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"It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could be relegated to anyone else if machines were used." Gottfried von Leibniz

Charles Babbage (1792-1871), the eleventh Lucasian Professor, was a polymath. According to the Dictionary of Scientific Biography , Babbage explored the areas of cryptanalysis, probability, geophysics, astronomy, altimetry, opthalmoscopy, statistical linguistics, meteorology, actuarial science, lighthouse technology and the use of tree rings as historic climatic records. He published in the fields of magnetism, biology, geology, religion, submarines, politics, economics and machinery.¹ These accomplishments were in addition to his main work in mathematics and his calculating engines. He was awarded the first Gold Medal presented by the Astronomical Society.²

Babbage was active in efforts to reform the education and science of his time which were in a sad state of affairs. He lived in times of great change. England was in the midst of a transformation from a rural society to one of city dwellers, and London was rising to become a great city of Europe. His destiny, for all his ability and achievements, was to have little impact on his own times. Babbage thought more like a man of the twentieth century than one of the nineteenth century. Like a great artist, his impact was felt more after his life was over.

Babbage entered Trinity College, Cambridge in 1810, transferred to Peterhouse College and graduated first in his class in 1814. He received his MA in 1817. The transfer to Peterhouse was reported to have been motivated by his fear of finishing third in his class behind John Herschel and George Peacock, his cohorts in the Analytical Society.³ But this is disputed by Enros who claims that Babbage did not meet Herschel or Peacock until after the transfer, and furthermore was in the class behind them so he would have been examined after they graduated. Babbage probably transferred to expand his freedom to do what he wanted, which is more consistent with the way he lived.⁴

In 1812, Babbage helped to found the Analytical Society at Trinity College to reform the notation and teaching of calculus.⁵ At the time, England had fallen behind the rest of Europe in mathematics, caused by two main choices made by British mathematicians. The first was the use of Newton's notation in calculus rather than Leibniz's notation. The second choice was a generalization of the first, to ignore mathematical developments on the Continent in favor of British mathematicians. Mathematics was moving ahead in France, but England did not try to learn or keep up. Babbage tried to change that with the founding of the Analytical Society and other efforts throughout his life.

Babbage was abroad when he learned of his appointment to the Lucasian Chair, and he immediately prepared a letter of refusal. He was talked into accepting the post by friends who convinced him that others had fought hard to get him the Chair, and it would have been a terrible blow to them if he refused. So the letter was never sent. Once appointed, however, he did not really participate as it was hoped he would. He did not give any lectures as the Lucasian Professor and spent his time in London working on his calculating machines. When he finally resigned from the professorship ten years later, it was to focus on his calculating machines. It appears that he considered the position simply a source of income of between eighty and ninety pounds per year. His main efforts were in examining for the Smith's prize.

A number of major changes occurred in mathematics during this period. Noteworthy was the shift of algebra as an arithmetic idea, to arithmetic as a specialized case of algebra. Robert Woodhouse had laid the groundwork for this development, but Babbage and Peacock made the final drive over the threshold. This was an important release for mathematicians. Algebra had been viewed mainly as arithmetic that used symbols instead of numbers, not as a branch of mathematics standing on its own. With the understanding of the symbolic nature of algebra as a system that worked with reference to any particular numerical value, algebra became the more general of the two. Now, arithmetic could be seen as a particular case of algebra. The release was the ability to assign meaning to symbols and manipulate them as symbols. Even the "=" sign took on a new meaning, it no longer meant "equal to," but rather "algebraically equivalent to." This advance is detailed in George Peacock's work, but it also appeared earlier in Babbage's work. Because the two mathematicians worked together for years and learned from Woodhouse, it is not surprising that both had something to say on the topic.⁶ A further development of Babbage was the invention of the calculus of functions.⁷ He also worked in the areas of geometry, number theorem probability, and infinite series.

A second major advance during this period was the extension of geometry past its Euclidean origins. Babbage showed an interest in mathematical notation throughout his career, beyond the calculus notation controversies. In order to progress in mathematics Babbage believed that good notation was a critical and fundamental characteristic of logical reasoning. He wrote several papers on notation beyond his Memoirs.⁸ He was ahead of his time when it came to symbolic manipulation. He even invented notation for mechanical drawing.

