Ten all-time most influential math books

I came across the following article recently, and
I thought it would be interesting to ask how others
think this list of ten most important math books
has withstood the 83-year passage of time since
its publication.

Dave L. Renfro

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Walter C. Eells, "The ten most important mathematical
books in the world", American Mathematical Monthly
30 #6 (Sept./Oct. 1923), 318-321.

This title is suggested by H. G. Wells's article 'The Ten
Most Important Books in the World' in the 'American Magazine'
for April, 1923. "What are the ten most important books in
the world?" an interviewer asked Mr. Wells, and in his
reply he says, "Absurd questions sometimes make the most
interesting discussions. ... Following the precedents,
I will first show how unreasonable a question it is,
and then give myself up to its insidious fascination."

The question, "What are the ten most important mathematical
books in the world?", is equally unreasonable, but to the
mathematician it may prove equally or even more fascinating.
I suggested a similar question to my class in the History
of Mathematics last term, with rather interesting results.
I suggest this question now for discussion on the part of
the readers of the MONTHLY who may be tempted to yield to
its "insidious fascination." To be sure, no two lists will
agree, but this very fact will give the discussion its
interest and value. What books, if any, will be common
to all lists suggested? [2]

[2] It is hoped that a number of readers will follow up
the suggestion made by the author of this discussion,
either by contributing short papers, as in this
instance, or by supplying lists of ten which may
be of use when the time comes for a final summing
up. --EDITOR

The result of such a discussion should be similar to that
stated by Mr. Wells in discussing the answer to the
question of the six greatest men in the world, as a
result of which he says "endless people were set
thinking, very profitably, and sent to their encyclopedias
and histories and biographies for refreshing and
stimulating reading."

A list of the ten most important mathematical books
is not necessarily synonymous with a list of the ten
greatest mathematicians. Archimedes or Leibnitz, for
instance, should doubtless be included in the latter
class, but it is difficult to pick out a single
outstanding work of either which nearly approaches
the importance and influence of Euclid's 'Elements',
or Newton's 'Principia'. Much, too, of the important
and influential work in mathematics of the modern
period has appeared in scattered articles in the
journals, not in books.

Neither is a list of the ten most important mathematical
books necessarily the ten most important ones for
present-day study, any more than is Mr. Wells's list
suitable for a similar purpose. In fact less than half
of his list does he recommend as valuable reading at
the present time.

It is interesting to note that four of Mr. Wells's
ten books are scientific, but none of them are
mathematical. The nearest he comes is when he
considers Newton's 'Principia', "which brought
the whole material universe under the domain of
natural law," but reluctantly he rejects it as
one of his ten.

As a first approximation toward a mathematical
list, and as a basis for discussion and suggestion
of other lists, I venture to offer the following
as my choice, arranged in chronological order,
accompanied, in some cases, by noteworthy
characterizations of the contents of these
books or of their influence which have been
made by others.

EUCLID'S "Elements" (Alexandria, c. 325 B.C.),
which "has been for nearly twenty-two centuries
the encouragement and guide of scientific thought"
(Clifford), which has passed through more than two
thousand editions and has exercised such profound
influence on the teaching and knowledge of geometry
for more than two thousand years, and which is
"still regarded by some as the best introduction
to the mathematical sciences" (Cajori).

APPOLLONIUS'S "Conic Sections" (Alexandria ?
c. 210 B.C.), the great systematic treatise
which developed the geometrical "theory of conic
sections, and was the prelude to the theory of
geometrical curves of all degrees -- and of the
geometry of form and position" (Cajori) as
distinguished from the geometry of measurement.

LEONARDO OF PISA'S "Liber Abaci" (Pisa, 1202),
which marked the first renaissance of mathematics
on Christian soil, introduced Arabian algebra,
and brought into general use in Europe the
labor-saving Hindu-Arabic numerals, and for
centuries was a storehouse of material for
later writers on arithmetic and algebra; among
others forming the basis for the first printed
work on arithmetic, algebra, and geometry,
Pacioli's, which was printed at Venice in 1494.

NAPIER'S "Mirifici Logarithmorum Canonis Descriptio"
(Edinburgh, 1614), which gave the world Napier's
great invention of logarithms with their miraculous
power in modern computation, than which "with the
exception of the 'Principia' of Newton there is no
mathematical work published in the country which
has produced such important consequences" (Glaisher,
in 'Encyclopaedia Britannica').

DESCARTE'S "Geometrie" (Leyden, 1637), which in spite
of its obscure style was of epoch-making importance
in giving to the world the powerful method of analytic
geometry "which far transcended everything that ever
could have been reached upon the path pursued by the
ancients" (Hankel), and than which "there cannot be
a language more universal and more simple, ... and
better adapted to express the invariable relations
of nature" (Fourier), and which contains in addition
the modern exponential and literal notation of algebra.

NEWTON'S "Principia" (Full Title: 'Philosophiae Naturalis
Principia Mathematica') (Lodon, 1687), which established
the mathematical foundation of the universe, "the greatest
production of the human mind" (Lagrange); "the brightest
page in the records of human wisdom -- and preëminent
above all the productions of human intellect" (Brewster's
Life of Newton); and which, Laplace says, will always be
assured "a preëminence above all the other productions
of human genius."

LAGRANGE'S "Mécanique Analytique" (Paris, 1788), "an
epoch-making work ... a most consummate example of
analytic generality" (Cajori), "a kind of scientific
poem" (Hamilton), the foundation of all later work
on analytic mechanics, in which Lagrange "impressed
on mechanics, as a branch of pure mathematics, that
generality and completeness toward which his labours
invariably tended" (Ball).

LAPLACE'S "Mécanique Céleste" (Paris, 5 vols., 1799-1825),
"the translation of the 'Principia' into the language
of the differential calculus" (Ball), which according
to the author was intended to "offer a complete
solution of the great mechanical problem presented
by the solar system."

BOLYAI'S "Science Absolute of Space" (Hungary, 1833),
which, although only the appendix of a two-volume
work by his father, is characterized by Halsted as
"the most extraordinary two dozen pages in the history
of human thought," and which, together with Lobachevski's
work, opened up the whole fascinating and broadening
field of non-Euclidean geometries.

HAMILTON'S "Lectures on Quaternions" (Dublin, 1852),
"the great discovery of our nineteenth century ...
(in which) there is as much real promise of benefit
to mankind as in any event of Victoria's reign"
(Thomas Hill), which is the foundation of all modern
developments in the field of vector analysis, with
its important applications in mathematical physics,
including electromagnetic theory and Einstein's
generalizations.

It is with much regret that the argibrary limit of
ten forbids the inclusion of such works as Diophantus's
"Arithmetic," Alkowarezmi's "Algebra," Cardan's "Ars
Magna," Euler's "Analysin Infinitorum," Legendre's
"Fonctions elliptiques" and "Théorie des Nombres,"
Gauss's "Disquisitiones Arithmeticae," Cantor's
"Geschichte der Mathematik" and others which
could easily be mentioned.

This list of ten important books is well distributed
among the great fields of mathematics, as well as
in time and in nationality. It ranges over twenty-two
centuries. In it are represented two Greeks, two (or one)
Italians (depending upon whether Lagrange is considered
Italian or French), one Scotchman, two (or three) Frenchmen,
an Englishman, a Hungarian and an Irishman -- a very
cosmopolitan group.

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