Date: May 17, 1995 12:02 AM Author: Chih-Han sah Subject: Re: order of operations, history 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, sah@math.sunysb.edu