And seeing the action, by which a strait line is made crooked, or contrarily a crooked line is made strait, is nothing but the bringing of its extreme points nearer to one another, or the setting of them further asunder, a crooked line may rightly be defined to be that, whose extreme points may be understood to be drawn further asunder; and a strait line to be that, whose extreme points cannot be drawn further asunder; and comparatively, a more crooked, to be that line whose extreme points are nearer to one another than those of the other, supposing both the lines to be of equal length. Now, howsoever a line be bowed, it makes always a sinus or cavity, sometimes on one side, sometimes on another; so that the same crooked line may either have its whole cavity on one side only, or it may have it part on one side and part on the other side. Which being well understood, it will be easy to understand the following comparisons of strait and crooked lines.
First, if a strait and a crooked line have their extreme points common, the crooked line is longer than the strait line. For if the extreme points of the crooked line be drawn out to their greatest distance, it will be made a strait line, of which that, which was a strait line from the beginning, will be but a part; and therefore the strait line was shorter than the crooked line, which had the same extreme points. And for the same reason, if two crooked lines have their extreme points common, and both of them have all their cavity on one and the same side, the outermost of the two will be the longest line.
Secondly, a strait line and a perpetually crooked line cannot be coincident, no, not in the least part. For if they should, then not only some strait line would have its extreme points common with some crooked line, but also they would, by reason of their coincidence, be equal to one another; which, as I have newly shown, cannot be.
Thirdly, between two points given, there can be understood but one strait line; because there cannot be more than one least interval or length between the same points. For if there may be two, they will either be coincident, and so both of them will be one strait line; or if they be not coincident, then the application of one to the other by extension will make the extended line have its extreme points at greater distance than the other; and consequently, it was crooked from the beginning.
Fourthly, from this last it follows, that two strait lines cannot include a superficies. For if they have both their extreme points common, they are coincident; and if they have but one or neither of them common, then at one or both ends the extreme points will be disjoined, and include no superficies, but leave all open and undetermined.
Fifthly, every part of a strait line is a strait line. For seeing every part of a strait line is the least that can be drawn between its own extreme points, if all the parts should not constitute a strait line, they would altogether be longer than the whole line.
The definition and properties of a plane superficies.
2. A plane or a plane superficies, is that which is described by a strait line so moved, that all the several points thereof describe several strait lines. A strait line, therefore, is necessarily all of it in the same plane which it describes. Also the strait lines, which are made by the points that describe a plane, are all of them in the same plane. Moreover, if any line whatsoever be moved in a plane, the lines, which are described by it, are all of them in the same plane.
All other superficies, which are not plane, are crooked, that is, are either concave or convex. And the same comparisons, which were made of strait and crooked lines, may also be made of plane and crooked superficies.
For, first, if a plane and crooked superficies be terminated with the same lines, the crooked superficies is greater than the plane superficies. For if the lines, of which the crooked superficies consists, be extended, they will be found to be longer than those of which the plane superficies consists, which cannot be extended, because they are strait.