In the configuration, illustrated in Figure 1, ABC is a triangle with I1; I2, I3 the excentres opposite A, B, C respectively. The triangles I1BC, I2CA, I3AB are denoted by T1, T2, T3 respectively. Ok, Hk, are the circumcentre, orthocentre respectively of triangle Tk, k = 1, 2, 3. Lk is the Euler line of Tk, k = 1, 2, 3. Define the point Ek (t) on Lk by the equation

In Classroom Notes No. 178 (Math. Gazette, 1968, pp. 376-380), a problem is set and formally solved on a body falling through a tube in the interior of the Earth. One can condone taking the ideal conditions of a frictionless tube and even overlook the fact that no such tube could withstand the stresses present in the Earth below some 30 kilometres depth. But when, as in this Note, it is further assumed (without even reference to any assumption) that g is proportional to r, the calculations, whatever mathematical interest they might have, cannot be regarded as having any relation whatever to the planet Earth.

At some time in the eighties or nineties of the last century the Rev. William Done Bushell, a mathematics master at Harrow School, bought a portrait painting at an auction sale at a house in Harrow. It does not seem to have had any pedigree but bore the inscription “Robt. Recorde M.D. 1556”. It was thus presumed to depict the Tudor physician and mathematician Robert Recorde (1510?–1558) who wrote some of the earliest books in English on mathematics and was the first to use the symbol “=” for equality. No other portrait was (or is) known.

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:

(a)

(b) , also for complex values of x;

(c) x → exp (x) is the unique solution to the initial value problem [4]

(d) x → exp (x) is the inverse of

(e)x → exp (x) is the unique continuous function satisfying the

functional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];

(f) Define dr for rational r, and then use a continuity/density argument [8].

All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.