For many years candidates (and teachers) involved in A-level work have quoted Newton’s experimental law of restitution as one of the principles enabling collision problems to be solved. It had always seemed to me that it is fortunate that such a simple law holds and also that it is a great pity that there is no simple relationship between the initial and final kinetic energies. Recently, however, I realised that, viewed in terms of invariance, the law is not at all surprising and also that there is a very simple way of dealing with the loss of kinetic energy.

Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.

In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Euler was impressed by his work on this topic and recommended his admission to the Berlin Academy of Science. Euler then generalised Fagnano's work on integrals. In addition, Fagnano was employed to assist in reinforcing the dome of Saint Peter's which was in danger of collapse. Pope Benedict IV rewarded him for his work by publishing his mathematical papers. Fagnano achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

I am continually surprised by the mathematical structure which surfaces in unexpected situations, and even more so when arranging a tenth birthday party for my elder son. The invasion of the home by some eight rather energetic boys on a cold February afternoon was viewed with some trepidation. The instinct of self preservation was uppermost in the search for activities which would allow the afternoon to pass with the minimum of discomfort and the maximum enjoyment. A family friend is very good at organising team games at parties, but with a space problem (our house does not have a large living room as so many modern houses have these days) it was important to split the children up for their activities, but yet devise a programme which was coherent.

§ 11. We are now in a position to understand how Napier completed his Tables.

We take, to begin with, his First Table. The logarithm of the first term is zero. From § 10, Theorem I., the logarithm of the second term lies between r-s and (r-s)r/s; that is, between 1.000 000 0 and 1.000 000 I. If this is taken as the A.M. of these two numbers, we have for the logarithm of the second term in that Table the value 1.000 000 05. The remaining terms in that Table form with these two a G.P., so that the logarithms have a common difference 1.000 000 05, and the logarithm of the last term can be taken as 100.000 005.

The polar equation of the parabola y2=8ax, referred to the origin of coordinates as pole is

r sin2θ = 8a cos θ

or

Inverting with the pole as centre of inversion and 4a as the radius of inversion we find that the equation to the inverse curve is

or r cos θ = 2a sin2θ

or r = 2a sin θ tan θ.

The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations

(1)

and

(2)

(Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a0, the smaller has perimeter b0. (Archimedes used hexagons with and b0 = 3, but regular polygons of any number of sides can begin the iterations.) Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated, from (2) and (3). (See [1] for a derivation or many explanations on the web.) Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon.