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BSHM Bulletin: Journal of the BritishSociety for the History of MathematicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tbsh20

ReviewsMassimo Mazzotti , Mark McCartney , Anthony V Piccolino ,Snezana Lawrence & Amirouche MoktefiPublished online: 22 Jun 2009.

To cite this article: Massimo Mazzotti , Mark McCartney , Anthony V Piccolino , Snezana Lawrence& Amirouche Moktefi (2009) Reviews, BSHM Bulletin: Journal of the British Society for the History ofMathematics, 24:2, 121-130, DOI: 10.1080/17498430902848789

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BSHM Bulletin

Volume 24 (2009), 121–130

Reviews

Antonella Cupillari, A biography of Maria Gaetana Agnesi, an eighteenth-centurywoman mathematician, with translations of some of her work from Italian intoEnglish, The Edwin Mellen Press, 2007, viiþ322 pp, $119.95, ISBN 0 7734 5226 5

Investigating the active involvement of women in scientific practices during theearly modern period is a relatively recent field of study. It is not coincidental thatmost of the materials analysed thus far are from northern Italy, a region in

which scientific life was characterized by the unusual presence of women in scientificacademies and universities. The best-known case is that of Laura Bassi (1711–78),professor of experimental physics at the University of Bologna and a member of thelocal academy of sciences. In the last twenty years a series of studies have focused onBassi’s life and work and then, enlarging their scope, on other less celebrated butequally significant women who were engaged in practicing and teaching experimentalsciences and mathematics. Such studies, pioneered by Marta Cavazza and PaulaFindlen, have enriched significantly our understanding of the gendering of scientificpractices in the sixteenth and eighteenth century and, more generally, have cast lighton the functioning of early modern scientific networks and institutions.

One of the main problems in this area of research is the paucity of informationavailable about these women, which often consists of little more than their publishedworks. At the root of this situation is the long-lasting assumption that women playeda necessarily marginal and passive role in the making of modern science. Such anassumption has functioned as a self-fulfilling historiographical prophecy, hidingfrom the historian’s sight the fact that there were indeed exceptional cases of womenwho established themselves as legitimated natural philosophers and mathematicians.Much evidence about the life and work of these women has thus been dispersed orforgotten, and what is available is often difficult to access. Furthermore, as anybodywho is teaching this topic would know, there are very few excerpts of their workavailable in English.

The situation is rapidly improving though, and in the last few years historicalanalyses and translations of relevant works have begun to appear. In this context,Antonella Cupillari’s book on Maria Gaetana Agnesi (1718–99) is a timely and veryuseful contribution, which will become an essential tool for those who do not readItalian. Agnesi’s is the story of a child prodigy born into a wealthy bourgeois familyof Milan, who became famous displaying her linguistic and mathematical skillsduring the evening gatherings at her family palazzo. She then turned to the study ofmodern mathematics and, in 1748, she published her main work, a remarkabletextbook designed to take young students from the first rudiments of algebra toanalytic geometry and the techniques of integral and differential calculus.The textbook became a standard reference in Italy, so much that Lagrange stillused and recommended it several years later. Already well known in northern Italy,with the publication of the book Agnesi became a prominent figure of the republic ofletters, and was invited to take up an honorary lectureship of mathematics at the

BSHM Bulletin ISSN 1749–8430 print/ISSN 1749–8341 onlinehttp://www.tandf.co.uk/journalsDOI: 10.1080/17498430902848789

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University of Bologna, where she would have joined Laura Bassi. She politelyrefused the offer though, as she felt she was called to devote herself entirely topedagogical and charitable activities in favour of the poor and the dispossessed.

