Counting over the ages
Counting over the ages
Mankind needed to count as soon as they became aware of each other. But it is remarkable how they handled it. There are still some tribes who can count only up to two. Everything more than that is 'plenty' or 'much' or 'many'. In thirty native Australian languages, the counting does not proceed beyond four. But there is a language community which can go up to fifteen or twenty. In Andaman's, there is a community which can count up to two only but with combinations and groups can go up to ten. the nose is struck with one finger, the little finger and 'ubatul' - one - is spoken. the next strike is counted as 'ikpar' - two. The third strike is 'anka' - one more. When the fingers of the second hand are also finished, the two hands are brought together to indicate ten.
In the earlier period, the object is important. We count the figures, we measure by fingers and hands. the measure 'foot' started in this way and Indians had, till recently, balisht - distance between thumb and finger when stretched - as a measure. Abstract numerals came much later in the history of mankind. But there are still languages in Malaysia and Indonesia where the numbers literally mean one stone, two stones, etc. Nieus of South Pacific use one fruit, two fruits, etc. When a Zulu counts beyond five, he uses the expression 'taking the thumb' - tatisitupa - while holding the thumb meaning he has already counted the fingers of the other hand. For seven he says 'u kombile' - he pointed - with the index finger.
It is to be noted that Englishman counts with his fingers, he starts with the fingers and ends with the thumb. The Frenchman starts with the thumb.
When the symbols came, they were two to start with. Early Syriac writing has the numbers as ۱, ۲, ۲۱, ۲۲. In that sense, we have returned to binary notation after twenty centuries though we count with 0 and 1. Natives of Queensland in Australia count as "one, two, two and one, two twos, much". African pigmies count a, oa, ua, oa-oa, oa-oa-a, oa-oa-oa and so on.
The scale of three is also exhibited at many places. The one, two, plenty sequence was popular and 'three' was used as a substitute. In English we still say 'thrice is he armed that has his quarrel just'. In French we have the phrase 'tres bien'; in Latin we have 'ter felix' - thrice happy meaning very happy; the Greek 'trimeg istos'. In Yahgan Fuegian tribe, the words are 'kaueli, kombai, maten, akokumbai (the other two), ako maten (the other three).'Old Phoenician system grouped their marks by threes indicating a ternary system.
In South America, some tribes count to a scale of four. After fur comes four and one, four and two etc. But five is the scale which was quite popular. Certain South American tribes use one, two, three, four, hand, hand and one, hand and two and so on. Blending of scales is also noted. In Siberia, the count proceeds as follows - one, two, three, three one, five, two threes, one more, two fours, ten with one missing, ten. In English also, we go up to twelve and then revert to scale of ten i.e. thirteen (three ten), fourteen (four ten) etc.
Scale of six is uncommon but is found amongst Bolans of West Africa. Scale of eight is also known. In India the weights went by rati, masha, tola, chhatank where scale of eight is known, there being eight ratis to a masha. The thickness of the wood still goes by 'soots;, there being eight soots to an inch. We have eight furlongs to a mile.
But ten is the usual scale adopted. It is mainly due to the reason that man has ten fingers on his hands. But there are scales above ten also. Twelve is common perhaps due to the fact that 12 is divisible by two, three, four and six. Twelve inches to a foot, twelve ounces to a pound, twelve pence to a shilling, twelve moths to a year (though that could be related to the moon circles) , twelve signs of zodiac and by the common counting unit of dozen. Sixteen is another scale known and is being continued in computer language as hexadecimal system. In India, we had sixteen chhatanks to a seer, sixteen girahs to a yard, sixteen annas to a rupees. We also have sixteen ounces to a pound. Scale of twenty is also known. In French, the counting system goes up to twenty (vingt), the next number being twenty and one(vingt et un) though there is hybrid for sixteen (dix-sis), seventeen (dix-sept) etc. We have quatre-vingt for eighty; quatre-vingt-dix for ninety. In the older version of French, it went up to quinze - vingt (fifteen twenties). In Danish, we have 'the mean between two twenties and two thirties' for fifty; in Welsh, we have 'one and fifteen over twenty' for thirty six. In English 'score' is used quite often. In Mayas of South America and Aztecs of Mexico, twenty was a popular counting unit.
How high the count went was determined by the needs of the community. It is easy enough to recognize the sheep who is missing without having to count them. So the corresponding numbers also were not necessary. In Papua New Guinea, the persons on the coastal belt counted up to ten but those inland needed numbers up to five only. The Kaffirs, of Africa, having a large number of cattle counted up to hundred or more. Nubians and Abyssinians went further up to thousand and even beyond. One Polynesian language has number names up to thousand. In India, the Vedic philosophy led to a practically unlimited number system. It was the most extensive found in the ancient people.
(From History of Mathematics Vol. I David Eugene Smith (Dover Publications New York - 1958)