How It Was Intended To Work
 

Babbage's First Difference Engine
How it was intended to work.

Introduction
 
In 1821 Babbage invented an Engine to manufacture error-free mathematical tables, Difference Engine No.1, the world's first programmable automatic digital calculating machine, in which the only human intervention was the setting of the machine at the start of the production of a table and the turning of its handle. It was a  machine embodying the mathematical principles of the Method of  Differences using only mechanisms for addition repeated many times over.
 
For over 12 years he laboured on its designs, securing Government financial backing to build a full-size version. This was eventually abandoned in 1833 uncompleted, after a row with Joseph Clement, the engineer appointed to construct it. During his lifetime he left no published full technical explanation of how it was intended to function. Based on Professor Bromley's and my own recent studies of the surviving drawings and manuscripts now in the London Science Museum, this lecture attempts to pay homage to Babbage on the occasion of his bicentennial by completing what he was unable to do then, by giving as comprehensive as possible, in the time and space available, a description of how its parts interacted to produce the desired result.
The version of Difference Engine No. 1 described here will be that developed in the latter years of the project, around 1832/33, as fairly complete particulars of this model are to be found in the archive. Details of the earlier versions, which are different, are somewhat scant.

Overview of Engine

The engine itself would have stood 9' high by 9' long and been 3'3" deep, having the appearance from the front of the shape of an "L". The main divisions to be found in the engine: the Calculating Part to the left, the Printing Part to the right, a part which interfaces the two and the framing and chassis running on castors which supports the whole. The principle material used in its construction for all moving parts and support was Gun Metal: a very strong bronze-like substance comprising approx. 85% copper, 12% tin and 3% lead or zinc for self-lubrication. Otherwise French steel was used in the manufacture of the shafts and axes. Approximate overall weight of the Engine, some 4 tonnes.

DE1 was very much more complex in its design than its later sister, DE2, having of the order of 25,000 parts, but would have been very much slower. Its rate of production would have been around 4 digits punched per minute. DE2 would have worked some 20 times faster. At the time of its abandonment some 12,000 parts had been made: the Government having spent £15,288-1s-4d on its development and construction excluding building works.

Each part had its own name. The names used in this paper, if known, are those used by Babbage himself or Clement. They have no equivalents in today's technology.

The intended produce of the engine was copper plates punched with all kinds of mathematical, astronomical, navigational or business tables, in a form suitable for producing stereotype.

Operating Cycles

For a better appreciation of the action of the parts in DE1, it is essential to have an understanding of its Operating Cycles. Difference Engine No. 1 can be considered as having four levels of basic operating cycle:

 1. Turns of the First Axis
 2. Result Cycles
 3. Line Cycles
 4. Block Cycles

These cycles are somewhat analogous to the second, minute, hour, day etc. cycles in a clock.

Turns of the First Axis
DE1 was intended to be operated manually by discrete, whole turns of the First Axis by means of its Driving Handle, the prime mover and source of power for all the parts in the Engine. The cue for the timing of their movement was to be taken from its rotation of the First Axis and the parts set along its length. A Turn of the First Axis was required to punch one digit.

In the timing diagram on the Mechanical Notation BAB.[U11] Babbage has further subdivided this cycle into tenths, to show the detailed timings of the movement and interaction of components in the engine. This was the smallest time unit he considered when designing the Engine.

Result Cycles
These are the fixed number of turns of the First Axis required to punch one result on the Stereotype Copper Plate and to calculate the next. The operations of punching and calculation may overlap to the following extent. DE1 wants to punch one digit on each and every turn of the First Axis. Calculation may also take place at the same time as punching subject to the following constraint: the reading off of numbers for punching from the Engine's Table or Result Axis during those turns of the First Axis when it was either being added to by the 1st Difference Axis or on the subsequent turn when the carriage of tens was being resolved on it was not allowed, as the Calculating Wheels were not necessarily stationary at these periods, which is a must for the punching process. These two turns, however, were not wasted. They are used to reset the printing mechanism to begin the punching of a new result, and also, if the engine has just come to the end of punching a line of results, to back the Copper Plate and feed a new line, which actions also require two turns of the First Axis. During these two turns DE1 completes the calculation of the new result.

The length of a Result Cycle is thus dependent on the number of digits per result being punched expresssed in the following formula

  The No. of Turns of the First Axis in a Result Cycle = No. of Digits Punched per Result + 2.

This, however, assumes that the number of turns required for each complete calculation within a Result Cycle, the Calculation Subcycle, is less than or equal to the number of digits being punched. In most instances of the use of the Engine this would have been the case. In all normal calculations using the method of differences the calculation subcycle would have been 4 turns of the First Axis. However  if only First Differences were being added to the Table Axis this is reducible to 2 Turns. Or, if special calculations are being performed, which DE1 was programmable for, say 2 additions of the 2nd Difference Axis to the First for every addition of the First to the Table, and the like, then the Calculation Subcycle may be more than 4 turns. If the number of turns for the Calculation Subcycle exceeds the number of digits to be punched then it determines the length of a Result Cycle.

DE1 was capable of being run using two different lengths for the Result Cycle: an extended and an abbreviated Result Cycle. This feature was incorporated in its 20-Axis and 3-Figure Motion Departments, which are described later. Its purpose was, for example a table being punched to 8 significant figures, to allow only the least 4 to be punched during most or abbreviated Result Cycles of 6 turns of the First Axis, but when at the beginning of a new line to use a longer or extended result cycle of 10 turns of the First Axis to punch all 8 digits, when and if any changes in the value of the first 4 had taken place. This procedure was designed to save a great deal of time in the punching of tables, roughly 30%, as well as making them more readable. This saving would have amounted to several weeks of work if the table being prepared took several months to complete.

Extended or Full Result Cycle
10 Turns of First Axis
Turn of First Axis Calculation Punching of Result
      1            No Calc.    1st Sign. Digit if Necessary or Space
      2            No Calc.    2nd Sign. Digit if Necessary or Space
      3            No Calc.    3rd Sign. Digit if Necessary or Space
      4            No Calc.    4th Sign. Digit if Necessary or Space
      5            No Calc.    5th Sign. Digit
      6            No Calc.    6th Sign. Digit
      7            Calc.}  Calculation 7th Sign. Digit
      8            Calc.}  Sub         8th Sign. Digit
      9           *Calc.}  Cycle       Reset
      10          *Calc.}              Reset
Note: * = Calculation on Table Axis

Abbreviated Result Cycle
6 Turns of First Axis
Turn of First Axis Calculation Punching of Result
      1            No Calc.      5th Sign. Digit
      2            No Calc.      6th Sign. Digit
      3            Calc.} Calculation 7th Sign. Digit
      4            Calc.} Sub         8th Sign. Digit
      5           *Calc.} Cycle Reset (and new line if at end of line)
      6           *Calc.} Reset       (and new line if at end of line)
Note: * = Calculation on Table Axis

Line Cycles
A line of results punched across the width of the Copper Plate constitutes a Line Cycle. In the printing of a table of logarithms 10 results might be considered a suitable format. Setting DE1 to punch one extended Result Cycle followed by 9 abbreviated Result Cycles, the last one containing instructions to back the Copper Plate to the beginning of the line and feeding a new line would achieve the necessary layout. Other kinds of table, say trigonometric functions, could be set for a different layout, of say 12 or 15 results across the page. Standard spaces between results are introduced whilst DE1 is punching. Larger spaces can be left after a specific number of results, say 5, have been punched by means of the pattern of studs on the stud wheel controlling the Zig-Zag Motion Crank.

Block Cycles
After a specific number of rows in the table have been punched, say 5, 6 or 10, DE1 can be set to feed a larger new line, to allow for a gap between blocks in the table. This constitutes a Block Cycle.

Note: DE1 had no End of Page Cycle. When the punching of a whole sheet of copper was finished the operator of engine would have had to reset the copper plate carriage manually to the top of a new copper plate.

