CALCULATING MACHINES. The evolution of num bering and of numerical calculation is embodied in that simple but great mechanical calculating instrument known as the abacus. (See ABACUS.) The efficiency of this instrument in expert hands has been recognized from early times, and it is still largely em ployed in India, China, Russia and Japan.
1617, John Napier of Merchiston published a description of his numbering rods, since known as "Napier's bones," for facilitat ing the multiplication of numbers. These were widely used during the 17th century, and various modifications have been made since they were introduced. The set shown in Pl. I., fig. 1, which be longed to Charles Babbage, is preserved in the Science Museum at South Kensington, where there are other sets, some of which have come from the Napier family. The strips bear the columns of numbers of the ordinary multiplication table. To multiply
by any number, the strips are selected from the case and arranged as shown. The "sevens" strip bears the numbers 7, 14, 21, 28, etc., but written in the form 0,i
etc. Horizontally oppo site 6 on the index to the left, the product 6 X 76S479 is indi cated. The rule is : For units, put down the lower right-hand figure, for "tens," etc., add diagonally in pairs and we get 4592874. If the multiplier consists of several figures, the product of each with the multiplicand is read off separately, written down as in ordinary multiplication and added up. Sir Samuel Morland in 1666 in vented a multiplying instrument, in which Napier's rods are re placed by rotatable disks, and the figures, instead of being in the two diagonal halves of squares, are arranged near the edge of the disk at opposite ends of diameters. For any given multiplicand, set by placing the appropriate disks on spindles, the partial pro ducts were at once indicated by means of a slot-marker, actuating the disks through a rack and pinions. Gaspard Schott in 1668 de scribed the cylindrical form of Napier's bones, which facilitated manipulation. A similar instrument, stated to have been the prop erty of John Napier, is preserved in the Science Museum. In 1885 M. Genaille made the further improvement of eliminating the slight but repeated mental effort required to add the pairs of adjacent figures before writing down the separate products. These ingenious devices, though in some cases embodying rudimentary mechanical arrangements, are not usually described as machines.
This, as the term is usually understood, was invented by Blaise Pascal in 1642. Pascal made a considerable number of machines, a few of which are pre served in the Conservatoire des Arts et Metiers at Paris. Figure wheels, each bearing the numbers o to 9, are mounted in a box and have parallel axes with a stylus or peg, a (P1. I., fig. 2) horizontal wheel at the front and immediately under the cover of the machine can be advanced one-tenth to nine-tenths of a complete turn, the same movement being transmitted by pin wheel gearing to the corresponding figure-wheel. The uppermost figure of each figure-wheel is seen through a sight-hole in the cover. During the movement of the figure-wheel from 9 to o, a "carry ing" device moves the next figure-wheel to the left through one tenth of a revolution. In machines made by Pascal for adding livres, sous and deniers, the wheels for sous and deniers were modified for recording o to 19 and o to 11 respectively. In 1666 Morland invented a compact little instrument for the same pur pose, measuring Sin. by 4in. by less than 4in. thick. It was oper ated by a stylus, but there was no tens-carrying device the num bers to be carried were registered on small counter-disks. Viscount Charles Mahon (afterwards the 3rd Earl Stanhope) in 1780 im proved Morland's instrument by providing a tens-carrying device. This acted simultaneously from the units wheel to those of higher denominations, and the addition of 1 to 999,999, for instance, was almost beyond the capacity of the instrument owing to the great force required. Dr. Roth in 1842 improved this type of stylus driven instrument, by arranging for the successive carrying of tens. Low-priced instruments of this general type are still being made in considerable numbers.
Multiplication is really repeated addition. For example, 7543 X
= ( 7543 + 7 543 + 7543 + 7543 ) + (
-I- 754300+754300+754300). Multiplication can therefore be per formed in a simple manner by all adding machines of the Pascal type, but the time occupied would be about the same as by the ordinary figuring method on paper. In the above example, for instance, there would be 52 separate operations by hand, each operation involving the placing of the stylus in its proper hole and by its means driving the figure-wheel through its proper arc.
