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I'm going through Euclid's Elements (Book 1) (Proposition 4). We're given two triangles (ABC, DEF) where two sides of the first triangle are equal to the two other sides (say AB=DE and AC=DF) of the second triangle. The angle between these two sides are also equal so (A=D).

In the proof, we place vertex A onto vertex D. Because the angles are equal and because we have the sides equal, we can conclude that the vertices B and C coincide with the vertices E and F respectively. This is fine so far.

Now, given that the three vertices coincide with the three other vertices, can we just use axiom 8 (magnitudes which coincide with one another or exactly fill the same space are equal) to conclude that the triangles are equal and so the remaining sides and angles are also equal?

In Byrne's version, we first use axiom 10 (two straight lines cannot include a space) to conclude that the sides BC and EF are equal. I was wondering if we really need this axiom or not?

nemo
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  • I have Heath's translation and also an 1885 edition from Conway. The Heath version of I.4 is purely an argument of superposition and appeal to equality, and what you call axiom 10 is not present anywhere. In the Conway version that axiom 10 is mentioned, and it's principally used in I.4 and XI.3 as I remember. If you read the prefaces to most annotated versions of Euclid, you'll see many caveats about later interpolations, the lack of division of common notions from postulates (axioms), etc. I'm OK with both formulations. – RobinSparrow Mar 28 '24 at 19:59
  • I suspect the main reason it ended up as an axiom in some translations is that in Book XI (XI.3), Euclid states (per Heath) that it is absurd for two straight lines contained between the same points to enclose an area, but this is an ungrounded statement. So axiom 10 is a patch in some sense. – RobinSparrow Mar 28 '24 at 20:12
  • I was just reading Oliver Byrnes's copy. https://www.c82.net/euclid/book1/ has the full text of it there – nemo Mar 28 '24 at 20:23
  • A stylish but unorthodox version. There are several good versions free on Google play if you want a more traditional copy to accompany the Byrne. – RobinSparrow Mar 28 '24 at 20:50
  • Thanks! I'll look now and see what I can find – nemo Mar 28 '24 at 22:16
  • You prompted me to check with HSM, which yielded this excellent response: the axiom is not native to Elements, but subsequent writers felt the axiom was needed to make the proof(s) by superposition rigorous. Also, if you do look at Google Books for Elements, you will find Conway's version, and his answer key, and his sequel to Elements, all of which are amazing resources. – RobinSparrow Mar 29 '24 at 00:53
  • That is excellent! thank you! – nemo Mar 29 '24 at 02:57
  • When you said Axiom 8, my immediate thought was "what the heck", because the Elements only has 5 axioms. It is well-known that several additional axioms are needed, including some way of justifying steps Euclid took in this very theorem. Many more modern treatments (say 500 years old instead of 2500) simply take this theorem (SAS) to be an axiom itself. – Paul Sinclair Mar 29 '24 at 22:40

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