VICTORIA AND ALBERT MUSEUM.
BABBAGE'S CALCULATING MACHINE;
OR
DIFFERENCE ENGINE.
This machine was invented by the late Charles Babbage, who was born on the 26th December 1791, and died on the 28th October 1871.
The first idea of a calculating machine was conceived by Mr. Babbage in 1812 whilst at Cambridge, and the first publication respecting it was in a letter addressed by him to Sir Humphry Davy, President of the Royal Society, dated 3rd July 1822.
In 1823 the Government took up the project and referred it to the Royal Society, who reported that the machine was fully adequate to the objects proposed. The construction of the machine was then commenced and was carried on for several years, Mr. Babbage giving his time in making the drawings and superintending the work, and the Government paying the workmen employed.
Early in 1833 the work was suspended, owing to the Engineer employed in the construction demanding exorbitant ternis for removing his.workshops to the buildings erected by Government for the reception of the machine. During this suspension of the works of the Difference Engine, Mr. Babbage conceived the-idea and commenced the drawings of the Analytical Engine.
After several fruitless attempts to make some suitable arrangement with the Engineer, Mr. Babbage in 1834 applied to the Duke of Wellington, then Prime Minister, for definite instructions as to the completion of the Difference Engine, pointing out the several plans which might be pursued, and at the same time announcing the invention of the Analytical Engine which he observed did not supersede the Difference Engine but added greatly to its utility.
The Duke of Wellington did not remain long in office, and no immediate decision was arrived at. Minister succeeded Minister, the position at issue was misunderstood or lost sight of amidst the strife of Politics, and it was not until nearly eight years had elapsed that any decision was arrived at. Finally, in November 1842, Mr Babbage was informed by Mr. Goulburn, Chancellor of the Exchequer in Sir Robert Peel's Government, that they regretted the necessity of abandoning the Machine, alleging the expense of its completion as the ground for their decision.
At the time of its suspension about £17,000 had been expended, by government upon its construction, and a large part of the machinery had been made. A small portion of it, now in the Victoria and Albert Museum, was put together in 1833 prior to the suspension of the work, in order to shew the action of the machinery.
The whole engine when completed was intended to have had 20 places of figures and 6 orders of differences.
OBJECTS OF THE DIFFERENCE ENGINE.
This machine was expressly designed for the purpose of calculating and printing tables, and not to perform single arithmetical sums.
If a single article is wanted, it is not generally speaking worth while to construct a machine to make it, but when large numbers are required, their production comes within the true province of machinery, and in this sense the Difference Engine is emphatically a machine for manufacturing tables.
EXPLANATION OF THE DIFFERENCE ENGINE.
The mode in which the Difference Engine calculates tables is by the continual repetition of the simultaneous addition of several columns of figures to other columns in the manner more particularly described below, and printing the result.
In the small portion put together, and now in the Victoria and Albert Museum, the figure opposite the index on the lowest wheel visible, in all cases represents units; the figure on the next wheel above, tens that on the one above it, hundreds, the next thousands, and so on.
The right hand column of wheels shows the result of the calculation or the tabular number; for instance, series of squares, cubes, or logarithms, etc., appear upon it, according to the nature of the calculation the machine is making.
The next or central column represents the First Difference, and the left hand column the Second Difference. At the bottom of the central column is a figure wheel, covered, which can be used as a third difference so as to enable this portion of the machine to calculate tables of which the Third Difference does not exceed 9. This will be better understood if this last wheel is supposed to represent the lowest wheel of a fourth column of figures standing beyond the left hand side of the machine, as it would be if it formed part of the complete machine.
This arrangement is effected by a movable platform with axles and gearing wheels upon them, which are used for adding from the third difference wheel at the bottom of the central column to the second difference which is shown on the left hand column. The effects capable of being produced by this mechanism, when the gearing is altered, and the loose wheels belonging to it are put into gear with certain figure wheels, is explained in Babbage's Ninth Bridgewater Treatise, together with the new views which it opened up to him upon the subject of natural laws.
The three upper wheels of the left hand column are separated from the rest of the machine and are employed in counting the natural numbers. In other words they register the number of calculations made by the machine, and give the natural numbers corresponding with the respective terms of the table.