Gottfried Wilhelm Leibniz, who in 1671 conceived the idea of a calculating machine which would perform multiplication by rapidly repeated addition. It was not until 1694 that his first complete machine was actually constructed. The figure-wheels on the fixed portion of the machine register the results of additions, rapidly repeated up to nine times of the multiplicand, which was set on a sliding portion, movable by steps into positions corre sponding to units, tens, hundreds, etc. This machine is still pre served in the Royal Library at Hanover, and examination has shown that the tens-transmission mechanism was never quite re liable in operation. A second machine was made in 1704, but this has disappeared.
a cylindrical wheel or drum having on a portion of its outer surface nine teeth of increasing length, from one to nine. This element is embodied in many subsequent machines which perform multi plication by repeated addition, and is in considerable use at the present time.
mathematicians and mechanists to evolve a satisfactory machine which could be made commercially. Among these machines may be mentioned those of Lepine (1725), Leupold (1727), Boistes sandeau (173o), Gersten (1735), Pereire (175o), Hahn
Mahon (Earl Stanhope) (1775 and 1777), and Muller. The chief difficulty, however, was the high degree of accuracy necessary in the construction of the details such as wheel teeth.
on a Commercial Scale.—This was that invented in 1820 by Charles Xavier Thomas, of Colmar in Alsace. This model forms a distinct type which has persisted up to the present day, with modifications and improvements in detail due to various makers. By 1865, 500 machines had been made, and i,000 more were made during the next 13 years. The machine is still made in Paris by Darras, the present successor of the original makers. The example shown in Pl. I., figs. 3, 4, was made about 1866.
concerned with setting, counting and recording respectively. These are arranged in order from front to back. Any number up to 999,999 may be set by moving the pointers in one or more of the six slots in the fixed coverplate, to the numbers o to 9 engraved on the cover plate to the left of each slot. The movement of any of these pointers slides a small pinion with ten teeth along a square axle, underneath and to the left of which is a Leibniz stepped wheel. This is driven from the main shaft by means of a bevel wheel, and the small pinion is thus rotated by as many teeth as the cylinder bears in the transverse plane corresponding to the digit set. This amount of rotation is transferred through one of a pair of bevel wheels carried on a sleeve on the same axis, to the "results" figure-wheel on the back row on the hinged plate. This plate also carries the figure-wheel recording the number of turns of the driving crank for each position of the hinged plate. Accord ing to whether the lever at the top left-hand corner of the fixed plate is set for "Addition and Multiplication" or "Subtraction and Division," one or other of the pair of bevel wheels is placed in gear with a bevel wheel underneath a "results" figure-wheel, which is thereby rotated anticlockwise in the direction o to 9, or clock wise respectively. The actual operation in multiplying 3,042 by 536, for example, may be performed as follows :—First lift the hinged plate, turn and release the two milled knobs so as to bring all the figure-wheels to show zero. Lower the hinged plate in its position to the extreme left. Set the number 3,042 on the four slots on the fixed plate. Set the lever on the left to "multiplica tion" and turn the handle, which can be turned only in a clockwise direction, six times. Lift the hinged plate, slide it one step to the right and lower it into position. Turn the handle three times. Step the plate again one point to the right and turn the handle five times. The product
will then appear on the top row, and the multiplier 536 on the next row of figures.
founded by Arthur Burkhardt, who commenced the manufacture of this type of machine, under the title of "Burkhardt Arith mometer." Amongst the machines of this form constructed by other firms are :-Saxonia (1895), Peerless (1904, the late key board models being known under the name Badenia), Gauss (1905; a small circular type, embodying a circular stepped plate in place of the Leibniz wheel), Archimedes (1906), Tim (1907, made with two slides under the name Unitas), Hermes (191 I), Record (1913), Rheinmetall (1924)-in Germany; Lay ton (1883, known also as the "Tate"), Edmondson (1885, a cir cular type)-in England; Graber (1905), Austria (c. 1906, known in Germany as the Austro-Germania), Bunzel-Delton (1908) in Austria; Fournier-Mang (1919)-in France; Kuhrt (1923) in Switzerland; Allen (1927) in the United States of America. Machines of the Odhner Type.-In 1875 Frank Stephen Baldwin patented a machine in which the Leibniz stepped wheel was replaced by a wheel from the periphery of which a variable number of teeth (I to 9) could be protruded. About the same time W. T. Odhner designed a machine embodying the same device. This type of machine has been made and developed ex tensively in Germany by Grimme Natalis and Company, since 1892, under the name "Brunsviga." By 1912, 20,000 machines of this type had been constructed by this firm alone. An early example made in 1892, is shown in Pl. I., figs. 5, 6.
Though the machine performs multiplication by repeated addi tion, as in the Thomas type, the use of the thin Odhner wheel instead of the Leibniz wheel, led to a more compact design. The Odhner wheels (nine in the example shown) fit very close together on the axle at the back. A setting lever, the end of which projects through a slot in the cylindrical portion of the cover plate, forms part of each wheel. If a lever is set against any figure (I to 9) of its slot, a corresponding number of teeth are made to project from its wheel. When the operating handle is turned, these teeth gear with small toothed wheels of the product register, which in turn gear with the number wheels in front. The product register is mounted on a longitudinally movable carriage arranged in front of the machine, which carries also a second counter for registering the multiplier in the case of multiplication and the quotient in division. For addition and multiplication, the handle is turned in a clockwise direction ; for subtraction and division, in the reverse direction, no change of gear being necessary as in machines of the Thomas type. The carriage is stepped to the right or left by pressing one or other of the projecting pieces in front. Zeroiza tion of the product and multiplier registers is effected by turning the butterfly nuts on the ends of the carriage through a complete revolution.
Under the original Odhner patents and since these have run out, this type of machine has been made under various names by many firms in different countries. Amongst these machines are the Brunsviga (1892, known in France as the "Rapide"), Monopol Duplex (1894), Berolina (19oI ), Triumphator (19o4), Thales 0910, Teetzmann (1912, known in England as "Colt's Calcu lator"), Lipsia (1914), Rema (1915), Hannovera (1921), Orga (1921), Monos (1923), Gauss (1923), Mira (1924), Hamann Manus (1925) in Germany; Dactyle (c. 19o5), Sanders (1912), Muldivo (19 24) in France; Marchant (191I) , Lehigh (1919) , Arrow (1921), Rapid (19 23) in the United States; Original Odhner (a model adapted for British currency is sold in England under the name "Lusid"), Facit (1918), Odhner Universal (1925), Mercur (1925), in Sweden; Demos (1923, recently sold in England under the name "Eos"), in Switzerland; Calco (1921) in Den mark ; and the Britannic (1922) in England.
The machine, which is made in various sizes and capacities, has been steadily modified and improved in details of construction up to the present time. In the latest model by the original German makers, which was issued at the beginning of 1927 under the name Nova Brunsviga (P1. I., fig. 7), there is a change of design. Amongst other new features is a device by which the result regis tered on the product dials can be instantly transferred to the setting levers. In a model previously introduced-the "Triplex" giving 20-figure results, the result register can be divided into two parts, enabling multiplication of two different numbers by the same multiplier to be carried out in one operation.
keyboard type of calculating machine, which originated and has been developed chiefly in the United States, may be divided into two distinct classes, key-driven and key-setting. In the former the energy necessary to drive the machine is provided by simply depressing the keys, without any auxiliary movements. The first key-driven adding machine which was patented in U.S.A. by D. D. Parmalee, in 185o, could add only a single column of digits at a time. Many others of this limited capacity were invented before 1887. In this year Dorr Eugene Felt patented his Comptometer, which was the first suc cessful key-driven multiple-order calculating machine. In the first models, each key had to be operated separately so as to ensure the proper carrying of the tens. In subsequent models many improvements were made which contributed to speed, ease and accuracy of operation. In the "Duplex" model introduced in 1903, simultaneous depression of keys in every column, without inter fering with the proper carrying of tens, became possible for the first time. This constituted a great advance, and an extremely high speed in operation was thereby attainable. Later, in order automatically to prevent the operator from overlooking any errors which might arise from imperfect operation, the "controlled-key" model was introduced. Interference guards at the side of the key tops prevented accidental depression of a key near the one being operated. If a key is not given its full downward stroke, the keys in all the other columns are immediately locked, and the numeral wheel in the column where the error is made shows a figure in the answer register standing out of alignment. The error can then be corrected by completing the unfinished stroke. By another auto matic block device, no key can be depressed again until it has completed its up stroke. In the most recent models (Pl. I., fig. 8), a short-pull zeroizing lever replaces the longer double-stroke lever, and at the beginning of a new calculation a clear register is indi cated to the operator by visible, audible and sense-of-touch signals.
1872 E. D. Barbour incorporated a printing device with an adding machine. Other inventions of this type were made by F. S. Baldwin (1875), H. Pottin (1883) and A. C. Ludlum (1888). The first really prac tical adding and listing machines were produced by D. E. Felt in 1889, and by W. S. Burroughs in 1892. Both these inventors had realized the wide field of work, in banking and accounting work generally, to which an efficient machine of this type could be applied, and it is largely due to their steady efforts towards modi fying, perfecting and advocating the adoption of such machines that the initial opposition to their use was gradually overcome. Up to the present time well over a hundred different models of the Burroughs machines have been designed and more than a million machines have been made.
