an analysis of his unpublished calculations.''
Hist. Math. (2019),
We discuss the number 2.7182818
mentioned cryptically by Leibniz in the private letter noted below without explanation of its
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).
More about the Catenary
Several talks resulted from an effort to find out
who first solved the catenary problem: find the curve of a freely
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
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
A YouTube video was prepared by IPAM's expert videographer, Kayleigh Steele.
This was a slight modification of an invited talk at Dartmouth
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 after all!
I learned after this talk that Leibniz explained his
solution in a private letter to Rudolph Von Bodenhausen of 1691.
In ignorance of Bodenhausen, this talk features an independent solution that
demonstrates directly how the hyperbolic functions can be easily discovered at the
heart of the catenary problem.
In the most recent talk at IPAM (June 30), I explained Leibniz's analysis in detail.
(For a discussion of the Bodenhausen letter, see
Previous Invited Talks
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 Problem—Or Reality vs the Continuum.
New Mexico Math Contest of
Archimedes' Law of the Lever and How He Used it to Deduce
the Volume of the Sphere:
(Repeated at Agilent Technologies Inc in Santa Clara at request of
Geront Owen, 8/12/2013)
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)
Innkeeper's Problem and Tale
of pi, with companion notes on Transcendentality
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
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.
–Program director for Research in Industrial Projects for Students (RIPS)
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
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,
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.
Photos and Slide
(Last modified July 10, 2019)