There are two reasons why the use of tangential coordinates should be introduced at an earlier stage than is customary into the teaching of analytical geometry and placed on a somewhat different footing. The educational advantages of encouraging the student to follow out a course of reasoning with the help of only a few symbols instead of grinding out hard examples by long meaningless algebra should not require much pressing, although they do not seem to be sufficiently recognised in this country; and on the other hand many theories receive their most appropriate expression in tangentials chiefly owing to the fact that the absolute has a less specialised form in tangential than in point coordinates, being (in two dimensions) the product of two distinct factors in the former and a perfect square in the latter.