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Small Adders Part 1


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1 Small Adders Part 1

Bob Otnes, Palo Alto, California

Vortrag, gehalten beim 2. Greifswalder Symposium zur Entwicklung der Rechentechnik 12. - 14. September 2003, erschienen in Girbardt/Schmidt 9-2003
Im Rechnerlexikon mit freundlicher Genehmigung des Verfassers

I am in the process of writing a book on "small adders", a term to be defined in the talk that follows. One of the reasons for giving this paper is that I need help in the form of "reality checks". While any book can end up being controversial to some, I would prefer to have my book, if it is ever finished, have most of such problems worked out in advance. I will mainly be talking about adders in the United States in the 19th and early 20th centuries. A large number of small adders were patented in the US in the 19th century: perhaps as many as 150. And this does not count the ones that were not patented. Of course, not all of those that were patented made it to market. On the other hand, there were some that were not patented that were manufactured. What caused all of this activity? I believe that some of the causes were as follows:

1.1 Acknowledgements

I am not going to include a formal bibliography in this paper. However, I do want to mention some of the many people who directly have helped me, or whose books have been used in my studies. These include, in no particular order, Dr. Peggy Kidwell of the Smithsonian Institution, Thomas Russo, George C. Chase, Conrad Schure, the people at the Arithmeum and at Paderborn, F. Diestelkamp, the translation of the book by "Martin", Dr. Michael R. Williams, Maurice d'Ocagne, Jean Marguin, etc.

Patents: I have put quite a few US Patent numbers into these notes. If you would like to get free copies of the patents, the URL for the US Patent Office for this purpose is as follows:  http://patft.uspto.gov/netahtml/srchnum.htm
Put this URL into whatever browser you use, and hit return. A page will come up with a location to insert the patent number of interest. Input it and hit return, and it will come back saying that you can only get an image or images. Click on the box that says "Images", and it will return the first page of the patent. The difficulty comes in printing these images. You must have a TIFF plugin as the pages are in that format, and many systems can display TIFF, but cannot print it. Plugins can be obtained on the internet, many of them as freeware.

1.2 Types of Adders

There many different types of small adders. The following is an attempt to categorize them: Carrying of Results The most critical characteristic of the majority mechanical adders is the process of carrying, that is, the means by which the overflow in addition is carried from one column of numbers to the next. For example, when we do the addition


\begin{matrix} & 24 \\ 
               & 19 \\
             = & 43 
\end{matrix}

we start from the right adding 4 and 9 which yields the result 13. We write a 3 under the column and carry the 1 standing for 10 to the top of the next column. The carry of 1 in this case might actually be written at the top of the column for later check, or it might simply be retained in the mind in one's own short-term memory. In any event, what seems so simple and natural when one is doing the problem mentally becomes a major difficulty when the process is implemented mechanically. Witness the problems that Pascal had with his design. Carrying was done in a variety of methods:

With the Bassett, when the bands it employs are turned, a special flag passes into view when a carrying is required. The user must immediately take the appropriate action to effect the carry. With the stop hitting devices, when one attempts to do an addition that will require a carry, the slide, bar or whatever will not go the distance that it should. This is the signal to the user to instead move in the opposite direction all the way. With the slide bar devices, the user must do the carry himself. With the Kummer machines, as will be seen, there is an aid to effecting the carry: rather than a simple slot to move the stylus in, it takes the form of a candy cane or shepherd's crook. This crook goes over one column to that of the next left digit. When the stylus is moved all the way to the top of the curve of the crook, and then left and down, the carry is performed. Note that there is a lot of variability in just how this is done among the different manufacturers. The user powered carry occurs as follows: suppose that the current sum on a dial machine such as the Lightning is 9, 999, and that the user adds 1 to the rightmost dial. The result should be 10, 000. In adding one unit into the sum, the user will cause 5 visible dials and 4 hidden internal gears all to turn one position. This is a total of nine dials and gears; if there is any dirt or grit in the machine, this operation may be difficult to do. Note also that there are a number of detents in these machines; these are springs or spring-loaded devices that cause the dials to stop centered on a number. The detents require force to overcome also. The net result is that a carry involving a number of decimal places can be difficult to do. The stored-energy carry in a sense winds a spring as a given dial is turned. When the dial is turned to position 9, the spring is fully compressed. When the dial is turned one more position to 0, the spring is released causing it to add one to the position to the left. With a long carry, such as the one just described, the carries will ripple across to the left one after another without a great deal of effort on the part of the user other than inputting the single unit on one dial. This method is better from the user's standpoint. There are a few difficulties: the mechanism is more expensive, the numerous springs make the device more prone to breakage, and true subtraction would be much more difficult to implement. This feature is not common on small adders. The Webb and Calcumeter are the major examples of devices using the stored-energy carry.

1.3 Complementary Subtraction

Otnes-00-goldmann.png
Goldman's ArithMachine.

The operation of complementary subtraction replaces the expression

x - y = z
where x and y are positive and, in the this example, 99999999 > x - y with the equivalent expression: x + (99999999 - y) + 1 - 10000000 = z The operation (99999999 - y) + 1 produces the complement of y. This is easily accomplished as 99999999 - y amounts to the following digit transformation
0 -> 9
1 -> 8
2 -> 7
3 -> 6
4 -> 5
5 -> 4
5 -> 3
6 -> 2
8 -> 1
9 -> 0

This is often printed directly on the adder in some form. Goldman's ArithMachine is an eight-decimal digit adder with carrying. It performs subtraction by complementary arithmetic. Input to the machine is accomplished by using a special stylus to pull down on the small horizontal bars shown in the middle of the machine. Each bar is part of a chain; as the bar is pulled down, it adds into the display in the column directly above it. The specific value added corresponds to the large number printed on either side of the starting row. As an example, suppose we desire to calculate 1234567 - 123456 = 1111111 using complementary subtraction as on the ArithMachine. That is,

x= 1234567
y=123456
99999999 - y=99876543
x + (99999999 - y)=1234567 + 99876543
 =101111110

In the ArithMachine, the leftmost "1" is lost, dropping o_ the left end. This drop of a high order "one" carry happens with all such machines when doing this kind of subtraction. Finally, we would add the required 1 into the right column to get the result 01111111, which is the correct answer. At first glance this seems complicated. However, with use it becomes straightforward. The ArithMachine (and many other adders) have aids to memory on them. There are two sets of numbers on the machine, one set on each side. Each set consists of two columns of ten numbers. In each case, the large numbers are for adding and are the same. The small numbers on the left set are for subtraction in the leftmost 7 columns. The rightmost set are employed only for subtracting in column one: they already have the extra "1" added into them.

Clearing Mechanisms Clearing a machine before starting is done "manually" with most simple machines. The operator starts on the right and clears the machine by adding into the first column whatever number brings it to zero, then proceeding to the the next column on the left, and so going across the whole machine. Actual clearing mechanisms seem to have appeared first in France with Dr. Roth's and the Thomas de Colmar Arithmometre. These in turn may have been influenced by the reset capability in stop watches.

1.4 Copyright

Alle Rechte beim Verfasser

Hinweis: Copyright 2003 Robert K. Otnes
Please do not copy or reprint

Dieses Material ist Bestandteil eines Buches von Bob Otnes, das voraussichtlich noch im Jahre 2004 erscheinen wird.

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