Rouse Ball Research Articles

A colouring problem for the dodecahedral graph

We consider vertex colourings of the dodecahedral graph with five colours, such that on each face the vertices are coloured with all the five colours. We show that the total number of these colourings is 240. All such colourings can be obtained from any given such colouring, by permuting the colours, and possibly applying central symmetry with respect to the centre of the regular dodecahedron. For any such colouring, the colour classes form the vertex sets of five regular tetrahedra. These tetrahedra together form one of the two compounds of five tetrahedra, inscribed in the regular dodecahedron. We give two proofs: a combinatorial one, and a geometrical one. Our result is related to the result in W. W. Rouse Ball -- H. S. M. Coxeter, stating that there are four such colourings, as follows. There are four such colourings, up to applying an arbitrary orientation-preserving congruence of the regular dodecahedron.

Alan Turing's First Cryptology Textbook and Sinkov's Revision of it

Seventeen-year-old Alan Turing received a copy of the tenth edition of W. W. Rouse Ball's Mathematical Recreations and Essays as part of a school award. Ball's book contained a chapter titled “Cryptographs and Ciphers.” This might have been Turing's first cryptology textbook. In this paper we will examine what Turing could have found in Ball's chapter, and we will examine changes made to later editions by Ball and by Abraham Sinkov.

NONE, ONE OR SEVERAL? PERSPECTIVES ON THE UK’S CONSTITUTION(S)

AN inaugural lecture is the occasion when the University of Cambridge can look its gift horse in the mouth, weighing the new professor in the balance against his or her distinguished predecessors. The Rouse Ball Professorship of English Law has been held in the past by a long series of distinguished scholars, from Sir Percy Winfield to my immediate predecessor, Sir Jack Beatson whom we are delighted to welcome back today. Their work has influenced generations of lawyers. They certainly influenced me. Before I encountered Criminal Law: The General Part, a great little volume by Professor Glanville Williams, Learning the Law, was my “Guide, Philosopher and Friend” (as it still says on the cover of the latest edition, now edited by my colleague Professor Tony Smith) as I approached the study of law. Another Rouse Ball Professor, the late Sir William Wade, had a formative effect on my understanding of land law and administrative law both through his famous books, Megarry and Wade on the Law of Real Property (now edited by a former Fellow of Downing College, Dr. Charles Harpum) and Administrative Law (now in the hands of my colleague Dr. Christopher Forsyth), not to mention the lectures that I attended as an undergraduate in (softly be it said) the University of Oxford.

Has the Common Law a Future?

I must begin with a few words about my predecessor in the Rouse Ball chair, Sir David Williams. David Williams has had a career of outstanding service to legal studies, to universities, in particular Cambridge, and to the wider public. After completing his studies, he became one of the formidable group at the University of Nottingham's Law Faculty. He went on to Oxford—he has told me that he went there as a missionary—and during his time there produced his pathbreaking books on official secrets and public order, Not in the Public Interest and Keeping the Peace. He was, it must be said, not the only Cambridge public lawyer-missionary in Oxford. Sir William Wade was also there. By 1967 it appears that two missionaries were no longer required, and David Williams returned to Cambridge. In 1982 he succeeded Wade—by now also back in Cambridge—as Rouse Ball Professor. He has been an important presence in the world of administrative law and his contribution to environmental issues has been enormous. We are delighted that now he has laid down the burdens of office as Vice-Chancellor he has returned to the Faculty— albeit to a different chair.

