At the thirteenth article you recite the demonstration by which I prove the perimeters of two circles to be proportional to their semidiameters; and with _esto_, _fortasse_, _recte_, _omnino_, noddying to the several parts thereof, you come at length to my last inference: _Therefore, by_ Chap. XIII. art. 6, _the perimeters and semidiameters of circles are proportional_; which you deny; and therefore deny, because you say it followeth by the same ratiocination, that _circles also and spheres are proportional to their semidiameters_. “_For the same distance_, you say, _of the perimeter from the centre which determines the magnitude of the semidiameter, determines also the magnitude both of the circle and of the sphere_.” You acknowledge that perimeters and semidiameters have the cause of their determination such as in equal times make equal spaces. Suppose now a sphere generated by the semidiameters, whilst the semicircle is turned about. There is but one _radius_ of an infinite number of _radii_, which describes a great circle; all the rest describe lesser circles parallel to it, in one and the same time of revolution. Would you have men believe, that describing greater and lesser circles, is according to the supposition (_temporibus æqualibus æqualia facere_) to make equal spaces in equal times? Or, when by the turning about of the semidiameter is described the plane of a circle, does it, think you, in equal times make the planes of the interior circles equal to the planes of the exterior? Or is the _radius_ that describes the inner circles equal to the _radius_ that describes the exterior? It does not, therefore, follow from anything I have said in this demonstration, that either spheres or planes of circles, are proportional to their _radii_; and consequently, all that you have said, triumphing in your own incapacity, is said imprudently by yourselves to your own disgrace. They that have applauded you, have reason by this time to doubt of all the rest that follows, and if they can, to dissemble the opinion they had before of your geometry. But they shall see before I have done, that not only your whole _Elenchus_ , but also your other books of the _Angle of Contact_ , &c. are mere ignorance and gibberish.