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Pubkeybreaker science forum Guru
Joined: 24 Mar 2005
Posts: 333

Posted: Mon Jul 17, 2006 7:33 pm Post subject:
Re: Ten alltime most influential math books



Dave L. Renfro wrote:
Quote:  Pubkeybreaker wrote:
I think, that with the exception of Euclid's Elements
the list is absurd. Especially absurd is the failure to
include Gauss' Disquisitiones.
Also, while it is debatable whether Knuth's TAOCP is a math
book, it is hard to choose a 20th century text that has been
more widely used and applied.
99.9% + of all the math books ever written were witten
in the 20th Century. It is totally absurd to think that a
'top 10' list would not include any of them. A top 10 list
should have the majority of its books from the 20th Century.
Not a single good book on calculus, differential equations,
modern algebra, etc. etc. was listed
The paper I typed and posted was written in 1923, 15 years
before Knuth was born.
Moreover, I seriously doubt any differential equations or
calculus books have had significant influence on mathematics,

I clearly missed the date.
How does one define 'influence'?
Influence on math education? Influence on mathematical theory?
Influence on mathematical applications?
How does one measure influence? By e.g. how widely the book
is read and used? Or by how important it was in furthering the
development of theory?
It is hard to envision any engineering student not being influenced
by a book on diffeq's (e.g. Carrier's or Boyce & DiPrima)
Has any book been more widely used (and had more influence on math
education)
than Thomas' book on calculus [yech]? It has not influenced THEORY at
all.
But I'll wager that more engineering/science students have learned
calculus
from it than any other book. And calculus is the basic tool of the
trade for
almost all of enginnering.
This is not a discussion about how GOOD a book is, but rather on
how much influence it has. One measure of influence is how much it is
used.
As an aside, allow me to ask how influential people think Whitaker &
Watson's
'Modern Analysis' is. It summarizes much of the 19'th century
development
in Complex Variables and special functions [e.g. gamma, zeta,
hypergeometric etc]
Most of the influence on mathemtical theory does not come from
textbooks at
all, but from research papers. I argue that that considering how much
influence
textbooks have had on theory is therefore the "wrong appraoch". 

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Dave L. Renfro science forum Guru
Joined: 29 Apr 2005
Posts: 570

Posted: Mon Jul 17, 2006 6:28 pm Post subject:
Re: Ten alltime most influential math books



Pubkeybreaker wrote:
Quote:  I think, that with the exception of Euclid's Elements
the list is absurd. Especially absurd is the failure to
include Gauss' Disquisitiones.
Also, while it is debatable whether Knuth's TAOCP is a math
book, it is hard to choose a 20th century text that has been
more widely used and applied.
99.9% + of all the math books ever written were witten
in the 20th Century. It is totally absurd to think that a
'top 10' list would not include any of them. A top 10 list
should have the majority of its books from the 20th Century.
Not a single good book on calculus, differential equations,
modern algebra, etc. etc. was listed

The paper I typed and posted was written in 1923, 15 years
before Knuth was born.
Moreover, I seriously doubt any differential equations or
calculus books have had significant influence on mathematics,
with the possible exception of L'Hopital's 1696 text.
As for algebra, Van der Waerden's "Algebra" might be
a contender. However, I think Gauss' "Disquisitiones"
should be on the list, maybe in place of Hamilton's
"Lectures on Quaternions". One of the problems with
picking 20'th century books is that not enough time has
passed for their influence to be properly accessed.
(I think this was the case with the 1923 list I posted,
when Hamilton's book was included.) Also, most of the
significant and groundbreaking publications in the
20'th century were not books (Hausdorff's 1914 topology
book being one of the rare exceptions).
Dave L. Renfro 

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Pubkeybreaker science forum Guru
Joined: 24 Mar 2005
Posts: 333

Posted: Mon Jul 17, 2006 5:23 pm Post subject:
Re: Ten alltime most influential math books



Dave L. Renfro wrote:
Quote:  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 83year passage of time since
its publication.

<snip>
Quote:  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 twentytwo
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.

