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By Jenny Ramsden by Jenny Ramsden William Oughtred is a mathematician of whom few understand today meaning a:b = c:d was usually written people have heard, yet to whom we owe a great deal as a–b–c–d. Oughtred expressed it as a.b:: c.d, using a dot in the mathematical world. He invented the slide rule, to denote division or a ratio. This book was very popular gave us the multiplication symbol and was responsible in its time and is believed to have been instrumental for the mathematical education of several of our famous (along with Descartes’ Géométrie ) in convincing Newton scientists. All of this from a very quiet, reserved clergyman to study mathematics instead of chemistry as his main whose brain was forever active. subject of study at Cambridge in 1661. Oughtred was born on The invention for which Oughtred is most famous is that 5th March 1574 in Eton, of the slide rule. Napier’s discovery of logarithms in 1614 Buckinghamshire. He had revolutionized the speed at which calculations in atten-ded Eton College arithmetic could be carried out. In 1620 Edmund Gunter and from there he went to plotted a logarithmic scale along a single straight ruler King’s College, Cambridge and by using a pair of dividers, could add and subtract in 1592 and became a lengths in order to carry out multiplication and division Fellow in 1595, gained a calculations. Oughtred extended this idea and used a BA in 1596 and an MA in pair of these logarithmic rulers together, dispensing 1600. Although Oughtred with the dividers, to perform similar calculations. He was passionate about mathematics, despite very little of it being taught at that Circles of Proportion and the Horizontal Instrument , time in either Eton or Cambridge, his main calling was for publishedϐ in 1632, and his rectilinear slide rule was described in an Addition to this work in 1633. As is often was in Shalford in Surrey in 1604; in 1610 he moved to the case, the ownership of the invention of the circular the ͳ͸Ͳ͵Ǥ ϐ parish of Aldbury in Surrey, where he remained for slide rule was disputed very strongly. One of his pupils, the rest of his life, passing away in his vicarage there on Richard Delamain, accused Oughtred of stealing his 30th June 1660. invention from him, since Delamain had published a booklet Grammelogia, or the Mathematicall ring in 1630, Oughtred indulged his passion for mathematics by describing a circular slide rule. Oughtred retaliated, studying well into every evening and by tutoring saying that Delamain had stolen the idea from him. The privately. His most famous pupils were the mathematician row was never settled and dogged Oughtred for the rest John Wallis, astronomer Seth Ward and the architect of his life. However, he did have sole claim to the invention Christopher Wren. He gave these lessons free of charge, of the rectilinear version, which he designed as early as with many of his pupils living in the vicarage while 1621. Immediately adopted by mathematicians, it was undergoing their studies. Oughtred was particularly used extensively (undergoing several adaptations and fond of using symbols in his mathematical writings augmentations) right up until the 1970s, when the hand- and in 1631 published a good systematic textbook on held, electronic calculator saw the slide rule’s demise. arithmetic, called Clavis Mathematicae , which contained almost everything known about the subject at that Oughtred published several other important books time. In it he used over 150 symbols, some of which including Trigonometrie , in 1657, in which he introduced were in common use at that time, others were of his abbreviations for cosine and cotangent, to complement own invention. Three of his symbols introduced in this those already in use since 1626 for sine, tangent and book are still in use today; these are: the multiplication secant due to Girard. This was a very important advance symbol ×, the symbol for absolute difference ~ (where in symbolism, but was mainly forgotten until Euler a~ b is the positive difference between a and b), and the reintroduced them in 1748. On the last two pages of his ‘in proportion’ symbol ::. Prior to this, the equation we work, Oughtred also used the colon : to denote a ratio. 