[CRLeedhamGreen]

Euclid, Book 5, Proposition 18. If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly; that is, if the first be to the second as the third to the fourth, then the first and second together shall be to the second as the third and fourth together to the fourth.

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.