- Shopping Bag ( 0 items )
"A truly enlightening and fascinating study for the mathematically oriented reader."
"Well researched. . . . This book is a rich resource for those involved in researching the history of computers."
The Mathematics Teacher
In this brilliant follow-up to his landmark international bestseller, The Universal History of Numbers, Georges Ifrah traces the development of computing from the invention of the abacus to the creation of the binary system three centuries ago to the incredible conceptual, scientific, and technical achievements that made the first modern computers possible. Ifrah takes us along as he visits mathematicians, visionaries, philosophers, and scholars from every corner of the world and every period of history. We learn about the births of the pocket calculator, the adding machine, the cash register, and even automata. We find out how the origins of the computer can be found in the European Renaissance, along with how World War II influenced the development of analytical calculation. And we explore such hot topics as numerical codes and the recent discovery of new kinds of number systems, such as "surreal" numbers.
Adventurous and enthralling, The Universal History of Computing is an astonishing achievement that not only unravels the epic tale of computing, but also tells the compelling story of human intelligenceand how much further we still have to go.
HISTORICAL SUMMARY OF ARITHMETIC,
NUMERICAL NOTATION, AND WRITING SYSTEMS
The writing of words and the writing of numbers show many parallels in their histories.
In the first place, human life was profoundly changed by each system, which allowed spoken language on the one hand, and number on the other, to be recorded in lasting form.
Further, each system answered marvellously to the universal need, felt by every member of every advanced society, for a visual medium to embalm human thought - which otherwise would inevitably dissolve into dust.
Again, everyone became empowered to create a persistent record of what he had expressed or communicated: of words which were otherwise long silent, or of calculations long since completed.
Finally, and most importantly, each system granted direct access to the world of ideas and thoughts across space and time. By encapsulating thought, and by inspiring it in others, the writing down of thought imposed on it both discipline and organisation.
Number and letter have often worn the same clothes, especially at times when letters were used to stand for numerals. But this is superficial: at a much deeper level there is still a close correspondence between the alphabet and the positional number-system. Using an alphabet with a fixed number of letters, every word of a language can be written down. Using our ten digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, any whole number whatever can be written down.
So we perceive a perfect analogy between these two great discoveries, the final stage in the development of writing and the final stage in the developmentof numerical notation. They are among the most powerful intellectual attributes of the modern human race.
The analogy is not limited to this, however. Throughout its history, each written number-system evolved in a very similar way to the verbal writing system that it grew up with. This can be seen in the way that both reflected the spoken language or the cultural traditions to which the language was adapted; or, again, in the mannerisms of local scribes, and in the influence of the very materials used for writing.
The main purpose of the chapter is to present a recapitulation of the history of numerical notation and arithmetical calculation. But so close is the parallel with writing just noted, that we shall incorporate notes on the history of writing at relevant points; for clarity, these notes will be flagged with [W].
For all that, the writing of language and the notation of numbers differ radically in one respect. For something to be called writing, its signs must be related to a spoken language; it must reflect a conscious effort to represent speech: "Writing is a system of human communication which uses conventional signs, which are well defined and which represent a language, which can be sent out and received, which can be equally well understood by sender and receiver, and which are related to the words of a spoken language." [J.G. Février, Histoire de lécriture (1959)]
By contrast, numerical notation needs no correspondence with spoken numbers. The mental process of counting is not linked to any particular act of speech; we can count to any number without speaking or even without thinking a single word. We need only create a "sign language for numbers", and in fact humankind devised many "number languages" before inventing even the word "number" and going on to use the human voice itself to measure concrete or abstract quantities.
While written characters correspond to the articulations of a spoken language, the signs in a numerical notation reflect components of thought, of a method of thinking much more structured than the sounds of speech. This method of thinking is itself a language (the language of numbers, no less), but to acquire this language we need first of all to have a concept of distinct units and the capacity to aggregate them. The language of number organises numerical concepts into a fixed order according to an idea which, on reflexion, we recognise as a general principle of recurrence or recursion [a principle according to which the evaluation of a complex entity is resolved by evaluating its component entities of a lower degree of complexity, which in turn . . . , until the simplest level is reached, at which each entity can be evaluated immediately. Transl.] It also makes use of a scale of magnitude (or base) according to which numbers can be distributed over successive levels called first-order, second-order, . . . units.