I think, that with the exception of Euclid's Elements
the list is absurd. Especially absurd is the failure to
include Gauss' Disquisitiones.
Also, while it is debatable whether Knuth's TAOCP is a math
book, it is hard to choose a 20th century text that has been
more widely used and applied.
99.9% + of all the math books ever written were witten
in the 20th Century. It is totally absurd to think that a
'top 10' list would not include any of them. A top 10 list
should have the majority of its books from the 20th Century.
Not a single good book on calculus, differential equations,
modern algebra, etc. etc. was listed 

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Tim Golden science forum Guru Wannabe
Joined: 12 May 2005
Posts: 176

Posted: Mon Jul 17, 2006 3:07 pm Post subject:
Re: Ten alltime most influential math books



Dave L. Renfro wrote:
Quote:  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 83year passage of time since
its publication.
Dave L. Renfro
##########################################################
Walter C. Eells, "The ten most important mathematical
books in the world", American Mathematical Monthly
30 #6 (Sept./Oct. 1923), 318321.
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
presentday 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 twentytwo 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
laborsaving HinduArabic 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 epochmaking 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
epochmaking 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., 17991825),
"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 twovolume
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 nonEuclidean 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 twentytwo
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.
##########################################################

There is a little math book by Boaz that stuck in my memory as a very
human approach at math. There is something likeable about it. I suppose
this is style points rather than importance. I forgot the title.
Tim 

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John Schutkeker science forum Guru Wannabe
Joined: 30 May 2005
Posts: 172

Posted: Mon Jul 17, 2006 11:16 am Post subject:
Re: Ten alltime most influential math books



Thomas  Calculus & Analytic Geometry. 

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Gerry Myerson science forum Guru
Joined: 28 Apr 2005
Posts: 871

Posted: Mon Jul 17, 2006 3:26 am Post subject:
Re: Ten alltime most influential math books



In article <1152894181.136881.253340@s13g2000cwa.googlegroups.com>,
"Dave L. Renfro" <renfr1dl@cmich.edu> wrote:
Quote:  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 twentytwo
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.

An Englishman, a Hungarian, and an Irishman walk into
a mathematics library....

Gerry Myerson (gerry@maths.mq.edi.ai) (i > u for email) 

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Karl M. Bunday science forum beginner
Joined: 01 May 2005
Posts: 46

Posted: Mon Jul 17, 2006 1:43 am Post subject:
Re: Ten alltime most influential math books



Dave wrote:
Quote:  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 83year passage of time since
its publication.
Walter C. Eells, "The ten most important mathematical
books in the world", American Mathematical Monthly
30 #6 (Sept./Oct. 1923), 318321.
EUCLID'S "Elements" (Alexandria, c. 325 B.C.),
which "has been for nearly twentytwo 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).

Euclid's Elements definitely still belong on this list. They are the inspiration
for great books like Robin Hartshorne's Geometry: Euclid and Beyond.
http://www.amazon.com/gp/product/0387986502/
(oddly, Amazon says the book was published in 2005, while reminding me that I
bought it in 2001)
And Heath's annotated English translation of the Elements
http://www.amazon.com/gp/product/1888009195/
http://www.greenlion.com/cgibin/SoftCart.100.exe/euclid.html?E+scstore
or
http://www.amazon.com/exec/obidos/tg/stores/series//43/paperback/
http://search.store.yahoo.com/cgibin/nsearch?followpro=1&vwcatalog=doverpublications&catalog=doverpublications&query=Euclid
is a delightful treasure, as much worth reading for its footnotes as for the text.
Quote:  APPOLLONIUS'S "Conic Sections" (Alexandria ?
c. 210 B.C.), the great systematic treatise
which developed the geometrical "theory of conic
sections.

This one is a keeper for the list, too.
But shouldn't the next book on the list be AlKhwarizmi's?
Quote:  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
laborsaving HinduArabic numerals.

This unquestionably belongs if the list is ten most influential WESTERN
mathematics books, but otherwise AlKhwarizmi really ought to get the honor,
since he influenced Leonardo.
Quote:  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.

