This is a seminar for faculty from colleges and universities
in the Philadelphia area who share an interest in the history of
mathematics, open to the public and also available via Zoom to
allow remote
participants. The seminar meets monthly during the academic
year, usually at 6:00 p.m. on a Thursday evening at Room 103 of
the Mendel Science Center at Villanova
University for a light meal (donation $10.00), followed by a presentation by the
seminar speaker usually one hour long starting at 6:30pm which is then is followed by
open discussion for up to about 30 minutes.

This group was established in January 2001 for persons in the greater Philadelphia
area to:

- share our common interest in history of mathematics,
- encourage one another in our research efforts,
- offer a forum for reporting on work in progress

and it currently directed by Alan Gluchoff.

To add your name to our emailing list, send information to alan.gluchoff@villanova.edu. Zoom links are sent via email from Alan in response to requests.

__
Click here for
general directions to Villanova University.
__The Mendel Science Center is at the west end of campus next to
the train station [

**Driving:
**Villanova University is located on route
30, Lancaster Avenue, just east of I-476. Set your GPS to

800 Lancaster Ave, Villanova, PA 19085

**Parking:**

If you drive to the
meeting, you may park on the
Ithan Avenue I-1 parking garage at the intersection of Lancaster
and Ithan Avenues free for 1 hour and then for a low hourly fee
after that.

To enter Villanova by the main gate on Ithan Avenue and park in the multilevel M-2 parking garage adjacent to St. Augustine Center, you must fill out an electronic form requested from Alan by email. The Mendel Science Center is on the other side of the St. Augustine Center.

**Public transportation:** Take SEPTA's Paoli -
Thorndale train to the Villanova station. If outbound from
center city go down through the tunnel to the inbound side. From
the inbound platform a few steps lead to a parking lot behind the Mendel
Science Center. Proceed
to the archway entrance on the right to go up to the first floor
and then down the hallway away from the city to Room 103 on the left
just before the stairwell.

During the Zoom session, please keep your Zoom participation muted unless you wish to make a comment or ask a question of the speaker, since background sounds in any participant window will switch the Zoom session recognition of the person speaking away from the seminar speaker and show the errant window instead of the seminar speaker. During the shared screen talk, you may select Speaker or Gallery View from the Zoom View menu: select Speaker view to see the speaker together with the presentation, with the vertical screen division moveable from left to right to maximize either one.

Some recent talks to give a flavor of the group [see archives for additional links to talk materials]:

September 12, 2024 Tom Drucker, University of Wisconsin, Whitewater:

**Exploring Felix Klein’s Contested Modernism**

ABSTRACT: One of the pleasures of history of mathematics is that it is often free from the jargon that one finds prevalent in much contemporary history of a broader sort. The article about which I'll be speaking has five authors, one from Israel, three from Germany, and one from the United States, and is laden with more jargon than is necessary. The number of directions in which the article sets off suggests that different contributors may have had different points they were trying to make. The primary target is the Marxism of the historian Mehrtens, but another target is Frank Quinn. Among the questions the authors hope to settle is whether Klein was antisemitic in behaviour or writings. There is also the issue of whether Klein was countermodernist, while Hilbert was recognizably modernist.
I

t's clearly impossible to tackle all these matters in one talk, but I'll address a few of those that seem least dependent on jargon. First, there's the issue of Klein's alleged antisemitism. Then there's the question of whether a Marxist historian can address such issues without any historical bias. Finally, there are the roles that Klein and Hilbert played in creating the Gottingen that was to fall victim to the Nazi view that only Deutschemathematik was worth pursuing. There may be time to look at Quinn in the questions that follow the talk.

April 18, 2024 David E. Dunning, History and Sociology of Science, University of Pennsylvania:**
From Notations to Neurons: Mathematical Logic, AI, and the Act of Writing**

ABSTRACT: In this talk I will discuss two episodes from my current book project, a history of the rise of mathematical logic and its connection to early computer science and AI. I focus on shifting and competing notational practices in order to show how methods of writing have shaped understandings of natural and artificial reasoning alike. After briefly introducing the larger project, I will share two cases. I’ll discuss Victorian self-taught mathematician George Boole’s efforts to rewrite logic in algebraic notation, revealing his specific vision for how and why his system should be learned. Many of his readers were very enthusiastic, but rather than straightforwardly adopting his system, they tended to follow him in taking notation itself as a fruitful arena for innovation. After presenting Boole, I’ll turn to the watershed 1943 paper “A Logical Calculus of the Ideas Immanent in Nervous Activity,” by Warren McCulloch and Walter Pitts. The authors drew on recent mathematical logic and navigated “typographical necessity” in order to present abstract neurons as a system for writing logical propositions. By taking seriously the centrality of writing in the project that launched neural network techniques, I aim to show how modelling the mind mathematically was not simply about understanding or imitating intelligence, but rather reimagining thought as a fundamentally notational phenomenon.

March 21, 2024 Daniel Otero, Xavier University:**
Barrow's Sum of the Secants**

ABSTRACT: There are a handful of methods used to determine the integral of the secant function. We examine in this session what may be the earliest attempt to handle this challenging integral in the Lectiones Geometricae (1670) of Isaac Barrow (1630—1677). Barrow operated in the days when analysis of curves and their properties were pre-eminent in geometry, a generation before Newton successfully employed these ideas to describe a theory of gravitation and Leibniz clarified the relation between derivative and integral, and about 100 years before Euler recast the calculus as a collection of powerful and versatile techniques for their study. The diagram above comes from the Appendix to Lectio XII of the Lectiones Geometricae. In seventeenth-century geometric language, Barrow provides a derivation of what we would interpret today as the integral of the secant function. This talk will invite the audience into a reading of Barrow’s text (and a wrangling of the nest of curves in the figure above!) to clarify this interpretation.

March 21, 2024 Daniel Otero, Xavier University:**
Barrow's Sum of the Secants**

ABSTRACT: There are a handful of methods used to determine the integral of the secant function. We examine in this session what may be the earliest attempt to handle this challenging integral in the Lectiones Geometricae (1670) of Isaac Barrow (1630—1677). Barrow operated in the days when analysis of curves and their properties were pre-eminent in geometry, a generation before Newton successfully employed these ideas to describe a theory of gravitation and Leibniz clarified the relation between derivative and integral, and about 100 years before Euler recast the calculus as a collection of powerful and versatile techniques for their study. The diagram above comes from the Appendix to Lectio XII of the Lectiones Geometricae. In seventeenth-century geometric language, Barrow provides a derivation of what we would interpret today as the integral of the secant function. This talk will invite the audience into a reading of Barrow’s text (and a wrangling of the nest of curves in the figure above!) to clarify this interpretation. [illustration]

February 15, 2024 Bonita Lawrence, Professor Emerita, Marshall University:**
Solving Dynamic Equations: Using Gifts from the Past**

The Marshall Differential Analyzer Project developed from an idea sparked by a visit to the London Science Museum’s display of historic differential analyzer machines. The primary goal of the project was to offer an alternative perspective, and hence an enhanced understanding, of the behavior of solutions to dynamic equations. As our study and the construction of our machines progressed, we found that the dynamic motion of the machine’s components and the sounds created offered observers an amazing physical connection to the “programed” mathematical equation. Through the years, our machines have been used to teach students about modeling physical systems with dynamic equations, for research studies, and to offer prospective students an alternative view of our mathematical structure.
This presentation will begin with some discussion of the history of the development of these machines and of our own project. A general overview of the primary components of the machine and the relationship between the mechanics and the mathematics being modeled will follow. The big finale will be a live streaming of a run of the large four integrator machine, complete with a discussion of the link between the physical connections between the components and the differential equation under consideration.

Integrator Machine Known as “Art”,
Mrs. Johnson, and her Calculus Students
from St. Joseph’s High School, Ironton, Ohio
(Marshall Differential Analyzer Lab)

January 18, 2024 Jeffrey Oaks, University of Indianapolis

**How to think like a medieval algebraist
**

There are many seemingly minor yet consistent differences in wording, procedure, and notation between what we read in pre-Vietan algebra and what we practice today. Together with how the earlier authors describe their work, they point to a radically different way of understanding monomials, polynomials, and equations. In this talk I describe this premodern algebra, focusing mainly on medieval Arabic algebra, though what I say applies equally to Diophantus, medieval Latin and Italian algebra, as well as the algebra of sixteenth-century Europe. Some attention will be given to delineating the role of algebra in premodern mathematics and to the concept of number upon which the concept of monomial was based.

December 7, 2023 Maryam Vulis, St. John’s University and Norwalk Community College :

**The History of Markov Chains**

This presentation will review the origins of Markov chains. Notably, Markov was inspired by poetry in his development of the chain link theory. His 1906 paper was the first to mention the idea of chains in which he stated that the that it was not necessary for “quantities” to be independent for the validity of the Law of Large Numbers. Markov’s interest in studying chain dependence stemmed from a dispute with Pavel Nekrasov from Moscow Mathematical School. Nekrasov, a deeply religious man, believed that the Weak Law of Large numbers was true for pairwise independent events only due to “free will”. Markov intended to refute Nekrasov’s statement and eventually proved that the Law of Large Numbers did not have to apply to independent events only. Why Markov chains or the chains of linked events are of interest now? The insight is that the current event depends only on the immediate previous event, not on the prior events and this process can be used in AI such as chatbots. Thus, 100 years after the Markov Chains were introduced they are used in chatbots which prominently emerged in popular culture. Markov’s work has a great impact on today’s technology.

November 16, 2023 Benjamin B. Olshin, University of the Arts, Bryn Mawr College

**Early Circular Maps: An Example of Perspectographic Imaging?**

A great deal has been written about map projections, and their development over the many centuries that human beings have engaged in map-making. From the time of Ptolemy, there have been various methods to carry out such projections, with the fundamental goal being the rendering of the globe or part of the globe onto a two-dimensional surface. Traditionally, history tells us that there were crude circular maps in the Middle Ages, and then a marked transition to Ptolemaic (mathematical) projections, followed by increasingly advanced projections, such as Mercator's. However, this lecture will present the conjecture that many of the circular maps from the late medieval and early Renaissance period were created using Albrecht Dürer's simple method of drawing a 3D object in 2D through the use of a device known as a perspectograph. This lecture will show how this method might have worked, and how it might have represented a way of creating 2D maps without formal projective geometry.

October 19, 2023 Benjamin B. Olshin, University of the Arts, Bryn Mawr College:

**Leonardo da Vinci and the Deconstruction of Perpetual Motion**

The engineering drawings of Leonardo da Vinci are famous for both their ingenuity and aesthetic qualities. But Leonardo used drawing in a rather unique way as a method of “visual thinking” or formal analysis, to investigate and work out problems in fields ranging from anatomy to mechanics to hydraulics. Interestingly, Leonardo used this same method to investigate the possibility ― or impossibility ― of perpetual motion. In many of his notebook folio pages, we find pictures and text dealing with a range of designs for perpetual motion machines powered by weights or water. One can actually find a thorough typology in his renderings of various schemes for perpetual motion machines: a classification scheme based upon the various mechanical elements, and motive forces employed, that Leonardo posited or analyzed in these devices. This presentation shows that these are not random sketches, but rather a systematic and even mathematical "deconstruction" of the myth of perpetual motion.

September 21, 2023 Lawrence D'Antonio, Ramapo College:

**Edmond Halley, Samuel Pepys, and the “Historia Piscium”**

In this talk, we will look at the remarkable life of Edmond Halley (1656 – 1742). Not just a predictor of comets, Halley produced the first star catalog of stars visible from the Southern Hemisphere. He also discovered an interesting root solving method, became the Savilian Professor of Geometry at Oxford after the death of Wallis, was a sea captain, and was named the second Astronomer Royal (succeeding John Flamsteed). Halley also played a major role in the Royal Society of London. It was a visit by Halley that encouraged Newton to write the Principia Mathematica and Halley used his own funds to publish the first edition. Halley also played a key role in the controversy surrounding the publication of Flamsteed’s star catalog (the controversy ended with a book burning!). Finally, we will discuss Halley’s election as the Clerk of the Royal Society which will help explain the title of this talk.