The work that took up Babbage's adult life was his compute engines. The two main engines, the Difference Engine and Analytical Engine, were different in purpose, design and success, but they are important steps in the history of computers. These engines were also a great source of frustration and difficulty for Babbage who never really profited from them. As mentioned before, they were also an avenue from which George Airy caused Babbage serious difficulties.

Babbage did not invent the mechanical calculator, but he added significant improvements with his new approach, including the idea of mechanically printing the results directly from the engine, an idea we take for granted today.⁹ Remarkably, he accomplished this over one hundred years before anyone else attempted the construction of modern electronic computers. One of the main drawbacks of the early mechanical calculators was the need for constant human intervention.¹⁰ Generally speaking, the term computer referred to a roomful of people performing computations at that time.

One must appreciate the state of calculations and the value of tables in performing complex calculations during the 1800s to grasp the significance of Babbage's proposal for a machine that would automatically produce tables. The order of achievement is equivalent to the invention of movable type for book printing. The greatest advance in calculations up to this point had been the discovery of logarithms by John Napier in 1614. Performing long, complex calculations by hand was a formidable task for even the best of minds and prone to error. Logarithms made it easier by reducing multiplication to addition and division to subtraction, but it was still an arduous and time consuming process. Tables were greatly needed by banks, sailors, the military, government bureaucrats, engineers and architects. George Airy, the Royal Astronomer, had spent much of his career producing tables for astronomical events. Some important tables constructed in ancient times took as many as fifty educated men twelve years to complete and were paid for by kings.¹¹ Babbage offered a great deal with his idea for a machine to produce such tables in a short time. One problem was of course that not many people of his time were capable of seeing what he saw.

In 1822, Babbage completed the construction of the first mechanical calculator capable of calculating tables of logarithmic and trigonometric functions to six decimal places.¹² With this proof of concept he began the work for the Difference Engine. The principle of differences is a straight forward method of determining the next number in a sequence of calculations which are described as a function or as a polynomial. It reduces the calculation to adding and subtracting to discover the next result rather than multiplying the function completely. some arbitrary number.

In 1834, he started to design the Analytical Engine, the true precursor of today's computers. This engine used punched cards, one set for operations and another for variables. The punched cards had three functions, to input a constant, to input a variable, and an instruction. The variable card could even store a variable and later refer to that stored variable so it would be placed into the processor.¹³ The instruction cards would be interpreted by the main processor, then instructions were fed to the sub-parts of the engine as appropriate. The engine never actually ran, but Augusta Ada Byron (Countess of Lovelace) wrote the first computer program for the engine to calculate Bernoulli numbers. The programming language ADA is named for her. She was Babbage's lifelong friend and collaborator after their initial meeting in 1833.

The Analytical Engine was a major leap forward in concept from the Difference Engine. It was not just a calculator of tables, but a processor (which Babbage called the Mill ), memory (which Babbage called the Store ), and a printer. He envisaged a control program that used punched cards, variables and operators, all the fundamental parts of today's computers, although we have advanced past the use of punched cards.¹⁴ One of the most interesting features was the engine's ability to make decisions based on calculations it had performed. It was not limited to sequential calculations to produce a static table, but could choose to perform the next operation depending on how the previous operation turned out. This feature made the engine appropriate for many more applications.

For the remainder of his life, Babbage concentrated on the design and test of the engine. Unable to obtain funding from the government for its development after Airy's assessment that it would be worthless, he nevertheless continued to think about it. His thinking led him to produce hundreds of excellent engineering design drawings of the engine and its various parts.¹⁵

Babbage's genius produced the Analytical Engine, an invention that could have revolutionized industry and science during his lifetime. Babbage himself was well aware of the problem, writing in Passages: "The discovery of the Analytical Engine is so much in advance of my own country, and I fear even of the age, that it is very important for its success that the fact should not rest on my unsupported testimony."¹⁶

It also appears that an important underlying feature of his work on the Analytical Engine was almost missed even by the small band of supporters. This point was the difference between numeric operations and symbolic manipulation. It is a crucial mathematical difference as well as a difference in the design of his two engines. He was moving towards an Algebra Engine later in his life. Ada Lovelace made the distinction in her notes to the Menabrea paper.¹⁷

Babbage devised a notation that simplified the process for engineering drawings. One of his great contributions resulted in the field known as operations research, a quantitative approach to management, especially in the areas of mass production.