The advance in calculating power led to many other advances in mathematics, so
this nomination seems warranted.
Quote:  DESCARTE'S "Geometrie" (Leyden, 1637), which in spite
of its obscure style was of epochmaking importance
in giving to the world the powerful method of analytic
geometry.

Too bad Fermat didn't write a book tying together all of his contemporary
investigations.
Quote:  NEWTON'S "Principia" (Full Title: 'Philosophiae Naturalis
Principia Mathematica') (Lodon, 1687), which established
the mathematical foundation of the universe.

Yep, for sure. And the University of California translation of this work into
English
http://www.amazon.com/gp/product/0520088174/
makes for very interesting reading, while the original is worth learning Latin for.
I'm surprised that nothing by Euler, e.g. Introduction to Analysis of the
Infinite, appears on the list.
Quote:  LAGRANGE'S "Mécanique Analytique" (Paris, 1788), "an
epochmaking work ... a most consummate example of
analytic generality".

Maybe LaGrange's onebook generality is what displaced Euler.
Quote:  LAPLACE'S "Mécanique Céleste" (Paris, 5 vols., 17991825),
"the translation of the 'Principia' into the language
of the differential calculus".

The English translator, Nathaniel Bowditch, is famous for noting "I never came
across one of Laplace's 'Thus it plainly appears' without feeling sure that I
have hours of hard work before me to fill up the chasm and find and show how it
plainly appears."
Quote:  BOLYAI'S "Science Absolute of Space" (Hungary, 1833),
which, although only the appendix of a twovolume
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 nonEuclidean geometries.

I might give Lobachevski's work priority on this list, although Bolyai was
probably more widely read in WESTERN Europe.
Quote:  HAMILTON'S "Lectures on Quaternions" (Dublin, 1852),
"the great discovery of our nineteenth century.

Interesting choice.
Quote:  It is with much regret that the arbitrary 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.

Yeah, I'd definitely mention most of those, especially Gauss's book.
How about top ten influential math books of the twentieth century, now that we
are in the twentyfirst?
Thanks for opening this interesting thread.

Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345
Learn in Freedom (TM) http://learninfreedom.org/
remove ".de" to email 

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bert science forum addict
Joined: 04 Jan 2006
Posts: 54

Posted: Fri Jul 14, 2006 5:13 pm Post subject:
Re: Ten alltime most influential math books



Dave L. Renfro wrote:
Quote:  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 83year passage of time since
its publication.
EUCLID'S "Elements" (Alexandria, c. 325 B.C.) . . .
APPOLLONIUS'S "Conic Sections" . . .
LEONARDO OF PISA'S "Liber Abaci" (Pisa, 1202) . . .
NAPIER'S "Mirifici Logarithmorum Canonis Descriptio"
(Edinburgh, 1614) . . .
DESCARTE'S "Geometrie" (Leyden, 1637) . . .
NEWTON'S "Principia" (Full Title: 'Philosophiae Naturalis
Principia Mathematica') (London, 1687) . . .
LAGRANGE'S "Mécanique Analytique" (Paris, 1788) . . .
LAPLACE'S "Mécanique Céleste" (Paris, 17991825) . . .
BOLYAI'S "Science Absolute of Space" (Hungary, 1833) . . .
HAMILTON'S "Lectures on Quaternions" (Dublin, 1852) . . .

I would have expected JORDAN'S "Course d'Analyse"
to be included, after G.H. Hardy's great tribute to it
in his autobiographical "A Mathematician's Apology":
"I shall never forget the astonishment with which I read
that remarkable work . . . and learned for the first time
what mathematics really meant."
 

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Dave L. Renfro science forum Guru
Joined: 29 Apr 2005
Posts: 570

Posted: Fri Jul 14, 2006 4:23 pm Post subject:
Ten alltime 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 83year passage of time since
its publication.
Dave L. Renfro
##########################################################
Walter C. Eells, "The ten most important mathematical
books in the world", American Mathematical Monthly
30 #6 (Sept./Oct. 1923), 318321.
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
presentday 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 twentytwo 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
laborsaving HinduArabic 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 epochmaking 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
epochmaking 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., 17991825),
"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 twovolume
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 nonEuclidean 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 twentytwo
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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