Galley Method

Division of the number 8 888 880 000 000 888 800 000 000 888 888 (in the center) by 99 999 000 000 009 990 000 000 009 999 (under it, the last nine is added later). Private: 88 (right), balance: 88 968 000 000 009 680 000 000 008 976 (above). 3, 1, 4, 6 in the cross are the residues when dividing these numbers by 7 (for checking). An example is taken from the textbook of arithmetic Niccolo Tartaglia ^[1] 1556.

Dividing by the galley method the number 965 347 653 486 (in the center) by 6 543 218 (below). The answer is written to the right: 147 534 , and the remainder above: 529 074 . An example is taken from an unpublished manuscript of the Venetian monk “Opus Arithmetica D. Honorati Veneti monachj coenobij S. Lauretig” of the 16th century ^[2] ^[3] . (There are several descriptions made by the artist in the picture).

The galley method (strikethrough method) is a division method that was most used in Europe until around the 1600s, and continued to be popular until the end of the 18th century ^[4] . The method arose on the basis of Chinese and Indian methods. The method is mentioned by Al-Khwarizmi in the works of 825 ^[4] , by Luca Pacioli in 1492 ^[3] .

Unlike previous methods, in this method the numbers were not erased, but crossed out ^[4] . It is similar to the modern method of dividing a column , however, in the galley method, subtraction of partial works was performed from left to right, and not from right to left, as in modern methods.

The method got its name for the similarity of the lines recorded in the calculation with the silhouette of the ship of the same name ^[4] ^[3] . At the same time, the slashes that were used to strike out the numbers resembled oars. Sometimes, to obtain similarities, the pattern must be rotated 90 ° ^[5] .

A similar method was also used to extract the roots. .

Content

1 History
2 The essence of the method
- 2.1 Example
- 2.2 Comparison with other methods
3 Options
- 3.1 Without strikethrough numbers
- 3.2 With the calculation of partial products
4 Division Check
5 Root Extraction
6 notes

History

Arithmetic operations with increasing bit depth of numbers become very laborious and sensitive to mechanical errors, and division is the most difficult of them. “Difficult matter is division” ( Italian: dura cosa e la partita ), the ancient Italian expression read ^[6] .

Although in Europe division was considered a complex operation until the 15th century, in China and India division was not considered to be something especially complicated ^[4] ^[7] . The division method is mentioned in “ Mathematics in Nine Books ” (II century A.D.) and is described in detail in the Sun Tzu (III — V century) ^[4] . Many Indian mathematics do not describe the division method, assuming it is known. For example, Ariabhata (499) does not write about the method of division, although, undoubtedly, the method of division was known to its readers, since Ariabhata describes a method for extracting roots that requires division. In Indian mathematics, a division method similar to Chinese was first mentioned by about (about 800 years). A detailed description of the method is given by in the X century ^[7] .

The Indian method was performed in sand or chalk on a blackboard. In the Chinese method, sticks were used as numbers. In both cases, the numbers were easy to erase. In these methods, the divisor is written under the dividend. As in the modern method of dividing by a column , partial products (that is, products of the divisor by each digit of the answer shifted by the corresponding number of digits) are subtracted from the dividend. However, unlike the modern method, the old dividend was erased, and the difference was written in its place, while the partial product itself was not recorded, and was not even calculated, and the subtraction occurred bitwise from left to right. After that, the divider was shifted one digit to the right (this operation in medieval Europe was called in Latin anterioratio ) ^[7] ^[4] . In Chinese (and possibly in the Indian method), the quotient was written over the divisor ^[4] .

This method became known to the Arabs, starting with the works of Al-Khwarizmi (825) ^[7] ^[4] . From there, this method came to Europe ^[7] . In Europe, the division was carried out with ink on paper, because of this, the division method has undergone a natural modification due to the fact that the numbers were not erased, but crossed out ^[3] ^[7] ^[4] . When subtracting partial products from the divisor, the result is written on top. It became impractical to write the quotient over the dividend; they began to write it on the right ^[4] . This modification began to be called the galley method ( galea, batello ) ^[7] , for the British this method was also called the strikethrough method ^[5] ( English scratch method ) ^[7] .

