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## Calculation

General

This section describes the operation of the mechanisms that execute the addition of numbers according to the method of finite differences. The engine operates with vertical stacks of figure wheels thirty one digits high i.e. the maximum value of a number is thirty one figures. The least significant number position is at the bottom of the stack and the most significant posrtion is at the top. Multi-digit numbers are represented by columns of figure wheels with one figure wheel for each digit-position of the number. During a calculating cycle 31-digit numbers on the figure wheel columns are added to 31-digit numbers on adjacent columns to the left in a process of repeated addition.

The column closest to the drive handle represents the seventh difference and the lower order differences are represented on the columns to the left as seen facing the engine. The column closest to the printer represents the result or tabular value. Babbage's notation identifies the nnt difference column as [Delta]n and the tabular value column as T (163 eg). The highest order difference column (extreme right) is therefore labelled Delta7 and successive lower differences are labelled with decreasing superscripts running right to left ending with the lowest order difference Delta1 on the last but one figure wheel column on the left. The tabular value is held on the extreme left hand figure wheel column, T (see 161, 163, 164, 176 and 177).

For calculations requiring seven orders of difference, column 1 contains the constant difference. In situations where fewer than seven differences are used the active columns are those closest to the tabular value column, and columns closest to the drive handle are inactive. Columns are rendered inactive by setting all its figure wheels to zero. The inactive columns remain a functioning part of the calculation cycle but have no effect on the calculation. In circumstances in which fewer that seven differences are used the right-most active column (lowest numbered) contains the constant difference.

Odd-even Phasing

When using the method of finite differences as a manual technique for subtabulation it is usual to form the table in a way that allows addition of the higher order difference to the next lower order difference in a fixed incremental sequence. The manual calculation proceeds from the highest order difference (the constant difference) to the tabular value by progressing from right to left column by column. However, the sequence of additions as executed by the engine does not proceed in a stepwise way from right to left as one might expect from the manual method. One complete calculating cycle consists of two symmetrical half-cycles. During the first half-cycle the number values on the odd-numbered axes are simultaneously added to those of the even-numbered axes to the immediate left i.e. axes 1, 3, 5, 7 are added to 2, 4, 6,and 8 respectively. Similarly, during the second half-cycle all the even-numbered axes are simultaneously added to the odd-numbered axes again to the immediate left. Provided this is taken account of when setting up at the start of a run by offsetting the initial values on alternate axes by a half-cycle, the end result is the same i.e. each full calculating cycle results in a new tabular value which is the cumulative sum of the number of active differences.

The advantage of the phased operation of odd-to-even and even-to-odd half-cycles is that it allows a form of parallel operation which shortens the calculating cycle. Adding each column to its neighbour in a stepwise way would require eight distinct identical operations involving only two of the eight columns i.e. 75% of the engine would be idle at any one time. Simultaneously adding four columns to four columns in each of two half-cycles is more time efficient and also allows only 50% of the machine to be idle at any time. A further feature of this 'pipelining' effect is that the cycle time of the engine is independent of the number of differences used in the calculation. An additional advantage is the comparative simplicity of the control mechanism. A serial step-wise addition sequence would require a control system that independently activates each column pair in turn. This is substantially more complex a design than required for the phased pairing of alternate axes.

Each addition half-cyde is itself divided into two distinct phases.

The mechanism caters for secondary carries. This situation arises when a figure wheel is set at 9 after the giving off phase and the next warning mechanism down is armed indicating that the 9 is to be incremented to 0 and a carry passed on. During the carry cycle the wheel set at 9 will be incremented to 0. This action will set the carry warning device in the next significant digit and the associated figure wheel will be incremented as the next action in the successive carry sequence. The worst-case carry condition arises if the column is set with nines in each digit position and a number between 1 and 9 is added to the least significant digit. In this case the carry ripples through the column as a series of secondary carries sometimes called 'domino carry'.

