Now seeing X _b_ cuts off from the point B the fourth part of the arch B F, let that fourth part be B _e_; and let the sine thereof, _f e_, be produced to F T in _g_, for so _f e_ will be the fourth part of the strait line _f g_, because as O _b_ is to O F, so is _f e_ to _f g_. But B T is greater than _f g_; and therefore the same B T is greater than four sines of the fourth part of the arch B F. And in like manner, if the arch B F be subdivided into any number of equal parts whatsoever, it may be proved that the strait line B T is greater than the sine of one of those small arches, so many times taken as there be parts made of the whole arch B F. Wherefore the strait line B T is not less than the arch B F. But neither can it be greater, because if any strait line whatsoever, less than B T, be drawn below B T, parallel to it, and terminated in the strait lines X B and X T, it would cut the arch B F; and so the sine of some one of the parts of the arch B F, taken so often as that small arch is found in the whole arch B F, would be greater than so many of the same arches; which is absurd. Wherefore the strait line B T is equal to the arch B F; and the strait line B V equal to the arch of the quadrant B F D; and B V four times taken, equal to the perimeter of the circle described with the radius A B. Also the arch B F and the strait line B T are everywhere divided into the same proportions; and consequently any given angle, whether greater or less than B A F, may be divided into any proportion given. But the strait line B V, though its magnitude fall within the terms assigned by Archimedes, is found, if computed by the canon of signs, to be somewhat greater than that which is exhibited by the Rudolphine numbers. Nevertheless, if in the place of B T, another strait line, though never so little less, be substituted, the division of angles is immediately lost, as may by any man be demonstrated by this very scheme. Howsoever, if any man think this my strait line B V to be too great, yet, seeing the arch and all the parallels are everywhere so exactly divided, and B V comes so near to the truth, I desire he would search out the reason, why, granting B V to be precisely true, the arches cut off should not be equal.