Before recounting what I learned from the late Professor Howard Eves today, there are some announcements to make. I am currently attending Professor Robert Cassanello’s Readings in Curation and Public History (HIS 5088) and the assignment for this week was to ask the life cycle of an object in their collection (basically what their policies are from the point someone wants to donate an object, to it being acquired, cataloged and then displayed). To my surprise, Ms. Rubin was a potential option for the class to use as a resource. How convenient! So, this Friday (when I will have a little bit more time) I will be learning about for then assignment and I have already asked Ms. Rubin’s permission about it during lunch (she agreed to this request). Also, I already know what my next assignment after finishing the Howard Eves Audio Collection is.
My next assignment is something I have already mentioned on the blog before and have helped process in some fashion. I will be working on the George Stuart Collection. Thankfully, John Settle (before he left) and a few others have already done much of the initial processing. The collection needs to be reorganized and that will be my task. But, for the present I am on the tail end of the Howard Eves Audio Collection.
Resuming from yesterday, Dr. Eves explained the second of the three stages of algebraic notation: syncopated algebra and reviewed syncopated abbreviations from the equations. During this review, Dr. Eves made a comment on how recent the present stage of algebra was developed (approximately four hundred years). He soon shifted subjects to about the man who helped develop syncopated algebra, Diophantus of Alexandria.
After briefly commentating on Diophantus’ life, the ancient Greek’s written works were discussed. Most prominent was Diophantus’ Arithmetica, which during the overview Dr. Eves made an error on the total number of books in the work (he was correct by student that there were 13 books out of 18, not 6 out of 13). Diophantus also wrote on polygonal numbers, but only a fragment of it exists in the present. Unfortunately, Diophantus’ The Porisms has been lost to time. Dr. Eves then shifted the discussion back to Arithmetica and Diophantine analysis.
The professor explained Diophantus looked for rational numbers as solutions and used examples from the Arithmetica. Dr. Eves then explained how Diophantus used syncopated algebra and developed syncopated abbreviations. During this description, Dr. Eves told a story of when he was at Princeton interacting with a student preparing to be minister and created a sermon on why mathematicians cannot find the unknown. Next, the professor examined symbols Diophantus used, especially the final sigma and x2. He paused his lecture to announce the next test was to be open book test due to the symbols.
Dr. Eves finished talking on the symbols with the symbol for units, then he illustrated the symbols used in equations. After the illustrations, Dr. Eves dismissed the class. I quickly switched to the next disc to continue the collection.
The recording began as Dr. Eves reminded the class what the next test would cover and how the papers were graded. He then introduced the topic of the class: The Life and Works of Pappus of Alexandria. Pappus wrote commentaries on Euclid’s Elements and Ptolemy’s Almagest, but he also wrote Mathematical Collection (unfortunately, Book I and part of Book II are lost). Dr. Eves briefly commented on the surviving contents of Book II, which contained Apollonius’ system of large numbers. Book III was more heavily discussed.
Book III was divided into four parts. The first was the Theory of Means, which Dr. Eves used at great detail some equations as examples of the theory in practice. After a lengthy display of these theories in practice, Dr. Eves shared he once believed as a child that everyone thought in pictures like he did. To wrap up his discussion on the first part of Book III, the professor used equations to demonstrate the harmonic mean, as Pappus would. He then briefly discussed Part Two: The Insertion of Two-Mean Proportionals, Part Three: Triangulating Postulates, and Part Four: Inscription of the Five Regular Polyhedra in A Sphere. Unfortunately, the recording ended not too long after.
The next disc began with Dr. Eves discussing how students had published papers based on the subjects chosen in the class. After giving overview of the previous class on Pappus, he announced the topic of the class would be The Commentators of the Post-Greek Era of Mathematics. He began with Theon of Alexandria, who became an administrative official at the Library of Alexandria at some point.
Dr. Eves compared Theon’s position as similar to becoming a Dean at a college, a position he did not think too highly of. Dr. Eves was reminded of Professor Armstrong of the College of Arts and Sciences who became a Dean but was returning to be a professor at the time. He briefly discussed Theon’s commentary of Claudius Ptolemy’s Almagest and his revision of Euclid’s Elements. The professor then noted Theon’s daughter, Hypatia, became one of the first known female mathematicians.
Dr. Eves then briefly discussed Hypatia’s accomplishments in mathematics, philosophy, and medicine. He noted Hypatia had aided her father in his revision of Euclid’s Elements and wrote commentaries on Diophantus’ Arithmetica and Apollonius’ Conics. The professor then paused to mention his outlook on the future of mathematics in the final chapter of his book, where he believed women would become more involved mathematics. He also took pride in that his history book gave attention to Hypatia.
However, the discussion would take a tragic turn. Dr. Eves noted Hypatia’s philosophical lectures on religion and how the fact she was a pagan ultimately lead to her demise. He briefly discussed that Alexandria’s Christian population developed into two factions: intellectuals and fanatics. The professor carefully differentiated acts conducted between those “by religion” and “because of religion” when talking about fanatics. In the end, fanatics killed Hypatia in a savage manner (dismemberment was involved). Dr. Eves closed the discussion by mentioning Charles Kingsley’s 1853 novel Hypatia contained the most graphic countenance of her murder.
The next contributor discussed was Proclus Lycaeus, Neoplatonic philosopher and mathematician. Dr. Eves made a point that though Plato was not a mathematician, Plato’s Ideal of State and his high value of geometry influenced mathematics. Among Proclus’ written work Proclus’ Commentary on Euclid: Book I and that it contained the Eudemian Summary. Other works included Proclus’ Commentary on Plato’s Republic and how Proclus is credited as the one who created the Trammel Construction of Ellipses. Dr. Eves then proceeded to demonstrate it.
Unfortunately, the audio cut out and was restored as Dr. Eves was discussing Proclus disagreement with Ptolemy’s attempts in Almagest (unfortunately, because of the missing audio, it is unknown what the subject was). Dr. Eves stated this disagreement formed the basis for a new form of geometry to be created. The next scholar was Simplicius of Cilicia, who created the Athenian School in Persia.
Simplicius mostly commentated on the works of Aristotle, but also commentated on Apollonius’ squaring circles (invented special curves to solve it), Hippocrates of Chios’ lunes, and Eudoxus’ concentric spheres. The next scholar was Eutocius of Ascalon.
Dr. Eves briefly discussed Eutocius’ Commentary on Archimedes’ On Sphere and Cylinder, Commentary on Archimedes’ Measurement of a Circle, and Commentary on Archimedes’ Plane Equilibriums. He also noted Eutocius wrote a commentary on Apollonius’ Conics. With these remarks, Dr. Eves noted that this marked the End of the Greek Mathematical Schools. The professor then quoted Edgar Allan Poe’s poem, “To Helen,” “the glory that was Greece and the grandeur that was Rome.” He then dismissed class and handed papers back (humorously, the student who recorded these tapes did not stop recording after this point).
On the next disc, Dr. Eves gave an overview of the previous class and noted the loss of creative mathematicians was a consequence of the closing of the Greek Schools of Mathematics. He remembered his mentor at Harvard University, Julian Lowell Coolidge, stating “There were giants in the land then…” as a reference to the Greek period of mathematics. He then stated there was a thousand year lull of mathematics in Western Europe as a result.
Dr. Eves then shifted the focus to the mathematical accomplishments of Asia and in the Islamic World as the topic for the class. Unfortunately, due to an abrupt interruption in the semester (something about a gas leak?), the class was actually behind on schedule and Dr. Eves had to shorten the mathematical discussions from the Middle East. The professor then began a short overview of the development of civilization in Asia.
The professor then explained the lack of primary sources were due two factors: the material the findings were written (bamboo in China, and the bark used in India) and the efforts by Shih Huang-Ti (also known as Qin Shi Huang) to burn books. He also began to list the major scholarship works in the History of Chinese Mathematics. However, he noted that few scholars have written about Chinese Mathematics until recently.
The first major work was The Development of Mathematics in China and Japan that was published by Yoshio Mikami in 1918. Next was Noel Joseph Terence Montgomery Needham’s Science and Civilisation in China that was published in 1959. The third volume, Mathematics and the Sciences of the Heavens and Earth, focused on mathematics.
This was where my own trouble with gathering information began. Dr. Eves stated n 1987, Shang Kan-shi (the spelling may not be correct) wrote An Introduction to the History of Chinese Mathematics (the name was inspired by Dr. Eves’ own book, An Introduction to the History of Mathematics). This Chinese scholar contacted Professor Eves to inquire if he would be able translate it into English. Dr. Eves felt he was not qualified for the job and he did not know anybody with the right knowledge of the sciences, but he was optimistic that book’s information would be worthwhile.
I would have continued to to listen and make notes of the recording, but I was bogged down in trying to find more information on this book by that Chinese scholar (to make sure I got his name spelled right among other things). Unfortunately, I ran out of time and did not get a concise answer. I will just have to move on from it tomorrow.
So, tomorrow will be the last “class” I have with Dr. Eves. I will be sure to post the final installment tomorrow. But, I still have to make the finding aid. Nevertheless, enjoy the evening and stay safe. Bye!