For a system of signs to constitute a written number-system, therefore, they must in the first place have a structure which its user can conceive, in his mind, as a hierarchical system of units nested each within the next. Then there must be a predetermined fixed number which gives the number of units on one level which must be aggregated together so as to constitute a single unit at the next level; this number is called the base of the number-system.
In summary: a notation for numbers is a very special human communication system using conventional signs called figures which have a well defined meaning, which can be sent and received, which are equally well understood by both communicating parties [in other words, a code. Transl.] , and which are attached to the natural whole numbers according to a mentally conceived structure which obeys both the recurrence principle and the principle of the base.
Over the five thousand years which have elapsed since the emergence of the earliest number-system, of course people have not merely devised one single number-system. Nor have there been an indefinite quantity of them - as can be seen from our Classification of the Written Number-systems of History in Chapter 23, which brings together systems so separated in space and time as to be effectively isolated from each other.
At the end of this chapter, we shall present the main conclusions of this Classification in a series of comparative systematic tableaux which will exhibit the mathematical characteristics of each number-system.
The order of succession of the number-systems in this series of tables will not be purely chronological, but will follow a path which traces their evolution in logic, as well as in time, from the most primitive to the most advanced.
The number-systems which are found throughout history fall into three main types, each divided into several kinds (Fig. 1.40):
A. the additive type of number-system. They are based on the additive principle and each of their figures has a particular value which is always the same regardless of its position in the representation of the number (Fig. 1.14 to 16). Basically, they are simply written versions of more ancient methods of counting with objects (Fig. 1.1 to 13);
B. the hybrid type of number-system. These use a kind of mixed multiplicative and additive principle ( Fig. 1. 28 to 32) , and are essentially transcriptions of oral number-systems of varying degrees of organisation;
C. the positional type of number-system. These are based on the principle that the value of a particular figure depends on its position in the representation of a number (Fig. 1.33 to 36) , and therefore need a zero (Fig. 1.37) . Number-systems of this type exhibit the greatest degree of abstraction, and therefore represent the final stage in the development of numerical notation ( Fig. 1.38 and 39).
In the tables at the end of this chapter, the letters A, B and C will therefore indicate the above types of additive, hybrid, and positional number-systems respectively. The kinds within these types will be indicated by numbers attached to these letters, such as A1 for an additive number- system of the first kind, and so on.
By classifying them in this way we shall be able to perceive clearly the true nature of our modern system of numerical notation, and therefore to understand why no essential improvement of it has been found necessary - or indeed possible - since the time it was invented in all perfection fifteen hundred years ago in India. Its birth, then and there, came about through the remote chance that three great ideas came together, namely: well-conceived figures representing the base digits (1, . . . , 9) ; their use according to a principle of position; and their completion by a sign 0 for zero which not only served to mark the absence of a base digit in a given position but also - and above all - served to denote the null number.
When we talk of "our modern number-system", by the way, we do not only mean the way we now write numbers worldwide, but also any of the other number-systems also used in the Near East, in Central Asia, in India and in Southeast Asia, which have identical structure and therefore identical possibilities. See "Indian Written Numeral Systems ( The mathematical classification of)" in the Dictionary of Indian Numerical Symbols (Chapter 24, Part II).
This "temporal logic" alone could demonstrate the deep unity of all human culture; but it is not merely a question of order and system revealed solely on lines of time - this would be to ignore both transmission within each culture and also, above all, transmission from one culture to another. It would also gloss over the true chronology of events in which, in our story, we see cultures both overlapping each other, and leapfrogging past each other. At certain times, some cultures have been far in advance of others. Some other peoples have clung to inadequate number-systems throughout their history, whether by failure to break out of the prison of an inadequate system, or through a conservatism which attached them to a poor tradition.
For these reasons, we now embark on a systematic chronological résumé which will trace out this logic of time. Fig. 1.41, to be found below, will show how from the different civilisations emerged our different classifications of number-systems (defined in Fig. 1. 14 to 16, 1. 28 to 32, and 1.38 to 39); and Fig. 1.41 is the inverse of Fig. 1.42, showing how the different number-systems emerged in order of time, according to civilisations.
Our chronological résumé traces the history of a graphical notation for numbers, whose prime function was to represent numbers obtained in the course of calculations or counts previously carried out, in order to compensate for the defects of human memory.