The famous Italian mathematician Niccolo Tartaglia (XVI century) in his famous textbook of arithmetic wrote the following about the method ^[6] :

The second way of dividing is called in Venice a boat or a galley because of some similarity of the figure obtained in this case, because when dividing some kinds of numbers, a figure similar to a boat is formed, and in others - to a galley, which really looks beautiful; the galley sometimes turns out to be well finished and equipped with all the accessories - it is laid out from the numbers so that it really appears in the form of a galley with aft and bow, mast, sails and oars.
Original text (Italian)
Il secondo modo di partire, è detto in Venetia per batello, ouer per galea per certe similitudine di figure, che di tal atto resultano, perche in la partitione di alcune specie di numeri nasce vna certa figura alla similitudine di vno batello, materiale, & in alcuni altri, vna figura simile a vna galea legno maritimo, perche in effetto il pare vna gentilezza a vedere, in alcune specie di numeri vna galea ben lauorata, & tratteggiata con li suoi depenamenti protratti tutti, per vn verso dispositione paiono veramente vna figura simile alla detta galea materiale, con la proua, poppa, albero, vela, & remi, come chenel processo si vedra manifesto ^[1] .

It is interesting to note that the ink galley method was brought back to China from Europe and published in a 1613 ^[4] .

In Russia, the galley method was used until the middle of the 18th century: in the "Arithmetic" by Leonty Magnitsky, it is described among the six methods of division proposed there and is especially recommended by the author; throughout the presentation of the material of his book, Magnitsky mainly uses the galley method, without mentioning the name itself ^[6] .

The so-called “Italian method” ^[3] (or “gold division” ^[5] ), which is now known as column division, competed with the galley method. This method appeared in print in 1491 in "Arithmetic" ^{[8] of} , although it was even earlier encountered in manuscripts of the 15th century ^[3] . In it, the partial product was clearly calculated and written under the dividend, then subtracted from the dividend, and the result was written below. Subtraction was carried out, as in the usual addition by a column , starting from the lower digits, which allowed saving on records, but it was necessary to remember the transfer of the discharge in the mind ^[3] . The main advantage of this method is that all actions are visible on its record - this makes it easier to check the calculations and quickly fix errors. However, the disadvantage of this method is that it needs to multiply multi-digit numbers by single-digit ones ^[5] .

Subsequently, the (“Austrian method”) appeared. It was similar to Italian, but, unlike him, in it, as in the galley method, partial works were not calculated explicitly - they were immediately subtracted bitwise. However, unlike the galley method, subtractions were performed starting from the lower digits, which allowed saving on recordings. Thus, this method combined the advantages of the galley method and the Italian method ^[3] . The disadvantage of this method is that the calculator needs to store more information in his mind.

All these methods competed in Europe with "iron division": the abacus division method described by the mathematician-monk Herbert (future pope Sylvester II) ^[5] .

Method Essence

The galley method, although more difficult to record, is similar to the modern method of dividing a column . Just as when dividing by a column, the quotient is calculated by numbers, starting with the highest digit: at each step, one digit of the quotient is selected. As the private digit, the largest digit is taken so that the partial product (the product of this digit by the divisor shifted by the corresponding number of digits) can be subtracted from the dividend, remaining in positive numbers. After this, the partial product is subtracted from the dividend, the divider itself is shifted one digit to the left, and the process is repeated. Unlike modern column division, in the galley method, the partial product is not calculated, and subtraction occurs from the digits from left to right. In addition, in the galley method, the result of the subtraction is written from above, not from below.

Example

Consider an example from the " " (1478), which divides 65284 into 594 ^[4] . The example is divided into several steps: at each step, the numbers that are added at this step are highlighted in bold and the numbers that are crossed out in italics are highlighted. For ease of perception, the numbers with which the actions are performed are highlighted in color, in reality, the method used ink of only one color.

