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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