We shall also follow the principal stages of development of arithmetical calculation, which evolved in parallel with the writing of numbers. It began with counting on the fingers, and with pebbles; continued through many- coloured strings, the abacus, checkerboards and abacuses traced in wax or dust or sand, and finally made contact with written numerical notation when our modern positional numerical notation, and the zero on which it depends, were discovered.
The dates of "first appearance" given in the following Figures are the results of archaeological, epigraphical and palaeographical research, to mention a few of the domains of study from which the information has been gleaned. They do not correspond to the definite date of an invention or a discovery. A date which we give below merely means the established date, according to these researches, of the earliest known documentary evidence for the system or concept in question. They are therefore only approximate.
Nevertheless we must maintain a distinction between the date of invention or discovery of something, and the date by which it came into common use; and the latter must in turn also be distinguished from the dates of the earliest instances of which we are currently aware.
Quite possibly, a discovery occurred many generations prior to its popularisation, and there may well be a delay between this and the date of the earliest evidence we possess today. There may be many reasons for this. Often, a discovery long remained the esoteric property of a closed sect, or of a specialised elite who jealously guarded their monopoly of an arcane art. In many cases, no doubt, documents which had existed prior to those of which we know have perished; or perhaps they have yet to be discovered. Archaeological or documentary discoveries which remain to be made may yet cause changes in the conclusions which we present below.
At undetermined dates, beginning in prehistory, the following sequence of developments occurred. Entries flagged with [W] refer specifically to developments in writing, rather than numbers.
* The human race was, in the earliest stages of its evolution, at the most primitive stage of the notion of number, which was confined to such number (up to four or five) as could be assimilated at a glance. This never- theless awoke in the human mind a realisation of the concrete aspects of objects which it directly perceived.
* By force of necessity, aided by native intelligence and by the capacity for thought, human beings little by little learned to solve an increasing range of problems. For numerical magnitudes greater than four, people devised procedures based on the manipulation of concrete objects which enabled them to achieve, up to a point, results which met their needs of the moment. These were simply based on a principle of counting by one-to-one correspondence, and amongst them may be found especially the methods of using the fingers or other parts of the body; thus they had simple methods which were always to hand. These methods came to be expressed in articulated speech, accompanied by corresponding gestures.
* By force of habit, counting according to these parts of the body (once adopted according to an invariable routine) slowly became part abstract, part concrete, thereby suggesting less and less the specific part of the body and more and more the concept of a certain corresponding number which increasingly tended to become detached from the notion of the body and to become applicable to objects of any kind. (For a detailed account of the above, see Chapter 1.)
* The resulting necessity to make a distinction between the numerical symbol itself and the name of the concrete object or image led people finally to make a clean break between the two, and the relationship between them disappeared from their minds. Thenceforth, people progressively learned to count and to conceive of numbers in an abstract sense, not related to specific concrete counting tokens. In particular, as they learned to employ speech sounds for the purpose, the sounds themselves took over the role of the objects for which they had been created. Day by day, the notion of successor became established in the human mind, and what had been a motley collection of concrete objects became a structured abstract system, at first based on gesture before assuming verbal or written forms. It became a spoken system when the names of the numbers were invented as abstractions out of custom, usage and memory. Much later, in a similar way written systems came about when all kinds of graphic symbols were brought into use - scratched or drawn or painted lines, marks hollowed out of clay or carved in stone, various figurative symbols, and so on (see Chapter 2).
* This proliferation of representations created problems, which were solved by the invention of the principle of the base of a number-system (the base 10 being the most commonly used throughout history). Using every kind of object and device (the fingers, pebbles, strings of pearls, little rods, . . .) people gradually arrived at the abstractions embodied in the procedures of calculation and in the operations of arithmetic (see Chapter 2).
From this point in the history, we are able to assign approximate dates to the successive stages of development.
35000 - 20000 BCE. The earliest notched bones of prehistory are the most ancient known archaeological objects which had been used for numerical ends. They are in fact graphical representations of numbers, though we do not know what precise purposes they served (see Chapter 4).
20000 BCE [W]. The first pictures on rock appeared in Europe: these, among the earliest known visual representations of human thought, were made by engraved or painted lines.
