John Napier was a Scottish physicist, mathematician, and astronomer who usually gets the credit for inventing logarithms. But his contributions to simplifying mathematics and building shorthand solutions didn’t end there. In the course of performing the many calculations he needed to practice these subjects in the 1500s, Napier invented a kind of computing mechanism for multiplication. It’s a physical manifestation of an old system known as lattice multiplication or gelosia.

Lattice multiplication makes use of the multiplication table in order to multiply huge numbers together quickly and easily. It is thought to have originated in India and moved west into Europe. When the lattice method reached Italy, the Italians named it gelosia after the trellised window covering it resembled, which was commonly used to keep prying eyes away from one’s possessions and wife.

Every generation thinks it has unique problems and, I suppose, sometimes it is true. My great-grandfather didn’t have to pick a cell phone plan. However, a lot of things you think are modern problems go back much further than you might think. Consider Kickstarter. Sure, there have been plenty of successful products on Kickstarter. There have also been some misleading duds. I don’t mean the stupid ones like the guy who wants to make a cake or potato salad. I mean the ones that are almost certainly vaporware like the induced dream headgear or the Bluetooth tag with no batteries.

Overpromising and underdelivering is hardly a new problem. In the 30’s The McGregor Rejuvenator promised to reverse aging with magnetism, radio waves, infrared and ultraviolet light. Presumably, this didn’t work. Sometimes products do work, but they don’t live up to their marketing hype. The Segway comes to mind. Despite the hype that it would revolutionize transportation, the scooter is now a vehicle for tourists and mall cops.

One of my favorite examples of an overhyped product comes from World War II: The Norden Bomb Sight. What makes the Norden especially interesting is that even today it has a reputation for being highly accurate. However, if you look into it, the Norden–although a marvel for its day–didn’t always live up to its press.

Horse racing has been around since the time of the ancient Greeks. Often called the sport of kings, it was an early platform for making friendly wagers. Over time, private bets among friends gave way to bookmaking, and the odds of winning skewed in favor of a new concept called the “house”.

During the late 1860s, an entrepreneur in Paris named Joseph Oller invented a new form of betting he called pari-mutuel. In this method, bettors wager among themselves instead of against the house. Bets are pooled together and the winnings divided among the bettors. Pari-mutuel betting creates more organic odds than ones given by a profit-driven bookmaker.

Oller’s method caught on quite well. It brought fairness and transparency to betting, which made it even more attractive. It takes a lot of quick calculations to show real-time bet totals and changing odds, and human adding machines presented a bottleneck. In the early 1900s, a man named George Julius would change pari-mutuel technology forever by making an automatic vote-counting machine in his garage.

In the late 1800s, a railway engineer named Philbert Maurice d’Ocagne was part of a group of men faced with the task of expanding the French rail system. Before a single rail could be laid, the intended path had to be laid out and the terrain made level. This type of engineering involves a lot of cut and fill calculations, which determine where dirt must be added or removed. The goal of earthwork is to create a gentle grade and to minimize the work needed to create embankments.

In the course of the project, d’Ocagne came up with an elegant, reusable solution to quickly solve these critical calculations. Most impressively, he did it with little more than a pen, some paper, and a straightedge. By developing and using a method which he called nomography, d’Ocagne was able to perform all the necessary calculations that made the gentle curves and slopes of the French railway possible.

If you’re completely landlocked like I am, you may dream of ocean waves lapping at the shore, but you probably don’t think much about the tides. The movement of the ocean tides is actually quite important to many groups of people, from fishermen to surfers to coastal zone engineers. The behavior of the tides over time is helpful data for those who study world climate change.

Early tide prediction was based on observed changes in relation to the phases of the Moon. These days, tide-predicting is done quickly and with digital computers. But the first purpose-built machines were slow yet accurate analog computation devices that, as they were developed, could account for increasing numbers of tidal constituents, which represent the changes in the positions of tide-generating astronomical bodies. One of these calculating marvels even saved the Allies’ invasion of Normandy—or D-Day— in World War II.

