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New edition of a text/reference which describes the development of arithmetic and calculation tools from Egyptian stone carving to the IBM/360 computer. It includes discussion of important stages such as the invention of the zero, logarithms, the Babbage machines, analog, and the Zuse machines, and provides interesting stories about both the machines and the people who produced them. Annotation c. by Book News, Inc., Portland, Or.Product Details
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A History of Computing Technology
By Michael R. Williams
John Wiley & Sons
ISBN: 0818677392Chapter One
In the Beginning ...1.1 Numeration
Our system of numeration, if not a machine, is machinery; without it (or something equivalent) every numerical problem involving more than a very limited number of units would be beyond the human mind.
One of the first great intellectual feats of a young child is learning how to talk; closely following on this is learning how to count. From earliest childhood we have been so bound up with our system of numeration that it is a feat of imagination to consider the problems faced by early humans who had not yet developed this facility Careful consideration of our system of numeration leads to the conviction that, rather than being a facility that comes naturally to a person, it is one of the great and remarkable achievements of the human race.
It is now impossible to learn the sequence of events that led to our developing a sense of number. Even the most backward tribe of humans ever found has had a system of numeration that, if not advanced, was sufficient for the tasks they had to perform. Our most primitive ancestors must have had little use for numbers; instead their considerations would have been more of the kind Is this enough? rather than How many? when they were engaged in food gathering, for example. When early humans first began to reflect about the nature of things around them, they discovered that they needed an idea of number simply to keep their thoughts in order. As they began to live a settled life, grow plants, and herd animals, the need for a sophisticated number system became paramount. How and when this ability at numeration developed we will never know, yet it is certain that numeration was well developed by the time humans had formed even semipermanent settlements.
It is very popular, in works dealing with the early history of arithmetic, to quote facts about the socalled primitive peoples and their levels of numeration. These facts generally note that the primitive peoples of Tasmania are only able to count one, two, many; or the natives of South Africa count one, two, two and one, two two's, two two's and one, and so on. Although often correct in themselves, these statements do not explain that in realistic situations the number words are often accompanied by gestures to help resolve any ambiguity. For example, when using the one, two, many type of system, the word many would mean, Look at my hands and see how many fingers I am showing you. This type of system is limited in the range of numbers that it can express, but this range will generally suffice when dealing with the simpler aspects of human existence.
The lack of ability of some cultures to deal with large numbers is not really surprising. Our own European languages, when traced back to their earlier versions, are very poor in number words and expressions. The translation of the Gospels made by Bishop Ulfilas in the fourth century for the Goths uses the ancient Gothic word for ten, tachund, to express the number 100 as tachund tachund, that is, ten times ten. By the seventh century the word teon had become interchangeable with the tachund or hund of the AngloSaxon language, and the Gospels of that period denote 100 as hund teontig, or ten times ten. The average person alive in seventhcentury Europe was not as familiar with numbers as we are today. The seventhcentury Statute of Shrewsbury laid down the condition that, to qualify as a witness in a court of law, a man had to be able to count to nine. To apply such a condition today would seem ludicrous.
Perhaps the most fundamental step in developing a sense of number is not the ability to count, but rather the ability to see that a number is really an abstract idea instead of a simple attachment to a group of particular objects. It must have been within the grasp of primitive humans to conceive that four birds are distinct from two birds; however, it is not an elementary step to associate the number 4, as connected with four birds, to the number 4, as connected with four rocks. Associating a number as one of the qualities of a specific object is a great hindrance to the development of a true number sense. When the number 4 can be registered in the mind as a specific word, independent of the object being referenced, the individual is ready to take a first step toward the development of a notational system for numbers and, from there, to arithmetic. As was noted by Bertrand Russell:
It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two.
Traces of the very first stages in the development of numeration can be seen in several living languages today. Dantzig, in his book Number, describes the numeration system of the Thimshian language of a group of British Columbia Indians. This language contains seven distinct sets of words for numbers: one for use when counting flat objects and animals, one for round objects and time, one for people, one for long objects and trees, one for canoes, one for measures, and one for counting when no particular object is being numerated. Dantzig conjectures that the last set of words is a later development while the first six groups show the relics of an older system. This diversity of number names is not confined to obscure tribal groupsit can be found, for example, in widely used languages such as Japanese.
