John Stillwell is a card-carrying algebraist. The reader can be in no doubt about this, right from the first paragraph of the Preface: ``Algebra is abstract mathematics ... yet it is also applied mathematics in its best and purest form ... because it reveals the simplest and most universal mathematical structures.'' Superficially this is a text in the elements of abstract algebra (groups, rings and fields) leading to Galois Theory, but there is more to it than that.
A strength of the book is the way in which abstract ideas are motivated by concrete questions, usually drawn from the history of mathematics (more about this below): the quite brief introduction sets the scene, and most mathematicians, even most algebraists, will find surprises in it. Moreover the level of abstraction is never overdone.
One of the author's clear aims is to present abstract algebraic ideas as the means of unifying several areas of mathematics. In this spirit, for example, he dwells on the classical geometrical problems (duplication of the cube, angle trisection etc.) more than is usual. On the other hand he does not make too much of the algebraist's case. For example he does not gloss over the problem of irrationals: we find the Dedekind definition of the reals, well-motivated, and proofs of the basic, analytic, properties of continuous functions, surely unusual in an algebra text.
The long Discussion sections at the end of each chapter are particularly enjoyable. Here the reader will find summaries of the history behind the content presented in the chapter, with explicit references to original works. It will be no fault of the author if readers pass by ``the door [he has opened] to the great works by Gauss, Abel, Galois and others''. On the other hand these Discussions are not simply backward looking: some references to current research are given, especially to works which complete some strand of the developing story of the theory of equations. Open problems are flagged. This text demonstrates the value of history in the pedagogy of mathematics and sets a reasonable balance between expounding a piece of modern mathematics and ignoring its provenance on the one hand, and dwelling on the past and playing down modern insights on the other.
To the extent that the author leans one way or the other on this issue, it is to the latter. This is clear in this book; his subtitle - Geometry, Numbers, Equations - does not include Galois Theory; and the discussion of insolubility precedes that of Galois Theory, and is in the pre-Galois, Ruffini-Abel, spirit. This is the order of texts of a previous period, such as, for example, Serret [2] rather than that of, say, Herstein [1]. An eloquent, more explicit, statement of the author's point of view can be found in his article [3].
A price is paid for weighting the balance so heavily in favour of historical flavour and motivation, and for this ordering. A couple of brief examples will suffice to make the point. Most innocent is the use of older terminologies: for example, a subnormal series in a group is termed a ``decomposition" of the group. There is a lack of precision regarding the nature of the roots of the equation under consideration in the lead-up to proving the non-existence of a solution by radicals for equations of degree n\geq 5: are they algebraically independent? It matters, if one is to extend the action of a permutation of them to an automorphism of the field they generate. To label them with the undefined term ``indeterminate" does not quite seem to meet the point. This is not to criticise the author's point of view, and none of this limits the usefulness of the book; in fact it may easily be turned to advantage in heightening students' understanding of the development of the ideas.
A more serious comment concerns the proof of Theorem 8.2. A permutation
\sigma of the roots x_1, x_2, ... , x_n is extended to an
automorphism of a radical extension:
Stillwell defines
\sigma e(x_1,... ,x_n):=e(\sigma x_1,... ,\sigma x_n) for a radical
expression e. Given
that radical expressions are multi-valued, which map from the set of values of
e(x_1,... ,x_n) to the set of values of e(\sigma x_1,... ,\sigma
x_n) is to be chosen as
extending \sigma? And which of the many radical expressions, of which a
given field element is a
value, should be used to determine the image of that element? For example,
the values of the
radical expression
e_1(x_1,x_2):=\sqrt{2x_1} are values of the radical expression
e_2(x_1,x_2):=\sqrt{(x_1+x_2)+\sqrt{(x_1+x_2)^2-4x_1x_2}}.
However, if \sigma interchanges x_1 and x_2, then
e_1(\sigma x_1,\sigma x_2)\ne e_1(x_1,x_2)
while
e_2(\sigma x_1,\sigma x_2)=e_2(x_1,x_2).
Stillwell's writing is clear and economical. There is too much material to fit comfortably into the 24 or so lectures of one semester that one might normally have available: the author suggests a subset which he covers in his course at Monash. As text for a course over two semesters, say straddling 2nd and 3rd year, the scope of the book would be ideal.
The author has succeeded in highlighting the unity of mathematics in a predominantly algebraic setting and of the unity of the past with the present.
