18. The sum of the squares on lines drawn from any point to one pair of opposite angles of a rectangle is equal to the sum of the squares on the lines from the same point to the remaining pair. 19. Divide the hypotenuse of a right-angled triangle into two parts, such that the difference between their squares shall be equal to the square on one of the sides. 20. From the extremities of the base of a triangle perpendiculars are let fall on the opposite sides; prove that the sum of the rectangles contained by the sides and their lower segments is equal to the square on the base. PROP. XLVIII.—Theorem. If the square on one side (AB) of a triangle be equal to the sum of the squares on the remaining sides (AC, CB), the angle (C) opposite to that side is a right angle. Dem.—Erect CD at right angles to CB [xi.], and make CD equal to CA [iii.]. Join BD. Then because AC is equal to CD, the square on AC is equal to the square on CD: to each add the square on CB, and we have the sum of the squares on AC, CB equal to the sum of the squares on CD, CB; but the sum of the squares on AC, CB is equal to the square on AB (hyp.), and the sum of the squares on CD, CB is equal to the square on BD [xlvii.]. Therefore the square on AB is equal to the square on BD. Hence AB is equal to BD [xlvi., Ex. 1]. Again, because AC is equal to CD (const.), and CB common to the two triangles ACB, DCB, and the base AB equal to the base DB, the angle ACB is equal to the angle DCB; but the angle DCB is a right angle (const.). Hence the angle ACB is a right angle. The foregoing proof forms an exception to Euclid’s demonstrations of converse propositions, for it is direct. The following is an indirect proof:—If CB be not at right angles to AC, let CD be perpendicular to it. Make CD = CB. Join AD. Then, as before, it can be proved that AD is equal to AB, and CD is equal to CB (const.). This is contrary to Prop. vii. Hence the angle ACB is a right angle. Questions for Examination on Book I. 1. What is Geometry? 2. What is geometric magnitude? Ans. That which has extension in space. 3. Name the primary concepts of geometry. Ans. Points, lines, surfaces, and solids. 4. How may lines be divided? Ans. Into straight and curved. 5. How is a straight line generated? Ans. By the motion of a point which has the same direction throughout. 6. How is a curved line generated? Ans. By the motion of a point which continually changes its direction. 7. How may surfaces be divided? Ans. Into planes and curved surfaces. 8. How may a plane surface be generated. Ans. By the motion of a right line which crosses another right line, and moves along it without changing its direction. 9. Why has a point no dimensions? 10. Why has a line neither breadth nor thickness? 11. How many dimensions has a surface? 12. What is Plane Geometry?