 But there is another weakness in the Elements that is also worth noting, although it drew less attention, and this is the nature of the plane. The plane has another sub-standard definition, evidently modelled on that of the line: “a plane surface is a surface which lies evenly with the straight lines on itself” (and, unsurprisingly, “a surface is that which has length and breadth only”). After that, the word ‘plane’ is not mentioned in the first four Books, although they are solely concerned with plane geometry. When Euclid turned to solid geometry in Book IX, he began with three theorems to show successively that a straight line cannot lie partly in a plane and partly not, that if two straight lines cut one another they lie in a plane and every triangle lies in a plane, and that if two planes meet then they do so in a line. However, he can only be said to claim these results and make them plausible, because he cannot use his definition of a plane to prove any of them. They do, however, form the basis for the next theorems: there is a perpendicular to a plane at any point of the plane, and all the lines perpendicular to a given line at a given point are on a plane.