and Babbage had a delicacy in pressing for the com pletion of the difference engine, as he had recently designed a new machine, the analytical engine, which, if completed, would entirely supersede it. The portion completed is in
King s College, London.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 consists 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 V 1? V 2 , &c., the numbers of the special problem or formula are placed ; the other set is called the "mill," and performs the required operations of multiplica tion, division, addition, or subtraction. Its working was directed by means of two sets of cards "operation" cards, which instructed the mill whether to multiply, divide, add, or subtract, and " variable" cards, which indicated to the mill the particular columns, i.e., numbers on which it was to perform this operation. An example will make this clear. Suppose we wish to solve the equations
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ax + by = c , dx +fy = g. On the wheels of V 1? the first column of the variables, the number a is placed, b on V 2 , c on V 3 , and so on. Six columns in all are required for this. It is evident that x = -j ~. Hence, to get x, we require the products of /and c, b and g, &c. To get these the superintendent intimates to the mill by means of an " operation " card that a multi plication 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 V 6 , V 3 , and V 7 respectively. By another opera tion card and another variable card, the mill would then be instructed to multiply the numbers on V 2 and V , and to place the result on V 8 . Similarly ca and bd would be obtained on V 9 and V 10 . The superintendent would then instruct the mill to subtract the number on V 8 from that on V 7 , to place the result (fc bg] on V n , and similarly, fa bd would be placed on V 10 . By a new operation card the mill would now be put into a " dividing " state, and a variable card would tell it that V n was to be divided by V 12 , and the result given on V 13 . This would be the value of x. Similarly for y. The number of cards can be greatly 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 holes, &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 which it multiplies (say) the numbers indicated to it. The variable cards act in a similar manner. When an operation card arid a variable card are given to the engine, the numbers on the assigned columns are transferred to the mill, the operation is per formed, and the numbers and the result are placed on the proper columns. The series of cards used for any one prob lem would enable the machine to solve 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 will calculate and print tho numerical results of that formula." In the construction of this engine he overcame one of the greatest difficulties in such an instru ment, that of effecting the carrying of tens. The engine was designed so as to foresee 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, apparatus 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 computes. The engine would com puts 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 con structing 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 this effect from the Royal Swedish Academy of Sciences, they endeavoured unsuccessfully to get orders for their machine. In 1853, with the aid of grants from the Swed ish Government, the Messrs 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 square pianoforte. It could calculate series with four orders of differences each of fifteen figures. It printed the results to eight figures, the last of which was capable of an automatic correction where neces sary for those omitted. " 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 in fallibility in certain respects ; but it is not infallible, except in very skilful hands. The weakest point is the printing apparatus, and that admits of evident improvement."
M. Staffel and M. Thomas (de Colrnar) have invented machines which can perform addition, subtraction, multipli cation, 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 Cyclopcedia, where many references will be found, and a translation of General Menambrea s article on Babbage s Analytical Engine m Taylor s Scientific Memoirs, vol. iii.
