th of the entire circumference; and therefore, if chords equal to GC [i.] be placed round the circle, we shall have a regular polygon of fifteen sides, or quindecagon, inscribed in it.
Scholium.—Until the year 1801 no regular polygon could be described by constructions employing the line and circle only, except those discussed in this Book, and those obtained from them by the continued bisection of the arcs of which their sides are the chords; but in that year the celebrated Gauss proved that if 2n + 1 be a prime number, regular polygons of 2n + 1 sides are inscriptable by elementary geometry. For the case n = 4, which is the only figure of this class except the pentagon for which a construction has been given, see Note at the end of this work.
Questions for Examination on Book IV.
1. What is the subject-matter of Book IV.?
2. When is one rectilineal figure said to be inscribed in another?
3. When circumscribed?
4. When is a circle said to be inscribed in a rectilineal figure?
5. When circumscribed about it?
6. What is meant by reciprocal propositions? Ans. In reciprocal propositions, to every line in one there corresponds a point in the other; and, conversely, to every point in one there corresponds a line in the other.
7. Give instances of reciprocal propositions in Book IV.