Napier's Sticks and the “Crib Sheet” – 17th-century pocket “calculators” for multiplying and dividing complex numbers

As part of a series of materials about various ancient calculating devices, we have already become acquainted with complex mechanisms, mainly intended for adding numbers. Multiplication operations in all cases required a number of complex actions, but in one of the devices (Wilhelm Schickard's counting clock) this process has been simplified thanks to devices called “Napier Sticks“, they are “Napier Bones“, which were invented in the 17th century by a Scottish mathematician John Napier. Today we will pay special attention to them, and also talk about the “Crib Sheet” – another Napier device to facilitate calculations.

John Napier

John Napier was born February 1, 1550 in Edinburgh (Scotland) in a wealthy noble family (laird, untitled nobility of Scotland): father laird Archibald Napier and mother Janet Boswell, daughter of the famous Scottish politician Francis Boswell. John was the 8th Laird of Merchiston of the Clan Napier, but despite this, very little information about his childhood and education has survived to this day.

John was probably home-schooled and lived at Merchiston Castle, where he was born. In 1563, John's father, Archibald Napier, sent his son to St. Salvaitor's College, University of St. Andrews. Moreover, in 1560, in his letter, Adam Boswell, brother of Janet Boswell, begged Archibald sent the boy to study at universities in continental Europe and expressed great doubts about the quality of Scottish education, including private education.

The year he entered college, John's mother died. Judging by the documents left after her death, young John was under the tutelage of the director of the educational institution while studying at college. But, despite this, Napier’s name does not appear on the lists of candidates for admission to bachelor’s and master’s degrees in 1566 and 1568. Most likely, Napier received only an elementary education in college, or at some point dropped out of school altogether and headed to continental Europe to receive an education on the advice of Uncle Adam.

Napier traveled throughout Europe, studying in the Netherlands, France and Italy. He returned to his homeland in 1571 and a year later married Elizabeth Stirling, thanks to which he received new estates, but he began to receive full income from them only in 1608 after the death of his father. In 1574 the couple moved to a castle in Gartness (Scotland). In 1579, Elizabeth died and a couple of years later John married her second cousin Agnes Chisholm. From his second marriage, John Napier had ten more children. In 1608 Napier returned to Merchiston Castle and lived there for the rest of his life. In 1617 he died of gout.

John Napier's life was marred by civil and religious wars. However, his family managed to avoid serious consequences from these conflicts. Napier studied philosophy and mathematics and even published several religious works. In particular, in 1593 he created, in his opinion, the most important work of his life – “A simple explanation of all the revelations of St. John” (A Plaine Discovery of the Whole Revelation of St. John), where in English (not Latin, unlike other works), using mathematical analysis of the Book of Revelation, he tried to establish the date of the Apocalypse (according to his assumption, in 1688 or 1700). There were also rumors that Napier was involved in the occult, but they were not confirmed over time – nothing was found in John’s notes and writings that would hint at a passion for the occult.

However, the key achievements and works of John Napier that interest us do not lie in religious treatises and disputes over the Reformation. In 1614, Napier published the essay “Description of the Amazing Table of Logarithms” (Mirifici Logarithmorum Canonis Descriptio) in two parts – a revolutionary work for its time. In the first part, the scientist describes logarithms and their properties, in the second – plane and spherical trigonometry. The appearance of logarithms greatly simplifies the processes of multiplication, subtraction and root extraction, which was the purpose of writing the work. Discoveries in plane and spherical trigonometry, such as Napier's analogy formulas, also made calculations easier.

It is believed that, when creating logarithms, Napier was inspired by the idea of ​​the German mathematician Michael Stiefel and his book “Complete Arithmetic” (Arithmetica integra, 1544). In it, Stiefel laid down the idea of ​​a logarithm, but ultimately did not reach practical implementation. It is not known for certain whether John Napier read this work, but he was the first to publish a clear and concise concept with tables for practical application.

“Convinced that there is nothing else… that would cause greater difficulties in mathematical practice, and would also disturb and annoy calculators than multiplication, division, and the extraction of square and cube roots, which operations, in addition to being a tedious waste of time, are the main source of numerous errors, I began to think about what reliable and easy way I could remove these obstacles. And, thinking over various means suitable for achieving this goal, I finally found wonderful short rules that can be used in the future. Among all these rules, none are more useful than those that … exclude from calculations numbers that must be multiplied, divided or turned into roots, and in their place other numbers are put in, with the help of which all calculations are performed only by addition, subtraction or division by two or three.” Source.

