I don't know if everyone knows the name of one of the outstanding mathematicians, Baron John Napier (1550-1617) - Scots by birth.
Here he is in person (c) Wikipedia: 
It is famous first and foremost for the fact that invented logarithms!
You can imagine how people suffered in those days when multiplying and dividing. multi-digit numbers. Napier came up with special tables in which a one-to-one correspondence was made between the geometric progression and the arithmetic one. And, naturally, geometric progression was the original one. Thus, Napier compared multiplication with much more easy folding, and division, accordingly, is subtraction.
For which all progressive humanity is grateful to him to this day.
But that’s not what I’m going to talk about now.
In 1617, Napier proposed another, non-logarithmic, method of multiplying numbers, for which he came up with a special device called the “Napere sticks.”
I am talking about it in connection with the notes on figured numbers. This is another way to visualize arithmetic. (Although, in fact, there is nothing else to do with curly numbers here).
I learned about Napier's sticks when I was preparing a presentation on the history of development computer technology. For the presentation I only needed one slide with brief information. Now I tried to find something more extensive and was horrified: Napier is mentioned everywhere, as a rule, just in the “history of computer technology” section, and a couple of absolutely identical paragraphs wander from article to article.
Here's what we managed to glean from all this.
This “computing tool” consisted of bars with numbers from 0 to 9 and their multiples printed on them. To multiply a number, the bars were placed side by side so that the numbers at the ends made up this number. The answer could be seen on the sides of the bars.
Look here: (this is the best picture I found): 
That is, as I told the students, this is a kind of three-dimensional multiplication table.
Now I understand that I got carried away with the three-dimensional one. It seems that we are talking about a flat representation (I thought these bars had numbers on all four sides, but it looks like they are only on one “front” side and on the end).
The stripes with numbers printed on them were also divided by diagonals so that the diagonals are tens on the left (above), and ones are on the right.
To obtain the products, summation is carried out “along the diagonals”. 
To be honest, I don’t fully understand HOW this happens. But from what I read, four-digit numbers were multiplied with these sticks as a joke.
In addition to multiplication, Napier's sticks made it possible to perform division and extract Square root.
Under the cut I will hide a quote from one site, which I am not able to comprehend)))
However, everything is explained there))
An exercise for inquiring minds:
J. Napier proposed special counting sticks (later called Napier sticks), which made it possible to perform multiplication and division operations directly on the original numbers. On the top of the grid, each cell is assigned the digits of an A-number, and on the right - the digits of a B-number. In each (k,j)-cell of the lattice the result of the product Rkj=xk*yj of the corresponding digits of the numbers is written. In this case, the number of tens is placed above the diagonal of the cell and the units - below the diagonal. After filling all the grid cells, S p is summed over the inclined bars of the grid from right to left with the transfer of the most significant digits.
The described principle of multiplication is illustrated by the example of multiplying the numbers 1942 and 54: 1942x54=104868. Napier's sticks (9 in number; they represent a kind of multiplication table in which numbers are written in the cellular form described above) were mainly used for multiplying large numbers and were used very rarely for division and root operations. Napier himself subsequently proposed sticks of a special design, designed specifically for extracting square roots; these were used in combination with regular Napier sticks. Along with sticks, Napier proposed a counting board for performing the operations of multiplication, division, squaring and square root in binary s.s., thereby anticipating the advantages of such a number system for automating calculations.
From here.
The first device for performing multiplication was a set of wooden blocks known as Napier sticks. They were invented by the Scotsman John Napier (1550-1617). A multiplication table was placed on such a set of wooden blocks. In addition, John Napier invented logarithms.
This invention left a noticeable mark on history with the invention of logarithms by John Napier, which was reported in a publication in 1614. His tables, which required a lot of time to calculate, were later “built into” a convenient device that greatly speeds up the calculation process - the slide rule; it was invented in the late 1620s. In 1617, Napier came up with another way to multiply numbers. The instrument, called “Napier's knuckles,” consisted of a set of segmented rods that could be positioned in such a way that by adding numbers in segments adjacent to each other horizontally, we obtained the result of their multiplication.
