Mechanical Calculators


From: Vagabondo <>
To: Ray McKay
Fri 11/7/2003 5:33 PM
Interesting information on Comptometer Model H


thanks for the link but is an 'overload' server that is linked to my Comptometer website (my ISP only allows 20 megabytes per server).

I had originally scanned the manual and had posted it as images and then Andreas De Man took on the task of OCRing all the pages and either he or I added the hyperlinks and cuts. Came out pretty well I think.

Always good to hear from you anytime for any reason.

Stay well,


From: Ray Mackay <>
To: Vagabondo
Fri 11/7/2003 3:42 PM
Interesting information on Comptometer Model H

Hi Brook & James

I think, if you don't already know of it, you will find the following web page interesting.

It contains a repair manual similar to the one I lost on Comptometer Model 'H' (including working drawings) 


From: Ray Girvan <ray@RAYGIRVAN.CO.UK>
Fri 11/7/2003 3:20 AM
Subject: Comptometer: Division (C. Nöring)

Nicholas Bodley <nbodley@WORLD.STD.COM> wrote:

> It's so nice to see the high-order carries create the quotient digits! I
> like that.

Yes, that's as elegant as it gets.

> Well, folks, is anyone willing to calculate square roots on a Comptometer
> by the fives method? (Is that the best method for a Comptometer, or am I
> just a little crazy?)

I had wondered if they used square root tables as with the Curta.

Ray Girvan

From: Gilbert Harrus <>
To: Ray McKay
Fri 11/7/2003 12:14 AM
Division on a Comptometer 


Thanks for your reply. First let me thank you for your notes "MEMORISED DESCRIPTION OF COMPTOMETER's ® MECHANISM". I have read and tried to understand fully the content (but it was too much time for me to spend, unfortunately). I came through them while trying to repair a M-model - one of the carry mechanism was not operating properly. After reading the notes along with a repair manual for the H-model, I came to the conclusion that I would probably not be able to repair without much luck and/or time.

Deep cleaning the machine, and re-oiling it, I could finally get it to operate.

About the division: I understand your point about operators, and fully agree that the division instruction is simply, and " contrary to what others say, the method was not prone to error" ; indeed, this is also what I was writing. I was quite surprised by the fact that, as of today we know what an algorithm is, that the few pages I found on the division method on Comptometer do not see and document that simplicity - they just give examples without any explanation.

I will take the time to write a summary and will let you know.

Best regards,


From: Nicholas Bodley <nbodley@WORLD.STD.COM>
Thu 11/6/2003 6:19 PM
Subject: Comptometer: Division (C. Nöring)


Please take a look at this web page: 

It's so nice to see the high-order carries create the quotient digits! I like that.

I remember working for a very experienced mechanical calculator tech, who showed me with lots of glee (excited happiness, I'd say) how an experienced Comptometer operator would shape her fingers into the pattern of numbers to be entered, and rapidly shake her hand up and down, poking at the keys.

Well, folks, is anyone willing to calculate square roots on a Comptometer by the fives method? (Is that the best method for a Comptometer, or am I just a little crazy?)

Left hand for the unchanging digits, and the right for the incrementing digit? Well, maybe not quite; there's that rightmost five to be subtracted every time.

Erez's simulator is neat! Try it, if only to see a very nicely-written applet. He's used a photo of a real machine, with clickable "hot spots" at every key, and even sound! You do need to aim your mouse pointer rather carefully.

I did try the fives method, but got lost (forgot how) in the procedure for calculating another digit. I need supper; brain fade.


Btw, that technician (Bill Morrison) was an expert on Moon-Hopkins billing machines. He said that the "table-lookup" fast multiplier mechanism was very reliable; in fact, iirc he said the whole machine was very reliable.

Best regards to all,

Nicholas Bodley
Waltham, Mass.