9th - 6th millennia BCE [W]. We see the simultaneous appearance in Anatolia (Beldibi), in Mesopotamia (Tepe Asiab), in Iran (Ganj Dareh Tepe), in Sudan (Khartoum), in Palestine (Jericho) and in Syria of the little clay tokens of various sizes and shapes (cones, discs, spheres, small balls, little rods, tetrahedra etc.). Some of them bear parallel lines, some crosses and other motifs, while others are decorated with carved figurines representing every kind of object (jugs, animal heads, etc.). These relief drawings (which surely had significance for their creators and users) are clear evidence of the development of symbolic thought. We do not know, however, whether these are the elements of some system nor whether this corresponds to one of the intermediate stages between a systematic purely symbolic expression of human thought and its formal expression in a spoken language (see Chapter 10).
9th - 2nd millennium BCE. The peoples of the Middle East (from Anatolia and Palestine to Iran and Mesopotamia, from Syria to Sudan) made their calculations using cones, spheres, rods and other clay objects which stood for the different unit magnitudes of a number-system. Such systems can be found, from the fourth millennium BCE onwards, in Elam and in Sumer, in a much elaborated form which will give rise, not only to written counts, but also to some extent to the graphical forms of the Sumerian and proto-Elamite figures (see Chapters 10 and 12).
6th - 5th millennia BCE [W]. The earliest ceramic artefacts, on which motifs have been painted, engraved, cut out or impressed on the raw clay, or engraved after firing, appear in the Middle East. These are evidently graphical representations emanating from some symbolic system, but we do not know their meaning or purpose.
At the same time, in Asia Minor (Çatal Hüyük) and later in Mesopotamia, there appear the earliest seals (carved objects which can be used to impress a relief design on soft material such as clay).
4th millennium BCE (?). The people of Sumer have an oral number- system, to base 60 (see Chapters 8 and 9) . This base 60 has come down to us via the Babylonians, the Greeks and the Arabs, and we use it yet for the minutes and seconds of time, and for the measure of angle in minutes and seconds of a degree.
3500 BCE [W]. The first cylindrical seals appear in Elam and in Mesopotamia. They are small cylinders of stone, precious or semi-precious, bearing an engraved symbolic design. Every man of a certain standing had one of these: it represented the very person of its bearer and therefore was associated with all economic or judicial aspects of his life. By rolling the cylinder onto any object of clay, the proprietor of the seal thereby impressed his "signature", or his right of property, upon it. The different designs did not constitute "writing" in the strict sense of the word; rather, they had a symbolic significance subject to every kind of interpretation.
3300 - 3200 BCE. The figures of the Sumerian number-system and the figures of the proto-Elamite number-system make a simultaneous appearance at this time. These are the most ancient written number-systems at present known (see Chapters 8 and 10).
3200 - 3100 BCE [W]. The writing signs of Sumer, the most ancient writing system known, make their appearance. These are pictograms which represent every kind of object, and they are found on clay tablets which seem to have been used for some economic purpose. However, this is still not a true writing system, since the signs are symbolic of objects rather than directly related to a spoken language. This latter step will occur only at the begin-ning of the third millennium BCE, at which time the Sumerian system will have become phonetic, will represent the various parts of speech, and will have become linked to spoken language, which is the most highly developed way of analysing and communicating reality (see Chapter 8).
3000 BCE [W]. In ancient Persia, the proto-Elamite writing signs appear (see Chapter 10).
3000 - 2900 BCE [W]. The signs of Egyptian hieroglyphic writing appear (see Chapter 14).
3000 - 2900 BCE. The figures of the Egyptian number-system appear (see Chapter 14).
2700 BCE [W]. The cuneiform characters (in the form of angles and wedges) of the Sumerian writing system appear on their clay tablets (see Chapter 8).
2700 BCE. The cuneiform figures of the Sumerian number-system appear (see Chapter 8).
2700 - 2300 BCE. For doing arithmetic, the people of Sumer now abandon their old calculi and invent their abacus, a kind of table of successive columns, ruled beforehand, which delimit the successive orders of magnitude of their sexagesimal number-system. By clever manipulation of small balls or rods on the abacus, they are able to perform all sorts of calculations (see Chapter 12).
2600 - 2500 BCE [W]. Egyptian hieratic writing appears, a cursive abbreviation of hieroglyphic writing and used alongside the latter for the sake of rapid writing on manuscripts (see Chapter 14).
2500 BCE. The Egyptian hieratic figures appear (see Chapter 14).
2350 BCE [W]. The Semites of Mesopotamia borrow the cuneiform characters of Sumer to write down their own speech. This is the beginning of the Akkadian script from which will emerge the Babylonian and Assyrian writing systems.