The key drawing is BAB [A] 171, particularly the section bottom left. The notation on 171 indicates that the view depicts the 1st and 2nd difference columns, [Delta]1 and [Delta]2 respectively (i.e. columns 6 and 7 in the new convention), with the intermediate even sector column (R3), odd warning axis (W1), and odd carry axis (C1).

In Babbage's design the relationship of figure wheels, sectors and carry mechanism is identical for odd and even axes. This can be seen in re is plan view shows the basic layout of a calculating unit viewed as a one-digit assembly that adds from nght to left. This scale view is diagrammatic i.e it is strictly neither a section view nor a plan but a combination of the two: the figure wheel supports, locks, zero stops and shafts are shown in section but the carry axis is shown in plan as though with the top plate removed. The elevation shown in the upper part of 171 is also diagrammatic in the sense that mechanism has been opened out and the horizontal spacing has been expanded for clarity. Other drawings containing information on the calculation mechanism are 161, 163, 164, 176 and 177.

The view on 171 lower left shows the basic mechanical unit of addition. The mechanism adds the 31digit number on the right hand column to the 31-digit number on the left hand column taking account of any carries. The view shows two figure wheels coupled by an intermediate sector wheel (the partial wheel with a wedge cut away), a carry axis with helically arranged carry arms, and the warning and carry mechanism. Only the parts that are active in a two-column addition are shown. The carry mechanism for the right hand column is omitted as are the remaining six figure wheel, warning and carry axes. In the original design the arrangement shown in 171 is repeated symmetrically for the eight columns i.e. sector axes couple adjacent figure wheel axes and each figure wheel axis has a carry and warning axis in the same relative positions. This can be seen clearly from the plan views in 161, 164, 176 and 177. The repetition of this basic two-column addition unit in 'its unmodified form constitutes a fundamental design flaw. This is described in detail below.

Each of the eight 31-digit numbers (seven differences and the tabular value) is represented by a column of 31 figure wheels with one figure wheel for each digit of the multi-digit number. The value of each digit is represented by the angular rotation of the associated figure wheel. Each figure wheel has forty teeth with the pitch of one tooth (9o) corresponding to a single decimal number 0-9. Each wheel therefore has four identical runs of decimal numbers and the same digit value can be represented by any one of four interchangeable angular positions of a wheel. The figure wheels rotate freely on the figure wheel shafts and rest on the figure wheel supports. These are comb-like pillars that run the full vertical length of the columns with the figure wheels resting on the equally spaced teeth of the comb. This is shown clearly on the extreme right top of 171 in the opened out view. In the lower view the supports are shown as rectangles with each column of figure wheels fitted with three figure wheel supports positioned roughly at 5-o'dock, 10-o'clock and 2-o'dock. The supports are not equally spaced around the figure wheel axis (see below).

The figure wheels are driven by drive arms keyed to the figure wheel drive shafts. The drive arms operate against internal nibs positioned at 90" intervals inside the barrel of each figure wheel. The figure wheels themselves remain on the same horizontal plane throughout i.e they do not move vertically but undergo circular motions within the cages formed by the combs of the figure wheel supports. The drive arms are engaged and disengaged by lifting and lowering the figure wheel axis at appropriate points in the cycle. The lifting action disengages the drive arms by raising them clear of the internal nibs. This occurs when the wheels are being added to and when manually setting the initial values at the start of a run of calculations.

As well as four internal nibs, each figure wheel has four external nibs integral with the barrel. The external nibs perform two separate functions: they run up against the zero stops (shown below the figure wheel supports shown at 2-o'clock) to ensure that the wheels do not overrun the zero position during giving off. They also set the carry warning rnechanisrn by operating the curved warning lirnb of the carry lever if the figure wheel is driven beyond 9 during addition.

Figure wheels in neighbouring figure wheel columns are coupled digit for digit by sector wheels assembled on axes that interpose between pairs of figure wheel axes. The sector wheels are free to rotate on the sector axes. They are driven by restoring arms which are keyed to the sector axes. The restoring arm rotates against a peg which protrudes above each wheel (see 171 upper right) and restores the sector to zero. The sector wheels and restoring arms are raised and lowered by the sector axes. Unlike the figure wheel drive arms which disengage from the internal nibs by being raised clear, the sector drive arms are fixed in the same plane as the peg and the drive arms and wheels are raised and lowered together with the sector axis.

The sectors can occupy three discrete vertical positions, fully disengaged, partially engaged and fully engaged.

The rotation of the sectors is limited by the zero stops shown at the bottom of the view in 171. The zero stop performs two separate functions. It stops over-rotation of the sector by directly obstructing the path of the cut-away section of the wheel. It also provides a locking action when the sector is fully disengaged i.e. in the fully raised position when the sector is in the zero position a lug on the zero stop engages with a slot in the sector wheel. This locking action prevents derangement when the sector is not meshed with either figure wheel. The only detail of the sector locking slot is given in 171 where a small gap is indicated between the end of the stop attached to M and the root of the slot on the sector. The sector zero stop comb-pillars are not shown in the opened out elevation above.

The wedge-shaped parts alongside the figure wheel supports at about 9-o'clock are locks. These are sword-like devices which run the full length of the columns of figure wheels. The upper and lower ends of the lock are angled and pass through angled slots in the frame. The top end passes through an angled slot in the upper framing plate; the lower end passes through an angled slot in framing member below the odd figure wheel racks. The T-shaped framing member spans the length of the engine and is supported at each end by two front-to-back framing members at each end. The T-shaped section of this member is shown as I in 160 right centre, and in plan, with the angled slots for the locks, in 161 running right/left across the centre. Raising the lock withdraws the wedge from the between the teeth of the figure wheel. Lowering the lock drives the wedge into the inter-tooth gaps. During entry the wedging action corrects small rotational derangements of the wheel and when fully home locks the wheels in valid integral-value positions.

The locks operate on all figure wheels in any given column at the same time. If a wheel is deranged to the extent of presenting a tooth to the front of the wedge, the path of the lock is obstructed and the engine jams. These jams are intended to warn that a figure wheel is in an indeterminate position and thereby halt an operation that would lead to an incorrect result The locks extend below the figure wheel columns and also lock the circular motion quadrants that impart the rotational motion to the figure wheel axes (171 lower right). (Locking the circular motion quadrants is shown in 160 left centre). The locks are engaged and disengaged by the vertical motion drive in accordance with the timing cycle.

The carry mechanism consists ofthe warning axis shown at 12-o'clock to the figure wheel axis, the three-limbed carry levers mounted on the warning axis, the spring-loaded warning mechanism with detent lever, the reset stops which are fixed to the figure wheel support shown at 2-o'clock to the figure wheel axis, and carry axis with the helical arrangement of carry arms.

The carry lever is the most complex single part in the engine and performs several different functions central to the operation of the calculating mechanism. Each carry lever has three limbs which project radially from a central boss with which it is forms a single piece. The carry levers do not pivot on the warning axis directly but are free to rotate on the boss attached to the detent support arm which is in turn keyed to the carry axis.

The carry lever can occupy four discrete positions the locations of which are determined by the position of four v-shaped detents in the detent limb of the lever (shown at 9-o'clock on 171). The spring-loaded detent lever which pivots on a spigot attached to the lever arm locates the lever in the four positions. These correspond to the four states,

Each figure wheel position has an associated carry lever keyed to the carry axis. The carry arms are arranged on a fixed vertical pitch equal to the pitch of the figure wheels and a fixed angular pitch of 22-1/2o with a single 45o gap between the 15th and 16th carry arm only (see below for explanation).

There are thirty carry arms in all, arranged in two runs of fifteen arms on a 22-1/2o pitch separated by a double-pitch gap between the two runs. The helical arrangement of carry arms is shown at the top of the lower view in 171. In the plan view each arm shown hides its counterpart in the run below i.e. C30 conceals C15, C29 conceals C14 etc. The uppermost carry arm (at 4-o'clock) is annotated 'C15 C30' indicating that the single arm drawn represents the upper and hidden arm below. For strict consistency each of the arms should have two notations though the first three have only one (C1 C2 C3). This is another respect in which the view in 171 is neither a true section view nor a plan but a diagrammatic hybrid. As the carry axis rotates, the carry arms sweep round and poll each figure wheel position in turn. Staggering the carry arms in the helical arrangement shown allows secondary carries to propagated i.e. a carry that results from a carry in the lower order decade can ripple upwards in what Babbage called 'successive carry' operation (see below).

