In the first place, therefore, if A V be drawn, cutting the arch B H D in X, and the side D C in Z, I desire some analyst would, if he can, give a reason why the strait lines T E and T C should cut the arch B D, the one in Y, the other in X, so as to make the arch B Y equal to the arch Y X; or if they be not equal, that he would determine their difference. Secondly, if in the side D A, the strait line D _a_ be taken equal to D Z, and V _a_ be drawn; why V _a_ and V B should be equal; or if they be not equal, what is the difference. Thirdly, drawing Z _b_ parallel and equal to the side C B, cutting the arch B H D in _c_, and drawing the strait line A _c_, and producing it to B V in _d_; why A _d_ should be equal and parallel to the strait line _a_ V, and consequently equal also to the arch B D. Fourthly, drawing _e_ K the sine of the arch B K, and taking (in _e_ A produced) _e f_ equal to the diagonal A C, and connecting _f_ C; why _f_ C should pass through _a_ (which point being given, the length of the arch B H D is also given) and _c_; and why _f e_ and _f c_ should be equal; or if not, why unequal. Fifthly, drawing _f_ Z, I desire he would show, why it is equal to B V, or to the arch B D; or if they be not equal, what is their difference. Sixthly, granting _f_ Z to be equal to the arch B D, I desire he would determine whether it fall all without the arch B C A, or cut the same, or touch it, and in what point. Seventhly, the semicircle B D _g_ being completed, why _g_ I being drawn and produced, should pass through X, by which point X the length of the arch B D is determined. And the same _g_ I being yet further produced to D C in _h_, why A _d_, which is equal to the arch B D, should pass through that point _h_. Eighthly, upon the centre of the square A B C D, which let be _k_, the arch of the quadrant E _i_ L being drawn, cutting _e_ K produced in _i_, why the drawn strait line _i_ X should be parallel to the side C D. Ninthly, in the sides B A and B C taking _g l_ and B _m_ severally equal to half B V, or to the arch B H, and drawing _m n_ parallel and equal to the side B A, cutting the arch B D in _o_, why the strait line which connects V _l_ should pass through the point _o_. Tenthly, I would know of him why the strait line which connects _a_ H should be equal to B _m_; or if not, how much it differs from it. The analyst that can solve these problems without knowing first the length of the arch B D, or using any other known method than that which proceeds by perpetual bisection of an angle, or is drawn from the consideration of the nature of flexion, shall do more than ordinary geometry is able to perform. But if the dimension of a circle cannot be found by any other method, then I have either found it, or it is not at all to be found.