We shall not consider any part of space or time as indivisible, or infinitely little; but we shall consider a point as a term or limit of a line, and… - Colin MacLaurin

And as this has been the occasion of my delay in publishing... I hope it will serve for an apology, if some mistakes have escaped me in treating of such a variety of subjects, in a manner different from that in which they have usually been explained.

He considered magnitudes as generated by a or motion, and showed how the velocities of the generating motions were to be compared together. There was nothing in this doctrine but what seemed to be natural and agreeable to the antient geometry. But what he has given us on this subject being very short, his conciseness maybe supposed to have given some occasion to the objections which have been raised against his method.

They proceeded therefore in another manner, less direct indeed, but perfectly evident. They found, that the inscribed similar polygons, by increasing the number of their sides, continually approached to the areas of the circles; so that the decreasing differences betwixt each circle and its inscribed polygon, by still further and further divisions of the circular arches which the sides of the polygons subtend, could become less than any quantity that can be assigned: and that all this while the similar polygons observed the same constant invariable proportion to each other, viz. that of the squares of the diameters of the circles. Upon this they founded a demonstration, that the proportion of the circles themselves could be no other than that same invariable ratio of the similar inscribed polygons; of which we shall give a brief abstract, that it may appear in what manner they were able... to form a demonstration of the proportions of curvilineal figures, from what they had already discovered of rectilineal ones. And that the general reasoning by which they demonstrated all their theorems of this kind may more easily appear, we shall represent the circles and polygons by right lines, in the same manner as all magnitudes are expressed in the fifth book of the Elements.