In her book, Cupillari sets out with various goals in mind. Firstly she introducesbriefly the figure of Agnesi drawing from the classic biographies available in Italian.She refers particularly to a significant and original book dated 1900 that, althoughoutdated in many respects, is still useful and factually reliable, and easily surpassesanything that was published in English in the following decades. Then Cupillaripresents her translation of the very first biography of Agnesi, which was publishedshortly after her death in 1799. The edition chosen for this translation is not theoriginal one, but an edited version of 1965, which includes some useful notes andbiographical information about the people cited in the book and a couple of relevantdocuments as appendices. To this, Cupillari adds her own substantial notes andcomments. The biography is an important document on the life of Agnesi, as it waswritten by a family friend who frequented the Agnesi palazzo, and is therefore rich ininsights into family life that would have been otherwise lost. Like most eighteenth-century biographies, it is an apologetic reconstruction, to be handled with care, andit does not do much to situate Agnesi’s life in its broader context, although the notesare helpful in this respect. The remaining sections of the book include some veryinteresting excerpts from Agnesi’s main book, the Instituzioni analitiche. In this waythe reader can have an idea of Agnesi’s style of writing and of her modus operandi.There are passages on the notion of infinitesimals, on the introduction of the rules ofdifferentiation and integration, as well as the descriptions of the properties of variouscurves—including the famous ‘Witch of Agnesi’. These materials will be particularlyvaluable for teachers interested in presenting the work of Agnesi and comparing itwith other textbooks of the time. Researchers, however, will need to integrate themwith other sources relating to Agnesi’s surviving manuscripts and letters, which areonly briefly mentioned in the book, and whose complex structure would requirea much more detailed description. Similarly, the book does not engage with thehistorical significance of Agnesi’s religious attitude and writings, or with the natureof her charitable work.

Cupillari’s book is an important contribution to what has become a thriving fieldof inquiry, and will hopefully be followed by more English translations of Agnesi’spublished and unpublished work, as well as of the works of many other eighteenth-century women philosophers and mathematicians whose voices we are only nowbeginning to hear.

Massimo Mazzotti

� Massimo Mazzotti

Alex D D Craik, Mr Hopkins’ men: Cambridge reform and British mathematicsin the nineteenth century, Springer-Verlag, London, 2007, 405 pp, £25.00,

ISBN 1848001320

Areader with an interest in the history of mathematics in Britain will nottravel far before reaching Cambridge, and once Cambridge is reached thereader will soon come upon the nineteenth-century figure of William

Hopkins. But neither Cambridge, nor Hopkins, are what they appear at firstglance.

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Today Cambridge mathematics has an almost mythical status; it recruits manyof the best students as undergraduates, and boasts many famous names among itsfaculty. And yet part of the picture painted by Craik is of a Cambridge at thevery beginning of the nineteenth century which contained more than its fair shareof lazy dons, had teaching of mixed standard, students of very wide-rangingability, no set academic criteria for the admission of undergraduates, anda haphazard examination system. However, we do not go far into the centurybefore reform begins. The undergraduate Analytical Society, formed in 1812, hadmembers such as George Peacock and Richard Gwatkin who went on to collegefellowships and thence to set and examine Tripos papers. They introducedCambridge students not only to Leibniz’s notation, but also to the wider work ofthe continental mathematicians. Further reform came in the late 1840s with thethreat of a Royal Commission. Among the many changes introduced at thisstage, King’s College surrendered the right of its students to obtain their BAdegrees without having to sit an examination.

When, in 1823, William Hopkins enrolled at the ripe age of 29 as anundergraduate at Cambridge, he was one of the first wave of students to feel thefull benefit of the changes instigated by men like Peacock. He graduated seventhwrangler in 1827, the same year that Augustus de Morgan was fourth. Hopkins’ firstwife had died before he enrolled at Cambridge, and while a student he married again.Hence, upon graduation, a college fellowship was not open to him, and so he turnedto private coaching. Thus we remember him as the great outsider to the collegesystem who became the university ‘wrangler maker’: the man who taught the menwho went on to be giants of Victorian mathematics and natural philosophy.Men such as G G Stokes, Arthur Cayley, William Thomson, James Clerk Maxwelland P G Tait.

But as Alex Craik shows, although Hopkins was indeed a wrangler maker andan excellent teacher, he was no outsider to Cambridge University. Upongraduation Hopkins was appointed mathematical lecturer at St Peter’s College(now Peterhouse) and Esquire Bedell of the university, holding the latter postfrom 1827 until his death in 1866. He served as one of the three secretaries of theCambridge Philosophical Society from 1839 to 1851 before serving as its Presidentfrom 1851 to 1853, the first President who was not a professor or head ofa college. When the 1850 Royal Commission asked questions of the university’smathematics provision, Hopkins was one of the respondents. Outside theuniversity he was, amongst other things, an FRS (elected 1837), president ofthe British Association for the Advancement of Science (1853) and President ofthe Geological Society of London (1851–52). To top it all, Hopkins’ salary asEsquire Bedell, plus his considerable fees from coaching, made him much betterpaid than the average Cambridge professor.

