Tuesday, October 05, 2021

Pascal's Triangle

 

So yes, this is me talking about math. Shocker, since I soooo love math. Did you hear the sarcasm? To finish my degree, I had to take one more math class. That was almost a deal breaker for me until I found out I could take a history of math course that would suffice. So this is a history paper on mathematics. Go me! 


The concept of Pascal’s Triangle has been around for centuries, although it wasn’t given that name until the 17th century. The Persians and the Chinese both appear to have discovered the application independent of the other in the eleventh century. In Persia, Omar Khayyam was extracting number roots with the triangle. Khayyam was a teacher of geometry and algebra, and studied astronomy and the Jalali calendar. He was also an “advisor to Malik Shah I” (Famous Mathematicians 1) until Shah was murdered. In 1070, Khayyam’s finished writing his treatise called Treatise on Demonstration of Problems of Algebra. Within it “he laid the foundation of the Pascal’s triangle with his work on triangular array of binomial coefficients” (Famous Mathematicians, 1). This treatise was not only highly influential in Persia, but made its way across Europe as well. 


 About the same time, circa 1050, Jia Xian who is also called Chia Hsien is also one of the first mathematicians known to have developed what would later be called Pascal’s Triangle. He lived between 1022 and 1054. Very little is known about his life, except that he was a eunuch, one of the emperor’s special guards who welded more influence as advisors than one would think. He wrote two books on mathematics, both which have been lost except the titles, The Yellow Emperor’s detailed solutions to the Nine Chapters on the Mathematical Art or Huangdi Jiuzhang Suanjing Xicao, and a collection of mathematical rules called Suanfa Xuegu Ji. After Jia Xian’s discovery, Chinese mathematicians continued to look at the triangle’s binomial coefficients, trying to take it further. A few centuries after Jia Xian, in 1261, one of these mathematicians, Yang Hui, (writer name of Qianguang) wrote an analysis of the mathematical rules in Huangdi Jiuzhang Suaniing Xicao in great detail (Beard, 1). Because of Hui, the work of Jia Xian has survived even though his book perished. In his preface, Hui explains his intention to make Jia Xian’s work better known, including his understanding of the triangle with his table “which records the coefficients up to the row 1 6 15 20 15 6 1” (Mac Tudor, 1). Very little is known about the life of Yang Hui, except for the works he left behind in reclassifying the ancient mathematical works of those mathematicians who came before him. Because of his efforts in preserving the ancient methods, in China the triangle is often referred to as the Yanghui triangle.

Later, an itinerant teacher, Zhu Shijie traveled across China in the later part of the 13th century. He was considered one of the greatest mathematicians of China. He is best “known for having unified the southern and northern Chinese mathematical traditions” (Horiuchi, 1). In 1303 he published Siyuan yujian or “Precious Mirror of Four Elements” which showed a diagram of the triangle, which was labeled the “Old Method”, proving that the concept was much older.





Zhu Shijie's illustration of Jia Xian's triangle. Image courtesy of Encyclopeadia Britannica, Beard

 

In Italy, another mathematician Niccolò Fontana, came onto the mathematics scene. In 1535, Bologna University held one of their public mathematics competitions where Fontana revealed a solution that had been considered impossible. He later  “devised a method to obtain binomial coefficients called Tartaglia’s Triangle” (16th Century, 1). Due to an injury, Fontana stammered and was called Tartaglia, which means “the stammerer”.  Even though he produced many formulas, he “died penniless and unknown” (16th Century, 1). Which brings us to Blaise Pascal.

Much later in 1654, French philosopher and mathematician Blaise Pascal wrote a treatise on the triangle named Traité du triangle arithmétique (Treatise on Arithmetical Triangle). It was published in 1655.


Blaise Pascal was both religious and a scientist. “He laid the foundation for the modern theory of probabilities, formulated what came to be known as Pascal’s principle of pressure, and propagated a religious doctrine that taught the experience of God through the heart rather than through reason” (Jerphagnon, 1). Syringes, hydraulic pressure, the barometer, and the first type of calculator, among many other contributions of science, can all be linked back to Pascal, including the triangle named after him. Although Pascal didn’t discover the triangle first, he “made the conceptual leap to use the triangle to help solve problems in probability theory” (17th Century, 1).

                                                  

                                           References

Beard, Andrea. Yang Hui: Chinese Mathematician. Encyclopaedia Britannica. https://www.britannica.com/biography/Yang-Hui

Famous Mathematicians. https://famous-mathematicians.com/omar-khayyam/

Horiuchi, Annick. “Zhu Shijie: Chinese Mathematician.” Encyclopaedia Britannica. https://www.britannica.com/biography/Zhu-Shijie

Hosch, William H. Encyclopedia Britannica https://www.britannica.com/science/Pascals-triangle.  

“Jia Xian” Mac Tudor. http://www-history.mcs.st-and.ac.uk/Biographies/Jia_Xian.html

Jerphagnon, Lucien, et al. “Blaise Pascal: French Philosopher and Scientist.” Encyclopaedia Britannica. https://www.britannica.com/biography/Blaise-Pascal

Kazimir, Jessica. “Pascal’s Triangle.” Montclair University.  http://pages.csam.montclair.edu/~kazimir/history.html

“16th Century Mathematics – Tartaglia, Cardano, and Ferrari” The Story of Mathematics. https://www.storyofmathematics.com/16th_tartaglia.html

“17th Century Mathematics – Pascal.” The Story of Mathematics. https://www.storyofmathematics.com/17th_pascal.html

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Pascal's triangle made its first appearance in Europe in Jordanus de Nemore's Arithmetic (13th century). Gersonides used the multiplicative formula to determine the binomial coefficients in the early 14th century. The whole triangle was published on the frontispiece of Petrus Apianus' (1495–1552) book on commercial calculations in 1527. In 1544, Michael Stifel published a section of the triangle (from the second to the center column in each row) as a table of figurate numbers. Tartaglia's triangle is the name given to Pascal's triangle in Italy, after the Italian algebraist Niccol Fontana Tartaglia (1500–1577), who published six rows of the triangle in 1556.

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