Zero - Charles Seife [47]
A Frenchman, not an Englishman, would be remembered for taking the first nibble at the mysterious zeros and infinities that suffused calculus; mathematicians learn of l’Hôpital’s rule when they first learn about calculus. Oddly enough, it was not l’Hôpital who came up with the rule that bears his name.
Born in 1661, Guillaume-François-Antoine de l’Hôpital was a marquis—and was thus very wealthy. He had an early interest in mathematics, and though he spent some time in the army, becoming a cavalry captain, he soon turned back to his true love of math.
L’Hôpital bought himself the best teacher that money could buy: Johann Bernoulli, a Swiss mathematician and one of the early masters of Leibniz’s calculus of infinitesimals. In 1692, Bernoulli taught l’Hôpital calculus. L’Hôpital was so enthralled by the new mathematics that he persuaded Bernoulli to send him all Bernoulli’s new mathematical discoveries for the marquis to use as he desired, in return for cash. The result was a textbook. In 1696, l’Hôpital’s Analyse des infiniment petits became the first textbook on calculus and introduced much of Europe to the Leibnizian version. Not only did l’Hôpital explain the fundaments of calculus in his textbook, he also included some exciting new results. The most famous is known as l’Hôpital’s rule.
L’Hôpital’s rule took the first crack at the troubling 0/0 expressions that were popping up throughout calculus. The rule provided a way to figure out the true value of a mathematical function that goes to 0/0 at a point. L’Hôpital’s rule states that the value of the fraction was equal to the derivative of the top expression divided by the derivative of the bottom expression. For instance, consider the expression x/(sin x) when x = 0; x = 0, as does sin x, so the expression is equal to 0/0. Using L’Hopital’s rule, we see that the expression goes to 1/(cos x), as 1 is the derivative of x and cos x is the derivative of sin x. Cos x = 1 when x = 0, so the whole expression equals 1/1 = 1. Clever manipulations could also bring l’Hopital’s rule to resolve other odd expressions: ?/?, 00, 0?, and ?0.
All of these expressions, but especially 0/0, could take on any value you desire them to have, depending on the functions you put in the numerator and denominator. This is why 0/0 is dubbed indeterminate. It was no longer a complete mystery; mathematicians could extract some information about 0/0 if they approached it very carefully. Zero was no longer an enemy to be avoided; it was an enigma to be studied.
Soon after l’Hôpital’s death in 1704, Bernoulli started implying that l’Hôpital had stolen his work. At the time the mathematical community rejected Bernoulli’s claims; not only had l’Hôpital proved himself an able mathematician, but Johann Bernoulli had a tarnished reputation. He had previously tried to claim credit for another mathematician’s proof. (The other mathematician happened to be his brother, Jakob.) In this case, though, Johann Bernoulli’s claim was justified. His correspondence with l’Hôpital backs his story. Alas for Bernoulli, the name for l’Hôpital’s rule stuck.
L’Hôpital’s rule was extremely important for resolving some of the difficulties with 0/0, but the underlying problem remained. Newton’s and Leibniz’s calculus depends upon dividing by zero—and on numbers that miraculously disappear when you square them. L’Hôpital’s rule examines 0/0 with tools that were built upon 0/0 to begin with. It is a circular argument. And as physicists and mathematicians all over the world were beginning to use calculus to explain nature, cries of protest emanated from the church.
In 1734, seven years after Newton’s death, an Irish bishop, George Berkeley, wrote a book entitled The Analyst, Or a Discourse Addressed to an Infidel Mathematician. (The mathematician in question was most likely Edmund Halley, always a supporter of Newton.) In The Analyst, Berkeley pounced on Newton’s (and Leibniz’s) dirty tricks with zeros.
Calling infinitesimals