Thursday, 3 October 2013

Division, Some Notes on the History of Division Procedures



A dozen years ago I began researching and collecting information to construct on line page with the origin, etymology and history of many of the mathematical terms and their etymology. Over the years I collected a little over 1000 of them. In the last few years I have not revisited and corrected the pages as much as I should have, but I have decided to pull out a couple of the major ones that might be interesting, and include them in my blog.. I hope you enjoy.

The word Divide shares its major root with the word widow. The root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse. A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel. The prefix, di, of divide is a contraction of dis, a two based word meaning apart or away, as in the process of division in which equal parts are separated from each other. Notice that the vi part of vidua is also derived from a two word, and is the same root as in vigesimal (two tens), for things related to twenty. An individual is one who can not be divided.
In a division problem such as 24 / 6 = 4 the number being divided, in this case the 24, is called the dividend and the number that is being used to divide it, the 6, is called the divisor. The four is called the quotient. If the quotient is not a factor of the dividend, then some quantity will remain after division. This quantity is usually called the remainder, although residue sometimes is used. The Treviso Arithmetic uses the word lauanzo for remainder. In Frank Swetz's book, Capitalism and Arithmetic he gives, "The term lauanzo apparently evolved from l'avenzo, meaning a surplus, or in a business context, a profit." Swetz also points out that in the 15th Century the term partition (partire in Latin) was synonymous with the word divisision.

In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,
"It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near N├╝rnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use."

I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10):
From the fourteenth century on, merchants from the north traveled to Italy, particularly to Venice, to learn the arte de mercadanta, the mercantile art, of the Italians. Sons of German businessman flocked to Venice to study...


Early algorithms for division:
By the middle ages there seem to have been five approaches to the process of division.

The first was called the Galley, galea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term “scratch” has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Recorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. " Addendum (I just found a note that says that the famous Arab mathematician,  Al-Khwarizmi, whose book laid the foundation for western introduction to the now-common Arabic numbers, used this method in his writing. The Hindu  mathematical calculations were usually done on a  dust covered tablet, and they would wipe out numbers instead of scratching through them so that in the final result, only results appeared.  The scratch-out method in Europe was simply an adaptation for paper or slates. )

Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda. The page the image is from has a nice step by step illustration of the process.


In 2005 I acquired a German student "copy book" from 1804 which seems to show the Galley division method and the student's illustration of the ship around the work. (below right)

A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from a popular Arithmetic in the US by Charles Davies, published in 1833. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others .

This method of subtraction grew into what is taught frequently as an alternative use to many modern classrooms, and I believe is the standard method at some levels for a program called "Everyday Math".  Instead of slowly subtracting one divisor at a time, the use of simple multiples is used to group subtract.
For example, to divide 227 by 8, it is easy to see that 10 x 8 or eighty can be subtracted, so they might take out ten groups of eight repeatedly until the remaining part was too small to divide by 80.  So after removing 20 groups of eight,  there would be 67 remaining.  At this point they might recognize that 8x8 is 64 so by removing  8 more sets of eight would leave only three remaining, so the quotient would be 28 with 3 remaining.   
A video of this method is shown here
(I have been informed that the "correct" term for this method is "partial quotients".)
I found an example of this method, with the name partial quotients in a 1773 Encyclopedia Britannica


A third method
was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.
divide 92467 by 168 or 4 x 6 x 7


4|92467

6|23116 …. groups of four and 3 units over

7| 3852 ….. groups of 24 (4x6) and 4 foursover

___550 groups of 168 and 2 twenty-fours over



The complete remainder is 2 (24) + 4(4) + 3 = 67


A fourth method is presented in the Liber Abaci, by Fibonacci in 1202. After introducing how to divide by numbers of one digit,
and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A composed fraction might look like image at right. Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right.
The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75.
When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four.

This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of 749 divided by 75 as

which would be 9 and 4/5 + 4/25 + 2/75 or 9 74/75.

A fourth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method is shown below from another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400’s up to the early 20th century.


The fifth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.


The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.

In a Greasham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."
I recently found a site called
The Algorithm Collection Project
. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one
interesting note
, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted. (More about division symbols at Symbols of Division
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