But some man may yet ask the reason why the strait lines, drawn from X through the equal parts of the arch B F, should cut off in the tangent B V so many strait lines equal to them, seeing the connected straight line X V passes not through the point D, but cuts the strait line A D produced in _l_; and consequently require some determination of this problem. Concerning which, I will say what I think to be the reason, namely, that whilst the magnitude of the arch doth not exceed the magnitude of the radius, that is, the magnitude of the tangent B C, both the arch and the tangent are cut alike by the strait lines drawn from X; otherwise not. For A V being connected, cutting the arch B H D in I, if X C being drawn should cut the same arch in the same point I, it would be as true that the arch B I is equal to the radius B C, as it is true that the arch B F is equal to the strait line B T; and drawing X K it would cut the arch B I in the midst in _i_; also drawing A _i_ and producing it to the tangent B C in _k_, the strait line B _k_ will be the tangent of the arch B _i_, (which arch is equal to half the radius) and the same strait line B _k_ will be equal to the strait line _k_ I. I say all this is true, if the preceding demonstration be true; and consequently the proportional section of the arch and its tangent proceeds hitherto. But it is manifest by the golden rule, that taking B _h_ double to B T, the line X _h_ shall not cut off the arch B E, which is double to the arch B F, but a much greater. For the magnitude of the straight lines X M, X B, and M E, being known (in numbers), the magnitude of the strait line cut off in the tangent by the strait line X E produced to the tangent, may also be known; and it will be found to be less than B _h_; Wherefore the strait line X _h_ being drawn, will cut off a part of the arch of the quadrant greater than the arch B E. But I shall speak more fully in the next article concerning the magnitude of the arch B I. And let this be the first attempt for the finding out of the dimension of a circle by the section of the arch B F. [Sidenote: The second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness.] 3. I shall now attempt the same by arguments drawn from the nature of the crookedness of the circle itself; but I shall first set down some premises necessary for this speculation; and First, if a strait line be bowed into an arch of a circle equal to it, as when a stretched thread, which toucheth a right cylinder, is so bowed in every point, that it be everywhere coincident with the perimeter of the base of the cylinder, the flexion of that line will be equal, in all its points; and consequently the crookedness of the arch of a circle is everywhere uniform; which needs no other demonstration than this, that the perimeter of a circle is an uniform line.