Control Mechanisms

Control Mechanisms: DE1 principle control mechanisms comprised the following:

Cams -an eccentric disk for putting parts in/out of gear. Acted upon by roller levers

Sine Qua Non Wheels - a pair of cams, one of which was movable the other fixed, which could be sandwiched together to extend or reduce the eccentricity as necessary.

Stud Wheels - wheels carrying an alterable pattern of studs acted upon by roller levers for putting various parts in gear at specific times.

Mechanical Logic: Much of DE1's control used the equivalent of an AND-gate. This is illustrated by the following where Part A drives Part B in which a control mechanism puts them in gear with one another.

Part A        Control      Part B
Moves         In-gear      Moves
Not Moving    In-gear      Not Moving
Moving        Out of Gear  Not Moving
Not Moving    Out of gear  Not Moving

CALCULATING PART

Drive for the Calculating Mechanisms
 

Running under the Engine beneath the General Platform is the Great Axis. It is driven by a wheel fixed on it called the Great Wheel. The Great Wheel is driven by a Pinion fixed on the First Axis placed in/out of gear by a stud wheel in the 2nd 20-Wheel Dept. The overall gear ratio is such that one half turn of the Great Axis is made for every turn of the First Axis. This provides all the calculating mechanisms in DE1 with all the power needed to move them, and, at the same time, gives them the correct amount of angular rotation, for most require a distinct movement of 180 degrees at a time each. The Great Wheel completes its movement at the end of the 9th tenth of a turn of the First Axis; the Pinion is out of gear with it during the 10th tenth of a turn of the First Axis.
 

Timing Diagram of the Drive for the Great Wheel

         PINION ON FIRST AXIS                   GREAT WHEEL
Tenths of  Starting                   Normal         Starting     Normal
Turn of    Teeth                      Teeth          Teeth        Teeth
First Axis

1         \                 } 72 deg    :       \     } 22.5 deg   :
           \                } Rotation  :        \    } Rotation   :
2            \              } Accel.    :         \   } Accel.     :
              \             }           :          \  }            :
3              :                      { |            :           { |
               :                      { |            :           { |
4              :                      { |            :           { |
               :                      { |            :           { |
5              :                      { |            :           { |
               :                      { |            :           { |
6              :   252 deg            { v            : 157.5 deg { v
               :   Rotation           { |            : Rotation  { |
7              :                      { |            :           { |
               :                      { |            :           { |
8              :                      { |            :           { |
               :                      { |            :           { |
9              :                      { |            :           { |
               :                      { |            :           { |
10             :   36 deg               :              Stationary
               :   Rotation             :              and Locked
 
 

A wheel on the Geat Axis drives Horizontal Bolting Axis. This latter rotates through 180 degrees for each turn of the First Axis in a Calculation Sub-Cycle.

A Sector Wheel on the Great Axis drives the Horizontal Adding Axis. The Sector is set to come into gear at the start of the first and third turns of the First Axis during a Calculation Sub-Cycle, driving the Horizontal Axis through 180 degrees on these turns, leaving it stationary at other times.

Another sector wheel drives the Horizontal Carrying Axis. The sector wheel is set to act during the second and fourth turns of the First Axis during a Calculation Sub-Cycle driving the Horizontal Axis through 360 degrees at each of these times, leaving it stationary at others.

On each of the horizontal axes are bevel wheels. Each of these is in gear with bevel wheels at the base of each of the vertical axes. When the horizontal axes turn these latter also rotate but these will run loose around their vertical axes unless otherwise acted upon.

Barrels Department

The Barrels control the placing in/out of gear in a fixed sequence, of the Vertical Carrying, Bolting, Adding Axes with the drive mechanisms. The Barrels carry the "program" for the calculation being undertaken by the Difference Engine. This "program" is set up at the start of any table to be produced.

On the front side of the Engine is found the Carrying and Intermediate Barrel Axis. This carries 14 Barrels (7 for the Carrying Axes and 7 for the Intermediate Axes). This controls the timing for the process of the carriage of tens and also the feedback operations via the Intermediate Axes, when the Engine is "eating its own tail". Connected to it by means of a Communicating Axis on the backside of the Engine is found the Bolting and Adding Barrel Axis. This latter carries 12 Barrels. It is responsible for the timing of the placing in/out of gear of the 6 Vertical Bolting Axes and the 6 Adding Axes. Each Barrel is a drum. Screwed to their curved surfaces are studs. These studs act on levers which via Pin Clutches at the base of each of the Vertical Axes puts those Axes in/out of gear with the bevel wheels on the horizontal driving axes. This is rather like the control mechanism found in a bell-tower carillon.

It is so arranged that in normal difference calculations the Barrel Axes are set to move forward to their next positions twice during every Calculation Sub Cycle of 4 Turns of the First Axis, alternately placing the odd and even vertical axes in and out of gear.  This takes place during the last 10th of a turn of the First Axis during the 1st and 3rd turns of the Calculating Cycle

Calculating Mechanisms

DE1 has 7 Calculating Axes each of which has 16 Digit Cells 1 for each decimal place, except for the Result Axis which has 18. Each Cell has mechanisms for adding and being added to. During the Calculating Sub-Cycle each Cell goes through the operations of Bolting and Locking, Half Turn and Adding, and the Carrying of Tens.

Bolting and Locking. Each Vertical Bolting Axis has 16 Fingers projecting from its shaft, one for each  Cell in the elevation of the Engine. These are arranged in a spiral around one side of the Axes. These same Axes also carry studs for locking the Lower Adding Wheels. During one Calculation Sub-Cycle each Axis goes through one bolting and one locking operation.

During bolting each of the Fingers successively will push the sliding part of a Bolt in-between the teeth on an Upper Adding Wheel. Where, however, the Figure Wheels stand at zero no bolting will take place in those cases.

During locking the studs 'lock' tight all the Lower Adding Wheels on the Difference Axis concerned. These remain locked while the digits are being added to the adajacent axis, after which they are released.

Adding Wheels. Coaxial but running loose on each of the Adding Axes in each Digit Cell are found a Lower and an Upper Adding Wheel. The Lower Wheels are permanently geared to Figure Wheels at the front of the Engine. These register the value of the digit held by the Lower Wheels. Inside each Lower Wheel are fixed two unbolting wedges diametrically opposite one another.

The Upper Adding Wheels have 20 downward-pointing teeth with gaps between them. These are permanently in gear with the corresponding Lower Adding Wheels on the adjacent, next lower Difference Axis.

In between the Upper and Lower Adding Wheel one finds the Bolt. This has a fixed part, which is attached to the Adding Axis, and a sliding part.  The Bolts mechanisms are centred on the Axes and, in their starting positions, are aligned horizontally front to back with the Engine.

Half Turn and Add. If on one turn of the First Axis the Bolts have been pushed forward and are now in-between the teeth on the Upper Adding Wheels, on the very next turn the operation of adding will take place. As the Vertical Adding Axis rotates so will the Bolts fixed on it. This drives the Upper Adding Wheels round. When a Bolt encounters one of unbolting wedges it will be become unbolted from its Upper Adding Wheel, no longer able to drive it round. The Bolt, however, continues to rotate with its Axis until it is aligned front to back with the Engine again. A 180 degrees rotation of the Adding Axis has been made: this action is termed a 'Half-Turn'.

In the meantime the corresponding Lower Adding Wheel on the adjacent Difference Axis will have turned through the same distance as the Upper Adding Wheel, with which it was geared, was pushed by the Bolt. A modulo-10 addition of the digits on the higher difference to those on the lower has taken place: this action is termed an 'Add'.

Carry Warning Mechanism. If any of the Figure Wheels pass through from 9 through to 0 this sets off a 'carry warning'.  A detent releases an arm pivoting on the Adding Axis, the Carrying Lever. This latter Lever is in conjunction with the Lower Adding Wheel in the Cell immediately above. It registers that a carry of 1 is owed.