An efficient algorithm for the Knight's tour problem

A knight’s tour is a series of moves made by a knight visiting every square of an n x n chessboard exactly once. The knight’s tour problem is the problem of constructing such a tour, given n. A knight’s tour is called closed if the last square visited is also reachable from the first square by a knight’s move, and open otherwise. Define the knight’s graph for an n x n chessboard to be the graph G = (V,E), where V={(i,j) ( 1 d i,j Gn}, and E={((i,j),(k,Q) ( {Ii-kl,Ijt[} = {1,2}}. That is, there is a vertex for every square of the board and an edge between two vertices exactly when there is a knight’s move from one to the other. Then, more formally, an open knight’s tour is defined to be a Hamiltonian path, and a closed knight’s tour is defined to be a Hamiltonian cycle on a knight’s graph. A knight’s graph has n2 vertices and 4n2 12n + 8 edges. The formal study of the knight’s tour problem is said to have begun with Euler [lo] in 1759, who considered the standard 8 x 8 chessboard. Rouse Ball and Coxeter [l] give an interesting bibliography of the history of the problem from this point. Dudeney [8, 91 contains a description of exactly which rectangular chessboards have knight’s tours; in particular, an n x n chessboard has a closed knight’s tour iff n Z 6 is even, ’ and an open knight’s tour iff n 3 5. It is not clear who first proved this fact, but it appears to be part of the folklore of the subject (see, for example, [2]). There exist several independently conceived linear time (i.e. O(n2)) algorithms for constructing knight’s tours (see, for example, [5, 191). Takemji and Lee [22, 231 recently proposed a neural network solution to the knight’s tour problem, although it appears to be of little use in practice (see [ 17, 181). We will describe in this paper a new, simple, and fast algorithm for constructing knight’s tours on square boards.

The prehistory of the Principia from 1664 to 1686

Since S.P. Rigaud’s pioneering Historical essay appeared in 18381 there have been many, from Rouse Ball, Cajori and Beth down to I.B. Cohen, A.R. Hall, J.W. Herivel and R.S. Westfall in our own day, who have explored how Newton’s Principia came to be. 2 Surely there can be nothing profoundly new to be said about its progress from first conception as an inchoate idea in its author’s mind to the maturity of its first publication in 1687? No and yes. There is now a broad balance of agreement over the main stages in its evolution: one no longer set greatly awry by the nuggets of Principia gold still (if with decreasing frequency and size) to be sieved from Newton’s papers by those willing laboriously to do the panning. Anyone not of the fraternity, however, would surely be surprised to see how much Newton scholars can still at times find to disagree upon in assessing what is now in itself known in such abundance, sometimes even at the most basic level of dating a manuscript. 3 As for the changes that must now be made in the accepted account, these only slowly filter through. How often am I still asked: 'Did Newton use calculus to obtain the theorems in his Principia ?’ How, without seeming to patronize, do you lay the groundwork on which you can reply that the question is ill-formed and therefore meaningless? I will not here go into the reasons why. 4 But I would like briefly to tell anew the tale of how Newton wrote his Principia , embellishing it with some of the freshnesses of insight that have come out of recent research.

Book Reviews: The preliminary manuscripts for Isaac Newton's 1687 Principia 1684-1685. Facsimiles of the original autographs, now in Cambridge University Library , with an introduction by D.T. Whiteside

The preliminary manuscripts for Isaac Newton's 1687 1684-1685. Facsimiles of the original autographs, now in Cambridge University , with an introduction by D.T. Whiteside. Cambridge University Press, 1989. Pp. xxi + 246. £60. ISBN 0-521-33499-3. How vastly Newtonian studies have changed in less than half a century! In 1947, a jejune research student, I made a first daring reconnaissance into those Newton manuscripts in the Cambridge University Library which, together with those dispersed in 1936 (and now largely in academic keeping), have since so vastly enriched our knowledge of the greatest of all scientific figures readily accessible to us, nay, the greatest of all. The most important of the manuscripts in this vast body of material are here reproduced in excellent photographic facsimile under the direction of Professor Whiteside. I do not know who before myself had troubled the peace of the Principia papers since they had been (sparingly) catalogued by a committee of distinguished Cambridge scientists in the 1880s. Certainly not Louis Trenchard More, whose biography of Newton appeared in 1934. Only W.W. Rouse Ball had long before looked into the Portsmouth Collection so that his Essay on Newton's Principia (1893) remained for many years the sole important addition to Principia manuscript studies since the publications of Rigaud, Edleston and Brewster.