Napier's theory of logarithms was destined to find wide application. However, its “knuckles” were soon supplanted by the slide rule and other computing devices - mainly of the mechanical type - the first inventor of which was the brilliant Frenchman Blaise Pascal.
Logarithmic ruler
The development of counting devices kept pace with the achievements of mathematics. Shortly after the discovery of logarithms in 1623, the slide rule was invented.
In 1654, Robert Bissacar, and in 1657, independently, S. Patridge (England) developed a rectangular slide rule - this is a counting tool to simplify calculations, with the help of which operations on numbers are replaced by operations on the logarithms of these numbers. The design of the line has largely survived to this day.
The slide rule was destined long life: from the 17th century to the present time. Calculations using a slide rule are simple, fast, but approximate. And, therefore, it is not suitable for accurate, for example financial, calculations.
Napier's Sticks was the beginning new era- “the era of science”, which replaced the previously popular trade business. Counting sticks are the invention of Scottish mathematician John Napier, who went down in history thanks to the invention of logarithms. With the help of the first computer technology, the development of arithmetic took a step forward, and Napier's sticks are still considered the prototype of the first computer technology, for example, such as a calculator.
John Napier is a Scottish mathematician, known as the inventor of a new type of computing tool - logarithms, the impetus for which was the “Napear sticks”. In the 16th century, science felt the need to conduct complex calculations, however, were not created at that time the necessary conditions for her further development. Therefore, John Napier suggested using the addition process instead of the complex multiplication operation, which he managed to compare using special tables. Thanks to this scheme, the time-consuming division process can also be replaced by a subtraction operation. This invention made it possible to significantly facilitate the work of computers.
Napier's sticks - what are they?
John Napier published a book in 1617 in which he proposed new method carrying out the multiplication operation using special sticks. At that time, the lattice multiplication method was very popular, so the scientist decided to create his own technique based on it.
“Napere's sticks” were a set of special sticks, consisting of a board with markings from one to nine and the rest of the sticks, on which a multiplication table with the same number markings was placed. At the top of each tablet were numbers in ascending order, and along the entire length of the laid out table Napier placed the actual results of multiplying numbers by numbers from one to nine. In other words, the table made it possible to perform operations of multiplying the number 123456789 by the number 123456789. The grid itself was divided by columns.
In order to obtain a result when multiplying, it was necessary to select sticks that would correspond to the digit of the multiplicand, and arrange them in a line, a series of numbers of which would indicate the number itself. Due to the fact that the digits in the multiplicand could be repeated, the set always included additional sticks responsible for each digit. A board with vertically arranged numbers from one to nine was placed on the left. Using it, it was possible to select the line corresponding to the digit of the multiplier.
John Napier decided that if he divided the cell into 2 parts using a diagonal line, then it would be possible to compactly write down the result of the operation: in the upper compartment, record the most significant digit of the resulting number, and in the lower compartment, the least significant digit. To obtain the final result of the operation, you need to add the numbers in the “table” from right to left - the sum of the numbers will be the necessary answer.
“Napier's sticks” could be used both for multiplication and division, and for calculating the square root of a number. If numbers could be divided according to a principle similar to multiplication, then in order to extract the square root, another stick consisting of three columns was added to the set. The first column contained the squared numbers that corresponded to the value of the tablet indicating the rows, the second - the numbers obtained by multiplying the row index by two, and the third column contained the numbers from one to nine.
Modernization of "Napere's sticks"
After the invention of this arithmetic method, many mathematicians tried to introduce some innovations into the mechanism developed before them. For example, in 1666, an English scientist-inventor made an attempt to transfer the entire table from sticks to disks. This experience was crowned with success, since such a technique simplified the work with the invention of its predecessor. And in the late 60s German mathematician Kaspar Schot put forward the idea of replacing the planks with cylinders, on two sides of which everything should be placed numeric values along with a multiplication grid from one to nine. If you put the cylinders in such a position that their upper side with numbers forms a multiplier, then the multiplication operation can be carried out according to the same principle as using “Napeer’s sticks”.
Already in the 19th century, in order to facilitate the use of the device, instead of ordinary flat boards, they began to make bars at an angle, with an angle of 65 degrees. As a result, the triangles containing the numbers for the operation could be used in order, since they were now located below each other. By the end of the century, some more changes were made related to the replacement of sticks with thin strips, fixed in a special case that resembled a notebook. The strips had to be moved using a sharp stick.
Napier's Sticks were in great demand at the time. This seemingly simple discovery made a big breakthrough in the development of arithmetic.
Napier's sticks were destined to have a long life. They are wide and for a long time used for calculations in astronomy, artillery and other fields. A wonderful film from the 70s about the 16th-century English philosopher Thomas More was called “A Man for All Seasons,” but if a film were being made about his compatriot who lived several decades later, then perhaps it would have been called “The Man Ahead of His Time.” . It's about about Sir John Napier, whose name can be safely placed on a par with, for example, the names Galileo Galilei or Nicolaus Copernicus, and maybe Leonardo da Vinci.
Napier - Scottish mathematician and Protestant theologian - was hereditary nobleman, was born in 1550 at Merchiston Castle near Edinburgh, and died there on April 4, 1617. He studied at the University of Edinburgh, and then traveled for a long time in search of knowledge throughout Europe. As a result of his wanderings, like most scientists of his time, Napier became a generalist, a generalist. Most Napier devoted his subsequent life to theology and actively participated in theosophical debates, where, like a true Scotsman, he was distinguished by his zeal.
As a theologian, he is known for publishing in 1593 A Simple Exposition of the Whole Revelation of John the Evangelist, the first interpretation Holy Scripture on Scots, but at the same time Napier was not alien to the then fashionable sciences - astrology and alchemy. Along with these hobbies, he was also an engineer, invented whole line machines for tillage and water pumps for irrigation. He also made several “secret” inventions, including a mirror for setting enemy ships on fire, a device for swimming under water (scuba gear), a cart that is not pierced by bullets (a tank), and something resembling an unguided rocket projectile.
However, it is quite possible that all this successful activity at that time, which was significant for his contemporaries, would have remained unknown to descendants if not for his main works, completed in his seventh decade, shortly before his death. Chronologically, the first of them was a mathematical work - a system of logarithms “Description of an amazing table of logarithms (Mirifici logarithmorum canonis descriptio, 1614)”, it proposed (without disclosing the method of its construction) the first table of logarithms, as well as the term “logarithm” itself. Later, the construction method was revealed in the essay “Construction of an amazing table of logarithms (Mirifici logarithmorum canonis constructio),” published in 1619, after the death of the author. Henry Briggs, a professor at Gresham College London, who later became Napier's publisher, successor and biographer, was directly related to the appearance of these works. It so happened that, having become acquainted with the "Description ...", Briggs became a faithful follower of Napier's ideas, therefore, driven by the desire to help him, he went to Scotland to personally meet with the author and subsequently devoted his life to bringing his work to end. His descendants played a significant role in preserving the memory of Napier.
Both of these works are of interest rather for the history of mathematics, and for the history of computers, the most important and at first glance very simple technical invention of the Scottish scientist, which later began to be called Napier’s sticks (or bones), is essential. It became the second practical device in the history of mankind, after the abacus, to facilitate calculations. To be fair, it should be said that there is an earlier drawing by da Vinci, which is considered to be an image of a calculating machine; there are even modern attempts to reconstruct it, but no documentary evidence about work and practical use I don't have a da Vinci calculator. And with Napier's sticks, despite all their apparent simplicity, a chain of devices began that ultimately led to the modern PC.
Apparently realizing the significance of his invention, Last year Napier devoted his life to preparing for the printing of the final creative path treatise - “Rhabdology, or Two Books on Counting with Sticks”, in the preface to which he wrote: “Now we have also found a much better variety of logarithms and intend (if God grants long life and good health) publish both the method for calculating them and the way to use them. But, due to our bodily weakness, we leave the calculation of these new tables to people experienced in this kind of work, and above all to the most learned husband Henry Briggs, professor of geometry and our dearest friend.