Carrying of Tens. On each Vertical Carrying Axis set in a spiral are 15 Fingers projecting from its shaft.  When the Axes rotate they do so in turns of 360 degrees. Each of the Fingers acts in succession from bottom to top of the Axis during that turn and picks up all the carriages of tens that are owed. If the Carrying Lever for a particular Cell has been released then the Finger will push it. In doing so the Lower Adding Wheel in the Cell above moves forward one digit. The Carrying Warning Mechanism is also reset. The carry owed has now been paid.

Carriage by Succession. If the Figure Wheel above registers 9, carriage of 1 to it would cause it to indicate 0. Consequently a 'Carry-Warning' would have been set in that Cell. This happens before its Carrying Finger reaches it. When it does catch up its Finger acts on the retracted Carrying Lever and adds 1 to the Cell above that. If there was a sequence of 9s before Carrying started and a carry-warning was set in the cage immediately below the lowest in the sequence, carriage would be transmitted via each Cell up the column by the rotation of the Vertical Carrying Axis. This is termed 'Carriage by Succession'.
 

CHART OF THE CALCULATION SUB-CYCLE OF DIFFERENCE ENGINE NO. 1

                                 Differences

TURN OF    | 6TH     5TH     4TH     3RD     2ND     1ST     TABLE
FIRST AXIS | DIFF.   DIFF.   DIFF.   DIFF.   DIFF.   DIFF.   RESULT
---------- | ------- ------- ------- ------- ------- ------- --------
           |
           |
     1     |         <---- CALCULATING PART QUIESCENT ---->
           |
           |
---------- | ------- ------- ------- ------- ------- ------- --------
           |
           |
     2     |         <---- CALCULATING PART QUIESCENT ---->
           |
           |
---------- | ------- ------- ------- ------- ------- ------- --------
           |
           | HALF-           HALF-           HALF-
     3     | TURN    ADD     TURN    ADD     TURN    ADD        -
           |
           | unlock          unlock          unlock
---------- | ------- ------- ------- ------- ------- ------- -------
           |         CARRY           CARRY           CARRY
           |         &               &               &
     4     |    -    BOLT       -    BOLT       -    BOLT       -
           |
           |         lock            lock            lock
---------- | ------- ------- ------- ------- ------- ------- -------
           |
           |         HALF-           HALF-           HALF-
     5     |    -    TURN    ADD     TURN    ADD     TURN    ADD
           |
           |
---------- | ------- ------- ------- ------- ------- ------- -------
           | (CARRY) unlock  CARRY   unlock  CARRY   unlock
           | &               &               &
     6     | BOLT       -    BOLT       -    BOLT       -    CARRY
           |
           | lock            lock            lock
----------   ------- ------- ------- ------- ------- ------- -------

NOTES

1. The above diagram illustrates the Calculation  Sub-Cycle (Turns 3, 4, 5 and 6) of a Short Result Cycle (6 turns). In a Long Result Cycle (10 Turns) the Calculating Part of DE1 would be quiescent during Turns 1, 2, 3, 4, 5 and 6 and its Calculation Sub-Cycle would be Turns 7, 8, 9 and 10 with the same operations as shown above.

2.  CARRY: Carriage of Tens by Succession. Represents a full or 360 degree turn of the Vertical Carrying (and Figure Wheel) Axes and their Fingers for the Difference concerned. Carrying precedes Bolting during the same phase for each individual 'Cage'.

3.  BOLT: A half or 180 degree turn of the Vertical Bolting and Locking Axis for the Difference concerned, in which the Fingers projecting perpendicularly and in a spiral from the shaft of the Axis act on Bolting Levers which lock the Bolts on the Adding/Calculating Axis to their Upper Calculating Wheels. Bolting succeeds Carrying for a particular 'Cage' during the same turn.

4.  ADD: In which the Lower Calculating Wheels on the Vertical Adding/Calculating Axis are in such a state as to receive addition (Modulo-10) from the digits stored on its immediately adjacent and higher Difference, via the latter's Upper Calculating Wheels. The Lower Calculating Wheels on this Axis are not 'locked' in this state, but free to move.

5. HALF-TURN: A 180 degree turn of the Vertical Adding/Calculating Axis and its Bolts. The Bolts have been locked or bolted in the previous turn to the Upper Calculating Wheels and will cause them to turn through the same angular distance as they themselves move until they are 'unbolted' by the Inner Inclined Planes fixed on the Lower Calculating Wheels in the same 'Cage'. The angular orientation of these indicate the digit stored by that 'Cage'. The angular displacement of this digit will therefore be transmitted to the Lower Calculating Wheel on the adjacent, next lower Difference to which the Upper Calculating Wheel is directly geared, and thence to the Figure Wheel on that Difference. After it has been 'unbolted' the Bolt will continue to rotate during this phase, free of the Upper Calculating Wheel until it is in its starting position front to back with the Engine again. A modulo-10 addition has been performed. During this phase the Lower Calculating Wheels on this Axis  have remained 'locked' tight, so that the Bolts do not push them round as well when they encounter the Inner Inclined Planes.

6.  lock: The Cams on the Vertical Bolting and Locking Axis come into action and lock the Lower Calculating Wheels on that Difference rigidly. Locking takes place for all 'cages' at the same time and commences when Bolting is complete. The Cams act throughout the next half-turn of the Bolting Axes, by which time the Bolting Fingers set around the other half of the shaft are restored to their starting position.

7.  unlock: The Cams cease to act on the ends of their Levers and the Lower Calculating Wheels on that Difference are once again free to move.

8. The 6th Difference shows the action of Carrying in parentheses, thus (CARRY). This is because, although its Vertical Carrying (and Figure Wheel) Axis has Fingers, in normal Difference calculations the digits it carries remain constant and these Fingers therefore do not act on anything. If an Oblique Axis  was connected to the 6th Difference and transfers of values from a lower Difference made via this Oblique Axis then Carriage of Tens might be carried out on it. Similarly the Table or Result Axis shows no action for Bolting. This is because it does not have a Bolting Axis, has no Bolts  and no Upper Calculating Wheels which to bolt to. Neither does it perform the operation of Half-Turn but simply Add and Carrying, as shown. It does have a Locking Axis [see 20-Axis Mechanisms].

CALCULATING PART: BARRELS

The purpose of the Barrels in DE1 is to control the placing in and out of gear, at the appropriate times and in a fixed sequence, of the Vertical Carrying, Bolting and Adding/Calculating Axes, [and also, when the Engine is set to perform feedback operations by "eating its own tail", the Intermediate Axes]. In this sense the Barrels carry the "program" for the calculation being undertaken by the Difference Engine. This "program" is set by the Mathematical Superintendent at the start of any table to be produced.

Each Barrel is a 5 inch diameter gun metal drum. The placing in gear of the various Vertical Axes is effected by means of Studs screwed to the curved surfaces of the Barrels. Each has 48 such fixing positions for Studs equally spaced around its circumference. Each acts on a sprung Roller Lever, the Roller of which runs around the surface of the drum as it rotates. When it encounters a Stud the other end of the Lever, which is connected with a Pin Clutch at the bottom of one of the Vertical Axes puts that Axis out of gear with its Mitre Wheel. If a blank position on the drum's surface is met that particular Vertical Axis is put in gear with its mitre wheel ready to be driven. In a way this process is similar to the control mechanism found in a bell-tower carillon, in which a set of levers act on the studs of a rotating drum. These pull on a set of ropes attached to the bells causing them to be played in a fixed sequence. This results in a tune which can be repeated over and over again*.

Across the front side of the Engine and fitted just above the General Platform of the Engine is found the Carrying and Intermediate Barrel Axis. This carries 14 Barrels of the type described above, on which 14 Roller Levers act (7 for the Carrying (or Figure Wheel Axes) and 7 for the Intermediate Axes). This controls the timing for the process of the carriage of tens [and also the feedback operations via the Intermediate Axes and Oblique Axes, if the Engine is "eating its own tail"]. Connected to it by means a set of Mitre Wheels and a Communicating Axis on the backside of the Engine is found the Bolting and Adding Barrel Axis. This latter carries 12 Barrels of the above type and is responsible for the timing of the placing in and out of gear of the 6 Vertical Bolting and Locking Axes and 6 Adding/Calculating Axes. Again 12 Roller Levers (one for each Vertical Axis) act on these. Because of the interconnections both Barrel Axes and their Barrels rotate in unison.

The Mitre Wheel connected to the Carrying and Intermediate Barrel Axis and which drives the Mitre Wheel on one end of the Communicating Axis carries a dial or Notice Wheel to indicate the general orientation of the Barrels to the operator of the Difference Engine.

In normal difference calculations the Barrel Axes and the Barrels fixed on them are set to move forward twice during every Calculation Cycle of 4 Turns of the First Axis, alternately placing the odd and even vertical axes in and out of gear. When they move they do so by one 48th of a full turn (or 7.5 degrees or 1 Stud position.) It is so arranged that this takes place during the last 10th of a turn of the First Axis during the 1st and 3rd turns of the Calculating Cycle [3rd and 5th turns of the Short Result Cycle, or 7th and 9th turns of the Long Result Cycle].

To effect this 4 large wheels (ca 11.5" in diam.) of 48 teeth fixed on two (intermediary) Axes lead the power from the First Axis or source of motion to a 5th 48-(toothed)-Wheel which drives the Carrying and Intermediate Barrel Axis (which in turn via the Communicating Axis drives the Bolting and Adding Barrel Axis at the back of the Engine). The First Axis drives the 1st 48-Wheel and its [intermediary] Axis by means of a wheel with a single tooth on it known as the 1st 48-Wheel Starting Tooth. For every turn of the First Axis this drives the 1st 48-Wheel a 48th of a revolution which in turn drives the 1st 48 (intermediary) Axis and a 2nd 48-Wheel fixed on it. The 2nd 48-Wheel then drives a 3rd 48-Wheel and its (intermediary) Axis which drives a 4th 48-Wheel fixed on it. This latter is in gear with the 5th 48-Wheel fixed on the Carrying and Intermediate Barrel Axis. The driving Wheels in this arrangement have square teeth, and each tooth on the driven Wheels is half-square and half-saw. The intermediary Axes are supported by brackets and plumbing blocks bolted to the underside of the General Platform of the Engine. The Axes run horizonatally to the long axis of the Engine under its Calculating Part.

This arrangement would drive the Barrel Axes forward one 48th every turn of the First Axis; such an arrangement could be made to work by a suitable pattern of Studs on the Barrels, but is not the one Babbage adopted in BAB.[U11]. The 2nd and 4th 48-Wheels have 48 removable teeth. In BAB.[U11] it seems that every second tooth on has been left off the 2nd 48-Wheel and the first in every group of three left off the 4th 48-Wheel. When the 2nd 48-Wheel is turned it only bites the 3rd 48-Wheel on every other turn of the First Axis, and when the 4th 48-Wheel is turned it only drives the 5th 48-Wheel and the Barrel Axes on the required turns. This arrangement gives the desired pattern of movement to the Barrels during the Short Result Cycles and the latter half of the Long Result Cycle (fifth to tenth Turns of the Cycle).

The Starting Tooth on the First Axis is connected to a Lever which acts on a Stud Wheel controlled by the 20-Axis. It is so arranged by horizontally sliding along the First Axis to come into gear with the 1st 48-Wheel when dictated by the Studs on the 20-Axis Stud Wheel on the 2nd tenth of a turn of the First Axis. BAB.[U11] shows this potentially happening on every turn of the First Axis (a dashed line at these times); i.e. the Starting Tooth for the arrangement shown on the chart remains in an active position to interact with the 1st 48-Wheel for all cycles, but that should it be required it can be put in or out of gear by the 20-Wheel Stud Wheel at these times.

The 1st 48-Wheel (intermediary) Axis can be driven by the First Axis by another means. There is a Pinion on the First Axis which has 30 teeth. This can interlock with the teeth of Sector Wheels fixed on the 1st 48-Wheel (intermediary) Axis. These latter Sector Wheels also have removable teeth which can be set to be from 0 to 144 in number. When this happens the First 48-Wheel Axis will be driven forward at a faster rate than that provided for by the Starting Tooth: 4.8 turns of the First Axis as compared with 48. BAB.[U11] shows this potentially happening [by dashed lines on the chart] during the first four turns of the Long Result Cycle. Babbage has, however, left no description of the purpose of this arrangement and we can therefore only speculate on this.

This speculation is made more difficult by the fact that I believe Babbage has made an error on BAB.[U11]. Each Line Cycle for a table of Logarithms, three Result Cycles for which I believe is what BAB.[U11] is trying to depict, is 64 turns of First Axis long: one Long Result Cycle of 10 turns and nine Short Result Cycles of 6 turns each. Whilst the arrangement of the teeth on the 2nd and 4th 48-Wheels described above is fine for the Short Result Cycles, during the Long Result Cycle the 4th 48-Wheel will get out of step, the length of the Long Result Cycle not being divisible by 3. It might be supposed that the Pinion and Sectors can be arranged to fast forward the 48-wheels and Axes over this problem during this particular Result Cycle bringing the arrangement back into step before the Short Result Cycles begin. This it cannot do as the Line Cycle is too long with respect to the number 48, the number of steps required to make a full revolution of the 1st 48-Wheel Axis and the Sectors fixed on it. The Sectors would be brought into action again for a second time long before the line was finished, which having brought the 4th 48-Wheel into step earlier would take it out of step again.

For DE1 to work in printing a table of Logarithms, a different arrangement from that described above is required: the 2nd 48-Wheel would again have every 2nd tooth missing and the 4th 48-Wheel would have the first in every group of three taken out of action. The Stud Wheel on the 20-Axis controlling the Starting Tooth of the 1st 48-Wheel would have its Studs so arranged as to take the Starting Tooth out of action during the first four turns of the Long Result Cycle, but putting it into gear for the last six turns and for all the Short Result Cycles. The Pinion and Sectors arrangement would be totally disabled, not being required.

The Barrels would be moved by this arrangement, as required, on every 3rd and 5th Turns of the Short Result Cycles and the 7th and 9th Turns of the Long Result Cycles. Their studs would be arranged in 24 adjacent pairs of equal patterns around the circumference, each pair of patterns putting the odd and even difference Vertical Axes in and out of gear alternately. The fact that the Barrels are not in step with the Line Cycle is irrelevant. During one Line Cycle the Barrels would rotate through 40 out of 48 steps; after 6 lines they would be back once again in step with the Line Cycles.

The arrangement described is for normal difference calculations. Different patterns of teeth on the 2nd and 4th 48-Wheels can be set, and, of course, many different arrangements of Studs on the Barrels are possible. For instance it would be possible to set it so that the 3rd Difference Axis could add its value to the 2nd Difference Axis 2 or 3 (or n, n being a whole number) times for every once the 2nd Difference Axis adds its value to the 1st Difference Axis. This flexibility gives DE1 the power to perform many different types of difference calculation other than the standard. In addition feedback calculations are possible by arranging for the Intermediate Axes to come into gear with the Figure Wheel Axes and by means of Oblique Axes transfer results from one set of Axes to others. The formulae for some of these are suggested in the paper by Lardner on DE1 (1834) [see footnote on P.73 Babbage's Calculating Engines].

Driven by the 3rd 48-Wheel (intermediary) Axis is a 6th 48-Wheel which too has 48 removable teeth. [The Origin of Motion Chart on BAB.[U11] actually shows this wheel being driven by the 5th 48-Wheel, but the Timing Diagram beneath it indicates that its source of motion must be otherwise and as I suggest.] This wheel drives a 7th 48-Wheel which carries a Stud Wheel with 24 positions for Studs. These Studs are responsible for the timing at which the Starting Teeth of the 1st and 2nd 20-Wheels are slid horizontally on the First Axis into or out of gear with the Wheels they drive. Presumably the pattern of teeth on the 6th 48-Wheel is set to be the same as that on the 4th 48-Wheel so that the 7th 48-Wheel and its Studs are moved in unison with the Barrels. As printing is forbidden during those turns of the Axis when the Engine is adding differences to the Table Axis or resolving carriages of tens on it, since the pattern of studs on the Barrels indicate the turns on which these operations are performed, and as no printing takes place if the 20-Axis is not moved, it might be thought that the 7th 48-Wheel Studs signal to the 20-Axis department the times when printing operations are not allowed. This is not the case. On BAB.[U11] the Starting Teeth of the 1st and 2nd 20-Wheels are shown being moved into gear by the studs on the 7th 48-Wheel every turn of the First Axis during the last 10th of a turn.

See 20-Axis department for further details.

Note: * Carillon (from Latin 'Quadrilionem'): originally 4 bells in a row struck by hand-held hammers xylophone style, later meaning extended to cover same 4 bells or up to 5 octaves of bells arranged in a chromatic (12 tone) scale. The Dutch in the 12th/13th century AD perfected a means of striking these bells automatically. They set up large revolving barrels several feet in diameter driven by water power or falling weights and a pulley mechanism. Around the circumference or surface of the revolving barrel or cylinder were set a series of pegs or pins (similar to those found in the later developed music boxes) which as the barrel turned would catch hold of a hammer or lever and release it to strike a bell. If the arrangement of pegs or pins was in the right order and the bells were caused to be struck at the right time intervals a tune would result. A different arrangement of pegs/pins would, of course, play a different melody. These tunes/melodies would be endlessly repeated as the barrels turned.

Calculating Part: Indicating Apparatus or Alarm Bells

This is sometimes called in the surviving Mss on DE1 'Apparatus for Pointing Out the Nines' or any other figures come to that. It comprises:

a) Indicating Wheels A wheel of this type is fitted to the socket at the base of the Drum of every Figure Wheel, where it is retained by stiff friction. It can be turned by hand to any orientation so as to be able to point out any particular digit (0 to 9) that that Figure Wheel might arrive at. A line inscribed on its circumference assists in aligning it with the Index or Pointer on the Shade or Blind fixed to the framing, which points out the digit currently held by the Figure Wheel. The Indicating Wheel has a recess cut out of its circumference for the purpose of allowing an Indicating Roller to fall into it.

b) Indicating or Alarm Arms and Alarm Axes These are arms which are fitted to an axis one above the other, each projecting perpendicularly from a hollow, cylindrical Boss or Spindle loose on the shaft of the axis. Each Boss has a small slit in its side which receives a screw with a capstan head, which fixes it to the Axis so as to allow the Boss to slide up and down on the head of the screw, but to prevent the Boss from turning on the Axis. The axes are called Alarm Axes: there is one for each Difference. These Axes pass vertically through the Framing Plates behind the pillars which support the framing plates on the front side of the Engine, on the opposite  side of the column of 'Cages' to the Carrying apparati. At the extreme end of each Arm is a notch made to receive a Roller (made of ivory?) which is retained on its arm by means of another small screw with a capstan head.

At the top of each Alarm Axis is an arm (fitted to the upperside of the top Framing Plate in the 1832 Fragment of DE1) which has and is acted upon by a Lever and Spring. This Spring presses all the Arms and their Rollers against the Indicating Wheels on that Axis. When all the Rollers drop into the notches in the Indicating Wheels (in an orientation when they are all directly behind the Figure Wheels when one is facing the Engine from its front side) then the hollow, cylindrical Bosses of the Arms all slip down on the head of the small screw which fixes them to the Alarm Axis; and when this happens, the topmost Arm or Lever lets loose a detent which pulls a wire attached to a Hammer, which, in turn, strikes a Bell. The purpose has been achieved: the Bell for that Difference is rung when all its Figure Wheels on that column reach the number which had been previously set by hand or 'indicated' by its Indicating Wheels. It is probable that Babbage intended the Bell on each Difference to have had a different tone, so that the Superintendent or Operator could know which Axis had rung and had thereby reached the desired number.

If all the Indicating Wheels on a particular Difference are set to indicate 9999 ... 99, then when all the Figure Wheels on that axis reach that number, its bell will ring -indicating that 'overflow' is about to take place and all the wheels are about to pass through to 0000 ... 00 (Zero or DE1's Infinity). Babbage had hoped this could be used for two important purposes:

(1) to solve equations by finding their roots using the Newton-Raphson method.

(2) to prevent the Engine from calculating beyond its range of digits. It is probable that a mechanism would have been included which let loose a lever when a bell rang putting the calculating part of DE1 out of gear, and stopped it continuing.

This apparatus suffers from two major defects in its operation and thus fails in its principal purpose. It only correctly tests for a number if the Figure Wheels have completed a calculation and are still. Bromley (1986) has shewn that the Alarm mechanism would only have worked if the indicated number appeared on the column of Figure Wheels immediately after the process of Carrying. But if the column of Figure Wheels passed through this number during Carrying then the Indicating Arms would miss it, and Bell not ring when it might be expected to have. More  than that the Alarm Bell would go off when not required to; that is after an Add process but before the Carriage of Tens; if the desired number was arrived at then the bell would ring, but, of course, the full process of addition has not been not completed, and the Engine has given us a false warning. It appears Babbage was ignorant of these defects.

Locking Mechanisms

Many of the parts in DE1 have been fitted with a Locking mechanism. Most of these consist of a wheel or cam with a scallopped edge around which the Roller of a sprung Roller Lever runs. The number of scallops on this wheel determines the number of 'locking' positions for the part concerned. This wheel is called a Locking or Roller Wheel.

There are two types of Locking:

(A) Rigid Locking. This involves the Roller of the Sprung Roller Lever being pressed solidly into one of the recesses in the scallopped edge of the Locking Wheel. In this instance the Roller Lever is often 'double-ended'. Rigid Locking is effected by means of a Locking Cam or stud acting on the opposite end of the Roller Lever from its Roller. This locks the part to which the wheel is connected tightly in the given position until the Cam or stud is moved and the Roller released.

(B) Loose Locking. In this case the Roller of the Roller Lever is not pressed hard against its Locking Wheel, but is only held there lightly by the  spring on the Lever. Levers of this type are usually only single-ended. Movement of the part to which the Locking Wheel is connected is allowed, but only in discrete steps determined by the positions of the recesses along the edge of the Locking Wheel. This effectively digitises the movement of the part concerned.

It is possible from the various manuscripts that have survived to identify most of those parts in DE1 which were supposed to have had a locking mechanism. A list of these follows. It is not necessarily complete, but as Babbage never left any overall account of them the existence of others can only be speculated upon. Itemized below are each of those parts known so far as having one, together with a description of its nature, the number of locking positions and any other relevant notes:

Great Wheel: Type B, 2 positions.

Horizontal Bolting Axis: Type B, 2 positions.

Horizontal Calculating Axis: Type B, 2 positions.

Horizontal Carrying Axis: Type B, 1 position.

Lower Calculating Wheels on 1st to 6th Differences: Types A and B, 20 positions. Type A locking occurs when the Locking Cams on the Vertical Bolting Axes act  on the ends of the double-ended Roller Levers (see Bolting and Locking); Type B happens when the active edges of these Cams are pointing away from the ends of the Levers. The Lower Calculating Wheels each have 20 D-shaped upward-pointing crown teeth. These provide the necessary scallopping effectively making each its own Locking Wheel.

Lower Calculating Wheels and Snails on Table or Result Axis: Types A and B, 20 positions. The same as for the Lower Calculating Wheels on the other Differences, but in this instance the Locking Cams are found not on a Bolting Axis but are found instead on an Axis placed in the same position had the Table Axis had one. This Axis is called the  Locking or Eccentrics Axis and is moved and controlled by the Lower 20-Axis and not by the Horizontal Bolting Axis. Type A locking occurs when the Locking Cams on the Vertical Eccentrics or Locking Axis act  on the ends of the double-ended Roller Levers (see Bolting and Locking); Type B occurs when the active edges of these Cams point away from the ends of the Levers. The Lower Calculating and Snail Wheels of the Table Axis also each have 20 D-shaped upward pointing crown teeth. These again provide the necessary scallopping making each its own Locking Wheel.

