It is possible to formally prove that 2+2=4 using Peano's five axioms for the natural numbers and elementary set theory (actually a long and tedious process). Is it possible to prove it based on the definitions and axioms for numbers given in Euclid's Elements?
On the issue of Euclidean Arithmetic, see by Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981 - Dover reprint).
All Ch.2 is devoted to this topic; see page 58 :
In books VII-IX Euclid develops the subject of arithmetic in almost complete isolation from the remainder of the Elements. [...] [In contrast to previous books, we] find no specifically arithmetic postulates in the Elements.
We have definitions regarding numbers in Book VII [see Euclid's Elements] :
Book VII : Definitions
Definition 1 : A unit is that by virtue of which each of the things that exist is called one.
Definition 2 : A number is a multitude composed of units.
Definition 3 : A number is a part of a number, the less of the greater, when it measures the greater;
Definition 4 : But parts when it does not measure it.
Definition 5 : The greater number is a multiple of the less when it is measured by the less. [...]
But there ara no Axioms; see Mueller, page 59 :
Euclid does not prove that 2+2=4 or that a 2 and a 2 combined yield a 4, nor does he even have the apparatus for doing so. Such facts, insofar as they are used in the Elements, are used without proof.