THE EMPTY COLUMN
- An Abridged Version (S.R)
Introduction:
In this essay Tobias Dantzig narrates the story of one of the three R's, namely 'Rithmetic, which though the oldest, came hardest to mankind. The essay opens with a brief epigraph of Laplace which says:
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols…Its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions.
The Origin of Written Numeration:
Written numeration probably originated in man's desire to keep a record of his flocks and other goods.
Notches on a stick or tree,
Scratches on stones and rocks,
Marks in clay,
these are the earliest forms of this endeavour to record numbers by written symbols. Archaeological researches trace such records to times immemorial, as they are found in the caves of prehistoric man in Europe, Africa and Asia.
The Oldest records of Written Numerals:
The oldest records of the systematic use of written numerals are those of the ancient Sumerians and Egyptians, both traced back around 3500 B.C. In spite of the distance that separated them , we are struck with the great similarity in the principles used.
The Distinct Cardinal Characters of Old Numerals:
Whether it be
the cuneiform numerals of the ancient Babylonians,
the hieroglyphics of the Egyptian papyri,
the queer figures of the early Chinese records, or
the tally-stick of the English.
We find everywhere a distinctly cardinal principles. Each numeral up to nine is merely a collection of strokes. The same principle is used beyond nine, such as tens, hundreds etc, being represented by special symbols.
The Art of Reckoning In The Middle Ages:
Neither the ordinal system of the Greeks nor the cardinal system of Rome was capable of creating an arithmetic which could be used by a man of average intelligence. That is why, from the beginning of history until the advent of modern positional numeration, so little progress was made in the art of reckoning. When a German merchant of the 15th century appealed to a prominent professor of a university for advice as to where he should send his son, the reply was that, if the mathematical curriculum was confined to adding and subtracting, he could obtain it in a German university, but for the art of multiplying and dividing, he should go to Italy, which was the only country where such advanced instruction could be obtained.
Multiplication and division in those days had little in common with the modern operations bearing the same names. Multiplication, for instance, was a succession of duplations, which was the name given to the doubling of a number. In the same way, division was reduced to mediation, i.e., 'having' a number. A clear insight into the status of reckoning in the middle Ages can be obtained from an example.
Using modern notations:
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Today / Thirteenth Century
46 / 46 x 2 = 92
13 / 46 x 4 = 92 x 2 = 184
138 / 46 x 8 = 184 x 2 = 368
46 / 368 + 184 + 46 = 598
598 /
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thus, computations which a child can now perform required then the services of a specialist.
Difficulties in Calculation before the Discovery of Modern Numeration and How Man Countered Them:
The growing complexities of life, industry and commerce, of property, taxation and military organization, all called for intricate calculations which were beyond the scope of the finger technique. The numeration of those times was incapable of meeting the demand. So man had to resort to mechanical devices which ultimately resulted in the invention of the counting board or abacus, which has been found in practically every country where a counting technique exists.
The abacus consists of a flat board divided into a series of parallel columns, each column representing a decimal class, such as units, tens, hundreds, etc. The board is provided with a set of counters which are used to indicate the number of units in each class. The many counting boards known, differ merely in the construction of the columns and in the types of the counters used. The Greek and Roman types had loose counters, while the Chinese Suan-Pan has perforated balls sliding on slender bamboo rods. The Russian Szczety, consists of a wooden frame on which are mounted a series of wise rods with sliding buttons for counters. The ancient Hindu “dust board” was also an abacus in principle, the counters here being laid by erasable marks written on sand.
To this day, the counting board is in daily use in the rural districts of Russia and throughout China, where it persists in open competition with modern calculating devices. But in Western Europe and America the abacus survived as a mere curiosity which few people have seen except in pictures.
The history of Reckoning: A Picture of Desolate Stagnation:
In the long period of 5000 years of civilized existence, which saw the rise and fall of many a civilization, there was a lack of achievement in the history of reckoning, the earliest art practiced by man. Thus the history of reckoning presents a peculiar picture of desolate stagnation.