The limb of the carry lever shown at 2-o'clock is the carry limb. The limb is drawn solid in the unwarned position and as dotted partial views in the other positions, warned, carried and disconnected (two positions). If the carry lever is in the unwarned position then the carry arm sweeps past without action. However, if the carry lever is warned then the locus of the lozenge at the end of the carry arrn intersects with the warned position of the lozenge at the end of the carry limb so as to advance the carry lever to the 'carried' position. The motion imparted to the carry lever nudges the figure wheel in the next highest position by one tooth i.e. increments the next most significant digit by one. The figure wheel nudged by a lug on the third limb.

The third and remaining limb of the carry lever (shown at 6-o'clock) has attachments at two vertical levels. The two attachments can be seen in elevation in the opened out view at the top of 171, centre. The upper attachment has two lugs and resembles an escapement, the lower is a curved warning limb. The vertical position of the limb and the spacing between the two attachments is arranged so as to straddle two vertically adjacent figure wheels. In this way the escapement attachment and the warning limb of the same carry lever interact with different decades of the same number and carry action on a figure wheel occurs as a result of warning from the figure wheel immediately below: the lower attachment is operated by the outer nib of a figure wheel, and the upper attachment of the same carry lever operates on the figure wheel of the decade immediately above. The curved warning limb engages with the external nib of a figure wheel as the wheel passes from 9 to 0. The passage of the nib past the warning limb nudges the warning lirnb and advances the carry lever from unwarned to warned. This occurs while the figure wheel is being added to. During this phase of the cycle the escapement is raised clear of the teeth of the figure wheel immediately above and has no direct effect

After the addition phase the carry lever is lowered from its raised position for the carry phase. The warning limb can be still be operated by the external nib of the lower wheel with the carry lever in the lowered position. (This allows the carry lever to be advanced from unwarned to warned by secondary carries which rnay occur during the carry phase of the cycle, see below.) In the lowered position the lugs on the escapement are in the plane of the figure wheel teeth above and rnesh with thern in different ways to perform three separate functions.

1. With the carry lever in the unwarned the v-shaped cutout on the left hand lug engages with the figure wheel tooth as the carry lever is lowered and locks the figure wheel to prevent derangement during the carry cycle i.e. when the figure wheel is otherwise unsecured.

2. the carry lever is warned when lowered, the left hand lug is clear of figure wheel teeth but the right hand lug interposes between adjacent teeth and nudges the figure wheel one position on when the carry lever is advanced from warned to earned by the sweep of the carry arm. A carry occurs in the figure wheel as a result of a warning from the figure wheel immediately below. The right hand lug has an additional function: while the carry lever is in the warned position and waiting to be sen/iced by the rotation of the helical arrangement of carry arms the right hand lug acts as a partial lock.

3. with the carry lever in the disconnected position, the right hand lug again interposes between two adjacent teeth and acts as a figure wheel lock.

During the carry phase of the addition half-cycle the carry axis is lowered. If the carry lever is in the unwarned position then the cutaway on the left hand lug on the escapernent engages with a tooth and locks the wheel. This ensures that the wheel does not derange when the locking wedges are withdrawn during the carry phase. If the position is warned then the right hand lug engages with the figure wheel and the motion imparted to the escapement by the rotation of the carry arm nudges the figure wheel one position on. The warning axis is raised during the addition half-cycle. In the raised position the warning limb can still be activated by the outer nib of the figure wheel but the escapement is lifted clear of the figure wheel above i.e. the lugs are disengaged leaving the figure wheel to free to be driven by the sector.