2350 BCE [W]. Appearance of the writing of Ebla (the capital of the Semite kingdom situated at Tell Mardikh, to the South of Aleppo in Syria), a cuneiform script cut into clay tablets, for their Western Semitic dialect which was close to Ugaritic, Phoenician and Hebrew.
2300 BCE [W]. Proto-Indian writing appears in the Indus valley at Mohenjodaro and Harappâ (in what is now Pakistan). This writing of the ancient Indus civilisation (25th - 18th centuries BCE) is separated by a hiatus of over two thousand years from the earliest written texts in any true Indian language and in true Indian writing. It is not known how to bridge this gap, nor, indeed, if it ever was bridged.
End of 3rd millennium BCE. The Semites of Mesopotamia are now slowly adopting a cuneiform decimal notation which has come down to them from their predecessors. In everyday use, this system will come to supplant the Sumerian sexagesimal system (of which, however, the base 60 will survive in the positional notation of the Babylonian scholars). At the same time, the ancient Sumerian abacus undergoes a radical transformation: instead of using beads or rods, they trace their cuneiform figures inside the ruled columns of a large clay tablet; in the course of calculation, these figures are successively erased according as the successive partial results are obtained (see Chapter 13).
2000 - 1660 BCE [W]. The hieroglyphic writing of the Minoan civilisation appears in Crete, found at Knossos and Mallia on bars and tablets of clay which appear to have been accountancy documents (see Chapter 15).
2000 - 1660 BCE. At the same time appear the hieroglyphic figures which they used for numbers (see Chapter 15).
1900 BCE [W]. The "Linear A" script of the Minoan civilisation appears in Crete, found at Haghia Triada, Mallia, Phaestos and Knossos on clay tablets which were undoubtedly inventories of some kind. Somewhat casual in style, this script occurs not only in administrative quarters but also in sanctuaries and probably in private houses too (see Chapter 15).
1900 BCE. At the same time appear the "Linear A" figures which they used for numbers (see Chapter 15).
1900 - 1600 BCE [W]. The cuneiform script of the Semites of Mesopotamia gradually supplants the Sumerian script and spreads across the Near East, where it will even become the official script of the chancelleries.
1900 - 1200 BCE. The decimal cuneiform number-system of the Semites of Mesopotamia spreads across the Near East.
1900 - 1800 BCE. The oldest known positional number-system comes on the scene: this is the cuneiform sexagesimal system of the Babylonian scholars, but it is not yet in possession of a zero (see Chapter 13).
17th century BCE [W]. The first known venture into an alphabetic script - the Semites who were in the service of the Egyptians in the Sinai made use of simple phonetic symbols derived from Egyptian hieroglyphics (the so-called "proto-Sinaitic inscriptions" of Serabit al Khadim).
17th century BCE. Notwithstanding the very rudimentary nature of their hieroglyphic and hieratic numerals, the Egyptians are able to make use of them for arithmetical calculations (see Chapter 14). These methods relieve the burden on the memory (since it is sufficient simply to know how to multiply and divide by 2), but they are not unified and they lack flexibility; they are time-consuming, and are very complicated in comparison with the procedures of our own day.
16th century BCE. By now, the Egyptian hieratic number-system has come to the end of its graphical evolution (see Chapter 14).
15th century BCE [W]. Desiring abbreviation, and keen to break away from the complicated Egyptian and Assyro-Babylonian writing systems then in use in the Near East, the Semites of the Northwest who were settled along the Syrian and Palestinian coasts develop the very first purely alphabetical writing system in history, thereby inventing the alphabet. This superior method of transcribing words, capable of being adapted to any spoken language, henceforth allows all the words of any language to be written by means of a small number of simple phonetic symbols called letters(see Chapter 17).
15th century BCE [W]. The hieroglyphic writing of the Hittite civilisation appears. This script will not only be used for religious and dedicatory purposes, but also - and above all - for secular purposes (see Chapter 15).
15th century BCE. At the same time, the Hittite hieroglyphic number-system appears (see Chapter 15).
1350 - 1200 BCE [W]. The Creto-Mycenaean script called "Linear B" appears in Crete (found at Knossos) and in Greece (Pylos, Mycenae, etc.); it is a modification of "Linear A", used to write an archaic Greek dialect (see Chapter 15).
1350 - 1200 BCE. At the same time, the "Linear B" numerals appear (see Chapter 15).
