The idea of attaching signs to certain geometrical magnitudes such as lengths, angles and areas is fundamental in analytical geometry. It has its uses, too, in pure geometry in simplifying certain theorems and constructions. Many theorems, for instance, such as “ the area of a triangle is half that of a rectangle with the same base and height ”, or “ the angle subtended at the centre of a circle is double that at the circumference”, have to be proved differently for two or more figures, the word “ add ” being in some cases changed to “ subtract ”. In such theorems the one proof will suffice for any figure provided the proper interpretation is made of the signs of the magnitudes.