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CALCULATING MACHINES
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CALCULATING See also:MACHINES . See also:Instruments for the See also:mechanical performance of numerical calculations, have in See also:modern times come into ever-increasing use, not merely for dealing with large masses of figures in See also:banks, See also:insurance offices, &c., but also, as See also:cash registers, for use on the counters of See also:retail shops. They may be classified as follows:—(i.) Addition machines; the first invented by Blaise See also:Pascal (1642). (ii.) Addition machines modified to facilitate multiplication; the first by G. W. See also:Leibnitz (1671). (iii:) True multiplication machines; See also:Leon Bolles (1888), Steiger (1894). (iv.) Difference machines; Johann Helfrich von See also: It contains See also:historical notes and full references. See also:Walther von Dyck's See also:Catalogue also contains descriptions of various machines. We shall confine our-selves to explaining the principles of some leading types, without giving an exact description of any particular one. Practically all calculating machines contain a " counting See also:work," a See also:series of " figure disks " consisting in the See also:original See also:form. of See also:horizontal circular disks (fig. I), on which the figures o, 1, 2, to 9 are marked. Each disk can turn about its See also:vertical See also:axis, and is covered by a fixed See also:plate with a hole or " window " in it through which one figure can be seen. On turning the disk through one-tenth of a revolution this figure will be changed into the next higher or See also:lower. Such turning may be called a " step," See also:positive if the next higher and negative if the next lower figure appears. Each positive step therefore adds one unit Add~tine See also: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 See also:side by side, their windows lying in a See also:row, then any number of six places can be made to appear, for instance 000373. In See also:order to add 6425 to this number, the disks, counting from right to See also:left, have to be turned 5, 2, 4 and 6 steps respectively. If this is done the sum 006798 will appear. In See also: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 See also:wheel with ten See also:teeth. Each figure disk has, besides, one See also: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 See also:simple, because the long teeth as described would See also: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 See also: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 See also:calcium compounds, especially when moistened with hydrochloric See also:acid, impart an See also:orange-red See also:colour we have an addition See also:machine, essentially of Pascal's type. In it each disk had to be turned by See also:hand. This operation has been
simplified in various ways by mechanical means. For pure
addition machines See also: 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 See also: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 See also: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 A-wheel. The covering plate MK has a slot above the axis EE allowing a See also:rod LL' to be moved by aid of a See also:button L, carrying the wheel D with it. Along the slot is a See also:scale of See also: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 See also:top See also:middle See also:part of Steiger's machine (fig. 3). Let now the handle See also:driving the axis Z be turned once See also:round, the button being set to 4. Then four teeth of the B-wheel will turn D and with it the A-wheel, 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 See also:fuller description of the manner in which a See also:mere addition machine can be used for multiplication and See also:division, and even for the extraction of square roots, see an article by C. V. Boys in Nature, ttth See also: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 See also:lever which appears at the top plate of the machine, its two positions being marked " addition " and " subtraction." The B-wheel, 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 A-wheels out of gear. When thus raised the figure gure disks can be set to any figures; at the same See also:time it can slide to and fro so that an A-wheel can be put in gear with any C.wheel forming with it one " See also:element." The number of these varies with the See also:size of the machine. Suppose there are six B-wheels 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 B-wheels to the right. If now the buttons belonging to the latter be set to 3 z 5 6, then on turning the B-wheels 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 B-wheels is performed simultaneously by the See also: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 A-wheels 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 B-wheels and twelve disks the product of two six-figure numbers can be obtained. Division is performed by repeated sub-See also:traction. The lever regulating the C-wheel is set to subtraction, producing negative steps at the disks. The See also:dividend is set up at the windows and the divisor at the buttons. Each turn of the handle subtracts the divisor once. To See also: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 See also: 2). In the Brunsviga, the figure disks are all mounted on a See also:common horizontal axis, the figures being placed on the rim. On the side of each disk and rigidly connected with it lies its A-wheel with which it can turn See also:independent of the others. The B-wheels, all fixed on another horizontal axis, gear directly on the A-wheels. 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 See also:special setting being required for subtraction or division. It is extremely handy, taking up much less See also:room than the others. Professor Eduard Selling of See also:Wurzburg has invented an altogether different machi,te, which has been made by Max Ott, of See also:Munich. The B-wheels are replaced by lazy-See also:tongs. To the See also: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 A-wheels, and the figures are placed on cylinders as in the Brunsviga. The carrying is done continuously by a See also:train of epicycloidal wheels. The working is thus rendered very smooth, without the jerks which the See also:ordinary carrying tooth produces; but the arrangement has the disadvantage that the resulting figures do not appear in a straight See also:line, a figure followed by a 5, for instance, being already carried See also:half a step forward. This is not a serious See also: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 lazy-tongs allow of an See also:extension See also:equivalent to five turnings of the handle, if the multiplier is 5 or under, one push forward will do the w etewhawe .waah.vizii E. 974 same as five (or less) turns of the handle, and more than two pushes are never required. The Steiger-Egli 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 See also: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 See also:shaft of square section on which a toothed wheel, the A-wheel, slides to and fro with the See also:index in the slot. Below these wheels again See also: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 See also: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 See also: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 See also:face divided into a See also: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 units-rod and the tens-rod. Instead of the number 63 we have thus a tens-rod 6 cm. and a units-rod 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 units-rods being opposite the ends of the racks. The See also: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 See also: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 units-rods 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 tens-rods 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 See also:quarter-turn the block moves forward, the units-rods coming into operation. During the second quarter-turn 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 tens-rods opposite the racks. During the next two quarter-turns 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 See also: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 Steiger-Egli 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 See also:state that the product of two 8-figure numbers can be got in 6–7 seconds, the quotient of a 6-figure number by one of 3 figures in the same time, while the square See also:root to 5 places of a 9-figure number requires 18 seconds. Machines of far greater See also: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 See also:hour hand, and with only the striking apparatus retained. Let the hand of the first See also:clock be turned. As it comes opposite a number on the See also: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 See also: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 See also: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 See also: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 See also: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 See also:constant difference on the first dial. A See also:function y = a -J-bx+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 See also:power. Imagine further that with the last clock is. connected a See also:kind of type-writer 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 See also:formula like the above, will automatically See also:print, or prepare stereotype plates for the See also:printing of, tables of the function without any copying or typesetting, thus excluding all possibility of errors. Of this " Difference See also:engine," as Babbage called it, a part was finished in 1834, the See also: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 See also:cards with holes punched in them into which levers could drop. It was long taken for granted that Babbage left See also:complete plans; the See also:committee of the See also:British Association appointed to consider this question came, however, to the conclusion (Brit. Assoc. See also:Report, 1878, pp. 92-102) 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 See also:present state of the See also: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, See also:General Babbage. Slide rules are instruments for performing logarithmic-calculations mechanically, and are extensively used, especially where only rough approximations are required. They are slide almost as old as logarithms themselves. See also:Edmund rates. See also:Gunter See also: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 ,: • See also:iIIi 8 r~ s U 9.8.9. u Bp ge u 0 9 . Additional information and CommentsThere are no comments yet for this article.
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