Barrels: Type B, 48 positions.

20-Axis and 1st 20-Wheel: Type A, 20 positions.

2nd 20-Wheel: Type A, 20 positions.

Vertical Bolting/Locking Axes: each Type A, 2 positions. Their locking is not provided for by Roller Wheels but by the Pin Clutches. Each of these has two studs in the upper half of their mechanism. They work in this fashion: when the Bent Lever acts on a stud on the relevant Barrel (see Calculating Part:Barrels) these two studs lock rigidly the Vertical Axis concerned to two recesses in the Upper Platform (a gun metal plate fixed ca. 7 inches above the General Platform of the Engine).

Vertical Adding/Calculating Axes: each Type A, 2 positions. Same mechanism as for Vertical Bolting/Locking Axes.

Vertical Carrying Axes: each Type B, 1 position. Unlike the other vertical axes in the calculating part of DE1 Locking Wheels with sprung Roller Levers are fitted to the base of each Vertical Carrying Axis to provide for their locking.

1st 60-Wheel and Axis: Type B, 60 positions.

Calculating Part: On the Intermediate and Oblique Axes

In Bromley's long review of Franksen's article in the Annals of the History of Computing October 1983 Vol. 5 No.4 pp 411-415, he states on page 412:

"In section 12 Franksen deals with Babbage's observation that the second difference of the sine function is itself proportional to the sine. He illustrates an implementation of this function within the conceptual framework of the Difference Engine. Babbage did provide facilities in the fragment of the Difference Engine [1833] whereby the engine could 'eat its own tail' by feeding back a digit of the tabulated function to the second difference, as alluded to in the note at the end of this review. The sine function could not be so generated, however, for there was no mechanism to effect the multiplication required. Collier has shown the critical importance of this example in the evolution of the Analytical Engine. To perform a multiplication requires the addition of many partial products; the 'anticipating carriage' was devised to speed these additions. The mechanical complexity of the anticipating carriage led, in turn, to the clear separation of the 'mill' (equivalent to the central processing unit of a modern computer) from the 'store', the axes of which relinquished to the mill all of the calculating power possessed by the axes of the Difference Engine."

There is a need to clarify the points he has made about DE1's power to produce a table of sines, and how Babbage intended to do it with special facilities he planned to incorporate in the machine. The key to understanding this was that the angles were to be measured in Radians, not Degrees. Babbage thought that DE1 did not need the full power of multiplication whilst 'eating its own tail' to do this, but hoped to have been able to produce the tables by a much simpler mechanism, the 'stepping' of powers of tens alone.

I quote with an extract from a letter from Babbage to JFW Herschel (Letter no.171, dated April 9th 1822. Herschel-Babbage Correspondence, Herschel Collection Vol 2: Royal Society, London):

"... Another idea concerning a table of sines to make by once setting an engine is not quite ripe but you shall have the embrio. You know that

D2sin Q = -(2.(sin h/2) 2 .sin(Q+h))  [See Below]
suppose for a moment that in the difference of any two arcs in the table is such that

2.(sin h/2) 2  = .0001

then it would be easy to make a machine in which the second difference should be made by transferring the preceding tabular number cutting off the four figures at the end and such as one would make without interruption a table of Sin h, Sin 2h etc. ..."

This is true as long as the arcs or angles are measured in radians. It was fortunate that Babbage got the principle right even though he had made a mistake in the above formula!

The correct formula for the above can be proved thus:-

Consecutive Values in a Table of Sines T-1 = Sin(x-h)    (where x is the base value of the table T0  = Sin(x)      and h its interval)

T1  = Sin(x+h)

First Differences are therefore D1(Sin)-1 = T0-T1 = Sin(x)-Sin(x-h) D1(Sin)0  = T1-T0 = Sin(x+h)-Sin(x)

Second Difference is therefore D2(Sin)-1 = D1(Sin)0 - D1(Sin)-1 = Sin(x+h) + Sin(x-h) - 2.Sin(x) = Sin(x)Cos(h)+Sin(h)Cos(x)+Sin(x)Cos(h)-Sin(h)Cos(x)-2.Sin(x) = 2.Sin(x).(Cos(h)-1)

But (Cos(h)-1) = -2.(Sin h/2) 2

Therefore D2(Sin)-1 = -4.(Sin h/2) 2 .Sin(x)

Let x-h = Q

D2(Sin(Q))-1 = -4.(Sin h/2) 2 .Sin(Q+h)  Q.E.D.

If -4.(Sin h/2) 2  = 0.0001 Then h = 0.010000041667 radians

This is close enough to 0.01 to allow a Table of Sines interval 0.01 radians to be constructed using this method: viz.  by making the second difference in the next calculation equal to the preceeding calculated value of the Sine function divided by 10,000 (ie. stepped down 4 decimal places). Babbage planned to achieve this by devising an apparatus which could be attached to the front of his DE1 which allowed it "to eat its own tail".

[The following has been adapted with additions from CB's own notes on the matter. Babbage's Quarto Scribbling Book No. 12, Science Museum, London] That apparatus comprises of sets of special gun metal wheels, one for each figure wheel, on 7 vertical axes (6' 4 3/4" long 9/32" diam.) parallel to each of the Figure Wheel Axes called "Intermediate Wheels and Axes", (situated just to the left of each Figure Wheel Axis on the engine so as not to obscure the Figure Wheels themselves). Each of these Intermediate Wheels carries spur teeth which are always in gear with the immediately adjacent Figure Wheel. The upper side of each of these wheels carries a bevel wheel which gears with an inner bevel wheel on the cross studs. This gearing in made or broken for all the wheels at once by raising or lowering the Intermediate Axes. The latter is controlled by levers acting on the studs or the absence of studs on the Intermediate and Carrying Barrel.

The Intermediate Axes are supported by additional parts projecting from every fourth framing plate. Through the extremities of all these projecting plates pass two collar bolts, which bolts pass through four pairs of sockets each pair connected by a cross bar. In each of these cross bars is screwed a horizontal stud on which turns a double bevel wheel attached to an Intermediate Wheel, the other end of which gears with a bevel on the Oblique Connecting Axis [called 'Oblique' as they cross the front of the machine obliquely].

The Oblique Connecting Axes turn in collars which screw into the horizontal studs and at their other end have bevel wheels which connect them with a similar set of apparatus on another Difference Axis.

Depending on the setting of the machine, by and large even numbered axes could be connected either additively [but not  subtractively, see below] in this fashion to other even numbered difference axes [Table, 2nd, 4th or 6th], or odd numbered axes to odd [1st, 3rd and 5th].

The formulae for the connection would not have been: Next value on 2nd Difference axis [or other even numbered axis] = value on Table axis/10,000 [or other power of 10] But rather Next value on 2nd Difference axis = Old value on 2nd Difference axis + (Value added to Table axis [from 1st difference axis]/10,000).

Mathematically: If D2Ux+1 = -(Tx/10**4) Then D2Ux+1 = D2Ux - (D1Ux/10**4)

Which is how Babbage intended DE1 to work

In fact two sets of steel oblique axes were made by Clement for DE1: [From accounts held in the Public Record Office, Kew] 4 short (6ft 6ins long  19 /32 ins diam.) and 4 long  (7ft 6ins long  19 /32 ins diam.)

Given this apparatus, if it had worked (see below), and the above formulae it would have been a very easy matter for DE1 to have produced a Table of Sines in Radian Measure for intervals of 0.01 radians, 0.001 radians, 0.0001 radians etc.. It would also have been a very easy matter, in its normal mode, for the engine to have produced a Table converting Degrees to Radians. (It would not have been so easy for it to produce the reverse, Radians to Degrees, as Degrees do not lend themselves readily to the Decimal System so well. In any case Babbage, being a first and foremost a mathematician, would probably have preferred his Sine Tables to have been in Radians anyway.) Anyone possessing a programmable pocket scientific calculator with at least three memories can demonstrate for themselves the principle upon which it is based.

