Dear Colleagues,
I shall try not to take up the cudgels for one side of this subject,
but I am not certain that this aim will be achieved whatsoever. Said
this, and at the risk of contravening potential copyright laws (which may
vary in accordance with the landscape), I will pour out large portions
of a juicy, insightful, enlightening, witty, provocative, and very harsh
review of a famous general history of mathematics. I presume (just my
intuition) the following piece is not well known (and not easy to get).
Still, both the author and the reviewer are household names to all (or
most) of us.
"A History of Mathematics", by [Z, I prefer to leave out the author's
name for the moment], New York: [again, I omit further details].
The extracts (of a very long review) follow:
<begin>
"It is a long time since an American work has been awaited with so
much anticipation by readers of mathematics as Professor Z's recent
history. The book had been extensively advertised, there was and is
a growing demand for such works, and the supply of material was
well-nigh inexhaustible. But while few books have ever enjoyed such
advantages, few books have ever so seriously failed to improve them.
This is a harsh statement and should neither be lightly made nor
lightly accepted. It is based upon the following facts, which are
stated as concisely as an adverse criticism allows.
_First_. The work is, in very considerable measure, merely a
paraphrase of portions of better works to be found in the libraries
of most readers. Witness the following extracts. (Bracketed clauses
are merely transposed.)
=========
Z: "Plato was born at Athens in 429 B.C., the year of the great
plague, and died in 348. He was a pupil and near friend of Socrates,
but it was not from him that he acquired his taste for mathematics.
After the death of Socrates, Plato travelled extensively... He went
to Egypt, then to Lower Italy and Sicily, where he came in contact
with the Pythagoreans. Archytas of Tarentum and Timaeus of Locri
became his intimate friends."
Gow: "Plato was born... at Athens in 429 B.C., the year of the great
plague. [He died ... in 348] He was a pupil of Socrates,... but
he did not derive from his teacher his enthusiasm for mathematics...
After the death of Socrates [Plato] went away from Athens... He went
certainly to Egypt... and lastly to Magna Graecia and Sicily, where
he [consorted with Pythagoreans... He] became a close friend of
Archytas and Timaeus of Locri."
=========
Z: "At the age of thirteen he is said to have been familiar with
as many languages as he had lived years. About this time he came
across a copy of Newton's _Universal Arithmetic_. After reading that
he took up successively analytical geometry, the calculus, Newton's
_Principia_, Laplace's _Me/canique Ce/leste_", etc.
Ball: "When thirteen he was able to boast that he was familiar with
as many languages as he had lived years. It was about this time that
he came across a copy of Newton's Universal Arithmetic; ... he soon
mastered the elements of the analytical geometry and the calculus.
He next read the _Principia_, and... Laplace's Me/canique Ce/leste",
etc.
=========
Z: "This problem was reduced to another, now generally known as
Malfatti's problem: to inscribe three circles in a triangle that (sic)
each circle will be tangent to two sides of a (sic) triangle", etc.
"Steiner gave without proof a construction, remarked that there were
thirty two solutions, generalised", etc (A reference is given to
Fink in a preceding sentence, but not on the part transcribed.)
Fink: "Diese Aufgabe reduzierte er auf die jetzt allgemein als
*Malfatti'sches Problem* bekannte Forderung, in ein gegebenes
Dreieck drei Kreise so einzubeschreiben, dass jeder Kreis", etc.
"Steiner gab (ohne Beweis) eine Konstruktion, fu"hrte an dass es
zweiunddreissig Lo"sungen gebe, und verallgemeinerte", etc.
=========
The above extracts are only specimens of many cases that might be
cited, in which the author seems to have copied, without giving due
credit, from Gow, Ball, Hankel, Cantor, [...]. Sometimes the extract
is from one author and the credit is assigned to another, but usually
no credit is given to any one. [...] Now and then a note refers to
one of these writers, but it rarely happens that, at such times, the
writer cited is followed more closely than on other occasions. One
cannot but wonder why Professor Z did not, at the beginning, frankly
say that he had copied _ad libitum_ from three or four authors,
instead of giving only occasional credit.
But it may be said, laying aside the ethics of the matter, that the
work claims (which it does not) to be merely a compilation; that good
authors have been selected, and their words carefully transcribed.
A single selection may be given in reply to such a suggestion. This
particular one is taken because, laying aside the seriousness of the
discussion for a moment, it may cause a pardonable smile.
=========
Z: Creditable work in theory of numbers and algebra was done by
_Fahri des Al Karhi_, who lived at the beginning of the eleventh
century. His treatise is the greatest algebraic work of the Arabs.
In it he appears as a disciple of Diophantus. He was the first to
operate with higher roots and to solve equations of the form
x^2n + ax^n = b."
Hankel: "Das gro"sste algebraische Werk der Araber, das wir
besitzen, der Fahrides Al Karhi aus dem Anfange des 11. Jarhhunderts,
welches ein genaues Studium des Diophant zeigt, geht u"ber die
a"ltere Algebra nur insofern hinaus als es auch mit ho"heren Wurzeln
als Quadratwurzeln operiren und Gleichungen von der Form
x^2p (+/-) bx^p = (+/-)a auflo"sen lehrt."
