Your first argument therefore is nothing worth, except you make good that which in your second argument you affirm, namely, that all plane angles, not excepting the angle of contact, are (_homogeneous_) of the same kind. You prove it well enough of other curvilineal angles; but when you should prove the same of an angle of contact, you have nothing to say but (p. 17, l. 15), “_Unde autem illa quam somniet heterogenia oriatur, neque potest ille ullatenus ostendere, neque ego vel somniare_:” “_Whence should arise that diversity of kind which he dreams of, neither can he at all show, nor I dream_;” as if you knew what he could do if he were to answer you; or all were false which you cannot dream of. So that besides your customary vanity, here is nothing hitherto proved, neither for the opinion of Peletarius, nor against that of Clavius. I have, I think, sufficiently explicated, in the first lesson, that the angle of contact is quantity, namely, that it is the quantity of that crookedness or flexion, by which a straight line is bent into an arch of a circle equal to it; and that because the crookedness of one arch may be greater than the crookedness of another arch of another circle equal to it; therefore the question _quanta est curvitas_, how much is the crookedness, is pertinent, and to be answered by _quantity_. And I have also shown you in the same lesson, that the quantity of one angle of contact is compared with that of another angle of contact by a line drawn from the point of contact, and intercepted by their circumferences; and that it cannot be compared by any measure with a rectilineal angle. [Illustration]