Dem.—The same construction being made, since the arc BC is equal to CG, the angle BOC is equal to COG. Hence the sectors BOC, COG are congruent (see Observation, Proposition xxix., Book III.); therefore they are equal. In like manner the sectors COG, GOH are equal. Hence there are as many equal sectors as there are equal arcs; therefore the arc BH and the sector BOH are equimultiples of the arc BC and the sector BOC. In the same manner it may be proved that the arc EJ and the sector EPJ are equimultiples of the arc EF and the sector EPF; and it is evident, by superposition, that if the arc BH is greater than, equal to, or less than the arc EJ, the sector BOH is greater than, equal to, or less than the sector EPJ. Hence [V. Def. v.] the arc BC : EF :: sector BOC : sector EPF. The second part may be proved as follows:— Sector BOC = rectangle contained by the arc BC, and the radius of the circle ABC [xx. Ex. 14] and sector EPF = rectangle contained by the arc EF and the radius of the circle EDF; and since the circles are equal, their radii are equal. Hence, sector BOC : sector EPF :: arc BC : arc EF. Questions for Examination on Book VI. 1. What is the subject-matter of Book VI.? Ans. Application of the theory of proportion. 2. What are similar rectilineal figures? 3. What do similar figures agree in? 4. How many conditions are necessary to define similar triangles? 5. How many to define similar rectilineal figures of more than three sides? 6. When is a figure said to be given in species? 7. When in magnitude? 8. When in position? 9. What is a mean proportional between two lines? 10. Define two mean proportionals. 11. What is the altitude of a rectilineal figure? 12. If two triangles have equal altitudes, how do their areas vary? 13. How do these areas vary if they have equal bases but unequal altitudes? 14. If both bases and altitudes differ, how do the areas vary? 15. When are two lines divided proportionally? 16. If in two lines divided proportionally a pair of homologous points coincide with their point of intersection, what property holds for the lines joining the other pairs of homologous points? 17. Define reciprocal proportion. 18. If two triangles have equal areas, prove that their perpendiculars are reciprocally proportional to the bases. 19. What is meant by figures inversely similar? 20. If two figures be inversely similar, how can they be changed into figures directly similar? 21. Give an example of two triangles inversely similar. Ans. If two lines passing through any point O outside a circle intersect it in pairs of points A, A′; B, B′, respectively, the triangles OAB, OA′B′, are inversely similar. 22. What point is it round which a figure can be turned so as to bring its sides into positions of parallelism with the sides of a similar rectilineal figure. Ans. The centre of similitude of the two figures.