Overview: What is a Difference Engine?
A difference engine is a specialpurpose mechanical calculator designed to automatically generate mathematical tables – sets of values for functions like logarithms, trigonometry, and astronomy. They get their name from computing tables using the method of finite differences. How‘s this work?
The method of finite differences takes advantage of the fact that polynomials and other smooth mathematical functions, when evaluated at equally spaced arguments, yield tables in which successive differences form predictable constant patterns.
For example, let‘s say we want to create a table of values for:
f(x) = x^2 + 2x + 3
Here‘s what that table would look like:
x  f(x) 

0  3 
1  6 
2  11 
3  18 
Now if we take the difference between each successive value of f(x), we get:
x  f(x)  Difference 

0  3  – 
1  6  3 
2  11  5 
3  18  7 
Notice how each difference increases by 2 every time! The differences form a constant pattern, allowing us to easily calculate further values.
Difference engines use geardriven mechanisms to perform these finite difference calculations automatically, with the results being printed mechanically as tables – eliminating errorprone manual work. Now let‘s see how one ambitious inventor sought to make this a reality…
Babbage‘s Breakthrough Vision
In early 19th century Britain, legions of human "computers" manually calculated essential mathematical, astronomical, and navigation tables vital for industry and trade. It was incredibly tedious work, taking years and costing the British government over £25,000 annually!
English mathematician Charles Babbage knew there had to be a better way. After spending his teenage years poring over computation manuals by mathematician Isaac Newton, he became one of few people who truly understood the dogged work of human computers.
In 1821, Babbage expressed his nowfamous desire to mechanize this drudgery:
"I wish to God these calculations had been executed by steam!"
He realized that the method of finite differences, combined with mechanical automation, could help. Babbage set to work developing plans for calculating machines, drawing on established clockwork technologies.
Within a year, he unveiled a small proofofconcept model able to calculate simple polynomial tables. Impressed scientists hailed it as the first step toward mechanized computation. With government funding secured, Babbage collaborated with engineer Joseph Clement to manufacture parts for a largescale difference engine.
However, the project severely underestimated complexities involved…
Meticulous Mechanization
Babbage‘s difference engine contained over 25,000 precision components – equivalent to five fullsized train locomotives!
Gears, levers, axes and wheels powered arithmetic calculation, information storage, and automatic printing. Here‘s a highlevel overview of how it functioned:

Number storage: Decimal digits were stored mechanically using the rotational positions of interlocking brass gears called "figure wheels" arrayed vertically on axes. Each axis stored a decimal number up to 31 digits long.

Calculation: Special mechanical adders automatically performed finite difference calculations by rotating figure wheels and propagating carries. Tables progressed via repeated difference calculation between axes.

Output: Results were output as printed tables using a mechanical printer actuated by cams, levers, and presses.
This was far more sophisticated mechanics than clockwork for mills or factories. Years passed as each custom component was painstakingly manufactured by hand. Frustrated with slow progress, the British government halted funding in 1833 with no complete engine built, having spent over £17,000.
While disappointed, Babbage went on to design Difference Engine No. 2 in 18471849, improving upon the incomplete first model. Let‘s compare their capabilities:
Feature  Difference Engine  Difference Engine No. 2 

No. of digits stored  31  31 
Max order of differences  6th order  7th order 
Calculation time  45 mins per table  15 mins per table 
Output  Digitbydigit print  Entire numbers printed simultaneously 
Size  14ft x 11ft x 5ft  14ft x 11ft x 5ft 
Weight  4 tons  4 tons 
Despite increased calculation speeds, Babbage failed to ever construct a complete working engine during his lifetime. Still, his singular passion set in motion innovators across Europe toward building sophisticated mechanical computing machinery.
Pioneering Impact on Early Computing
Babbage‘s unfinished but captivating difference engine designs directly inspired Swedish inventors Georg and Edvard Schuetz to construct simplified versions from 18431853. Their engines successfully calculated tables and even won awards at the 1855 Paris Exposition Universelle. One commissioner reporting back to Emperor Napoleon III noted:
“What distinguish it from all other instruments which have been employed for similar calculations, are the perfection it realizes the exactness, certainty and rapidity of its operations."
High praise! While limited compared to modern computers, these mechanical wonders proved automatic advanced calculation feasible, showcasing innovative applications of precision engineering.
Babbage‘s ambitious dreams similarly provided technological inspiration for early computing innovators in the 1840s1930s, including inventors Martin Wiberg, Ramón Verea, and Percy Ludgate, as well as computing pioneer Alan Turing.
Modern Reawakenings
In 1985, London Science Museum curators Doron Swade and Allan Bromley began work toward actually constructing Babbage‘s Difference Engine No. 2 to original 19th century specifications – aiming to finally bring Babbage‘s unfinished legacy to life.
It took a grueling 17 years requiring engineers to rediscover longlost machine tooling and assembly techniques…but in 2002, the fiveton fully operation machine was complete! In 2004, it successfully printed tables automatically according to Babbage‘s exact 1847 programming – over 150 years after originally envisioned.
Swade later initiated building Difference Engine No. 1 as well. Unveiled in 2012, this replica faithfully captures Babbage‘s original 1820s breakthrough vision for automated computation. Better late than never!
So while he tragically never lived to see them in action, Babbage‘s ingenious engines now whir, click, and print just as intended…standing as monuments both technological and philosophical to one eccentric inventor‘s epic quest to mechanize mental labor!
Let me know if you liked this engaging little history lesson! I‘m happy to elaborate anything more in detail.