The history of written numbers contains one instance in which a system of numerals derived from the counting board achieved universal validity and recognition. Since the era before the birth of Christ; about 500 B.C., the Chinese civilization used little bamboo or wooden sticks as calculating pieces (chou) on their counting board. To form the numbers, the Chinese arranged them in a special manner, like this :
A new form of abacus saw the light of day in the tenth century. A man named Gerbert introduced it. He travelled to Spain and got acquainted with the Hindu-Arabic numerals. He developed an abacus based on that. He has left a description on how to do arithmetic calculations, but there is very little information about how the abacus itself was. The board was similar to the ancient decimal Roman abacuses.
At the time they thought that Phytagoras invented the abacus. Therefor a part of the abacus was named after him. This was all wrong.
As earlier mentioned he based a part of his abacus on the ancient Hindu-Arabic numeral system. He introduced counters on which Hindu-Arabic numbers were written. They were called apices.
Gerberts abacus did not get of with a good start. It was a disaster, but later it got accepted, and used in merchant and traders.
Reckoning on the lines was developed in the twelfth century and became the normal method of computation for trade and money exchange generally. That of horizontal decimal lines replaced the principle of decimals columns. Plain counters were now placed actually on the lines, which might be drawn on paper, embroidered on cloth, or engraved on wooden tables. Occasionally a plain cloth would be used and counters or other objects placed one below the other at one side to indicate where lines would have been had they actually been marked out. Up to 4 counters were placed on any one line, the space between the lines being used for counters representing 5 of those on the lines on the line immediately below.
Sometimes the lines would be designated with symbols representing a specific coinage. Often they would be completely unmarked.
There was one advantage of this new orientation, it was now easy to represent several numbers side by side. However, the parallel with the new place-value notation, gradually being introduced, was lost, and those who practised reckoning on the lines became fierce opponents of the Hindu-Arabic numerals and the new arithmetic associated with them.
There are many textbooks that describe exactly how calculations were performed on the lines. The operations include numeration, addition, subtraction, duplation, mediation, multiplication, division, elevation and resolution.
Numeration simply meant placing counters on or between the lines. Duplation and mediation were the operation of doubling and halving. Elevation was the process of grouping counters together so as to replace them with counters higher up the board. Resolution was the reverse of elevation, it involved the decomposing of a counter into either 2 or 5 counters.
In England, a special kind of counting board came into common use in government offices. It was used in addition to the usual form of reckoning on the lines. The board was divided horizontally and vertically. The vertical divisions representing pence, shillings, pounds, and twenty-pounds. Up to 19 counters could therefore be required for any of the columns except the one on the right, for which up to 11 counters could be required. A clever system was adopted whereby a counter placed above the left-hand end of a group of counters in any particular cell, other than the pence cell, meant 10, and one placed at the right-hand end meant 5. In a pence cell, a counter placed above the right hand end of a group meant 6. Patterns of counters were copied down directly into documents, though this seems to have happened only rarely. The main reason for this is that money-boards using counters in this particular way, were introduced in the sixteenth century.
By the close of the seventeenth century, the counting board had almost completely disappeared, so much that a Russian abacus brought to France after the Napoleonic campaigns was regarded as a remarkable curiosity. The abacus and reckoning on the lines did, however, leave behind a permanent heritage of words.
Although the Roman hand abacus never became widely popular in the West, it nevertheless penetrated eastwards, and modern forms of it are still extensively used in the Far East. It was adopted in China, probably in the twelfth century, and passed to Japan in a modified form some four centuries later. It then returned westwards and linked up with an old tradition in the Slavic countries where it still can be found. Displaced persons brought the Slavic form back into western Europe, where it can still occasionally be seen in practical use. We are probably most familiar with it from the classroom, where it has frequently been used for teaching elementary arithmetic to young children. Special forms of abacus have in fact been developed for teaching purpose, of which the abacounter is an example. The Roman hand abacus had grooves in which small spheres could be moved up and down. However, the Chinese form always seems to have consisted of beads on bamboo rods. It is possible that this change was suggested by the knot-numbers, which were widely used in the Far East. The suan-pan, as the Chinese abacus is called, usually has 2 beads, representing fives, on the upper part of each wire and 5 beads on the lower part. Beads are moved towards the dividing bar, which is said to separate heaven (the fives region) from hell (the units region). There are usually sufficient rods to enable at least two 4-digit numbers to be entered side by side. The Japanese soroban usually has even more rods than the suan pan. Generally, there will be 5 beads for units and only 1 for fives. A little of the versatility of the suan-pan is therefor lost. However, the Japanese have developed an incredible dexterity in using the soroban.