At first, the divisor ( 594 ) was written under the dividend ( 65284 ):

  65284
 594

Step 1: in 652, the divider 594 is included only 1 time. So the first digit is private 1 . We write it on the right, and subtract from the dividend 1 × 594 (shifted by two digits). In the galley method, this is done from left to right: first the first digit (5) is deducted, then the second digit (9), and at the end the last digit (4) from the corresponding digits.

 
 
 652 84 |  one
 594

Step 1: 594 is included
at 652 once.

 
 one
 6 5284 |  one
 5 94

Step 1a: 6 - 5 = 1

 
 1 6
 6 5 284 |  one
 5 9 4

Step 1b: 15 - 9 = 6

  5
 1 6 8
 65 2 84 |  one
 59 4

Step 1c: 62 - 4 = 58

Step 2: We shift the divider one digit to the right ( anterioratio ). Since the resulting shifted divisor ( 594 ) is more than what remains of the divisible ( 588 ...), we cannot subtract the divisor even once, which means that the second digit of the quotient is 0 :

  5
 16 8
 652 8 4 |  1 0
 594 4
  59

Step 2: 594 enters
588 zero times.

Step 3: We shift the divider one more digit to the right. Now we need to subtract 594 from 5884 . This can be done 9 times. We write 9 in quotient and subtract from the dividend 9 × 594 . In this case, we do not calculate 9 × 594 , but simply subtract 9 × 5 , 9 × 9 and 9 × 4 from the corresponding digits.

 
  5
 16 8
 652 84 |  10 9
 5944 4
  59 9
   5

Step 3: 594 enters
5884 nine times.

  one
  5 3
 16 8
 652 84 |  10 9
 5944 4
  59 9
   5

Step 3a: 58 - 9 × 5 = 13

  1 5
  5 3
 168 7
 652 8 4 |  10 9
 5944 4
  59 9
   5

Step 3b: 138 - 9 × 9 = 57

  1 5
  53 3
 168 7 8
 6528 4 |  10 9
 5944 4
  599
   5

Step 3c: 74 - 9 × 4 = 38

Answer: dividing 65284 by 594 gives a quotient of 109 and 538 in the remainder.

  1 5
  53 3
 1687 8
 65284 |  109
 59444
  599
   5

Full calculation result

Comparison with other methods

For comparison, we give the same division, performed with erasing numbers, as well as Italian and methods ^[3] . As mentioned above, these methods differ in the way they subtract the partial product. For example, in the last step, a partial product of 9 × 594 is subtracted. In the Italian method, first 9 × 594 = 5346 is calculated, and then the result is subtracted. In the galley method and in the method with erasing numbers, the product is not calculated, but subtracted sequentially: 9 × 500, 9 × 90, 9 × 4. In this case, in the method with erasing the numbers, the result is written in the place of the deductible, and in the galley method - on top, and the old numbers are crossed out. Finally, in the Austrian method, the product is also not calculated, but subtracted sequentially: 9 × 4, 9 × 90, 9 × 500. Since subtractions start from the lower digits, only one digit is recorded at each step, and the senior digit is transferred , which allows you to shorten the record, but requires remembering the transfer in the mind.

Digit Erase Method

  65284 |  594
 594 |  109
  5884 
  5346
   538

Italian method

  65284 |  594
 5884 |  109
   538

Austrian method

Options

Without strikethrough numbers

Sometimes the numbers are not crossed out. In this case, only the top and bottom digits were considered. In this case, instead of strikethrough, zeros were written on top of the column. See the illustration at the beginning of the article.

With the calculation of partial works

Sometimes partial works were calculated. This option is practically no different from modern division by a column. The only difference is where the numbers are written: the galley method uses less paper, so the numbers are written more compactly, with no empty space between them. But when dividing by a column, the calculations are better visible and easier to check.

As an example of this option, consider the division of 44977 by 382 ^[2] . One figure corresponds to obtaining one decimal place of the quotient.

