Algebraic geometry is thriving these days.
To substantiate this claim, it suffices just to visit briefly the ``Front for the xxx Mathematics Archive'' maintained at UC Davis http://front.math.ucdavis.edu, this very recent and already hugely successful development on the road to the global information space finally overcoming the `tyranny of distance.' The Archive brings together a number of previously disparate and independent electronic preprint archives (31 of them already), standardises them, and mirrors the entire repository at eight different locations all over the world, so as to protect this priceless collection in case of any local disaster, be it an eight-point earthquake or something even more destructive, like the rm * command executed by a UNIX system manager. Out of 31 categories the Front is subdivided into, the category AG -- Algebraic Geometry has the largest number of submissions stored: a formidable 1561 of them as of November 11. Only the Quantum Algebra (QA) category, with its 1516 e-prints, rivals Algebraic Geometry, while the rest of categories are lagging far behind -- the third largest one, Functional Analysis (FA), can only boast of 438 submissions, and a typical specimen would rather be something like Rings and Algebras (RA), with the modest total of 33 submissions. Assuming the degree of conservatism of mathematicians and thus the percentage of all new preprints submitted in the electronic form is uniform across the spectrum of mathematical disciplines, those figures give a reasonably accurate picture of the level of current research activity in all the branches of mathematics. (Though one should bear in mind that not all existing e-print archives have merged into the Front yet, one sizeable exception being the Topology Atlas repository.)
It is however no wonder that algebraic geometry is so important, if one recalls for the moment that the objects studied in this simultaneously very classical and very modern area of mathematics are of the most basic nature: algebraic geometry is dealing, in essence, with sets of solutions to systems of polynomial equations. Other objects of mathematics may come and go, drifting along with fashion, but solutions to polynomial equations are always here: basically all of the classical figures and shapes in low dimensions, from triangle and hyperbola to sphere and cone, are either loci of solutions to polynomial equations, or else intersections of finitely many loci of this kind. Surely one can somehow survive without the Cantor perfect set, but what about trying to get along without a circle? This is also why algebraic geometry provides the focal point where geometry and algebra meet each other and interact in a remarkably fruitful fashion.
Among all branches of pure mathematics, algebraic geometry is nearly unique in that it maintains a fine and healthy balance between the abstract and the concrete. On the one hand, it might be that no other area of modern mathematics requires such a consistent and determined effort over a long period of time to master as algebraic geometry apparently does. Before one starts producing new results in modern algebraic geometry, one needs to achieve, among all other things, a good working knowledge of sheaf theory, which in itself is a formidably abstract area. A typical object one has to work with in sheaf theory would be a functor from some small category to a category of algebraic objects, such as groups or vector spaces or algebras. This is something that really takes one's time to develop a good intuitive feeling for. At the same time, the sheaf-theoretic foundation of algebraic geometry, laid down to a large extent by Grothendieck, has not obscured the hard core of the discipline, and computations remain as difficult as they ever were during the classical Italian period one hundred years ago. There is no outward sign of degeneration in sight, unlike some other mathematical disciplines better left unnamed where an excessive level of abstraction has taken its toll, turning once healthy and still vitally important areas into a veritable `gallery of monsters.'
Among other areas of knowledge that have benefitted from contacts with algebraic geometry in a most spectacular fashion are number theory (Fermat's Last Theorem), physics (mirror symmetry), and cryptography (elliptic curve cryptosystems). The book under review is devoted to computational algebraic geometry, that is, the design of algorithms for solving systems of polynomial equations. The authors suggest that the text can be used as a basis for a number of University courses taught at various levels, from undergraduate courses in algebra through first graduate course in algebraic geometry. My own clear impression would be that in New Zealand Universities, the book could very well serve as a textbook for a nice 300-level course in algebra plus an Honours course in algebraic geometry.
To begin with, the book handles the subject matter very gently, that is without referring to `abstract nonsense' such as sheaf theory at all. For example, an affine variety is defined, very simply, as the common locus of zeros of a finite system of polynomial equations. From the fundamentalist Bourbakist viewpoint, this locus just forms the underlying set, or set of points, of an affine variety, which is a much more complex and intricate object than a set; what one has to do, is to put the so-called Zariski topology on the set of points (which topology is different from the one induced by the standard product topology on the finite-dimensional vector space, and more difficult to work with), and then to equip the resulting topological space with a sheaf of rings of polynomial functions. The approach adopted in the book is, in a sense, `naïve.'
