> 1. Exactly how and for what purpose did Blaise Pascal come up with his
>"arithmetic triangle," which, I believe, he called what now has his name?
>
> 2. Who first denoted this triangular array of numbers as Pascal's
> triangle?
>
> 3. Before 1654, were there other mathematicians who or cultures that
> used Pascal's array of numbers?
>
> 4. Besides books, are their any cultural artifacts that display Pascal's
> triangle?
>
> 5. Besides shortest path problems, what other problems have solutions that
> yield Pascal's triangle?
When looking for the first occurrence of a mathematical term, a good place
to check is Jeff Miller's Web page at
http://members.aol.com/jeff570/mathword.html
(one reason to check there is that a number of members of this mailing list
are contributors to Jeff's page)
The entry for "Pascal's Triangle" reads
PASCAL'S TRIANGLE appears in 1886 in _Algebra_ by G. Chrystal. Blaise Pascal
used the term "arithmetical triangle" (triangle arithmetique). In Italy it
is called Tartaglia's triangle and in China it is called Yang Hui's
triangle.
Who was Yang Hui? He was a Chinese mathematician who flourished from circa
1261 to 1275 and who may also have been known as "Qianguong".
On the Web Page
http://math.truman.edu/~thammond/history/YangHui.html is the following:
Biggs, N. L. The roots of combinatorics. Historia Math. 6 (1979), no. 2,
109--136. (Reviewer: J. Dieudonne'.) SC: 05-03 (01A15 01A20 01A25 01A30 01A32
01A40 01A45), MR: 80h:05003.
(1) As the author explains, the most ancient problem connected with
combinatorics may be the house-cat-mice-wheat problem of the Rhind Papyrus
(Problem 79), which occurs in a similar form in a problem of Fibonacci's
Liber Abaci and in an English nursery rhyme. All are concerned with
successive powers of 7. (2) The first occurrence of combinatorics per se may
be in the 64 hexagrams of the I Ching. (However, the more modern binary
ordering of these is first seen in China in the 10th century.) A Chinese monk
in the 700s may have had a rule for the number of configurations of a board
game similar to go. In Greece, one of the very few references to
combinatorics is a statement by Plutarch about the number of compound
statements from 10 simple propositions; Plutarch quotes Chrysippus,
Hipparchus, and Xenocrates on the subject, so all apparently had some
interest in the subject. (Plutarch's statement is also discussed in a recent
article in the Monthly.) Boethius apparently had a rule for the number of
combinations of n things taken two at a time. The author discusses interest
in combinatorics in the Hindu world, by the Jainas, Varahamihira, and
Bhaskara (the latter in the Lilavati). The work of Brahmagupta should be
relevant, but is not currently available in English. The Arabs seem to have
adopted their combinatorics from the Hindus. The author also briefly
discusses some interest in combinatorics in the Jewish mathematical
tradition; two examples are Rabbi ben Ezra and Levi ben Gerson. (3) Magic
squares may first occur in the lo shu diagram, which is often linked with the
I Ching. The author discusses how the idea of magic squares may have entered
the Islamic world, was then improved, appeared in the work of Manuel
Moschopoulos, and possibly through him entered the Western world. What
happened in China is less clear. As the author suggests, the the work of Yang
Hui suggests that there had been a Chinese tradition of work in magic
squares, already dead by Yang Hui's time. For example, the squares Yang Hui
gives are not of types found elsewhere. In addition, Yang Hui seems unclear
on the techniques for construction. It is interesting that De la Loube\re
learned of a simple method for constructing magic squares in Siam. The author
also discusses: the possibility of a Hindu study of magic squares; the
presumably Arab source of Western magic square mysticism; and later
developments, such as Euler's questions on orthogonal Latin squares. (4) The
author discusses how questions in partitions arose in gambling, such as the
throwing of astrogali (huckle bones, which can land 4 ways) or dice (which
can land in 6 ways). An early systematic study is in the late Medieval Latin
poem De Vetula, which gives the number of ways you can obtain any given total
from a throw of 3 dice. Cardano and Galileo examined the subject in more
depth. (5) Combinatorial thinking in games and puzzles. Discusses the
wolf-goat-cabbage, attributed to Alcuin. [Similar puzzles also occur in a
variety of other cultures, but are not discussed in this article.] Also
discusses the Josephus problem, based on a process similar to the childhood
process of "counting-out". The Josephus problem is named for the Jewish
historian Josephus of the 1st century AD, who supposedly saved his life with
a correct solution. This problem unexpectedly turned up in Japan. (6) The
author discusses how "Pascal's" triangle was possibly known to Omar Khayyam
in the context of taking roots. The Hindu scholar Pingala may have known a
method, but the case is more cryptic. At any rate, it was known by the time
of Halayudha, who may have lived in the 900s AD. A more clear-cut reference
occurs in the work of Nasir al-Din al-Tusi in 1265. In China, the triangle
appears in the work of Chu Shih-Chieh (1303), but may have been very ancient
by then. The triangle was used by Pascal and Fermat to resolve the "problem
of points". This problem had the goal of determining how to distribute stakes
when a game ends early
James A Landau