Calculating Machines. From early times the necessity of aids to calculation has been felt and has been met by simple contrivances, such as the Abacus, Shwanpan, Napier's Bones, the Vernier, Sliding Rules, and so on. Many of these are very interesting, but they cannot properly be classed with calculating machines. The earliest with which we are acquainted is one invented by Pascal about 1642. The addition of each place of figures was performed separately, and subtraction was effected by the addition of the numerical complements. It is not likely that even Pascal himself found this machine of any practical use. In 1666 Sir S. Moreland invented a machine, about 3 by inches in size, for adding sums of money. In this machine, as in that of Pascal, the addition of each place or order was made separately by hand. Each place had a pair of wheels unconnected with its neighbours, and the carriages were transferred by hand. The multiplication machine of the same inventor was really nothing more than Napier's Bones on circles instead of strips. Some years later, Leibnitz conceived the idea of an enlarged machine which seems to have been designed for important astronomical calculations, the necessity for which had arisen from the discoveries of Galileo, Kepler, and the other astronomers of that century. He spent many years of his life, and more than 24,000 thalers (£3600) on this project, but all that remains is a model still preserved at Göttingen. In 1775 Viscount Mahon, afterwards Earl Stanhope, invented a machine which performs the four rules of arithmetic. It is about 6 by 8 by 18 inches in size. The operator, using both hands, pushes a frame containing twelve prisms of ten sides each, mounted side by side, to and fro in the line of their axes, performing in one direction addition, and in the other subtraction. Each face of the prism has a rack, which engages a toothed wheel, and turns it as many teeth as there are teeth in the rack—0, 1, 2, &c. as required. There is a simple hand adjustment which works like the 'points' of a railway, and shunts the prisms out of the way on each return motion, during which the carriage is done. The set of prisms can be moved laterally for stepping. Multiplication and division are done by successive addition and subtraction. In 1777 a second machine on the same lines was constructed. In 1779 a German named Hahn invented a circular machine; followed in 1784 by another by Müller, likewise circular and worked by a handle at the centre. Subtraction was not done directly, but the arithmetical complement was read off instead of the sum. Stepping was done by shifting the upper plate, multiplication and division by successive addition and subtraction. It was about 12 inches outside diameter.
The next machine invented was one of a very different type. About 1812 Charles Babbage (q.v.), then a student at Cambridge, conceived the idea that logarithmic, trigonometric, and other tables could be calculated by the method of 'differences' by a machine capable of performing only simple addition. An immense range of nautical and astronomical tables lies within these limits, and can be produced by calculating the first few differences, and setting them in the machine. About 1822 he had made a small trial piece, and soon after undertook to superintend, without payment, the construction for the government of a machine to calculate and print such tables. It would have been, when completed, about 6 feet high by 3 broad and 1 deep, and would have had six columns of differences of eighteen or twenty places each. It was never completed, and the design was abandoned in 1842. It is enough here to say that the engagement was to the inventor a disaster, harassing him continually through twenty years; and to government all that remains is a fragment of a beautiful machine which does its work with unerring accuracy, but is useless. It stands in the South Kensington Museum.
About 1850 M. Thomas of Colmar produced a calculating machine of a high degree of excellence, which performs the four rules of arithmetic with surprising speed, and is in extensive use. In this machine, instead of the prism with racks, as in Stanhope's machine, there is a cylinder, on the different sections of which are successively 0, 1, 2, 3, &c. teeth; with each cylinder is a toothed wheel sliding on its axis, and this being brought by a finger-knob opposite the proper section of the cylinder, 0, 1, 2, 3, &c., as required, is added. These cylinders are all moved simultaneously in one direction. The carriage is made afterwards successively. One multiple is produced by each turn of the handle. The stepping is done by raising a long frame or lid on hinges, moving it down one place and letting it fall into the new position. There are very beautiful contrivances for reversing the motion for subtraction, for destroying the momentum of the moving parts, and for locking them, which can only be alluded to here; unfortunately, the whole is complicated and somewhat delicate. It requires careful handling, as indeed do all such machines.
In 1883 Mr J. Edmonson of Halifax patented a circular machine on the general lines of Thomas', but modified to suit the circular arrangement. Besides the beauty and ingenuity of its contrivance, it has several considerable recommendations: the computer is able to deal with one result after another, without the necessity of repeated transfers, every one of which, done by hand, is a fertile source of error. Another machine by Mr Tate follows closely the details of Thomas' machine; but a multitude of minor parts are discarded, and friction is relied on to destroy the momentum of the moving parts the instant the driving power ceases.
The analytical engine of Charles Babbage does not exist except on paper, but is too remarkable to be passed over. There are about 400 detailed drawings to scale; many volumes of notes and rough sketches and elaborate notations. In 1833, when the fragment of the difference engine was put together, Babbage found that several surprising results which had not been anticipated could be easily produced by causing the table to influence the last difference in various ways. Some of these results are used as illustrations in his book, the Ninth Bridgewater Treatise (Lond. 1838). He was thus led to the idea of placing the axes round a large central wheel. He called this arrangement 'the engine eating its own tail.' Then came the thought of governing the engine by quite independent means, and the general idea of the analytical engine was soon complete. Briefly, it may be described as a machine to calculate the numerical value, or values, of any formula, or function, of which the mathematician can indicate the method of solution. It was to be governed by cards very similar to the cards used in Jacquard's loom; these having been supplied would guide the engine to make the right operation, at the right time, on the right quantities. Each function would require its own set of cards, which would make the engine special for the development of that particular function, with any constants that might at any time be desired. It was to be absolutely automatic, the slave of the mathematician, and relieving him of the drudgery of computing. It was of course to print the results.