Motion can be imparted to the carry lever in four separate ways:

1. by the outer nib of the figure wheel as the wheel is driven past 9 during addition. This advances the lever from the unwarned to the warned position

2. by the sweep of the carry arm during the carry phase. This advances the lever from warned to carried

3. by the detent support arrn. This returns the detent lever to the unwarned position by rotating the carry lever against the reset stops and driving the detent lever along the notched detent lirnb in the last detent

4. by hand to the disconnected position. This last operation requires that the reset stops are raised to allow the lever to pass under the stop. The stops are then lowered to trap the lever. The stops are freed for raising and lowering by releasing the fixing screws which pass through slotted holes in the support bars for the stops. This is shown dearly in 171 top left Disabling the carry rnechanisrn in this way prevents carries from propagating past the disabled position. This has the effect of isolating the figure wheel colurnn above the disabled position frorn the calculation below. Sections of the figure wheel columns can then be used for different purposes, as a cycle counter, for example (see below).

The operation of the calculating rnechanisrn will be explained with reference to the layout of the basic addition unit shown in 171 lower left At the start of the cycle the right hand figure wheel colurnn will in general store a non-zero number. In the rnost general case each of the 31 figure wheels in a colurnn will register a different angular displacement from the zero position. Similarly, the left hand column will in the general case have a non-zero number represented by appropriate initial settings of the figure wheels. The calculating half-cycle adds the 31-digit number from the right hand column to the 31-digit number on the left hand column and takes account of any carry. The transfer of numbers is non-destructive i.e. at the end of an addition halfcycle the number on the right hand column at the start-of-cycle is restored to that column, and the left hand column ends up with the sum of the two numbers.

The view shown in 171 lower left can be taken as showing the positions of the figure wheel drive arms and the sectors at the start of the cycle. Both figure wheel axes are rotated fully anticlockwise and the drive arms, in the raised position, are clear of the internal nibs. The figure wheel zero stops, which move with the figure wheel axes, are also clear of the external nibs. (The zero stop comb is shown n 171 top right (S). The coupling of the figure wheel zero stops and the figure wheel axes is shown in 160, left view, right A yolk on the zero stop support engages with a colar on the figure wheel axis. This allows the axis transmit vertical motions to the zero stops but not circular motions.) The left and right hand figure wheels can be regarded as being shown set to zero. This is a special case and in not generally true i.e. in general the figure wheels will register a non-zero number at the start of the cycle and will be offset anticlockwise from the zero position shown, by a variable number of teeth (9o intervals) corresponding to the digit value for that number position. The internal nib shown at 5-o'clock will therefore in general be somewhere in the quadrant between 5-o'clock and 2-o'clock. This is true of both right and left figure wheels i.e. in general each of the 31 wheels in a column will be displaced anticlockwise from the position shown by a variable displacement corresponding the digit value. In the special case shown in 171 both figure wheels can be regarded as set at zero at the start of the cycle i.e. in the next addition half-cycle, zero (to 31 places) on the right hand column will be added to zero (to 31 places) on the left hand column. The figure wheel locks are full engaged as shown and both wheel columns are immobilised.

At the start of the half-cycle the sectors are against the zero stops (i.e. full clockwise), fully raised and with the restoring arms full anticlockwise. In the fully raised position the sectors are fully disengaged from the figure wheels. With the sectors in the home position (i.e. against the zero stops) raising the sectors locks them - this by engaging the locking lug on the zero stop with the slot in the sector. The sectors are against the zero stop at the start of cycle regardless of the numbers set on the right and left figure wheels at the start of cycle.

The warning axis is in the raised position as shown in the opened out elevation. In this position the warning lirnb of the carry lever can be acted on by the outer nibs of the figure wheel but the escapernent lugs are clear of the teeth. The carry lever is in the unwarned position i.e. with the toprnost detent engaged.

The cycle commences with the lowering of the sectors into full engagement with the two figure wheel columns. Lowering the sectors unlocks them by disengaging the locking lug on the zero stop from the slot in the sector. In the same interval the right hand figure wheel axis and zero stops are lowered so that the drive arms are in the plane of the internal nibs and the zero stops are in the plane of the external nibs. The left hand figure wheel axis remains in the raised position with the drive arms and zero stops disengaged. This allows the external nib to pass the zero stop position which will occur if the number on the figure wheel exceeds 9.

