History of Computers and Computing, Babbage, Differential engine. Charles Babbage, 1. Portrait from the Illustrated London News, Nov. The Differential Engine of Charles Babbage. A numerical table is a tool designed to save the time and labour of those engaged in computing work. The oldest tables which are preserved, were compiled in Babylon in the period 1. B. C. They were intended to be used for the transformation of units, for multiplication and division, and they were inscribed in cuneiform on pieces of clay. Charles Babbage (Londra, 26 dicembre 1791 – Londra, 18 ottobre 1871)![]()
During the first century B. C. Claudius Ptolemy in Alexandria created his theory about the motions of the heavenly bodies in a work which later came to be known by the name of Almagest. They were to form one of the Ancient World's most important astronomical documents and they contained all the necessary tables for the calculation of eclipses as well as various kinds of ephemeris, that is to say tables which specified the positions of the heavenly bodies during a particular period, e. During the first half of the thirteenth century the Ptolemy's tables caught the attention of King Alphonso the Wise of Castile. He then gathered together a great number of scholars in Toledo who were given the task of calculating a new collection of astronomical tables. The reason for this endeavor was said to be that King Alphonso, who was interested in astronomy, had discovered many errors in Ptolemy's tables. The work began some time in the 1. The tables produced were later known as the Alphonsine Tables. The vast costs involved were paid for by the king, whose name soon spread with the copies of the tables throughout the European scientific world. Besides the Babylonian tables, Ptolemy's work and the Alphonsine Tables, a great deal of toil went into the production of many other numerical tables of different kinds during this period. With the introduction of the art of printing throughout Europe during the latter half of the 1. The Alphonsine Tables for example, were printed in Venice in 1. At the end of the sixteenth century, several famous arithmetical and trigonometric tables were published. In order to simplify multiplication work, multiplication tables were published. A real revolution in the table business happened after John Napier's discovery of logarithms in 1. With a table of logarithms at hand, the computational effort could be greatly reduced. Charles Babbage was born in London on December 26, 1792, the son of Benjamin Babbage, a London banker. As a youth Babbage was his. English mathematician and inventor who is credited with having conceived the first automatic digital computer. In 1812 Babbage helped found the Analytical Society. CHARLES BABBAGE, ADA LOVELACE, AND THE BERNOULLI NUMBERS “Was Ada LoveLace ! In 1. 61. 7 Henry Briggs published the first table of logarithms. Two hundred years later, at the beginning of the 1. Europe. The sole alternatives were Napier's Bones and the slide rule. ![]() Mechanical calculating machines were extremely rare and at most a handful of very select individuals can ever have used them for serious calculations. Most of them were simply remarkable gadgets illustrating man's scientific progress, rather than genuine aids to calculation. For the normal calculator or scientist who had to carry out complex computations which demanded great accuracy, Napier's Rods and the slide rule were of little help. In effect, his tools were pen, paper and tables. There were tables for mathematics, astronomy, navigation, physics, engineering, statistics, trade and finance, in the army and in many other areas. ![]() The publication of such tables however required a lot of manual calculating work and the final product was full of errors. Some time in 1. 82. Charles Babbage (biography of Charles Babbage) got the idea about the mechanical computation. He has provided us with two versions of the origin of his ideas about machines, but the one written in 1. According to the first story, in 1. Astronomical Society assigned Babbage and his friend Herschel one of the tasks for improving the tables of the navigational book Nautical Almanac. They constructed the appropriate formulas and assigned the arithmetic to clerks. To diminish errors, they had the calculations performed twice, each by a different clerk. Then they compared the two sets for discrepancies. In the course of their tedious checking, Herschel and Babbage found a number of errors, and at one point Babbage said I wish to God these calculations had been executed by steam. It is quite possible, remarked Herschel. But in his autobiography Babbage remembered another version of the story, which must have happened either in 1. I was sitting in the rooms of the Analytical Society, at Cambridge, my head leaning forward on the table in a kind of dreamy mood, with a table of logarithms lying open before me. Another member, coming into the room, and seeing me half asleep, called out, . He made a small model consisting of 9. The machine was ready by the end of the spring of 1. June was announced publicly and has been examined by several of the members of the Astronomical Society. It seems Babbage must have known very little about