# Definition:Pascal's Triangle/Historical Note

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## Historical Note on Pascal's Triangle

- The earliest reference to Pascal's triangle seems to date from between the $5$th and $2$nd centuries B.C.E. by the Hindu writer Pingala.

- The earliest known detailed discussion on it was by Halayudha in his
*Mṛtasañjīvanī*from around $1000$ C.E. This was a commentary on Pingala's*Chandaḥ-sūtra*, in which it was referred to as*meru-prastaara*.

- In Iran it is known as the Khayyam Triangle after Omar Khayyam discussed it in ca. $1100$ C.E. It had been discussed even before that by al-Karaji a hundred years previously.

- In India it was discussed at length by Bhaskara II Acharya in his ca. $1150$ work
*Līlāvatī*.

- In China it is known as Yang Hui's Triangle after Yang Hui, who himself (in $1261$) credited it to Chia Hsien in a work (ca. $1000$ C.E.) now lost.

- It also appears in Chu Shih-Chieh's
*The Precious Mirror of the Four Elements*, published in $1303$.

- While the binomial coefficients for small arguments appear in works of the ancient Greeks and Romans, the first actual record of Pascal's triangle in Europe seems to be when Petrus Apianus published it on the frontispiece of his $1527$ book on business calculations
*Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen*.

- It is also known (particularly in Italy) as Tartaglia's Triangle, after Niccolò Fontana Tartaglia.

- It was used by Michael Stifel, Tartaglia and Gerolamo Cardano to calculate binomial coefficients. Tartaglia in particular used it to calculate the coefficients of the expansion of the $12$th power.

- It was Pascal's $1665$ treatise
*Traité du Triangle Arithmétique*(written in $1653$) which was perhaps the first time the main properties of this triangle were documented in one place.

- The name Pascal's Triangle was assigned by Pierre Raymond de Montmort in $1708$, and Abraham de Moivre in $1730$.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $24$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $35$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.16$: Pascal ($\text {1623}$ – $\text {1662}$) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $24$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $35$

- Weisstein, Eric W. "Pascal's Triangle." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/PascalsTriangle.html