3. Of straitness, whether it be in lines or in superficies, there is but one kind; but of crookedness there are many kinds; for of crooked magnitudes, some are congruous, that is, are coincident when they are applied to one other; others are incongruous. Again, some are ὁμοιομερεῖς or uniform, that is, have their parts, howsoever taken, congruous to one another; others are ἀνομοιομερεῖς or of several forms. Moreover, of such as are crooked, some are continually crooked, others have parts which are not crooked. [Sidenote: Definition and properties of a circular line.] 4. If a strait line be moved in a plane, in such manner, that while one end of it stands still, the whole line be carried round about till it come again into the same place from whence it was first moved, it will describe a plane superficies, which will be terminated every way by that crooked line, which is made by that end of the strait line which was carried round. Now this superficies is called a CIRCLE; and of this circle, the unmoved point is the _centre_; the crooked line which terminates it, the _perimeter_; and every part of that crooked line, a _circumference_ or _arch_; the strait line, which generated the _circle_, is the _semidiameter_ or _radius_; and any strait line, which passeth through the centre and is terminated on both sides in the circumference, is called the _diameter_. Moreover, every point of the radius, which describes the circle, describes in the same time its own perimeter, terminating its own circle, which is said to be _concentric_ to all the other circles, because this and all those have one common centre. Wherefore in every circle, all strait lines from the centre to the circumference are equal. For they are all coincident with the radius which generates the circle. Also the diameter divides both the perimeter and the circle itself into two equal parts. For if those two parts be applied to one another, and the semiperimeters be coincident, then, seeing they have one common diameter, they will be equal; and the semicircles will be equal also; for these also will be coincident. But if the semiperimeters be not coincident, then some one strait line, which passes through the centre, which centre is in the diameter, will be cut by them in two points. Wherefore, seeing all the strait lines from the centre to the circumference are equal, a part of the same strait line will be equal to the whole; which is impossible. For the same reason the perimeter of a circle will be uniform, that is, any one part of it will be coincident with any other equal part of the same. [Sidenote: The properties of a strait line taken in a plane.]