At the seventh article, where I define both an _angle_, simply so called, and an _angle of contingence_, by their several generations; namely, that the former is generated _when two straight lines are coincident, and one of them is moved, and distracted from the other by circular motion upon one common point resting, &c._; you ask me “_to which of these kinds of angle I refer the angle made by a straight line when it cuts a crooked line_?” I answer easily and truly, To that kind of angle which is called simply an angle. This you understand not. “For how”, will you say, “can that angle which is generated by the divergence of two straight lines, be other than rectilineal? or how can that angle which is not comprehended by two straight lines, be other than curvilineal?” I see what it is that troubles you; namely, the same which made you say before, that if the body which describes a line be a point, then there is nothing which is not moved that can be called a point. So you say here, “If an angle be generated by the motion of a straight line, then no angle so generated can be curvilineal;” which is as well argued, as if a man should say, the house was built by the carriage and motion of stone and timber, therefore, when the carriage and that motion is ended, it is no more a house. Rectilineal and curvilineal hath nothing to do with the nature of an angle simply so called, though it be essential to an angle of contact. The measure of an angle, simply so called, is a circumference of a circle; and the measure is always the same kind of quantity with the thing measured. The rectitude or curvity of the lines, which drawn from the centre, intercept the arch, is accidentary to the angle, which is the same, whether it be drawn by the motion circular of a straight line or of a crooked. The diameter and the circumference of a circle make a right angle, and the same which is made by the diameter and the tangent. And because the point of contact is not, as you think, nothing, but a line unreckoned, and common both to the tangent and the circumference; the same angle computed in the tangent is rectilineal, but computed in the circumference, not rectilineal, but mixed: or, if two circles cut one another, curvilineal. For every chord maketh the same angle with the circumference which it maketh with the line that toucheth the circumference at the end of the chord. And, therefore, when I divide an angle, simply so called, into rectilineal and curvilineal, I respect no more the generation of it, than when I divide it into right and oblique. I then respect the generation, when I divide an angle into an angle simply so called, and an angle of contact. This that I have now said, if the reader remember when he reads your objections to this, and to the ninth article, he will need no more to make him see that you are utterly ignorant of the nature of an angle; and that if ignorance be madness, not I, but you, are mad: