CALCULATING MACHINES. Instruments for the mechanical performance of numerical calculations, have in modern times come into everincreasing use, not merely for dealing with large masses of figures in banks, insurance offices, &c., but also, as cash registers, for use on the counters of retail shops. They may be classified as follows:—(i.) Addition machines; the first invented by Blaise Pascal (1642). (ii.) Addition machines modified to facilitate multiplication; the first by G. W. Leibnitz (1671). (iii:) True multiplication machines; Leon Bolles (1888), Steiger (1894). (iv.) Difference machines; Johann Helfrich von Muller (1786), Charles Babbage (1822). (v.)Analytical machines; Babbage (1834). The number of distinct machines of the first three kinds is remarkable and is being constantly added to, old machines being improved and new ones invented; Professor R. Mehmke has counted over eighty distinct machines of this type. The fullest published account of the subject is given by Mehmke in the Encyclopadie der mathematischen Wissenschaften, article " Numerisches Rechnen," vol. i., Heft 6 (1901). It contains historical notes and full references. Walther von Dyck's Catalogue also contains descriptions of various machines. We shall confine ourselves to explaining the principles of some leading types, without giving an exact description of any particular one.
Practically all calculating machines contain a " counting work," a series of " figure disks " consisting in the original form. of horizontal circular disks (fig. I), on which the figures o, 1, 2, to 9 are marked. Each
disk can turn about its vertical axis, and is covered by a fixed plate with a hole or " window " in it through which one figure can be seen. On turning the disk through onetenth of a revolution this figure will be changed into the next higher or lower. Such turning may be called a " step," positive if the next higher and negative if the next lower figure
appears. Each positive step therefore adds one unit Add~tine
mach/es.
to the figure under the window, while two steps add
two, and so on. If a series, say six, of such figure disks be placed side by side, their windows lying in a row, then any number of six places can be made to appear, for instance 000373. In order to add 6425 to this number, the disks, counting from right to left, have to be turned 5, 2, 4 and 6 steps respectively. If this is done the sum 006798 will appear. In case the sum of the two figures at any disk is greater than 9, if for instance the last figure to be added is 8 instead of 5, the sum for this disk is 11 and the T only will appear. Hence an arrangement for " carrying " has to be introduced. This may be done as follows. The axis of a figure disk contains a wheel with ten teeth. Each figure disk has, besides, one long tooth which when its o passes the window turns the next wheel to the left, one tooth forward, and hence the figure disk one step. The actual mechanism is not quite so simple, because the long teeth as described would gear also into the wheel to the right, and besides would interfere with each other. They must therefore be replaced by a somewhat more complicated arrangement, which has been done in various ways not necessary to describe more fully. On the way in which this is done, however, depends to a great extent the durability and trustworthiness of any arithmometer; in fact, it is often its weakest point. If to the series of figure disks arrangements are added for turning each disk through a required number of steps,
scapolites (qq.v.).
Detection and Estimation.—Most calcium compounds, especially when moistened with hydrochloric acid, impart an orangered colour
we have an addition machine, essentially of Pascal's type. In it
each disk had to be turned by hand. This operation has been
simplified in various ways by mechanical means. For pure
addition machines keyboards have been added, say for each disk
nine keys marked 1 to 9. On pressing the key marked 6 the disk
turns six steps and so on. These have been introduced by
Stettner (1882), Max Mayer (1887), and in the comptometer by
Dorr Z. Felt of Chicago. In the comptograph by Felt and also
in " Burrough's Registering Accountant " the result is printed.
These machines can be used for multiplication, as repeated
addition, but the process is laborious, depending for rapid execu
tion essentially on the skill of the operator.' To adapt
Modified an addition machine, as described, to rapid multipliaddlelon
meehlaee. cation the turnings of the separate figure disks are
replaced by one motion, commonly the turning of a handle. As, however, the different disks have to be turned through different steps, a contrivance has to be inserted which can be " set " in such a way that by one turn of the handle each disk is moved through a number of steps equal to the number of units which is to be added on that disk. This may be done by making each of the figure disks receive on its axis a tentoothed wheel, called hereafter the Awheel, which is acted on either directly or indirectly by another wheel (called the Bwheel) in which the number of teeth can be varied from o to 9. This variation of the teeth has been effected ih different ways. Theoretically the simplest seems to be to have on the Bwheel nine teeth which can be drawn back into the body of the wheel, so that at will any number from o to 9 can be made to project. This idea, previously mentioned by Leibnitz, has been realized by Bohdner in the " Brunsviga." Another way, also due to Leibnitz, consists in inserting between the axis of the handle bar and the Awheel a " stepped " cylinder. This may be considered as being made up of ten wheels large enough to contain about twenty teeth each; but most of these teeth are cut away so that these wheels retain in succession 9, 8, . . . 1, o teeth. If these are made as one piece they form a cylinder with teeth of lengths from 9, 8 . times the length of a tooth on a single wheel.
In the diagrammatic vertical section of such a machine (fig. 2) FF is a figure disk with a conical wheel A cn its axis. In the covering plate HK is the window W. A stepped cylinder is shown at B. The axis Z, which runs along the whole machine. is turned by a handle, and itself turns the cylinder B by aid of conical wheels. Above this cylinder lies an axis EE with square section along which a wheel D can be moved. The same axis carries at E' a pair of conical
L , r\ r
B
Vii. J i i w i i w, ~~ r i i e t Yi e t FIG. 2.
wheels C and C', which can also slide on the axis so that either can be made to drive the Awheel. The covering plate MK has a slot above the axis EE allowing a rod LL' to be moved by aid of a button L, carrying the wheel D with it. Along the slot is a scale of numbers o t 2 . 9 corresponding with the number of teeth on the cylinder B, with which the wheel D will gear in any given position. A series of such slots is shown in the top middle part of Steiger's machine (fig. 3). Let now the handle driving the axis Z be turned once round, the button being set to 4. Then four teeth of the Bwheel will turn D and with it the Awheel, and consequently the figure disk wi!l be moved four steps. These steps will be positive or forward if the wheel C gears in A, and consequently four will be added to the figure showing at the window W. But if the wheels CC' are moved to the right. C' will gear with A moving backwards, with
'For a fuller description of the manner in which a mere addition machine can be used for multiplication and division, and even for the extraction of square roots, see an article by C. V. Boys in Nature, ttth July 1701.the result that four is subtracted at the window. This motion of all the wheels C is done simultaneously by the push of a lever which appears at the top plate of the machine, its two positions being marked " addition " and " subtraction." The Bwheel, are in fixed positions below the plate MK. Level with this, but separate, is the plate Kid with the window. On it the figure disks are mounted.
This plate is hinged at the back at H and can be lifted up, thereby throwing the Awheels out of gear. When thus raised the figure gure disks can be set to any figures; at the same time it can slide to and fro so that an Awheel can be put in gear with any C.wheel forming with it one " element." The number of these varies with the size of the machine. Suppose there are six Bwheels and twelve figure disks. Let these be all set to zero with the exception of the last four to the right, these showing 1 4 3 2, and let these be placed opposite the last Bwheels to the right. If now the buttons belonging to the latter be set to 3 z 5 6, then on turning the Bwheels all once round the latter figures will be added to the former, thus showing 4 6 8 8 at the windows. By aid of the axis Z, this turning of the Bwheels is performed simultaneously by the movement of one handle. We have thus an addition machine. If it be required to multiply a number, say 725, by any number up to six figures, say 357, the buttons are set to the figures 725, the windows all showing zero. The handle is then turned, 725 appears at the windows, and successive turns add this number to the first. Hence seven turns show the product seven times 725. Now the plate with the Awheels is lifted and moved one step to the right, then lowered and the handle turned five times, thus adding fifty times 725 to the product obtained. Finally, by moving the plate again, and turning the handle three times, the required product is obtained. If the machine has six Bwheels and twelve disks the product of two sixfigure numbers can be obtained. Division is performed by repeated subtraction. The lever regulating the Cwheel is set to subtraction, producing negative steps at the disks. The dividend is set up at the windows and the divisor at the buttons. Each turn of the handle subtracts the divisor once. To count the number of turns of the handle a second set of windows is arranged with number disks below. These have no carrying arrangement, but one is turned one step for each turn of the hand.ie. The machine described is essentially that of Thomas of Colmar, which was the first that came into practical use. Of earlier machines those of Leibnitz, Muller (t7A2), and Hahn (1809) deserve to be mentioned (see Dyck, Catalogue). Thomas's machine has had many imitations, both in England and on the Continent, with more or less important alterations. Joseph Edmondson of Halifax has given it a circular form, which has many advantages.
The accuracy and durability of any machine depend to a great extent on the manner in which the carrying mechanism is constructed. Besides, no wheel must be capable of moving in any other way than that required; hence every part must be locked and be released only when required to move. Further, any disk must carry to the next only after the carrying to itself has been completed. If all were to carry at the same time a considerable force would be required to turn the handle, and serious strains would be introduced. It is for this reason that the Bwheels or cylinders have the greater part of the circumference free from teeth. Again, the carrying acts generally as in the machine described, in one sense only, and this involves that the handle be turned always in the same direction. Subtraction therefore cannot be done by turning it in the opposite way, hence the two wheels C and C' are introduced. These are moved all at once by one lever acting on a bar shown at R in section (fig. 2).
In the Brunsviga, the figure disks are all mounted on a common horizontal axis, the figures being placed on the rim. On the side of each disk and rigidly connected with it lies its Awheel with which it can turn independent of the others. The Bwheels, all fixed on another horizontal axis, gear directly on the Awheels. By an ingenious contrivance the teeth are made to appear from out of the rim to any desired number. The carrying mechanism, too, is different, and so arranged that the handle can be turned either way, no special setting being required for subtraction or division. It is extremely handy, taking up much less room than the others. Professor Eduard Selling of Wurzburg has invented an altogether different machi,te, which has been made by Max Ott, of Munich. The Bwheels are replaced by lazytongs. To the joints of these the ends of racks are pinned; and as they are stretched out the racks are moved forward o to 9 steps, according to the joints they are pinned to. The racks gear directly in the Awheels, and the figures are placed on cylinders as in the Brunsviga. The carrying is done continuously by a train of epicycloidal wheels. The working is thus rendered very smooth, without the jerks which the ordinary carrying tooth produces; but the arrangement has the disadvantage that the resulting figures do not appear in a straight line, a figure followed by a 5, for instance, being already carried half a step forward. This is not a serious matter in the hands of a mathematician or an operator using the machine constantly, but it is serious for casual work. Anyhow, it has prevented the machine from being a commercial success, and it is not any longer made. For ease and rapidity of working it surpasses aft others. Since the lazytongs allow of an extension equivalent to five turnings of the handle, if the multiplier is 5 or under, one push forward will do the
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974
same as five (or less) turns of the handle, and more than two pushes are never required.
The SteigerEgli machine is a multiplication machine, of which fig. 3 gives a picture as it appears to the manipulator. The lower Multi part of the figure contains, under the covering plate, a p&ation carriage with two rows of windows for the figures marked machines. i and gg. On pressing down the button W the carriage
can be moved to right or left. Under each window is a figure disk, as in the Thomas machine. The upper part has three
sections. The one to the right contains the handle K for working the machine, and a button U for setting the machine for addition, multiplication, division, or subtraction. In the middle section a number of parallel slots are seen, with indices which can each be set to one of the numbers o to 9. Below each slot, and parallel to it, lies a shaft of square section on which a toothed wheel, the Awheel, slides to and fro with the index in the slot. Below these wheels again lie 9 toothed racks at right angles to the slots. By setting the index in any slot the wheel below it comes into gear with one of these racks. On moving the rack, the wheels turn their shafts and the figure disks gg opposite to them. The dimensions are such that a motion of a rack through I cm. turns the figure disk through one " step " or adds 1 to the figure under the window. The racks are moved by an arrangement contained in the section to the left of the slots. There is a vertical plate called the multiplication table block, or more shortly, the block. From it project rows of horizontal rods of lengths varying from o to 9 centimetres. If one of these rows is brought opposite the row of racks and then pushed forward to the right through 9 cm., each rack will move and add to its figure disk a number of units equal to the number of centimetres of the rod which operates on it. The block has a square face divided into a hundred squares. Looking at its face from the right—i.e. from the side where the racks lie—suppose the horizontal rows of these squares numbered from o to 9, beginning at the top, and the columns numbered similarly, the o being to the right; then the multiplication table for numbers o to 9 can be placed on these squares. The row 7 will therefore contain the numbers 63, 56, . 7, 0. Instead of these numbers, each square receives two " rods " perpendicular to the plate, which may be called the unitsrod and the tensrod. Instead of the number 63 we have thus a tensrod 6 cm. and a unitsrod 3 cm. long. By aid of a lever H the block can be raised or lowered so that any row of the block comes to the level of the racks, the unitsrods being opposite the ends of the racks.
The action of the machine will be understood by considering an example. Let it be required to form the product 7 times 385. The indices of three consecutive slots are set to the numbers 3, 8, 5 respectively. Let the windows gg opposite these slots be called a, b, c. Then to the figures shown at these windows we have to add 21, 56, 35 respectively. This is the same thing as adding first the number 165, formed by the units of each place, and next 2530 corresponding to the tens; or again, as adding first 165, and then moving the carriage one step to the right, and adding 253. The first is done by moving the block with the unitsrods opposite the racks forward. The racks are then put out of gear, and together with the block brought back to their normal position; the block is moved sideways to bring the tensrods opposite the racks, and again moved forward, adding the tens, the carriage having also been moved forward as required. This complicated movement, together with the necessary carrying, is actually performed by one turn of the handle. During the first quarterturn the block moves forward, the unitsrods coming into operation. During the second quarterturn the carriage is put out of gear, and moved one step to the right while the necessary carrying is performed ; at the same time the block and the racks are moved back, and the block is shifted so as to bring the tensrods opposite the racks. During the next two quarterturns the process is repeated, the block ultimately returning to its original position. Multiplication by a number with more placesis'performed as in the Thomas. The advantage of this machine over the Thomas in saving time is obvious. Multiplying by 817 requires in the Thomas 16 turns of the handle, but in the SteigerEgli only
turns, with 3 settings of the lever H. If the lever H is set to 1 we have a simple addition machine like the Thomas or the Brunsviga. The inventors state that the product of two 8figure numbers can be got in 6–7 seconds, the quotient of a 6figure number by one of 3 figures in the same time, while the square root to 5 places of a 9figure number requires 18 seconds.
Machines of far greater powers than the arithmometers mentioned have been invented by Babbage and by Scheutz. A description is impossible without elaborate drawings. The following account will afford some idea of the working of Babbage's difference machine. Imagine a number of striking clocks placed in a row, each with only an hour hand, and with only the striking apparatus retained. Let the hand of the first clock be turned. As it comes opposite a number on the dial the clock strikes that number of times. Let this clock be connected with the second in such a manner that by each stroke of the first the hand of the second is moved from one number to the next, but can only strike when the first comes to rest. If the second hand stands at 5 and the first strikes 3, then when this is done the second will strike 8; the second will act similarly on the third, and so on. Let there be four such clocks
with hands set to the numbers 6, 6, 1, o respectively. Now set the third clock striking I, this sets the hand of the fourth clock to 1; strike the second (6), this puts the third to 7 and the fourth to 8. Next strike the first (6) ; this moves the other hands to 12, 19, 27 respectively, and now repeat the striking of the first. The hand of the fourth clock will then give in succession the numbers 1, 8, 27, 64, &c., being the cubes of the natural numbers. The numbers thus obtained on the last dial will have the differences given by those shown in succession on the dial before it, their differences by the next, and so on till we come to the constant difference on the first dial. A function
y = a Jbx+cx2+dxa+ex4
gives, on increasing x always by unity, a set of values for which the fourth difference is constant. We can, by an arrangement like the above, with five clocks calculate y for x=1, 2, 3, . to any extent. This is the principle of Babbage's difference machine. The clock dials have to be replaced by a series of dials as in the arithmometers described, and an arrangement has to be made to drive the whole by turning one handle by hand or some other power. Imagine further that with the last clock is. connected a kind of typewriter which prints the number, or, better, impresses the number in a soft substance from which a stereotype casting can be taken, and we have a machine which, when once set for a given formula like the above, will automatically print, or prepare stereotype plates for the printing of, tables of the function without any copying or typesetting, thus excluding all possibility of errors. Of this " Difference engine," as Babbage called it, a part was finished in 1834, the government having contributed £17,000 towards the cost. This great expense was chiefly due to the want of proper machine tools.
Meanwhile Babbage had conceived the idea of a much more powerful machine, the " analytical engine," intended to perform any series of possible arithmetical operations. Each of these was to be communicated to the machine by aid of cards with holes punched in them into which levers could drop. It was long taken for granted that Babbage left complete plans; the committee of the British Association appointed to consider this question came, however, to the conclusion (Brit. Assoc. Report, 1878, pp. 92102) that no detailed working drawings existed at all; that the drawings left were only diagrammatic and not nearly sufficient to put into the hands of a draughtsman for making working plans; and " that in the present state of the design it is not more than a theoretical possibility." A full account of the work done by Babbage in connexion with calculating machines, and much else published by others in connexion therewith, is contained in a work published by his son, General Babbage.
Slide rules are instruments for performing logarithmiccalculations mechanically, and are extensively used, especially where
only rough approximations are required. They are slide
almost as old as logarithms themselves. Edmund rates.
Gunter drew a " logarithmic line " on his " Scales "
as follows (fig. 4) :—On a line AB lengths are set off to scale to represent the common logarithms of the numbers 1 2 3 . . . to, and the points thus obtained are marked with these numbers.
0 o A 3
• 1 0 0
O Q 2 0 2 i 0.
2 dl
0 3 3 3 3 3 3 3 3 /
O! 4 1 4 ! + 4 V p
{s eg
,: • iIIi 8 r~ s U 9.8.9. u
Bp ge u
0 9 . ©®®®O ® O
®
a 7 s 3+ 3 2 if i 2 3 4 3 6 7 8
End of Article: CALCULATING MACHINES 

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