Ancient Multiplication
Methods





   One of the earliest records of multiplication is the method shown in the Rhind Papyrus.  The method used a process of doubling that as in this brief example of 22 x 44= 968.  First we write down the numbers 1 and 44, then we double each number and write the results under the originals.  This is continued until the next number on the left side would exceed  22.

1------44
2-------88+
4-----176+
8-----352
16----704+

   At this point we start down the left side looking for a total of 22. Each time we can add the number without exceeding our goal of 22, we put a check mark by the number opposite (I have used a plus sign instead).  To total 23 we take the 16 first. Since 16+8 > 22 we omit 22 and move to the 4, and two. Since 16+4+2=22, we mark, and then sum, the values adjacent to these numbers.  The result, 88 + 176 + 704 = 968.
    A copy of the Rhind Papyrus with the Egyptian method can be found at the University of St. Andrews Math History web site.  It includes how the same method is used for division and additional material about Egyptian fraction techniques.
  According to Bea Lumpkin Algebra Activities from Many Cultures, Egyptians also would use a multiple of ten and then doubles of that if needed.  For instance in the problem above they may have started by putting down 10---440 and 20---880 and then continuing with 1---44 and 2---88 and found 22*44 by adding 880 + 88.
  The method is often confused with a similar approach known as Russian Peasant Multiplication.  In the Russian method the two factors are written, and then one is doubled as the other is halved.  This method is sometimes called duplation and mediation (doubling and halving) in some early arithmetic texts.  The same 22*44 example from above would look like this:
22----44
11----88+
5----176+
2----352
1----704+
   In this procedure the doubled number (right column) is marked each time the halved column is an odd number.  Note that when there is a remainder or fraction it is ignored.  Notice that the marked numbers are identical to the ones marked in the Egyptian method above.  You can find a nice article with explanations of the binary nature of this method at the Dr. Math FAQ  on the topic.

   A third method that seems to have been created by Hindu mathematicians is called lattice, cell,  or gelosie (from the name of a type of shutter, now spelled jalousie) multiplication.  This method was known at least as early as 1010 when the Persian scholar, Karaji (ka-ra-yee) demonstrated it in his Kafi fil Hisab, (Book of Satisfactions).  This method is very similar to the modern algorithm, except that the alignment is in a rectangle, and the additions are carried out along a diagonal row.
   This figure shows an example of one method using 132 x 247.
  The factors are written along the top and right side.  In many early Arabic and other examples the numbers were frequently written right to left.  The multiplication of 7 times 2 is shown in the bottom right corner with the tens moved to the shaded column.  This is repeated in 7x3 = 21 in the next column to the left. Notice that with this method it made no difference if you started from units to hundreds or hundreds to units since each product had two spaces for an answer.  The results of the multiplications were added along the diagonals which are shaded to make alignment easier in the figure.  In the figure the shaded diagonal at the bottom right shows 8+1+1=10.  The ten is written by spreading the number over two columns. The arrow shows the carry digit.   The numbers in the boxes are final results with arrows indicating when a ten value was carried to the next column. Many teachers use this today as an alternative to the traditional algorithm, but it seems to offer few advantages in terms of understanding the nature of multiplication. You can find additional examples and explanation here.
A method that should look very familiar to most people was used by the Indian mathematician Brahmagupta during the eleventh century. The method is referred to as gomutrika, which seems to translate to "trajectory of a cow's urine". Brahmagupta wrote one of the factors horizontally and repeatedly along a skew line for each digit of the second factor. This second factor he wrote down the page. Here is how he might have written out 315 x 452.


The notations in italics at the right are added to show the steps, and were not part of the original. Although the alignment in this example goes right to left, he also wrote variations with the shift moving to the left from row to row.

Another type of multiplication used in earlier days is shown in the first arithmetic book published in North America, the Sumario compendioso de las quentas de plata y oro que in los reynos del Piru son necessarias a los mercaderes y todo genero de tratantes Los algunas reglas tocantes al Arithmetica. The title translates to "Comprehensive Summary of the counting of silver and gold, which, in the kingdoms of Peru, are necessary for merchants and all kinds of traders". The author was Brother Juan Diez, a priest who arrived in Mexico with Cortez in 1519 but the book was not published until 1556. I have attempted to duplicate in type a figure from an article Spanish colonial Mathematics: A Window on the Past by Ed Sandifer that I found on Dr. Sandifer's webpage. I will show several steps of the development to facilitate the readers understanding.

The problem is to multiply 875 by 978. Multiplication starts from the left, and the 800 is multiplied by the 900.

Notice that the zeros are left off, much as we leave them off now when we start from the other side. Next the 800 is multiplied by the 70 in 978. Since this has one less power of ten than the previous multiple, it will start one column to the right, so the 5 of the 56 goes under the 2 of the 72, but since there is nothing in the third column, the 6 goes at the top of this column, next to the 2 of the 72.

The next multiplication is between the 800 and the 8 in 978, which is again one less power than the previous multiple, and so the 6 in the 64 must go in the column under the 6 of the previous 56. The 4 will go at the top of the next column since there is nothing in that column yet.

