The theorem in question is that if two of the diagonals of a spherical quadrilateral be quadrantal arcs, the third diagonal is also a quadrantal arc. (Fig. 31.)

Denote the direction cosines of the radius to the point 1 by l1, m1, n1, & c., and l1l2 + m1m2 + n1n2 by 12.

It is evident in the first place as is pointed out by Steiner that the conic will be a parabola if, and cannot be a parabola unless the point at infinity on one range correspond to the point at infinity on the other, that is, the two ranges must be similar. This is the converse of the well-known proposition that a movable tangent to a parabola divides two fixed tangents similarly.

At a recent meeting of the Royal Society of Edinburgh, Professor Tait proposed and solved the following problem:—

To calculate the number of Partitions of any number that can be made by taking any number from 2 up to another given number.

Let us denote by the number of partitions of r obtained by taking any of the numbers 2, 3, 4,……(n − 1), n. In the particular case n = 7, r = 10, the actual partitions are 3 + 7, 4 + 6, 5 + 5; 2 + 2 + 6, 2 + 3 + 5, 2 + 4 + 4, 3 + 3 + 4; 2 + 2 + 2 + 4, 2 + 2 + 3 + 3; 2 + 2 + 2 + 2 + 2; ten in all. Hence =10.

§ 1.— The following account of this problem is taken from the Journal des Débats for December 27th, 1883.

La poste nous a apporté ces jours-ei une petite boîte en carton peint, sur laquelle on lit: la Tour d' Hanoï, véritable casse-tête annamite, rapporté du Tonkin par le professeur N. Claus (de Siam), mandarin du collège Li-Sou-Stian. Un vrai casse-tête en effet, mais intéressant. Nous ne saurions mieux remercier le mandarin de son aimable intention à l'égard d'un profane qu'en signalant la, Tour d' Hanoï aux personnes patientes possédées par le démon du jeu.

In the first volume of Gergonne's Annales de Mathématiques (1810–11), there is a paper by Lhuilier, in which he gives properties of the right-angled spherical triangle, analogous to the following properties of the right-angled plane triangle:

1. The square on the hypotenuse is equal to the sum of the squares on the other two sides;

2. If a perpendicular be drawn from the right angle to the hypotenuse, the square on each side is equal to the rectangle contained by the hypotenuse and the adjacent segment of the hypotenuse;

3. The squares on the sides are to one another as the adjacent segments of the hypotenuse;

4. The square on the perpendicular is equal to the rectangle contained by the segments of the hypotenuse;

5. The hypotenuse, the sides, and the perpendicular are in proportion.