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. [Sidenote: The first attempt for the finding out of the dimension of a circle by lines.]