However, it makes the book immensely readable, and concrete examples, which abound throughout the text and are often supplemented with good pictures, help the reader to develop a good intuitive understanding of the objects involved. And since the book is addressed in a large degree to undergraduates, this approach is the only one feasible. Actually, after having taught at Victoria for seven years, I cannot possibly imagine a course in modern algebraic geometry, Grothendieck-style, being offered to our Honours students, who usually lack any background in topology by the time they start graduate studies, and only have a very limited knowledge of algebra. However, the book under review would serve the purpose perfectly.
What is perhaps equally (or even more) important, is the remarkably close rapport with computational aspects of the theory maintained throughout the treatise. Many concrete algorithms are written out in pseudocode, and useful suggestions are given on how to implement them in all the major mathematical software packages, namely Maple, Mathematica, AXIOM, REDUCE, and even a number of others. This makes the text both very attractive for the modern-day students (just imagine a variety of assignments that can be set using software, and the amount of creativity and inventedness required on the part of the students to do them), and very useful -- after all, we have to think of making our mathematics students employable, and computing jobs is what more and more of them have to take after graduation. (Indeed teaching such a course will add employability to any one of us, the Lecturers as well --- which is not something to be dismissed lightly either.)
Concepts of algebraic geometry are illustrated with the help of examples from robotics, computer aided geometric design, automatic geometric theorem proving, and other exciting disciplines where methods of algebraic geometry are presently being applied.
However, and this is important, the book would also make perfect reading for someone uninterested in computing and any other `practical applications' of algebraic geometry. The contents of the book cover in great detail seemingly all of the by now classical topics in commutative algebra and algebraic geometry, such as basic polynomial algorithms, affine and projective varieties, ideals, Nullstellensatz, Groebner bases, invariant theory, elimination and extension, irreducibility and factorization of polynomials, the ideal-variety correspondence, quadric hypersurfaces, Bezout theorem. The dimension of an affine variety is treated in great detail (because of its obvious relevance to the computational aspects), and singular points of a variety appear near the end of the book together with the tangent cone. Even the Zariski topology is introduced at some stage through the closure operator, which is no doubt makes the concept much more accessible than a purely abstract head-on approach. This leads to irreducible varieties vs prime ideals and minimal decomposition. What is more, functions (polynomial and rational) on varieties are introduced and studied as well, and it turns out that one does not need to frighten the reader with sheaves to communicate essentially every basic fact there is to know about those functions. In such a way, the coordinate ring of an affine variety is introduced, and the notion of birationally equivalent varieties appears. The presentation always makes a point of consistently stressing interactions between algebra and geometry, and the basics of commutative algebra are taught alongside very geometric concepts.
Every subsection is generously supplied with exercises. The book concludes with four Appendices listing the basic concept from algebra (rings and fields for a layman), computing (pseudocode for the computationally challenged among us), comments on usage of the most common software packages for the needs of this study, and a list of possible independent projects for the students.
One side of this all is that a course taught along the lines of the reviewed book, if advertised properly, might attract computer science graduate students as well as mathematicians. However, mathematics students should be particularly encouraged to study algebraic geometry, through stressing both its importance in pure mathematics where it forms the area of bubbling research activity, and its unusually high relevance for applications of mathematics, in particular those in computing.
If the present reviewer ever decides to resume his study of algebraic geometry, he will do so through offering an Honours course, and the present book, with its hands-on approach, would make for about the most attractive recommended textbook --- almost certainly more attractive than many a classical text based on sheaf theory, because of capturing the Zeitgeist so well.
Vladimir Pestov Victoria University of Wellington
Topology, Geometry and Gauge Fields
Gregory L Naber
Texts in Applied Mathematics, 25 Springer-Verlag New Yourk, 1997, 396pp, DM 78.00. ISBN 0-387-94946-1.g
Mathematics and Physics have enjoyed a close relationship over the centuries. The study of physical and geometrical problems has often been a strong motivating factor in the development of new mathematics which in turn has contributed to the understanding of physical phenomena. In the last century the two disciplines have proceeded more independently with mathematics pursuing its fascination with abstraction. Nevertheless from time to time mathematicians and physicists discover that they are following similar lines of thought and the resulting interaction enriches both disciplines. A well-known example is the development of differential geometry and general relativity. More recently the work of physicists on the problem of quantizing classical field theory using gauge fields has provided an application of the theory of fibre bundles. The two groups, working independently, used different nomenclature, but once the links were realised the resulting activity and interactions produced mathematics of great depth and beauty along with profound insights into the structure of fundamental physical theories.
This book provides the mathematics needed to begin to appreciate this amazing parallel development of ideas. Its goal is to weave together notions from the classical gauge theory of physics with the topological and geometric concepts which become the mathematical models of these notions. Essentially it presents certain aspects of topology, algebra and differential geometry which form the foundation and vocabulary for describing gauge theories.
The author begins with a Chapter 0, designed to provide the physical and geometric motivation for the abstract mathematics which follows - ``an initial aerial view of the terrain'' as he says. He traces the notion of a gauge field using, as an example, Dirac's magnetic monopole and the classical quantum mechanical description of the motion of a charged particle in its field. This points to the need for a fibre bundle and a path lifting procedure to keep track of the particle's phase. We then get informal descriptions of the Hopf bundle, connections on principal bundles and non-Abelian gauge fields and the moduli space of the bundle. All of this is punctuated with references forward to the main text.
The following chapters deal with the mathematics - topological spaces, homotopy groups, principal bundles, differentiable manifolds and matrix Lie groups. Here Naber assumes only a solid background in analysis, linear algebra and some of the terminology of modern algebra. Exercises are liberally scattered through the text, gently leading the reader to assist in proofs or to explore particular examples pertinent to the understanding of the concept. When the going gets tough there are references back to the motivational chapter, or forward to assure the reader that this further leap into abstraction is exactly what will be needed at a later stage.
In some sections of the final chapter on gauge fields and instantons Naber again reverts to a less formal survey, providing more ``excursions into the murky waters of physical motivation'', and outlining results which require deeper mathematics.
Inevitably the book selects from topology and geometry only those topics (and they are substantial) which are needed for its purposes though there are often signposts and references to the broader fields. It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics. It is very self contained, all the definitions are here along with references for any results which are not proved. It is also very readable. Naber combines a deep knowledge of his subject with an excellent informal writing style. I recommend this book for graduate students and others with interests in mathematical physics, topology or differential geometry.
Gillian Thornley Massey University
Optimization: Algorithms and Consistent
Elijah Polak Applied Mathematical Sciences, 124, Springer-Verlag, Berlie-New York-London, 1997, 785pp, DM 118.00. ISBN 0-387-94971-2.
Contents: (Chapters and major sections)
The author reports that the book grew out of two sets of his graduate class notes, the first being an analysis of optimization algorithms in current use forming chapters 1 and 2. The second is a more advanced course on semi-infinite optimization and optimal control that has been transformed into chapters 3 and 4. The unifying thread is the use of optimality conditions and the description of algorithms in terms of conceptual algorithm models and then specific implementable algorithms. Chapter 5 provides mathematical background. From the outset it was clear that this book is best viewed as two books in one cover; Chapters 1 and 2 on standard optimization, chapters 3 and 4 on optimal control, with chapter 5 as a common appendix.
The goal in chapters 1 and 2 of providing a ``unifying framework'' for ``most existing optimization algorithms'' is ambitious. After all, optimization is an umbrella term for a broad range of problems with distinct characters. For example, model fitting often results in few-variable problems with severely non-linear objective and constraints, travelling salesperson problems typically have hundreds of variables with no constraints but a disconnected state space, image deconvolution requires million-variable optimization of weakly non-linear functions with relatively few constraints, while trajectory optimization with physical constraints requires semi-infinite optimization. Each of these problems is tackled using methods that are natural to that class but not really any other. An attempt to provide a unifying theory is not much use if it adds more baggage to the analysis than value. Not surprisingly, the framework developed is not at all a balanced analysis of current optimization but rather a catalogue-like recasting of local optimization methods (that are global for convex functions) in a light that proves to be useful for analysing optimal control and semi-infinite programming. I note that linear programming is not even mentioned while sequential quadratic programming is developed in detail.
I found the development of the various optimization methods in chapters 1 and 2 quite hard going and could not help thinking that there has to be an easier way of introducing the algorithms. The use of abstract optimality conditions for unconstrained optimization, in chapter 1, is an unnecessary obfuscation of the simple idea of local increase away from a minimum and also hinders the development of finitely constrained optimization in chapter 2. Many of the ideas are simply catalogued rather than developed. For example, on page 35 the section on the Trust Region Model begins with ``There are a number of trust region algorithm models in the literature...'' followed by a small list of references to seminal papers and formal statements of algorithm models. Those references, and the whole section, were of considerable interest to me because I know what a trust region is and why one is likely to use one. But the whole section would be virtually useless for the uninitiated or as a text.
