The author's take on V.18 is interesting. Clarke has the proposition here: https://mathcs.clarku.edu/~djoyce/java/elements/bookV/propV1... It's not three pages long in Clarke (nor in Heath), and either way, the argument is only 10 statements long. > But the reader who takes the trouble to decode the proposition will see that it is trivial primary school arithmetic. This is also not super obvious from Clarke or Heath.…
This is from the Todhunter edition of Euclid's elements. the proof occupies precisely three pages. in post-Euclidean notation, if a/b = c/d then (a+b)/b = (c+d)/d. For young children replace the letters by small positive integers.
To convert Euclid's formulation to the symbolic formulation requires turning a magnitude into a real number, and defining the division of two real numbers. Euclid's proof avoids these difficulties. His magnitudes are the lengths straight line sections. He can decode a/b = m/n where m and n are positive integers as meaning na = mb. And similarly he can define a/b > m/n and m/n > a/b. So now the statement a/b > c/d can be decoded as the existence of positive integers m and n such that a/b > m/n > c/d. Finally the statement a/b = c/d is decoded as asserting that both a/b > c/d and c/d > a/b are false.
In modern jargon, Euclid is using the Dedekind cut definition of a real number. Using this definition, and working from first principles, the proof unsurprisingly requires three pages.