Note that "D" is used in place of a capital letter "delta" in this transcription. Also, the layout of some equations and table borders differ.

Encyclopædia Britannica With New American Supplement, 9th ed. (1898). S.v. "Calculating Machines." by P. R. S. Lang, M. A., Professor of Mathematics, University of St. Andrews.

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 calcuation 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 astronomer 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 difference 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 questions, 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.

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.

 Table of Square Numbers, N First Differences, D1 Second Differences, D2 1 3 2 4 5 2 9 7 2 16 9 25

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 D 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 D2. As the different series were obtained by subtraction, it is quite evident that by reversing the process we shall obtain the original table. Suppose we are given the first terms of N, D1, and D2, i.e., 1, 3, 2. If we add 3, the first term of D1, to 1, the first term of N, we get 4, the second term of N; and if we add 2, the first term of D2, to 3, the first term of D1, we get 5, the second term of D1; 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, designed and partially constructed by Babbage, are these:— It consisted of several vertical columnds of figure-wheels like large "draught men" one above another, to the number of six in each column. The natural numbers from 0 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 5706 would be represented on the wheels of a column as in the margin.

 5 7 0 6

The different columns were to represent the successive series of differences above referred to, and were called the table column, the first difference column, &c.

"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 upon 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.

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

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.

It will be noticed that the use of the difference engine was limited to the working of such problems as can be solved by successive additions or subtractions. The analytical engine, on the other hand, was designed to work out any problem that the superintendent knew how to solve. It consits of two parts, each of a number of vertical columns of figure wheels, similar to those of the difference engine; on the one set called the "variables," which we shall designate by V1, V2, &c., the numbers of the special problem or formula are placed; the other set is called the "mill," and performs the required operations of multiplication, division, addition, or subtraction. Its working was directed by means of two sets of cards—"operation" cards , which instructed the mill whether ot multiply, divide, add, or subtract, and "variable" cards, which indicated to the mill the particular columns, i.e., the numbers on which it was to perform this operation. An example will make this clear. Suppose we wish to solve the equations
 ax + by = c , dx + fy = g .
On the wheels of V1, the first column of the variables, the number a is placed, b on V2, c on V3, and so on. Six columns in all are required for this. It is evident that x = ( fc - bg ) / ( fa - bd ) . Hence, to get x, we require the products of f and c, b and g, &c. To get these the superindentendent intimates to the mill by means of an "operation" card that a multiplication is to be performed, then points out by a "variable" card what are the two numbers, i.e., the two columns to be multiplied, and on what column the result is to be placed. In the first case the columns indicated would be V5, V3, and V7 rspectively. By another operation card and another variable card, the mill would then be instructed to multiply the numbers on V2 and V6, and to place the result on V8. Similarly ca and bd would be obtained on V9 and V10. The superintendent would then instruct the mill to subtract the number on V8 from that on V7, to place the result ( fc - bg ) on V11, and similarly, fa - bd would be placed on V12. By a new operation card the mill would now be put into a "dividing"e; state, and a variable card would tell it that V11 was to be divided by V12, and the result given on V13. This would be the value of x. Similarly for y. The number of cards can be gratly diminished. Thus, for the four multiplications one card would suffice. The cards are of pasteboard (say) and have holes punched in them,— a "multiplication" card having a certain number of holes bored in it and arranged in a particular way, a "division" card a different arrangement of holds, &c. The cards are so placed in the machine that certain levers drop through these holes, while others are unaffected, and the machinery in connection with the levers is put out of gear or not as is desired. In this way the mill is put into a condition in whicyh it multiplies (say) the numbers indivated to it. The variable cards set in a similar manner. When an operation card and a variable card are given to the engine, the numbers on the assigned columns are transferred to the mill, the operation is performed, and the numbers and the result are placed on the proper columns. The series of cads used for any one problem would enable the machine to solved any other similar problem. Babbage says of the engine, "The analytical engine is therefore a machine of the most general nature. Whatever formula it is required to develop, the law of its development must be communicated to it by two sets of cards. When these have been placed the engine is special for that particular formula. The numerical constants must then be put on the wheels, and on setting the engine in motion it wil calculate and print the numerical results of that formula." In the construction of this engine he overcame one of the greatest difficulties in such an instrument, that of effecting the carrying of tens. The engine was designed so as to forsee these carriages, and act upon that foresight, and thus a great reduction of the time necessary to make a given calculation was at once obtained by effecting all carriages simultaneously instead of in succession. He says of it, "The analytical engine will contain—1°, aparatus for printing on paper, one, or if required, two copies of its results; 2°, means for producing a stereotype mould of the table or results it computs. The engine would compute all the tables it would itself require. It would have the power of expressing every number to fifty places of figures." It would multiply two numbers of fifty figures each, and print the result in one minute. Its construction was never begun, but babbage left complete plans of every part of it.

In the Edinburgh Review for July 1834 appeared an account of the principles of Babbage's difference engine. Herr George Scheutz, a printer at Stockholm, read it, and shortly afterwards he and his son Edward set about constructing a calculating machine. By 1843 they produced one capable of calculating series with terms of five figures, with three orders of differences, also of five figures each, and of printing its results. Provided with a certificate to thsi effect from the Royal Swedish Avademy of Sciences, they endeavoured unsuccessfully to get orders for their machine. In 1853, with the aid of grants from the Swedish Government, the Messsrs. Scheutz finished a second machine which was exhibited in England, and at the Paris exhibition of 1855. It eventually went to America. It was about the size of a small pianoforte. It could calculate series with with four order of differences each of fifteen figures. It printed the results to eight figures, the last of which was capable of an automatic correction where necessary for those ommitted. "It could calculate and stereotype without a chance of error two and a half pages of figures in the same time that a skilful compositor would take to set up the types for one single page."

A new machine by the Messrs Scheutz was constructed about 1860 by Messrs Donkin for the Registrar-General for the sum of £1200. It has been used in the calculation of some of the tables in the English Life Table, published in 1864. Dr Farr says of it, "The machine has been extensively tried, and it has upon the whole answered every expectation. But it is a delicate instrument, and requires considerable skill in the manipulation. It approaches infallibility in certain respects; but it is not infallible, except in very skilful hands. The weakest point is the printing aparatus, and that admits of evident improvement."

M. Staffel and M. Thomas (de Colmar) have invented machines which can perform addition, subtraction, multiplication, division, and extraction of the square root. M. Thomas's machine is extensively used.

Sir William Thomson has recently invented an instrument (no description of which has yet been printed) which is able to solve any linear differential equation with variable coefficients.

Professor Tait has also invented the principle of a machine, which, if constructed, will integrate any linear differential equation of the second order with variable coefficients.

See the article "Calculating Machines" in Walford's Insurance Cyclopædia, where many references will be found, and a translation of General Menambrea's article on Babbage's Analytical Engine in Taylor's Scientific Memoirs, vol. iii.
 (P.S.L.)

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