"Like the proofs of al-Karaji and ibn al-Haytham, al-Samaw'al's argument contains the two basic components of an inductive proof. He begins with a value for which the result is known, here ''n'' = 2, and then uses the result for a given integer to derive the result for the next. Since al-Samaw'al did not have any way of stating the general binomial theorem, however, he cannot be said to have proved it, by induction or otherwise. What he had done was provide a method acceptable to his readers for expanding binomials up to the twelfth power..."