Three 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 three 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 April 13 I present
his analysis as disclosed in a private letter to Rudolph Von
Bodenhausen made known to me by Sigmund Probst of the Leibniz Archive
at the Göttingen Science Academy.
In reverse-chronologic order:
2017 (April 13)
The Leibniz Catenary, an invited
talk for the
Dartmouth Mathematics Colloquium. In a private letter,
Leibniz explained the analysis he used to arrive at his geometric
construction of the catenary. 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 but it doesn't really matter!
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) Talk for RIPS Program at IPAM:
How did Leibniz Solve the
Catenary Problem? A Mystery Story.
Turned out not to be a mystery after all! As I learned before the
talk of January 4, Leibniz explained his
solution in a private letter to Rudolph Von Bodenhausen of 1691 noted in the talk of Jan
4, 2017 and explained fully in the talk of March 13.
This talk sets forth an independent solution that
demonstrates directly how the hyperbolic functions can be easily discovered at the
heart of the catenary problem.
Previous Invited Talks
2014 A straightforward
technique using integration to derive 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
2011 I participated in review panels for the NSF and for
the S. -T. Yau High School Mathematics Awards.
2007–present, Instructor at the
LACES Calculus Camp (four days in April), the creation of Robert
Vriesman, Chairman of the Math Department at the LACES magnet school in
Los Angeles. See the
2012 Calculus Camp video by LACES student Blake Simon.
Photos and Slide
(Last modified May 16, 2017)