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.
Phases of the Addition Cycle: Giving off and Carry
Each addition half-cyde is itself divided into two distinct phases.
- The first phase is 'giving off'.
This refers to the process of transferring a multi-digit number on one of the columns, to the
column of figure wheels immediately to the left via intermediate wheels called sectors.
During giving off the number value on the right hand column is added to that on the left
hand column. All thirty one digit positions in a column are transferred simultaneously.
The addittion is effected in a tooth-by-tooth transfer in each digit position.
- The second phase takes account of any carries. Each figure wheel has a spring-loaded
warning device which is 'armed' if a figure wheel exceeds nine during giving off. This warns
that a carry to the next decade is required for a correct result. Digit positions are polled in
sequence starting with the least significant digit at the bottorn of the column. If a warning is
armed or set then the next figure wheel up is incremented by one position during the carry
cycle. If the warning device is not set then no action is taken. Each position is polled in turn
and Babbage calls the action 'successive carry'.
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'.
Drawing References
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.
Description of the Adding Mechanism
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.
Figure Wheels
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.
Sectors
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.
- When fully raised the sectors are lifted clear of the figure wheels and
disengage from both columns. In this position the two figure wheel columns are uncoupled
from each other. This is the position shown in the opened out view in 171 top right
- When
half-lowered the sectors engage with the right hand column only and are disengaged from
the left column. This partial engagement occurs because the teeth that engage with the right
hand figure wheels are have double vertical depth. This is shown clearly on 171 top right In
the partially engaged position rotation of the sector is imparted to the figure wheel with
which it meshes and this rotation is not transferred to the figure wheels to the left.
- Finally,
when fully lowered the sectors mesh with both left and right hand figure wheels. In this fully
engaged position rotation of the right hand figure wheels is transferred to the left hand figure
wheels, wheel for wheel and tooth for tooth, via the intermediate sectors.
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.
Locks
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.
Description ofthe Carry Mechanism
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,
- 'unwarned',
- 'warned',
- 'carried',
- and 'disconnected'.
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.
-
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.
-
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.
-
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:
- 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
-
by the sweep of the carry arm during the carry phase. This advances the lever from
warned to carried
-
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
-
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 Calculating Cycle
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.
Initial Conditions
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.
Giving off and Addition
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.
Warning
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.
Design Error
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.
Restoration
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.
Carry
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
Secondary Carries
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.
Resetting
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.
The Two Half-cycles
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.