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Topic: Re: order of operations, history
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Chih-Han sah

Posts: 75
Registered: 12/3/04
Re: order of operations, history
Posted: May 17, 1995 12:02 AM
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When it comes to history, discussions become difficult.

Sir Joseph Needham (dec. March, 1995) wrote on p. 53 of
volume 3 of his work: Science and Civilization of China,
Cambridge University Press, 1959.

.... By some obscure play upon words, it was thought that
the Europeans admitted that their algebra had come from the
East, and although the intense nationalism of some later
Chinese writers in this respect has been much castigated
by 19-th century Europeans, the fact remains that whatever
future research may reveal about transmission, algebra
was just as essentially Indian and Chinese as geometry was
Greek. Actually, there is some evidence of transmissions
from the Arabs to the Chinese in the 13-th and 14-th centuries,
and much more from the Chinese earlier to India and Europe.
A great deal of careful historical work will be needed before
any final conclusions can be drawn.

One should now refer to p. 1 of this treatise:

There is a very large literature on the history of mathematics
in East Asia, though unfortunately (for most Westerners) by
far the greater part of it is in the Chinese and Japanese
languages. Those who are debarred from this original material
inevitably to the well-known histories of mathematics in
Western languages, such as those of Cantor, Loria, Cajori,
D. E. Smith, and Karpinski. Cantor's famous work is now
old (1880); he had to rely on the still earlier translations
of E. Biot, while his other main source was a paper by
Biernatzki of 1856. This, however, was a mere translation
of the work of Wylie of 1852--an excellent account which can
still be read with profit today. The most succinct modern
description of Chinese mathematics in historical context
is that of Cajori, while the fullest is that in Smith, who
arranged his two volumes choronologically in the first and
according to subject in the second.
The work of all these scholars was vitiated by the
fact that none of them possessed sufficient Chinese to permit
any first-hand contact with the texts themselves. This
criticism bears least forcibly upon D. E. Smith, who himself
spent some time in China and Japan, made collections of mathematics
books there, and had the advantage of intimate collaboration
with Asian mathematicians, notably Mikami Yoshio. Such
collaboration is also evident in the distinguished recent
contribution of A. P. Yushkevitch in Russian.

For those who have access to the book:

Robert Temple, The Genius of China, 3,000 years of
science, discovery, and invention, Simon and Schuster, 1986.

There is an introduction by Needham plus a preface by Temple about
Needham. Unlike most of the western historians, Needham was alreay
a distinguished Biochemist when he became interested in China. At
the age of 37, he began learning Chinese, by 1942 (5 years later),
he was sent to Chunking, China, and literally scoured the countryside
to talk to scientists, engineers, and collecting books and materials
which were shipped back to Cambridge university. Needham applied the
scientific method to his investigations and checked against all kinds
of sources. A diligent reader can check Needham's sources. This is
in sharp contrast with a number of western texts on history of mathematics
where the author furnished very few references and the readers are
forced to accept the claims of the author as *truth*.

Since Needham concentrated on China, but gives Smith a good
rating (D. E. Smith's two volumes on History of Mathematics are
reprinted by Dover Publications), one may go to vol. 2, p. 378, where
he discusses the Nature of Algebra. Here, I quote:

If by algebra, we mean the science which allows us to solve
the equation ax^2 + bx + c = 0, expressed *in these symbols*,
then the history begins in the 17-th century; if we remove
the restriction as to these particular signs, and allow for
other and less convenient symbols, we might properly begin
the history in the 3rd century; if we allow for the solution
of the above equation by geometric methods, without algebraic
symbols of any kind, we might say algebra begins with the
Alexandrian School or a little earlier; and if we say that
we should class as algebra any problem that should now solve
by algebra (even though it was at first solved by mere guessing
or by some cumbersome arithmetic process), then the science
was known about 1800 BC, and probably still earlier.

One can now be immersed in Smith's treatment. However, one should
keep in mind that some of the rough estimates on dates related to the
ages of some of the Chinese texts as given in Smith may be a bit too
early. A more reliable source is the book:

Chinese Mathematics, A concise history, by Li Yan and Du Shiran,
translated by J. N. Crossley and A. W-C. Lun, Oxford Science
Publication, 1987.

In any event, D. E. Smith provides quite a bit of information on the
evolution of various arithmetic operations.

As an example of the problem of dating, the earliest Chinese text that
survived is Chou Pei Suan Ching. It was primarily a book on astronomy.
But it contains a picture and a dialogue describing the theorem of
Pythagoras. One can argue: the Babylonians had recorded many Pythagorean
triples; thus, they *must* have known the theorem of Pythagorus. There is
another scenario: they could have discovered that certain pairs of
integers have the properties that the sum of their squares were perfect
squares and then discovered a rule that would generate such pairs.
Unless they drew some pictures involving right triangles, we would not
be on safe ground to make the claim that they knew the theorem of
Pythagoras. At this point, one faces the problem of dating the text.
After citing Needham (who tentatively set the date around 300 BC),
Boyer's History of Mathematics went ahead and listed Chou Pei Suan
Ching in the chronological table as (1100 BC ?). In contrast, Yan Li's
Chinese Mathematics estimated the surviving edition of Chou Pei Suan Ching
to between 100 BC to 100 AD (see p. 27 for a description of the cross
checking process).

Incidentally, the original Chinese edition of Li Yan's book
first appeared around 1960. However, the English translation incorporated
some of the archeological discoveries in the 1970's. These seem to have
confirmed a number of "myths" or "folklores".

I have to confess that I do not know any of the Chinese texts
*first hand*. From all the secondary sources and from reading the
work and bibliography of Needham, it seems safe to put more trust in
Needham's work than the others. In history, there are no "absolute"
proofs. For an interesting account on the priority of the theorem of
Pythagoras, there is the little pamphlet:

Frank J. Swetz and T. I. Kao, Was Pythagoras Chinese?
The Penn. State Univ. Press, and NCTM. 1977.

Happy history reading!

Han Sah,

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