The locks on both figure wheel columns are then raised i.e. disengaged, and the right hand figure wheel axis rotates clockwise from its rest position though its full travel of 81o i.e. 9o (one tooth pitch) short of a full quadrant During this sweep the right hand wheel is reduced to zero and the zero stop prevents any overshoot by blocking the path of the outer nib. The number held of the figure wheel at the start of the cycle (in the case shown in 170 this is zero) is transferred tooth for tooth to the sector wheel which in turn drives the left hand figure wheel. The number on the left hand figure wheel is increased by the number on the right hand wheel and this is the basic operation of addition.

If during the addition the left hand figure wheel passes 9 then the external nib operates the curved warning lirnb and advances the carry lever to the warned position i.e. with the second detent engaged. The warning action is part of the giving off/addition phase and occurs only in digit positions that exceed 9 during addition and at different tirnes within the addition window depending on the nurnber value of the particular figure wheel. The process of giving off, addition and warning occurs simultaneously for all 31 digit positions in the columns.

The locks on both figure wheel axes then engage to correct minor derangements. At the same time the sectors are raised to their partially engaged positions i.e. they remain meshed with the right hand figure wheels and disengaged from the left hand figure wheels.

In the description so far the right hand figure wheel rotates clockwise when giving off, the sector is driven anticlockwise and the left hand figure wheel clockwise during addition. These directions of rotation are inferred from the position of the sector restoring arm which is displaced anticlockwise to the drive peg. This indicates that the restoring arm provides active drive to restorthe sector by clockwise rotation of the sector. It follows that rotation of the sector during giving off must be counter-clockwise from the zero stop as this is the only degree of rotational freedom possible. By this reasoning the right hand figure wheels rotates clockwise when giving off and, since it is meshed to the left hand figure wheel via the sectors, the left hand figure wheel therefore also rotates clockwise when being added to. This would indicate that the warning mechanism should be armed by clockwise rotation of the outer nib of the figure wheel during addition. However, with the warning rnechanisrn shown in 171 the curved warning limb will be correctly operated by antidockwise rotation of the figure wheel not clockwise rotation. If the figure wheel were to rotate clockwise with the arrangement as drawn, the curved warning lirnb would foul the outer nib of the figure wheel and act as a stop. Clockwise rotation of the figure wheel during addition is therefore inconsistent with correct warning action which requires advancing the carry lever from unwarned to warned and the mechanism well not work if made as drawn.

The directional arrows drawn on the five axes in 170 are little help as these are inconsistent however interpreted. So as to use 171 to complete the description of the calculating cycle it will be assumed that the left hand column rotates anticlockwise during addition for the correct operation of the warning mechanism. The measures taken to remedy this apparently serious design flaw and alternative interpretations of the layout in 171 will be explored in the section describing the modem implementation.

The end of the giving off and addition phase leaves the left hand figure wheel with the sum (without carry) of the two initial values of the two figure wheels and the warning mechanisms armed in the digit positions in which the figure wheels exceeded 9 during addition. The right hand figure wheels are at zero with the drive arms rotated fully clockwise. The right hand figure wheels have therefore lost their initial values. However, the sectors are displaced from zero by the number of teeth equal to the number value on the right hand figure wheels at the start of the half cycle and the values are restored to the right hand figure wheels in the next phase of the half-cycle.

The locks disengage from the right hand figure wheels and the sector restoring arms rotate in a clockwise sweep. During the sweep the restoring arms engage with the pegs on the sectors and restore the sectors to zero. The sector zero stops prevent any overshoot by obstructing the cutaway section of the sector. Since the sectors are disengaged from the left hand figure wheels these remain unaffected by the restoration of the sectors. The right hand figure wheels are driven by the sectors and the restoration of the sectors to zero restores the figure wheels to their positions at the start of the cycle before giving off. During the restoration process the right hand figure wheel axis rotates anticlockwise to the position shown in 171 i.e. returns the drive arms to the position at the start of cycle. This does not affect restoration as the figure wheel axis is in the raised position i.e. the drive arms are lifted clear of the internal nibs. The figure wheel locks then engage to correct minor displacements and prevent spurious derangement.