Babbage was a bit of radical in his day, one way of expressing it was to help in the founding of several learned societies, among them were the Astronomical Society in 1820, the British Association in 1831, and the Statistical Society in 1834.¹⁸ At the same time as he also criticized the Royal Society, of which he was also a member. In 1830 he wrote Reflections on the Decline of Science in England in which he describes scientific fraud.¹⁹ When he was at Cambridge, he had participated in the founding of the Analytical Society which was intended to reform English mathematics. His history in this regard is long and clear. He did work for business and industry by bringing in scientific methods to manufacturing and actuarials into the insurance industry. Babbage's mathematical interests were varied, but he was concerned with mathematical tables, which led him to the machines to produce tables, as well as to groundbreaking work in life insurance. Mortality tables are still tables of a mathematical nature.²⁰ The life insurance industry owes Babbage a major debt. He did groundbreaking work in what is known today as operations research or quantitative management. His main work is his book Economy of Manufactures and Machinery.²¹ It is inconceivable that anyone would try to run a corporation today without the benefits of the mathematics of operations research. The Great Western Railroad hired Babbage as a consultant, performing work to help improve rail safety and efficiency.²² Although England did not pay a lot of attention to his work on lighthouses, the United States Government did. Money was actually spent to try out his theories, with success.²³

The Ninth Bridgewater Treatise is the main source of Babbage's views on religion. The title is taken from the bequest of the Earl of Bridgewater, who left money for the publication a series of books on natural religion. Babbage was not one of the eight selected to receive money to write these books. He decided to publish his thoughts anyway.²⁴ Babbage, in the tradition of the Lucasian professors, believed in the use of science to understand religion and to further the understanding of God. He saw the universe as a great machine, but a step beyond the mere mechanistic view that was so common. He explained that a machine could be calculating something simple, then suddenly do something different, then go back to the simple operation. It was as though the machine were incrementing a series by one for several thousand iterations, then calculated a cube root, then returned to the incremental work. For Babbage this was the source of miracles, it was programmed into the universe already by God, and we needed to only to discover it. This also explained evolution.²⁵

Footnotes

1. Dubbey, Charles Babbage, 221.
2. Philip Morrison, Charles Babbage and his Calculating Engines (New York: Dover, 1961), xiii.
3. Morrison, xii.
4. Philip Enros, "The Analytical Society(1812-1813): Precursor to the renewal of Cambridge Mathematics," Historia Mathematica 10 (1983): 24-47.
5. Enros, 26.
6. Dubbey, Charles Babbage, 93.
7. Dubbey, Charles Babbage, 51.
8. Dubbey, Charles Babbage, 154.
9. Dubbey, Charles Babbage, 175.
10. Anthony Hyman, Charles Babbage, Pioneer of the Computer (Princeton: Princeton University Press, 1982) 48.
11. Michael Lindgren, Glory and Failure: The Difference Engines of Johann Muller, Charles Babbage and Georg and Edvard Scheutz (Cambridge: MIT Press, 1990), 28.
12. Lindgren, 71.
13. Hyman, 168.
14. Lindgren, 63.
15. Hyman, 167.
16. Charles Babbage, Passages from the Life of a Philosopher (London: Dawson's of Pall Mall, 1864), 309.
17. Hyman, 210.
18. Dictionary of Scientific Biography.
19. Parkinson.
20. Morrison, xxii.
21. Morrison 1961 xi.
22. Morrison 1961 xxviii.
23. Morrison 1961 xxviii.
24. Lindgren 1990 266.
25. Lindgren 1990 267.