j. higgins, · analysis is fully justified. scientific journals were then only beginning to be...
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BOOK REVIEWS
P. J. Higgins, An Introduction to Topological Groups, Cambridge
University Press, 1974, 109 pages.
This is a beautiful presentation of well chosen material. It has
sufficient generality to be worth doing but is restricted enough to
allow uncomplicated elegance. Such blends of elementary algebra,
analysis and topology are rare indeed and it would be an asset to any
program to include a one term course taught from this book during the
first or second year of post-graduate work. The book does not introduce
all of topological group theory, but rather develops the algebra,
analysis and topology necessary to present integration on locally
compact groups and to introduce complex representation theory.
Except for some very basic algebra, the book is self contained
(until the final chapter on representations). I do think though, that
unless the student has had some previous course work in abstract
algebra and topology and has had a good treatment of some integration
theory, the beauty of this scene will be lost in the crowd of necessary
concepts. The material is well chosen and is efficiently presented in
a "learn as you go" program instead of the old "learn everything, then
use a little of it". One of the most delightful features of the book
is that, despite the obvious efficiency of the presentation, the author
has provided many well chosen examples and explanations that make the
text even more informative than much longer presentations. There are
certainly parts in which I would prefer different presentations on
material, but overall I think it is an excellent book. Moreover it
would be very easy to substitute or supplement at various times without
destroying the virtues of the book.
Chapter I provides a brief history of the subject of topological
groups and establishes the basic notation followed throughout the book.
It is only 15 pages long and could be covered quite quickly. Chapter II
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takes the reader on a carefully guided trip through much of elementary
point set topology (including a most interesting proof of the Tychonoff
product theorem) but always interwoven with the main theme, namely the
fundamental properties of topological groups. The chapter ends by
describing the relationship between compact and profinite groups and a
discussion of locally compact groups. All the usual examples of
topological groups (real numbers, quaternions, 1-sphere, 3-sphere,
matrix groups, p-adic topologies on rational numbers) are discussed and,
in fact, discussed in more detail than most longer texts. This chapter
is only 45 pages long, but, depending on the sophistication of the
students, could take a long time to cover. Chapter III treats Haar
integration on locally compact groups. The existence and uniqueness
are proved, the Fubini theorem is proved and the modular function is
treated. It is a very readable, efficient and complete presentation.
I prefer a slightly different development of the Haar integral from
the "approximate Haar integral" but the algebraic approach used here
is interesting also. It would be very easy to substitute a different
approach here. The chapter is only 22 pages long and should be
straight forward to cover. The final chapter is the coup de gr£ce.
First the author gives explicit and illuminating formulas for the HaarYL i
integral on many examples of topological groups @R , S1, finite,
profinite, discrete, general linear) (unfortunately less explicitly on
0 ). Next the author proceeds to state a very elegant introduction to
complex representations of finite groups and to outline the proofs of
the analogous results about complex representations of compact topo
logical groups. Of course, the representation theory is much more
demanding mathematically than the rest of the book but its purpose is
to demonstrate one way in which the basic material can be used. This
purpose is well carried out. My only regret is that the author did not
include similar introductions to some other topics (e.g. Fourier trans
forms or dual groups).
The book seems to be quite free of errors of all types. On page 15
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the author mishandles a minor problem on quotient maps and products but
the point is minor and irrelevant to the main topic. On page 73 the
author claims that the proof of Urysohn's lemma depends on the axiom of
choice. I rather doubt that it does but again the statement is
irrelevant. Sometimes, in the interest of continuity of presentation,
the author has included definitions and exercises in the body of
discussions and proofs. This makes them hard to relocate later. The
index is fairly complete though. The discussion of "ff-modules over C"
and "C G-modules" on pages 92-93 is rather confusing also. Especially
for the non-algebraist, it would be nice if this were done a little
more carefully.
Tom Price
Joseph E. Hofmann, Leibniz in Paris 1672-16763 Bis Growth to Mathematical
Maturity, translated from the German by A. Prag and D. T. Whiteside,
Cambridge University Press, Cambridge, 1974 (xi + 372 pages, £8.50).
Joseph E. Hofmann was principal editor of the current Berlin
Academy edition of Leibniz’ works, and the German edition of this book
was published in 1949. This English version was revised by Hofmann in
collaboration with the translators, until his death in 1973. The Royal
Society's portrait of Leibniz (1646-1716) makes a fine frontispiece.
The book is a minutely detailed analysis of Leibniz' mathematical
progress from 1672 to 1676; and since he invented his version of the
differential and integral calculus during that period, this detailed
analysis is fully justified. Scientific journals were then only
beginning to be published, and mathematicians communicated their
results largely through letters. The printing of mathematical bocks
was very costly, and accordingly few mathematical books got printed,
especially in Great Britain. An enormous mass of documents written by
Leibniz has survived, and there are also a vast number of relevant
writings by his contemporaries. The primary manuscripts (in Latin,
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English, French, Dutch and German) relating to Leibniz' 4 years in Paris
were published together with this 1974 edition, as the first volume
(edited by Hofmann) of Leibniz' Mathematisch-naturwissensdhaftlich-
technischer Briefuechselj in the third series of the current Berlin
Academy edition of his Sccmtliche Schriften vend Briefe.
When Leibniz arrived in Paris on a diplomatic mission in 1672 (at
the age of 26), he had only a slight knowledge of mathematics. In 1666
he had published his Dissertatio de arte combinatoriat which has since
become renowned for its advocacy of the mechanization of logic, but its
mathematics did not go beyond some rather rudimentary operations on
combinations. When an un-authorized reprint appeared in 1690, Leibniz
published his disclaimer of that work as a "schoolboy essay". His
logical studies in Paris resulted in his first major mathematical
discovery, of his simple but very important theorem on the summation of
differences. He showed the theorem to Huyghens, who encouraged his
young friend by setting for him a problem of exactly the right degree
of difficulty, to sum the reciprocals of the triangular numbers:
1 1 1 1 1 1 1 0•7 * + ■ ■ - + ■■■■" ' ^ 11 + , , , SS —— + -r r + - r + , , , = 2 •1 1+2 1+2+3 1 3 6 10
Leibniz solved that problem by his theorem, and triumphantly generalized
it by summing the reciprocals of the higher figurate numbers, e.g. (p. 19)
1 1 1 1 31 + 1+3 + 1+3+6 + 1+3+6+10 + = 2 *
(Later, he learned that Mengoli had published these results in 1650).
This was done at a time when the leading mathematicians of Europe were
fumbling clumsily with the summation of simple geometric series. However,
the modern reader will shudder at Leibniz' conclusion that
1 + 1 + 1 + . . . = 1/0 = 0 ( ! ) .
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Leibniz visited London for the first time in 1673 (January 24th to
February 20th) and displayed the initial model of his celebrated
calculating machine to the Royal Society, which promptly elected him a
Fellow. He soon became embarrassed by his ignorance of contemporary
mathematical work, especially in Great Britain. Back in Paris, he
undertook a serious study of contemporary mathematics; and he quickly
found how to simplify, clarify, unify and generalize many of the
isolated results of other mathematicians. In particular, his theorem
(of 1673) on a transformation of areas (p. 55) could well be regarded
as the invention of the calculus, since it expresses integration by
parts in a manner which links tangents with areas. Since the start of
the 17th century, many European mathematicians had been grappling
confusedly with the concept of infinitesimals, in problems concerning
tangents, areas, arc-lengths and moments. It is part of Leibniz’
triumph that his forerunners seem to us to have been clumsily attempting
problems which he resolved with his general approach to the calculus.
Hofmann demonstrates the truth of Leibniz1 assertion that it was Pascal's
writings which were the major stimulus on him, rather than the abstruse
works of Isaac Barrow, as has been claimed by many people ever since
Tschimhaus in 1678.
Leibniz applied his transformation theorem to the rectangular
hyperbola, producing an improved treatment of Nicolaus Mercator's power
series (1668) for log (1+^) = fdx/(1+x). Applying his transformation
to the circle, he produced his famous series:
Only in later years did anyone notice that James Gregory's series
expansion of arctan x (in letters of 1671) gave ^/4 as arctan 1 :
however, Gregory's writings were so obscure (even to his contemporary
mathematicians) that this oversight is understandable. Moreover (a
point not mentioned by Hofmann), it was not until recent years that
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European mathematicians learned that Indian mathematicians in the 15th
century had been familiar with infinite series for several trigonometric
and inverse trigonometric functions, including the so-called "Leibniz
series" as a simple instance.
In his correspondence with the Royal Society, Leibniz gradually
gained more information about the results obtained by several British
mathematicians, especially many of the series expansions produced by
Gregory and by Newton. However, he was unsuccessful in his attempts
to learn how the results had been obtained, or how they could be
proved. In particular, Newton did not choose until decades later to
disclose his calculus of fluxions, which he used to discover many
results involving areas and tangents. Hofmann explains (p. 142) "that
it was then common practice to keep general methods of solution back
when communicating results and that higher analysis was then passing
through a severe crisis with regard to its rigorous foundation because
of the forceful intrusions of the ill-defined concepts and techniques
of indivisibles, so that no adequate methodological presentation of
its advances would in any case have been at all feasible". There were
intense rivalries between various scientists, national, professional and
conceptual, and Newton in particular attached extreme importance to
priority in invention. (It is interesting to speculate on what Newton's
reaction would have been if he had learnt of the Indian work on infinite
series, which he had duplicated.) Many years later, when the acrimonious
dispute between Newton and Leibniz (each urged on by his nationalistic
partisans) culminated in that notorious book the Cormieroiirn epistol'Lcum
of 1712 (and its even more distressing second edition, in 1722), Newton
accused Leibniz of having plagiarized the calculus from his work, on
the basis of the information which he had acquired through the Royal
Society. Hofmann shows convincingly that Leibniz' invention of his
calulus, in the form in which we use it today, was indeed an independent
invention by Leibniz, stimulated by the results reported from Gregory
and Newton, and that Newton should have acknowledged that fact if he
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had been more charitable in his presentation of the evidence available
to him, instead of consistently interpreting that evidence in the manner
most unfavourable to Leibniz.