Alex Craik uses Hopkins as a focus for a wide-ranging and fascinating book. Inthe first 100 pages he gives a history of Cambridge, its colleges, students, foibles,failings and reform during the first half of the nineteenth century. Next is a chapterfocusing on William Hopkins’ life and work, and then comes the visual centre of thebook, a set of colour portraits of some of Hopkins’ best students, painted by T CWageman. In the second section of the book Craik looks in detail at the lives andcareers of some of the wranglers. Slight licence is taken here, in that some of thewranglers discussed were not coached by Hopkins. But what author writing ofVictorian wranglers could resist retelling the story of George Green, the miller and

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self-taught mathematician who, five years after publishing his Essay on theapplication of mathematical analysis to the theories of electricity and magnetism,and still diffident and uncertain of his own ability, entered Caius College aged 40? Orwho would not give in to the temptation to retell the story of John Couch Adams andthe controversy surrounding the discovery of Neptune? However, beside these better-known parts of mathematical history, Craik also takes us down less travelled paths,delineating the roles and influences of wranglers in the then ‘new’ universities inLondon and Durham, or in Owen’s College, Manchester, or much further afieldin Sydney and Melbourne. The symbiotic relationship between Cambridge and theancient Scottish universities is also highlighted, with the rise and reform ofCambridge mathematics leading to its graduates being appointed to chairs north ofthe border. This led, with the encouragement of their new professors, to some of thebest of Scottish talent travelling down to the Fens to enrol for the Tripos after theyhad taken their first degree in Scotland.

Many Cambridge graduates became clergy in the Church of England, and Craikdiscusses the careers of a number of them both at home and abroad. Perhaps themost interesting in this category is J W Colenso (second wrangler in 1836), whoeventually became Bishop of Natal. Amongst other parts of his colourful life, hefound himself in debt to the tune of £5000 before he was 30 (managing to regainfinancial balance by writing a bestselling textbook, Arithmetic for schools) and wasdeposed for a time from his Bishopric on grounds of heresy. Finally in the lastchapter of the book, entitled ‘Achievements in the mathematical sciences’, AlexCraik gives what amounts to an excellent free-standing essay on the development andapplications of mathematics in Britain up to 1880.

Mr Hopkins’ men is a book that takes the reader on a hike across nineteenth-century mathematics in the British Isles. It takes in university reform, the lives ofgreat mathematicians, cultural influences and religious controversies. The authorprovides an engaging combination of historical colour, breadth of scope andfascinating detail in his narrative. It was a joy to read.

Mark McCartney

� Mark McCartney

Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts,

Tools of American mathematics teaching 1800–2000, John Hopkins University

Press, 2008, 416 pp, £46.50, ISBN 978 0 8018 8814 4

As an undergraduate mathematics major in the early 1960s, I was presentedwith the opportunity to read and enjoy a book written by my mathematicsprofessor and advisor Edmond R Kiely, entitled Surveying instruments and

their classroom use, published by the National Council of Teachers of Mathematics(1947). I was fascinated by the descriptive and historical account of these surveyinginstruments down the ages and how they made their way into the mathematicsclassroom. Regrettably, this book has long been out of print and only a few copiessurvive. One copy of this book is archived in the National Museum of AmericanHistory.

Indeed, the time has come for mathematics educators and other interested partiesto have available a single volume devoted to the tools that mathematics educatorshave utilized over the past two centuries. This need has been admirably addressed by

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Kidwell, Ackerberg-Hastings, and Roberts in Tools of American mathematicsteaching 1800–2000. However, this volume is more than a historical account of thephysical tools utilized in the teaching of mathematics (geometric models, measuringinstruments, calculators, computers, and so on), but also tools such as standardizedtests, programmed instruction, textbooks, and others.

The volume is subdivided into four broad topical areas:

. Tools of presentation and general pedagogy

. Tools of calculation

. Tools of measurement and representation

. Electronic technology and mathematical learning

The first area, tools of presentation and general pedagogy, addresses the use of themost common and perhaps most important tools in promoting the teaching andlearning of mathematics, namely, the blackboard, textbooks, and standardized tests.The early chapters trace the European origins of the blackboard and textbook andthe rise of standardized tests in school systems across America during the earlytwentieth century. Another full chapter is devoted to the introduction and growthin popularity of the overhead projector. Interestingly, the overhead projector asa teaching tool first gained widespread use in military circles before its popularityspread to the educational community. The augmentation of federal funding duringthe late 1950s and 1960s contributed to the growth in popularity and use of teachingmachines and programmed instruction. Their influence on educational programs andon the early use of computers in the classroom serves as a concluding topic for thisfirst part of the book.

