Methods of Counting and Their Uses

 As mentioned earlier, primitive people had various ways in which they counted objects.  In many cases, the tools used to keep track of various objects depended on the availability and ease of use of different materials.  For example, some cultures used collections of shells while others used tally sticks.  No matter what aid was used for the counting process, the underlying concept was the same.  It was all about making a correspondence between the objects of interest and the counting aids.  What is interesting about of this type of counting are the different methods used.   

Body Counting

A very common form of counting was by using the hand.  The hand was an obvious choice for the use of correspondence counting, or reckoning, as it is an instantly accessible device.  As noted by Aristotle, the widespread use of counting to ten is not the result of a calculated choice, but rather the anatomical accident that most humans are born with ten fingers and ten toes.  So man learned to count using his fingers and perhaps at this stage, without names for the numbers that he was counting.  Nevertheless, this was an effective method of counting that is still used all over the world.  Even in schools today, young children are taught to count on their fingers.

"The Universal History Of Numbers", Ifrah, Georges p22

Finger counting was such an effective method that it was used by some people to count to numbers greater than ten.  For example, there exist certain South African people that employ the use of three people to count to numbers larger than one hundred.  This is done by the first person, beginning with the little finger on their left hand, counting continuously until ten is reached.  Then when the first person reaches ten, the second person raises their left little finger and the first person starts again.  Each time ten is reached, the second person raises a finger.  Similarly, when one hundred is reached the third person raises their left little finger and the process continues until the desired number is reached.  Therefore, by using three people, the first representing units, the second representing tens and the third representing hundreds, these South African people were able to count to numbers above one hundred.

The idea of counting on the fingers was extended by some cultures to reach numbers higher than ten.  For example, a method used in China and India enabled a person to count to 28 using the finger joints on each hand.  This can be done as illustrated (left).
"The Universal History Of Numbers", Ifrah, Georges p50
This idea of using a part of the body to count was taken to a different level by other cultures.  People from places like Papua New Guinea and the Torres Straits Islands used their whole body to count.  By taking various points of their body, for example including the eyes, nose, hips, etc, as well as the fingers, these people were able to count to numbers as high as 41.  The diagram at right illustrates one particular method of body counting.
"The Universal History Of Numbers", Ifrah, Georges p16

Using the body as a counting tool, people were able to communicate information between tribes.  For example, a messenger could be sent to another tribe with the message that they wished to trade 20 baskets of food for 15 pearl necklaces, say.  This could have been done by simply indicating the point on his body that corresponded to the correct number of objects.

Unfortunately, counting on any part of the body was not permanent and in many cases permanent records needed to be kept.  So while body counting was extremely useful, another method would be needed if any type of permanent record were to be kept.


Collection Counting

Another type of method of keeping a record of the number of objects is by having a collection of smaller objects that can be used to make a one-to-one correspondence with the objects to be counted.  This method was briefly explained earlier with the example of the shepherd, his sheep and a collection of pebbles.    Keeping a record of the number of objects to be counted is as simple as placing one of the smaller objects in a pile for each of the objects to be counted.  In this way, as explained earlier, there is one smaller object for each of the counted objects.  Now if we were to come back and check our counted objects, all that is required is to repeat this method.  It can be quickly seen if there are any objects missing, as there would be some of the smaller objects left over.

A slight variation on the example of the shepherd is the following.  For each animal let out of the enclosure, place one pebble in a collection.  Then at the end of the day, remove a pebble for each animal that returns to the enclosure.  This way you quickly see if you have lost or gained any animals throughout the day.

Many primitive people in varying ways used this method of keeping tally.  Some cultures used collections of shells, pearls, pebbles, elephant teeth, sticks, or even coconuts to keep a tally.  These were quite useful methods that were employed frequently by early people; however, in some cases it could be quite impractical.  If the shepherd had a very large flock of sheep; it would not be very convenient to be carrying around a large, heavy bag of pebbles.  So from this we move to another method of keeping a tally of objects.


Tally Sticks

Tally sticks are yet another way of counting a group of objects.  A tally stick consists of a piece of wood, bone or any object in which it is possible to cut small notches.  Each notch in the tally stick corresponds to an object being counted.  Primitive man would have used this method of counting to make a record of anything of interest or importance, for example, the number of animals slain by a hunter.  Tally sticks were a much more efficient way of keeping track of the number of objects being counted, as one needed to carry around only a small piece of wood or bone.  The following picture illustrates some very primitive tally sticks made from bone.

