Definition 4

Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.

Guide

 This definition limits the existence of ratios to comparable magnitudes of the same kind where comparable means each, when multiplied, can exceed the other. The ratio doesn’t exist when one magnitude is so small or the other so large that no multiple of the one can exceed the other. This definition excludes the ratio of a finite straight line to an infinite straight line and the ratio of an infinitesimal straight line, should any exist, to a finite straight line.

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.

Definition 4 as an axiom of comparability

This definition is used repeatedly as a axiom for magnitudes rather than a definition. It is frequently invoked in this book, starting with proposition V.8 but also required for more fundamental properties, and elsewhere, such as the important proposition X.1. In the proofs of these propositions one magnitude is less than another, and it is asserted that some multiple of the smaller is greater than the larger. Euclid implicitly assumes that the magnitudes he discusses, except horn angles, are all comparable. Straight lines, rectilinear angles, plane figures, and solids are all comparable to any other of the same type.

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.