From: Ray Mackay <>
To: James Redin
Thu 11/6/2003 4:02 PM
Division on a Comptometer

Dear Gilbert

Terms like algorithms etc would have made the process disconcerting to students in the Calculator School. In the Comptometer teaching and examples manual the process is described somewhat simplisticly. Remember that each keytop has the whole number to the rights and a smaller complimentary number to the left on each key. The Students were simple taught


Add Divident into machine.
Use small figures and one less for divisor.
Repeat the following three rules until remainder is less than divisor.

1> Catch up figure on left
2> Make Less
3> Move over. (One place to right)

This simple instruction was taught to literally thousands of Comptometerist and, contrary to what others say, the method was not prone to error. Every work sheet was double checked by a second operator and if there was any question of an error the calculation was sent to scrutinising operators. This only happened very infrequently and usually where less experienced operators were involved.

Balance, Cost and Markup sheets were calculated non stop over the thirty years I was involved and I only started with Comptometer halfway through their history in that time there were negligable reports of errors in outsources work performed in our offices.

Ray Mackay

PS James you may like to post this note to the Cal-list on my behalf. If anyone is interested in the complete training manual (circa 1950) I am will to scan one it and put it up on my web site for collection in PDF format


From: Derek Peschel <dpeschel@ESKIMO.COM>
Thu 11/6/2003 2:18 PM
Division algorithm for Comptometer (and similar machines)

With ordinary subtraction you pad your number at the left with 9s (on the Burroughs key-driven machines) or use one of the carry cut-offs (on the Comptometer). With division you intentionally do not pad the number, so that the carries will build up in the next column over which holds the current quotient digit.

The manual says you can move to the next column if the current quotient digit equals the number of additions. There may be other short cuts to use in certain division problems (I haven't looked lately). Those short cuts are the most interesting part of the algorithm, but I haven't looked for the principles they are based on.

-- Derek

From: Christofer Nöring <christofer.n@TELIA.COM>
Thu 11/6/2003 7:55 AM
Subject: Division Algorithm


Please take a look at this web page: 

with a short description of quatuor species for the Comptometer (Burroughs Calculator)

Best regards

Christofer Noering

From: Ray Girvan <ray@RAYGIRVAN.CO.UK>
Thu 11/6/2003 7:06 AM
Subject: Division Algorithm

Gilbert Harrus <Gilbert.Harrus@XRCE.XEROX.COM> wrote:

> I am sure that the (quite simple) maths behind were documented at
> some point ... Has any of you come to such a precise description?

The basic method is called "complementary subtraction". You subtract X by adding the complement of X (which is the little number on the keys) and suppressing the carry. There's some discussion of how the algorithm works here ... (This page)

... and a description of it in computing context here ...

Division would be a repeated application of this sustraction method.

Ray Girvan


From: Ronald A. Kriss <>
To: <>
Date: Thursday, August 19, 1999 8:08 AM
Subject: Burroughs Calculator

Just got a Burroughs Calculator, 9 rows and 9 columns with an extra 9 in the upper left hand corner.  Serial no. 5-476631.  Says Burroughs Calculator on front and Burroughs on the back.

What are the little numbers for?
And what are the little knurled knobs at the bottom near the back for?
Can you use the machine to subtract?
Do you know about how old it is?
What are they worth?  (I paid $16 in a store in Massachusetts - I would have paid MUCH more!)

Hope you can help with these answers.

Ron Kriss

From: Nicholas Bodley <nbodley@WORLD.STD.COM>
Subject: Burroughs Calculators
Date: Thursday, August 20

What are the little numbers for?

I'll jump in, here, at the risk of putting out wrong information.

I'd make a semi-educated guess that this is a Comptometer (TM, Felt and Tarrant)-like mechanism that can internally add, only, but when you add the complement of numbers, it effectively subtracts. The Curtas use complementary subtraction. (Set all slides to zero, and  pull out the crank. Turn once. Watch all accumulator dials make exactly one turn. Good test of the carry mechanism.)

If the little numbers, added to the big ones, add up to 9 (10?), then this must be the case.

