Your fifth proposition is, “_the spiral line is equal to half the circle of the first revolution_.” But what spiral line? We shall understand that by your construction, which is this: “_The straight line M A_ [in your figure which I have placed at the end of the fifth lesson] _turned round (the point M remaining unmoved) is supposed to describe with its point A the circle A O A, whilst some point, in the same M A, whilst it goes about, is supposed to be moved uniformly from M to A, describing the spiral line_.” This therefore, is the spiral line of Archimedes; and your proposition affirms it to be equal to the half of the circle A O A; which you perceived not long after to be false. But thinking it had been true, you go about to prove it, “_by inscribing in the circle an infinite multitude of equal angles, and consequently an infinite number of sectors, whose arches will therefore be in arithmetical proportion_;” which is true. “_And the aggregate of those arches equal to half the circumference A O A_;” which is true also. And thence you conclude “_that the spiral line is equal to half the circumference of the circle A O A_;” which is false. For the aggregate of that infinite number of infinitely little arches, is not the spiral line made by your construction, seeing by your construction the line you make is manifestly the spiral of Archimedes; whereas no number, though infinite, of arches of circles, how little soever, is any kind of spiral at all; and though you call it a spiral, that is but a patch to cover your fault, and deceiveth no man but yourself. Besides, you saw not how absurd it was, for you that hold a point to be absolutely nothing, to make an infinite number of equal angles (the radius increasing as the number of angles increaseth) and then to say, “_that the arches of the sectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.” For you make the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0, 0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence of this absurdity reacheth to the end of the eighteenth proposition. So many are therefore false, or nothing worth. And you needed not to wonder that the doctrine contained in them was omitted by Archimedes, who never was so senseless as to think a spiral line was compounded of arches of circles.