The fact is, however, the Oblique and Intermediate Axes apparatus would have worked in the production of Hyperbolic Sines as positive values of differences would have been added to one another. But as the production of a Table of Sines requires the subtraction of the 2nd Difference from the 1st, which could only be performed by DE1 by the adding of negative complements, the proposed apparatus would not have worked properly. It is probable Babbage was not aware of this defect: I am grateful to Allan Bromley for pointing out this fault in his reasoning.

PRINTING DEPARTMENT

20-Axis (or Printing Axis) Department

The 1st 20-Wheel and the 20-Axis. The 1st 20-Wheel is a 20 cog-toothed wheel fixed on a shaft called the 20-Axis parallel to the 1st Axis. It can be locked in 20 possible orientations. Its job is to count the number of digits in a result that have been stamped, and to select the next or to report that the punching of a result has been completed, and that it is time to calculate the next.  The 20-Axis has two motions.

Digit by digit motion: for each turn of the handle, it is driven forward one position by a single tooth located on the 1st Axis. This latter is placed in gear with the 20-Wheel by a lever acting on a stud wheel in the Barrels Department. This tooth is oriented to act on the 2nd tenth of each turn of the First Axis.

Fast forward motion: this is required to wind the 20-Wheel mechanisms back to their starting position. On the 20-Axis are a pair of adjustable sector wheels, one fixed the other movable, which are driven by a pinion on the First Axis. 2 turns of the First Axis are allowed for fast forward motion, which is usually set to occur on the 5th and 6th turns of the First Axis in each result cycle of 6 turns. During the turn just before extended result cycles have to occur the movable sector is slid out of contact with the pinion on the First Axis.  This allows the 4 most significant digits of a result to be selected and punched when they need to be.

The 1st 20-Wheel dept directly controls: (a) The orientation of the Spiral Axis hence selects digits to be punched on that turn. And the locking of the Calculating Wheel bearing the value being punched. (b) A Stud Wheel to put the Barrels Department in/out of gear. (c) The orientation of various Sine-Qua-Non Wheels Two which put the Returning and Zig-Zag Motion Cranks in/out of gear. And one which controls the timing of the swinging in/out of the Type Sector.

2nd 20-Wheel Department. On the 20-Axis, running loose around it, is another component called the 2nd 20-Wheel. This too has 20 cog-teeth and 20 possible orientations, and also its own locking mechanism. It too is driven forward by a single tooth on the First Axis. This tooth is set behind that which drives the 1st 20-Wheel, and is arranged to come into action on the 8th tenth of a turn of the First Axis. The 1st 20-Wheel dept acts on each turn of the First Axis before digits are punched, whereas the 2nd 20-Wheel dept deals with those actions required afterwards. Both single teeth are placed in/out of gear at the same time as they are controlled by the same stud wheel in the Barrels Department. This takes places on the 9th tenth of a turn of the First Axis.

The job of the 2nd 20-Wheel is to control (a) A stud wheel which determines the time when the Calculating Part comes into action. (b) The format and spacing of the punching of results by means of two stud wheels. (c) The timing of the engagement of the Friction Cones which pull the Copper Plate along to the next digit position.

When the 1st 20-Wheel is zeroed by means of fast forward motion so too is the 2nd 20-Wheel. There are a pair of arms to effect this.

Selection and Punching of Results

Punching. DE1 wants to punch a digit on each turn of the First Axis. For each turn the Forcer of the Punches bobs up and down once. This happens whether a punch is underneath it or not. If a punch is in position then the digit it bears will be stamped on the Copper Plate, otherwise the Forcer continues to act but without effect. The Forcer is driven by a cam turned by a direct train of shafts geared with the First Axis

The Spiral Axis. The 20-Axis described above drives the Lower 20-Axis parallel to it at the rear of the machine. This axis drives a Communicating Axis. This latter passes right underneath the machine driving the Spiral Axis on the front side of the Engine. The Spiral Axis is a vertical shaft around whose body are set, protuding in a spiral, 18 fingers and two blank positions, one for each digit on the Result Axis. Thus for each turn of the First Axis the Spiral Axis moves round one twentieth. As a conseqence of its physical arrangement one of its fingers ends up being closer to the Type Sector than the others. This is the finger that is opposite that digit on the Result Axis to be stamped on that particular turn.

Type Sector and Axis. The Type Sector is DE1's daisy-wheel-like mechanism. It is a 45â frame for holding 11 vertical punches, one for each digit 0-9 and the decimal point. It swings around a vertical axis in a horizontal plane. On its vertical shaft are 18 arms set one above the other, one opposite each digit on the Result Axis. At the end of each is a Drop Pin which can click up and down like a retractable biro.

On the First Axis is a Cam which causes the Type Sector and its Axis to swing in underneath the Forcer and out again towards the Spiral Axis. This happens on each turn of the First Axis when a digit has to be punched. Otherwise a detent controlled by a SQN wheel holds it back out.

On the First Axis is yet another Cam. This lifts the Spiral Axis up and down once for each turn of the First Axis. That finger on the Spiral Axis which is closest to the Type Sector Axis' arms when the latter swings out lifts that Drop Pin closest to it. All the remainder remain down. When the Type Sector Axis swings back in again this Drop Pin is up, whilst all the others are down. On swinging back out again the Spiral Axis pushes this Pin back down again when it is lowered by the Cam.

Selection of Value of the Digit to be punched. Fitted to each Lower Calculating Wheel on the Result Axis is a Snail Wheel.  These are cam-like devices with two limbs, shaped rather like a double spiral galaxy. They have been specially designed to present a different radius for each value of the digits borne by the Lower Calculating Wheel. When the Type Sector Axis swings in, that Drop Pin which is up strikes against its corresponding Snail Wheel, causing the swinging to halt. Depending on the Snail Wheel's orientation so the Type Sector adopts the appropriate angle corresponding to the value of the digit required to be punched.

Figure to Figure Motion of the Copper Plate

On the First Axis are two cranks. For each turn of the First Axis these pull ratchet wheels fixed to an axis at the rear of the Engine. The amount the Ratchet Axis is turned depends on the number of teeth that are exposed on the ratchet wheels. Each has a shade which covers or reveals a different number of its teeth. There are two stud wheels, one for each ratchet, on the 2nd 20-Wheel, which control the movement of the shades.

The Ratchet Axis drives the Endless Screw Wheel of the Figure to Figure Motion department. If two pairs of Cones have been pressed together then, by friction, the Endless Screw Wheel will turn the Ribbon Axis. This in turn drags the Copper Plate in its frame along the Upper Slider to the next figure stamping position. The timing of the pressing together of the Cones is controlled by a SQN Wheel on the 2nd 20-Wheel.

In turning the Ratchet Axis the Cranks act additively. In switching the ratchets' shades on/off the Stud Wheels on the 2nd 20-Wheel determine the spacing of the digits in and between results. In this way DE1 can deal with different size typefaces.

The 60-Wheels Department

Its job is to count how many results have been calculated and punched, and then to determine certain actions after a particular number of these have taken place. It comprises 3 large wheels called respectively the 1st, 2nd and 3rd 60-Wheels set on a shaft called the 60-Wheel Axis. The 1st and 3rd 60-Wheels are fixed on the axis and rotate in unison together. The 2nd runs loose, but is driven by movable teeth on the 1st 60-Wheel. A locking mechanism holds the 1st 60-Wheel in place. The 2nd 60-Wheel has its own separate locking mechanism. Each of the wheels has 60 orientations. The number 60 was chosen as it has a large number of divisors suitable for specifying the number of columns in tables. For each turn of the 20-Axis, i.e. one result, a single tooth on the 20-Axis drives the 1st 60-Wheel forward one position.

The 1st 60-Wheel has two sets of studs. One set determines when the Returning Crank is to be placed in/out of gear: this identifies how many results across the page are to be stamped (2, 3, 5, 6, 10, 12 etc.). The other side determines when the Zig-Zag Motion Crank is to be placed in/out of gear. The 3rd 60-Wheel has 30 studs. These determine which of the result cycles are abbreviated ones and which are extended. Its studs  place in/out of gear the movable portion of the Sector Wheel which fast forwards the 20-Axis.

