and when an angle is comprehended between a straight and a crooked line (if I may compute the same angle as comprehended between the same straight line and the point of contact), that it is consonant to my definition of a point by a _magnitude not considered_. But when you, in your treatise, _De Angulo Contactus_ (chap. III. p. 6, l. 8) have these words: “_Though the whole concurrent lines incline to one another, yet they form no angle anywhere but in the very point of concourse_:” you, that deny a point to be anything, tell me how two nothings can form an angle; or if the angle be not formed, neither before the concurrent lines meet, nor in the point of concourse, how can you apprehend that any angle can possibly be framed? But I wonder not at this absurdity; because this whole treatise of yours is but one absurdity, continued from the beginning to the end, as shall then appear when I come to answer your objections to that which I have briefly and fully said of that subject in my 14th chapter. At the twelfth article, I confess your exception to my universal definition of parallels to be just, though insolently set down. For it is no fault of ignorance (though it also infect the demonstration next it), but of too much security. The definition is this: _Parallels are those lines or superficies, upon which two straight lines falling, and wheresoever they fall, making equal angles with them both, are equal_; which is not, as it stands, universally true. But inserting these words _the same way_, and making it stand thus: _parallel lines or superficies, are those upon which two straight lines falling the same way, and wheresoever they fall, making equal angles, are equal_, it is both true and universal; and the following consectary, with very little change, as you may see in the translation, perspicuously demonstrated. The same fault occurreth once or twice more; and you triumph unreasonably, as if you had given therein a very great proof of your geometry. The same was observed also upon this place by one of the prime geometricians of Paris, and noted in a letter to his friend in these words (Chap. XIV. art. 12): “_The definition of parallels wanteth somewhat to be supplied_.” And of the consectary he says, “_It concludeth not, because it is grounded on the definition of parallels_.” Truly and severely enough, though without any such words as savour of arrogance, or of malice, or of the clown.