In Babbage's times there was a really high error rate in the calculation of math tables, when Babbage planned to find a new method that could be use to make it mechanically, removing the human error factor. This idea started to tickle his brain very early, in 1812.
Three different elements influenced him in this decision: he disliked untidiness and unprecision; he was very able with logarithmical tables; he was inspired from an existing work on calculating machines produced by W. Schickard, B.Pascal, and G. Leibniz.
He discussed the main principles of a calculating engine in a letter he wrote to Sir H. Davy in the early 1822.
Difference engine
Babbage presented something that he called "difference engine" to the Royal Astronomical Society on Jun 14, 1822 and in a paper entitled "Note on the application of machinery to the computation of astronomical and mathematical tables."
It was able to calculate polynomials by using a numerical method called the differences method.
The Society approved the idea, and the government granted him £1500 to construct it, in 1823. Charles Babbage converted one of the rooms in his home to a workshop and hired Joseph Clement to oversee construction of the engine. Every part had to be formed by hand using custom machine tools, many of which Babbage himself designed. He took extensive tours of industry to better understand manufacturing processes. Based on these trips and his experience with the difference engine, Babbage published On the Economy of Machinery and Manufacture in 1832. It was the first publication on what we would now call operations research.
The death of Georgiana, Babbage's father, and an infant son interrupted construction in 1827. Work had already taxed Babbage heavily and he was on the edge of a breakdown. John Herschel and several other friends convinced Babbage to take a trip to Europe to recuperate. He passed through the Netherlands, Belgium, Germany, and Italy visiting universities and manufacturing facilities.
In Italy he learned he had been named the Lucasian Professor of Mathematics. He initially wanted to turn down the position but several friends convinced him to accept. He moved to 1 Dorset Street upon returning to England in 1828.
The difference engine project had come under fire during Babbage's absence. Rumours had spread that Babbage had wasted the government's money; that the machine did not work; and that it had no practical value if it did. John Herschel and the Royal Society publicly defended the engine. The government continued its support, advancing £1500 on April 29, 1829, £3000 on December 3, and £3000 on February 24, 1830. Work continued, but Babbage would have continual difficulty getting money from the treasury.
Babbage's problems with the treasury coincided with numerous disagreements with Clement. Babbage had built a two-story, 50 foot long workshop behind his house. It had a glass roof for lighting, and a fireproof, dust-free room to contain the machine. Clement refused to move his operations to the new workshop and demanded more money for the difficulty of travelling across town to oversee construction. In response, Babbage suggested that Clement draw his pay directly from the treasury. Before then, Babbage would get money from the government that he would use to pay Clement. He often had to pay Clement out of his own pocket when the bureaucracy lagged behind Clement's pay schedule. Clement refused the request and stopped working.
Clement further refused to turn over the drawings and tools used to build the difference engine. After an investment of £23000, including £6000 of Babbage's own money, work on the unfinished machine ceased in 1834. Charles wrote, "The drawings and parts of the Engine are at length in a place of safety—I am almost worn out with disgust and annoyance at the whole affair." In 1842 the government officially abandoned the project.
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