Pascal and Fermat

Fermat and Pascal exchanged correspondence discussing the problem of points.

Here is a loose sketch of the problem:

Two players toss a fair coin. Player A gets a point, if it comes up heads, while player B gets a point, if it comes up tails.

They repeat this, until one of the player gets to 10 points.

At the start of the game, each player wagers $50 for a total pot of $100.

Suppose the game is interrupted at a certain point due to unavoidable reasons (say player A has 8 points and player B has 7 points).

How should the pot be divided?

At the start of the game, the odds are even, since the coin is fair. However, when the game is interrupted, player A has a higher chance of winning. How can we systematically take this condition into account?

A wonderful exploration of this problem written using "modern" terminology is available here. It outlines the problem, sketches Fermat's and Pascal's approaches, and generalizes the problem and solution.

The original correspondence (translated) is available here (pdf).

Diffusion: A Historical Perspective

The paper (pdf) "One and a half century of diffusion: Fick, Einstein, Before and Beyond" by Jean Philibert traces the history of diffusion phenomena.

It starts with Thomas Graham (of dialysis fame) who perhaps made the first systematic observations, which were integrated into phenomenological law by German physiologist Adolf Fick in 1855, at the age of 26.

Fick observed the analogy between mass diffusion and heat conduction (now considered obvious), and piggy-backed on Fourier's law of conduction (1822). The paper cites the opening lines of Fick's work:

A few years ago, Graham published an extensive investigation on the diffusion of salts in water, in which he more especially compared the diffusibility of different salts. It appears to me a matter of regret, however, that in such an exceedingly valuable and extensive investigation, the development of a fundamental law, for the operation of diffusion in a single element of space, was neglected, and I have therefore endeavoured to supply this omission.

Next, the paper talks about the contributions of  W. C. Roberts-Austen (an assistant to Thomas Graham, and successor as Master of the Mint) to quantification of diffusion in solids.

In 1905, Einstein integrated Robert Brown's observations of random zig-zag trajectories and Fick's phenomenological laws, with the crucial observation that it was the mean-squared displacement, and not the mean displacement that was related to diffusion.

Following Einstein's paper, the experimental work of Perrin was responsible helping the world accept the link between the microscopic (MSD is proportional to diffusivity and time) and macroscopic worlds (flux is proportional to concentration gradient).

It is always interesting to look at the chronological development of (now familiar) ideas. These uncontroversial ideas were once strongly wrestled with. It took centuries for scientists to come up with a comprehensive understanding, and to develop interesting applications based off of it.

A Brief History of Runge-Kutta Methods

Butcher's "A History of Runge-Kutta Methods" contains a beautiful account of the early days of this now ubiquitous method for solving ordinary differential equations.

Carl Runge, a German mathematician and spectroscopist, toyed with the idea of improving upon Euler's simple time stepping approximation for solving ODEs. In a 1895 paper, he wrote three down three different "Runge-Kutta" schemes to solve a particular differential equation that arose in the study of atomic spectra.

He used so much mathematics in his research that physicists thought he was a mathematician, and he did so much physics that mathematicians thought he was a physicist.

Two other German mathematicians, Karl Heun and Martin Kutta made additional contributions. In 1900, Heun wrote down RK methods up to order 4. In 1901, Kutta extended it to order 5, and systematized most of the prior work. His contribution included the classical 4th order RK method, which is the ODE analog of Simpson's method for numerical integration.

Numerical Recipes has this to say about this century old algorithm:

For many scientific users, fourth-order Runge-Kutta is not just the first word on ODE integrators, but the last word as well. In fact, you can get pretty far on this old workhorse, especially if you combine it with an adaptive stepsize algorithm.

By 1925, Nystrom had extended and cleaned up most of the RK methods developed previously.

Over the years, higher order methods were developed, stability properties were studied, and implicit RK methods capable of solving stiff problems were developed.

On History Education

I am a big fan of Dan Carlin's "Hardcore History" podcast, which is (partially) available on iTunes. I recently re-listened to the "Wrath of the Khans", which provides an amazing overview of the rise and fall of the Mongol empire.

He wrote a short-piece for edutopia on how he would restructure history curriculum at the K-12 level. Here's is the key message:

Were I anointed History Czar, I would ditch the curricula entirely. These things are holdovers from another era of history instruction. My goal would be to get kids to love the study of the past by connecting to their affinities. Into music? It's got a history. Motorcycles? Fashion? Entertainment? Sports? Getting them to explore the history of a subject they already love is a great way to teach historical knowledge and how the current reality came to be. In the 21st century, this is the greatest practical value the study of the past provides.

As somebody whose interest in history increased rapidly, after I stopped learning it formally, I do see considerable value in this proposition.

Here is a reddit AMA that he participated in.

NB: It should be pointed out that Dan Carlin's standing among professional historians is mixed.