Solution of the Grazing Goat Problem: A Conflict between Beauty and Pragmatism

What is the ideal solution of a problem in mathematics? It depends on your nerd ideology. Pure mathematicians worship the beauty of a mathematics result. Closed form solutions are particularly beautiful. Engineers and applied mathematicians, on the other hand, focus on the result independent of its beauty. If a solution exists and can be calculated, that's enough. The job is done. An example is solution of the grazing goat problem. A recent closed form solution in the form of a ratio of two contour integrals has been found for the grazing goat problem and its beauty has been admired by pure mathematicians. For the engineer and applied mathematician, numerical solution of the grazing goat problem comes from an easily derived transcendental equation. The transcendental equation, known for some time, was not considered a beautiful enough solution for the pure mathematician so they kept on looking until they found a closed form solution. The numerical evaluation of the transcendental equation is not as beautiful. It is not in closed form. But the accuracy of the solution can straightforwardly be evaluated to within any accuracy desired. To illustrate, we derive and solve the transcendental equation for a generalization of the grazing goat problem.

Gauss and the Mathematical Background to Standardisation

Our aim is to explore the links between standardisation, the quantifying spirit, and the discipline mathematics. To do so, we consider the work of Gauss, renowned as a pure mathematician, but professionally an astronomer, and one heavily engaged with all kinds of measuring and precision initiatives. He contributed to the mathematical correction of data with the method of least squares; to observations of high precision in his geodetic work; to the introduction of absolute measures in his collaborations with Weber on terrestrial magnetism; and to the rationalisation of weights and measures in the state of Hannover. Ultimately, the question is to what extent such precision and standardisation activities may have been rooted in the mathematical way of thinking. Mathematics in our tradition has had a strong contemplative bias (theory, theorein in Greek means to contemplate), but it’s a fact that mathematics has always had a non-eliminable technical side.

Jacob Bronowski: a humanist intellectual for an atomic age, 1946–1956

Jacob (‘Bruno’) Bronowski (1908–1974), on the basis of having examined the effects of the atomic bombing of Japan in late 1945, became one of Britain's most vocal and best-known scientific intellectuals engaged in the cultural politics of the early atomic era. Witnessing Hiroshima helped transform him from pure mathematician–poet to scientific administrator; from obscurity to fame on the BBC airwaves and in print; and, crucially, from literary intellectual who promoted the superior truthfulness of poetry and poets to scientific humanist insisting that science and scientists were the standard-bearers of truth. A cornerstone of Bronowski's humanist ideology was that Hiroshima and the bomb had become symbols of the public's distrust of science, whereas, in reality, science was merely a scapegoat for society's loss of moral compass; more correctly, he stressed, science and scientists epitomized positive moral values. When discussing atomic energy, especially after 1949, Bronowski not only downplayed the bomb's significance but was deliberately vague regarding Britain's atomic weapon development programme; this lack of candour was compounded by Bronowski's evasiveness regarding his own prior involvement with wartime bombing. The net effect was a substantial contribution to British scientific intellectuals' influential yet frequently misleading accounts of the relations between science and war in the early atomic era.

The contributions of J.J. Sylvester, F.R.S., to mechanics and mathematical physics

A survey is made of the papers written by J.J. Sylvester (1814–1897) on mechanics and mathematical physics. Some relate to aspects of his professional career. They form only a small part of the output of this largely pure mathematician, but are of variety and intrinsic merit. Their limited total exemplifies the limited measure of interest that physical applications sustained among some mathematicians during a period when the preference for pure mathematics was increasing worldwide.

The young Arthur Cayley

Arthur Cayley, F.R.S. (1821–1895) is widely regarded as Britain's leading pure mathematician of the 19th century. From his start as a Cambridge prodigy, he built up a formidable record of publication, from incidental notes to extensive memoirs, on a wide range of mathematics. His emergence as a mathematician, before he became the sober–minded eminence of the 1860s and 1870s, is largely unknown. The primary guide to his whole life remains, more than a century later, the obituary written by his student and successor at Cambridge, A.R. Forsyth. The folklore surrounding Cayley's life is dominated by his collaboration with James Joseph Sylvester, F.R.S. (1814–1897). This partnership, tempered by Sylvester's obsession with the apportionment of credit, began in earnest in the 1850s. In this paper, I attempt to signpost Cayley's formative years, when he was at the beginning of his long mathematical journey, the period which ends on 3 June 1852, the date of his election to the Royal Society, when he formally came of age as a Victorian man of science.

Bakerian Lecture, 1975: Global geometry

It must be many years since a pure mathematician was asked to address this society on such an occasion. Certainly none appears in the records going back to 1940. This is of course no reflection on the impartiality of the Council of the Royal Society, but simply an acknowledgement of the wide gap that separates pure mathematicians from other scientists and of the serious difficulties of communicating across that gap. Fortunately we have our intermediaries - the applied mathematicians - who extract from the body of mathematical knowledge the most useful parts and bring them to bear on recognizable scientific problems in a wide variety of fields, all the way from the traditional areas of physical science right through to the biological and social sciences. Many distinguished applied mathe­maticians have indeed spoken to the society on problems such as aerodynamic noise or stellar evolution, which are heavily dependent on mathematical analysis, but which can be explained in physical terms readily understood by a wide audience. On such occasions the mathematical techniques involved will quite properly have been relegated to decent obscurity. As a result scientists at large probably have only the vaguest ideas about mathematical research per se . They will understand mathematical work related to their particular field, frequently, I may add, better than the mathematicians themselves, but they must find it hard to visualize mathematics in the abstract. It is therefore perhaps worthwhile for a pure mathematician to attempt to explain how we view our subject and what motivates our research in the absence of any particular scientific interpretation or application. If I might summarize the situation diagrammatically, consider mathematics as some kind of giant computer with a large number of terminals on its periphery, representing fields of application. A practising scientist is like the terminal user. He is primarily interested in the output and will know something about what the computer can do for him, but he is not involved in what goes on inside the heart of the computer. In the early days of computers, users and designers were frequently the same people, but with their rapid growth and sophistication this is now the exception rather than the rule. Similarly it is the increasing sophistication of mathematics which has led to the large gap between ‘users’ and ‘designers’.