To a topologist, a knot is a simple closed curve in 3-space, and the task of knot theory is to classify all knots. This problem may be stated more generally in terms of studying how one geometric object may embed in another, but the original version (``classical'' knot theory) is already nontrivial, and reflects the most significant features of the generalization. Examples of these features are the focus on invariants of the knot complement, such as the knot group, the Alexander polynomial, the use of surfaces spanning the knot as a basis for calculations, sums of knots, the knot concordance group and linking. On the other hand, classical knots may be studied via knot projections, which are diagrams in the plane, and this notion does not generalize well.
Knot theory has been a testing ground for techniques and a rich source of examples in almost every phase of topology, from the taxonomy of the mid nineteenth century to the development of the notions of fundamental group and combinatorial group theory early in this century, and to the growth of high-dimensional differential topology and surgery in the 1960s. In recent years low-dimensional topology (the study of spaces of dimension at most 4) has once again come to the fore, and here the connections with knot theory are particularly strong. The most convenient ways of constructing 3- and 4-manifolds involve surgery on knots; moreover, some developments in 3-manifold theory were anticipated surprisingly early in knot theory. Conversely, new ideas arising from 3-manifold theory have lead to the solution of one of the fundamental early problems of knot theory, namely whether a knot is determined by its complement. The great progress in 4-manifold theory in the 1980s has also had major consequences for knot theory; in particular, Freedman found the precise topological meaning of the condition that the Alexander polynomial be trivial, and Witten used gauge theory to suggest the right context for the new polynomial invariant discovered by Jones. The latter was a by-product of work on operator algebras, and initially it was thought to be a disguised version of the Alexander polynomial. The novelty and power of the Jones polynomial was soon realised, and distilling away the ideas from operator algebras lead to a revival of interest in combinatorial aspects of knot theory.
Current work on knot theory is concentrated o n connections with 3-manifold theory and on understanding the new families of invariants (Jones, HOMFLY, Vassiliev). There is also some interesting work being done on concordance of classical links, which is motivated by higher-dimensional considerations. On the other hand, several major problems of the pre-Jones era such as the ribbon conjecture and the structure of the classical knot concordance group remain unsettled and are at present out of fashion, for want of new ideas.
The Knot Book is written at a level accessible to an enthusiastic undergraduate. The emphasis is on geometric and combinatorial ideas, and the only algebra used is that of polynomials (with the exception of a brief discussion of groups in Chapter VII, in connection with braids). The choice of topics and emphasis is very good, within the constraints set by the presumed background of the reader. A novel feature is the chapter on applications of knot theory to biochemistry and chemistry - large molecules, such as DNA and other complex organic chemicals can have the form of closed loops, and thus can be knotted. There are more than 200 exercises (some quite challenging) and over 50 unsolved problems dispersed through the text.
In a tradition which goes back to the first book on the subject, Knotentheorie (Ergebnisse der Mathematik 1, Springer-Verlag, Berlin (1932)) by Reidemeister, the appendix gives a list of diagrams of the simplest nontrivial knots and links (the prime knots and links having diagrams with at most 9 crossings), accompanied by selected invariants. Here the diagrams are augmented by the Conway shorthand notation, the Jones polynomial and the volume of the knot complement, when it is hyperbolic, i.e., has a geometry of constant curvature -1 (as is the case for most of the knots listed). The bibliography is organized by chapters, and is a mixture of books and research articles, with a leavening of articles from Scientific American and the Mathematical Intelligencer . Each item is given a brief description.
I think that a proof of the uniqueness of factorization of knots (only mentioned in passing, at the top of page 10) could have been included. (See the article ``The factorization of knots'' by C. Kearton, in Low-Dimensional Topology , London Mathematical Society Lecture Notes 48, Cambridge (1982), for an account in the spirit of this book). In the final chapter, on higher-dimensional issues, it would have been natural to consider also slice knots, ribbon knots and concordance, and then show that the set of concordance classes of knots forms an abelian group. To go much further one must introduce the knot group and covering spaces, and address readers with a more sophisticated background.