A section of one of these machines is shown in Pl. I., fig. 9. The essential element is a lever pivoted near the middle, and carrying at one end a set of type figures for printing, and at the other end a segmental rack, with which an adding wheel is alternately in and out of gear. The depression of a numeral key brings the end of a stop wire into a position which limits the possible downward travel of the rack. When the operating handle is actuated, this downward travel, during which the adding wheel is out of gear, raises the other end of the lever so as to bring the proper type figure into position for printing. After the print ing is effected, the wheel comes into gear with the rack which during its ascent turns the adding wheel through a number of teeth equal to the number of the depressed key.
These machines may be grouped as follows: (a) Single counter adding machines, in which subtraction is performed by the comple mentary method. (b) Single counter adding machines, with direct subtraction. Where direct subtraction is provided for, the item is set up on the keyboard as if for addition, but the depression of a "subtraction" control key causes the item to be subtracted. The provision of this latter feature enlarges the scope of performance so as to include accounting work, where balances are obtained by the addition and subtraction of various amounts. (c) Duplex and multiple counter adding machines, without direct subtraction. The provision of two or more separate counting mechanisms en ables operations to be carried out together, which on single counter machines have to be performed separately. (d) Duplex and mul tiple counter adding machines, with direct subtraction. (e) Billing, accounting and book-keeping machines. These are designed to deal with such work as the preparation of invoices, reports, busi ness forms and allied documents, where typewriting is combined with arithmetical computations and recording. The first practical machine of this type was the Moon-Hopkins billing machine, in vented by Hubert Hopkins. This machine, which embodies a direct multiplication mechanism of the Bollee and Millionaire type, is the only example in this series equipped for direct multiplication.
In accordance with this broad classification, the groups into which the many machines made in the United States at the pres ent time would fall are indicated in the following list (the larger firms make both hand and electrically operated machines, in various capacities) : Add-Index (a), Allen-Wales (a, b, c), Barrett (a), Burroughs (a, b, c, d, e), Corona (a), Dalton (a, b, c, d, e), Elliott-Fisher (e), Ellis (a, c, d, e), Federal (a, b), Gardner (a, b, c, d), Hayes (a), National (e), Peters (a), Remington (e), Sundstrand (a, b, c, e), Underwood (e), Victor (a).
In 1916 the design and construction of the American type of machine was also taken up in Germany, where it is being consider ably developed. Amongst these machines, made in various styles and capacities, are the Continental (1916), Adma (1919), Goerz (1921), Astra (1922), Naumann (1922), Votam (1922), Tim Add (1923). Typewriters with adding mechanism are represented by the Urania-Vega (1920) and the Mercedes-Elektra (1924). There are also several adding mechanisms made for use in combi nation with a standard typewriter.
Of all the different types of machines em bodying adding mechanisms, the cash register, used in most retail stores, is the most familiar to the general public. Up to the present time some three million examples of the "National" cash register have been made since 1883 by the pioneer firm of Patterson Brothers in Dayton, Ohio. Standard cash registers of the same general design and principle of action are made by the American, Federal, Remington and other firms.
Though its original object was the prevention of dishonesty in retail stores, the machine has been developed so as to provide also, in many of its forms, an automatic record of cash trans actions, together with the issue of duplicate receipts to customers. In the most highly developed model of the "National," 29 indi vidual totals, corresponding with the sales of each clerk or each class of object, may be accumulated, as well as three grand totals. The machine has also been adapted for the purposes of accounting and book-keeping.
Since 1919 a combination type of cash register has been devel oped, in which a cash drawer is combined with an adding and listing machine. This possesses the advantage that the latter can be used independently when required for general purposes. When used in combination, provision is made for the automatic opening of the drawer as each item is dealt with.
Attempts to transform Napier's rods into mechanically operative form were made in the United States by Edmund D. Barbour (187 2) and Ramon Verea (1878), who patented their devices, which, however, never went beyond the first model stage. The first machine to perform multiplication successfully by a direct method, and not by re peated addition, was invented by Leon Bollee in 1887. The es sential feature of the mechanism is the multiplying piece, which consists of a series of tongued plates, representing in relief the ordinary multiplication table up to "nine times." Though ex cellent in action, few of these machines were made, chiefly because the inventor soon became fully occupied with his work in connec tion with automobilism.
