Brusky Joffe - multiplying tool based on the Slonimsky theorem
In the XIX century there were interesting tools for multiplication, built on the basis of the Slonimsky theorem. This is the “Snonad for multiplication” of Slonim and Ioffe bars. This article is devoted to the second of them, proposed in 1881 by Hirsch Zalmanovich Ioffe (variant Ioffe).
The materials on this tool in runet are very scarce, however, I think I managed to restore their appearance. In any case, the option I attached below is close to the original and is suitable for use as intended.
This article is for those who, like me, are interested in the history of computing. When I wrote an article [3] about the principle of constructing the Slonimsky table and using it for multiplication, my eyes became blurred and I did not pay enough attention, so to speak, to the material part. In addition, then I did not have the necessary clues in order to restore the appearance of the bars.
Therefore, when I was asked about the practical side of the issue, and I had some clues for it, I decided to restore the appearance of Ioffe bars and write an article about them.
The article is devoted, albeit ancient, but still computer technology. Consequently, it is suitable for Habr on the subject, and the hub "History of IT" is, as you know, on Hiktayms. Giktayms is well indexed, and I want that anyone interested in this calculating tool, it was easy to find information on it.
Ioffe bars are designed to quickly compile a table of works of a given number into a series of numbers from 2 to 9. To do this, a column of numbers is written on each face of each bar, and the table you are looking for is created by folding several bars together in the right order.
This is what has been found on the Internet:
From a source [1]:
From source [2]:
It is a pity that I did not have a second source when I opened the algorithm for working with the Slonimsky table. There was also a picture illustrating the principle of multiplication:
This picture was the key to understanding what was drawn on the bars.
The Slonimsky table (described in more detail by me in the article [3] ) consists of 280 columns, as Slonimsky proved, this is enough to add a plate with the works of any given number to a series of one-digit numbers 0 ... 9 from them (columns).
To select the desired column is the "key" - in Ioffe this is the pair "Roman numeral" - "Latin letter", and the digit of the multiplied number. Joffe used seven numbers for the key, written in Roman numerals, and four letters — that is, all keys are 28. And the digits in the decimal system, as is known, are 10.
28 * 10 = 280.
As you can see in the picture above, in each column Joffe wrote one key at the top, and another key at the bottom. For convenience, we call them the upper and lower keys. The top key is used to identify the column itself, and the bottom key is used to select the column for the next digit. In addition, the column at the top has a number — it is a multiplicative digit; it also serves to identify the column.
The algorithm can be described as automaton, where the input string is a multiplied number, readable from right to left (from low-order bits to high-order), and the state is the key of the previous column. At each stage, we need to find a column whose upper key is equal to the lower key of the previous column, and the number is the next input digit. The initial state is the key IA, the final state is also IA, provided that the number is read completely. To avoid surprises, you should add a leading zero to the number.
Now the same thing on the fingers and on the bars. Girsh Zalmanovich grouped columns of 4 on the edges of their bars, and the bars themselves - 7 in boxes. Out of 10 boxes. It is not difficult to guess that the number of the box should be simultaneously the number of all columns in it. Bars in boxes can have 7 numbers - obviously, this is the meaning of the Roman numeral. Further, 4 letters, as appears from the description, designate four sides of a bar.
On the picture from the source [2] the plate multiplying the number 325 by the row 2 ... 9 is added. As I advised, the leading zero is assigned to the number for prevention. I will repeat the picture in order not to scroll:
We look in the reverse order: we must consistently find the columns for the numbers 5, 2, 3, 0.
We start from state IA.
Go:
We take from box 5 bar I and put it with side A. We read its lower key: IC. Our mental machine goes to the IC state.
We take from box 2 bar I and put it to the left of the previous side C. We read its lower key: IB.
We take from box 3 bar I and put it with side B. We read its lower key: II-B.
We take from box 0 bar II and put it with side B. The number is over, check the lower key of the last bar: IA, which was to be proved.
→ Scanning of all PDF bars
In the attached file, every four of the columns is a scan of one bar. In horizontal groups - seven bars for one box.
1. Counting bars Ioffe Apokin I. A., Maistrov L. E. “The history of computing technology”. M .: Nauka, 1990. - pp.112-116 ...
2. A. Apokin, L. Maistrov. “The development of computers”. M .: Science, 1974. - p.98-99.
3. A multiplying tool based on the Zonitchik Slonimsky theorem, Habr, 2014 :)
The materials on this tool in runet are very scarce, however, I think I managed to restore their appearance. In any case, the option I attached below is close to the original and is suitable for use as intended.
Purpose of writing an article
This article is for those who, like me, are interested in the history of computing. When I wrote an article [3] about the principle of constructing the Slonimsky table and using it for multiplication, my eyes became blurred and I did not pay enough attention, so to speak, to the material part. In addition, then I did not have the necessary clues in order to restore the appearance of the bars.