In “Rabdology...” Napier described a method of multiplying numbers using special stick-bars with numbers printed on them; they look like domino bones, but with a large number fields on each of them. The idea of automation using pre-marked sticks clearly goes back to one of the oldest methods of multiplication, called gelosia. Today no one thinks about the internal complexity of this arithmetic action, even the phrase “method of multiplication” sounds somehow strange, because the only algorithm known to most, “in a column,” is taught in the third grade. And in those distant times, multiplication was a science to which entire treatises were devoted. The most famous is Luca Pacioli's work Summa de arithmetica, where, among others, this method of gelosia, invented in India and in the 14th century came to Europe through the mediation of the Persians and Arabs, is described. For those interested in multiplication methods, I recommend the article Multiplication Methods ( www.ex.ac.uk/cimt/res2/trolfg.pdf), where various ancient techniques are beautifully described.
The gelosia algorithm is very elegant in its own way; its essence is that the factors are written to the right and above of a special counting matrix consisting of square fields, each of which is divided by a diagonal, and the triangles located together along the diagonal form “oblique” rows and columns. So, the factors are written on the top and on the right, and the intermediate products of each pair of digits, from ones to the highest, are written in squares, separating the ones and tens within each, the ones in the lower triangle, and the tens in the upper. When summing “obliquely”, the result is obtained; it must be read from top to bottom and from left to right. Napier's own idea was at first glance very simple: you need to cut the table into columns and perform actions, selecting the necessary sticks in accordance with the composition of the number. Naturally, to “enter” a number, there must be more sticks in the set; the numbers can be repeated. Thus, multiplication becomes a trivial task, but this does not exhaust the potential of the sticks; with them you can perform division, exponentiation, and root extraction, based on the addition and subtraction of logarithms.
The idea of sticks was developed in Germany. Ten years after the publication of “Rhabdology...”, Professor oriental languages Wilhelm Schickard of the University of Tübingen invented a mechanism that simplified the work with sticks, which he described in correspondence with Johannes Kepler. As you know, letters were at that time the only form publications. It is difficult to say now whether this machine was built or not, but in any case it was the first mathematically substantiated model of a calculator. Now in Germany several working examples of the Schickard mechanism have been recreated. The history of the creation of the calculator and the biography of the author are successfully described in the article by Yuri Polunov ( http:// museum.iu4.bmstu.ru/ firststeps/ letters.shtml).
Napier's sticks were destined to have a long life. They have long been widely used for calculations in astronomy, artillery and other fields; sticks influenced the creation of the slide rule, which has become a classic engineering tool XIX and XX centuries, and in Great Britain until the mid-60s, Napier sticks were used to teach arithmetic to schoolchildren.
14. 6th grade students read a poem by N.P. Konchalovskaya and argued.
Marina claimed that she did not read anything new in this poem compared to the text about Naum the Grammar. And Yura said that the poem contains important new information.
Which student do you agree with? Write down your answer and provide justification.
15. During the lesson, students were asked to come up with their own signature for a painting by the artist B. M. Kustodiev. Which of the proposed captions most accurately reflects the content of the picture? Write down the number of the correct answer.
1) “They teach the alphabet - they shout at the whole hut.”
2) Lesson at the school of Ancient Rus'.
3) Teaching is light.
4) Reading lesson.
16. How much time passed in the old days from the beginning school year before the initiation ceremony? Write down the number of the correct answer.
2) 2 months
3) 3 months
4) 6 months
17. What signs existed in the Old Russian school? Write down two signs.
18. Teacher's Day was celebrated as one of the first professional holidays in Russia. And in modern Russia Teacher's Day is a national holiday. Why do you think this holiday has survived centuries? Write down words (justification) from the text that support your opinion.
NEPER STICKS
Read the text and complete tasks 19-27
I always tried my best
and abilities, to free people from difficulty and
boredom of calculations, the tediousness of which
usually scares away a lot of people from
studying mathematics.
John Napier
Scottish theologian and amateur mathematician iki
John Napier
In 1617, Napier published a treatise entitled “Rhabdology, or the Art of Counting with Sticks” (Fig. 1). In it, he described a method by which numbers could be multiplied without difficulty. Today, no one thinks about the complexity of this arithmetic operation; even the phrase “method of multiplication” sounds somehow strange, because the only multiplication algorithm known to most is “in a column”, they are taught in the third grade. And in those distant times, multiplication was a science to which entire treatises were devoted.