40 Mathematics in School, November 2017 The MA website www.m-a.org.uk Others of his published works were on watch-making, mathematical world owes so much. His slide rule made solving spherical triangles by the planisphere and arithmetical calculations very swift to perform and his methods to determine the position of the Sun. two main texts, Clavis Mathematicae and Trigonometrie , are responsible for the methods and treatment of The cause of Oughtred’s death is apocryphally attributed arithmetic, algebra and trigonometry with which today’s to the excitement and delight he felt on hearing that the mathematicians are so familiar. House of Commons had voted for the King’s return to power (the Restoration) – this is somewhat discredited by the date of his death. More plausible is the idea Keywords: History of mathematics. that he died of old age, since he was 86 years old at his demise. He is perhaps one of the lesser known of English Author Jenny Ramsden mathematicians, yet it is to Oughtred that the present e-mail: [email protected] REVIEWS • REVIEWS • REVIEWS • REVIEWS • REVIEWS to contemporary methods are dealt with principles for teachers to use in class. The Mathematics of Secrets embracing the mathematics of factorials, This includes organizing progression into Joshua Holden permutations, probability and variance. topics and enhancing the mathematical John Wiley and Sons Limited, The By Chapter 4, Holden considers the use knowledge of pupils. Atrium, Southern Gate, Chichester, of computers in cryptology. This requires The next couple of chapters look at West Sussex PO19 8SQ explanations of number bases and binary alternative approaches for pupils rather www.wiley.com codes. Interesting is why, for example, than memorizing a whole series of rules 978-0-691-14175-6 56 bit codes were used instead of 128 that, to most, don’t make much sense. 373 pages, hardback bit ones. The reader then engages with There is an assessment tool for teachers £24.95 recurrence relations to understand stream to identify pupils’ stages of unit measure ciphers, and then modular arithmetic and coordination and is similar to the bars Fermat’s little theorem to understand method from the previous chapter (often I really enjoy those whodunit stories codes. This leads on to public encryption we hear about the Singapore bar method). set in Tudor times. In the one I have just and the RSA (Rivest, Shamir, and Adleman) These are set out in tasks which can be finished the hero is living in Paris and process. Holden finishes by discussing used directly in a classroom. wishes to convey a message to Sir Francis the concepts of quantum computers and The remaining chapters are the Walsingham to inform him of a Catholic lattice based cryptology which gives rise developmental progress from the notion threat against Elizabeth I. The cipher he to concerns about internet security of the of early number concepts through to uses replaces each letter of the message future and the importance of finding new multiplying and dividing fractions finishing by ones in different books. This is a system mathematical techniques. with algebraic fractions. The chapters which Holden refers to early in his account This book is a very comprehensive, if follow a similar pattern being split into of cryptography (the writing of ciphers and not exhaustive, review of the history and three sections – an overview of the topic, codes) and cryptanalysis (the breaking of development of cryptology. It is written with assessment tasks to find out exactly those codes). a real sense of passion and enthusiasm. where a pupil is in their learning and finally The early chapters give a gentle Though the style is relaxed and readable five instructional activities to improve the introduction. The simple process of some of the detail requires some in- learning of that topic. In all, there are replacing letters by shifting them along depth thought by the reader. For anybody 39 assessment tasks and instructional the alphabet is familiar to us all and interested in codes and ciphers this is activities so plenty for all teachers! dates to Julius Caesar. Holden explains an excellent book. To mathematicians, it The book concludes with further how the strength of the cipher depends provides some real world applications of references and an index. There is also a on the factors of the number of letters in number theory and probability. three-page glossary which I have to admit the alphabet and so concepts of modular to referring to a couple of times as some arithmetic and composite numbers are John Sykes terms were different from terms I would introduced. Very quickly he makes use use. The full colour throughout enhances of Euclid’s algorithm to find greatest the diagrams to make the question, task or common divisors and how these relate concept clear. Everything is broken down to the strength of the cipher. The reader in clear and concise explanations and no learns about polyalphabetic ciphers in Developing Fractions Knowledge matter the mathematical background of which more than one letter may have the Amy Hackenberg, Anderson Norton a teacher, this book is accessible to all, same cipher. Frequency analysis of letters and Robert J. Wright primary and secondary. in a language is explained and how this, Sage Publications Ltd, 1 Oliver’s Yard, Chapters can be read individually, but combined with the use of lowest common 55 City Road, London EC1Y 1SP I would advise having a pencil and paper denominators, leads to more efficient www.sage publishing.com close by to take notes as many other cryptanalysis.
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