Textbook of Criminal Law. Second edition. By Glanville Williams, Q.C, LL.D., F.B.A., Honorary and Emeritus Fellow of Jesus College, Cambridge; Honorary Bencher of the Middle Temple; formerly Professor of Public Law and Quain Professor of Jurisprudence in the University of London, and Rouse Ball Professor of English Law in the University of Cambridge. [London: Stevens & Son. 1983. xlvii, 987

Textbook of Criminal Law. Second edition. By Glanville Williams, Q.C, LL.D., F.B.A., Honorary and Emeritus Fellow of Jesus College, Cambridge; Honorary Bencher of the Middle Temple; formerly Professor of Public Law and Quain Professor of Jurisprudence in the University of London, and Rouse Ball Professor of English Law in the University of Cambridge. [London: Stevens & Son. 1983. xlvii, 987 and (Index) 19 pp. Hardback £28·00, paperback £16·50 net.] - Volume 43 Issue 2

Textbook of Criminal Law. By Glanville Williams, q.c., ll.d., f.b.a., Rouse Ball Professor of English Law in the University of Cambridge; Honorary Bencher of the Middle Temple. [London: Stevens & Sons. 1978. xl, 962 and (Index) 11 pp. Hardback, £16.00; paperback, £10.00 net.

Textbook of Criminal Law. By Glanville Williams, q.c., ll.d., f.b.a., Rouse Ball Professor of English Law in the University of Cambridge; Honorary Bencher of the Middle Temple. [London: Stevens & Sons. 1978. xl, 962 and (Index) 11 pp. Hardback, £16.00; paperback, £10.00 net.] - Volume 38 Issue 2

The third revolution in 20th century physics

“Twentieth century physics already has undergone two breathtaking revolutions—in relativity and in quantum mechanics. We may be standing on the threshold of a third.” Thus spoke Geoffrey F. Chew on the occasion of the Rouse Ball Lecture which he delivered at the University of Cambridge on March 12, 1963. The title of this lecture is “The Dubious Role of the Space‐Time Continuum in Subatomic Physics”.

A Short Account of the History of Mathematics, by W. W. Rouse Ball. Dover Publications Inc. , New York, 1960, xxiv + 522 pages. $2. 00 - The History of the Calculus and its Conceptual Development, by Carl B. Boyer. Dover Publications Inc., New York, 1959. 346 pages. $2. 00.

A Short Account of the History of Mathematics, by W. W. Rouse Ball. Dover Publications Inc. , New York, 1960, xxiv + 522 pages. $2. 00 - The History of the Calculus and its Conceptual Development, by Carl B. Boyer. Dover Publications Inc., New York, 1959. 346 pages. $2. 00. - Volume 5 Issue 2

REVIEWS

Book reviewed in this article:The Sanctity of Life and the Criminal Law. By Glanville WilliamsThe Danish Criminal Code, With an Introduction by Dr. Knud Waaben, Professor A.I. in the University of Copenhagen.Introduction to the Law of Property. By F. H. Lawson, Professor of Comparative Law in the University of Oxford.The Law of Wills. By S. J. Bailey, m.a., ll.m., Barrister‐at‐Law, Rouse Ball Professor in the University of Cambridge. Fifth edition.The Survival of General Average. By Knut S. Selmer.Die faktischen Vertragsverhältnisse als Ausdruck der gewan‐delten sozialen Funktion der Rechtsinstitute des Privatrechts. By Spiros Simitis.An Introduction to American Administrative Law. By Bernard Schwartz, Professor of Law and Director of the Institute of Comparative Law, New York University.The Inter Se Doctrine of Commonwealth Relations.The Universal Declaration of Human Rights. Its origins, significance, application and interpretation. By Nehemiah Robinson.Termination of Membership of International Organisations. By Nagendra Singh.Green's Death Duties. Fourth edition. By C. D. Harding, ll.b. (Lond.) of the Estate Duty Office.The Civil Practice of the Magistrates' Courts in South Africa. By Jones and Buckle. Sixth edition by the Hon. Mr. Justice Joseph Herbstein, P. W. E. Baker, b.a., ll.b., Advocate of the Supreme Court of South Africa and S. Aaron, b.a., ll.b., Advocate of the Supreme Court of South Africa.

The Law of Wills, including Intestacy and Administration of Assets. Fifth Edition. By S. J. Bailey, m.a., ll.m., of the Inner Temple, Barrister-at-Law, Fellow of St. John's College, Cambridge; Rouse Ball Professor of English Law in the University of Cambridge. [London: Sir Isaac Pitman & Sons, Ltd.1957. lii and 306 and (index) 25 pp. 50s. net.