Let's say,

5 - 3 = 2

Take the tens complement of 3, which is seven. Add 5 and 7 to get 2, but with a carry out the high end.

87 - 34 = 53

Now, you'll see that you need a tens complement in the least-signif. digit (units) position, but the nines complement in the others. (Curtas do this.)

Units, first:

Tens complement of 4 is 6, so add 7 and 6 to get 3 and a carry. Nines complement of 3 is 6, so add 6 and 8 to get 4 with a carry out the high end, then add in the carry from the units digit to get 5.

You should get a bell or some indication that there was a carry out the high end, if the result has a positive sign.

It's basic, but I don't recall how you'd subtract a small number; you probably have to enter (nines complements of zero) in all higher-order columns.

Afaik, Comptometer operators would have entered a 4 and then a 5 into those higher-order columns.

1) Hope this helps
2) Hope I didn't make any silly goofs. (Mailing-list addict, here!)

|*  Nicholas Bodley    *|*  Autodidact & Polymath * Electronic Tech. (ret.)
|*   Waltham, Mass.   *|*   -----------------------------------------------
nbodley@WORLD.STD.COM  *|*   Before 1960 or so: $100. Later: 100$
|*  Amateur musician  *|*  Before 1990 or so: 100%. Later: %100

From: Brian Borchers
Subject: Burroughs Calculators
Date: Thursday, August 20

The nines complement technique can be used to subtract on any machine that can add.  Suppose for convenience that we have a 4 digit adder, and that we want to compute x-y.  We can write x-y as

  x-y = x + (9999-y) + 1

Note that 9999+1=10000, and the 10000 simply carries off of the end of our four digit adder.

Computing 9999-y is easy- just replace each 0 in y with a 9, each 1 with an 8, each 2 with a 7, and so on.  Adding x+(9999-y) is easily done with the four digit adder.   Adding an extra 1 at the end is also easy.

For example, suppose we want to compute 7853-5678

The 9's complement of 5678 is 4321.  So we add

          +     1

Because we've got a four digit adder, the left most 1 never appears.

I've seen a number of calculators (Such as the Hoffritz and Swift Handy Calculator)   that have the 9's complement digits labeled on the calculator.

Note that if you want to subtract a relatively small number, you'll have to include all of the leading 9's to get a correct answer.

You can also use this scheme to represent negative numbers.  -y is represented as 10000-y for y between 1 and 5000.  In this system, any number that begins with the digits 0-4 is positive, while any number that begins with the digits 5-9 is negative.   For example,

  number                     is represented as
  -1                              9999
  -4000                         6000
  -5000                         5000

Note that the range of representable negative numbers is -5000 to -1, while the nonnegative numbers range from 0 to 4999.

Some terminology:  The number (9999-y) is called the 9's complement of y.  The number (9999-y)+1 is called the 10's complement of y.

It's interesting that this same technique is used in binary form in modern digital computers.  In that case, you use the ones complement (all ones replace by zeros and vice versa)  for subtraction, and negative numbers are written in twos complement form.

Brian Borchers                    
Department of Mathematics
New Mexico Tech                          Phone: 505-835-5813
Socorro, NM 87801                        FAX: 505-835-5366

From: Nicholas Bodley <nbodley@WORLD.STD.COM>
Subject: Burroughs Calculators
Date: Thursday, August 20

(Somewhat off-topic:)

As a bit of a follow-up to Brian's nice exposition, it might be of interest to note that "business" computers, back in the earlier days, used several bits to represent each decimal digit; they were decimal-oriented, although binary in detail. Sometimes this was called BCD, or binary-coded decimal. (IIrc, IBM had to be different, and called it DCB.)

Converting to natural binary on input, and converting back to decimal on output came later.


What prompted this message was that some ways of defining the ten digits in binary were cleverly planned so that reversing every bit created the code for the nines complement.   One was called excess-three, or "XS3", perhaps. Using 8-4-2-1 weighting, the binary number was three greater than the digit. 0 was 0011, 1 was 0100, etc. There were others; iirc, 2-4-2'-1 was another; no combination of bits could represent a digit greater than 9.