14th century BCE [W]. The oldest known entirely alphabetic script appears, found on tablets from Ugarit (Ras Shamra, near Aleppo in Syria). This is a cuneiform script whose alphabet has only thirty letters; it was used to write a Semitic language related to Phoenician and Hebrew (see Chapter 17).
End of the 14th century BCE [W]. One of the oldest known specimens of archaic Chinese writing appears at Xiao. It is found on inscriptions made on bones or on tortoise shells, and its main purpose was to enable communication between the world of the living and the world of the spirits by means of various divinatory and religious practices (see Chapter 21).
End of the 14th century BCE. At the same time, we find the oldest known Chinese numerals (archaic Chinese number-system, in inscriptions on the bones and tortoise shells at Xiao dun; see Chapter 21).
End of the 12th century BCE [W]. The earliest known specimens of the Western Semitic "linear" alphabet, a precursor of all modern alphabets (see Chapter 17), are used by the Phoenicians. Since they had dealings with a great variety of peoples, these notable merchants and bold navigators diffused their alphabet far and wide. In the East, they will pass it first to their immediate neighbours (Moabites, Edomites, Ammonites, Hebrews, etc.), including the Aramaeans who in turn will spread it from Syria to Egypt and to Arabia, and from Mesopotamia to the Indian sub-continent. From the ninth century BCE, it will spread also round the whole Mediterranean seaboard and be progressively adopted by the Western peoples who will adapt it to their own languages, and modify it by the addition of some further symbols.
9th century BCE [W]. Ancient I raelite inscriptions in palaeo-Hebraic characters, derived directly from the 22 Phoenician characters, appear (see Chapter 17).
9th century BCE. The Hebrews adopt the Egyptian hieratic numerals, which they especially make use of in correspondence (see Chapter 18).
9th - 8th centuries BCE [W]. The alphabetic script of Phoenician origin spreads across the near East and the Eastern Mediterranean (Aramaeans, Hebrews, Greeks, etc.).
End of 9th century BCE [W]. The Greeks perfect the principle of the modern alphabetical system by adding symbols for the vowels to the consonants of the original Phoenician alphabet. This is the first alphabet to have a strict and integrated notation for the vowels (see Chapter 17). In turn, this alphabet will inspire the Italic alphabets (Oscan, Umbrian, Etruscan, etc.) and then Latin, later giving rise to the Coptic, Gothic, Armenian, Georgian and Cyrillic alphabets.
8th century BCE [W]. Egyptian demotic script appears, a cursive script, arising from a local branch of Egyptian hieratic script but more abbreviated, which it will later supplant in everyday use (see Chapter 14).
8th century BCE [W]. Appearance of the Italic alphabets, in particular the Etruscan (see Chapter 17).
8th century BCE. The earliest clearly differentiated forms of Egyptian demotic numerals appear (see Chapter 14).
8th century BCE. Italic numerals (Oscan, Umbrian, and especially Etruscan) appear (see Chapter 16).
8th century BCE. This is the period of the earliest known Western Semitic numerals; Aramaean numerals appear (see Chapter 18).
7th century BCE [W]. Archaic Latin writing appears.
6th century BCE. Archaic Latin numerals appear (see Chapter 16).
6th century BCE. Greek acrophonic numerals appear in Attica (see Chapter 16).
End of 6th century BCE. The earliest known Phoenician numerals (see Chapter 18).
5th century BCE [W]. The earliest known specimens of Zapotec writing in pre-Columbian Central America.
5th century BCE. The Zapotecs use an additive number-system with base 20, which can be found in use amongst all the pre-Columbian peoples of Central America (Mayas, Mixtecs, Aztecs, etc.) to within minor graphical variations (see Chapter 22). For purposes of calculation, these peoples certainly do not make use of this ill-adapted number-system; on the contrary, they make use of calculating instruments. Although Central-American archaeology has yielded up nothing relevant to this subject, we may nevertheless get an idea of it by appealing to ethnology and to history which provide us with numerous analogies. We can therefore suppose that, on the example of certain African societies, they made use of rods, each corresponding to an order of magnitude, along which they slid pierced pebbles. They possibly proceeded in the same way as the Apache, the Maidu and the Havasupai of North America who threaded pearls and shells onto coloured threads; or, perhaps more plausibly, like the Incas of South America who distributed pebbles, beans or grains of maize onto the squares of a checkerboard on a kind of tray made of stone, pottery or wood, or drawn on the floor (see Chapters 12 and 22).