The Law of Wills, including Intestacy and Administration of Assets. Fifth Edition. By S. J. Bailey, <scp>m.a.</scp>, <scp>ll.m.</scp>, of the Inner Temple, Barrister-at-Law, Fellow of St. John's College, Cambridge; Rouse Ball Professor of English Law in the University of Cambridge. [London: Sir Isaac Pitman &amp; Sons, Ltd.1957. lii and 306 and (index) 25 pp. 50s. net.

Who Made Notable Contributions to the Development of English Law or its Literature

These notes on thirty-six judges and chancellors, prompted by memory of my own requirements fifty years ago, were prepared for distribution on stencilled sheets to the students attending my lectures on legal history at the Inns of Court. My aim was to provide both indications of the principal achievements of each of the lawyers named, and also references to readily accessible sources of further knowledge.The editor of this journal has kindly suggested that it would be useful to its readers to have my notes available in print.It is not nearly as difficult as it used to be for beginners to find out about the great legal figures of the past. Sir William Holdsworth, Vinerian professor at Oxford from 1922 to 1944, placed all lawyers in his debt by his book, Some Makers of English Law, published in 1938. It was based on the Tagore lectures which he had given in Calcutta.Sir Percy Winfield, Rouse Ball professor at Cambridge from 1926 to 1943, gave detailed information as to the principal law books of the past and their editions in his manual The Chief Sources of English Legal History (1925) based on lectures given at the Harvard Law School. Twenty-four of my judges and chancellors have entries in his book as authors.By far the most numerous of my references are to Holdsworth's monumental History of English Law, in thirteen volumes, cited as H.E.L. The other works most referred to are The Dictionary of National Biography cited as D.N.B.; Fourteen English Judges (1926) by the first Earl of Birkenhead, L.C. 1919–1922; and The Victorian Chancellors (1908) by J. B. Atlay.

REVIEWS

Book reviewed in this article:The Legal Aspect of Money. By F. A. Mann. 2nd ed.The Queen's Peace. By Sir Carleton Kemp Allen, q.c., f.b.a., Fellow of University College, Oxford: Being the fifth series of Hamlyn Lectures, delivered in the University College of Wales, Aberystwyth, 1953.French Administrative Law and the Common‐Law World. By Bernard Schwartz.The Machinery of Justice in England. By R. M. Jackson, ll.d. 2nd edition by the author.English Courts of Law. By H. G. Hanbury, d.c.l. 2nd edition by the author.De Quelques Piliers des Institutions Britanniques. By M. Jean Duhamel and M. J. Dill Smith.McNair's The Law of the Air. 2nd Edition. By Michael R. E. Kerr, m.a., Barrister‐at‐Law and Robert E. MacCrindle, ll.b., Barrister‐at‐Law.The Law of Tort in Local Government. By Mary Bell Cairns, ll.m., ph.d.The Law of Trusts. Sixth Edition. By George W. Keeton, m.a., ll.d., of Gray's Inn, Barrister‐at‐Law; Professor of English Law in the University of London; Dean of the Faculty of Law, University College, London.The Law of Wills. Fourth Edition. By S. J. Bailey, m.a., ll.m., of the Inner Temple, Barrister‐at‐Law, Rouse Ball Professor of English Law in the University of Cambridge.The Meaning of Income in the Law of Income Tax. By F. E. LaBrie, Associate Professor in the School of Law, University of Toronto.Hans Würdinger. Freiheit der persönlichen Entfaltung, Kartell‐ und Wettbewerbsrecht. Juristische Studiengesellschaft, Karlsruhe. Schriftenreihe, Heft 8.Securities for Bankers' Advances. By J. Milnes Holden, ll.b., PH.D., a.i.b., of Lincoln's Inn, Barrister‐at‐Law.Gilbart Lectures on Banking, 1953. The Law relating to the Collection of Cheques by Bankers for their Customers. By Lord Chorley, m.a., Barrister‐at‐Law.Problems of Parliamentary Government in Colonies. A Report prepared by the Hansard Society.Pakistan Constitution: An Essay. By C. M. Shafqat. Foreword by Mr. Justice Kayani.Le Conseil d'Etat: Livre jubilaire.The Business Law Review: a Journal of Commercial Law and Practice. Edited by Clive M. Schmitthoff. Vol. I, 1954.

