In Euclid I. 47, it is proved that if regular tetragons be described on the sides of a right-angled triangle, that described on the hypotenuse is equal to the sum of those on the sides containing the right angle. This proposition, as is shown in Euclid VI. 31, is only a particular case of a more general one; and the object of this Paper is to establish the corresponding result in the case of regular trigons, pentagons, hexagons, and generally regular figures of any number of sides, and finally, in the case of any similar rectilineal figures, without the use of ratio and proportion.