Intermixed with the development of a number sense is the development of an ability to count. Counting is not directly related to the formation of a number concept because it is possible to count by matching the items being counted against a group of pebbles, grains of corn, or the counter's fingers. These counting aids must have been indispensable to very primitive people who would have found the process impossible without some form of mechanical aid. Such aids to counting, although in different form, are still used by even the best educated professionals today, simply because they are convenient. All counting ultimately involves reference to something other than the things being counted. At first it may have been grains of corn but now it is a memorized sequence of words that happen to be the names of the integers. This matching process could have been responsible for the eventual development of the various number bases that came into existence because the act of counting usually takes the form of counting into small groups, then groups of groups, and so on.
The maximum number of items that is easily recognizable by the human mind at one glance is small, say five or less. This may be why initial groups consisted of about five items, and would account for the large number of peoples whose number systems were of base five. The process of counting by matching with fingers undoubtedly led to the development of the different number systems based on ten.
By the time that a number system has developed to the point where a base such as 5,10, 20, or even 60 has become obvious to an outside observer, the eventual development of its use in higher forms of arithmetic and mathematics has probably become fixed. If the base is too small, then many figures or words are required to represent a number and the unwieldy business of recording and manipulating large strings of symbols becomes a deterrent to attempting any big arithmetical problems. On the other hand, if the number base is too big, then many separate symbols are required to represent each number, and many rules must be learned to perform even the elementary arithmetical operations. The choice of either too small or too large a number base becomes a bar to the later development of arithmetic abilities.
A study of various forms of numeral systems yields the fact that a great variety of bases have been used by different peoples in various places over the globe. By far the most common is, of course, the number systems based on ten. The most obvious conclusion is that a very large percentage of the human race started counting by reference to their ten fingers. Although 5, 10, or 20 was the most popular choice for a number base, it is not uncommon to find systems based on 4, 13, or even 18. It is easy to advance a possible explanation of systems based on 5 and 20 because of the anatomical fact of there being 5 fingers on each hand and a total of 20 such attachments on both hands and feet. It is not quite so easy to see how the other scales may have developed in such a natural way.
If modern linguistic evidence can be trusted, the scale of 20 must have been very widespread in ancient times. It has been pointed out by several authors that some Eskimo people have a welldeveloped system which uses terms such as oneman for the number 20, twomen, for 40, and so forth. This obviously implies that a base 20 number system was in use at one time, but it is difficult to see how an early Eskimo could have had access to his toes for counting, unless he was inside a relatively warm igloo. This could imply that the Eskimos brought their number system with them when they moved into North America. In fact, it is possible to trace remnants of various number systems based on 20 all the way back to the very early people living in the tropical regions of southern Asia.
The European languages also show a marked preference for the grouping together of 20 objects. It is entirely possible that the general prevalence of base 10 numbers among the IndoEuropean peoples developed later from an original base 20 system. The most modern example of this is the French use of quatrevingt (fourtwenty) instead of huitante for 80, and quatrevingtdix (fourtwentyten) instead of nonante for 90. If we are willing to look at forms of the language which died out in the thirteenth century, then the terms sixvingt (120) septvingt (140), huitvingt (160), and even quinzevingt (1520 for 300) show that the base 20 system was no stranger. The English word score, as in the Biblical three score and ten years or Lincoln's famous Four score and seven years ago ... is another modern example of Europeans dealing in groups of 20. The term score will be discussed further in the section dealing with tally sticks.
Once the sense of number, as an entity separate from the thing being counted, has developed, there arises a need to assign names to the integers. This sequence of names, once decided upon, often becomes one of the most stable parts of a people's language. Thus, although languages may evolve so that almost all the words acquire vastly different forms, the sequence of integer names remains almost intact. The original meanings of the names of the integers are likely, therefore, to be quite obscure. The similarity of number names across related languages is easily seen by examining a table of these names in the family of IndoEuropean languages. Although the actual names differ from one language to another, the basic form remains constant (languages are Ancient Greek, Latin, Sanskrit, French, German, and English).