The second area, tools for calculation, not only chronicles the historical evolutionof well-known calculating devices such as the abacus and the slide rule, but alsoaddresses perennial debates on issues such as the efficacy of utilizing variousmanipulatives and calculating devices in the teaching and learning of mathematics,the role of electronic technology in the mathematics classroom, and the effectivenessof textbooks which incorporated innovative teaching methodologies such as those ofJ H Pestalozzi (1746–1827) and Warren Coburn (1793–1833) versus more traditionaltextbooks which emphasized a ‘mental discipline’ approach. The most vocal amongthe traditionalists was the renowned Boston minister, Hubbard Winslow (1799–1864) who decried teachers who embraced innovative programs and discarded ‘theexperience of thousands of years far too hastily’. Winslow contended that‘innovations in teaching [. . .] might produce a precocious exhibition of large andsplendid acquisitions of knowledge. Yet the world did not need a luxurious growth ofmushroom scholars.’ He argued that recent textbooks were ‘adapted to please ratherthan to profit’.

The third area of this volume, tools for measurement and representation, tracesthe evolution of devices used for measuring and representing mathematicalstructures, with special emphasis on the introduction and increased use of two-and three-dimensional geometric models and linkages during the latter half of thenineteenth century. Many of the more complex models were imported into Americanschools from Europe, especially Germany. Several American educators proposedreplacing proofs in geometry entirely with models, arguing that a visual demonstra-tion would have greater impact than a logical demonstration. Most notable amongthese advocates was Isaac Harrington of Connecticut, who in 1873 patented a set ofwooden forms, which could be transformed into other shapes. This emphasis on

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models in the teaching and learning of mathematics was not generally accepted bymathematics educators. Nevertheless, even the most ardent supporters of rigorousmathematical thinking recognized the importance and potential of physical objectsand models in the mathematics classroom. At the turn of the century, E H Moore(1902), outgoing president of the American Mathematical Society, vigorouslyadvocated the use of ‘laboratory methods’ in the teaching of mathematics.

The fourth and final area of this book, electronic technology and mathematicallearning, surveys the explosive growth of programmable calculators and computersin the school mathematics curriculum. This phenomenal growth in the availabilityand increased sophistication of electronic technology has fundamentally altered ourthinking about what should be included in the mathematics curriculum and howmathematics should be taught. Classroom access to the world wide web hassignificantly reshaped school mathematics curricula and the resources available tomathematics educators and students alike. Since this historical survey terminateswith the year 2000, perhaps a later edition will address the impact of recenttechnological advances such as Smartboards, and also goals for future technologicaluses in the classroom as envisioned in the NCTM Principles and Standards forSchool Mathematics.

In summary, the authors of this volume have provided mathematics educators andother interested parties with a wealth of information and insights into an area ofmathematics education that has been largely ignored. This volume is certainly anexcellent and welcomed addition to the mathematics educator’s library.

Anthony V Piccolino

� Anthony V Piccolino

Nathalie Sinclair, The history of the geometry curriculum in the United States,Information Age Publishing, 2008, 116 pp, £25.95/ £45.95, ISBN 978 1 59311 697

2/978 1 59311 696 5

The history of the geometry curriculum in the United States is a small bookof just over 100 pages, including the bibliography. The introduction by theauthor explains why she decided to write a short book rather than a long

article on the topic: the subject matter and the time span make the task of writingsuch a history one which cannot feasibly fit into an article. This book is thereforeconceived as an attempt to explain the history of the geometry curriculum in theStates without explaining everything that happened on the way, although the‘chronological sequence of notable events’ gives the story its pace. At the sametime, the author states the intention that this work should offer enoughinformation and historical data to inform the mathematics education communityabout the pathways that geometry education took in the States.

There are very many good points about this work: you can, if you know a bitabout the history of mathematics yourself, ‘fill in the gaps’ while reading it, as itgives the names and sources which a British scholar would find difficult to access.Having said that, though, it is not an exhaustive study; and, more importantly,the accuracy and the precision of the narrative are uneven. For example, Euclid’sProposition I.47 (Pythagoras’ Theorem) is given the number 48 (I.48 is theconverse of the theorem), and although this may just be a typo, it is a worryingone nonetheless. Again, some secondary sources are quoted far too often (for

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example Gonzales and Herbst 2006, ‘Competing arguments for the geometrycourse: Why were American high school students to study geometry in thetwentieth century?’), eroding the reader’s confidence that this book is basedlargely on original research.

