# math

# We know the lion by his claw

Nine years after Isaac Newton had published his *Philosophiæ Naturalis Principia Mathematica*, Johann Bernoulli, a friend of Gottfried Wilhelm von Leibniz, challenged the mathematicians of the world to solve two problems. The two problems required an extensive knowledge of calculus at the time. Some mathematicians and scholars have suspected that the contest was a test by Bernoulli and Leibniz to see how much Newton truly knew and to resolve the ongoing dispute about who had invented calculus.

Bernoulli gave the mathematicians six months to solve the problems. Bernoulli received no response for six months until he had received a letter from Leibniz. Leibniz had asked for the contest to be extended by a whole year to Christmas Day, 1698.

Meanwhile in England, Isaac Newton, the Warden of the Royal Mint, finally received Bernoulli’s challenge in January, 1697. According to his nephew, Newton immediately started working on the two questions and solved both in 12 hours. Newton then had Charles Montagu, the then President of the Royal Society, publish his solutions anonymously, possibly because he had suspected that it was all a ploy by Leibniz’s defenders.

When Bernoulli announced the winners of his contest, he named Leibniz, the Marquis Guillaume de l’Hopital, and the one anonymous winner. Bernoulli recognized the anonymous winner in public with the phrase, “*tanquam ex ungue leonem*,” Latin for “we know the lion by his claw.”

Sources:

- Richard Feynman,
*Feynman Lectures on Physics*, vol. 2, ch. 19, - L. T. Moore,
*Isaac Newton: A Biography*, Dover, NY, 1934.

# Graham’s number

“Graham’s number” is perhaps the largest known finite number. Graham’s number is so vast, there aren’t enough atoms in the entire *observable* universe to represent it in numbers. However, Graham’s number can be described via shorthand techniques, like Knuth’s up-arrow notation: