6. Two lines are said to _touch_ one another, which being both drawn to one and the same point, will not cut one another, though they be produced, produced, I say, in the same manner in which they were generated. And therefore if two strait lines touch one another in any one point, they will be contiguous through their whole length. Also two lines continually crooked will do the same, if they be congruous and be applied to one another according to their congruity; otherwise, if they be incongruously applied, they will, as all other crooked lines, touch one another, where they touch, but in one point only. Which is manifest from this, that there can be no congruity between a strait line and a line that is continually crooked; for otherwise the same line might be both strait and crooked. Besides, when a strait line touches a crooked line, if the strait line be never so little moved about upon the point of contact, it will cut the crooked line; for seeing it touches it but in one point, if it incline any way, it will do more than touch it; that is, it will either be congruous to it, or it will cut it; but it cannot be congruous to it; and therefore it will cut it. [Sidenote: The definition of an angle, and the kinds thereof.] 7. An angle, according to the most general acceptation of the word, may be thus defined; _when two lines, or many superficies, concur in one sole point, and diverge every where else, the quantity of that divergence is an_ ANGLE. And an angle is of two sorts; for, first, it may be made by the concurrence of lines, and then it is a _superficial angle_; or by the concurrence of superficies, and then it is called a _solid angle_. Again, from the two ways by which two lines may diverge from one another, superficial angles are divided into two kinds. For two strait lines, which are applied to one another, and are contiguous in their whole length, may be separated or pulled open in such manner, that their concurrence in one point will still remain; and this separation or opening may be either by circular motion, the centre whereof is their point of concurrence, and the lines will still retain their straitness, the quantity of which separation or divergence is an _angle_ simply so called; or they may be separated by continual flexion or curvation in every imaginable point; and the quantity of this separation is that, which is called an _angle of contingence_. Besides, of superficial angles simply so called, those, which are in a plane superficies, are plane; and those, which are not plane, are denominated from the superficies in which they are. Lastly, those are _strait-lined angles_, which are made by strait lines; as those which are made by crooked lines are _crooked-lined_; and those which are made both of strait and crooked lines, are _mixed angles_. [Sidenote: In concentric circles, arches of the same angle are to one another, as the whole circumferences are.]