machine design, mechanical calculating and the history of such machines at that time, because he started by considering the use of sliding rods, instead of the more natural use of wheels in the adding mechanism. This kind of mechanism which was . In fact this seems to have been such a revelation to him, that in November of 1. The working model had a section of the calculating mechanism, including two orders of difference, but no printing mechanism. He successfully calculated the first thirty values, arising from the formula x. The machine produced correct results at the rate of 3. Later on same year Babbage wrote a note to the Society and an article . The arrangements are such that.. In this letter, Babbage pointed out the advantages such a machine would have for the Government in producing the lengthy tables for navigation and astronomy, and proposed to construct a machine on an enlarged scale for the Government's use. The Astronomical Society received Babbage's proposal with the highest enthusiasm, and the Royal Society reported favorably on his project for building what he called a Difference Engine, a specialized calculating machine for calculation of tables, using the method of differences. Babbage was not the first to suggest a printing calculator, nor was he the first to propose the method of differences as a suitable principle on which to base a mechanized calculation. This distinction goes to the german engineer and master builder Johann Helfrich M. There is an evidence, that at some point, Babbage learned about M. Let's consider the same formula, used by Babbage: T=x. It generates a sequence of values for T, which happen to be prime numbers, as seen in the table in the nearby figure, in which with D1 is noticed the first difference column, while with D2—the second difference column. If we take the differences between successive values of T, these so called first differences follow quite a simple rule. If we take the differences between the differences, known as the second differences, the result is even more striking- the second difference, is a constant. With this knowledge, the table can be built up in a very simple way, as shown by the box in the table. Take the second difference, and add it to the first difference to form a new first difference, 4+2=6. The process can be generalized. In our example the second difference is constant because the function T is a quadratic. If the function T were a cubic, such as T = x. In general a polynomial of degree n will have a constant nth difference and each successive new value of the function can be obtained by n simple additions. The usefulness of difference techniques is greatly increased by the fact that any section of a well- behaved continuous function can be approximated by a polynomial. The shorter the section and the higher the degree of the polynomial the closer the approximation. So if we wished to tabulate a function, such as a sine or the time of sunset, it is only necessary to divide the function into short enough intervals and find a suitable approximating polynomial for each interval. The method of differences can then be used to tabulate the function throughout the interval. This process is known as sub- tabulation. Babbage realized that a machine could carry out this sub- tabulation process. First, he needed a mechanism for storing, separately, the numbers corresponding to the values of the tabular value, the first difference, the second difference, etc. While using these drawings, he felt that they did not fully and adequately describe the mechanism. For a machine. with many parts moving in various ways, the static drawings could only show the shape and arrangement of the parts. So. Charles devised a system of mechanical notation that would also indicate how the parts moved—their speeds and interconnections. Unlike the usual drawings, the notation did not picture the shapes of the parts. Rather, it was a table of numbers, lines, and symbols to describe the machine’s actions. It was a general system that could be used to describe any machine. Charles published a description of his mechanical notation in the Philosophical Transactions of the Royal Society in 1. Laws of mechanical notation). However, this mechanical notation did not ever come into widespread use. In an interview held in 1. Babbage and the Chancellor of the Exchequer, a rather vague verbal agreement was made whereby the Government would grant funds for the enterprise which was expected to take three years. His own Astronomical Society was so impressed by the machine, that it awarded him its first gold medal in 1. In the same year the British government advanced Babbage a fee of . Babbage needed a small factory and competent workers, although initially two rooms in Babbage's house were converted into workshops and a third into a forge. He hired a good engineer—Joseph Clement, to maintain the mechanical works in his shop. By 1. 82. 8, Charles had spent more than . After a supportive report from Charles’s friends in the Royal Society, the government agreed to make up the difference. But the work went rather slowly. The whole project was taking much longer than anyone had anticipated.
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