5th century BCE [W]. Aramaean script becomes generally adopted for international correspondence in the Middle East, henceforth supplanting the Assyro-Babylonian script for this purpose.
5th century BCE. The Aramaean numerals, which have already reached their final form, spread over the Middle East (Mesopotamia, Syria, Palestine, Egypt, Northern Arabia, etc.).
5th century BCE. Earliest archaeological evidence for the use of the Greek abacus: tables made out of wood or marble, pre-set with small counters in wood or metal suitable for the mathematical calculations (see Chapter 16). The Persians in the time of Darius were to use this type of abacus, and after them the Etruscans and Romans. The Western Christian world was to inherit the use of this abacus, which they were to continue until the French Revolution (see Chapters 16, 25 and 26).
5th century BCE. The Greek acrophonic number-system spreads across the Hellenic world (see Chapter 16).
5th century BCE. The acrophonic numerals of Southern Arabia appear in the inscriptions of the kingdom of Sheba (see Chapter 16).
End of 4th century BCE. The earliest known records of the Greek alphabetic number-system appear in Egypt, showing that the Greek letter numerals are in general use by this time (see Chapters 17 and 18).
3rd century BCE. The first known case of the use of zero comes on the scene, as used by the Babylonian scholars. This was a cuneiform character. Although used in the positional Babylonian number-system to signify the absence of a sexagesimal unit, it is nevertheless still not perceived as a number in its own right (see Chapter 13).
3rd century BCE. The Greek alphabetical number-system spreads across the Middle East and the Eastern Mediterranean (see Chapter 17).
3rd century BCE [W]. The Aramaeo-Indian Kharoshthî writing appears in the edicts of the Emperor As'oka. This is a cursive script derived from Aramaean writing, and used in the Northwest of India, as well as in the territories which are now Pakistan and Afghanistan (see Chapter 24).
3rd century BCE. At the same time, Kharoshthî numerals appear in Northwest India and in the various countries now subsumed in Pakistan and Afghanistan.
3rd century BCE [W]. Brâhmî script appears in the edicts of the Emperor As'oka. (see Chapter 24). This is derived from the ancient alphabetic scripts of the Western Semitic world, no doubt with an intermediate Aramaean form of which no specimens have been found. This will become the earliest truly Indian script, and will be the origin of all of the alphabetic scripts of the Indian sub-continent and of Southeast Asia. Over the centuries, it will undergo many changes which culminate in many distinctly different types of writing, such as Gupta, Bhattiprolu and Pâlî . Gupta in turn will split into Nâgarî , Siddham, and ´ Shâradâ from which will descend all the current scripts to be found in Central and Northern India, in Nepal, in Tibet, and in Chinese Turkestan. Bhattiprolu will give rise to the scripts of Southern India and Ceylon, while Pâlî will give rise to the scripts of Southeast Asia. The apparently considerable differences between all these scripts are due either to the natures of the languages and traditions to which they have been adapted, or to regional differences between scribes and differences between writing materials. (See "Indian Styles Of Writing" and "Indian Styles of Writing (The materials of)" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
3rd century BCE. Brâhmî numerals appear in the edicts of the Emperor As´oka, which can be found throughout the Maurya Empire. These, the earliest truly Indian numerals, occur more and more frequently in later inscriptions (Shunga, Ândhra, Shaka, Kshatrapa etc.); and they are the prototypes of all the numerical notations which flourished later in India, Central Asia, and Southeast Asia. Although the number-system did not at the time follow a principle of position, the figures which correspond to the first nine digits are clear precursors of the digits 1 to 9 in our own number-system and in the modern Arabic number-system (see Chapter 24).
3rd century BCE to 4th century CE [W]. Greek manuscript writing splits into three types: "book" script, official script, and the script used for private documents.
2nd century BCE. In Plutarch, we find mention of the sand abacus alongside the abacus with tokens. A board, with a raised border, is filled with fine sand on which lines are drawn to mark out the columns, and the numbers are written in, using an iron stylus. The same type of abacus is later found amongst the Christian population of the mediaeval West, and they carry out their calculations using either Roman numerals or the Greek alphabetic numerals (see Chapter 16).