The suan-pan and the soroban are still to be seen today. They are used in both elementary and secondary schools, where the children make surprising progress in arithmetic. They are used in shops, in banks, in accountants offices and in government finance departments. The abacus has remained the hand-calculator for the individual in some places, but it remains to see for how long. The Slavic form of the abacus is normally used with the wires in the horizontal rather than the vertical position. There are ten beads on each wire, the middle two beads being coloured differently from the rest. This proves an even more effective way of allowing for variations in representing numbers during calculations.
John Napier was the man to invent the forerunner to the modern slide rule, which was used in schools until the middle of the 20th-century. Back in the days it when was invented, it was used to make calculations easier in engineering and in science. At that time it was also referred to as Napier`s Rods or Napier`s bones. The rods principle is based on lattice multiplication.
Napiers rod became very popular in
the17th century. It was as easy to carry out a division as it was to carry out a
multiplication. The rod did go through several changes in design like some were designed
to be placed side by side and others were made round.
A guy called Willian Oughtred invented the slide rule, but he waited three years after
the discovery before he went public with it. It was pretty simple to perform subtractions
and additions, but the porpoise with the slide rule was also to perform multiplications
and divisions. Logarithms were also a very important thing to use a slide rule.
The next obvious stage in the development of aids to calculation was that they should become mechanised. The first machine for performing the four basic arithmetical operations was designed and constructed by Blaise Pascal in 1942 when he was only nineteen. We have a full description of this machine and a number of authentic models have been preserved. Essentially it was an adding and subtracting device; multiplication and division had to be carried out by successive addition and subtraction.
The machine consisted of a number of geared wheels set inside a box in whose top there was a row of windows for reading off the result of any calculation. Numbers were set into the machine on a row of wheels below the windows. The crucial part of the invention was the automatic carrying mechanism. A ratchet was introduced between adjoining wheels so that whenever a wheel passe from reading 9 to reading 0, the wheel next to it was moved on one digit. To enable this to be effective, the wheels had to rotate in one direction only; hence, subtraction was carried out by the addition of complements. Thus to subtract 8 we simply add 2 and deduct from the result.
Leibniz achieved the mechanisation of multiplication and division about 30 years after Pascal’s invention of his machine. We have a full description of this later machine and some authentic models also. Leibniz’s machine had two parts. One part, that for carrying out addition and subtraction, was similar to Pascal’s, the other part was new. There were three kinds of wheel: wheels for addition, wheels for multipliers, and wheels for numbers to be multiplied. The crucial feature here was the introduction of gear wheels on which the number of operative teeth could be varied. These stepped wheels continued to be used in mechanical calculators until only recently, when the manufactore of such calculators ceased because of the competition from electronic machines
Small and not so small desk calculators began to be manufactored on a commercial scale early in the nineteenth century, and by the end of the century they were commonplace in engineering and accountancy. Various refinements were introduced, including additional mechanism for printing the results of calculations. However, these machines were not in any true sense automatic. The various operations had to be effected by hand, often in several stages, and calculation was therefor a comparatively slow progress.
During the nineteenth century attempts were made to design automatic machines, that is, machines that would carry out the basic arithmetic processes automatically once the numbers to be processed had been entered into their registers. The first person to propose such a machine was the Cambridge mathematician, Charles Babbage.
Babbage called his proposed machine a "difference engine". Its main purpose was the calculation of various specialised mathematical tables, wellknown at the time to be often very inaccurate. The machine was to have six registers which were so interconnected that it would be possible automatically to add the content of one register to another, and to transfer numbers between registers. A small prototype model was built in the early 1920s, and, as a result, Babbage received a substantial government grant to continue his work and produce a full-sized fully operative machine. This was never completed, and the projekt was eventually abandoned, largely because Babbage began to be inspired by the concept of a much more ambitious project. Babbage’s ideas were, however, put to good effect by a Swiss engineer, Georg Scheutz, who constructed a working machine, which was eventually used in the preparation of life tables.
Babbage’s new inspiration was the creation of what he called an "analytical engine". Today, we would call such a machine a "general-purpose computer". As with his earlier project, the building of this was never completed. It was to consist of an intricate system of gears, rod, and linkages making up a store for retaining numbers, a processing unit for performing arithmetic operation, a control unit for regulating the correct sequence of operations, and input and output devices. The scale invisaged was much larger than anything which had been attempted before. For example, the store was intended to retain up to 1000 numbers, each up to 50 decimal places. The control unit was the nervecentre of the whole contraption. It was this that effectively replaced the human operator essential to earlier machines.