The 2nd 60-Wheel counts the number of lines that have been punched. After a particular number of these it specifies when a wider line spacing is to be used. Its studs control the shade of the teeth on the Line to Line Motion Ratchet Wheel.

Cranks Department

These are DE1's "carriage return line feed" mechanisms. They return the Copper Plate to the start of lines and move the frame down so that new lines of results can be punched. This action takes two turns of the First Axis.

Returning Motion and Zig-Zag Motion Cranks. A wheel on the First Axis drives another wheel and a shaft perpendicular to it. On the shaft is a spur wheels which drives  two other wheels, one which drives the Returning Crank Clutch and the other the Zig-Zag Clutch. If the Clutches have been placed in gear, determined by levers acting on the studs on the 1st 60-Wheel, then at a time in the Result Cycle specified by the SQN wheels on the 20-Axis the First Axis will drive the Returning and Zig-Zag Motion Cranks. These two mechanisms work in opposite directions. The Returning Motion Crank pulls the Upper Slider on which the Copper Plate rides back to the start of lines. The Zig-Zag Crank pulls the frame forward along a line and is responsible for general column layout of tables.

Line to Line Motion Crank. At the same time that the Copper Plate is backed a wheel on the Returning Crank Axis drives the Line to Line Motion Axis. This drives a crank which pulls a ratchet wheel with a shade. This shade controls the number of teeth exposed on the Ratchet and is itself governed by the studs on the 2nd 60-Wheel. It determines when single or 1½ line spacing is to be used. Through a train of mechanism this drives an Endless Screw which pulls the Lower Slider giving Line to Line Motion to the Copper Plate.

3-Figure Motion Department

To save time, paper, size of table to be punched etc., about 30%, to make the tables produced by DE1 more readable Babbage devised a special formatting device which enabled his Engine to punch two different layouts for results:-

    (a) Extended or Full Result
    (b)  Normal or Abbreviated Result

Extended results were used when all the digits of a result were to be punched. Normal results were used when only the last n (n=4, say) least significant digits were to be stamped.  Extended results were intended to occur at the beginning of each line of results, but only when the significant digits of a result had changed. Otherwise normal or abbreviated results were to be used. Babbage's manually produced Logarithm Table (1826) illustrates this layout.

The timing of extended result cycles is determined by the 3rd 60-Wheel. At the beginning of each line the first result to be stamped always requires an extended result cycle (of say 10 turns of the first axis). Whether or not a full or abbreviated result is stamped, however, is decided by the 3-Figure Motion Department.

One of the Lower Calculating Wheels (say the 5th significant digit's) is screwed to the Result Column's Calculating Axis. All the others run loose around it. Note that there are no bolts on the Result Column and thus no conflict of interest occurs. At the base of the Calculating Axis are fixed a pair of arms. These arms adopt the orientation of the fixed Calculating Wheel. Whenever that Calculating Wheel passes over from 9 through to 0 one of these arms, via a series of levers, triggers the release of the 1st detent of the 3-Figure Motion Bar. This indicates that there has been a change in the value of say the 4th significant figure on the Result Axis. The 2nd detent of the 3-Figure Motion Bar is released by the Line to Line Crank at the end of a line of results. When the Bar is released this causes the movable part of the 3rd Sine-Qua-Non Wheel (part of the 1st 20-Wheel department) to be slid out of action. When both detents have been released this allows the first few significant figures of result to be stamped. Otherwise the Type Sector is held out of action are not permitted to swing in for punching during these first few digits. DE1 will go through the motions of attempting to punch a full result with all its digits at the beginning of the line, that is require the same number of turns of the first axis of an extended result cycle, but without effect, only stamping an abbreviated result.

Upon completion of its action the 3-Figure Motion Bar and its detents are reset by an inclined plane acting on the 3rd Sine-Qua-Non Wheel.

Selection and Punching of Results

(a) Punching

DE1 wants to punch a digit on each and every turn of the First Axis. For each turn of the First Axis the Forcer of the Punches bobs up and down once. This happens on every turn whether a punch is underneath the Forcer or not. If one of the punches from the Type Sector is in position underneath the Forcer then the digit it bears will be stamped on the Copper Plate. If not then the Forcer will still continue to perform its action but without effect.

On the First Axis is fixed a mitre/bevel wheel which drives another perpendicular to it fixed on a vertical shaft, called the Vertical Punching Axis. This connects in turn to a horizontal shaft, called the Horizontal Punching Axis, via yet another pair of mitre/bevel wheels. On the Horizontal Punching Axis is a cam which drives the top end of the Forcer down. It does this at the same time as controlling the timing of this action.

On the Vertical Punching Axis is another cam which operates a sprung roller lever which steadies the Type Sector during the Punching process.

(b) 1st 20-Wheel, 20-Axis, Excentrics Axis and Spiral Axis On the First Axis is a single tooth which drives a 20 cog toothed wheel called the 1st 20-Wheel round through one of its 20 orientations for each complete turn of the First Axis. The 20-Axis to which the 1st 20-Wheel is fixed via a pair of spur pinions drives the Lower 20-Axis parallel to it at the rear of the machine. This axis in turn drives a Communicating Axis via yet another pair of bevel wheels. The Communicating Axis passes right underneath the Machine at an oblique angle. This axis is directly connected via another pair of bevel wheels to the base of the Spiral Axis on the front side of the Engine and drives it. The Spiral Axis itself is a vertical shaft around whose body are set protuding in a spiral 18 fingers and two blank positions, one for each digit on the Result Axis. Thus for each turn of the First Axis the Spiral Axis moves round one twentieth of a turn.

As a conseqence of this motion and of the physical arrangement of the fingers on the Spiral Axis, one of the fingers on it ends up being closer to the Type Sector when it swings out. This is the finger that is opposite to the cage of the digit of the Result Axis to be stamped on that particular turn of the First Axis.

At the same time the Lower 20-Axis turns the Excentrics Axis through one of its 20 orientations. It has a spiral of 18 Locking Studs set around it shaft. One of the studs becomes active and locks the Lower Calculating Wheel (via a lever acting on the Ds of the Lower Calculating Wheel in a similar fashion to the locking action of the Vertical Bolting Axes). It, in particular, locks that one which carries the digit which is to be selected for punching on that turn of the First Axis.

A locking mechanism on the 20-Axis with 20-orientations holds all the above mechanism tight in position after moving. There is a lever on the First Axis which releases this locking mechanism when necessary.

(c) Type Sector and Axis, Cam for causing Type Sector to Swing, Cam for lifting Spiral Axis

The Type Sector is DE1's daisy-wheel-like mechanism. It is a 45 degrees frame for holding 10 vertical punches, one for each digit. It swings around a vertical axis in a horizontal plane. On the vertical shaft of the the Type Sector Axis are 18 arms set in a vertical plane one above the other, one opposite each digit cage on the Result Axis. At the end of each arm is a Drop Pin which can click up and down like a retractable ball-point pen.

On the First Axis is a Cam which causes the Type Sector and its Axis to swing in underneath the Forcer of the Punches and out again towards the Spiral Axis. This happens once for each turn of the First Axis but only on those turns when a digit is to be punched. Otherwise a detent controlled by the 2nd SQN wheel holds it back out.

On the First Axis is yet another Cam. This lifts the Spiral Axis up and down once for each turn of the First Axis. The finger on the Spiral Axis which is closest to the Type Sector Axis' arms when the latter swings out lifts that Drop Pin on the arm closest to it up. All the remainder remain down. When it swings back in again the Type Sector this Drop Pin is up, whilst all the others are down. On swinging back out again the Spiral Axis pushes this Pin back down again when it is lowered by the Cam.

(d) Lower Calculating Wheels, Snail Wheels, Selection of Digits for punching

Each Lower Calculating Wheel on the Result Axis has attached to it in the space where a Bolt would reside on another Difference Axis a Snail Wheel.