CHAPTER IX.
§ 1. ON THE NATURE OF ELEMENTS.
It would not be easy to find a more lucid explanation of the terms
element and
elementary, and of the distinction between them, than is found in Proclus
1, who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of
elements; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (
στοιχεῖα).
The term
elementary, on the other hand, has a wider application: it is applicable to things “which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the
elements, because their investigation is not of general use in the whole of the science, e.g. the proposition that in triangles the perpendiculars from the angles to the transverse sides meet in a point.”
“Again, the term
element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)
2, and
vice versa. Such an element is like a
lemma. But the term
element is otherwise used of that into which, being more simple, the composite is divided; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the nature of principles are elements of those which stand to them in the relation of results, as postulates are elements of theorems. It is
