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).



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

Friday, September 03, 2021

The Things They Carried


I really related to the matter-of-fact tone of “The Things They Carried” by Tim O’Brien. There have been times when the only way to get through talking about something had to be very clipped and matter-of-fact in order to keep my vocal chords from closing off. I had a terminally ill child and so many times I’d be in the emergency room rattling off previous medications and surgeries and what brought us there this time with the seemingly emotion of a stone all the while my heart was racing and I held back my fear because once I let it loose, I’d be weeping uncontrollably. The stark tone as the supplies they carry are listed with things interspersed such as “They carried the sky. The whole atmosphere, they carried it, the humidity, the monsoons, the stink of fungus and decay, all of it, they carried gravity” (O’Brien 1184) felt so familiar as when I rattled off the weight of my son’s mortality that I was trying futilely to bear. It also weaved in the setting of the jungles of Viet Nam in a way that made the life of the soldier so bleak and enduring to the point that the horribleness of it became mundane, which makes it all the more awful seeming as a reader.

Yet in the end Lieutenant Cross strips the weight of the world and home and love from his rucksack because to be a true soldier, a responsible leader of soldiers.  He had to let that all go in order to bear the heavier artillery of the burden soldiers bear. Everything else is gone, stripped away. “Commencing immediately, he’d tell them, they would no longer abandon equipment along the route of march. They would police up their acts” (O’Brien 1189) “He would not tolerate laxity. He would show strength, distancing himself” (O’Brien 1190). ­­­­­­

In comparison, the hateful, dark tone of The Cask of Amontillado came out gleeful in a macabre sense that matched the setting of the underground chilly vaults where wine was kept amid the family tomes and stacked bones of the dead. My takeaway of the theme? Never trust someone you have wronged, especially if they are offering gifts and are overly cheerful. When Fortunado is told “we will go back; your health is precious. You are rich, respected, admired beloved: you are happy, as once I was. You are a man to be missed” (Poe, 1128) he should have run for the hills instead of going further into the depths of the vault. I’m kind of shaking my head at this one as Fortunado should have known better. I feel about as sorry for him as the woman in a horror show who goes down into the basement when she hears a noise.


Work Cited

O’Brien, Tim, “The Things They Carried.” Abcarian, Richard, Marvin Klotz, and Samuel Cohen, eds. Literature: The Human Experience: Reading and Writing. New York: Bedford/St. Martin’s, 2016, pp. 1177-1190

Poe, Edgar Allen, “The Cask of Amontillado.” Abcarian, Richard, Marvin Klotz, and Samuel Cohen, eds. Literature: The Human Experience: Reading and Writing. New York: Bedford/St. Martin’s, 2016, pp. 1126-1131


"The Things They Carried by Tim O'Brien" by manhhai is licensed under CC BY 2.0


Thursday, August 19, 2021

The Future is Dead: The Absence of Hope in Things Fall Apart and “Death Constant Beyond Love”


There comes a time when the years behind number more than the years ahead. Were goals accomplished? Was fame found? Or did it all wither away? Is there anything left to look forward to? This is the circumstance that Okonkwo in Things Fall Apart and Senator Onésimo Sánchez in “Death Constant Beyond Love find themselves in. Their youth has passed them. They are both staring at the end of their lives. For Okonkwo this end is figuratively seen as the end of his society's way of life where he has no place in the adopted way of life anymore. For Sánchez, his impending death is literal as he has been diagnosed with only months left to live. With their lost futures, everything Okonkwo and Sánchez worked for in their youth, their fine reputations and standings in their communities has become unattainable. The theme of youth can be found in both Things Fall Apart and Death Constant Beyond Love as Okonkwo and Senator Sánchez feel that they have no future so there is nothing left to hope for, and that any of the respect they once held no longer has any meaning for them at the end of their lives.

The very first line of Things Fall Apart states, "Okonkwo was clearly cut out for great things" (Achebe 5.10). That is the view Okonkwo holds of himself. In the village of Umuofia, living to an old age was respected, but achieving great feats, wealth, and skill in battle was revered. Okonkwo had all of these things and was revered across all the nine villages. (Achebe 5.10). He is already highly respected and wants to continue that upward trend in his life.