Another problem for me when reading this work was the sweeping descriptionsthat are often used for those areas of the history of geometry education that are notthe main focus. The author understandably wants to describe a big picture beforelooking at the more minute details. So, for example, the period between the inventionof non-Euclidean geometries and the 1930s is described in a few sentences as follows:

[. . .] with the mathematical and philosophical doubts now surrounding geometry,many mathematicians turned to other areas of mathematics for foundationaltexts that could restore and perhaps even surpass the standards of rigorestablished by the ancient Greeks. This eventually prompted the decision tofound the field of analysis – an exploding and vast area of mathematics – onarithmetic instead of geometry, and would set the scene for the emergence of theBourbaki group in the 1930s (p. 25).

While one may agree with some aspects of this description, the looseness and theall-encompassing nature of the statements—made without any references—makeit inappropriate in what is intended as an academic book.

Once we get to the end of the nineteenth century, a wealth of information existsabout contemporary textbooks. Reading further, into the history of the twentieth-century geometry curriculum, it becomes clear that this is where the author’sexpertise lies. Here the reader can learn a great deal about the theories of learningand teaching that spread throughout and from the United States; the software andthe dynamic geometry environments and how they were employed; and the first‘math laboratory’ developed by Lore Rasmussen in Miquon School in the 1960s.This section of the book is well written and researched, with a wealth of detail andsophisticated discussion.

Particularly from 1955, when the ‘era of reform’ began, the data and theirinterpretation become most engaging. The geometry textbooks, their treatment ofgeometrical constructions, and the introduction of transformations as the drivingforce in the learning of geometry in the 1970s are explained well, with a wealth ofstatistics to back up the explanations on influential trends in mathematics education.Here the author is obviously ‘at home’ with the subject matter. As well as using theAmerican sources, she has also done a good deal of research on comparativedevelopments in other countries, most prominently France.

All of this is no accident: the organization of the work itself shows where thefocus is. Euclid’s Elements have nine pages devoted to them; the section on ‘Steps inthe history of the geometry curriculum’ has about ten pages on anything before 1902and the (English) Perry movement; the rest of the book is the real ‘history of thegeometry curriculum in the United States’, from the 1920s to our own time.

I wonder whether the author, with hindsight, thought again about the wisdom ofmaking this study into a book. A long article on the history of the US geometrycurriculum in the twentieth century might have been the better solution. But if youare interested in that era and the topic, you would be well advised to have this bookin your library.

Snezana Lawrence

� Snezana Lawrence

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Robin Wilson, Lewis Carroll in numberland: his fantastical mathematical logicallife. An agony in eight fits, Allen Lane (an imprint of Penguin Books), 2008, xiiþ237

pp, £16.99, ISBN 9780713997576.

This book offers the best and most comprehensive overview yet of the

mathematics and logic of Charles L Dodgson (1832–1898), alias LewisCarroll. Readers of the Bulletin who are familiar with Carroll’s literary

works, notably the Alice tales, will probably know that he was an Oxfordmathematician. Robin Wilson asks ‘[. . .] what mathematics did [Carroll] do? How

good a mathematician was he, and how influential was his work?’ (p. vii). In orderto provide objective answers, he discusses Carroll’s main contributions to various

mathematical disciplines.Eight pieces of mathematical humour from Carroll’s literary works provide

strong evidence that the author of the Alice tales, The hunting of the Snark, Sylvie and

Bruno, and other works of fiction, was a mathematician or at least had mathematicalinterests. Carroll’s fame as an author has eclipsed his mathematical abilities, amongother talents, for a long time. Indeed, the numerous biographies say very little about

Carroll’s mathematical achievements and acquaintances. Until the 1970s, only a fewauthors ventured to look seriously at his mathematical works. One critical

assessment, which has been frequently quoted, was made by the Americanmathematician Warren Weaver in the 1950s. Weaver’s authority was strengthened

on the one side by his status as a first-rank mathematician, and on the other by hisaccess (for the first time) to Carroll’s mathematical manuscripts (now in Princeton

Library). In a paper in Scientific American (April 1956), Weaver described Carroll’smain mathematical works and asserted that ‘he was not an important mathemati-

cian’ (p. 120). He concluded severely that ‘[s]o rare and so great were Carroll’s truetalents that we need not be condescending about the shortcomings of his formal

mathematical writings’ (p. 128).Though a first-rank mathematician, Weaver seems not to have been a first-rank

historian of mathematics. Indeed, he neglected to study Carroll’s works within theircontexts, and thus failed to understand fully their historical and mathematical value.

