Math is Hard: Thomas’ Arithmometer

We’ve all been there…It’s late on a Sunday night before that Monday homework is due, the white paper with the little black numbers seems to be making fun of you. No matter how hard you stare at that page, the arithmetic just doesn’t make sense. Now imagine you have a job, a job that requires you to do math all day! But this vision gets a bit worse. Imagine that at this job there is no calculator and you have to do math with very big numbers! In your head! What do you do? Well, Thomas de Colmar was in the same situation in the early 1800s.

Thomas de Colmar, or Charles Xavier Thomas de Colmar (we’re going to stick with “Thomas,” as he’s known today), was in the insurance business (Garfinkel and Grunspan 52).

Between the claims, premiums, and algorithms Thomas had quite a bit of math on his hands, and like most, he probably didn’t love math. Yet in the midst of this mathematical environment, a single small error could cost huge sums of money! So, driven by a desire to do less math, perform error-free calculations, and simply succeed in the general needs of his job, Thomas de Colmar set about inventing the first commercially practical calculator (Garfinkel and Grunspan 52)!

Thomas’ invention, the Arithmometer, has a long, complex, and winding history stretching decades. If you’re interested in the most minute of details, consider reading this piece by Stephen Johnston: https://www.mhs.ox.ac.uk/staff/saj/arithmometer/, for the other 99% of my readers, our journey begins in 1820 with the first patent of the Arithmometer (Kostka, “Discovering the…”).

Although Thomas was an insurance man, he seemed to have an immense knowledge of mechanics and physics, because his invention calculated addition, subtraction, multiplication, and division via purely mechanical means. To fully understand the main piece of the Arithmometer we must quickly visit 1672, roughly 135 years prior to Thomas’ patent (Garfinkel and Grunspan 52).

Gottfried Leibniz was a German mathematician and, interestingly enough, also a philosopher. In an event that illustrates the diversification of many inventors at the time, Leibniz was inspired to invent a device for mechanical calculation. The result was the stepped reckoner, “a new type of gear that could advance a 10-digit dial exactly 0 to 9 places” (Garfinkel and Grunspan 52). However, given the advanced nature of the reckoner, Leibniz could not get it to work, in addition, given that carrying (999 rolling to 1,000) was impossible, the invention proved impractical for practical applications.

Returning to the early 1800s, Thomas de Colmar pulled Leibniz’s reckoner from the dusty shelf of forgotten inventions. The stepped reckoner was now called a Leibniz wheel, and it sits as the centerpiece for the Arithmometer (Garfinkel and Grunspan 52).

Quickly explained, the Leibniz wheel is a type of gear formed from a barrel on which are raised, bumped lines of differing lengths.

Another gear slides up and down the barrel interacting with the raised lines depending on how high or low the gear is moved. The higher the number selected, the higher the gear will travel on the Leibniz wheel, thus engaging more raised, bumped lines (MechanicalComputing, “How the…”). For a more illustrative example, please see Figure 1, or watch this video from MechanicalComputing: https://youtu.be/nyCrDI7hRpE.

While the Leibniz wheel is the centerpiece of the Arithmometer, it is by far the simplest piece contained within. It is surrounded by numerous gears and chains, levers and dials, and a special rounding mechanism that allows the machine to roll numbers, a crucial piece missing in the 1670s version (MechanicalComputing, “How the…”).

So, the internal mechanism is complex, that much is certain, but how did a user interface with the calculator? It was incredibly simple, especially compared to other methods of calculation.

Although the internal design was constantly changing, the external user end remained relatively simple and unchanged.

The mechanism was housed in a wooden box. Upon opening the box, the user was greeted by a row of dials showcasing the numbers selected by the user and the result of the final calculation. Below these rows were sliders. Each slider was a place in the number. The user would select the inputs via the sliders. For example, the number 825 would be selected by using the last three sliders and selecting “8,” “2,” and “5” (MechanicalComputing, “How the…”). Next to the sliders was a crank, one turn of the crank entered the number into the mechanism. The next input could then be selected. After inputting the second number, followed by one crank, the problem could be solved by turning the crank a set number of times, depending on the operation and complexity of the problem.