One important feature of the proposed analytical engine was the use of punched cards. These had been invented in 1801 by the french loom-maker, Joseph Marie Jacquard, and were used for controlling the threads so that intricate patterns could be woven automatically. This principle has become a feature of the modern computer.
In 1890, an American statistician, Hermann Hollerith, introduced punched cards into a mechanical computer designed both to count and to sort which was used in the United States census. The machine is reputed to have saved the American government some two million pounds, a considerable amount in those daus even for a comparatively rich country. Hollerith’s cards had to be punched by hand; thus the setting up of an operation on his machine was a slow process. This problem was partially solved some 26 years later by the invention of a small hand machine, which would punch up to nine holes simultaneously.
Punched cards can be read automatically either by automatical or by electrical means. The information received at any particular location on the card is of a yes-no character: either there is a hole or there is not. This is immediately suitable for handling logical information; we can record directly whether or not some particular object or property exists, whether a particular statement is true or false, whether an aperture is open or shut, and so on. However, the objekt and using the punched cards is primarily to handle numerical information, and in order to do this the information has to be translated into binary form.
The binary numerial system should not be confused with counting by twos. Two-counting represents man`s first attempts at building up a system of number words and is a part of our pre-history. Leibniz first suggested binary numerials. His reasons were, curiously enough, theological. He believed that the two numerals
required 0 and 1, were symbolic of the creativity of God – God (represented by 1) created the universe out of nothing (represented by 0).
We do not present any detailed discussion of computers. However, having taken the story of calculating machines as far as the Hollerith machine, we will bring it up to date with a brief survey of the main developments in the twentieth century.
Mechanical machines are by their very nature comparatively slow in performance. For speed of operation, electrification was essential. The first automatic general-purpose computer was built at Harvard University by the IBM Corporation, one of the largest manufactures of punched-cards machines. It was completed in 1944 and contained more than 500 miles of wire. By today’s standards it was very slow, taking 3/10ths of a second for addition and 4 seconds for multiplication.
The first electronic machine appeared 2 years later. This was the ENIAC computer at the University of Pennsylvania. Valves replaced the moving parts, the switches, of the earlier computer and addition time was reduced to 1/5000th of a second. Since that time, speed of performance has been considerably improved and the invention of modern integrated circuits has meant that computers that once occupied the whole of a large room can now be housed in what is little more than a large piece of furniture. As moderntechniques of microprocessnig develop, it will be possible to have computers, having enormous capabilities, small enough to be inserted in a pocket. The basic components of the modern computer have remained very much as Babbage conceived them. There is an input, a memory, a control, a processing unit, and an output. The operations to be carried out are controlled according to programs, which have to be prepared beforehand; though once prepared they can be stored in a computer’s memory and called upon as and when required.
Preparing a program involves breaking down an overall operation into a sequence of simpler operations, which the computer can carry out. Suppose that we wish to devide 1311 by 57. The computer may well be programmed to carry this out by a process of successive subtraction. The overall program, when written out, will look something like this.
Assuming that, as is usually the case, the computer automatically converts decimal numbers to their binary equivalents we can now see just how this program deals with our division. We enter 1311 and 57 and instruct the computer to devide. The answer to the first question "is divisor all zeroes"? is "no", so the operation proceeds. The lenght of the divisor is six binary digits, and the lenght of the quotient will be five digits. The dividend is now moved so that subtraction may start. We can represent what now begins (in decimal notation) as follows:
Quotient
57 1311
- 44 Negative remainder, so add 57;
74 Positive remainder so add 1 to quotient
The subtraction is now repeated, and the calculation proceeds. Eventually, the answer to the question "was that the last digit?" will be "yes" and the computer will cease operation, printing out the final quotient as 23.
An important part of this operation was the repetition of certain cycles of operations. These are included in a program as a loop.
Programs have to be very carefully designed so that there is always a possible path out of a loop; otherwise some operations would never finish. The idea of a program loop, clearly a crucial concept in computer operation, goes back to the daughter of Lord Byron, the Countess of Lovelace. She was a friend of Babbage, and had acquired a thorough grasp of the principles underlying the analytic engine. She concentrated her attentions principally on what we now call programming and is said to have acquired at least as good a grasp of this as Babbage himself, and to be much better than he in explaining them to other people.
The possibilities inherent in the developments in computers which are now taking place are almost beyond comprehension. It is for this reason, that they are regarded with suspicion, if not fear, by many lay people.