The "Millionaire" machine (Pl. II., fig. 1), patented by Otto Steiger in 1893, was first made and marketed by Egli in 1899. It embodies the mechanical multiplication table invented by Bollee, by the operation of which only one turn of the driving handle is required for each figure of the multiplier. The carriage, or "re corder," being moved to the extreme right, the multiplication lever is set successively to one of its positions o to 9 in accordance with the figures of the multiplier, starting with the figure of the highest order. At each setting of the multiplication lever, the driving crank is turned once; and during the second quarter of each turn the carriage is automatically "stepped" to the left.
The mechanism includes nine parallel toothed racks, the ends of which are in line successively with either the tens or the units group of the tongues of a tongue plate. During each rotation of the crank the tongue plates are twice thrust against the ends of the racks; during the first thrust the tens tongues operate, and the units tongues during the second thrust. In the machines as now made the slot-markers have been replaced by a keyboard.
This machine (Pl. II., fig. 3) was designed by Ch. Hamann and marketed in 1910. Externally, in the general disposition of its main parts, it re sembles the Thomas machine, but internally there are many in novations. The Leibniz wheel is replaced by a series of ten paral lel racks, actuated from the driving handle by means of a connect ing rod and proportion lever. The carriage with its mechanism slides on rollers along guides in the frame of the machine, and can be "stepped" longitudinally without the lifting which is neces sary in machines of the Thomas type. The pushing of the car riage to the right stretches a spiral spring, the contraction of which supplies the force for "stepping," which is controlled by the de pression of a key. The left-hand member of the pair of levers at the top left-hand corner is placed at the bottom of its slot and the right-hand member at the top. The handle is now turned until it locks, which is the signal for reversal of the levers ; the handle is turned again until it locks, when the levers are again reversed. This cycle of operations is repeated until the carriage has returned to its normal position, when the quotient is given in the front row and the remainder in the middle row of figures. Multiplica tion is performed as with the Thomas machine, starting with the carriage to the extreme right.
In more recent examples the slot markers are replaced by a keyboard, and both multiplication and division can be performed automatically.
This machine (Pl. II., fig. 2), first introduced by Hans W. Egli in 1908, resembles the Thomas type. The operations of addition, multiplication and subtraction are performed in the usual way, but there is additional mechanism which enables division to be performed quite automatically, after setting the dividend and divisor. The ringing of a bell announces that the quotient and remainder are recorded.
This was introduced in 1911 by Jay R. Monroe and Frank S. Baldwin. It embodies a keyboard setting mechanism, combined with a slide at the back for "step ping," in the operations of multiplication and division.
The slide contains wheels, which are actuated by a crank handle turned in a clockwise direction for addition or multiplication, and in the reverse direction for subtraction and division, as in machines of the Odhner type. The wheels for adding are made in two co-axial parts, one with five equal teeth and the other with four, arranged in steps. The setting of a particular figure ad justs the two parts towards each other so as to enable 1, 2, . . . 9 teeth to gear with the counting wheel when the handle is turned. Division is by repeated subtraction in the ordinary way, but is rendered semi-automatic by the ringing of a bell when each "won't go" stage is reached; the handle is then turned once forward, and the slide is moved one step.
In the "full automatic" motor-driven model introduced re cently, division is quite automatic, and multiplication is automatic to the extent that only successive depression of keys on the secondary keyboard is required, corresponding with each digit of the multiplier.
In connection with the U.S. Census Bureau, an automatic system (known as the Hollerith system) was invented for dealing analytically and sta tistically with the enormous mass of information obtained. This system, which was also applied to the results of the British census of 1911, has been modified and developed to meet the needs of large commercial firms. By this system, many operations which, if performed by the ordinary mental and manual methods, would be economically impracticable, are carried out quickly and ac curately by automatic machines.
The basis of the system is a Jacquard card, in which each fact or item is indicated by a hole punched in a certain position. These cards, which are printed with vertical columns of figures from o to 9 or o to 12, are prepared by means of a special punching machine; as many holes may be punched as are required to regis ter every detail of each item. The cards are of two forms, dual and single ; the former bearing written information corresponding with the punched-hole record, the latter bearing only punched hole records corresponding with information recorded separately in some other form.
In the Hollerith sorting and tabulating machines the principle of electrical contacts is adopted, circuits being closed when steel brushes pass the holes in the cards. The controls are set by means of wires plugged into a switchboard in such a manner as to connect each punched position on the cards with a column in the counting register, or a sector of the printing mechanism. Changes in the circuit arrangements may be made by the operator in accordance with the particular sorting or tabulating which is being dealt with.
