Again, let the strait line B F be divided in the midst in Q, and the arch B P in the midst in R; and describing the quadrant B Q S (whose arch Q S is a fourth part of the arch of the quadrant B M D, as the arch B R is a fourth part of the arch B M, which is the arch of the semiquadrant A B M) let the chord S _e_ equal to the chord B R be set off from the point S in the arch S Q; and let B _e_ be drawn and produced to the arch A N in _f_; which being done, the strait line A _f_ will be quadruple to the chord B R or S _e_. And seeing the crookedness of the arch S _e_, or of the arch A _c_, is double to the crookedness of the arch B R, the excess of the crookedness of the arch A _f_ above the crookedness of the arch A _c_ will be subduple to the excess of the crookedness of the arch A _c_ above the crookedness of the arch A N; and therefore the arch N _c_ will be double to the arch _c f_. Wherefore the arch _c d_ is divided in the midst in _f_, and the arch N _f_ is ¾ of the arch N _d_. And in like manner if the arch B R be bisected in V, and the strait line B Q in X, and the quadrant B X Y be described, and the strait line Y _g_ equal to the chord B V be set off from the point Y in the arch Y X, it may be demonstrated that the strait line B _g_ being drawn and produced to the arch A N, will cut the arch _f d_ into two equal parts, and that a strait line drawn from A to the point of that section, will be equal to eight chords of the arch B V, and so on perpetually; and consequently, that the strait line A _d_ is equal to so many equal chords of equal parts of the arch B M, as may be made by infinite bisections. Wherefore the strait line A _d_ is equal to the arch B M or A N, that is, to half the arch of the quadrant A B D or B C A. Coroll. An arch being given not greater than the arch of a quadrant (for being made greater, it comes again towards the radius B A produced, from which it receded before) if a strait line double to the chord of half the given arch be adapted from the beginning of the arch, and by how much the arch that is subtended by it is greater than the given arch, by so much a greater arch be subtended by another strait line, this strait line shall be equal to the first given arch.