Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/726

anger. After the return of the Greeks from Troy he is said to have retired to Colophon. According to the story, his death was due to chagrin at being surpassed in a trial of soothsaying skill by one Mopsus. It had long been predicted that he should die whenever he met his superior

in divination.

CALCULATING MACHINES. Mathematicians and astronomers have felt in all ages the irksomeness of the labour of making necessary calculations, and this has led to the invention of various devices for shortening it. Some of these, such as the Abacus, Napier s Bones (invented by the father of logarithms), and the modern Sliding Rule, are rather aids to calculation than calculating machines. Pascal is believed to have been the original inventor of a calculating machine ; its use was limited to addition, multiplication, &c., of sums of money, and as it required the constant intervention of a human operator the results were subject to the ordinary errors of manipulation. After him came the celebrated Leibnitz, Dr Saunderson, who, blind from his childhood, became professor of mathematics in Cambridge, and others. But all their machines were completely cast into the shade by the wonderful inventions of the late Charles Babbage. He knew well the immense value that absolutely correct tables possess for the astro nomer and the navigator, and that a machine which could produce them with speed was a very great desideratum. The first calculating machine he invented he called a differ ence engine, because it was to calculate tables of numbers by the method of differences. By setting at the outset a few figures the attendant would obtain by a mechanical operation a long series of numbers absolutely correct. The difference engine was not intended to answer special ques tions, but to calculate and then print numerical tables, such as logarithm tables, tables for the Nautical Almanac, &c. An interesting account of some of the errors which are found in what are considered reliable tables is given in a paper by Babbage in the Memoirs of the Astronomical Society, 1827.

Table of First Second Square Xumbers, Differ ences, Differ ences, N A 1 A a 1 3 2 4 5 2 9 7 2 16 9 25

Every numerical table consists of a series of numbers which continuously increase or diminish. As an example take the squares of the natural numbers, 1, 4, 9, 16, 25, 36, &c. Designate this series by N. If we subtract each term from the one following it we get a new series, 3, 5, 7, 9, &c., which is called the series of first differences ; designate this by A 1 . If in the same way we subtract each term of this series from the succeeding term, we get what is called the series of second differences, every term of which is in this instance 2. Designate this series by A 2 . As the different series were obtained by subtraction, it is quite evident that by revers ing the process we shall obtain the original table. Sup pose we are given the first terms of N, A 1 , and A 2 , i.e., 1, 3, 2. If we add 3, the first term of A 1 , to 1, the first term of N, we get 4, the second term of N ; and if we add 2, the first term of A 2 , to 3, the first term of A 1 , we get 5, the second term of A 1 ; and this added to 4, the second term of N, gives us 9, the third term of N. Similarly we obtain 16 by adding 9, 5, and 2 together, and 25 by adding 16, 7, and 2. Hence, given 1, 3, 2, we can, by a process of additions, obtain the series of square numbers. All numerical tables can be calculated entirely by this method or by repetitions of it.

The main characteristics of the difference engine, de signed and partially constructed by Babbage, are these : It consisted of several vertical columns of figure-wheels like large "draught men" one above another, to the number of six in each column. The natural numbers from to 9 were cut on the rims of the figure wheels ; hence each figure-wheel in a column could represent a digit. Thus the lowest wheel gave the units digit, the second wheel the tens digit. The number 5703 ^ would be represented on the wheels of a column as in the margin. The different columns were to represent the successive series of differences above referred to, and were called the table column, the first difference column, <fcc.

" The mechanism was so contrived that whatever might be the numbers placed respectively on the figure wheels of each of the different columns, the following succession of operations took place as long as the handle was moved. Whatever number was found upon the column of first differences, would be added to the number found upon the table column. The same first difference remaining Hpon its own column, the number found upon the column of second differences would be added to that first difference." Similarly for all the other columns. For example, suppose we are calculating the cubes of the natural numbers. At a certain stage of the work we would find 125 shown by the wheels of the table column, 91 by those of the first difference column, 36 by those of the second difference column, and 6 on the lowest wheel of the third difference column. On making a turn of the handle the 91 would be added to the 125, which would then show 216 ; at the same time 36 would be added to the 91, so that the first difference column would then show 127; moreover 6 would be simultaneously added to the 36, which would thus become 42, and the 6 would remain unaltered. Another turn and we would get 343, 169, 48, 6 on the different columns. Had the engine been completed it would have had columns for six orders of differences, each of twenty places of figures, whilst the first three columns would each have had half a dozen additional figures.

Table Column. First Difference Column. Second Difference Column. Third Difference Column. 1 2 r 3 5 1 6 6 After one Turn ... 2 ] 1 2 4 6 7 2 After two Turns... 3 1 4 G 4 3 9 8 6

It was about 1822 that Babbage having constructed

a small model of his engine sent an account of it to Sir Humphrey Davy, then president of the Royal Society of London. Government heard of the invention, and, having received from the Royal Society a favourable report on the merits and utility of the engine, advanced money towards its construction. Sums of money were at irregular intervals voted for this purpose ; but so great were the difficulties to be overcome, so entirely new even were many of the tools necessary, so much time was occupied in testing the value of each proposed contrivance, that in 1834 only a portion was completed. The construction of the machine here stopped, although the Royal Society had again, in 1829, reported most favourably on the engine as regards its practicability, immense utility, and the progress it had made. The Government had already advanced 17,000 (over and above what Babbage had spent, besides giving his personal superintendence without any remuneration),

and they saw no definite limit to the amoun it would cost ;