One of the keys to nuclear fission is sustaining a chain reaction. A slow chain reaction can provide clean power for a city, and a fast one can be used to create a weapon that will obliterate a city. These days, kids can learn about Uranium and Plutonium in high school. But just a few generations ago, the idea of splitting the atom was just a lofty goal for the brightest physicists and mathematicians who gathered at Los Alamos National Laboratory under the Manhattan Project.

Decoding the mysteries of nuclear fission required a great deal of experimentation and calculations. One bright physicist in particular made great strides on both fronts. That man was [Enrico Fermi], one of the fathers of the atomic bomb. Perhaps his greatest contribution to moving the research beyond the Manhattan Project was creating a handheld analog computer to do the math for him. This computational marvel is known as the FERMIAC.

What is Fission?

Nuclear fission occurs when a nucleus is split into fragments, a process that unleashes a great deal of energy. As a handful of neutrons travel through a reactor pile or other fissionable material, a couple of outcomes are possible. Any one neutron collision might result in fission. This means there will be some number of new neutrons whose paths must be tracked. If fission does not occur, the neutrons may simply scatter about upon collision, which changes their speed and trajectory. Some of the neutrons might be absorbed by the material, and others will simply escape it. All of these possibilities depend on the makeup of the material being bombarded and the speed of the neutron.

Every event that happens to a neutron comprises its genealogical history. If this history is recorded and analyzed, a statistical picture starts to emerge that provides an accurate depiction of the fissility of a given material. [Fermi]’s computer facilitated the creation of such a picture by performing mathematical grunt work of testing different materials. It identified which materials were most likely to sustain a reaction.

Before he left Italy and the looming threat of fascism, [Fermi] led a group of young scientists in Rome called the Via Panisperna boys. This group, which included future Los Alamos physicist [Emilio Segrè], ran many experiments in neutron transport. Their research proved that slow neutrons are much better candidates for fission than fast neutrons.

During these experiments, [Fermi] ran through the periodic table, determined to artificially irradiate every element until he got lucky. He never published anything regarding his methods for calculating the outcomes of neutron collisions. But when he got to Los Alamos, [Fermi] found that [Stanislaw Ulam] had also concluded that the same type of repeated random sampling was the key to building an atomic weapon.

The Monte Carlo Method: Shall We Play a Game?

[Ulam], a Polish-born mathematician who came to the US in 1935, developed his opinion about random sampling due to an illness. While recuperating from encephalitis he played game after game of solitaire. One day, he wondered at the probability of winning any one hand as laid out and how best to calculate this probability. He believed that if he ran through enough games and kept track of the wins, the data would form a suitable and representative sample for modeling his chances of winning. Almost immediately, [Ulam] began to mentally apply this method to problems in physics, and proposed his ideas (PDF) to physicist and fellow mathematician [John von Neumann].

This top-secret method needed a code name. Another Los Alamos player, [Nick Metropolis] suggested ‘Monte Carlo’ in a nod to games of chance. He knew that [Ulam] had an uncle with a propensity for gambling who would often borrow money from relatives, saying that he just had to go to Monte Carlo. The game was on.

The Tricky Math of Fission

Determination of the elements most suitable for fission required a lot of calculations. Fission itself had already been achieved before the start of the Manhattan Project. But the goal at Los Alamos was a controlled, high-energy type of fission suitable for weaponization. The math of fission is complicated largely because of the sheer number of neutrons that must be tracked in order to determine the likelihood and speed of a chain reaction. There are so many variables involved that the task is monumental for a human mathematician.

After [Ulam] and [von Neumann] had verified the legitimacy of the Monte Carlo method with regard to the creation of nuclear weaponry, they decided that these types of calculations would be a great job for ENIAC — a very early general purpose computer. This was a more intensive task than the one it was made to do: compute artillery firing tables all day and night. One problem was that the huge, lumbering machine was scheduled to be moved from Philadelphia to the Ballistics Research Lab in Maryland, which meant a long period of downtime.