Chapters 3 and 4 contain the brunt of the valuable material in the book. These chapters deal with recent developments in semi-infinite programming for control via the ideas of consistent approximation and epi-convergence. I am certainly not an expert in these areas, but the expository style made the material easy to read and digest.
Chapter 5 is a collection of ``essential'' mathematical facts that are included since ``no (other) book available contains all of these results''. But do we really need to reprove the assertion that a continuous function attains its infimum on a compact set? I think not. The nearly 100 pages taken up by this chapter could have been greatly reduced by referencing other texts throughout the other chapters with enhanced derivations where absolutely needed.
There were many often-unconsidered trifles throughout chapters 1 to 4 that I enjoyed snapping up. The 11 pages on conjugate gradient methods contained insights into the relative merits of the varieties of conjugate-gradient algorithms --- which algorithms also worked for non-quadratic or non-convex functions, or even for finding stationary points, and so on. Sections 2.7 and 2.8 on penalty functions and augmented Lagrangians for dealing with equality-constrained optimization gave an enjoyable consistent description of some of the theoretical results that are not often dealt with.
One generally good feature is the ``Notes'' appearing at the end of each major section, containing a reasonably informal discussion of the historical progression of ideas and issues surrounding the algorithms. I particularly valued those that dealt with some part of the development in which Polak had played a part. These provide a kind of a scientific gossip as commentary on the mathematics. But one needs to be careful with the validity of the information in the purely historical notes; For example the note on page 87 regarding local Newton methods state that Ralphson [sic passim] had his name associated with the methods because of his translation of Newton's algorithm published in 1720. Apart from the misspelling of Raphson, the note is misleading in terms of Raphson's contribution to the Newton-Raphson-Kantorovich method. In fact Raphson was a mathematician in his own right and a friend of Newton. Raphson certainly knew of Newton's laborious method of 1669 for finding the zeros of a cubic by repeated linearization with residuals combined as the last step, and its application to functions other than polynomials. But it was Raphson who extended and polished the method to use the derivative of the function with each residual added at the end of the iteration. The resulting work was published in 1690. (Further details may be found in the reference given below.) Presumably the proof-reading left something to be desired as other misspellings occur throughout the book; Fréchet is consistently misspelled as Frechet, etc. I am tempted to mention that the author is Australian, except that that might be an unnecessary slur.
In summary, one really needs to consider the two books (chapters 1, 2 on local optimization, chapters 3, 4 on optimization for optimal control) separately. The combined appendix, chapter 5, is of little value. The first of these books catalogues standard methods with background notes but is too turgid to replace the many excellent books on algorithms for optimization. Consequently, this book would not be useful as a text, besides that fact that no graduate course in New Zealand is likely to require students to buy a 500 page book costing NZ $140. The second of the books, chapters 3 and 4, is an altogether different case and could be of great interest to researchers in optimal control. In my opinion these 278 useful pages should have been published as the whole book, that would then have been a digestible and informative work on a topic of contemporary interest. If you happen to be looking for a valuable up-to-date development of the theories of consistent-approximation and epi-convergence for optimal control, I suggest you augment the price by a few dollars for a razor blade to remove the rest. You may also be interested in the 10 page errata from http://diva.EECS.Berkeley.EDU/~polak/errata.ps that I intend to contribute further to.
I'd like to thank Andy Phillpott (who is also acknowledged by the author for comments he made on an early draft of the book) for useful comments and also Garry Tee who provided the definitive information regarding Raphson.
Reference: N. Bicanic and K.H. Johnson Who was `-Raphson'? Int. J. Num. Meth. Engng. 1978.
Colin Fox University of Auckland
A film by Darren Aronofsky, starring Sean Gullette.
Winner, Directing Award, Sundance Film Festival. http://www.pithemovie.com/
Maximilian Cohen is riding hunched up on the New York subway, cradling a black go stone in his hand. It's a circle. Pi r^2. The empty go board, its rules unseen, represents the blank universe; a played game, its state. The empty board becomes strangely appealing: clean and symmetric, large enough to suggest an infinite lattice, yet finite, truncating its symmetry to a groupoid. Its cells are not quite square, the game tree is complete. Why break that symmetry? A go master once spent five of his ten hours contemplating the empty board.
Once in play, the rules don't determine the structure. But like the islands of order in our sea of increasing entropy, there are patterns. Maximilian is looking for that emergent structure. He's a mathematician.