Because Okonkwo was greatly respected, he was given charge over a boy, Ikemefuna, who was given to their village as a sacrifice from another village in an exchange to keep the peace. Okonkwo raises him with his own son, Nwoye, for many years. It was a great honor but came at a cost to his own standing in his son's eyes and how his son would view their village traditions at a later time when it counted. The day comes when the village decides it is time to kill Ikemefuna. An elder of the village warns Okonkwo to "not bear a hand in his death" (Achebe 5.10), yet fearing that his people would think him weak, Okonkwo makes the final blow in Ikemefuna's death (Achebe 5.10). That act marks the beginning of his own son's path away from the traditions of their people. In an ironic twist of fate, at the funeral of the elder who had warned him about killing Ikemefuna, Okonkwo accidentally kills the deceased elder's teenage son when Okonkwo's gun explodes and shrapnel fatally hits the teenager. Even though the death was accidental, it was viewed as an offense against the earth goddess, which required Okonkwo with his family to leave the land for seven years. To cleanse the land from the offense, his houses were burned, and animals killed.

Throughout those seven years of exile, Okonkwo accumulated more wealth and dreamed and planned to go back to his village triumphant; "He was determined that his return should be marked by his people. He would return with a flourish, and regain the seven wasted years" (Achebe 5.12). However, the village had undergone a transformation during his time away. To his mind, the Christian "church had come and led many astray" (Achebe 5.12). The church claimed that their old traditions were wrong. It was to be expected that outcasts and villagers of lower stations might flock to it, but worthy and high esteemed men of the village had also joined it. Men "who had taken two titles, and who like a madman had cut the anklet of his titles and cast it away to join the Christians" (Achebe 5.12). Everything has changed and all the goals Okonkwo has set to gain respect for himself were futile.

As a warrior, Okonkwo falls back on the course of fighting back and driving the white men out. However, his friend explains that since "our own men and our sons have joined the ranks of the stranger. They have joined his religion and they help to uphold his government" (Achebe 5.12). At this point Okonkwo's own son, Nwoye, has given up the traditions of his father and joined the Christians. If they did uprise to drive the foreigners out, their own people who have been converted would send for more soldiers. "How do you think we can fight when our own brothers have turned against us?" (Achebe 5.12).

Everything that Okonkwo spent his entire youth yearning for and striving for have become unimportant. Everything that was once respectable has become something that is against the law. Mighty men are now put in prison for adhering to their traditions and made to work at menial tasks like fetching wood. "Some of these prisoners were men of title who should be above such mean occupation" (Achebe 5.12). The white men convinced their people that their customs are wrong so they cannot "fight when our own brothers have turned against us" (Achebe 5.12). Okonkwo mourns that "our clan can no longer act like one. He has put a knife on the things that held us together and we have fallen apart" (Achebe 5.12). There is nothing left for Okonkwo, no respect or esteem to gain. Everything he has worked for has come to nothing. There is no longer a place for him in this new future. His final act is an unthinkable abomination against the traditions he had once held dear in the taking of his own life. Compared to Okonkwo’s journey of seeking fame and titles of respect to an absence of hope for a good future, Senator Onésimo Sánchez takes a similar downward spiral to having no future at all to hope for.

Death Constant Beyond Love explores the theme of youth as the dying Senator Onésimo Sánchez realizes that there is nothing left of his life, no prize to seek, nothing worthwhile to look forward to. This explores the theme of youth as Senator Sanchez, who knows his time is up, moves from seeing the world around him from the viewpoint of what can be gained in his career to seeing things as an average person. It also allows him the freedom to speak plainly and do things that he never would have before.

During a campaign tour of the dying town Rosal del Virrey, Senator Sánchez, who only has a few months to live, sees things as they really are. He knows that like himself, the town will never flourish. He helps a few people he wouldn't have bothered with previously, speaks plainly to the "important" people of town, and allows himself to have an affair with a young woman, ruining his political legacy when he dies. Márquez frames the story with a mixture of pragmatic actions that revolve around deep fears. For example, Sánchez takes an ordinary nap, but is plagues with thoughts of his death. This sentence captures pragmatism blended with fear beautifully: “Then he put the electric fan close to the hammock and stretched out naked for fifteen minutes in the shadow of the rose, making a great effort at mental distraction so as not to think about death while he dozed” (Márquez 3.13). These types of sentences are sprinkled throughout the story, reminding us of the senator’s impending death, while he goes about his business, trying to pretend there is nothing to fear, which makes it stand out all the more. This shows that Sánchez really has nothing left to hope for as he has no future for himself and doesn’t really seem to care about what he is leaving behind.

