## L’Hospital’s Rule? … think again!

March 17, 2011 8 Comments

March 17 celebrates the anniversary of one of the most lopsided business deals in mathematical history. Before the days of companies and colleges employing mathematicians, one of the few ways for a mathematician to earn money was to be employed by a nobleman. You might earn your keep by tutoring, consulting on finances, surveying the lands or anything else involving numbers.

In the late 17th century, the great mathematician Johann Bernoulli was looking for a way to earn money while doing what he loved, mathematics. Enter the Marquis de L’Hospital. L’Hospital was a nobleman who was fascinated by mathematics, particularly calculus, which was in its infancy at the time. And he knew that Johann had worked with Gottfried Leibniz with the development of calculus.

However, unlike the great ones, L’Hospital lacked the skill necessary to understand the finer points of this new field of calculus. So, he had an idea. If he hired Bernoulli as a sort of mathematical consultant, Bernoulli would be available to help L’Hospital with any difficulties he might encounter.

So, on March 17, 1694, The Marquis de L’Hospital and Johann Bernoulli entered a financial relationship in which L’Hospital would pay Bernoulli an annual salary to be available as a mathematical consultant. Bernoulli would answer any questions L’Hospital might have and, here’s the big one, send any new mathematical discoveries directly to L’Hospital without announcing these discoveries to the world. Basically, L’Hospital believed that he should have some sort of ownership over Bernoulli’s ideas since it was he who was paying Bernoulli to research mathematics. In other words, any great breakthroughs would be credited to L’Hospital instead of Bernoulli. Sound strange? It was. But remember the times. It was almost impossible to get a paying job as a mathematician. So Bernoulli saw this as his only opportunity to earn a living as a mathematician.

What was the result of this arrangement? One of the first books in calculus … Analyse des infiniment petits … written by … wait for it … L’Hospital! Included in the book were many of Bernoulli’s ideas. However, since L’Hospital was the author, he was viewed as the mathematician who made the discoveries.

Now, L’Hospital knew full well that he was essentially ‘stealing’ the ideas of other mathematicians so he included the following statement in the book: “I have made free use of their discoveries, so that I frankly return to them whatever they please to claim as their own.” As history will show, his book became quite popular. In fact, one of the most famous and important ideas from the book became known as L’Hospital’s Rule … a great rule for evaluating limits when the limit yields the indeterminant form 0/0 … known by calculus students worldwide! Poor Bernoulli.

If you are interested in reading more about this scandal and many others, there is a great book on the subject … Mathematical Scandals by Theoni Pappas. Included on pages 16 – 21 is a more detailed account of the scandal involving L’Hospital’s Rule.