Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

Heiberg would reject VII. Def. 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XI. Def. 11) constitutes a difficulty. The use of the word ἐπιφάνεια suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words γραμμῶν, γραμμαῖς) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation: but it is not so easy to reject the first part up to and including “otherwise” (ἄλλως). No difficulty need be felt about the definitions of “oblong,” “rhombus,” and “rhomboid,” which are not actually used in the Elements; they were no doubt taken from earlier elements and given for the sake of completeness.

As regards the axioms or, as they are called in the text, common notions (κοιναὶ ἔννοιαι), it is to be observed that Proclus says¹ that Apollonius tried to prove “the axioms,” and he gives Apollonius' attempt to prove Axiom I. This shows at all events that Apollonius had some of the axioms now appearing in the text. But how could Apollonius have taken a controversial line against Euclid on the subject of axioms if these axioms had not been Euclid's to his knowledge? And, if they had been interpolated between Euclid's time and his own, how could Apollonius, living so comparatively short a time after Euclid, have been ignorant of the fact? Therefore some of the axioms are Euclid's (whether he called them common notions, or axioms, as is perhaps more likely since Proclus calls them axioms): and we need not hesitate to accept as genuine the first three discussed by Proclus, viz. (1) things equal to the same equal to one another, (2) if equals be added to equals, wholes equal, (3) if equals be subtracted from equals, remainders equal. The other two mentioned by Proclus (whole greater than part, and congruent figures equal) are more doubtful, since they are omitted by Heron, Martianus Capella, and others. The axiom that “two lines cannot enclose a space” is however clearly an interpolation due to the fact that I. 4 appeared to require it. The others about equals added to unequals, doubles of the same thing, and halves of the same thing are also interpolated; they are connected with other interpolations, and Proclus clearly used some source which did not contain them.

Euclid evidently limited his formal axioms to those which seemed to him most essential and of the widest application; for he not unfrequently assumes other things as axiomatic, e.g. in VII. 28 that, if a number measures two numbers, it measures their difference.

The differences of reading appearing in Proclus suggest the question of the comparative purity of the sources used by Proclus, Heron and others, and of our text. The omission of the definition of a segment in Book I. and of the old gloss “which is called the circumference” in I. Def. 15 (also omitted by Heron, Taurus, Sextus