E T Bell, in "Men of Mathematics" tells a story
about the Irish mathematician William Rowan Hamilton
1803-1865, and his discovery of the algebra of
quaternions (later called vectors), in particular 
of the statement:
         A x B = - B x A 
where the commutative law of multiplication does 
not hold.

"Hamilton himself was so impressed by the magnitude 
of what suddenly dawned on his mind (after fifteen 
years of fruitless thought) one day (16 October 1843) 
when he was out walking with his wife that he carved 
the fundamental formulae of the new algebra in the 
stone of the bridge on which he found himself at 
the moment."

Is this true? I have been told it was Brueen Bridge
over the Grand Canal, Dublin but have never been
able to find this bridge or anyone who can confirm 
the story.

Hello doxeyman:

Thanks for the fascinating question. 

This account is apparently true, if we can believe Hamilton's own
letters describing the discovery. Unfortunately, the engraving itself
has long since faded away with the ravages of time.

The bridge in question is actually "Brougham Bridge" (or Broome
Bridge), not Brueen Bridge. The following sources describe the
incident in more detail:

Letters describing the Discovery of Quaternions - Letter from Sir W.
R. Hamilton to Rev. Archibald H. Hamilton.
URL: http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html
Quotes:
"Nor could I resist the impulse - unphilosophical as it may have been
- to cut with a knife on a stone of Brougham Bridge, as we passed it,
the fundamental formula with the symbols, i, j, k; namely,
i2 = j2 = k2 = ijk = -1 
which contains the Solution of the Problem, but of course, as an
inscription, has long since mouldered away."
"They started into life, or light, full grown, on [Monday] the 16th of
October, 1843, as I was walking with Lady Hamilton to Dublin, and came
up to Brougham Bridge, which my boys have since called the Quaternion
Bridge."

William Rowan Hamilton
URL: http://www.wikipedia.org/wiki/William_Rowan_Hamilton
Quote: "According to a story he told, he was out walking one day with
his wife when the solution in the form of the equation i2 = j2 = k2 =
ijk = -1 suddenly occurred to him; he then promptly carved this
equation into the side of the nearby Brougham bridge."

A drawing of the event can be found at 

http://www-gap.dcs.st-and.ac.uk/~history/Diagrams/Hamilton9.jpeg

There is a plaque marking the spot. A picture of that can be found at

http://www-gap.dcs.st-and.ac.uk/~history/Diagrams/Quaternion.jpeg


I hope this information helps with your research.              
              
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Search Strategy (on Google):             
   
quaternions hamilton bridge

Thanks for such a quick and full answer. It gave me such a thrill to
find it was indeed true, read Hamilton's description of his discovery,
and see pictures of the bridge and the plaque. In 1990 I bicycled
1,600 miles around Ireland, and once back to Dublin spent a whole day
cycling along the Grand Canal trying to find the Bridge concerned
without luck. Pity Google Answers hadn't been thought of 13 years ago!

Anti-commutative law actually holds: <a
href="http://www.hypercomplex.us/Hypercomplex/comparison.htm">Comparison
of Hypercomplex Number Systems</a>