Geometrie, geometria

Geometry has more ἀρχαί than does arithmetic. The objects of geometry are λαμβανόμενα ἐκ προσθέσεως (cf. Post. An. I, 27, 87a35f.), “they are gained from what is determined additionally, through θέσις.” Πρόσθεσις does not simply mean “supplement.” What is the character of this πρόσθεσις in geometry? (…) Aristotle distinguishes the basic elements of geometry from those of arithmetic. The basic element of arithmetic is μονάς, the unit; the basic element of geometry is στιγμή, the point. (GA19RS:71)

The geometrical consists of a manifold of basic elements — point, line, etc. — which are the πέρατα for the higher geometrical figures. But it is not the case that the higher figures are put together out of such limits. Aristotle emphasizes that a line will never arise out of points (Phys. VI, 1, 231a24ff.), a surface will never arise out of a line, nor a body out of a surface. For between any two points there is again and again a γραμμή, etc. This sets Aristotle in the sharpest opposition to Plato. Indeed, the points are the ἀρχαί of the geometrical, yet not in such a way that the higher geometrical figures would be constructed out of their summation. One cannot proceed from the στιγμή to the σῶμα. One cannot put a line together out of points. For in each case there is something lying in between, something that cannot itself be constituted out of the preceding elements. This betrays the fact that in the οὐσία θετός there is certainly posited a manifold of elements, but, beyond that, a determinate kind of connection is required, a determinate kind of unity of the manifold. (GA19RS:76)

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