If ABCDE … is a polygon of an odd number of sides, O any point in its plane, and if AO, BO, CO, … cut the sides opposite A, B, C, … respectively, AB in X, BC in Y, CD in Z, … then

Let PQ be the side opposite A. Then AB will be the side opposite Q.

In a letter to Stirling, dated July 27, 1738, Euler mentions having happened on the infinite product, “satis notatu dignam,” the numerators being the odd primes in their natural order, the denominators the multiples of 4 nearest to those primes. He says he can prove that the limit of this product is His proof was probably as follows:

Girard enunciated in 1625 the following celebrated theorem, which is associated with the name of Fermat:

Every prime of the form 4m + 1 is the sum of two squares in one way only, and no prime of the form 4m - 1 is a factor of the sum of two squares that are not both multiples of that prime.

Theorem.

If a circle cut all the sides (produced if necessary) of a regular polygon, the algebraic sum of the intercepts, on the sides, between the vertices and the circle is zero.

(1) If the straight line joining two points subtends equal angles at two other points on the same side of it, the four points are con cyclic.

Let O be the centre of the circumscribing circle of ΔABC, A1 the middle point of BC, and EA1OF the diameter at right angles to BC. Draw AX perpendicular to BC and produce it to meet the circle in K. Let H be the orthocentre of ΔABC; join OH and bisect it in N, the centre of the nine-point circle.