The arithmetic triangle before Pascal

What in Italy is called Tartaglia’s triangle and elsewhere Pascal’s triangle is in fact one of those mathematical objects discovered several times, independently, by cultures far apart from one another. The triangular arrangement of the binomial coefficients

$$\begin{array}{ccccccc} & & & 1 & & & \\ & & 1 & & 1 & & \\ & 1 & & 2 & & 1 & \\ 1 & & 3 & & 3 & & 1 \end{array}$$

in which every number is the sum of the two above it — the rule $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ — was known to Chinese, Persian and Arab mathematicians centuries before Blaise Pascal devoted his 1654 Traité du triangle arithmétique to it.

In the Islamic world the construction already appears around the year 1000. Al-Karaji, active in Baghdad, proved the triangle’s formation rule by induction and used it to state the expansion of $(a+b)^n$; a century later the Persian poet and mathematician Omar Khayyam referred to it as a by-then standard method for extracting $n$-th roots, so much so that in the Iranian tradition the triangle is still linked to his name (Katz, Boyer).

In China the same table is documented in the Xiangjie Jiuzhang Suanfa of Yang Hui (1261), who nonetheless does not claim authorship: he traces it back to the mathematician Jia Xian, who lived around the middle of the 11th century. There too the purpose was practical — extracting square and cube roots and expanding powers of a binomial — rather than the combinatorial counting we now associate with the coefficients $\binom{n}{k}$. This is why in Chinese the triangle still bears Yang Hui’s name (Boyer, Katz).

The historical lesson is that one and the same array of numbers can arise from different problems: root extraction for the Arab and Chinese mathematicians, the problem of points and the calculus of probabilities for Pascal. Only with Pascal, and later with Isaac Newton, would the triangle be read systematically as the table of the coefficients in the binomial expansion and as a tool of combinatorics.

Links

Topics: Combinatorics
Concepts: Binomial coefficient · Tartaglia’s triangle
People: Yang Hui · Al-Karaji · Omar Khayyam · Jia Xian