than to expect, that if ever the circle be squared, the square to which it is equal must be commensurable with other known rectilineal spaces; for those who are skilled in geometry know that many lines and spaces may be exhibited with the greatest accuracy, geometrically, though they are incapable of being expressed arithmetically, without an infinite series. Agreeable to this, Tacquet well observes (in lib. ii. Geom. Pract. p. 87.) “Denique admonendi hic sunt, qui geometriæ, non satis periti, sibi persuadent ad quadraturam necessarium esse, ut ratio lineæ circularis ad rectam, aut circuli ad quadratum in numeris exhibeatur. Is sane error valde crassus est, et indignus geometrâ, quamvis enim irrationalis esset ea proportio, modo in rectis lineis exhibeatur, reperta erat quadratura.” And that this quadrature is possible geometrically, was not only the opinion of the above mentioned learned and acute geometrician, but likewise of Wallis and Barrow; as may be seen in the Mechanics of the former, p. 517 and in the Mathematical Lectures of the latter, p. 194. But the following discovery will, I hope, convince the liberal geometrical reader, that the quadrature of the circle may be obtained by means of a circle and right-line only, which we have no method of accomplishing by any invention of the ancients or moderns. At least this method, if known to the ancients, is now lost, and though it has been attempted by many of the moderns, it has not been attended with success. In the circle _g o e f_, let _g o_ be the quadrantal arch, and the right-line _g x_ its tangent. Then conceive that the central point _a_ flows uniformly along the radius _a e_, infinitely produced; and that it is endued with an uniform impulsive power. Let it likewise be supposed, that during its flux, radii emanate from it on all sides, which enlarge themselves in proportion to the distance of the point _a_ from its first situation. This being admitted, conceive that the point _a_ by its impulsive power, through the radii _a n_, _a m_, &c. acting every where equally on the arch _g o_, impells it into its equal tangent arch _g r_. And when, by its uniform motion along the infinite line _a_ φ, it has at the same time arrived at _b_, the centre of the arch _g r_, let it impel in a similar manner the arch _g r_, into its equal tangent arch _g s_, by acting every where equally through radii equal to _b r_. Now, if this be conceived to take place infinitely (since a circular line is capable of infinite remission) the arch _g o_ will at length be unbent into its equal, the tangent line _g x_; and the extreme point _o_, will describe by such a motion of unbending a circular line _o x_. For since the same cause, acting every where similarly and equally, produces every where similar and equal effects; and the arch _g o_, is every where equally remitted or unbent, it will describe a line similar in every part. Now, on account of the