Mass points were first used by Augustus Ferdinand Möbius in 1827. They didn't catch on right away. Cauchy was quite critical of his methods and even Gauss in 1843 confessed that he found the new ideas of Möbius difficult. This is found in little mathematical note by Dan Pedoe in Mathematics Magazine [ 1]. I first encountered the idea about 25 years ago in a math workshop session entitled ``Teeter-totter Geometry" given by Brother Raphael from Saint Mary's College. He apparently always taught one course using only original sources, and that year he was reading Archimedes with his students. It was Archimedes' ``principle of the lever" that he used that day to show how mass points could be used to make deductions about triangles. For a very readable account of the assumptions Archimedes makes about balancing masses and locating the center of gravity, I recommend the new book Archimedes: What Did He Do Besides Cry Eureka?[ 2] written by Sherman Stein of U.C.Davis.

My next encounter with mass points was in the form of an offer about twenty years ago from Bill Medigovich, who was then teaching at Redwood High School and helping Lyle Fisher coordinate the annual Brother Brousseau Problem Solving and Mathematics Competition. He offered to come to a math club and present the topic of Mass Points, if the students would commit to several sessions. I was never able to get my students organized enough, so we missed out on his wonderful presentation. Many years later I asked him for any references he had on the subject and he sent me a packet of the 30 papers [ 3] he used for his presentation. I also found the topic discussed in the appendix of The New York City Contest Problem Book 1975-1984 [ 4] with a further reference to an article The Center of Mass and Affine Geometry [ 5] written by Melvin Hausner in 1962. Recently, Dover Publications reissued a book published by Hausner [ 6] in 1965 that comprised a one year course for high school teachers of mathematics at New York University. The first chapter is devoted to Center of Mass, which forms the basis for the entire book. In the preface he credits Professor Jacob T. Schwartz, an eminent mathematician at the Courant Institute of Mathematical Sciences, ``who outlined the entire course in five minutes". While we are bringing out big names let me mention Jean Dieudonné, the world famous French mathematician, who went on record saying ``Away with the triangle". He wrote a textbook in the 60's for high schools in France which introduces the geometry in the plane and Euclidean space via linear algebra. The axioms are the axioms in the definition of a vector space over a field and no diagrams are given in the book. I looked at the book fifteen years ago and found it very interesting, but I cannot imagine it being used in public schools in the United States. The reason for bringing up Dieudonné at this point is another of his inflammatory comments, "Who ever uses barycentric coordinates?", and the response by Dan Pedoe is to by found in an article by Pedoe entitled Thinking Geometrically [ 7].

As I was preparing for this talk, I was going through old issues of Crux Mathematicorum and found a key paper on the this subject, Mass Points [ 8], that was originally written for the NYC Senior "A" Mathletes. The authors are Harry Sitomer and Steven R. Conrad. The latter author may be familiar to you as the creator of the problems for the past 25 years used in the California Mathematics League as well as all the other affiliated math leagues around the country. I will be using this paper and most of their examples as my main guide for this talk.