simplicity of the impulsive motion, such a line must either be straight or circular; for there are only three lines every where similar, i. e. the right and circular line, and the cylindric helix; but this last, as Proclus well observes in his following Commentary on the fourth definition, is not a simple line, because it is generated by two simple motions, the rectilineal and circular. But the line which bounds more than two equal tangent arches cannot be a right line, as is well known to all geometricians; it is therefore a circular line. It is likewise evident, that this arch _o x_ is concave towards the point _g_: for if not, it would pass beyond the chord _o x_, which is absurd. And again, no arch greater than the quadrant can be unbent by this motion: for any one of the radii, as _a p_ beyond _g o_, has a tendency from, and not to the tangent _g x_, which last is necessary to our hypothesis. Now if we conceive another quadrantal arch of the circle _g o e f_, that is _g y_, touching the former in _g_ to be unbent in the same manner, the arch _x y_ shall be a continuation of the arch _x o_; for if _γ x κ_ be drawn perpendicular to _x g_, as in the figure, it shall be a tangent in _x_ to the equal arches _y x_, _x o_; because it cannot fall within either, without making the sine of some one of the equal arches, equal to the right-line _x g_, which would be absurd. And hence we may easily infer, that the centre of the arch _y x o_, is in the tangent line _x g_. Hence too, we have an easy method of finding a tangent right-line equal to a quadrantal arch: for having the points _y_, _o_ given, it is easy to find a third point, as _s_; and then the circle passing through the three points _o_, _s_, _y_, shall cut off the tangent _x g_, equal to the quadrantal arch _g o_. And the point _s_ may be speedily obtained, by describing the arch _g s_ with a radius, having to the radius _a g_ the proportion of 6 to 4; for then _g s_ is the sixth part of its whole circle, and is equal to the arch _g o_. And thus, from this hypothesis, which, I presume, may be as readily admitted as the increments and decrements of lines in fluxions, the quadrature of the circle may be geometrically obtained; for this is easily found, when a right-line is discovered equal to the periphery of a circle. I am well aware the algebraists will consider it as useless, because it cannot be accommodated to the farrago of an arithmetical calculation; but I hope the lovers of the ancient geometry will deem it deserving an accurate investigation; and if they can find no paralogism in the reasoning, will consider it as a legitimate demonstration. [Illustration] [Illustration]