In _c f_ produced take _f g_ equal to _c f_; and draw _g m_ parallel to _d_ U, cutting I U in _m_, and _d o_ in _n_; and let the intersection of the two lines _a c_ and _d o_ be in _r_; which being done, the triangles _m n_ T, _r c_ T will be like and equal. Therefore _m n_ and _r c_ are equal; and consequently the straight line I _m_ T U shall pass through _c_. Dividing therefore _a c_ in the midst at _t_, and S N in the midst at _l_, and joining _t_ N, L _l_, the lines L _l_, _t_ N, and _c_ T produced, will all meet in one and the same point of B S produced; suppose at _q_. Therefore the point _q_ being given by the two known points T and I, the lines drawn from _q_ through equal parts of the sine of the arch B I, (for example through the points P, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O, O I. And this is enough to make good that problem, as to your objection. The straight line therefore B U, for any thing you have said, is proved equal to the arch B I, and the division of any angle given into any proportion given, the quadrature of any sector, and the construction of any equilateral polygon is also given. And though in this also I should have erred, yet it cannot be denied but that I have used a more natural, a more geometrical, and a more perspicuous method in the search of this so difficult a problem, than you have done in your _Arithmetica Infinitorum_. For though it be true that the aggregate of all the mean proportionals between the radius, together with an infinitely little part of the same, and the radius wanting an infinitely little part of the same; and again, between the radius, together with two infinitely little parts, and the radius wanting two infinitely little parts, and so on eternally, will be equal to the quadrant (a thing which every mean geometrician knew before); yet it was absurd to think those means could be calculated in numbers by interpoling of a symbol; especially when you make that symbol to stand for a number neither true nor surd; as if there were a number that could neither be uttered in words, nor not be uttered in words. For what else is surd, but that which cannot be spoken?