The vertical sorting machine will sort at the rate of about 25o cards a minute ; in a later horizontal model the rate is as high as
to 400 a minute. The sorting and tabulating machines are electrically driven and the latter are of two main types, in one of which the results are indicated by counters, while the other gives also a printed record.
cards are used in a similar manner in the Powers tabulating and sorting machines, which, however, function me chanically by means of pins passing through the holes in the cards.
towards the completely automatic calculating machine for rapidly performing with equal facility all the ordinary opera tions of addition, subtraction, multiplication and division once the numbers have been set, is at an interesting stage. The application of the electric motor drive to many machines, replacing the turn ing of a handle, has been an important development during the last 20 years. From the advantage which it gives of greater rapid ity in setting the figures to be operated on, the keyboard is gradu ally replacing the slideboard in machines of the Thomas, Odhner and other types designed primarily for multiplication and division —speeding up also the operations of addition and subtraction on such machines. In the Comptometer, and in the other key-driven non-printing adding machines of the same type introduced in more recent years (Burroughs Calculator, Mechanical Accountant) ad dition is extremely rapid and quite automatic. Multiplication and division are also performed with great rapidity, but in the absence of a slide, greater mental strain is involved and greater dexterity in operation is required. In key-set motor-driven adding and list ing machines of the Burroughs and allied types, automatic control is made as complete as possible by the provision of various auxil iary keys (add, non-add, subtract, repeat, sub-total, total, non print, etc.).
Automatic multiplication was almost completely attained in the Bollee and Millionaire machines, but the figures of the multiplier have to be applied successively. The same has resulted in recent years, so far as automatic control is concerned, in certain machines (Fournier-Mang, Monroe, Marchant, Ensign (Pl. II., fig. 5), Peer less, Record, Kuhrt) which perform multiplication by repeated addition. By depressing one of an additional row of keys (usually termed a secondary or auxiliary keyboard) to the left or right of the main keyboard, the number set on the main keyboard is rapidly added one to nine times according to the number on the key depressed.
In 1910-1912 Alexander Rechnitzer of Vienna patented a ma chine in which after the two numbers to be multiplied or divided had been set up, operation was performed automatically. In 192o Torres exhibited and designed an electrical arrangement which provided complete automatic control of the operations of multi plication or division performed by a machine of the Thomas type, after the numbers had been set up on a typewriter. Neither of these machines has taken commercial form. The only machine on the market which performs both multiplication and division en tirely automatically, once the figures are set, is the Mercedes Euklid.
II., fig. 6), "United accounting machine," made in St. Louis, Michigan, U.S.A., is being marketed which claims to perform multiplication automatically. The keyboard is in two equal portions, the multiplicand being set on the left and the multiplier on the right. On depressing the motor-bar, the result is both obtained and printed in three seconds. The carrying of tens in this machine is not by one unit at a time, but any number of units up to nine may be carried simultaneously from one column to the other.
In the "Barbel" system, developed by M. Barr and R. A. Bell, the rapid counting of small steel balls introduced a new feature into the design of electrically-controlled calculating machines. Though machines with this particular feature have not yet reached the commercial stage, it would be unwise to assume that develop ment in the future will be confined to improvement and elabora tion of existing types of machine. Invention and construction in the calculating machine industry are very much alive at the pres ent time, especially in the United States and in Germany.
Charles Bab bage (1792-1871) conceived the idea of a calculating machine of a different type from those previously described. The object of the machine was to calculate and print mathematical tables such as tables of logarithms. The machine worked on the method of differences and was known as a "difference engine." The principle underlying the method may be understood by taking a table such as the table of cubes of successive numbers 1, 2, 3, etc., and sub tracting each tabular number from the following one, obtaining another column of figures, called the first order of differences. Treating the numbers in this column in the same way, a column of second differences is obtained; on differencing a third time (in this particular case) a constant difference (6) is obtained. By re versal of the process, knowing the constant third difference and the numbers shown at the top of the columns, it is possible to obtain all the rest of the numbers by simple addition. It is the function of a difference engine to effect these additions succes sively in the proper order so as to obtain the desired series of tab ular numbers automatically, once the initial numbers are set.
Babbage's difference engine was commenced in 1823 by au thority and at the cost of the Government. The work was sus pended in 1833, and in 1842 the Government decided to abandon the machine on the ground of the estimated expense of its comple tion. The whole engine was intended to have 20 places of figures and six orders of differences. In 1833 a large part of the engine had been made and a small portion had been assembled in order to show the action of the mechanism.
From 1833 Babbage devoted his energies and resources to the design and construction of an "analytical engine," the object of which was to evaluate automatically any mathematical formula. Features of the difference engine were to be embodied in this new engine and the various operations were to be controlled by punched cards of the Jacquard type. The scheme proved to be too ambi tious, and the machine was left unfinished when Babbage died. Portions of the machine, with all his notes, drawings and notations are preserved in the Science Museum.
From 1834 to 1853 George Scheutz of Stockholm and his son Edward designed another difference engine, the first complete ex ample being constructed by C. W. Bergstrom. It was exhibited in operation in Paris and London, and finally purchased for the Dudley Observatory, Albany, U.S.A. A second example was made by Bryan Donkin in 1858 for the General Register Office, Somerset House, and used during the next few years for compu tations in connection with the preparation of the English life tables. Dr. Farr, the author of this book, states: "This volume is the result; and thus—if I may use the expression—the soul of the machine is exhibited in a series of tables which are submitted to the criticism of the consummate judges of this kind of work in England and in the world." Other engines of this type were designed and made by Martin Wiberg in Sweden, and G. B. Grant in the United States; others were designed by Leon Bollee in France, and Percy E. Ludgate in Ireland, but were not con structed.
In all the machines previously described the arithmetical results obtained are correct to the last figure indicated on the dials. There are many calculations in engineering, physics, etc., where an approximate result, rapidly obtained, is frequently desirable, and the logarithmic slide rule provides for this require ment in a very efficient manner.
The invention of logarithms in 1614 by John Napier of Mer chiston, and the computation and publication of tables of loga rithms, made it possible to effect multiplication and division by the more simple operations of addition and subtraction. (See LOGARITHMS.) In 1620 Edmund Gunter plotted logarithms on a two-foot straight line. With such scales, multiplication and divi sion were performed by addition and subtraction of lengths by a pair of dividers.
William Oughtred, according to his own statement (1633) con structed and used as early as 1621, two of these Gunter's lines sliding by each other so as to do away with the need for dividers. The lines were used in both the straight and circular forms. In the former the scales were held against one another by the hands; in the latter, dividers were replaced by an "opening index"—really a pair of dividers fixed centrally on the circular scale. Richard Delamain in 1630 gave the first published description of a circular slide rule, both in the flat and cylindrical forms. Thomas Brown introduced the spiral logarithmic line in
The first known slide rule in which the slide worked between parts of a fixed stock was made by Robert Bissaker in
Others were due to the enterprise of Seth Partridge (16S7), Henry Coggeshall (1677)—a slide in a 2 f t. folding rule adapted to timber measure, and Thomas Everard (1683) for gauging purposes. The one for gauging purposes, approximated in dimensions and arrange ment of scales, to the present-day loin. slide rule, and many thousands were made and sold during the period 1683 to 1705. The usefulness of the slide rule for rapid calculation became in creasingly recognized, especially in England, during the 18th century, and the instrument was made in considerable numbers, with slight modifications.
Improvements in the direction of increased accuracy in gradu ation, etc., were initiated by Boulton and Watt from about
in connection with calculations in the design of steam engines at their works at Soho, Birmingham. The rule evolved, which was the first designed for engineers, became known as the "Soho" rule. It was made by W. and S. Jones, Rooker, Bate and Nairne and Blunt.
The runner or cursor, though its advantages had been pointed out by Robertson (1778) and Nicholson (1787), was not added by instrument makers until Tavernier-Gravet introduced the Mann heim type of slide rule in 1850. This slide rule was much used in France and since about 1880 was imported in large numbers into other countries. Up to this period the rule had been constructed usually of boxwood and occasionally of brass or ivory, but a great improvement was introduced in 1886 by Dennert and Pape in Germany by dividing the scales on white celluloid, which gave a much greater distinctness in reading. This material is now almost universally adopted, and the slide rule made by such firms as Nestler, Faber (Germany), Keuffel and Esser (U.S.A.), Tavernier Gravet (France) and Davis (England) attain a high degree of perfection.
The disposition of the scales in the Mannheim rule (P1. II., fig. 7a) is the arrangement still adopted in the great majority of rules made at the present time. The A and B scales are double lines as in the Everard, Coggeshall and Soho rules, but the C and D scales are single lines like the D scale of the Soho rule. At the back of the slider are scales giving the sine and tangent scales and a scale of equal parts. Applied in conjunction with the scales on the face of the stock these are used for reading the values of sines, tangents and logarithms respectively, and in computations involving these factors.
