CHAPTER XX.
OF THE DIMENSION OF A CIRCLE, AND THE
DIVISION OF ANGLES OR ARCHES.
[1.] The dimension of a circle never determined in numbers by Archimedes and others.—[2.] The first attempt for the finding out of the dimension of a circle by lines.—[3.] The second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness.—[4.] The third attempt; and some things propounded to be further searched into.—[5.] The equation of the spiral of Archimedes with a strait line.—[6.] Of the analysis of geometricians by the powers of lines.
The dimension of a circle never determined in numbers by Archimedes and others.
1. In the comparing of an arch of a circle with a strait line, many and great geometricians, even from the most ancient times, have exercised their wits; and more had done the same, if they had not seen their pains, though undertaken for the common good, if not brought to perfection, vilified by those that envy the praises of other men. Amongst those ancient writers whose works are come to our hands, Archimedes was the first that brought the length of the perimeter of a circle within the limits of numbers very little differing from the truth; demonstrating the same to be less than three diameters and a seventh part, but greater than three diameters and ten seventy-one parts of the diameter. So that supposing the radius to consist of 10,000,000 equal parts, the arch of a quadrant will be between 15,714,285 and 15,704,225 of the same parts. In our times, Ludovicus Van Cullen and Willebrordus Snellius, with joint endeavour, have come yet nearer to the truth; and pronounced from true principles, that the arch of a quadrant, putting, as before, 10,000,000 for radius, differs not one whole unity from the number 15,707,963; which, if they had exhibited their arithmetical operations, and no man had discovered any error in that long work of theirs, had been demonstrated by them. This is the furthest progress that has been made by the way of numbers; and they that have proceeded thus far deserve the praise of industry. Nevertheless, if we consider the benefit, which is the scope at which all speculation should aim, the improvement they have made has been little or none. For any ordinary man may much sooner and more accurately find a strait line equal to the perimeter of a circle, and consequently square the circle, by winding a small thread about a given cylinder, than any geometrician shall do the same by dividing the radius into 10,000,000 equal parts. But though the length of the circumference were exactly set out, either by numbers, or mechanically, or only by chance, yet this would contribute no help at all towards the section of angles, unless happily these two problems, to divide a given angle according to any proportion assigned, and to find a strait line equal to the arch of a circle, were reciprocal, and followed one another. Seeing therefore the benefit proceeding from the knowledge of the length of the arch of a quadrant consists in this, that we may thereby divide an angle according to any proportion, either accurately, or at least accurately enough for common use; and seeing this cannot be done by arithmetic, I thought fit to attempt the same by geometry, and in this chapter to make trial whether it might not be performed by the drawing of strait and circular lines.
The first attempt for the finding out of the dimension of a circle by lines.
2. Let the square A B C D (in the [first figure]) be described; and with the radii A B, B C, and D C, the three arches B D, C A, and A C; of which let the two B D and C A cut one another in E, and the two B D and A C in F. The diagonals therefore B D and A C being drawn will cut one another in the centre of the square G, and the two arches B D and C A in two equal parts in H and Y; and the arch B H D will be trisected in F and E. Through the centre G let the two strait lines K G L and M G N be drawn parallel and equal to the sides of the square A B and A D, cutting the four sides of the same square in the points K, L, M, and N; which being done, K L will pass through F, and M N through E. Then let O P be drawn parallel and equal to the side B C, cutting the arch B F D in F, and the sides A B and D C in O and P. Therefore O F will be the sine of the arch B F, which is an arch of 30 degrees; and the same O F will be equal to half the radius. Lastly, dividing the arch B F in the middle in Q, let R Q, the sine of the arch B Q, be drawn and produced to S, so that Q S be equal to R Q, and consequently R S be equal to the chord of the arch B F; and let F S be drawn and produced to T in the side B C. I say, the strait line B T is equal to the arch B F; and consequently that B V, the triple of B T, is equal to the arch of the quadrant B F E D.
Let T F be produced till it meet the side B A produced in X; and dividing O F in the middle in Z, let Q Z be drawn and produced till it meet with the side B A produced. Seeing therefore the strait lines R S and O F are parallel, and divided in the midst in Q and Z, Q Z produced will fall upon X, and X Z Q produced to the side B C will cut B T in the midst in α.
Upon the strait line F Z, the fourth part of the radius A B, let the equilateral triangle a Z F be constituted; and upon the centre a, with the radius a Z, let the arch Z F be drawn; which arch Z F will therefore be equal to the arch Q F, the half of the arch B F. Again, let the strait line Z O be cut in the midst in b, and the strait line b O in the midst in c; and let the bisection be continued in this manner till the last part O c be the least that can possibly be taken; and upon it, and all the rest of the parts equal to it into which the strait line O F may be cut, let so many equilateral triangles be understood to be constituted; of which let the last be d O c. If, therefore, upon the centre d, with the radius d O, be drawn the arch O c, and upon the rest of the equal parts of the strait line O F be drawn in like manner so many equal arches, all those arches together taken will be equal to the whole arch B F, and the half of them, namely, those that are comprehended between O and Z, or between Z and F, will be equal to the arch B Q or Q F, and in sum, what part soever the strait line O c be of the strait line O F, the same part will the arch O c be of the arch B F, though both the arch and the chord be infinitely bisected. Now seeing the arch O c is more crooked than that part of the arch B F which is equal to it; and seeing also that the more the strait line X c is produced, the more it diverges from the strait line X O, if the points O and c be understood to be moved forwards with strait motion in X O and X c, the arch O c will thereby be extended by little and little, till at the last it come somewhere to have the same crookedness with that part of the arch B F which is equal to it. In like manner, if the strait line X b be drawn, and the point b be understood to be moved forwards at the same time, the arch c b will also by little and little be extended, till its crookedness come to be equal to the crookedness of that part of the arch B F which is equal to it. And the same will happen in all those small equal arches which are described upon so many equal parts of the strait line O F. It is also manifest, that by strait motion in X O and X Z all those small arches will lie in the arch B F, in the points B, Q and F. And though the same small equal arches should not be coincident with the equal parts of the arch B F in all the other points thereof, yet certainly they will constitute two crooked lines, not only equal to the two arches B Q and Q F, and equally crooked, but also having their cavity towards the same parts; which how it should be, unless all those small arches should be coincident with the arch B F in all its points, is not imaginable. They are therefore coincident, and all the strait lines drawn from X, and passing through the points of division of the strait line O F, will also divide the arch B F into the same proportions into which O F is divided.