Four half turns of the handle, two backwards and two forwards, are required for each calculation, and the words "calculation complete" come round upon a wheel at the top of the central column to show when this is done. This wheel also shows by the word "adjust" in what position of the handle the figure wheels may be freely moved by hand in order to introduce different numbers or a different table.
PRINCIPLE OF THE DIFFERENCE ENGINE.
The mathematical principle upon which the machine is constructed is called the Method of Differences, and as it was intended to have had six orders of Differences when the whole machine was put together, the mathematical form in which it might have been expressed was
and by allowing additional time for the calculations
In practice, however, a machine possessing six orders of differences would be able to calculate any tables that might be required.
The mathematician will see at once, that an engine capable of numerically tabulating this formula would command the whole range of tables, but for the non-mathematical reader a further explanation is necessary.
If any table, such as the table of the cubes of numbers given below is taken, and each of the tabular numbers is successively subtracted from the following one, another column of figures will be obtained which is called the first order of differences.
If again the column of the first order of differences thus obtained is subjected to a similar operation, each number in it being in turn subtracted from the one immediately below it, a second order of differences is arrived at, and by repeating the process a third order of differences may be obtained, and so on.
It will be observed, however, that the number in the third order of differences, or the Third Difference as it is called for brevity, is in the table of cubes given below, always six; in fact that a constant difference has been arrived at, and that consequently the fourth difference will be 0 so that the general law of such tables put into mathematical language is
The following is a table of the cubes of numbers treated in this manner, which is arranged for the sake of clearness, so that the columns of figures may represent, as nearly as may be, then actual position, in this portion of the Difference Engine. The natural numbers are represented by three upper wheels on the left hand axis. The column of third differences is represented by the bottom wheel on the central axis, that of second differences by the three lower wheels on the left hand axis, that of first differences by the wheels on the central axis, and that of tabular numbers (in this instance, cubes) by the wheels on the right hand axis.
This table has been obtained from a table of cube numbers by continual subtraction in the manner explained above. It is evident that by reversing the process, and by taking the figures in any line of this table and adding instead of subtracting, the table might be continued, and this is exactly what the Machine does.
If it is desired to set the Machine to calculate this or any other tables all the wheels must first be brought to zero, and this is most readily accomplished as follows:-
Draw the movable platform forwards so that the wheels upon it may be out of gear with the wheels on the figure axes, and move the handle until the word "adjust" comes round. Next turn the unit wheel on the lefthand axis to zero, and the two wheels above it to 9. Turn the bottom wheel of the central axis to zero, leaving the other wheels at any figures they may happen to stand at. Now work the handle until the words "calculation complete" come round, and all the nines will be reduced to zero. Next do the same for the central wheels, make another calculation and they too will have turned to zero, finally treat the wheels on the right hand axis in the same way, and having thus reduced all the wheels to the zero point except the counting, make one or two complete calculations to see that no change takes place. In all cases previous to setting the Machine to make a table, one calculation at least should be made with the wheels all at zero to ascertain that every thing is right, which is shown by the wheels remaining at zero. Unless this is done the figures directed below to be set in the machine will not produce a table of cubes.
All this time the counting part of the Machine will have continued registering the number of times "calculation complete" has come round. This must now be brought to zero as directed for the other parts, and the Machine will stand at zero ready to operate upon any figures that may be put into it. If a table requiring a third order of differences is wanted the movable platform must be pushed back into its place, and everything will then be ready.
If it is desired to make the Machine calculate the above table of cubes, bring the handle to "adjust " and put 6 upon the lowest covered central wheel which represents the third difference. Also 6 upon the left hand lowest wheel or second difference, 1 upon the lowest uncovered central wheel and leave the right hand wheels or table standing at zero, now move the handle until "calculation complete" comes round; you will observe the counting machine has registered 1, and the 1 upon the first difference has been added to zero upon the table, making it 1, and the machine will show as follows:-
| Natural number | 3rd difference. | 2nd difference. | 1st
difference. |
Table |
| 0 | 6 | 6 | 1 | 1 |
The next backward and forward stroke of the handle will add the second difference to the first, the backward stroke adding the figures on the wheels, and the forward stroke making the carriages, if any, and the result will be
| 1 | 6 | 6 | 1 | 1 |
and after two more strokes when "calculation complete" will come round again,
| 2 | 6 | 12 | 7 | 8 |
Thus in the first two turns after the words "calculation complete" The second difference is added to the first, and in the next two the third is added to the second and at the same time the first is added to the table, the counting part registering another calculation.
