|Virgil.GRiffith:: WebLog #1 Topic : 2008-02-26 12.53.27 virgil : Philosophical implications of Malfatti's problem||[Changes] [Calendar] [Search] [Index]|
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|How to arrange three non-overlapping circles of greatest total area inside of a triangle?|
The mathematical details of this problem are not relevant for making the philosophical point, but if you want the messy details, read the background information as well as the "Mathematical Notes" at the bottom.
In essence, here is the timeline.
The important thing here is that between 1803 and 1929 for all intents and purposes it was a mathematical fact that Malfatti's solution was correct. His proof was not part of some obscure branch of mathematics that few mathematicians understood, this is a Euclidean geometry problem! Yet it took 126 years before anyone noticed the problem. This wasn't an small oversight mistake either. Malfatti's solution was not simply wrong some of the time, but it turned out that Malfatti's solution was wrong *every* time -- there is absolutely no triangle in which Malfatti's accepted solution is correct. Yet to merely notice the holes in this proof of Euclidean geometry took over a century.
I received this wonderful response from Michael Fox. He could very well be right. It seems odd that people simply would've forgotten about Malfatti's original problem, but it's certainly possible. He writes:
I saw your comments on Malfatti and his 1803 solution to his circle problem. I thinnk the history of this problem is rather more complicated. Let's call the problem of putting three circles in a given triangle so that each touches the other two and two sides of the triangle 'the circle problem', and the problem of fitting in three non-overlapping circles so that they have the greatest possible area 'the marble problem'. (Malfatti wanted to determine the three circular cylinders of greatest volume that could be cut from a triangular prism of marble. This is equivalent to finding the circles of greatest area ... .)
Malfatti solved the circle problem correctly, but his assumption, as you say, was wrong. But independently, a French mathematician, Gergonne, had hit on the circle problem in around 1800, and took about 10 years finding a solution. Then, in 1810/11, he published the problem in the Annales de Gergonne, and a little later published his solution. He had also had a letter from a Professor Bidone of Turin telling him about Malfatti's solution, but with no mention of the marble problem. Other mathematicians, such as Steiner, heard of the problem through Gergonne's journal, and almost certainly were not aware of Malfatti's original marble problem. In the various mentions of Malfatti's problem that I have seen in 19th and early 20th century books, none link it to any other problem. My belief is that the marble problem was simply forgotten until more research was done early last century.
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