the other side is 6 The area of any parallelogram is equal to that of a rectangle having the same base and altitude. The area of a triangle is half that of a rectangle having the same base and altitude, or to that of a rectangle having the same altitude and a base of half the size. All rectangles are equal in area when the products of the numbers that represent their sides are equal. we car name. a. EXERCISES. [The numbers within the brackets indicate that no propo sition is to be used in the Exercise except some one or more of those which are so specified.] 1. A B is a finite right line divided into any two parts at C; D is the middle point of AC; E the middle point of BC. Find the middle point of A B. (I. 1-3.) 2. ABC is an equilateral A. D is a point in B C. Find a point E in A C, and a point F in A B, such, that if these three points be joined, D E F shall be an equilateral triangle. (I. 1-4.) 3. A B is a diameter of a circle, C is its centre, D is a point on the circumference. C and D, A and D, and B and D are joined. Show that in the A ABD the < at D is equal to the sum of the Zs at A and B. (I. 1-5.) 4. In the figure on p. 26, let O be the point where AL and F C intersect each other. Prove that A O and C O are equal, and that if B O be drawn, it bisects the < A B C. (I. 1-6.) 5. Construct an isosceles A, having given the base and the length of one of the sides. (I. 1-3.) 6. A B is a finite right line; Cis the middle point of it; C D is a line at right angles to A B. Prove that every point* in CD is equidistant from the points A and B. (I. 1-4.) 7. Construct an isosceles C., the base and the sum of the two sides being given. (I. 1-11.) 8. In the A ABC (p. 26), lines A K and CK are drawn bisecting the <s BAC and BCA, and meeting at K. Prove that A KC is an isosceles A. (I. 1-6.) * When every point in a line satisfies certain conditions, the line is said to be the locus of the point that satisfies those conditions. Thus, in this exercise, C D is the locus of a point equidistant from A and B. 9. A is a point in a given right line; B is a point outside the line. Find a point in the line equidistant from A and B. (I. 1-11.) 10. A B C is an angle; D is a point situated anywhere you please. Draw through D a line that shall cut off equal parts of A B and BC. (I. 1-11.) 11. Draw the line along which a rectangular leaf must be folded in order that the corner may coincide with a given point on the page. (I. 1-11.) 12. A B is a right line. A line, CD, is drawn from the point C in it in any direction. Show that the lines which bisect the As A CD and D C B are perpendicular to each other. (I. 1-13.) 13. Find a point which is equidistant from the three vertical points of a A. (I. 1-12.) 14. A ray of light, emitted from a given point, A, is reflected by the mirror B C. Show where the ray must strike the mirror in order to reach another given point, D.* (I. 1-15.) 15. Two isosceles triangles stand on the same base and on the same side of it. Prove that the line which joins their vertices, when produced, will bisect the base and be at right angles to it. (I. 1-8.) 16. The perpendicular is the shortest line that can be drawn from a given point to a given straight line. (I. 1-19.) 17. Of all the lines that can be drawn from a given point to a given straight line, those which make equal angles with the perpendicular are equal to each other, and of any other pair of lines that is the shorter which is the nearer to the perpendicular. (I. 1-19.) 18. Having given the segments of the base of a A formed by a line which bisects the vertical Z, and the difference between the other two sides, construct the A. (I. 1-9.) 19. A ray of light emitted from a given point is reflected by a mirror to another given point. Prove that the path which it takes is the shortest that it could take. (I. 1-20.) 20. A point passes from a given position, first to a given * The angle of incidence is equal to the angle of reflection. straight line, and then to a circle on the same side of that line. Determine the shortest path that it can take. (I. 1--20). 21 The two diagonals of any four-sided figure are together less than the sum of the four sides, and greater than half their sum. (I. 1-20.) 22. Construct a A, having given the base, one of the <s at the base, and the sum of the other two sides. (I. 1--23.) 23. If a right line be drawn cutting off any part of a polygon, the perimeter of the part that remains is less than that of the original polygon; and if one polygon be described within another, the perimeter of the inner polygon is less than that of the outer. (I. 1-20.) 24. Two equal straight lines bisect each other. Show that the two extremities of either are at the same distance from the other. (I. 1-26.) 25. If two exterior angles of a A are equal, the A is isosceles. (I. 1-13.) 26. If two straight lines are parallel to two others, the Zs which the lines form with each other are either equal or supplementary. (I. 1-29.) 27. Trisect a right angle. (I. 1–32.) 28. Take the figure in Prop. I. (p. 16). Produce A B to meet the right-hand circle in F. Join C and D, C and F, and D and F, and prove that CDF is an equilateral A. (I. 1-32.) 29. If lines be drawn bisecting two Zs of a A, the point where they meet is equidistant from all the three sides. (I. 1-26.) 30. If right lines be drawn through the vertices of a A parallel to the opposite sides, a figure will be formed which is divided into four s equal to each other in all respects. (I. 1-34.) 31. Construct an equilateral A, the sum of the three sides of which shall be equal to a given right line. (I. 1-32.) 32. If two parallel right lines be intersected by a right line, the lines which bisect the two interior angles on the same side are perpendicular to each other. (I. 1–32.) 33. Describe a rhombus in which two opposite angles shall be double the other two. (I. 1-34.) 34. Find a point in the base of a A, such that the two lines drawn from it parallel respectively to one side and meeting the other, shall be equal. (I. 1–34.) 35. Draw a straight line from one side of a A to another, so that it shall be parallel and equal to a given straight line. (I. 1-34.) 36. Prove that the diagonals of a parallelogram bisect each other. (I. 1-34.) 37. In a right-angled parallelogram the diagonals are equal. (I. 1-34.) 38. If the opposite sides of a quadrilateral are equal, it is a parallelogram. (I. 1-34.) 39. Describe a parallelogram, the area and perimeter of which shall be equal to those of a given A. 40. Of all As on the same base and between the same parallels, the isosceles A is that which has the least perimeter. (I, 1-37.) 41. If from any point in the diagonal of a parallelogram right lines be drawn to the opposite corners, the two As on one side of the diagonal are respectively equal to the two As on the other. (I. 1-38.) 42. The line joining the points of bisection of two sides of a Ais parallel to the base. (I. 1-39.) 43. If one side of a A be divided into any number of equal parts, and lines be drawn from the points of section parallel to the base, and meeting the other side, that side will be divided into the same number of equal parts as the other. (I. 1-34.) 44. Hence, show how to divide a given finite right line into any number of equal parts. 45. Construct a A, having given the base, the sum of the other two sides, and the vertical angle. (I. 1-32.) 46. A point is equally distant from two parallel straight lines. Prove that any line drawn through that point from N |