"The Universal History Of Numbers", Ifrah, Georges p62

As tally sticks were an extremely convenient way of keeping tally, they were still in use much later in time.  However, in some cases, the tally sticks evolved to become more than just notches cut into wood and had more varying uses.  Not only could they be used to count a group of objects, but they could also be used to keep a record of debts or credits owed by a person, the number of days, weeks, months or years gone by, or even a mark of ownership. 

An example of a tally stick that contained numerical symbols, rather than notches, are the Swiss water tallies. These tally sticks showed the amount of time that a particular farming family had water rights to on any given day.

"Number Words and Number Symbols", Menninger, Karl p235

Tally sticks could also be used to keep a record of credits.  For example, if a baker had given a number of loaves of bread to a woman over a week and was to settle the account at the end of the week, then a tally could be kept in the following way.  The baker would have one tally stick and the woman another.  When a loaf of bread was received, the tally sticks would be placed together and notches would be cut in both, corresponding to the number of loaves (in the presence of both parties).  This would continue during the week and at the end of the week the account could be settled without argument.  No dispute would occur as the tally sticks would be placed together and compared.  If the baker tried to add notches it would be obvious and it would be very difficult for the woman to remove any notches, therefore there would be no dispute about the account.

This method could be used for more than one customer and in many more situations.  To differentiate between tally sticks, they would be marked with the owner’s mark, or signature.  With the use of an identifying mark, tally sticks could be used for more accounting type purposes.  Even until the 1800’s, tally sticks were in use for taxation purposes.  The illustration to the left is of a Russian ‘tax book’.  Each tally stick in the collection contains a family mark, or signature, the number of family members and the amount of tax to be paid.
"Number Words and Number Symbols", Menninger, Karl p239


Knotted Strings

We now move on to another type of counting that involves the use of knotted strings.  In their simplest form, knotted number strings are much the same as the simple tally sticks.  That is, one knot in the string has the same meaning as one notch in the tally stick.  Counting with the use of knotted strings has been found all over the world.  For example, Tibetan prayer-strings and the rosary are both simple knotted number strings, which are used to prescribe the number of religious exercises to be undertaken. 

Like the tally sticks, the knotted number strings gradually evolved into a slightly more complicated device. Combinations and different types of knots replaced the simple single knot.  Different types of knots carried different values and the relative positions of the knots on the string came to denote the order of magnitude of the number. 

This can be seen clearly in the case of the Peruvian quipu, a number string used by the Incas to record, not only numbers, but also financial transactions, laws and even history.  The following is an illustration of a Peruvian quipu.

"Number Words and Number Symbols", Menninger, Karl p253

Each string coming from the main strand at the top contains knots which represent a particular number.  The knots have been grouped together along the string in such a way that the higher orders of magnitude are above lower orders of magnitude.  That is, higher orders of magnitude are closer to the main strand.

For example, the number 3,643 would be represented on one of these strands as in the illustration to the left.
"The Universal History Of Numbers", Ifrah, Georges p68
Another detail that has been added to these quipus is the use of three different types of knots.  The image aside shows these different knots.  First there was the single knot (illustrated as number 1), then the double or figure-of-eight knot (illustrated as number 2) and finally the slip knots, which could have from 2 to 9 loops (illustrated as number 3 with three loops). However, the double knot and slip knots were used only to represent the units in a representation.
"Number Words and Number Symbols", Menninger, Karl p254
Possibly a descendant if the quipu was the chimpu of the Bolivian and Peruvian Indians.  The chimpu was a very similar device to the quipu, however, in this case number were represented using more than one string.  The order of magnitude of a group of knots was not only determined by the relative position of the knots on the strings, but also by the number of strings the knots held together.  For example, the number 4,456 would be represented as illustrated to the left.
"Number Words and Number Symbols", Menninger, Karl p255

Knotted strings could be used to record any number of things such as taxes, wages owed, calculations and even the weights and types of bagged produce.  In Germany, millers had their own system of knots that were used to inform the bakers of the weight and type of flour in a bag.  These knots were tied in the ends of the strings that were used to seal the bags of flour.  The following knots were used to represent the weights of the bags of flour.

"Number Words and Number Symbols", Menninger, Karl p255

And these knots were used to identify the types of flour in the bag.

"Number Words and Number Symbols", Menninger, Karl p255

A slight variation on the knotted number strings is strings of beads.  Strings of beads can most commonly be found used for religious purposes, for example Christian rosaries.  Even today, many religious persons such as Muslims, Buddhists and Catholics use prayer beads to help in the recitation of certain prayers.  These strings of beads consist of the correct number of small beads and in some cases larger or coloured beads separate these smaller beads as markers.  As recitation begins one moves their fingers over each bead as the prayers are said until no beads remain.  Below we can see a string of Muslim prayer beads.

"The Universal History Of Numbers", Ifrah, Georges p11