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
