ad = cb.
But from this it follows, according to the last theorem, that
a : b = c : d.
Hence we conclude that the quotient a/b and the ratio a : b are different forms of the same magnitude, only with this important difference that the quotient a/b would have a meaning only if a and b have a common measure, until we introduce incommensurable numbers, while the ratio a : b has always a meaning, and thus gives rise to the introduction of incommensurable numbers.
Thus it is really the theory of ratios in the fifth book which enables us to extend the geometrical calculus given before in connexion with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions.
§ 64. Prop. 17. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and conversely, is only a special case of 16. After the problem, Prop. 18, On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure, there follows another fundamental theorem:
Prop. 19. Similar triangles are to one another in the duplicate ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. This is generalized in:
Prop. 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.
§ 65. Prop. 21. Rectilineal figures which are similar to the same rectilineal figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures may be said to be equal in “shape” but not in “size,” we may state it also thus:
“Figures which are equal in shape to a third are equal in shape to each other.”