Therefore, when I was asked about the practical side of the issue, and I had some clues for it, I decided to restore the appearance of Ioffe bars and write an article about them.
Why was the location of the article chosen
The article is devoted, albeit ancient, but still computer technology. Consequently, it is suitable for Habr on the subject, and the hub "History of IT" is, as you know, on Hiktayms. Giktayms is well indexed, and I want that anyone interested in this calculating tool, it was easy to find information on it.
Purpose and Description
Ioffe bars are designed to quickly compile a table of works of a given number into a series of numbers from 2 to 9. To do this, a column of numbers is written on each face of each bar, and the table you are looking for is created by folding several bars together in the right order.
This is what has been found on the Internet:
From a source [1]:
Counting bars were offered to Ioffe in 1881. In 1882, they received an honorary review at the All-Russian Exhibition. The principle of working with them is based on the Slonimsky theorem.
Ioffe device consisted of 70 tetrahedral bars. This made it possible to place 280 columns of the Slonimsky table on 280 edges. Each bar and each column was labeled, which used Arabic and Roman numerals and letters of the Latin alphabet. Latin letters and Roman numerals served to indicate the order in which it was necessary to place the bars in order to obtain the product of a multiplicand by a one-bit factor. The resulting works (and there are as many as the digits in the multiplier) were added (just like when using the Slonimsky multiplier) using pencil and paper.
From source [2]:
Iof's device consisted of a box with ten compartments, numbered 0, 1, 2, ..., 9. Each compartment contained seven four-sided bars, marked on the four sides of one of the numbers: 0, 1, 2, etc., and below the numbers I, II, etc. and the letters A, B, C, D respectively on each side. Then, after these symbols, there were columns of numbers from the Slonimsky table, one column on each face (there are 280 columns on the 70 four-sided bars that make up the complete Slonim table). Below this, Roman numerals and the same letters A, B, C, and D. Roman numerals and letters served to indicate the order in which the bars should be placed in order to obtain products of a given number by single-valued factors.
It is a pity that I did not have a second source when I opened the algorithm for working with the Slonimsky table. There was also a picture illustrating the principle of multiplication:
This picture was the key to understanding what was drawn on the bars.
Theory
The Slonimsky table (described in more detail by me in the article [3] ) consists of 280 columns, as Slonimsky proved, this is enough to add a plate with the works of any given number to a series of one-digit numbers 0 ... 9 from them (columns).
To select the desired column is the "key" - in Ioffe this is the pair "Roman numeral" - "Latin letter", and the digit of the multiplied number. Joffe used seven numbers for the key, written in Roman numerals, and four letters — that is, all keys are 28. And the digits in the decimal system, as is known, are 10.
28 * 10 = 280.
As you can see in the picture above, in each column Joffe wrote one key at the top, and another key at the bottom. For convenience, we call them the upper and lower keys. The top key is used to identify the column itself, and the bottom key is used to select the column for the next digit. In addition, the column at the top has a number — it is a multiplicative digit; it also serves to identify the column.
The algorithm can be described as automaton, where the input string is a multiplied number, readable from right to left (from low-order bits to high-order), and the state is the key of the previous column. At each stage, we need to find a column whose upper key is equal to the lower key of the previous column, and the number is the next input digit. The initial state is the key IA, the final state is also IA, provided that the number is read completely. To avoid surprises, you should add a leading zero to the number.
Practice
Now the same thing on the fingers and on the bars. Girsh Zalmanovich grouped columns of 4 on the edges of their bars, and the bars themselves - 7 in boxes. Out of 10 boxes. It is not difficult to guess that the number of the box should be simultaneously the number of all columns in it. Bars in boxes can have 7 numbers - obviously, this is the meaning of the Roman numeral. Further, 4 letters, as appears from the description, designate four sides of a bar.
On the picture from the source [2] the plate multiplying the number 325 by the row 2 ... 9 is added. As I advised, the leading zero is assigned to the number for prevention. I will repeat the picture in order not to scroll:
We look in the reverse order: we must consistently find the columns for the numbers 5, 2, 3, 0.
We start from state IA.
Go:
We take from box 5 bar I and put it with side A. We read its lower key: IC. Our mental machine goes to the IC state.
We take from box 2 bar I and put it to the left of the previous side C. We read its lower key: IB.
We take from box 3 bar I and put it with side B. We read its lower key: II-B.
We take from box 0 bar II and put it with side B. The number is over, check the lower key of the last bar: IA, which was to be proved.
Application:
→ Scanning of all PDF bars
In the attached file, every four of the columns is a scan of one bar. In horizontal groups - seven bars for one box.
Literature:
1. Counting bars Ioffe Apokin I. A., Maistrov L. E. “The history of computing technology”. M .: Nauka, 1990. - pp.112-116 ...
2. A. Apokin, L. Maistrov. “The development of computers”. M .: Science, 1974. - p.98-99.
3. A multiplying tool based on the Zonitchik Slonimsky theorem, Habr, 2014 :)