The History of Pascal's Triangle"

This history of Pascal’s triangle does not begin with Pascal, but at least many centuries earlier. Most sources cite Chu Shi Chien’s Triangle, in his book from 1303, as the first time Pascal’s Triangle was used but many equivalent triangle were constructed long before then.

200 BC

In India the interest in combinatorics began with prosody. Prosody is the study of syllable patterns in Sanskrit. Pingala formed an arithmetic triangle known as the meru prastara (the holy mountain). His triangle began with a row of 1’s. The next row was a sum of two values from the previous row, and formed the triangular numbers. This process continued forming the tetrahedral numbers, and so on through the figurate numbers. (Willson, R., 2015)

10th Century

In Indian poetry, a line consisted of a pattern of long and short syllables. In an effort to find the best sound quality, Halayudha needed to find all the combinations of long and short syllables. He constructed his arithmetic triangle using the same method we do now: each row begins and ends with 1 and the middle values are a sum of the two values above it. (Schwarts, 2010)




11th Century

Abū Bakr al-Karajī, a mathematician from Baghdad, used an arithmetic triangle to solve for binomial coefficients. (Swchwartz, 2010)

Jia Xian, from China, is credited with writing the triangle out to the 6th row and identified the rule used for construction, as addition of the two values above the number (the rule referred to as the inverted triangle rule). (Humphries, 2003)


12th Century

In what is now Spain, the Moors also developed a triangle. Mathematician and physician Ahmad ibn Ibrāhīm ibn ‘Alī ibn Mun‘im al-‘Abdarī, proposed what is known as the silk problem. His question was in how many ways could he combine 3 out of 10 colors to form a tassel. Since he didn’t have the combination formula that we have today, Ibn Mun’im solved this problem by considering a smaller problem:


Ways to get three colors out of 10:
If i take color 3 and the two colors below it
Color 4 and any 2 of the 3 colors below it
Color 5 and any 2 of the 4 colors below it

…

Color 10 and any of the 2 colors below it.

Since he did know how to find C(n, 2), he was able to find the number of ways for each of these scenarios and then sum them to find the total number of ways to get 3 colors out of 10.
He then generalized his solution to find the number of combinations for any lower order and constructed an arithmetic triangle that showed all of the ways to make a tassel out of ten strands. Compare his results to Pascal’s Triangle. (Schwarts, 2010)






13th Century


Yang Hui, a Chinese mathematician, wrote out the triangle in 1261 AD to the 8th row. (Humphries, 2013)


14th Century





The arithmetic triangle appeared in Zhu Shijie’s book Siyuan yujian (Jade Mirror of Four Elements) in 1303 (Humphries, 2003). The triangle was used for algebraic purposes to begin with, but later mathematicians applied to principles to of finding the sums of series to combinatorics (Wilson, R, 2015).


By Yáng Huī (楊輝), ca. 1238–1298) - w:en:Image:Yanghui_triangle.gif, Public Domain, Link





European Contributions 15th-17th Century

In Europe, the triangle appears in works by Tartaglia (1370), Stifel (1544), Cardano (1570), and Mersenne (1635) before Pascal’s work was published it in he Traite du Triangle Arithmetique in 1665.


Blaise Pascal began working on the arithmetic triangle when he was approached by a friend and asked how to split the winnings in an interrupted game of chance (Bardzell, 2004). He addresses this problem in the final chapter of his Traite. In his Traite, Pascal introduces the triangle for binomial coefficients, then figurate numbers and finally combinations. He refused to use n! In his paper, proposing the triangle as an easy method. (Wilson, R, 2015)

Many more mathematicians have worked on Pascal’s Triangle, both before and after Pascal, and as you can see it comes from a diverse background. Each person who studied it began with their own purposes in mind. You can solve some ancient problems yourself here.
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