Throughout the article we use areal (or barycentric) coordinates. The side lengths of BC, CA, AB are denoted as usual by a, b, c and for brevity and ease of reading we write a2 = p, b2 = q, c2 = r. The symmedian point K then has coordinates (a2, b2, c2 ) replaced by (p, q, r), it being the isogonal conjugate of the centroid, and the circumcentre O then has coordinates (a2(b2 + c2 − a2), b2(c2 + a2 − b2), c2(a2 + b2 − c2)) replaced by (p(q + r − p), q(r + p − q), r(p + q − r)). The construction of the triplicate ratio circle (also known as the Lemoine circle) and the Brocard circle is replicated but starting with the circumcentre O rather than the symmedian point K. It is found that the triplicate ratio circle is then replaced by a conic cutting internally each side of ABC twice and having centre X the midpoint of OK. The Brocard circle is replaced by a conic passing through O and K and also having centre X the midpoint of OK. These we define as the twin conics.

The proportion angle of a non-square rectangle R, written ‘Pa (R)’, is defined to be the angle r < between a diagonal and the longer attached side. Circumscribe around R any other rectangle M that carries the corners of R on its sides, and let its proportion angle be m, which is also < . Then the theorem states that r < m.

This article concerns the evaluation of the ‘logsine’ integral

We shall encounter it in several guises. Indeed, standard integration techniques used below readily show that (1) has the same value as the following integrals:

En passant, it is worth noting that forms (6) and (7) are the best behaved for numerical integration.

I first met the logsine integral as a callow youth in that strange hinterland of results that you may not have met at school but were not guaranteed to meet later on either.

A common problem in modelling population change, whether of humans or animals, is estimating the values of the various parameters in the mathematical functions used to describe the population trajectory. Typically both the population's rate of growth and ultimate maximum size are unknown and so various numerical approximation methods have to be used (see [1, 2 and 3]). In this article a mathematical method of population projection is presented which avoids these difficulties and takes the baseline case that the only data available for computation are the populations from just three censuses. It is a method indicated, but not developed by Keyfitz [4].

The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. Much to his surprise, however, the author recently came across the following result and theorem by Sylvester (1814-1897) in [1] that involves an interesting application of forces that relate to the Euler line (segment). This result is also mentioned in [2] without proof or reference to Sylvester.

Traditionally ‘Applied Mathematics’ meant mathematics applied to mechanics, physics and the hard sciences, leaving the application of mathematics to social sciences fairly neglected. This is both a challenge and an opportunity.

An instance of this is in the science of voting.

Let's go down to Chuddlehampton-upon-Severn to see how mathematics can be applied in a ‘real-life situation’. The good folk of this small village have to elect their local councillor, the person to represent them at the Regional Council. “Go and elect someone” was the only instruction they were given. So how were they to go about it?

In the midst of reading Robert Graves' I, Claudius [1], I was surprised to encounter a description of an interesting cipher used by the fictional Augustus. I therefore thought it would be worthwhile to write a short article to draw attention to what is perhaps not a particularly well-known cipher appearing in fiction. I will assume that the reader is familiar with the principle behind the famous Vigenère cipher (see [2] or [3]).

In the novel, Claudius, as narrator, relates how, upon Augustus' death, he is succeeded somewhat reluctantly by his stepson Tiberius, due to the machinations of Livia, who was Augustus' widow and Tiberius' mother. As her son's imperial reign begins, it is clear that Livia intends to hold on to the influence that she wielded during her husband's.

In Catalan numbers with applications [1], Thomas Koshy gives a clear and accessible account of some of the important characteristics of Catalan numbers. The author also discusses the significance of these numbers for mathematicians, scientists, engineers and amateur enthusiasts alike. The general contents of his book are reviewed in [2]. Catalan numbers, which are named after the mathematician Eugene Catalan (1814-1894), have very interesting and sometimes surprising properties.

Catalan numbers occur in combinatorial mathematics, in particular, where they show up as a natural number sequence that arises in counting problems that usually involve recursively-defined objects.

Here is a result which will surely make one sit up. We start with a sequence t1 = (1, 1, 2, 2, 4, 4, 8, 8, 16, 16, … ) in which each power of 2 occurs twice in succession, and produce a second sequence whose n th term is the sum of the n th and (n + 1) th terms of the original one; we get the sequence t2 = (2, 3, 4, 6, 8, 12, 16, 24, 32, … ). We repeat the same operation on the resulting sequence, and continue this, iteratively. Here are the sequences t3, t4, t5:

t3 = (5, 7, 10, 14, 20, 28, 40, 56, 80, 1l2, 160, 224, 320, 448,…),

t4 = (12, 17, 24, 34, 48, 68, 96, 136, 192, 272, 384, 544, 768,…),

t5 = (29,41, 58, 82, 116, 164,232, 328, 464, 656, 928, 1312,…).

If we compute the ratio of the second term to the first term for each ti, we get the following sequence of rational numbers,

which converges to the square root of 2. (These fractions turn out to be successive convergents to the simple continued fraction for √2; see Section IV for details.)

In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.

The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.

In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.

have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].

We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

Given two rational points on the graph of a cubic polynomial with rational coefficients, it is well known that one can construct another rational point by finding the third intersection point of the curve with the chord through the two points. For example, consider the two points A (2, 15) and B(0, 3) on y = 2x3 − x2 + 3.

The chord through A and B has equation y = 6x + 3 and this intersects the curve again at the point P, where

This cubic equation has three roots corresponding to A, B and the new point P,

P is the point

Step is a public examination administered by the admissions tests division of Cambridge Assessment (which is the parent body of the OCR examination board). It used as a basis for conditional offers by Cambridge and Warwick.

A grade one on a STEP paper is awarded to candidates who produce nearly complete solutions to four questions. The duration of the paper is three hours, so each question is designed to take a good candidate about 45 minutes to complete. Not surprisingly, then, the questions can be quite profound, often having depths far beyond what is required for the examination. One such question haunted me, on and off, for many years. It turns out that the underlying idea is elementary and far from new; but it is nevertheless very striking and not, I suspect, well known. A pleasing by-product is a relatively easy proof of the AM–GM (algebraic mean–geometric mean) inequality.