No expert on the history, I have been brushing up using Wikipedia, for 
better or for worse.

I was surprised to learn that L'Hospital's famous calculus textbook 
appeared in 1694, only a few years after Newton's Principia, and I hope 
you all know that Newton & Leibniz had been involved in a continuing 
dispute over priority for the development of calculus -- surely one of 
the most important of all priority disputes.

Mason Gaffney's statement that the famous Rule was Bernoulli's is surely 
true, but the statement that Bernoulli named the rule after his rich 
patron does not agree very well with the wiki

http://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital

which is detailed enough to have some credibility, and indicates that 
L'H  published the rule without giving credit to B, which made B unhappy.

Certainly this Rule uses the ghosts of departed quantities, but Berkeley 
was attacking Newton, so I took it that it was the definition of the 
derivative that was in the line of fire, not L'H's rule.  I still do not 
see any trace of ghostly "fingers" in the discussion from before 
Berkeley's 1734 book.

Most calculus teachers of my acquaintance don't like L'Hospital's rule, 
because students use it as a trick without understanding it, much less 
appreciating the subtleties of its hypotheses.  But we all are familiar 
with it, and a textbook writer is pretty much obliged to include it, to 
avoid having the book blackballed (for the omission) by its enthusiasts.

The thread below was so garbled in my email client that I have deleted 
most of it.  Hope there is no objection!

On 7/30/2014 11:04 AM, mason gaffney wrote:
> Perhaps one of our more numerate colleagues will check me on this, but
> calculus books used to give at least a footnote to "l'Hospital's Rule"
> giving a determinate value to a ratio that appears to collapse into 0/0 when
> one of its arguments falls to 0. (It was the work of one of the Bernoulli's,
> who named it for his rich patron.)  It is useful in the math of finance.
>
> Thus the problem of "ghostly fingers" was dispelled, at least in this case.
> Is this analogy acceptable math? Will it let Newton and Berkeley rest easier
> in their graves?
>
> Mason Gaffney
>    

-- 
Martin C. Tangora
tangora (at) uic.edu