From: Frank Lindauer
Subject: Burroughs Calculators
Date: Thursday, August 20

Hi Ron: I picked up one of these machines within the last year and paid about what you did, perhaps a few dollars less. I thought it was a bargain. I believe it is a class 5 calculator, ca. 1927. I'm not an expert on these, but it may function like a Comptometer, and I would almost be sure that it can subtract, but I haven't put it through its paces.

The little knurled knobs are unscrewed to help remove the top cover. Others on the Calc List know more than I do about this machine, but this is a starter.

Frank Lindauer

From: Ray Mackay
Subject: Burroughs Calculators
Date: Thursday, August 20

Although I did not receive Ron's enquiry regarding Burroughs' and therefore do not know of which model or system he speaks I feel I should stick my bib in.

Firstly 'Key Driven machines' of the Comptometer type consisting of a number of columns of nine keys were completely capable of Multiplication and Division plus extracting a square root.

They were work to apportion amount and calculate percentages and mark-ups. AFAIK any machines of this type and any machines capable of additon and subtraction are fully capable of Multiplication and Division as these functions are merely repeated addition and subtraction moving to the next higher or lower order as dictated by the result.

Models were constructed that work in Decimal, Sterling, Weights and Measures plus other specialised numbering systems.

Other adding machines with a ten (or more) numeral keypad and a series of function keys entered data into a pin box or other type of register from where it performed additions or other function via repeated addition and subtraction.  Some wrere more sophisticated than others and to speed up a process were capable of 'flagging' if a number was higher or lower than five and changed the method of calculation accordingly.  Thus reducing the number of repetitive cycles.

Burroughs' made machines of several types, the parallel to the Comptometer introduced to the market close to or maybe even before the time the Comptometer was released,  A largish listing machine with glass side panels and a side actuating lever (I recall some of these had print  outs, but memories grow weak with the years) plus other developments. They produced both mechanical and electronic machines.

James Redin's site contains mechanical details of some of these machines. Vagabond's site further details.

If any one wished to ask specifica question I will answer to the best of my ability to recall.  Messages to this site are restricted in site so I can not ramble on and get the message through

Divisions on adding machines.

Some machines had cut offs called subtraction cut offs these precluded the requirement of entering all the nines to the left of the calculation.  All this mechanism did was lose the carry. 

It should be noted that in Division on 'Key Driven' machines the negative carry was most important as it made its own specific noise as the carries ran across the machine. This told the skilled operator to add back the divisor and move her figures one position to the right before starting to subtract once again.

This process was repeated across the machine until the right hand columns were reached.  Its interesting to note that some models of Burroughs' actual had a single nine key in the left hand column specifically for running nines across the register.

The Burrough's 'Key Driven' Comptometer type machine did all the functions of the Comptometer. This included Addition, Subtraction, Multiplication and Division plus the extraction of square roots.

The keydrive mechanism was totally different from that used in the Comptometer as also was the Carry Over Mechanism.  The carry over mechanism being located inside the numeral wheel. As the number increased to the left of a wheel two small pinion rotated inside the next wheel to the right.  When the lower order wheel reached 10 it triggered the pinion mechanism rotating the 'skin' or outer numeral wheel to add in the one.  It was very fast and afaik was not subject to errors caused by duplication of carries ie

The actual keydriven mechanism was the typical parallel bars being pushed down and operating a segment which engaged the register pinions. Although I worked on them from time to time I did not convert them to decimal currency.  Basically this means the full adjustment process was never embedded in my mind. Walthers and Comptometers I know from A to Z

multiplying 334 continuously  or 375 continuously.  These tests were mandatory before machines were sent out to clients and once machines performed these calculations through all keyboard position many time they were considered suitable to send out to the client.

regards Ray Mackay

X-Number World
Revised: June 21, 2004.