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János Bolyai

A portrait, allegedly* of János Bolyai, by Mór Adler. Image: Pataki Márta and Szajci, via Wikimedia Commons.

One of my favorite things that we’ve been able to learn about this semester has been the different mathematicians that we’ve studied. It fascinates me to hear of their interaction with each other and how they affected one another’s work. As much as I love the math, the historical aspect is something that I had never heard and something that I love learning about. One of those mathematicians that I wanted to know more about was János Bolyai. Of him Gauss would say, “I regard this young geometer Bolyai as a genius of the first order.” Coming from someone like Gauss, that is quite the compliment. I wanted to know what made Gauss say that about such a young mathematician.

János Bolyai was born in Kolozsvár (which is now the city of Cluj in Romania) to Zsuzsanna Benkö and Farkas Bolyai, who was also a great mathematician, physicist and chemist at the Calvinist College. Like many fathers, Farkas wanted his son to follow in his footsteps and perhaps to achieve more than even Farkas himself had achieved in the field. So, he raised János with that goal in mind. However, Farkas was a firm believer that a strong body would lead to a strong mind, so in János’s younger years, most of the attention was spent developing his physical body. (O’Connor & Robertson, 2004)

János quickly became a child prodigy. According to Barna Szénássy in History of Mathematics in Hungary until the 20th Century, “… when he was four he could distinguish certain geometrical figures, knew about the sine function, and could identify the best known constellations. By the time he was five [he] had learnt, practically by himself, to read. He was well above the average at learning languages and music. At the age of seven he took up playing the violin and made such good progress that he was soon playing difficult concert pieces.” (Szenassy, 1992). Bolyai’s childhood and adolescence were fascinating. His father wanted to send him to live with Gauss as a student in order to accelerate his mathematical education, but Gauss would not agree to it. Because the Bolyai family didn’t have the financial assets to send János to an expensive university, they made the decision to send him to the Royal Academy of Engineering at Vienna to study military engineering. He truly was a “jack of all trades.” He finished the seven year engineering program in just four years, became an excellent sportsman and even performed as a violinist in Vienna. He was in the military for eleven years, where he became known as the greatest swordsman and dancer in the Austro-Hungarian Imperial Army. (O’Connor & Robertson, 2004) It wasn’t until 1820 that he began intense study on Euclid’s parallel postulate and the development of hyperbolic geometry. One of János Bolyai’s most recognized quotations comes from a letter that he wrote to his father when he said that he had, “created a new, another world out of nothing.”

The story is well-known of the publication of Bolyai’s work on hyperbolic geometry. During János’s military service, his father read the mathematical work that his son had sent him previously and then went to where János was stationed. Farkas then encouraged his son to publish his work. János later said, “Had my father not happened to urge or even force me at Marosvásárhely, on my way to duty in Lemberg, to immediately put things to paper, possibly the contents of the Appendix would never have seen the light of day.” When Farkas sent a copy of his son’s work to his old friend, Gauss, Gauss responded by saying, “To praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.”

Bolyai’s work on the parallel axiom led to the development of what would be known as a “pseudosphere,” which is an object that extends infinitely, but has a finite volume. This object was created by Beltrami many years later, but now is seen as an embodiment of hyperbolic geometry.

The story of János Bolyai ends as a sad one. He did not manage his money very well, gave very little care or attention to the family estate he had inherited, and finally left his wife and children. Years after his work on hyperbolic geometry, he found the works of another geometer named Lobachevsky, who he thought was fictional; a cover that Gauss had created in order to steal his work on hyperbolic geometry. He quit working on mathematics entirely and focused on “a theory of all knowledge.” (O’Connor & Robertson, 2004) Although he may not have felt like he received the credit that he deserved for his work, János Bolyai was indeed, as Gauss called him, “a genius of the first order.” He gave the world of mathematics a new way of understanding the concept of parallelism and the way in which mathematics relates to our natural world.

*Editor’s note: The portrait here, which also appears on postage stamps honoring János Bolyai, has long been associated with the mathematician but is not authentic. For more information, see “The Real Face of János Bolyai” by Tamás Dénes.

Works Cited

O’Connor, J., & Robertson, E. (2004, March). János Bolyai. Retrieved from MacTutor History of Mathematics:

Szenassy, B. (1992). History of Mathematics in Hungary until the 20th Century. New York: Springer-Verlag Berlin Heidelberg.

“Euclid – Master of us all”

Statue of Euclid in the Oxford Museum of Natural History, Courtesy of Lawrence OP on Flickr.

Statue of Euclid in the Oxford Museum of Natural
History, Courtesy of Lawrence OP on Flickr.

After talking about Euclidean and non-Euclidean geometry in class, I wanted to know more about Euclid and his life. I went to a couple different sources, and found an awesome biography from the MacTutor History of Mathmatics archive that gave me a lot of interesting information.

Euclid of Alexandria was born around 325 BC, but we don’t know exactly where he was born. We do know, however, that he died about 265 BC in Alexandria, Egypt. We refer to him as Euclid of Alexandria to avoid confusion with Euclid of Megara, who lived nearly 100 years earlier and was a student of Socrates. (Wikipedia, 2014) We don’t know a whole lot about Euclid’s life. One source of information that exists (although not believed to be very credible) comes from some Arabian authors who claim that Euclid was the son of Naucrates and that he was born in a city called Tyre (present day Lebanon). But, as mentioned, most mathematical historians believe that the claim was invented by the authors. (O’Connor & Robertson, 1999)

Math historians have varying opinions as to the existence of Euclid the mathematician. The most widely held is that Euclid actually was a real person, and that he really did write “The Elements” and the other works published in his name. The second idea is that he was a leader of a team of mathematicians in Alexandria that all contributed in writing the works attributed to Euclid. Some think that this team even continued to publish “The Complete Works of Euclid” in Euclid’s name after his death. The third hypothesis is that Euclid of Alexandria was a creation of this team of mathematicians in Alexandria who used the name Euclid, having derived it from Euclid of Megara. (O’Connor & Robertson, 1999) There exists a great deal of evidence suggesting that Euclid, whether that be an individual person or a team of mathematicians, founded a prestigious mathematics school in Alexandria.