It will be seen that the sequence of operations is as follows: in the first two turns of the handle the second difference is added to the first and a new first difference obtained; in the second two turns the second difference, and the first at the same time to the tabular number; thus a new second difference as well as a new tabular number is obtained, whilst at the same time another number is added to the natural number and the words "calculation complete" come round. The Machine has in fact registered another cycle of operations. The same operations again recommence until another tabular number is obtained, and may be continued until the limit of the number of figures in the machine is reached.
This order of proceeding would have been reversed in the Difference Engine as intended to be when completed, as the cycle, of operations would have been, first to add all the figures on the even columns (counting the table column) to those in the odd ones to their right, and then to add all the odd columns to the even ones. During the second operation as the Table column would have been at rest, the tabular numbers upon it were to have been printed.
It now only remains to describe the operation of carriage. Part of the second operation in the above table consists of adding six from the third difference to the second difference upon which six already stood. If the units wheel of the second difference be watched as this operation is effected by the handle being moved backwards, it would be observed that the 6 successively becomes 7, 8, and 9, and that as it passes from 9 to 0 click is heard, this is the sound of a warning being given that a carriage is to take place to the tens wheel above; the unit wheel still continues its movement and 1 and lastly 2 appear upon it, and the wheel then rests whilst the stroke of the handle is completed. The return or forward stroke of the handle brings round the carrying arms which pick up any carriages of which warning may have been given, and the 0 upon the tens wheel becomes 1 and the addition is completed, six and six making twelve. If the action of the Machine is carefully watched as the successive operations described are being performed, it will be seen that a similar process takes place whenever a carriage occurs, the order of events being first a click or warning, followed in the next stroke of the handle by the picking up. When however the picking up of the carriage adds 1 to a wheel above already standing at 9, if the handle be moved slowly a warning of carriage will be heard when the carriage from the wheel below is being picked up, which warning will be followed almost immediately by the actual carriage upon the wheel above. If this wheel also happens to be a 9, a warning, followed by the carrying of 1 to the wheel next above the last, takes place almost directly after, and however many nines there may happen to be one above another by the time that the handle has reached the end of its stroke, warning will have been given and the carriage completed successively from below upwards.
CALCULATION OF TABLES HAVING NO CONSTANT DIFFERENCE. .
There are many Tables which have no constant difference, that is, however often the working out additional columns of differences is continued, a constant difference will never be arrived at; it will however be found that the greater the number of difference columns that is worked out, the nearer a constant difference is approached. The mode of calculation of such tables may be seen by taking a table of logarithms as an example. The logarithms of numbers are usually given to seven places, of figures, that being. sufficient for ordinary purposes. As there is no constant difference to a table of this kind, it becomes necessary to use such a number of difference columns as will ensure the logarithms being correct to seven places. The Difference Engine in order to make such a table would be set to calculate to twenty places, but would only print seven (or whatever number was required for the table in hand) making the printed result true to the last figure by a special contrivance. The Engine would also be set to stop itself as soon as it had completed a number of calculations as would be true to the last figure printed, this number having been ascertained by the operator beforehand: it would then ring a bell to draw attention to its need of a fresh difference, and throw itself out of gear so as to stop the work and prevent the possibility of any inattention on the part of the operator allowing an error to creep in.
CAUTION IN USING THE MACHINE
The handle should be moved steadily and brought home at the end of each stroke, when a ratchet will be heard to click. Should anything happen to interfere with the proper action of the machine, this click at the end of each stroke will not be heard, and the person operating should at once report it to the superintendent. Also if at any time there should be a difficulty in moving the handle, force must on no account be used but the circumstance should be reported that the cause may be inquired into, and the obstacle removed.
B. HERSCHEL BABBAGE.
1 Dorset Street,
28th September 1872
Cover
Reference
B. H. BABBAGE (1872), Babbage's Calculating Machine; or Difference Engine, Science and Art Department, South Kensington Museum, London (1872).
| Title | Babbage's Calculating Machine; Or Difference Engine: A Description of a Portion of this Machine Put Together in 1833 and Now Exhibited in the Science Collections for Teaching and Research, Victoria and Albert Museum |
| Author | Benjamin Herschel Babbage |
| Publisher | H.M. Stationery Office, 1872 |
| Length | 8 pages |
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