=========
Fahri des Al Karhi! Does not every reader of the history of
mathematics know that Fahri (or Al-Fakhri^) was the name of the
_book_ that Alkarkhi^ wrote? It is all explained on p. 245 of
Hankel: "...". A glance at C (vol I, p 655), or at H (p. 24) would
have saved the author from this most awkward blunder.
_Second_. The work is weak in bibliography, where it should be
exceptionally strong. One has a right to expect a rich set of
references to the standard literature of the day. Such references
are offered by other histories, however humble, and every student
needs them. Yet in the work there is not a single reference by volume
and page. The bibliography is unscientific and meagre, and the use
made of it may be called fictitious. It consists of a few standard
histories, a few text-books, a few periodical articles not paged,
and a number of works of no special value. [Further mistakes are
pointed out hereafter, JGC]
...
[I]t is surprising that many works of so much more value than most
of those mentioned are ignored. [...] One misses all reference to
the many biographical contributions by Battaglini, Cremona, Beltrami,
Bertrand, Brioschi, de Comberousse, Darboux, [...] and to the numerous
historical memoirs of Boncompagni, Cantor, Chasles, Curtze, ... (an
impressive list of names follows, JGC). These men are not unknown,
nor are their works rarities. They have written extensively, and
their contributions are valuable [...]. Why should the only reference
on Pascal, for example, be Madame Perier's Life, which was translated
into English in 1744, while the valuable contributions by (again,
here, 17 distinguished authors), and others, are unmentioned? [...]
_Third_. But it may be said, and indeed it has been said, that this
work is especially strong in relation to modern mathematics. While
this will not excuse the errors in the treatment of the earlier
development of the subject, to be mentioned hereafter, nor the
weakness of the bibliography relating to that development, the claim
should be considered. But however charitable the reader may be, he
will close the final chapters with even greater disappointment than
he experienced in reading the earlier ones. What, for example, does
the work tell of the growth of the theory of substitutions and of
groups? Say fourteen lines, all told. Might not one expect some
mention of the contributions of Frobenius (and another list of
mathematicians, JGC), and, in general, a good re/sume/ of the
developments of the subject?
[...]
It strikes an American pleasantly (aha!, JGC) to see mentioned the
names and labors of over # [number] of his countrymen. While the
number is disproportionate, and while # American mathematicians
could not be found who would wish to be mentioned in a work which
ignores the names of so many world-known promoters of the science,
the effect in our own country may possibly be of value. The names
or labors of from fifty to seventy-five men (not including
contemporaries) who are much more entitled to mention than many
who are given place, are wanting, while the selection of living
mathematicians can scarcely be called a happy one.
_Fourth_. A final reason why the work is disappointing is apparent
from the first page: the work is carelessly written. One who consults
a history of any subject may reasonably expect to find the common
facts of that subject, together with the names, dates, nationalities,
and principal works of its leading contributors set forth in compact
form. But when he reads of Metius without his value of \pi, of Nonius
without mention of the nonius, of Napier's 'analogies' and 'rods' two
pages after the discussion of Napier; when he finds the Christian name
incorrectly given or frequently omitted; when he finds no dates
assigned to a large number of writers, including men of the prominence
of Galileo, Malfatti, Viviani, Bu"rgi, Cramer, and others equally well
known, -- may he not reasonably affirm that the greatest care has not
been taken?
[...]
Of the other errors in the book it is necessary to speak at no great
length. A few may be mentioned to show that, like all first editions,
the work is not free from them.
[...]
There are a number of errors in chronology, however, that may annoy
the student. Such are the date of Tschirnhausen's birth and the date
of Ivory's death [...]. In many cases about which there is much doubt
the dates are given which apparent certainty, while in cases about
which there is little or no doubt the dates are frequently omitted
entirely or given with a question-mark.
[...]
From this effort to call attention, fairly and without exaggeration,
to the chief deficiencies in Professor Z's work, it should not be
inferred that the book is without merit. Far from it. It tells in a
popular way the general story of the growth of mathematics. It is
well printed and is altogether an attractive piece of book-making.
[...]
Moreover, it makes an effort in the way of tracing the recent
developments of mathematics. For all this the work deserves credit,
and Professor Z deserves thanks. But in view of what has been said,
it seems only a plain statement of the truth to add that as a
scientific treatise the work cannot be regarded as an authority."
<end>
History of mathematics (HoM) is a hybrid field. Pedagogy of Mathematics
(PoM) is also a hybrid. Therefore, in a close inspection of the unknown
'X' in
HoM + PoM = X
we run the risk of finding neither (real) mathematics nor history. It
is easy to fall in the trap of just telling *stories* (not history)
for motivating (entertaining?) purposes. This could be a legitimate
attitude (we may discuss it); but, in *my* opinion, we ought to be
careful (at least of not 'matrixing' a wrong pattern). Right now I
cannot help thinking of "Why Johnny can't add" (Y). Most members may
well remember the equation
NM + PoM = Y
where NM stood for "New Math". Do not get me wrong. I am NOT suggesting
that we should avoid teaching math history for undergraduates. Rather, I
am concerned with (the so-called) "revivals", and inflation of (the so-
called) math-history (text-)books. Thanks.
By the way, any clues as to the author and reviewer? ...
With warmest regards from 90F Montevideo,
Julio Gonzalez Cabillon