  1) 67 (Multiplication: 1 x382 = 382 )
    382 |  449 77 |  1 (Difference: 449 - 382 = 67 )
          382

  2) 29 (Multiplication: 1 x382 = 382 )
           67 5 (Difference: 677 - 382 = 295 )
    382 |  449 7 7 |  1 1
          382 2
           38

  3) 2 (Multiplication: 7 x382 = 2674 )
           29 8 (Difference: 2957 - 2674 = 283 )
           67 5 3
    382 |  4497 7 |  11 7 Answer: Private 117 , remainder 283 .
          3822 4
           38 7
           26

Division Check

There was a method for checking the remainder of dividing by a small number. The most commonly used , since the remainder when dividing by 9 is very easy to find: just find the sum of the digits of the number. However, this verification method did not catch common errors when a digit fell into the wrong category. Therefore, more reliable, but sophisticated methods were also used: checking balances for 7 or 11.

The essence of the method is as follows. Let when dividing the number $A$ ${\ displaystyle A}$ on $B$ ${\ displaystyle B}$ it turned out incomplete private $Q$ ${\ displaystyle Q}$ and the remainder $R$ ${\ displaystyle R}$ . It means that $A=BQ+R$ ${\ displaystyle A = BQ + R}$ . To check this equality, the remainders from $A$ ${\ displaystyle A}$ , $B$ ${\ displaystyle B}$ , $Q$ ${\ displaystyle Q}$ and $R$ ${\ displaystyle R}$ by a small number (for example, 9). Let these residues be respectively equal $a$ ${\ displaystyle a}$ , $b$ ${\ displaystyle b}$ , $q$ ${\ displaystyle q}$ and $r$ ${\ displaystyle r}$ . Then $a$ ${\ displaystyle a}$ and $bq+r$ ${\ displaystyle bq + r}$ must have the same balance.

These residues were written in the form of a “flag”: ${\begin{array}{c|c}q&r\\\hline b&a\end{array}}.$ ${\ displaystyle {\ begin {array} {c | c} q & r \\\ hline b & a \ end {array}}.}$ Sometimes instead of the cross + , the cross × was used .

For example, Niccolo Tartaglia ^[1] when dividing 912 345 by 1987 received 459 and 312 in the remainder. To check this, he took the remainder of these numbers from division by seven: 912 345 gives the remainder 0, 1987 gives 6, 459 gives 4, 312 gives 4. Tartaglia writes this as ${\begin{array}{c|c}4&4\\\hline 6&0\end{array}}.$ ${\ displaystyle {\ begin {array} {c | c} 4 & 4 \\\ hline 6 & 0 \ end {array}}.}$ Then it checks that $4\cdot 6+4$ ${\ displaystyle 4 \ cdot 6 + 4}$ divided by seven with a remainder of 0. So the result has passed the test ^[9] .

Root Extraction

Extraction of the square root by the galley method from the number 96 837 278 . Received reply

\approx 9840{\frac {11678}{19680}}.

{\ displaystyle \ approx 9840 {\ frac {11678} {19680}}.}

From the textbook of Niccolo Tartaglia in 1556 ^[10] .

A similar method was used to extract the roots . As in the division, the answer was in bits.

To extract the square roots at each step, the square of the partial answer already received was subtracted from the number. In this case, the formula was used $(a+b)^{2}=a^{2}+2ab+b^{2}$ ${\ displaystyle (a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2}}$ . Namely, if at some step towards a partial answer $a$ ${\ displaystyle a}$ attributed figure $b$ ${\ displaystyle b}$ (i.e. a new partial answer $10a+b$ ${\ displaystyle 10a + b}$ ), then we need to subtract from the original number $(10a+b)^{2}$ ${\ displaystyle (10a + b) ^ {2}}$ . But $(10a)^{2}$ ${\ displaystyle (10a) ^ {2}}$ we have already subtracted in the previous step. Therefore, we have to subtract $(20a+b)b$ ${\ displaystyle (20a + b) b}$ . To do this, in the galley method, the number $20a+b$ ${\ displaystyle 20a + b}$ written below, the figure $b$ ${\ displaystyle b}$ recorded on the right, and then the partial product was subtracted, as in the usual method ^[11] .

When extracting roots of higher degrees, the Newton bin was used , which was known even before Newton ^[12] .