By contrast, recent work by historians of mathematics has paid more attention to thepersonal and social contexts in which Carroll’s writings were produced, including the

educational debates taking place on the mathematical scene of the time and Carroll’sown style of writing for a large audience. Wilson has benefited greatly from this

‘historical turn’ in Carroll studies, conducted by such authors as Francine F Abeles,William W Bartley III, Duncan Black, Edward Wakeling, and Eugene Seneta. He

explains in his preface that: ‘[m]uch work has been done on [Carroll’s] contributionsto all these areas; my aim here is to make this material accessible to a wider

readership’ (p. vii).The reader of this review will forgive my insistence on the historical and

biographical perspective, and its new understanding of Carroll’s mathematicalinterests. My excuse is that that perspective is the strong point of Wilson’s book, and

allows it to challenge Weaver’s influential view. One simple instance is the discussionof Carroll’s seventy-second ‘pillow problem’: ‘A bag contains 2 counters, as to which

nothing is known, except that each is either black or white. Ascertain their colourswithout taking them out of the bag’ (Pillow problems, 1895, 18). It is obvious that this

problem cannot be solved without further data. But Carroll offered an apparently

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serious mathematical solution, concluding that there should be one black counter

and one white counter in the bag. Curiously, Weaver considered this a sign of

‘limitations in his mathematical thinking’ and added that Carroll’s solution was

‘good Wonderland, but very amateurish mathematics’ (Weaver 1956, 119). Wilson,

on the contrary, treats the problem simply as a ‘deliberate joke’ (p. 203), an

interpretation which is more compatible with Carroll’s ironic comment on this

problem in the preface of the book: ‘If any of my readers should feel inclined to

reproach me with having worked too uniformly in the region of Common-place, and

with never having ventured to wander out of the beaten tracks, I can proudly point

to my one Problem in ‘‘Transcendental Probabilities’’—a subject in which, I believe,

very little has yet been done by even the most enterprising of mathematical explorers.

To the casual reader it may seem abnormal, and even paradoxical; but I would have

such a reader ask himself, candidly, the question ‘‘Is not life itself a paradox?’’’

(Pillow Problems, p. xvii).The most remarkable feature of Carroll’s mathematical endeavours, as displayed

by Wilson, is their variety. Carroll was first of all a mathematical teacher; most of his

work concerned mathematical education. He not only wrote several textbooks and

syllabuses, but also participated in the educational debates of his time, notably on

the adequacy of Euclid’s Elements as a geometry textbook. But in addition to his

teaching Carroll undertook original, and sometimes successful, research on several

purely mathematical problems in geometry, algebra, trigonometry, cryptology,

probability, and symbolic logic. He also extended his mathematical inquiries to

several practical problems in the theory of elections, sports tournaments, astronomy,

and other subjects. It is of course needless to mention here the use he made of

mathematics for recreational purposes. It suffices to say that Wilson provides plenty

of mathematical games, puzzles and satires to make his book a source of amusement

as well as instruction.Wilson has chosen not to give a conclusion making strong claims about Carroll’s

mathematical ability. Thus, he doesn’t answer directly the question of how good

a mathematician Carroll was, and lets the reader judge for himself based on a clear

and objective presentation of Carroll’s mathematical achievements. In doing so, he

runs the risk of leaving some readers lost, particularly those not acquainted with the

historical method or the Victorian mathematical scene. I imagine Wilson would not,

in fact, make a claim for Carroll as a forgotten first-rank mathematician. But the

book makes it obvious that Carroll was an original, creative and (at least

historically) important mathematician, making real contributions to several

mathematical disciplines. Moreover, this book shows the benefit that historians of

Victorian mathematics may derive from studying second-rank authors in addition to

the usual leaders.In this book Wilson has undertaken the risky task of combining a biographical

sketch with a puzzle book and a mathematical study. The result is successful,

however, and the reader will surely appreciate its fair balance between instruction

and entertainment. The author’s ability to make mathematical issues accessible to

a wide audience is once again shown in this book, where some complex historical and

technical matters (the Oxford examination system, the calculus of determinants . . .)

are clearly set out. In addition, the book is written in a lively style and richly

illustrated, with numerous excerpts from Carroll’s own works.

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This book is an essential source for the readers of this Bulletin who are interested

in Victorian mathematics in general and Lewis Carroll’s work in particular. It will

also be enjoyable and instructive reading for anyone interested in the history of both

‘serious’ and recreational mathematics.Amirouche Moktefi

� Amirouche Moktefi

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