13. What portion of plane geometry forms the subject of the “First Six Books of Euclid’s Elements”? Ans. The geometry of the point, line, and circle. 14. What is the subject-matter of Book I.? 15. How many conditions are necessary to fix the position of a point in a plane? Ans. Two; for it must be the intersection of two lines, straight or curved. 16. Give examples taken from Book I. 17. In order to construct a line, how many conditions must be given? Ans. Two; as, for instance, two points through which it must pass; or one point through which it must pass and a line to which it must be parallel or perpendicular, &c. 18. What problems on the drawing of lines occur in Book I.? Ans. ii., ix., xi., xii., xxiii., xxxi., in each of which, except Problem 2, there are two conditions. The direction in Problem 2 is indeterminate. 19. How many conditions are required in order to describe a circle? Ans. Three; as, for instance, the position of the centre (which depends on two conditions) and the length of the radius (compare Post. iii.). 20. How is a proposition proved indirectly? Ans. By proving that its contradictory is false. 21. What is meant by the obverse of a proposition? 22. What propositions in Book I. are the obverse respectively of Propositions iv., v., vi., xxvii.? 23. What proposition is an instance of the rule of identity? 24. What are congruent figures? 25. What other name is applied to them? Ans. They are said to be identically equal. 26. Mention all the instances of equality which are not congruence that occur in Book I. 27. What is the difference between the symbols denoting congruence and identity? 28. Classify the properties of triangles and parallelograms proved in Book I. 29. What proposition is the converse of Prop. xxvi., Part I.? 30. Define adjacent, exterior, interior, alternate angles respectively. 31. What is meant by the projection of one line on another? 32. What are meant by the medians of a triangle? 33. What is meant by the third diagonal of a quadrilateral? 34. Mention some propositions in Book I. which are particular cases of more general ones that follow. 35. What is the sum of all the exterior angles of any rectilineal figure equal to? 36. How many conditions must be given in order to construct a triangle? Ans. Three; such as the three sides, or two sides and an angle, &c. Exercises on Book I. 1. Any triangle is equal to the fourth part of that which is formed by drawing through each vertex a line parallel to its opposite side. 2. The three perpendiculars of the first triangle in question 1 are the perpendiculars at the middle points of the sides of the second triangle. 3. Through a given point draw a line so that the portion intercepted by the legs of a given angle may be bisected in the point. 4. The three medians of a triangle are concurrent.