Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations ‘sin’, ‘cos’ and ‘tan’ for the trigonometric functions.

Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)

Girard also showed how the area of a spherical triangle depends on its interior angles. If the angles of a triangle on the unit sphere are *A, B* and *C*, then

Area of triangle = *E* = *A* + *B* + *C* – *π*

The amount *E* by which the sum of the angles exceeds *π* radians is called the *spherical excess*. This result is called Girard’s theorem.

Girard was also the first to observe – in 1632 and without a proof – that each prime of the 4*n *+ 1 is the sum of two squares. In 1640, Pierre de Fermat (1601-1665) claimed that he had a proof of this result. Since the proof was announced by Fermat in a letter to Marin Mersenne dated Christmas Day (25 Dec. 1640), it is often known as Fermat’s Christmas Theorem.

**The Christmas Theorem.**

The Christmas Theorem is a result in *additive number theory*. It states that any odd prime *p* is expressible as

*p* =* r *² + *s *²

where *r* and *s* are integers, if and only if *p* is congruent to 1 modulo 4; that is, the remainder on dividing *p* by 4 is 1, written *p* = 1 (mod 4).

For example, the primes 5, 13 and 17 are congruent to 1 modulo 4. They can be expressed as sums of two squares:

5 = 1² + 2² , 13 = 2² + 3² , 17 = 1² + 4² .

On the other hand, the primes 3, 7 and 11 are congruent to 3 modulo 4. Clearly, none of these can be expressed as a sum of two squares.

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To all who visit *thatsmaths.com*, we wish you a **Very Happy Christmas**.

Peter Lynch.

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