Find the path taken by a ship which pursues, and always directs its course towards, another ship (supposed moving in a straight line), their speeds being constant and in fixed ratio, and their initial directions at right angles.

In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem: Of two unequal angles, the larger has the shorter bisector (see [1, 2]).

Sturm passed the problem on to other mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask.

We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.

In Mathematical Note 1911, this journal, Vol. XXX, No. 290, A. H. MacColl forms the determinant Δ of a pan-magic square of order 4 and states that Δ = 0. He also remarks that “the property appears to be peculiar to pan-magic squares of order 4”. Below we give our demonstration of the above theorem and extend it to the determinants of all pan-magic squares of even order. We shall show that the determinants of all pan-magic squares of even order have zero as their value. In the proof that follows below we make use of a property of pan-magic squares of even order.

The following activity is one of a series developed for junior high school students (age 12-15). Each of the activities involves a historical discussion of a subject related to a certain curriculum topic, and takes the form of a teacher directed presentation/discussion with accompanying transparencies and worksheet.

The choice and development of the activities was guided by certain general principles. The connection between the regular curriculum and the selected subject is meant to give relevance to the history and also, to motivate and deepen student understanding of subject matter. The structure of the activity enables the teacher to change, omit, or to add if he wishes. Points for discussion and questions to be worked are deliberately inserted at intervals to activate students, because in previous trial versions which were closer to the lecture format we found that students lost the thread and hence, also interest. Reactions to the present format were much more encouraging.

The classical problem of the knight’s tour consists of moving a knight over a chess board in such a manner that it moves successively on to every possible square once and only once. If the initial and final squares of this tour are a knight’s move away from each other, then for obvious reasons the tour is termed re-entrant. The problem has a long and interesting history. Solutions due to De Moivre, Euler, Vandermonde, Warnsdorff and Roget, together with further references can be found in [1].

In Cundy and Rollett’s invaluable book on Mathematical Models, the authors begin their Chapter on Polyhedra by remarking that “The most suitable, and in many ways the most attractive, subject for an experiment in the construction of mathematical models is a set of polyhedra.” Various methods are in general use to produce finished models of polyhedra for the showcase, and similar methods form the basis of those considered suitable for constructing the simpler polyhedra in the classroom. That most commonly adopted is to draw the net of the required solid on a sheet of cardboard, allow for tabs as necessary, cut it out, score the creases half-through, fold up, and stick the tabs with some suitable quick-drying cement. For class-room use the major drawback to model making on these lines is the time-consuming and potentially messy process of sticking; and it is the object of this article to develop a method which dispenses with the use of paste or cement altogether. In fact I shall show how, using paper and scissors only, firm, neat models can be made, which in the case of the simpler solids could easily be produced in the classroom, and which in the case of the more complex polyhedra reveal by their construction in a striking manner some of the geometrical relationships of these solids.

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞

In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.

We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l by

where the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.

The evaluation of moments of inertia of simple solids is frequently regarded merely as a convenient source of problems in the integral calculus. From the point of view of the student of dynamics it is, however, perhaps even more instructive to make these calculations without any actual integration, for in this way he may obtain a deeper insight into the physical significance of the underlying concept. Whether this be so or not it seemed to the writer an amusing task (performed during a moment of inertia caused by a bacteriological infection) to see how many of the well known formulae could be derived by such methods.