The Discovery of the Principle of Position:
The achievement of the unknown Hindu who sometime in the first centuries of our era discovered the principle of position assumes the proportions of a world-event. We know for sure that, without it, no progress in arithmetic was possible. The principle is so simple that today the dullest school boy has no difficulties in grasping it and it is particularly puzzling to us that even the great mathematicians of Greece did not stumble on it.
The 'Anatomy' of Our Modern Numeration:
The principle of position consists in giving the numeral a value which depends not only on the number of the natural sequence it represents, but also on the position it occupies with respect to the other symbols of the group.
Thus the same digit two has different meaning in three numbers -
342, 725, 269 ;
in the first case it stands for two;
in the second case it stands for twenty,
in the third case it stands for two hundred.
but that is precisely the scheme of the counting board, where 342 is represented by
___ ___ ___
True! But there is one difficulty. Any attempt to make a permanent record of a counting-board operation would meet the obstacle that such an entry as may represent any one of several numbers: 32, 302, 320, 3002 and 3020 among others.
In order to avoid this ambiguity it is essential to have some method of representing the gaps, i.e., what is needed is a symbol for an empty column.
No progress was possible until a symbol was invented for an empty class, a symbol for nothing, our modern zero. The concrete mind of the ancient Greeks could neither conceive the void as a number nor endow the void with a symbol.
The Indian term for zero was sunya, which meant empty or blank, but had no connotation for 'void' or 'nothing'. and so, from all probability, the discovery of zero was an accident brought about by an attempt to make a permanent record of counting -board operation.
The Evolution of the Indian Sunya into the Zero of Today:
How the Indian sunya became the zero of today constitutes one of the most interesting chapters in the history of culture. When the Arabs of the 10th century adopted the Indian numeration, they translated the Indian sunya by their own sifr which meant ‘empty’ in Arabic. Sifr was Latinized into zephirum. This happened at the beginning of the 13th century and in the course of the next hundred years the word underwent a series of changes which culminated in the Italian zero.
About the same time, Jordanus Nemerarius was introducing the Arabic system into Germany. He kept the Arabic word changing it slightly to cifra. In the English language the word cifra has become ‘cipher’ and has retained its original meaning of zero.
The essential part played by zero in the new art of reckoning did not escape the notice of the masses. Indeed, they identified the whole system with its most striking feature, the cifra.
This double meaning, the popular cifra of the masses standing for the numeral and the cifra of the learned, signifying zero, caused considerable confusion. The learned had to yield to popular usage, and the matter was eventually settled by adopting the Italian zero in the sense in which it is used today.
The same interest attaches to the word algorithm which today refers to any mathematical procedure consisting of an indefinite number of steps, each step applying to the result of the one preceding it. The word ‘algorithm’ is merely a corruption of Al Kworesmi, the name of the Arabian mathematician of the 9th century whose book (in Latin translation) was the first work on this subject to reach western Europe.
The Period of Transition:
The transition from the ancient unwieldy numeration to the positional numeration was not immediate, but extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century, and went through all the usual stages of obscurantism and reaction. In some places Arabic numerals were banned from official documents; in others the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging. Indeed very little of essential value or lasting influence was contributed to the art of reckoning in these transition centuries. Only the outward appearance of the numerals went through a series of changes. In fact, the numerals did not assume a stable form until the introduction of printing. So great was the stabilizing influence of printing that the numerals of today have essentially the same appearance as those of the fifteenth century.
Conclusion:
Conceived in all probability as the symbol for an empty column on a counting board, the Indian sunya was destined to become the turning point in a development without which the progress of modern science, industry, or commerce is inconceivable. By paving the way to a generalized number concept, it played a fundamental role in practically every branch of mathematics. In the history of culture, the discovery of zero will always stand out as one of the greatest single achievements of the human race.
It is indeed a great discovery which has profoundly affected the life of the race, not the reward of painstaking research, but a gift from blind chance.
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