9. Nothing is contributed towards the quantity of an angle, neither by the length, nor by the equality, nor by the inequality of the lines which comprehend it. For the lines A B and A C comprehend the same angle which is comprehended by the lines A E and A F, or A B and A F. Nor is an angle either increased or diminished by the absolute quantity of the arch, which subtends the same; for both the greater arch B C and the lesser arch E F are subtended to the same angle. But the quantity of an angle is estimated by the quantity of the subtending arch compared with the quantity of the whole perimeter. And therefore the quantity of an angle simply so called may be thus defined: _the quantity of an angle is an arch or circumference of a circle, determined by its proportion to the whole perimeter_. So that when an arch is intercepted between two strait lines drawn from the centre, look how great a portion that arch is of the whole perimeter, so great is the angle. From whence it may be understood, that when the lines which contain an angle are strait lines, the quantity of that angle may be taken at any distance from the centre. But if one or both of the containing lines be crooked, then the quantity of the angle is to be taken in the least distance from the centre, or from their concurrence; for the least distance is to be considered as a strait line, seeing no crooked line can be imagined so little, but that there may be a less strait line. And although the least strait line cannot be given, because the least given line may still be divided, yet we may come to a part so small, as is not at all considerable; which we call a point. And this point may be understood to be in a strait line which touches a crooked line; for an angle is generated by separating, by circular motion, one strait line from another which touches it, as has been said above in the 7th article. Wherefore an angle, which two crooked lines make, is the same with that which is made by two strait lines which touch them. [Sidenote: The distinction of angles, simply so called.] 10. From hence it follows, that _vertical angles_, such as are A B C, D B F in the second figure, are equal to one another. For if, from the two semiperimeters D A C, F D A, which are equal to one another, the common arch D A be taken away, the remaining arches A C, D F will be equal to one another.