The authors of the article, JJ O’Connor and EF Robertson, point out that while the third hypothesis is unlikely, we see the example of Bourbaki in the 20th century. However, the members were renowned mathematicians in their own right. If “Euclid” was a secret team of competent mathematicians, we don’t know who they were.

I find the argument that Euclid led a team of mathematicians most convincing. I believe Euclid was a person because of the stories and history associated with the man, but I believe that the work he performed was too much for any one man to produce without a team behind him. The Elements became a textbook that was used for centuries after his death, and I just think it’s unlikely that a work that comprehensive is something that came from just one person.

Other pieces of evidence that lead me to believe in the idea that Euclid was an actual person was the different accounts of Euclid’s relationship with others, especially mathematicians. Pappus, known as “the last of the great Greek geometers”, (O’Connor & Roberton, Pappus of Alexandria, 1999) said that Euclid was “… most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.” The fact that Pappus gave such a specific description makes it seem unlikely that Euclid was merely a creation of other mathematicians of that time period. One other story that the authors recounted was originally told by Stobaeus, who was a compiler of works from many ancient Greek authors (Wikipedia, 2014). He said, “…someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid “What shall I get by learning these things?” Euclid called his slave and said “Give him three pence since he must make gain out of what he learns.” My favorite story, and one that I had actually heard before in several other circles, is that of the interaction between Euclid and Ptolemy. Proclus says, “they say that Ptolemy once asked him if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry.” I loved that. I’ve often heard that same phrase applied to other subjects, athletics, and religious pursuits. Indeed, there is no “royal road” to anything that is of worth.

The Elements, like we’ve discussed in class, is Euclid’s greatest claim to fame. In the article, Robertson and O’Connor quote Sir Thomas Heath, who was responsible for translating the works of many ancient Greek authors, including Euclid to English. Heath said, “This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time. … Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency…” (O’Connor & Robertson, Euclid of Alexandria, 1999) Indeed, Euclid’s name will always be reverenced as one of the formative thinkers in all mathematics for his work on one of the greatest textbooks ever produced by men. However, I maintain my belief that The Elements was not the work of one man, but the work of many. I also believe that it should in no way detract from the respect and praise given to Euclid, the individual, as one of the greatest mathematicians of all time.

Mayan Mathematics

After we talked about Babylonian mathematics and how they used a base-60 system, it got me thinking about different ancient cultures and the numbering systems that they used. My little brother is currently on an LDS mission in Guatemala, and he sent me some pretty cool pictures of Mayan ruins. Also, I remember back to 2012 when many people were hysterical about the “end of the world” because the Mayan calendar had stopped on a specific day in December. But man, if I was part of a civilization from thousands of years ago, I would NOT have made a calendar that went even that far. Had I been in charge, people might have thought the world was ending around the time Columbus made it to the Americas. A lot of what we know about the Mayan people was lost when they were invaded by the Conquistadors from Spain. A Spanish missionary, Diego de Landa, had a great respect for the Mayan people, but despised their religious customs. He ordered most of their religious icons, texts, and other documents to be destroyed, but of those that survived remain the Dresden Codex, the Madrid Codex and the Paris Codex. (Side note: I think it’s interesting that these documents are named after those that found them rather than those that created them.) The Dresden Codex, which will be discussed hereafter, is probably the most well-known. The Mayan people were a city-building and innovative society. There were 15 large cities (some of 50,000 or more) in the Yucatan peninsula. The people were governed by “astronomer-priests” that manipulated others with their religious instructions. (True case of where math and knowledge is power!)

The Maya number system. Image from MacTutor History of Mathematics.

Image from MacTutor History of Mathematics.

The Mayans had one of the most advanced number systems in the world at its time. It was a base-20 system (kind of) that also relied quite heavily on the number five. Some think that this is due to five fingers and five toes on each hand and foot. Their system relied on three different symbols. A “pebble” (small black circle) was used to represent the ones place. A “stick” (straight black line) was used to represent the fives place, and a shell was used to represent the number zero. In a report about the Mayan numbering system, JJ O’Connor and EF Robertson explained why the system was not exactly a base twenty system, but one that had been slightly modified: “In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20’s up to 19, the next the number of 400’s up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20’s up to 19, it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 202, the next the number of 18 × 203 and so on.” (O’Connor & Robertson, 2000)

The Dresden Codex contains written evidence of the use of this numbering system. Ifrah claims that the system was used in astronomy and calendar calculations. He states, “Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.” (Ifrah, 1998)

One of the reasons the Mayan numbering system was not a true base-20 system was because it was partially a base-18 system as well. The reason for this seems to be the calendars that they maintained. Robertson and O’Connor state: “The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 “months” of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short “month” of only 5 days that was called the Wayeb…” (O’Connor & Robertson, 2000).

It amazes me what the Mayans were able to do with the knowledge that they had available to them. Much of what we know of the Mayans comes from the few documents that have been preserved and the ruins that have been interpreted, but I wish that more information was available on how they used these systems in their societies and the other ways in which they used mathematics.

Ifrah, G. (1998). A universal history of numbers: From prehistory to the invention of the computer. London.
O’Connor, J., & Robertson, E. (2000, November). Mayan mathematics. Retrieved from The MacTutor History of Mathematics Archive: