## Mike Raugh

Mathematician

Email: auranteacus at gmail com

Brief Professional Bio

## Connections between Calculus and the Logarithms of John Napier and Henry Briggs

NEW August 25, 2023: I gave an invited presentation to the 11th International Leibniz Congress in Hannover, Germany, July 31 - August 4, 2023. The slides, accompanying texts, and movie (mp4) of the presentation are now freely available online at, https://doi.org/10.5281/zenodo.8093063.

The presentation and text are based on my earlier study, John Napier and Henry Briggs at the Threshold of Analysis (2022), https://doi.org/10.5281/zenodo.6402702.

The point of the work is that Napier and Briggs foreshadowed calculus using merely arithmetic. Napier's method itself explains why the integral ∫ dx/x (with lower bound 1) gives the natural logarithm, and his tables afford a means of evaluating the integral. It may be the most elementary introduction to that integral and to logarithms in general.

Another point of interest is that a method used by Briggs implies a simple recursion formula that can be used to compute natural logarithms using Bernoulli numbers. The algorithm demonstrates how closely the Bernoulli numbers are affiliated with the natural logarithm.

The logarithms of Napier and Briggs, created for practical applications in astronomy and navigation, turned out to be major milestones in the development of mathematics.

## Who Discovered e?

Raugh, M., Probst, S., ``The Leibniz catenary and approximation of e an analysis of his unpublished calculations.'' Hist. Math. (2019), https://doi.org/10.1016/j.hm.2019.06.001. We discuss the number 2.7182818 mentioned cryptically by Leibniz in the private letter noted below without explanation of its derivation. As far as we can tell, Leibniz was the first (by decades) to compute e accurately, identify it as the basis for the natural logarithm and exponential function and use the value to graph a catenary (now better known as a hyperbolic cosine).

Siegmund Probst gave the introductory talk Some early implicit and actual appearances of the number commonly known as e, at The First International Virtual Symposium Honoring the "Eulerian" Number e = 2.71828...., February 7, 18:28 Amsterdam time.

My talk following Probst is linked below at 2023 under Invited Talks.

Several talks resulted from an effort to find out who first solved the catenary problem: find the curve of a freely hanging chain. Leibniz and Johann Bernoulli independently published the first solutions in Acta Eruditorum 1691. Leibniz presented his solution as a classical Euclidean construction without an explanation of how he discovered it. The following talks about his construction represent the evolution in my attempts to understand how Leibniz could have arrived at his solution. In the culminating talk of June 30, 2017, I presented his analysis as disclosed in a private letter to Rudolph Von Bodenhausen made known to me by Sigmund Probst of the Leibniz Archive Hanover (Göttingen Academy of Sience). This is a showcase example from the time when geometry was being eclipsed by analysis as a standard for defining mathematical objects.

In reverse-chronologic order:

2017 (June 30) This talk, given for RIPS17 at IPAM, follows one year after the first talk for RIPS16: Leibniz used Calculus to solve the Catenary Problem, but he presented it as a Euclidean Construction without Explanation. In a private letter, Leibniz explained his analysis. He wrote: ``Let those who don't know the new analysis try their luck!'' This talk presents his elegant construction and analysis. Paradoxically, the construction isn't possible as strictly Euclidean, but it doesn't really matter! I present the analysis in Leibniz's own idiom of differential calculus. A YouTube video was prepared by IPAM's expert videographer, Kayleigh Steele.

This was a slight modification of an invited talk at Dartmouth for the Dartmouth Mathematics Colloquium, April 13.

2017 (January 4) The Leibniz Catenary Construction: Geometry vs Analysis in the 17th Century, an invited talk for the Special Session on the History of Mathematics at JMM 17 in Atlanta. This talk positions the publication of Leibniz's construction at the time when mathematicians were turning away from Descarte's dictate to present curves as geometric constructs toward analytic presentations. Leibniz played it both ways: he published a construction that could only have been derived using calculus but did not disclose the derivation publicly.

2016 (July 6) An invited talk for RIPS16 at IPAM: How did Leibniz Solve the Catenary Problem? A Mystery Story.

Turned out not to be a mystery at all! Leibniz explained his solution in a private letter to Rudolph Von Bodenhausen of 1691. In ignorance of Bodenhausen, this talk demonstrates a discovery path for the hyperbolic functions at the heart of the catenary problem. The talk at IPAM (below at June 30, 2017) explains Leibniz's analysis in detail. (For a discussion of the Bodenhausen letter, see supplementary notes.)

## Invited Talks

2023 Leibniz's unpublished calculation of e: The First International Virtual Symposium Honoring the "Eulerian" Number e = 2.71828...., February 7, 18:28 Amsterdam time.
2017 (June 30) Video of the presentation at IPAM, "Leibniz used Calculus to solve the Catenary Problem, but he presented it as a Euclidean Construction," YouTube.
2014 A simple integration technique for deriving the Bernoulli Summation Formula, for the 29th LACC High School Math Contest, March 22.
2013 The Real Numbers are Not Real:, The Innumerable Infinities of Georg Cantor, March 16 at Los Angeles City College Math Contest.
Also presented to UNM Math & Stats Club on 3/8/2013. See the related Problematic Four Bugs ProblemOr Reality vs the Continuum.
2012 For New Mexico Math Contest of Feb 4: Archimedes' Law of the Lever and How He Used it to Deduce the Volume of the Sphere: Poster, Talk
(Repeated at Agilent Technologies Inc in Santa Clara at request of Geront Owen, 8/12/2013)
2012 What was on Top of Archimedes' Tomb?, Mar 2 at 27th Los Angeles City College Math Contest (abreviated version of the New Mexico Math Contest talk)
2010 The Innkeeper's Problem and Tale
2009 Irrationality of pi, with companion notes on Transcendentality of e
2008 How do you know what time it is?
2007 Hey, who really discovered that theorem!
2006 Eigenvalues and Eigenvectors, a chalk talk, first talk in a series for the Los Angeles City College High School Math Contest

## 2001–2015 Institute for Pure and Applied Mathematics (IPAM) at UCLA –Program director for Research in Industrial Projects for Students (RIPS)

On August 21, 2015 I concluded my fifteenth and final summer as director of the RIPS program at IPAM, a National Science Foundation's institute located at UCLA.

I worked with IPAM staff and the late Robert Borrelli of Harvey Mudd College to create the RIPS program in 2001, and then to continue developing the program over the fifteen summers of my directorship. I enjoyed working with the many students and academic mentors who participated in RIPS over all those years.

My approach to managing the RIPS program, was presented to the panel Starting and maintaining a student industrial research progam in the mathematical sciences at the MAA's MathFest of Aug 4, 2007 in San Jose, CA.

RIPS continued in the summer of 2016 under the directorship of the talented Spanish mathematician and teacher Prof. Susana Serna of the Autonomous University of Barcelona. Prof. Serna had been an academic mentor for RIPS teams for the previous eight summers.

## Some write-ups of mathematical topics

A miscellany to include lecture notes, drafts and reminiscence

## Some activites in mathematics

2007–2013, Instructor at the LACES Calculus Camp (four days in April) described by its creator, Robert Vriesman, who before retirement in 2014 chaired the Math Department at the LACES magnet school in Los Angeles. See the 2012 Calculus Camp video by LACES student Blake Simon.

2011 Participated in review panel for the NSF and for the S. -T. Yau High School Mathematics Awards.