During the carry phase the warning mechanisms are serviced in turn and the requisite carries propagated upwards i.e. the next decade up is conditionally incremented by one depending on whether the associated warning mechanism is armed or not.

The carry phase commences with the withdrawal of the locks from the left hand figure wheels which occurs immediately after the addition phase in the first half-cycle. At the same time the warning axis is lowered. This lowers the carry lirnbs into the same plane as the carry arms (the lozenges of the warned lirnbs are now in the path of the lozenges of the corresponding carry arrns), and the escapement limbs are lowered into the same plane as the figure wheel teeth. If the position is unwarned then the left hand escapement lug is active i.e. the v-shaped cutaway locks the figure wheel during the period in which it is otherwise unsecured. If the position is warned then the left hand lug is clear of the teeth and the right hand lug is active i.e. lowered so as to partially interpose between two adjacent figure wheel teeth. This acts as partial lock while the position awaits its turn in the successive carry sequence.

The rotation of the carry axis follows and with it the rotation of the helical array of carry arms. The direction of rotation is anticlockwise as indicated by the direction arrow drawn in the circle representing the carry shaft Each carry limb is polled in sequence by the corresponding carry arm starting from the below and working upwards. If a carry lever is unwarned then the trajectory of the carry arm lozenge does not intersect with the position of the carry limb lozenge. The lozenges pass without contact and no action to carry is taken. However, if the position is warned, the locus of the two lozenges intersect The outer face of the carry arrn lozenge wipes past the inner face of the cany lirnb lozenge and pushes the carry limb aside as it slides past. The action of the carry arm nudges the carry limb one position on i.e. into the earned position which is the next discrete position marked by a detent. The clockwise rotation of the carry lever advances the right hand lug of the escapement and this nudges the figure wheel one tooth pitch on i.e. increments the number value by one. The helical arrangement of carry arms services each warning rnechanisrn in turn and increments the figure wheel depending on whether or not the mechanism is warned. The net result of this conditional action is that mechanisms warned by figure wheels during addition increment by one the figure wheels of the next higher decade as required for correct multi-digit addition with carry.

With the carry lever in the disconnected position the right hand lug interposes between two adjacent figure wheel teeth when lowered and acts as a figure wheel lock during the carry phase.

Both sets of carry axes (i.e. odd and even axes) rotate together. During the even carry the odd carry levers rotate freely without encountering any warning limbs and vice versa. This is because during the carry phase of one set of axes the warning axes of the other set are raised and the warning limbs and swiriing carry arms are in different planes and do not interact

A primary carry is one that occurs as a result of a warning mechanism being armed during addition as described above. A secondary carry occurs when a figure wheel exceeds 9 as a result of primary carry. During a primary carry, phase the figure wheel at 9 will be nudged on to zero by the right hand lug of the escapement as its position is polled by the sweep of the rotating carry arm. As the figure wheel passes 9 the outer nib of the figure wheel passes the curved warning limb, and arms the next higher warning rnechanisrn as before. The newly armed warning rnechanisrn is immediately polled by the next carry lever in the sequence and the carry is propagated as before. Ripple carries can be propagated up the stack in this way. The worst case ripple condition occurs in a column of figure wheels all set to 9 and a warning rnechanisrn arrned in the least significant position. The successive carry operation described will correctly propagate the carry up the stack leaving each figure wheel set to zero and an overflow carry at the top of the stack.