This stability of number names has been a good tool of the philologists. They have made extensive collections of the number names in many languages in an attempt to see which languages have common roots. Care must be taken in any exercise of this sort because a common set of names could have resulted from simple mixtures of languages, trade between two peoples, or simple coincidence.
George Peacock, in his Arithmetic, points out the situations that can arise by simple coincidence. He compares the number words of the Nanticocks (a tribe of Indians who used to reside on the south bank of the Chesapeake) with those of the Mandingoes (of Africa). The similarity is striking but can only be attributed to coincidence because any mixing of the languages due to geographic proximity or cultural contacts via trade is ruled out.
The opposite side of the coin can be shown by comparing the number words from languages which are known to be related. In Europe, the languages spoken by the Finnish and Hungarian people can be traced back to common root, yet the number words in these languages have very little in common.
The ability to count must soon have led to the need to record the results of counting. This probably first took the form of setting aside the pebbles used in the matching process, but would soon lead to the use of tally sticks, knotted cords, and the actual invention of numerals.
1.2 Written Number Systems
There appear to be two distinct stages in the development of written record keeping systems, the pictorial stage and the symbolic stage. The inscriptions done by people in the pictorial stage will correspond with the earliest era in the development of a number sense. The inscriptions will have a picture of ten cattle to represent ten cows or ten tents may indicate ten family groups. This stage is most easily seen in some records made by North American Indians just about the time they came into contact with Europeans.
The symbolic stage, on the other hand, will show a picture of a cow followed by five strokes to represent five cattle. At this stage the strokes no longer represent objects but have become a determinative (a sign used to qualify the meaning of the object sign). In other words, the strokes have become adjectives rather than nouns. In many societies this step must have come at the same time as the development of numbers as abstract ideas. This stage is best illustrated by some early stone carvings left by the Egyptians.
Once a particular society has reached the symbolic stage of recording numbers, the notation system which it develops will be almost entirely dependent on cultural influences. The use of fingers or pebbles for counting, the number of items counted into a group before a new group is started (the number base), and the materials available for making a lasting record of the result are all culturally determined. Once these variables have been set, the result will lead to one of two different types of notational systems: the additive system or the positional system.
1,2.1 The Additive Number System
In this system there is a distinct symbol for each kind of group made during the counting process, and this symbol is repeated as often as necessary to indicate how many of each group are needed. The Egyptian number system is the classical example of an additive system; however, most people are more familiar with the additive system used by the Romans. In the Old Roman system (before the subtractive forms of IV for 4 and IX for 9 were in use) it was possible to express any number less than 5,000 by a sequence of symbols in which no individual sign need be repeated more than four times. For example, the number 2,976 would be represented as MMDCCCCLXXVI. Although it was the custom to write down the symbols in decreasing order of value (M = 1,000, D = 500, C = 100, L = 50, X = 10, V = 5,1 = 1), this is by no means necessary. The value of a number in an additive system has nothing to do with the position of any symbol within the string representing the number. It is quite unambiguous to write the number 2,976 as ILVCMCXDMXCC but, of course, ease of reading always precluded its being written otherwise than in order of descending value.
The pure additive system is quite easy to use for simple calculations, although it does not appear so at first glance. Addition involves the twostep process of simply writing down the group symbols from each number, then collecting the sequences of smaller valued symbols to make larger valued ones so that the number regains its canonical form. For example:
2319 = MM CCC X VIIII +821 = D CCC XX I 3140 = MMDCCCCCCXXXVIIIII
The second step now takes over and, because IIIII = V, VV = X, CCCCC = D, DD = M, the final result is written as MMMCXXXX.
Multiplication, although slow, is not really difficult and only involves remembering multiples of 5 and 10. For example:
28 = XXVIII x12 = XII 336
XXVIII times I = XXVIII XXVIII times I = XXVIII XXVIII times X = CCLXXX CCLXXXXXXXVVIIIIII
which would be written as CCCXXXVI.
(Continues...)
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