Hofmann traces the crucial developments of the invention of the
calculus in Leibniz' papers of 1675. In the course of tackling a
problem in double integration, he replaced the usual abbreviation arm.
by / , so that he first wrote I y dx as z = fy, with y - z/d; and a
few days later he replaced that clumsy z/d by dz . In tackling a
differential equation he approximated the solution by Euler's method,
as he had first done more than 2 years previously (p. 197). He had
begun this investigation by summarizing and simplifying various results
obtained previously by himself and others, but the outcome was a
powerful new calculus, in which it gradually became clear that
differentiation and integration are inverse operations. The advantages
of Leibniz' ^-notation over Newton's dots have been universally
acknowledged (at least, since 1830), but it is interesting to note that
some of their contemporaries found Leibniz' notation much more difficult
than the customary geometrical diagrams, and that Newton professed to
attach little importance to notation.
When Leibniz visited London for the second time, in October 1676,
he demonstrated a much improved model of his calculating machine (whose
working he never was able to perfect), and he was shown some of Newton's
papers at the Royal Society. His memoranda show that he did not confine
his studies to Newton's profound results in analysis, but that he also
found it worthwhile to discuss at length (with John Collins) problems
in algebra, interpolation and compound interest which seem so elementary
to us. In addition to his own work on the calculus and the calculating
machine, Leibniz had also been working at Paris on the design of
accurate clocks, number theory, algebra (especially cubic equations),
complex numbers, mechanics, elasticity, vis viva (= 2 x kinetic energy),
fluid motion, chemistry, the history of science, a universal language,
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philosophy, diplomacy, et cetera. He had also been trying to find a job,
preferably a research appointment in Paris. However, his efforts failed,
and he had reluctantly to accept the post of Librarian to the Duke of
Hanover, under whose orders he left Paris in October 1676, never to
return. When he died at Hanover in 1716, embittered by his controversy
with Newton, only his secretary attended his burial.
Criticism can be made of some details of this book. Several non
trivial Latin passages are left un-translated (e.g. p. 7 and p. 241,
n. 43). What is the point in giving titles of Greek books and quotations
from Euclid in Latin, rather than in Greek or English? There are some
passages which read awkwardly, as though they had been translated
literally from the German (e.g. on pp. 44, 141 and 295); and tenses are
sometimes mixed confusingly within a sentence (e.g. on pp. 71 and 295).
It is surprising that Hofmann’s reference (p. 63) to the "English"
mathematicians Wallis and Gregory (!) was not emended to read "British".
Some of the mathematical arguments are very difficult to follow (e.g.
on pp. 87, 110, 113, 216 and 269), and the notation is sometimes
obscure (e.g. p. 89, n. 43 and p. 176, n. 69). On p. 99, "Diophantus"
is printed as "Diophant". Hofmann claims (p. 101) that "During the
Dark Ages little more than the name of Archimedes survived; only when
a Greek text came to light again in the age of Humanism were his
thoughts resurrected in a major way". The inadequacy of that assertion
is amply demonstrated by Marshal Clagett's huge book on Archimedes in
the Middle Ages: Volume 13 the Arabo-Latin Tradition (University of
Wisconsin Press, Madison, 1964). On p. 106, Figure 21 does not
correspond to the text. The references to Georg Mohr could usefully
have identified him as the author of Euclides Danicus (1676), which
has become celebrated in this century as the first book on Euclidean
geometrical constructions by compass alone. In view of the great
significance attached to Leibniz' notation, the use of modern notation
almost everywhere can sometimes be misleading to a modern reader (e.g.
on pp. 237 and 269), where an indication of the original notation
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would have been informative. Hofmann remarks (p. 254, n. 22) that
"Wallis failed to see the difference between the result of an iterative
process and that of a series expansion"; whereas a series expansion is,
of course, simply a compact representation of that special form of an
iterative process in which the successive iterates are generated as
partial sums by adding successive terms of the series. The logarithmic
interpolation method proposed (by John Collins) on p. 256 for solving
a pair of equations in x and y could work only when the solution is
x = y = 0. In the index of names, early dates are given with no
explicit indication that they are negative. On p. 361, the title of
John Pell's manuscript Cribrum Eratosthenia is printed as ’Cribrum
syntheticum'. However, these criticisms all concern minor details.
Hofmann provides footnotes to support almost every significant
statement in his text, and the book is equipped with elaborate indexes
of Collected editions, correspondence, manuscripts, journals, names
and books, and a general index. He has provided an admirable presenta
tion of a crucial stage in the development of mathematics, and the
translators have performed a valuable service by making the book
available to the English-speaking reader.
G. J. Tee
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