Cor.—The square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter. Exercises. 1. Given the difference of the squares on two lines and their rectangle; find the lines. 2. Divide a given line, so that the rectangle contained by another given line and one segment may be equal to the square on the other segment. Questions for Examination on Book II. 1. What is the subject-matter of Book II.? Ans. Theory of rectangles. 2. What is a rectangle? A gnomon? 3. What is a square inch? A square foot? A square perch? A square mile? Ans. The square described on a line whose length is an inch, a foot, a perch, &c. 4. What is the difference between linear and superficial measurement? Ans. Linear measurement has but one dimension; superficial has two. 5. When is a line said to be divided internally? When externally? 6. How is the area of a rectangle found? 7. How is a line divided so that the rectangle contained by its segments may be a maximum? 8. How is the area of a parallelogram found? 9. What is the altitude of a parallelogram whose base is 65 metres and area 1430 square metres? 10. How is a line divided when the sum of the squares on its segments is a minimum? 11. The area of a rectangle is 108.60 square metres and its perimeter is 48.20 linear metres; find its dimensions. 12. What Proposition in Book II. expresses the distributive law of multiplication? 13. On what proposition is the rule for extracting the square root founded? 14. Compare I. xlvii. and II. xii. and xiii. 15. If the sides of a triangle be expressed by x2 + 1, x2 − 1, and 2x linear units, respectively; prove that it is right-angled. 16. How would you construct a square whose area would be exactly an acre? Give a solution by I. xlvii. 17. What is meant by incommensurable lines? Give an example from Book II. 18. Prove that a side and the diagonal of a square are incommensurable. 19. The diagonals of a lozenge are 16 and 30 metres respectively; find the length of a side. 20. The diagonal of a rectangle is 4.25 perches, and its area is 7.50 square perches; what are its dimensions? 21. The three sides of a triangle are 8, 11, 15; prove that it has an obtuse angle. 22. The sides of a triangle are 13, 14, 15; find the lengths of its medians; also the lengths of its perpendiculars, and prove that all its angles are acute. 23. If the sides of a triangle be expressed by m2 + n2, m2 − n2, and 2mn linear units, respectively; prove that it is right-angled. 24. If on each side of a square containing 5.29 square perches we measure from the corners respectively a distance of 1.5 linear perches; find the area of the square formed by joining the points thus found. Exercises on Book II.