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.
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.
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.
Secondly, and consequently: if two unequal arches of the same circle be made by the bowing of two strait lines equal to them, the flexion of the longer line, whilst it is bowed into the greater arch, is greater than the flexion of the shorter line, whilst it is bowed into the lesser arch, according to the proportion of the arches themselves; and consequently, the crookedness of the greater arch is to the crookedness of the lesser arch, as the greater arch is to the lesser arch.
Thirdly: if two unequal circles and a strait line touch one another in the same point, the crookedness of any arch taken in the lesser circle, will be greater than the crookedness of an arch equal to it taken in the greater circle, in reciprocal proportion to that of the radii with which the circles are described; or, which is all one, any strait line being drawn from the point of contact till it cut both the circumferences, as the part of that strait line cut off by the circumference of the greater circle to that part which is cut off by the circumference of the lesser circle.