The senator knows he is dying so there isn't anything left for him to care about. He knows he'll win the political race, but it is pointless since he won't be around. He also doesn't care about helping people while he can or leaving behind something good as a legacy in his last few months. In his speech he says, "'We are here for the purpose of defeating nature,' he began against all his convictions" (Márquez 3.13). He has no hope for anything good. "The erosion of death was much more pernicious than he had supposed, for as he went up onto the platform he felt a strange disdain for those who were fighting for the good luck to shake his hand" (Márquez 3.13). Although he seems indifferent, his impending doom also gives him the freedom to do things that normally wouldn't be in his best political favor and the liberty to say things that he normally would hold back on.

When the senator makes his rounds after the speech to meet the people of the village, he normally would find "some way to console everybody without having to do them any difficult favors" (Márquez 3.14). There is a woman he meets. She says if he really wants to help that he will provide a donkey so that it can carry the burden of water from the well to her home as she no longer as a husband to help her. Because at the state of mind he is in, he decides on the spot to get her a donkey and carries through with his promise that same day. "A short while later an aide of his brought a good pack donkey to the woman's house and on the rump it had a campaign slogan written in indelible paint so that no one would ever forget that it was a gift from the senator" (Márquez 3.14). This is possibly a hint that he does want to be remembered well in at least a small way, although he isn’t making any large contributions for a lasting impression at the end of his life. Later that evening when he has a meeting with the people of importance of the town, he is so tired of it all, that he veers from his normal platitudes and speaks plainly to them, saying, "my reelection is a better piece of business for you than it is for me, because I'm fed up with stagnant water and Indian sweat, while you people, on the other hand, make your living from it" (Márquez 3.14). In the height of his career, before he knew his life was going to be so short, he most likely would have told them what they wanted to hear, rather than the truth as he was able to see it. This breakdown of his hope for himself is seen with his interactions with Nelson Farina.

Nelson Farina had been trying to get a favor out of Senator Sánchez for years. Nelson noticed that his daughter, Farina, caught the senator's eye so he sent her to him that evening. She tells Sánchez she has come for her father. The senator is married and has never tarnished his reputation with a scandal, yet Farina's "beauty was even more demanding than his pain, and he resolved then that death had made his decision for him" (Márquez 3.15). This marks his final resolve on caring about the future he has left. However, Farina is wearing an iron chastity belt, which her father has the key to which he'll release once Sanchez provides him with a promise in writing that the senator will help him out with his predicament. At first the senator is angry, "then he closed his eyes in order to relax and he met himself in the darkness. Remember, he remembered, that whether it's you or someone else, it won't be long before you'll be dead and it won't be long before your name won't even be left" (Márquez 3.15). He reflects for a few moments and then lays aside what he normally would do and decides that it doesn't matter anymore. All hope is gone. He says to Farina that he'll grant her father's request.

Farina is ready to run to her father and get the key to her chastity belt, but the senator asks her to not worry about it just yet and to "sleep awhile with me. It's good to be with someone when you're so alone" (Márquez 3.15). This is a pivotal moment in his life as he is facing impending death. He finally allows himself to give into the fear that he has been facing all on his own. This begins an affair he has with Farina that within six months ruins his reputation and everything he had once worked so hard to achieve, yet his biggest regret upon his deathbed was that he would no longer be with Farina. Nothing else at that point mattered. Sánchez did have children that could carry on his legacy. He could have done something important to solidify the greatness of his political career, yet none of that mattered to him in his final days. In both Things Fall Apart and Death Constant Beyond Love the absence of hope wears both Okonkwo and Sánchez to the despair of nothing left for them.

Near the end of Okonkwo's and Sánchez's lives, these characters feel that they have nothing to hope in as the future is gone, which has made them do something which ruined their reputations that once meant so much to them. Okonkwo takes his own life which is an abomination to his clansman. They won't touch his corpse even to bury him. Senator Sánchez lived the last six months of his life having an affair that was "debased and repudiated because of the public scandal" (Márquez 3.15). The absence of hope for the future touches everyone at one point or another, whether that absence stems from the changing of society values in Okonkwo's case, or the stark realization that there will be no future as a person ages or is given an expiration date as shown by what Senator Sánchez faced. It's a universal condition that all humans will eventually have to come to terms with one way or another.


Works Cited

Achebe, Chinua. "Things Fall Apart." Modern World Literature. Soomo Learning, 2016, pp. 5.10-5.13.

Márquez, Gabriel García. “Death Constant Beyond Love.” Modern World Literature. Soomo Learning, 2016, pp. 3.13-3.15.

image: "Things Fall Apart, by Chinua Achebe" by elycefeliz is licensed under CC BY-NC-ND 2.0