To secure an additional significant figure, the length of the logarithmic scale has to be increased ten times. To keep the dimen sions of a slide rule bearing such a scale within reasonable limits four different types of design have been evolved :—(a) The flat spiral form. Examples have been made at various times since the invention of the slide rule, but this type has never been much used. (b) The cylindrical helix. Fuller's slide rule, originally de signed in 1878, has been in considerable use up to the present time. Amongst other rules of this type are the Otis King calcu lator (1922) and the R.H.S. calculator. (c) The flat gridiron type, in which the scale is cut up into strips mounted parallel to each other; examples designed by Everett (1866), Scherer (1892), Hannyington, Rieger (1920), Gladstone and others have been in considerable use. (d) The cylindrical gridiron type. The parallel strips are arranged longitudinally on the surface of a cylinder; examples made and in use at the present time are those of E. Thacher (1881) and the Rouleau calculator.
In 1815 Dr. Peter M. Roget invented his "log-log" slide rule for performing the involution and evolution of numbers. The fixed scale, instead of being divided logarithmically, is divided into lengths which are proportional to the logarithm of the logarithm of the numbers indicated on the scale; the sliding scale is divided logarithmically. By employing this new method of graduation, the value of any expression of the form yx may be obtained by the same mechanical process as that by which yx is obtained in the ordinary slide rule. Since log (yx) =x log y, log (log yx) =log x+log (log y). Hence if division 1 on the slider be set opposite y on the fixed scale, the value of yx will be read off on the fixed scale opposite x on the slider. All problems such as those of compound interest, increase of population, etc., are solved in this manner by mere inspection. The lower log-log scale on the stock is numbered 1.0024 to 1.2 5, and the upper log-log scale is numbered 1.25 to
The log-log scale was reinvented and applied to the slide rule by Captain J. H. Thomson in 1881, and by Prof. John Perry in 1902. In Perry's log-log slide rule (P1. II., fig. 7c) there are two log-log lines, one above and the other below the four scales of the ordinary loin. slide rule. The upper scale reads (left to right) from 1.1 to io,000, and the lower one the reverse way from o•000 1 to 0.91. The scales, which are reciprocal with each other, are used in conjunction with the B scale on the slider. Lieut.-Col. H. G. Dunlop and C. S. Jackson arranged the log-log lines on a spare slider (Pl. II., fig. 7b) used with the D scale of an ordinary slide rule.
P. Babbage, Babbage's Calculating Engines Bibliography.-H. P. Babbage, Babbage's Calculating Engines (1889) , brings together information from various sources, relating to the calculating machines of Charles Babbage; R. Mehmke, "Numer isches Rechnen," in Encyklopiidie der Mathematischen Wissenschaften, vol. i., part 2 ; pp.
; M. d'Ocagne, Le Calcul Simplifie, 2nd ed. (1905) ; F. Cajori, A History of the Loga rithmic Slide Rule, with full bibliography from 162o-1909 (1909) ; L. Jacob, Le Calcul Mecanique (191I) ; E. M. Horsburgh, Handbook of the Exhibition at the Napier Tercentenary Celebration (1914) ; Bulletin de la Societe d'Encouragement pour l'Industrie Nationale (Sept.—Oct. 192o), vol. cxx. No. 5, commemorating the centenary of the invention of the Thomas de Colmar calculating machine, and including papers by M. d'Ocagne, Paul Toulon, M. L. Torres y Quevedo ; bibliographies by L. Malassis, E. Lemaire and R. Grelet ; reprints of important papers in previous numbers (1822-95) relating to the machines of Thomas de Colmar, Bollee, etc. ; illustrated catalogue of the early and modern machines in the exhibition held in Paris (June 1920) in connection with the centenary. J. A. V. Turck, Origin of Modern Calculating Machines (Chicago, 1921) , deals more particularly with the evolution and development of keyboard adding machines of the Comptometer and Burroughs types; E. M. Horsburgh, "Calculat ing Machines," Glazebrook's Dictionary of Applied Physics (1923) ; E. Martin, Die Rechenmaschinen and ihre Entwicklungsgeschichte (Pappenheim, 1925) (all types of calculating machines, described in order of date of introduction) ; D. Baxandall, Mathematics I., Calculat ing Machines and Instruments (H. M. Stationery Office, 1926) ; a catalogue of the calculating machines and instruments exhibited in the Science Museum, South Kensington, with descriptive and historical notes and illustrations; L. J. Comrie, "On the Application of the Brunsviga-Dupla Calculating Machine to double Summation with finite Differences," Monthly Notices of the Royal Astronom. Soc., vol. lxxxviii. (1928), pp.