While the boys at Los Alamos waited for ENIAC to be operational again, [Enrico Fermi] developed the idea forego ENIAC and create a small device that could run Monte Carlo simulations instead. He enlisted his colleague [Percy King] to build the machine. Their creation was built from joint Army-Navy cast off components, and in a nod to that great computer he dubbed it FERMIAC.

FERMIAC: Hacking Probabilities

FERMIAC was created to alleviate the necessity of tedious calculations required by the study of neutron transport. This is something of an end-run around brute force. It’s made mostly of brass and resembles a trolley car. In order to use it, several adjustable drums are set using pseudorandom numbers. One of these numbers represents the material being traversed. A random choice is made between fast and slow neutrons. A second digit is chosen to represent the direction of neutron travel, and a third number indicates the distance traveled to the next collision.

Once these settings are dialed in, the device is physically driven across a 2-D scale drawing of the nuclear reactor or materials being tested. As it goes along, it plots the paths of neutrons through various materials by marking a line on the drawing. Whenever a material boundary is crossed, the appropriate drum is adjusted to represent a new pseudorandom digit.

FERMIAC was only used for about two years before it was completely supplanted by ENIAC. But it was an excellent stopgap that allowed the Manhattan Project to not only continue unabated, but with rapid progress. FERMIAC is currently on display at the Bradbury Science Museum in Los Alamos, New Mexico alongside replicas of Fat Man and Little Boy, the weapons it helped bring to fruition. [Fermi]’s legacy is cemented as one of the fathers of the atomic bomb. But creating FERMIAC cements his legacy as a hacker, too.

After Los Alamos, [Stanislaw Ulam] would continue to make history in the field of nuclear physics. [Enrico Fermi] was opposed to participating in the creation of the exponentially more powerful hydrogen bomb, but [Ulam] accepted the challenge. He proved that Manhattan Project leader [Edward Teller]’s original design was unfeasible. The two men worked together and by 1951 had designed the Teller-Ulam method. This design became the basis for modern thermonuclear weaponry.

Today, the Monte Carlo method is used across many fields to describe systems through randomness and statistics. Many applications for this type of statistical modeling present themselves in fields where probabilities are concerned, like finance, risk assessment, and modeling the universe. Wherever the calculation of all possibilities isn’t feasible, the Monte Carlo method can usually be found.

UPDATE: Commentor [lwatchdr] pointed out that the use of the FERMIAC began after the Manhattan Project had officially ended in 1946. Although many of the same people were involved, this analog computer wasn’t put into use until about a year later.

When I was young the first “computer” I ever owned was an analog computer built from a kit. It had a sloped plastic case which had three knobs with large numerical scales around them and a small center-null meter. To operate it I would dial in two numbers as indicated by the scales and then adjust the “answer” by rotating the third dial until the little meter centered. Underneath there was a small handful of components wired on a terminal strip including two or three transistors.

In thinking back about that relic from the early 1970’s there was a moment when I assumed they may have been using the transistors as logarithmic amplifiers meaning that it was able to multiply electronically. After a few minutes of thought I came to the conclusion that it was probably much simpler and was most likely a Wheatstone Bridge. That doesn’t mean it couldn’t multiply, it was probably the printed scales that were logarithmic, much like a slide rule.

Science Fair Analog Computer

Old meets new: Analog and digital computation

Did someone just ask what a slide rule was? Let me explain further for anyone under 50. If you watch the video footage or movies about the Apollo Space Program you won’t see any anyone carrying a hand calculator, they didn’t exist yet. Yet the navigation guys in the first row of Mission Control known aptly as “the trench”, could quickly calculate a position or vector to within a couple of decimal places, and they did it using sliding piece of bamboo or aluminum with numbers printed on them.