Knot Theory assumes somewhat greater mathematical maturity of its readers, and is more algebraic in its approach than The Knot Book , although the formal prerequisites are about the same: linear algebra and a smattering of group theory. This algebra is used and developed further, so that knot groups, (Wirtinger) presentations, the free differential calculus, Seifert matrices and signatures are all discussed in sufficient detail to justify exercises involving computations with these notions. The choice of topics is standard, but covers most of what I would expect to see in an introduction to knot theory. One novelty is a chapter on symmetries of knots, including a statement of the Murasugi conditions and of the results of Edmonds derived via equivariant 3-manifold topology. The chapter on higher-dimensional knot theory concentrates on 4-dimensional phenomena, and includes an extensive discussion of slice knots and the knot concordance group. On the other hand, although the knot group and Alexander polynomial are considered, covering spaces are not mentioned, and although linking number is defined and used, links are only mentioned very briefly, and 3-manifold issues (branched covers, Dehn surgery) are not mentioned at all. There are again over 200 exercises, but few open questions are raised. The appendices give the standard diagrams and the Alexander polynomials of the prime knots with diagrams of at most 9 crossings. The bibliography consists of seven texts, four survey articles and 33 research articles (grouped under these headings). Knot Theory is in a smaller format than The Knot Book ; I estimate that the latter has more than twice as many words.
These books are introductions to knot theory, rather than texts for graduate students or encyclopedic references, and are not directly comparable with the standard texts Knots and Links (Publish or Perish, Inc., Berkeley (1976)) by Rolfsen (whose purpose was to present knot theory as an introduction to the wider area of geometric topology) and Knots (W. de Gruyter, Berlin (1985)) by Burde and Zieschang. The latter books assume a knowledge of some algebraic topology, and predate the revival of interest in combinatorial aspects of knot theory. Knot Theory is more conventional than The Knot Book , and leads more directly to either of these texts. The books of Kauffman are probably closest in spirit to The Knot Book ; someone who started with it could then read On Knots (Annals of Mathematics Study 115, Princeton (1987)) by Kauffman and then (after acquiring some knowledge of homology) Knots and Links . As someone already familiar with the standard (pre-Jones) ideas of knot theory I found The Knot Book the more interesting to read of these two books, because of its unusual approach and success in covering much with little algebra.
Adams hopes that his book ``will excite people about mathematics". While I am sceptical of his finding a large lay readership, I would recommend it to anyone favourably disposed towards mathematics. Livingston's book is terser, and more likely to appeal to someone already committed to mathematics (as an undergraduate, teacher or scientific worker). Most undergraduate mathematics courses are algebraic or analytic in content. Knots and surfaces provide a rich variety of geometric phenomena immediately accessible to the novice, and are a natural topic for a course which does not make unrealistic demands on technique and yet touches upon issues of current research interest.
Either of these books could be used profitably as the basis for such a course.
The University of Sydney.
Hypatia of Alexandria is almost the first female mathematician of whom we know, and she is certainly the first of whom we have anything approaching detailed and reliable data. She is also to date the only woman ever to have been the leading mathematician of her day. She was, besides being a mathematician (and astronomer), a leading philosopher and religious thinker, a Neoplatonist who suffered martyrdom at the hands of a band of Christian fanatics.
Rather surprisingly, in view of all this, there is very little material on her that is at once accessible and reliable. In particular, no booklength biography has appeared in English between the 1753 reprinting of Toland's pamphlet and this present work.
Thus there was much to look forward to in this new work, and there is much of interest and much good sense between its covers. However the enterprise leaves one ultimately disappointed, as the discussion of Hypatia's mathematics is inadequate and insufficiently informed.
What we have begins with a full chapter devoted to literary legends invoking Hypatia's name and making (often very free) use of her story. Next there is a full and closely argued attempt to trace as many of her pupils as possible. This chapter includes passim some account of her thought, as does the third and final one (on her life and death).
Much of this succeeds as biography. The account of the historical background is good. There is a full and sensible discussion of Hypatia's likely date of birth (though not of her death; the date of 415 is accepted uncritically, when the arguments for 416 are in fact stronger). The so called ``menstrual-pad incident'' (whereby she used one such to repel a sexual advance) is discussed with neither puritanism nor prurience.
Beyond this, there is a good attempt to reconstruct the philosophical system she espoused. The reconstruction seems right and moreover it proceeds by a sound methodology. That is to take the writings of her best-known pupil, Synesius of Cyrene, many of which survive, and to proceed from there. Those writings include some seven letters to Hypatia, and these make it reasonably clear what sort of philosophy he learnt from her. This is the place to begin and Dzielska is at her best when discussing this aspect of the matter. Many previous discussions have adopted tortuous methodologies and have come up with all manner of sillinesses; this book provides a much-needed corrective.
When it comes to Hypatia's mathematics, most authors skirt the issue, which is a bit like writing of Goethe and hardly stopping to analyse Faust ! At least Dzielska avoids this; she recognises the importance of the topic. Rather the trouble is that her own skills are not commensurate to the task.
The era was in many ways a bad one for intellectual endeavour. The major academic institution of Alexandria was the `Museum', founded in about 300 BC and home in its time to Euclid, Ptolemy, Apollonius, Diophantus and Pappus (to name but a few). By about 400 AD, this institution was all but dead. Its last attested member was a mathematician, Theon of Alexandria, Hypatia's father. Theon was a minor talent, but his editorial and teaching work helped ensure the survival of major texts, including Euclid's Elements and Ptolemy's Almagest . Much of his own writing has come down to us.
Her father's work was continued by Hypatia who is known to have produced now lost commentaries (essentially annotated editions) on the work of Diophantus and of Apollonius, as well as an astronomical work of some sort. She is also known to have become interested (where her father was not) in the speculative (religious) elements of Neoplatonism and to have seen mathematics as a route to the One (i.e., the deity). The mathematics she taught was solid mainstream stuff, and this is what Synesius and his fellow-pupils learned.
In those dying days of the Museum, however, mathematics had acquired a bad name as it was pursued by charlatans - in our terms astrologers and numerologists. Both the civil and the (Christian) ecclesiastical authorities tried to ban mathematics, by which they meant, of course, the attempt to divine the future by means of astrology or numerology. Hypatia clearly practised good mathematics, not this rubbish; but if one is ignorant of the discipline, then such judgements are hard to make. When Hypatia was murdered, her murderers justified their actions by exactly such an elision of meaning.
Regrettably Dzielska falls into precisely this same error. Much is known of Theon and his work (see for example Toomer's article on him in Dictionary of Scientific Biography ), and this scholarship makes it clear that some of the early sources have conflated his output with that of other men of the same name. The picture of Theon that emerges is that of a specialist mathematician: true he is described at times as a philosopher, but that word could carry a general meaning of ``learned one'', and should certainly be so interpreted here.
There is very little evidence that Theon practised ``mathematics'' as well as mathematics, and some important evidence that he did not. (Although he wrote commentaries on all of Ptolemy's major astronomical works, he produced nothing on that author's Tetrabiblios , devoted to astrology.) However, because she shows no acquaintance with Toomer's work, Dzielska gets this quite wrong and has Theon dabbling in all manner of arcana.
Similarly, and for related reasons, she gives us the wrong idea of Synesius. Synesius was an interesting figure in his own right as a Christian bishop who espoused a form of Neoplatonism at a time when that body of thought was beginning to influence the formulation of Christian dogma, notably the doctrine of the Trinity. It is highly unlikely that such a man believed in the ``Chaldean mysteries''; true he knew a lot about them, but this is not the same thing.
Nor was the hydrometer that he asked Hypatia to have made for him (for medical purposes in his last illness) at all likely to have been intended for ``hydromancy''; more probably it was either used in brewing or distilling an alcohol-based medicine or else employed as a urinometer.
When we come to Hypatia herself, the same lack of discernment on matters mathematical and scientific characterises the discussion. Hypatia's mathematics have been the subject of a detailed attempt at reconstruction by Knorr ( Textual Studies in Ancient and Medieval Geometry , Boston: Birkh\"auser, 1989) and a more popular exposition by myself ( Amer. Math. Monthly 101 (1994), 234-243). Again there is no discussion of this work at all.
Rather we have sentences like this (p. 71): ``Apollonius' work The Conic Sections was in trigonometry; Perl has attempted to reconstruct Hypatia's commentary on it.'' Readers will know how judge the first clause; as to the second, Teri Perl's Math Equals is an imaginative high school text written from a feminist perspective. It integrates well-chosen and well-presented didactic mathematical material with brief derivative biographies of famous women mathematicians. It has nothing to do with the ``reconstruction'' of Hypatia's lost work!
Nor could Hypatia possibly have held astronomy to have been ``higher'' than arithmetic or geometry. These latter branches of mathematical science were, long before her time, thoroughly platonised and had dropped virtually all reference to material instantiations; astronomy by contrast has never taken this road. Geometry and arithmetic were spoken of by Ptolemy (in a sentence quoted approvingly by Synesius) as ``a fixed standard of truth'', a standard that guaranteed the cosequent predictive success of quantitative astronomy.
And regrettably, one could go on. But I won't. What could have been a good biography, what in many ways is a good biography, ultimately fails because it lacks the ability to assess the principal concerns of its subject.
Monash University