The central role of the resolvent cubic in the solution of the quartic was first appreciated by Leonard Euler (1707-1783). Euler's quartic solution first appeared as a brief section (§ 5) in a paper on roots of equations [1, 2], and was later expanded into a chapter entitled ‘Of a new method of resolving equations of the fourth degree’ (§§ 773-783) in his Elements of algebra [3,4].

In Euler derives the approximate summation formula

This primitive result does not appear to be well known to modern readers, and we will try to show by examples that it deserves more attention. After translating this paper into English and reflecting on its contents, we thought it desirable to present Euler’s work in a form convenient for today’s readers. Because of inadequate notations for functions, sums, and definite integrals, the modern reader is likely to find Euler’s writing difficult, even in translation. Also, in Euler’s time, mathematicians were familiar with properties of the conic sections that have largely been ignored today.

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

A ballistic missile trajectory consists of at least three sectors:

• 1. Powered flight from launch until engine cut-off, the point we shall refer to as the burnout point.

• 2. Free flight, which covers most of the trajectory.

• 3. Re-entry, which begins at some point where atmospheric drag forces become a significant factor in determining the subsequent motion of the missile, this sector lasting until impact with the surface of the earth which we shall refer to generally as the ground.

During the free flight sector, the trajectory is part of a conic section—invariably a sector of an elliptic orbit which intercepts the ground—the subject of this article. Figure 1 illustrates some of the important parameters in the analysis. We are concerned here with the free flight sector of the trajectory, which can be treated without any knowledge of the details of the powered flight phase.

In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating series

The ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by

The concept of the ordinary binomial coefficient

can be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers , where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

In the problem proposed by A. G. Ioachimescu in 1895, it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu’s problem in the literature (see, for example, [2], [3], [4, Theorem 1, parts (a) and (b)], [5, problem P2, parts (i) and (ii)],).

The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.