Select Legal Essays. By Sir P. H. Winfield, q.c., f.b.a., ll.d., Emeritus Professor of English Law, Cambridge, Fellow of St. John's College, Cambridge; of the Inner Temple, Honorary Bencher and Barrister-at-Law. With an Editorial Preface by S. J. Bailey, Rouse Ball Professor of English Law in the University of Cambridge. [London: Sweet &amp; Maxwell, Ltd. 1952. xv and 287 and

Select Legal Essays. By Sir P. H. Winfield, q.c., f.b.a., ll.d., Emeritus Professor of English Law, Cambridge, Fellow of St. John's College, Cambridge; of the Inner Temple, Honorary Bencher and Barrister-at-Law. With an Editorial Preface by S. J. Bailey, Rouse Ball Professor of English Law in the University of Cambridge. [London: Sweet &amp; Maxwell, Ltd. 1952. xv and 287 and (index) 5 pp. 37s. 6d. net.] - Volume 11 Issue 3

A Textbook of the Law of Torts. Fourth edition. By P. H. Winfield, k.c., f.b.a., ll.d., Emeritus Rouse Ball Professor of English Law in the University of Cambridge, of the Inner Temple, Honorary Bencher and Barrister-at-Law. [London: Sweet &amp; Maxwell, Ltd.1948. xxxvi and 666 and (index) 15 pp. 40s. net.] - Cases on the Law of Tort. Fourth edition. By

A Textbook of the Law of Torts. Fourth edition. By P. H. Winfield, k.c., f.b.a., ll.d., Emeritus Rouse Ball Professor of English Law in the University of Cambridge, of the Inner Temple, Honorary Bencher and Barrister-at-Law. [London: Sweet &amp; Maxwell, Ltd.1948. xxxvi and 666 and (index) 15 pp. 40s. net.] - Cases on the Law of Tort. Fourth edition. By the same author [London: Sweet &amp; Maxwell, Ltd.1948. xii and 331 pp. 30s. net.] - Volume 10 Issue 2

English Legal Authors before Blackstone. By H. A. Hollond, Rouse Ball Professor in the University of Cambridge, Reader to the Council of Legal Education [London: Stevens &amp; Sons, Ltd.1947. 42 pp. 3s.

English Legal Authors before Blackstone. By H. A. Hollond, Rouse Ball Professor in the University of Cambridge, Reader to the Council of Legal Education [London: Stevens &amp; Sons, Ltd.1947. 42 pp. 3s.]. - Volume 10 Issue 1

A Text-Book of the Law of Tort. Third Edition. By P. H. Winfield, K.C., F.B.A., LL.D., Emeritus Rouse Ball Professor of English Law in the University of Cambridge, of the Inner Temple, Honorary Bencher and Barrister-at-Law. London: Sweet &amp; Maxwell, Ltd.1946. xxxii and 652 pp., and Index 667 pp. (35s. net.)

A Text-Book of the Law of Tort. Third Edition. By P. H. Winfield, K.C., F.B.A., LL.D., Emeritus Rouse Ball Professor of English Law in the University of Cambridge, of the Inner Temple, Honorary Bencher and Barrister-at-Law. London: Sweet &amp; Maxwell, Ltd.1946. xxxii and 652 pp., and Index 667 pp. (35s. net.) - Volume 9 Issue 3

Pollock's Principles of the Law of Contract. Twelfth Edition. By P. H. Winfield, K.C., F.B.A., LL.D., Emeritus Rouse Ball Professor of English Law in the University of Cambridge and Honorary Bencher of the Inner Temple. London: Stevens &amp; Sons, Ltd.1946. xl and 571 pp. and Index 602 pp. (40s. net.)

Pollock's Principles of the Law of Contract. Twelfth Edition. By P. H. Winfield, K.C., F.B.A., LL.D., Emeritus Rouse Ball Professor of English Law in the University of Cambridge and Honorary Bencher of the Inner Temple. London: Stevens &amp; Sons, Ltd.1946. xl and 571 pp. and Index 602 pp. (40s. net.)