Notes

↑ ¹ ² ³ Nicolo Tartaglia . Book One // General trattato di numeri, et misure. - Vinegia : Curtio Trojano de i Navo, 1556.
↑ ¹ ² Carl B. Boyer, Uta C. Merzbach. A History of Mathematics . - John Wiley & Sons, 2011-01-25. - 680 s.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ Leland Locke. Pure Mathematics // The science-history of the universe / Francis Rolt-Wheeler (managing editor). - New York: Current Literature Pub. Co .. - Vol. Viii. - 354 p. - P. 48-52.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ Lam Lay-Yong. On the Chinese Origin of the Galley Method of Arithmetical Division (English) // The British Journal for the History of Science. - 1966/06. - Vol. 3 , iss. 1 . - P. 66–69 . - DOI : 10.1017 / S0007087400000200 .
↑ ¹ ² ³ ⁴ ⁵ Encyclopedia for children . T. 11. Mathematics / Chap. ed. M. D. Aksyonova. - M .: Avanta +, 1998 .-- S. 132-134. - ISBN 5-89501-018-0 .
↑ ¹ ² ³ Perelman Ya. I. Entertaining arithmetic. - 8th ed. - M .: Detgiz , 1954.- 100 000 copies.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ , AN Singh. Part I: Numerical Notation and Arithmetic // . - 1962 .-- S. 150.
↑ Filippo Calandri. Aritmetica : [ ital. ] / Lorenzo Morgiani e Johann Petri. - 1491.
↑ Florian Cajori. A History of Mathematical Notations . - Courier Corporation, 2013-09-26. - S. 260—261. - 865 s.
↑ Nicolo Tartaglia . Book Two // General trattato di numeri, et misure. - Vinegia : Curtio Trojano de i Navo, 1556. - S. 28.
↑ Graham Flegg. Numbers: Their History and Meaning . - Courier Corporation, 2013-05-13. - S. 133. - 307 p.
↑ David E. Smith. History of Mathematics . - Courier Corporation, 1958-06-01. - S. 148. - 739 p.

[Тарталья-1] ¹ ² ³ Nicolo Tartaglia . Book One // General trattato di numeri, et misure. - Vinegia : Curtio Trojano de i Navo, 1556.

[Boyer-2] ¹ ² Carl B. Boyer, Uta C. Merzbach. A History of Mathematics . - John Wiley & Sons, 2011-01-25. - 680 s.

[rolt-wheeler-3] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ Leland Locke. Pure Mathematics // The science-history of the universe / Francis Rolt-Wheeler (managing editor). - New York: Current Literature Pub. Co .. - Vol. Viii. - 354 p. - P. 48-52.

[lay-yong-4] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ Lam Lay-Yong. On the Chinese Origin of the Galley Method of Arithmetical Division (English) // The British Journal for the History of Science. - 1966/06. - Vol. 3 , iss. 1 . - P. 66–69 . - DOI : 10.1017 / S0007087400000200 .

[:1-5] ¹ ² ³ ⁴ ⁵ Encyclopedia for children . T. 11. Mathematics / Chap. ed. M. D. Aksyonova. - M .: Avanta +, 1998 .-- S. 132-134. - ISBN 5-89501-018-0 .

[:0-6] ¹ ² ³ Perelman Ya. I. Entertaining arithmetic. - 8th ed. - M .: Detgiz , 1954.- 100 000 copies.

[hindu-7] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ , AN Singh. Part I: Numerical Notation and Arithmetic // . - 1962 .-- S. 150.

[8] Filippo Calandri. Aritmetica : [ ital. ] / Lorenzo Morgiani e Johann Petri. - 1491.

[9] Florian Cajori. A History of Mathematical Notations . - Courier Corporation, 2013-09-26. - S. 260—261. - 865 s.

[10] Nicolo Tartaglia . Book Two // General trattato di numeri, et misure. - Vinegia : Curtio Trojano de i Navo, 1556. - S. 28.

[11] Graham Flegg. Numbers: Their History and Meaning . - Courier Corporation, 2013-05-13. - S. 133. - 307 p.

[12] David E. Smith. History of Mathematics . - Courier Corporation, 1958-06-01. - S. 148. - 739 p.