EUCLID TEXTS

GMR’S INTRODUCTION

Virtually nothing is known about Euclid, and it is most unlikely that the text we now have is exactly as it left his hands. He probably taught in Alexandria around 300 BC. The first printed edition was a Latin translation of an Arabic translation, printed in 1482. A Greek version was printed in 1533, and this remained the standard edition for over three centuries. Until very recently it was the text used for teaching geometry, with only minor amendments and additions. However, it has to be remembered that by no means every educated person knew any geometry. Hobbes, for example, managed to go through school and university without coming across it at all, and he first studied it while in exile in Paris, when he was around 50.

I give a few extracts from the very beginning of Book 1, so that you can see how closely Spinoza’s Ethics is modelled on it.

The fundamental idea is to start with definitions of all the technical terms which will be used. Then there is a list of postulates, which are assumptions which might not be provable by reason. [It was the denial of a version of postulate 5 (the confusingly named ‘axiom of parallels’) which first gave rise to non-Euclidean geometries. Spinoza doesn’t have any postulates in Book 1, since he is claiming that everything is provable by reason. However, he does allow postulates in the other books, where there is less certainty.] After the postulates come the axioms, which are so obvious that every rational being will naturally assent to their truth. They are also called ‘common notions’, which is another term for ‘innate ideas’. Finally there are the propositions to be derived from the definitions, postulates, and axioms. The word ‘proposition’ is used in its original sense of something which is proposed, and not in the logicians’ sense of that which is true or false. In Euclid, there are two kinds of propositions: problems and theorems. A problem is how to draw a particular construction, and a theorem is a geometrical truth to be proved. In Spinoza, there are only theorems, since he is concerned with establishing the truth.

I have translated from Euclidis Elementa, Vol. 1, ed. I.L. Heiberg (Leibzig, Teubner, 1883).

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