We turn our attention now to the scientific revolution. While \(8\) entered this enlightened time admired by bunny lovers and chocolate aficionados with a penchant for gathering in groups of five, it had not yet joined the ranks of the truly great numbers. But that was about to change, in large part due to its collaboration with two giants of the age: Isaac Newton and Leonhard Euler.

No one embodies the scientific revolution more than Sir Isaac Newton. Newton was born on Christmas Day in 1642, two months after his father died. He survived a premature birth and the plague, no small feat in mid 17th century England. Indeed, in 1665 Cambridge colleges closed their doors due to an outbreak of plague. Sent home, where he had no access to the internet, Newton experienced his *anni mirabilis*, 20 miraculous months of intense creativity during which he formulated the key ideas of all his major discoveries: gravitation, optics, and, central to our story today, the calculus.

In calculus one investigates change, typically of some process over smaller and smaller intervals. As such, very small, nebulous positive quantities such as \(\Delta x\) and \(\Delta y\) came to steal the spotlight from many larger numbers such as three. How did \(8\) manage in this age of infinitesimals? By standing on the shoulders of many tiny numbers, as we shall see.

Newton was an astute observer of the natural world, although he occasionally sat beneath over-ripe apples with little regard for his personal safety. For instance, in his youth he put forth a serious effort to quantify his happiness. He even developed a unit of measure for happiness: the warmfuzzy. Newton ran controlled experiments involving bread puddings from which he established a relationship between the rate at which his happiness changed and the number of bread puddings he had consumed. Although the original notes have not survived to the present day, this very paragraph points to the following result of Newton's observations:

\begin{equation*} H^\prime = 6p(2-p), \end{equation*}where \(H^\prime\) represents the rate at which his happiness level is changing (in units of warmfuzzies per bread pudding), and \(p\) is the number of bread puddings consumed on a given day.

As Newton eats his bread pudding his happiness level will rise as long as \(H^\prime\) is positive. According to the model, this will be the case as long as \(0 \lt p \lt 2\text{.}\) In fact, his happiness will be increasing fastest when \(p = 1\) (at which point it is increasing at a whopping rate of 6 warmfuzzies per bread pudding!), which no doubt makes it difficult not to have a second bread pudding. His happiness will continue to rise until \(p = 2\text{,}\) after which \(H^\prime\) is negative. If he were to keep on eating bread puddings after having two, his happiness level would begin to fall toward what he called grumpiness.

So, Newton's brain told him what his gut already knew: He maximizes the happiness gained from bread pudding consumption by eating exactly two of them. The natural question, of course, and the one that I dare say motivated the development of integral calculus, is this: How much happiness does he actually gain by consuming two bread puddings?

As calculus students know today, the net change in happiness level corresponds to the integral

\begin{equation*} \int_0^2 6p(2-p)~dp \end{equation*}which evaluates to \(8\) warmfuzzies.

So it came to pass that in the pursuit of happiness Newton found \(8\text{.}\) (As an aside, some etymologists believe the phrase “No, thanks. I just ate” has its origin in this saying of Newton's: “No more! I'm up eight!” This may explain why eight is one of the few numbers that is also a verb, phonetically.)