Now that all three digits of 978 have been multiplied by the 8(hundred), we move on to the 7 in 875 and multiply it across. Since the 70 times 900 will have only one less place than the 800 times 900, it will start in the second column also, so the 6 of the 63 will go under the 2 and 5 in the second column, and the 3 will go under the pair of sixes in the third column.

From here we hope the reader can follow on his own, and we show the completed problem.

the reader might wish to compare the appearance of this method to the Galley division method. It is not hard to imagine that this multiplication was even easier to perform on a sand tray or counting table where numbers were simply added and adjusted as the work progressed

Certainly one of the first counting methods involved the use of the hands, and probably simple multiplication methods were likewise commonly performed on the hands. The image at right shows the method of finger counting as used in the Summa Arithmetica by Pacioli (1532). Similar figures are described and illustrated back to the birth of Christ. In Menninger's Number Words and Number Symbols(pg 211) he shows an image of 1st Century Roman Markers used in a game which show the numbers eight and nine both in Roman Numerals and in finger gestures. The counters are located in the British Museum. Pliny the Elder, who died in 79AD, wrote of a statue of Janus in Rome which showed the finger sign for 300 on the right hand and 65 on the left, to indicate the number of days in the year. The representation of numbers greater than 100 on the right hand led to the reference by the Roman Author Juvenal, around 100 AD, about Nestor, the oldest of the Greek kings at the Battle of Troy; "Happy is he who so many times over the years has cheated death and now reckons his age on hte right hand."

Counting on the fingers is not the same as multiplying, but there can be no doubt that it was around by the time that Leonardo of Pisa wrote, "multiplication with the fingers must be practices constantly, so that the mind like the hands becomes more adept at adding and multiplying various numbers. (from Menninger as above, pg218)

One simple hand multiplication is now so wide-ranging that it is hard to believe it was not known prior to the first written description. The method is used to replace learning the multiplication tables from six to nine. To use the system, raise both hands before the face and think of the digits as the excess over five. To multiply 7 x 8, turn down two digits on the left hand (thumb and first finger) since 7 is 2 in excess of five. Now turn down the first three fingers leaving only the little finger and ring finger extended on the right hand. To find the product of 7 and 8, count the number of fingers turned down. Two and three are five, so the ten's digit is five for fifty. now multiply the fingers extended on each hand; two times three is six so the units digit is six, and the answer is fifty-six.

Essentially the same method is explained in Robert Recorde's Grounde of Artes(about 1543). The image on the right is from Cajori's A History of Mathematical Notations . The instructions are to multiply the compliments on the right to get the units digit, then take either of the original factors (7 or 8) and subtract the compliment of the other joined by a line, for example 8-3, to get the ten's digit. Some suggest this is the origin of the use of an X for multiplication.

Another old method of hand multiplication still found in use is the multiply by nine device. Hold both hands in front of the body with the palms toward you. To multiply nine by any number less than nine, for instance four, count in on the fingers from left to right and turn down the fourth digit. The three still standing on the left are the number of tens, the six on the right of the turned digit are the ones. The method also can be extended to multiples of 9 up through 19 by the same motions. For example, to multiply 9 x 14, bend the same finger as in 9 x 4. Now instead of three tens, add nine to the digits shown to get 12 tens or 120, and the six still standing on the right make 126. If that explanation is not clear, you can see a much longer explanation with photos of the hands here .

In the 15th Century Multiplication was presented in various, often similar, algorithms. In his Capitalism and Arithmetic, Frank Swetz lists eight different methods that appear in the 1494 work of Luca Pacioli:




The first is very similar to the gelosie method demonstrated above. It probably was originally modeled on a method using markers on a squared grid similar to a chess board. The difference was that the multiplier was written down the side on a diagonal, thus forcing the indexing of the partial products, and the carry digits were transferred from column to column mentally. This may have often been used with the "by the table" method so that partial products could be formed from values looked up in the tables. I have not seen the "little castle method of multiplication, but suspect it was similar to the method illustrated on page 441 of Menniger. This is essentially an upside down version of the method above from the "Sumario", the early Spanish language arithmetic. "Per Quadrilatero" was another chessboard method, usually written in squares and was essentially the gelosie method but the carries were done mentally so the diagonal lines were not found.

Per Crocetta is another candidate for the origin of the x as a symbol for multiplication. The first illustration shows how the figures were often written and shows the multiplication of 34 and 62. The lines shown should be familiar to any algebra student who has memorized the "FOIL" method for finding a product. At first all four products were probably written out below and indexed as needed to be added in columns. Later the idea of carrying from one product into another probably produced the current version. Many books printed the same operation with only the cross shown in the second figure according to Swetz. He also mentions that Pacioli extended this method to more than two digits in the factors.

Per Repiego was a multiplication that broke one of the multipliers into its factors and then multiplied by these. 35 x 14 might be multiplied by breaking 14 into 2x7 and then multiplying each of these times 35 and adding the two results. I have also not seen an example of the scapezza method, but assume it simply was a version of one of the chessboard methods which was seperated out instead of kept in the framework of the chessboard.