Rice. 1. One of the first
editions of Napier's treatise
The set for calculations described by Napier (Fig. 2) included: one stick with numbers from 1 to 9 (this is a line indicator) and sticks with a multiplication table for all numbers from 1 to 9 (digits of the multiplicand). Numbers from 1 to 9 were written on top of each stick, and along the entire length the results of multiplying this number by numbers from 1 to 9, and to record the result, the cell was divided diagonally into two parts: the tens place was written in the top, and the units place in the bottom ( Fig. 3).
The sticks looked like domino bones, and ivory was often used to make them.
For multiplication, sticks corresponding to the digit values of the multiplicand were selected and laid out in a row so that the numbers on top of each stick made up the multiplicand. A line index was placed on the left - the lines corresponding to the digits of the multiplier were selected from it. The numbers were then summed along a diagonal line. The summation was carried out bitwise with the overflow being transferred to the most significant digit.
For example, to multiply 187 by 3, you need to select three sticks corresponding to the numbers 1, 8 and 7, and line them up as shown in Figure 4. The third line shows the following:
Let's sum up two numbers, one of which is under the diagonal, and the other is above the diagonal, but not of this square, but of the one adjacent to the right (Fig. 5).
These sums give us the digits of the product: 561.
Napier based his calculating device on the principle of lattice multiplication, which was widespread in his time. For lattice multiplication, a table was drawn containing as many columns as there were digits in the multiplicand, and as many rows as there were digits in the multiplier. The multiplicand was written above the columns of the table so that the digits of the number were each above its own column. The multiplier was written to the right of the table (Fig. 6).
Lattice multiplication
Then the cells of the table were filled with the results of multiplying the digit of the multiplicand located above this cell and the digit of the multiplier located to the right of this cell. It was these actions that Napier simplified by putting the multiplication table on sticks. Then the products were summed up, as in the case of sticks.
Napier's rods were destined to have a long life: for several centuries they were used for calculations in the most different areas human activity. They influenced the creation of the slide rule, which became a classic engineering tool of the 19th and 20th centuries, and happily survived into the era of computers and calculators.
Tasks
19. What was the main goal of John Napier when working on the creation of a calculating device that received his name? Write the correct answer number.
1) attract people to study mathematics;
2) make a start new science - computational mathematics;
3) free people from the difficulty of calculations;
4) develop new way calculations other than columnar multiplication.
20. How Napier's sticks are arranged is discussed in the second paragraph of the text. Read it again and answer the question: what number should be written in the upper square of the stick shown in the picture? Write down the resulting number.
21. Using Napier sticks you need to multiply: 4169·5. The sticks corresponding to which numbers should be chosen? Write down the numbers of the corresponding sticks.
22. The second name of the described counting device is Napier's bones. What does this name mean? Find in the text those words that contain the answer to this question and write them down.
23. Using Napier sticks, multiply 187 by 4. Using Figures 4 and 5, complete tasks A-B.
A. Which line should I choose?
B. Write down all the required amounts.
IN. Write down the result.
24. Imagine what you have to say younger brother- for a third grader, how to multiply by hash marks two-digit number to the unambiguous. The individual steps of this algorithm are described below. Using Figure 6 and the description in the text, write down for each step its serial number. The first step is already indicated: D-1
A. Write down the resulting number.
B. Multiply the units digit of the multiplicand by the factor, write the result in the second cell.
C. We sum up the numbers in the cells along the diagonal, bit by bit.
D. Draw a table with two columns and one row.
E. Multiply the tens place of the multiplicand by the factor, write the result in the first cell.
F. We divide each cell of the table diagonally into two cells.
25. How did you multiply numbers that had a 0 in their place? How would you multiply 1807 by 3 using Napier sticks? Draw a diagram and write down the answer: 1807·3=
26. Tanya read in the encyclopedia that Napier's sticks have long been used for calculations in astronomy, artillery and other fields, and in the author's homeland - Scotland - for several centuries they were used to teach arithmetic to schoolchildren. She is trying to understand why this method was so attractive in those days. She has several guesses.