TITLE: Problem of Mary Somerville and Hypatia
QUESTION [6 upvotes]: I have a friend who's an amateur mathematician.  He's run across the story of Mary Somerville, (died 1892) who was one of the first female mathematicians.  According to the online articles, she solved a Diophantine problem (I think posed by Hypatia in the 5 century) and won a prize and some recognition.  This was enough recognition that the establishment would have called her a "man of science", but she wasn't a man.  So they coined the term "scientist" for her.  She's the first "scientist."
He, and now I, would really like to know what this Diophantine problem was.  Article after article mentions her success, but no one states to the actual problem.  So that's my question:  What was the problem Mary Somerville solved?

REPLY [7 votes]: In Mary Somerville's own words, as cited on page 79 of the book Personal Recollections, from Early Life to Old Age, of Mary Somerville, published by her daughter:

I became acquainted with Mr. Wallace, 
  who was, if I am not mistaken, mathematical 
  teacher of the Military College at Marlow, and 
  editor of a mathematical journal published there.
  $\cdots$
  I had solved some of the problems contained in 
  it and sent them to him, which led to a correspondence, as Mr. Wallace sent me his own solutions 
  in return. Mine were sometimes right and sometimes wrong, and it occasionally happened that we 
  solved the same problem by different methods. At 
  last I succeeded in solving a prize problem! It was 
  a diophantine problem, and I was awarded a silver 
  medal cast on purpose with my name $\;\ldots$

As a contest problem published in a $19^{th}$ century math journal, it is unlikely that the diophantine equation in question had any filiation going all the way back to Hypatia. More likely, someone along the way may have  got the names conflated while reading any of several articles/books where both Hypatia and Mary Somerville figure as women pioneers in science, for example Hypatia's Heritage.

[ EDIT ]   Credit goes to Robert Israel for the posted reference, which traces to the New series of The mathematical repository, which points in the end to the following solution "by a Lady" ;-)