There is a feature of the geometry of the arrangement of carry arrns that has yet to be explained. In the helical arrangernent of carry arrns the vertical separation of the carry arms corresponds to the pitch of the figure wheels and the angular pitch is fixed at 22-1/2o. There is one exception to this i.e. the spacing between the 15th and 16th arms (counted from below) is 45o not 22-1/2o (gap shown at 5-o'clock in 171, between plan of C1 and C30. The function of the gap is to prevent the carry arm and carry limb from fouling in one of two conditions. During giving off the warning axis raised as shown in the elevation 171 top left In this position the lozenges at the end of the carry arms are not in the same plane as the lozenges at the end of the carry limbs and there is no danger of the carry limb and arm if a carry lever advances from the unwarned to warned positions. However, during the carry cycle the warning axis is lowered. If the carry arrns were on a fixed 22-1/2o pitch then the lozenge of the 16th carry limb, if warned, would foul the lozenge of the corresponding carry arm when the axis lowered. In addition to axial fouling of this kind there is the danger of rotational fouling in the case of secondary carries. If the 15th carry limb was unwarned at the start of the carry cycle but received a secondary carry then the lozenges would foul as the 16th carry limb advanced to the warned state. This was foreseen in the original design and Babbage allowed a double-pitch gap between the 15th and 16th carry arms to allow sufficient clearance to avoid axial or radial fouling.

The end of the carry phase coincides with the end of the restoration of the right hand figure wheels to their initial values at the start of the half-cycle. Both left and right hand figure wheel locks then engage and the sector axis is raised into full disengagement from both left and right figure wheels. The end of the first half cycle leaves the right hand figure wheels with their start-of-cyde values restored, the left hand figure wheels with the sum of the left and right hand initial values, and the sectors against their zero stops, fully disengaged and locked.

There are two last actions needed to restore the mechanism shown in 171 to the start-of-cyde condition in preparation for the next repetition of the first half cycle: returning the sector restoring arms to their anticlockwise home position's; and resetting the armed warning mechanisms to their unwarned positions. Both these actions occur during the second half-cycle. The sector restoring arrn is returned to the home position at the start of the second half-cycle during giving off. Resetting the left-hand warning rnechanisrns to the unwarned positions occurs at the end of the second half cycle while the warning axis is raised and the escapernent is out of the plane of the figure wheel teeth. The warning axis rotates clockwise and with it the detent support arrns which are keyed to the axis. The spring-loaded detent lever, mounted on the detent support arrn, is earned clockwise and the carry lirnbs are driven towards the reset stops by the detent lever. The carry lirnbs in the earned position will be driven against the reset stops and their rnotion halted. The detent support arrn continues clockwise and the detent lever is driven in turn from engagement with the carried detent to the warned detent and finally to the unwarned detent as it rides along the notched outer curve of the detent lirnb. The travel of the detent support arrn is such that the carry levers already in the unwarned position at the start of the reset process do not bear on the reset stops and remain unaffected. The clockwise rotation of the detent support arm is followed immediately by a counter clockwise return rnotion which restores the unwarned carry levers to their home positions as shown in 171. The carry levers in the disconnected position are not reset to the unwarned state: these are trapped behind the reset stops and simply wave back and forth during the movement of the detent support arm. The extremities of travel of the disconnected carry limbs are shown by the two dotted positions bracketed opposite the 'Disconnected' label. The clockwise/anticlockwise waving motion of the warning axis resets the carry levers to the unwarned position. This is the only function of the circular motion of this axis and Babbage refers to the axis as the 'Unwarning Axis' in the timing diagram BAB [F] 385 la.

During the second half cycle the left hand figure wheels give off to the column to the immediate left (not shown in 171) in a repetition of the sequence described. As noted earlier, the progression from column 1 (or the lowest numbered active column) through to the tabular value on column 8 does not occur in a stepwise sequence right to left from column to column in seven distinct repetitions of the same sequence. Rather, all the odd-numbered columns are siulutaneously added to the even-numbered columns in the first half cycle and all the even-numbered columns are added to the odd-numbered columns in the second half cycle. The phasing of the odd-to-even and even-to-odd half-cycles and the interleaving of the various actions within a cycle are shown in the timing diagram BAB [F] 385/1a. This diagram is fraught with inconsistencies and errors but will serve for the meanwhile as a general guide to the timing of the events in the two half cycles.

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