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Suppose that a mysterious philanthropist provided fundings for Charles Babbage's intricate projects and the Analytical Engine was finished by 1850.
Surely there would be a lot of raptured academicians and some fuss in newspapers, but what are some problems from 19th century the Engine could solve and thus provide substantual economic boost?
"What are some problems from 19th century the Engine could solve and thus provide substantual economic boost?"
For those readers who don't know what we are talking about, Babbage's envisioned Analytical Engine would have had about the same capabilities as a typical ordinary programmable calculator, such as the famous HP-42S (of which a free open source re-implementation is available as Free42). Of course the HP-42S was very very very much faster, but speed is not of the essence. But what is of the essence is that the HP-42 was also very very very much cheaper than the Analytical Engine, and it was mass produced.
There are literally thousands of programs available for the HP-42, and any of those could, in principle, be implemented on a functional Analytical Engine.
But the competitors of the Analytical Engine were not mysterious programmable calculators from the future, but human computers. The Analytical Engine was to be a mechanical programmable calculator, with ordinary mechanical speed; certainly not any faster than 100 human computers.
Let's try to understand the economics.
In the 1850, 50 real gold pounds sterling per year would have been a fantastic wage for a human computer; that would be about 7000 debased paper pounds sterling of 2022. Mr. Babbage burned through more than 17,000 pounds sterling (about 2,500,000 British pounds in 2022 money) before Her Majesty's Treasury pulled the plug. With that money the Treasury could have employed a hundred human computers for three years, computing whatever they wanted to be computed. With the advantage that the human computers did not require any huge capital investment, but rather they would have been pay-as-you-go.
The point is that one fantastically expensive, slow-as-molasses Analytical Engine would not have provided the British Empire with any competitive advantage. It could not do anything that a team of human computers could not do, it would not have been any faster, and it would have costed a lot more.
It may be instructive to compare Babbage's visionary but utterly impractical dream with the severely limited but eminently practicable inventions of Herman Hollerith, who concentrated on designing and building electromechanical tabulating machines actually useful in real life.
Hollerith's machines were first used to tabulate the results of the 1890 U.S. Census; while the results of the 1880 census took eight years to tabulate, the results of the 1890 census were completed in six years, although the population had increased by 25%. Hollerith and other inventors continued to develop and refine unit record equipment, which became ubiquitous in commercial and governmental organizations throughout the first half of the 20th century; general-purpose digital computers only replaced them in the late 1960s or early 1970s.
In 1911, Hollerith's Tabulating Machines Company joined four other companies to form the Computing-Tabulating-Recording Company, which, in 1924, changed its name to International Business Machines, better known as IBM.
As stated on the Wikipedia page for the Analytical Engine
Babbage developed some two dozen programs for the Analytical Engine between 1837 and 1840, and one program later. These programs treat polynomials, iterative formulas, Gaussian elimination, and Bernoulli numbers
Gaussian elimination can be used to invert matrices, that is for example in solving systems of linear equations.
One possible application that comes to mind is the building design, where solving the frame requires solving such systems.
Babbage himself foresaw the development of Computer Science after the usage of his machine:
Babbage understood that the existence of an automatic computer would kindle interest in the field now known as algorithmic efficiency, writing in his Passages from the Life of a Philosopher, "As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise—By what course of calculation can these results be arrived at by the machine in the shortest time?"[
Almanacs
While chronometers help with determining longitude for navigation, they must be recalibrated fairly frequently. The moon, planets, and their moons are excellent natural clocks, but their motions don't perfectly conform to Kepler's laws. Corrections for their gravitational interaction were well understood in the 19th century, but the calculations are difficult. An almanac giving the positions of celestial objects versus time was very useful to navigators. Even in the 21st century, almanacs are important official publications. Automating the calculations allows the use of more accurate models and has improved the reliability of the predictions. Navigation was, of course, a critical activity for a worldwide empire funded by oceanic trade.
Firing Tables
Firing tables are used to estimate artillery range given elevation, charge, projectile type, etc. Tedious to calculate, they are of obvious use to an empire.
Tide Prediction
Later in the 19th century, Lord Kelvin invented a special-purpose analog computer for predicting tides. That was, of course, very useful to the Empire. A general-purpose digital computer could have done this, and Babbage's work was a couple of decades earlier.
Sure, the Analytical Engine is a bit of a toy. But now everyone knows that switches can be used to create thinking machines. Every artillery expert realises that faster computers = more accurate ballistics. Just having one computer incentivises the development of every technology that feeds into them.
As soon as relays get invented, they immediately get optimised for computers. People start trying to make vacuum tubes 40 years earlier. WW1 features WW2 style encryption and WW2 has semiconductors.
This doesn't scream competitive advantage, but what happens when Keynes comes along with the General Theory of Inflation and there's 80 years of computers being used to calculate things? Or when pervy old Kinsey shows up but the rates of sexual normbreaking is already known and tidily tabulated (and kills the scandal, as he's immediately disregarded as wrong)? Maybe the Soviets actually manage their economy quite a bit better with the aid of computers. The possibilities are endless.
No matter how well-oiled, there will be some resistance when turning the cogs. With a long enough chain of cogs, it may become impossible to turn the cogs without damaging them. This was a major problem for Babbage's actual Analytical Engine.
Cogs don't fit exactly together, or they wouldn't be able to move. If you have a series of cogs driving each other, the last cog will be behind the first by some small, but significant amount. This introduces imprecision, and the longer the chain of cogs, the greater imprecision.
