A theorem of Fermat and Girard (proved by Euler) states that a positive integer can be written as a sum of two squares if and only if no prime of the form 4k +3 occurs in its prime factorization with odd power. We will discuss a short proof of this due to Zagier. We will then discuss an analogous theorem of Lagrange that proves that every positive integer is a sum of four squares. We will discuss an elementary proof and time permitting, another proof using quaternions.