# Euclid’s Proposition I.3 overused?

[ Question copied from https://math.stackexchange.com/questions/2541170/euclid-s-proposition-i-3-overused ]

Although the references to postulates, axioms, and previous propositions are not part of the original text of Euclid's Elements, all the editions I have seen contain them.

In many propositions, it is needed to make adjacent segments, say $$PR$$ and $$PQ$$ equal. Examples include:

All the editions linked from this page and more (I do not have enough reputation to post the links to them): John Casey’s English translation, Richard Fitzpatrick’s edition, Johan Ludvig Heiberg's edition, and Józef Czech's Polish translation refer in those cases to Proposition I.3, which is about making arbitrary segments equal (“To cut off from the greater of two given unequal straight lines a straight line equal to the less”).

As I understand it, for making adjacent segments equal, Postulate 3 (“Let it be postulated to draw a circle with any center and radius”) is sufficient. Why do all those editions of Elements refer to Proposition I.3 instead of Postulate 3 in the propositions I listed above? The only reason I can think of is editors' inertia.

[I am not asking about uncontroversial uses of Proposition I.3, like Proposition I.6, where $$DB$$ is made equal to $$AC$$.]

I think Euclid does refer to previous propositions by echoing the words. Compare the alleged use of I.3 in I.5, given with a word-for-word translation of the Greek words:

δύο δοθεισῶν εὐθειῶν ἀνίσων         % Of two given straights unequal,
ἀπὸ τῆς μείζονος                  % from the greater,
τῇ ἐλάσσονι ἴσην                  % to the lesser equal
εὐθεῖαν                           % (a) straight
ἀφελεῖν.                          % to take

ἀφῃρήσθω                            % let there have been taken
ἀπὸ τῆς μείζονος τῆς ΑΕ           % from the greater AE
τῇ ἐλάσσονι τῇ ΑΖ ἴση             % to the lesser AF (ΑΖ) equal
ἡ ΑΗ                              % the (straight) AG (ΑΗ)


To be more pedantic one could interpret the references to the figure as appositives: for example, "ἀπὸ τῆς μείζονος τῆς ΑΕ..." could be read "from the greater, the (straight) AE,...." The main difference in I.5 is the omission of an explicit statement of what is given and the position of the verb. The rest is word-for-word the same except for the appositives, which clarify the application to the proposition. In this way, Euclid is referring to I.3 and no other proposition or postulate. Throughout the Elements, Euclid refers to the relevant definition, postulate, etc. by echoing the words. At first such references are word-for-word as above. Later he sometimes shortens it.

• (+1) Very nice! Dec 25 '18 at 19:43

I suppose that you are right. Besides, what you suggested (using postulate 3 when the segments are adjacent) is exactly what Euclid does in his proof of proposition 2. At this point, he could not possibly use proposition 3, of course. Besides, the proof of proposition 3 uses proposition 2.