In the small field of films featuring mathematicians---only Hollywood's Good Will Hunting and Jurassic Park and the touching Italian Death of a Neapolitan Mathematician come to mind---Pi is a winner. It's a movie of ideas, with intersecting streams from Rudy Rucker on cyberpunk, Jorge Luis Borges and Umberto Eco on conspiracies and hidden meaning, George Steiner on prodigies and obsession (maths, music, and chess) and Nabokov on chess and madness, all up there in high-contrast grainy black and white with a cool electronic soundtrack. Hey, I spent three years obsessed with go, I'm a mathematician, my teacher had a stroke just like Max's. This is my movie! Plus, I'm seeing it in Berkeley, where the streets are filled with real crazy people, partly crazy people who've read too many popular science books, and Fields medallists, in equal proportions. That fat guy next to me with sweaty feet is probably Ted Kaczynski's brother.
That stroke---there's a suggestion that Max's teacher stumbled on something hidden in mathematics that humans just weren't meant to know. Some kind of Gödel sentence that fused his mind. Now Max, who wants to model the stock market on a home-built supercomputer that fills his apartment (shades of the Chudnovsky brothers), is getting close also. His full-complex migraines and extensive self-medication don't help much either, although perhaps the pre-fit euphorias spark creativity. He's getting close, and Wall Street devils and Hasidic angels want a piece. The Hasids bail him out, but they want the number in his head---the Name of God. Max realizes that it's not the number itself that's important, it's its meaning, and the number was given to him.
This Name-of-God stuff could be seen as traditional science fiction, or one can read the whole Kaballah sequence as a figment of Max's imagination: religious and persecution mania triggered by overwork. These things happen. This is a film about doing science, meaning, and mania. After all, where does meaning reside? It's partly intrinsic, and partly (lest mathematics degenerate to a list of consequences of axioms) imposed by us. To do science, you have to believe in it and give it meaning; worse, it's usually difficult and likely to fail. Occasionally you have to suspend your critical faculties and just push ahead with some weird idea. You have to hang on to it and not let go. The Pythagoreans believed all was number, and lo, all was number. The potential rewards are great, but there's a risk of becoming a crackpot, or of starting to see too much meaning and too many connections---mania.
The maths in this film is pretty simple: the stock market time series, the digits of pi, Archimedes' spiral and the golden ratio. I think that's much better than hiring experts to fill blackboards with the latest jargon. Coincidences and connections and hidden patterns are the stuff of mathematics. And, although it may give the field a bad name, some mathematicians really do study the digits of pi. The Name of God lends a suggestion of weird science. But think of Gödel undecideability, nonstandard real numbers, even standard reals supporting things like the Mandelbrot set, multidimensional string theory, infinite state quantum computing,...A suggestion of weird science is allowed.
Pi is Aronovsky's first film, the traditional low-budget debut. I hope he hasn't used up all his ideas. I can't wait for his next one.
Robert McLachlan Massey University
Dissections: plane & fancy Greg N. Frederickson, Cambridge University Press, 1997, 324pp, $31.95,
How can a geometrical object like a square be dissected so that the reassembly of the obtained pieces have a certain desired property such as being three squares? Questions like this have been of interest to mankind for a long time. For example, in the translation of Book I of Euclid's Elements of Geometry by the Arabian astronomer and mathematician al-Sabi al Harrani Thabit ibn Qurra (836 - 901 AD) a proof of the Pythagorean Theorem in terms of dissecting two squares to one can be found (p.29).
Dissections (e.g.in the disguise of tesselations) are a well
studied area of modern mathematics. However, dissection problems need
not only interest the expert working in the field, but can also be
fascinating to the interested amateur. Dissections: plane &
fancy is a book which admirably addresses these problems. Its
declared aim to contribute to the area of recreational mathematics has
been more then reached. The way in which fairly complicated techniques
and dissections (e.g. Theobald's dissection of a hexagon to a
square p.129) are developed and presented turn it into an easy to
digest and enjoyable book to read. Also, applications of dissection
techniques such as the butterfly expanding keyboard of the IBM
ThinkPad 701C minilaptop computer (p.68) are given.
The material presented is very clearly structured: objects, finally universe'' dissections of squares are discussed. Their natural generalizations to polygons and polygrams are studied next. A short chapter is devoted to dissecting curve figures before the author moves on to dissections of solids (with not necessarily orientable surface) in the final chapters.