Napier Sticks and “Cribs”

In 1617, after the death of John Napier, his son posthumously published his father’s work “Rhabdology”, which describes two inventions: already known to you “Napier Sticks“, and “Cheat sheets“(Promptuary). Both inventions, like the essay “Description of an Amazing Table of Logarithms,” had the main goal of simplifying complex mathematical calculations. In addition, they were so easy to manufacture that absolutely anyone could make them from scrap materials, unlike the computers and arithmometers that we talked about earlier.

“Crib” included:

— narrow long vertical stripes with large numbers at the top and five squares, inside of which numbers are located in 9 sectors;

— a set of perforated strips (horizontal);

– a board for performing calculations (solely for convenience, its absence did not interfere with the calculations);

– storage box.

Each square had 9 sectors for 9 different digits in the multiplier respectively. That is, when a horizontal strip was applied to the perforation, a number corresponding to the multiplier sector was revealed, as in the picture shown.

The plural number was laid out from vertical stripes. That is, the numbers at the top of the stripes made up the same multiplicable number. In turn, the multiplier was made up of horizontal stripes and superimposed on top of the multiplicand. As a result, the perforated strips overlapped the lower vertical stripes, leaving only part of the numbers exposed.

Further calculations are carried out from the rightmost row (oblique, highlighted in red) in the perforation to the left. All visible numbers are added up. If the number is greater than 10, the tens are carried over to the next line for addition. The rightmost row corresponds to the rightmost digit in the final number, the second row to the second digit, and so on.

In the example below, the 1st row contains only the number 2, the 2nd row – 8, 1 and 2, that is, we add them together, write 1 and transfer the ten to the next row as when adding in a column. The result should be the number 305712.

This method required the person to perform the most simple and understandable addition actions, and in the end, even when multiplying large numbers, the final result was obtained as quickly and simply as possible.

“Napier Sticks” have a similar concept, but can be used to multiply, divide, and even take square roots. The set included:

• a board for performing calculations (solely for convenience, its absence did not interfere with the calculations);

• a set of 10 sticks (a straight parallelepiped), on which, similar to the stripes from the “cheat sheet”, large numbers are applied on top, but the lower blocks look completely different, we will return to this later;

• 11th stick with numbers from 1 to 9;

• 12th card for operations on calculating square roots.

Each of the 10 calculation sticks had the shape of a right parallelepiped. At the top there were numbers from 0 to 9 – one number for each stick. Below are 8 more blocks, each of which is divided diagonally. All 9 blocks correspond to the result of multiplying digits:

1st – 1×1 = 1, respectively, at the top is one;

2nd – 1×2 = 2, in the second block from the top there will be 0 / 2;

3rd – 1×3 = 3, will be 0 / 3;

and so on.

For numbers, when multiplied, a two-digit number is obtained, the following picture will appear:

8×1 = 8, respectively, at the top 8;

8×2 = 16, that is, 1/6 will be written on the stick;

8×3 = 24, that is, 2/4;

And so on.

The multiplication table was marked on all four sides of each stick. But in later versions of the device, the tablets already became flat with the multiplication table on only one side. Initially, the sticks were made of wood, metal or ivory, but soon they began to use ordinary paper, which did not spoil the calculation results at all. Options appeared with the application of tables on rollers, both separately and as part of counting devices, in particular in the same counting clocks of Wilhelm Schickard.

For multiplication It was necessary to first place the first stick with numbers from 1 to 9. Sticks are laid out on the right side of it, depending on the number being multiplied. That is, if the multiplicand is, say, 856, then the first stick with the number 8 in the upper sector comes first, then with 5 and the leftmost one with 6.

The multiplier is determined by the first stick. If it contains only one digit, then we take only the horizontal row that corresponds to this digit. Let's say we multiply 856 by 7. So we take the seventh row, and we get 5 / 6, 3 / 5 and 4 / 2. They need to be folded according to the example of the cheat sheet, that is, diagonally and with a carry of ten. In this case, nothing borders 2; we leave it as the rightmost digit in the final number. Then we add 4 and 5, we get 9. Then 6 and 3, also 9. There is nothing to add five with. As a result, we get – 5992.

If we need to multiply 856, for example, by 324, then we take several lines in accordance with the multiplier. In this case, lines 3, 2 and 4. For each line, we calculate the number according to the previous example, we get 2568, 1712 and 3424. We add them into a column, at the top is the rightmost number corresponding to the rightmost digit of the multiplier, and add to the bottom numbers at the end zeros. That is: 3424 + 17120 + 256800. Add in a column with tens carried over and get 277344. Try experimenting with your own examples. 🙂

For divisions the sticks were first laid out in the same way as for multiplication, but with one change. The vertical sticks now denoted the divisor. Let's divide 277344 back by 324. We need vertical sticks with the numbers 3, 2 and 4, we make up a divisor from them. Here you need to summarize the result for each row.

The resulting numbers are obviously smaller than the number being divided. We ignore the “extra” bits of the multiplier and choose the largest number taking into account this total bit depth. Or you can add digits (using 0) to numbers from the table, this method is also used for convenience, but it’s more convenient for me not to write extra zeros. It's easier to show with an example.

Let me remind you that we are dividing 277344 by 324. Since the dividend is six digits, and all the numbers from the table are up to 4 digits, for the first iteration we omit the last digits 4 and 4 and subtract the closest number from the dividend. We get 181, return two “extra” digits to it and get 18144. We omit one digit and, by analogy with the first iteration, carry out the second. We do this until the result of such subtractions becomes less than the divisor, in our case less than 324.

We get the result based on the line number. We took 2592 from the 8th line, which means the first digit of the result is 8. 1620 is taken from the 5th line, which means we write 5. 1944 – from 6th line. There were no further iterations, and as a result we get 856. 277344 : 324 = 856.

To get tenths of the result, we increase the remainder by 10 times and repeat all the operations until we get the required accuracy or the calculations lead you to 0.

Square Root more complicated, will require a special stick and several stages of calculations. Let's take the number 466489 as an example. For clarity, this example can be seen in the video analysis below. We divide the numbers into pairs, starting from the right edge: 46 64 89. In this case, we begin the calculations from the first pair of digits.

Take 46 and a square root stick. In the first row of the stick we look for the number closest to 46 by analogy with division, in our case it will be 36. We write it down as the result 6The th line (as the first digit), subtract from 46 – 36 and get 10. Write the 10 in the column.

For the next operation, look at the number to the right of 3/6 – 12. We take ordinary sticks and lay out the number 12, and place the stick for extracting the square root as the far right one. We calculate the values ​​from the first two sticks and the first row of the third, according to the principles from previous times.

We get the following picture.

When we subtracted the first pair of digits of the original number, we got the result 10. To these 10 we add the second pair of digits and we get 1064. Now we are looking for the closest (but not greater) to 1064 from the numbers that we just calculated. This will be 1024, taken from 8th line. As a result, we already have two numbers from the result – 68. 1064 – 1024 = 40, write it down.

Next, we also take the second row of the eighth line on the stick to extract the square root. In this case it will be 16. But we do not change 12 to 16, but add it up, adding digits to the first number. That is, in the end, the first three sticks add up to the number 136. We write down the values ​​​​from three sticks in a row by analogy with the previous example. We add the remaining pair of digits to 40 and get 4089. In this example, we find that the resulting values ​​contain the same number, it is in 3 line. Let's write it down. 4089 – 4089 = 0. The calculations are finished, there is nothing more to calculate with. As a result, the square root of 466489 is 683.

By analogy with division, if it is necessary to determine tenths, 00 is added to the total. For example, if you have 13 left, after adding it will be 1300. We repeat the steps according to the specified algorithm until you come to 0 or get the required accuracy.

Analyzing the principle of operation of Napier's sticks, I was surprised at how simple and convenient everything turned out to be. To multiply, divide and extract roots from complex multi-digit numbers, I only needed basic knowledge of the multiplication table, the principles of subtraction and addition, which any elementary school student knows.

Of course, now we have various calculators for calculations, but Napier sticks are a very convenient thing that will always be with you, even if you don’t make these sticks. The table is quite easy to fill out and reproduce, as is remembering the principles of calculations. If you have something to write on and with what, you can always count something difficult. Even now, Napier's sticks seem worthy of attention, not to mention the 17th century, when they were truly needed and in demand.