The result on horn angles in proposition III.16 excludes ratios between horn angles and rectilinear angles. That proposition states that a horn angle is less than any rectilinear angle, hence no multiple of a horn angle is greater than a rectilinear angle. The situation of horn angles is much worse than that, however, since horn angles of different sizes aren’t even comparable.
This principle of comparability should be explicit in order to justify the principle of comparability for magnitudes of these kinds. One solution is to make it a postulate that straight lines are comparable. From that postulate comparability of each of the other kinds of magnitudes could be proved.
Several of the propositions, stated and unstated, depend on this principle. Without it, some are simply false for kinds of magnitude that have infinitesimals. If x and y are two magnitudes of the same kind, then x is infinitesimal with respect to y, or y is infinite with respect to x, if no multiple of x is greater than y. For example, horn angles are infinitesimal with respect to rectilinear angles. Although this definition excludes ratios between horn angles and rectilinear angles, it allows a ratio between a rectilinear angle B and the sum of a horn angle A and the rectilinear angle, and, according to the next three definitions, the two ratios B : (A + B) and B : B do not satisfy the law of trichotomy, that is, they aren’t the same ratio but neither is greater than the other, either. Examples involving infinitesimals can be useful to show which propositions require treating this definition as an axiom.