The Binomial Theorem and a Silver Stater from Aegina

Coins dating back to the 4th century BCE are proof of the high level of mathematical culture achieved by the Greek civilization

Raise your hand if you, as a novice algebra student, have not made the mistake of expanding the square of a binomial into the sum of two squares at least once:

Unfortunately for those who think that that is the correct answer, 𝑎²+𝑏² is incomplete. The right expansion of the binomial is:

Probably for a student of ancient Greece to get this formula wrong would have been more difficult than for our contemporary. In fact, the geometric representation of mathematical relations such as that expressed by the square of a binomial was rooted in the Greek spirit.

For example, take Proposition 4 of Book II of Euclid’s Elements. It reads verbatim:

If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments.

Just draw the figure described in the proposition to immediately see that the square constructed on the sum of the…