The arguments in your twelfth and thirteenth chapters are grounded all on this untruth, that an angle is that which is contained between the lines that make it; that is to say, is a plane superficies, which is manifestly false; because the measure of an angle is an arch of a circle, that is to say, a line; which is no measure of a superficies. Besides this gross ignorance, your way of demonstration, by putting N for a great number of sides of an equilateral polygon, is not to be admitted; for, though you understand something by it, you demonstrate nothing to anybody but those who understand your symbolic tongue, which is a very narrow language. If you had demonstrated it in Irish or Welsh, though I had not read it, yet I should not have blamed you, because you had written to a considerable number of mankind, which now you do not. In your last chapters you defend Vitellio without need; for there is no doubt but that whatsoever crooked line be touched by a straight line, the angle of contingence will neither add anything to, nor take anything from, a rectilineal right angle; but that it is because the angle of contact is no angle, or no quantity, is not true. For it is therefore an angle, because an angle of contact; and therefore quantity, because one angle of contact may be greater than another; and therefore heterogeneal, because the measure of an angle of contact cannot (_congruere_) be applied to the measure of a rectilineal angle, as they think it may, who affirm with you that the nature of an angle consisteth in that which is contained between the lines that comprehend it, viz., in a plane superficies. And thus you see in how few lines, and without brachygraphy, your treatise of the angle of contingence is discovered for the greatest part to be false, and for the rest, nothing but a detection of some errors of Clavius grounded on the same false principles with your own. To return now from your treatise of the angle of contact back again to your _Elenchus_ . The fault you find at art. 18, is, that I understand not that Euclid makes a _plane angle_ to be that which is contained between the two lines that form it. It is true, that I do not understand that Euclid was so absurd, as to think the nature of an angle to consist in superficies; but I understand that you have not had the wit to understand Euclid.