To the untrained, non-engineer’s eye, the mechanical mechanism appeared incredibly complex, and in many ways it was, but compare the Arithmometer to the Difference Engine (which, keep in mind was never completed in Babbage’s lifetime).

The Difference Engine stood eight feet tall and eleven feet long. Weighing four tons, the Difference Engine consisted of 25,000 individual parts (Rhodes, “Calculating to…”). The Arithmometer fit in a compact wooden box, could sit on a worker’s desk, and could even be conveniently carried between locations. The rapidity of technological development is truly staggering.

Thomas initially patented his Arithmometer in 1820, however, due to manufacturing limitations and even some legal battles, Thomas couldn’t effectively use and build the device to the scale he wished (Johnston, “Making the arithmometer…”). Thomas did try to convince his insurance business partners to help commercialize the device, but they took little interest in the new invention (Garfinkel and Grunspan 52).

Over the decades, Thomas sought recognition for the device. He entered it into a number of exhibitions and competitions throughout Europe, first in 1844, then in 1849 and 1851 (Garfinkel and Grunspan 52). Although often passed over for the respect he sought, Thomas didn’t give up and by 1851 he had simplified the device, perfected the mechanism, and even added capabilities to the Arithmometer (Garfinkel and Grunspan 52). Thomas’ 1851 version featured “six sliders for setting numbers and 10 dials to display results” (Garfinkel and Grunspan 52). Additionally, given the thirty years of mechanical and technical progress, mass-manufacture of the complex device was now possible (Garfinkel and Grunspan 52).

Thomas de Colmar was convinced his new calculating machine would be a hit. He formed a company and began to sell the Arithmometer.

By the time of Thomas’ death, in 1870, over one-thousand Arithmometer calculators had been sold (Garfinkel and Grunspan 52).

Given the behemoth of competing calculators and the Arithmometer’s 7” x 6” (Garfinkel and Grunspan 52) size, it became and remained popular for ninety years (Swaine and Freiberger, “Arithmometer.”).

Thomas de Colmar understood the importance of math, correct accurate math, and its role in society. Today, in the age of calculators that are always with us, combined with a general societal distaste for math, few understand that math is the language of our world. Math rules the universe, our lives, and our countries. Thomas understood this importance and saw a common problem and he addressed it, building the first practical commercialized calculator.

Thomas de Colmar has been recognized as a genius of invention and mathematics for his role in computing and the true power of the device he created, he has earned a very important space in the history of computing.

And for its place as the very first practical calculator, the Arithmometer is the twentieth major milestone in the history of computing.

The Arithmometer is a complex device, for a beautiful illustration of its operation, I’d invite you to watch the video I mentioned above. MechanicalComputing, a popular YouTube channel, produced an educational 3D model of the device, check it out here: https://youtu.be/nyCrDI7hRpE.

Works Cited

Garfinkel, Simson, and Rachel H. Grunspan. The Computer Book: from the Abacus to Artificial Intelligence, 250 Milestones in the History of Computer Science. Sterling, 2018.

Johnston, Stephen. “Making the Arithmometer Count.” History of Science Museum, 29 Oct. 2019, www.mhs.ox.ac.uk/staff/saj/arithmometer/.

Kostka, Tim. “Discovering the Arithmometer.” Cornell CIS, 2005, www.cs.cornell.edu/boom/2005/ProjectArchive/arithometer/background.html.

MechanicalComputing. “How the Arithmometer Works.” YouTube, uploaded by MechanicalComputing, 5 Apr. 2014, https://youtu.be/nyCrDI7hRpE.

Rhodes, Benjamin. “Calculating to Perfection: The Difference Engine.” Medium, Tech Is A Tool, 27 June 2020, https://www.medium.com/tech-is-a-tool/calculating-to-perfection-the-difference-engine-6afe26542ca6.

Swaine, Michael R., and Paul A. Freiberger. “Arithmometer.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 28 June 2017, www.britannica.com/technology/Arithmometer.