Abstract

The author's purpose is to read the main work of Euclid “with modern eyes” and to find out what knowledge a mathematician of today, familiar with the works of V. D. Waerden and Bourbaki, can gain by studying Euclid's “theory of magnitudes”, and what new insight into Greek mathematics occupation with this subject can provide. The task is to analyse and to axiomatize by modern means (i) in a narrower sense Book V. of the Elements, i.e. the theory of proportion of Eudoxus, (ii) in a wider sense the whole sphere of magnitudes which Euclid applies in his Elements. This procedure furnishes a clear picture of the inherent structure of his work, thereby making visible specific characteristics of Greek mathematics. After a clarification of the preconditions and a short survey of the historical development of the theory of proportions (Part I of this work), an exact analysis of the definitions and propositions of Book V. of the Elements is carried out in Part II. This is done “word by word”. The author applies his own system of axioms, set up in close accordance with Euclid, which permits one to deduce all definitions and propositions of Euclid's theory of magnitudes (especially those of Books V. and VI.). In this way gaps and “tacit assumptions” in the work become clearly visible; above all, the logical structure of the system of magnitudes given by Euclid becomes evident: not “ratio” — like something sui generis — is the governing concept of Book V., but magnitudes and their relation “of having a ratio” form the base of the theory of proportions. These magnitudes represent a well defined structure, a so-called “Eudoxic Semigroup” with the numbers as operators; it can easily be imbedded in a general theory of magnitudes equally applicable to geometry and physics. The transition to ratios — a step not executed by Euclid — is examined in Part III; it turns out to be particularly unwieldy. An elegant way opens up by interpreting proportion as a mapping of totally ordered semigroups. When closely examined, this mapping proves to be an isomorphism, thus suggesting the application of the modern theory of homomorphism. This theory permits a treatment of the theory of proportions as developed by Eudoxus and Euclid which is hardly surpassable in brevity and elegance in spite of its close affinity to Euclid. The generalization to a “classically” founded theory of magnitudes is now self-evident.