Most of the standard textbooks and many teachers of Applied Mathematics make use of a velocity-time graph to facilitate the solution of elementary kinematical problems including vertical motion under gravity, but it is unusual to extend the method to projectiles and allied problems, in spite of the ease with which the more usual type of problem can be solved. Such problems may be solved by simple calculations based upon the properties of similar triangles and the areas of rectangles, triangles and trapezia, that is, by the normal velocity-time graph procedure.

The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative. We shall often ignore the fact that A3, A5, .. vanish, thereby refusing to complicate formulae unnecessarily.

In the development of the theory of complex numbers, it is important to give a definition of them dependent only upon real numbers. In the usual algebraic treatments the product is postulated in the form

or obtained from a matrix representation. The object of the present note is to show by elementary methods that the assumption that the multiplication of complex numbers is distributive, associative, and such that the vanishing of a product implies the vanishing of at least one factor, is sufficient to define the product. This is indeed merely a special case of results well known in the theory of linear algebras.