As to the mensuration of the circle it is well for us to take a broad view before coming down to details. There are only four leading propositions necessary for the mensuration of the circle and the determination of the value of [pi]. These are as follows: (1) The inscribing of a regular hexagon, or any other regular polygon of which the side is easily computed in terms of the radius. We may start with a square, for example, but this is not so good as the hexagon because its side is incommensurable with the radius, and its perimeter is not as near the circumference. (2) The perimeters of similar regular polygons are proportional to their radii, and their areas to the squares of the radii. It is now necessary to state, in the form of a postulate if desired, that the circle is the limit of regular inscribed and circumscribed polygons as the number of sides increases indefinitely, and hence that (2) holds for circles. (3) The proposition relating to the area of a regular polygon, and the resulting proposition relating to the circle. (4) Given the side of a regular inscribed polygon, to find the side of a regular inscribed polygon of double the number of sides. It will thus be seen that if we were merely desirous of approximating the value of [pi], and of finding the two formulas _c_ = 2[pi]_r_ and _a_ = [pi]_r_^2, we should need only four propositions in this book upon which to base our work. It is also apparent that even if the incommensurable cases are generally omitted, the notion of _limit_ is needed at this time, and that it must briefly be reviewed before proceeding further. There is, however, a much more worthy interest than the mere mensuration of the circle, namely, the construction of such polygons as can readily be formed by the use of compasses and straightedge alone. The pleasure of constructing such figures and of proving that the construction is correct is of itself sufficient justification for the work. As to the use of such figures in geometric design, some discussion will be offered at the close of this chapter. The first few propositions include those that lead up to the mensuration of the circle. After they are proved it is assumed that the circle is the limit of the regular inscribed and circumscribed polygons as the number of sides increases indefinitely. This may often be proved with some approach to rigor by a few members of an elementary class, but it is the experience of teachers that the proof is too difficult for most beginners, and so the assumption is usually made in the form of an unproved theorem. The following are some of the leading propositions of this book: THEOREM. _Two circumferences have the same ratio as their radii._