Euclid’s Book on the Regular Solids: Its Place in the Elements and Its Educational Value

In his commentary on the first book of the Elements, Proclus tells us no less than four times that one of two aims of the entire work is the investigation of the cosmic figures, the five regular solids.¹ This is a remarkable claim, and it is one not easy for modern readers of the Elements to swallow. In his own commentary on the Elements, written a millennium and a half after Proclus, Thomas Heath says in this regard Proclus is “obviously incorrect,” and Heath goes on to explain that, “It is true that Euclid’s Elements end with the construction of the five regular solids; but the planimetrical portion has no direct relation to them, and the arithmetical no relation at all; the propositions about them are merely the conclusion of the stereometrical division of the work” (Heath, 1956, I, p.2). Heath’s skepticism² cannot be dismissed causally. It is indeed difficult to see how Proclus can put aside the immense wealth of other mathematical work in the Elements and single out so pointedly one of its shortest books, Book XIII on the regular or Platonic solids—Book XIII, in this view, must take precedence over Book X with its 115 propositions on incommensurables, or Book V which sets out the theory of proportion, or Books VII-IX on the theory of numbers, not to speak of Books I, III, IV, and VI, which make up every modern student’s own elements of geometry. And Heath is right to cast doubt on Book XIII as in any way culminating the deductive structure of the Elements: the propositions in Book XIII do rely widely on propositions from other parts of the Elements, but not from all parts, as Heath points out, and not in a very striking way.³