A major part of the book is devoted to impressive examples such as Varsady's dissection of six pentagons to one (p.59), and older problems such as the dissection of the crescent to the cross (p.167f) and the Smart Alec Puzzle (p.270) are not missing. The reader is also actively involved in the development of the material through a great variety of exercises, whose solutions are given in the final chapter of the book. Moreover, an index of all the dissections discussed as well as a very comprehensive literature list are given at the end. A great number of biographical notes and anecdotes of the people involved in the development of the theory such as H.Lindgren and S.Lyod are spread through the whole text which adds an interesting touch.
The manuscript and the figures are done with great care. However, the missing of one of the dissection figures in Reid's hexagram (Figure 9.22) and a couple of minor typos (e.g.on page 73, the roles for x and y have to be swapped when passing from Fibonacci's formula to Diophantus' formula) are genuine misprints. Dissection problems that can be reformulated as the problem of solving an algebraic equation are mostly discussed in chapters eight and nine. In this context it would have been of help to label the dissection figures involved by the variables in the describing relation. In the same chapters, the various different dissection methods given in the context of Fibonacci's formula undoubtedly have an important part to play in the theory. However, a shortening of this chapter or a summary of these methods in form of a table at the end, probably would not have been of harm to the reader.
Summing up, the material is very well presented. This together with the agreeable writing style made the book a pleasure to read.
Katharina Huber Massey University
The Book of Numbers,
by John H. Conway and Richard K. Guy. Springer-Verlag, New York, 1996, 310pp, DM 58.00. ISBN 0-387-97993-X.
This is a book unlike any others says the dust jacket, and with justification. Its coverage is immense, from common words involving numbers, through number patterns and recurrence equations, prime numbers, fractions, algebraic and complex numbers and transcendental numbers to ordinal and cardinal infinities and surreal numbers, which can be interpreted as fractional ordinals. Along the way there are such topics as a naming system for zillions, number walls, phyllotaxis, Farey fractions and Ford circles, repeated card shuffling, continued fractions, ruler and compass constructions, constructions for regular polygons, some depending on angle trisections, calculations for pi, harmonic numbers and the games of Hackenbush and Nim. There are also many historical references. This broad sweep means that each topic is much abbreviated, and anyone familiar with any of the topics covered will find that some favourie result has been omitted.
But it is the approach which makes this book different. Informality predominates, though the statements of definitions are accurate and the statements of results precise. There is scarely a traditional proof in the whole book. Instead, maximum use is made of suggestive diagrams and tables. By looking at these and constructing the next few cases, readers can convince themselves that the results claimed are true. Not all of the illustrations are easy. The concentration on this approach will have determined the approach of the authors and limited the topics they can cover to some extent, for not everything even in this area lends itself to such an approach.
The main question a reviewer needs to answer is Who would this book be useful for? Certainly it would find a useful place in the library of anyone lecturing in discrete mathematics, particularly in the area of recurrence relations, where the diagrammatic approach works best. These pictures are well adapted to the needs of a lecture, and the material should extend, but not often defeat, an ordinary second-year university student. A few results would probably be beyond all but the best students. It could be a useful exercise to get students to turn these figures into traditional proofs. Besides these, there are many statements given without proof, ranging from the easy to very difficult which suggest problems that could be set. There are no collections of exercises.
The book could be used by a person working alone, but it is not ideal for that purpose. A mathematical background of roughly what a second-year university stuent is generally assumed to have is required. The use of diagrams also allows the solo reader to see the overall shape of the argument rather better than a sequence of definitions and proofs.
In any book with so many statements, it is easy to find a few quibbles. For example, early in the book is a section on words connected with number, where it is suggested that ``astronomy'' derives from counting the stars, whereas in fact it means ``the law of the stars''. A decent compromise would be ``the arrangement of the stars''. The origin of complex numbers is related by the authors to the solution of quadratics whereas, according to Boyer, the beginning was Bombelli's insight into the curious fact that when all the roots of a cubic are real, Tartaglia's solution gives real numbers in terms of expressions involving the square roots of negative numbers. But these are trivial objections. Of more significance is the treatment of the laws of algebraic operation in special arithmetics, such as that of ordinal numbers. It is demonstrated that addition and multiplication are not commutative and powers are defined, but the question of the associativity of addition and multiplication and the distributive laws are not mentioned.
In sum, this book lives up to its advertisement; a valuable resource for any lecturer in number systems or discrete mathematics, especially in the area of recurrence relations, illustrating the essentials of the subject and full of fascinating byways.
David Robinson University of Canterbury
Information has been received about the following publications. Anyone interested in reviewing any of these books should contact g g&# David Alcorn& Department of Mathematics& University of Auckland& (email: firstname.lastname@example.org)g
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