2nd century BCE [W]. This epoch sees the Chinese invention of paper. (According to some, it was invented by Cai-Lun who achieved it by boiling up unravelled tissues and old fishing nets.) They also invent xylography: the text to be reproduced is written on the polished surface of a wooden board, and then the wood surrounding the writing is cut away so as to leave it in relief; ink is applied, and a sheet of paper is pressed on. (See further information about paper in the course of the entry "Pâtîganita" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
2nd century BCE. The earliest known documents which refer to use of the Chinese abacus, and to "calculation with rods" (suan zí) in which small bamboo sticks are placed in successive squares of a checkerboard (see Chapter 21).
2nd century BCE. The earliest known documents which affirm the use of a positional decimal notation by the Chinese. This system does not however have a zero (see Chapter 21), and in fact is simply a written counterpart of the method of "calculation with rods" (see above).
2nd century BCE [W]. The earliest documentary reference to "square Hebrew": Hebrew writing in its modern form, but whose squat and massive letters are derived from the cursive Aramaean script (see Chapter 17).
2nd century BCE. The earliest documents in which we can see the use of modern Hebrew alphabetic numerals (see Chapter 17).
2nd century BCE to 3rd century CE. Indian arithmeticians perform their calculations by tracing their nine digits in Brâhmî notation on the floor, within consecutive columns already delineated, with a pointed rod. A similar procedure will later be used by the Arabs, especially those of the Maghreb and of Andalusia. Boards covered with fine sand, with flour or with some other powder are also used, with a stylus whose point is used to trace the figures and its flat end to erase them. This board might be placed on the floor, on a stool or on a table, or might be furnished with legs like those used much later in the Arab, Turkish and Persian administrations. The board might be made in a small version which could be kept in a case. (See Chapters 24 and 25, and also the entry "Dhûlîkarma" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
2nd century BCE to 2nd century CE [W]. The reform of Chinese writing, and the emergence of the lì shu graphics which will evolve towards the modern system of Chinese characters (see Chapter 21).
1st century BCE. Horace notes the use of the wax abacus, as well as the abacus with rods, by the Romans: a real "portable calculator" which could be hung over the shoulder, this consists of a board made of bone or wood, covered with a thin layer of black wax, on which the lines for the columns, and the figures, are drawn with an iron stylus (see Chapter 16).
Start of the Common Era [W]. A cursive branch of the ancient Aramaean evolves to give rise to Arabic script.
1st century CE. At this period we find the earliest archaeological evidence of the Roman "pocket abacus". This is a small metal plate with parallel slots along which mobile beads can be slid; each is associated with a numerical order of magnitude. It is therefore very similar to the bead abacus which in modern times still holds an important place in the Far East and in certain Eastern countries (see Chapters 16 and 21).
2nd - 3rd centuries CE [W]. The Roman script undergoes a change which will give rise to two new forms of Latin script: the New Common Writing and the Uncial.
End of 3rd century CE [W]. The oldest known specimens of Maya writing in pre-Columbian Central America (see Chapter 22).
End of 3rd century CE. The earliest examples of use of the "Long Count" for dates by Mayan astronomers (see Chapter 22).
3rd - 4th centuries CE [W]. The earliest known cases of runic script used by Germanic peoples (Futhark alphabet).
Beginning of 4th century CE [W]. Pharnavaz, first king of the country which lay between Armenia and the Caucasus, is inspired by Greek to invent the Mkhedrouli alphabet, ancestor of the Georgian alphabet (see Chapter 17).
4th century CE. Appearance of Ethiopian numerals in the inscriptions from Aksum in the kingdom of Abyssinia (see Chapter 19).
4th century CE [W]. The first appearance of the Chinese kai shu writing, a form of modern Chinese writing (see Chapter 21).
4th century CE [W]. Bishop Wulfila draws on the Greek alphabet to invent the Gothic alphabet, for the purpose of recording the Germanic language of the Goths (see Chapter 17).
4th - 5th centuries CE [W]. The earliest forms of the Indian Gupta alphabet appear, from which all the alphabetical scripts of central India, Nepal, Tibet and Chinese Turkestan will be derived.
4th - 5th centuries CE. The earliest forms of the Indian Gupta numerals appear, from which all the numerical notations of central India, Nepal, Tibet and Chinese Turkestan will be derived (see Chapter 24, and also the entry "Gupta Numerals" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II).
4th - 5th centuries CE. The first nine digits of the Indian system, derived from the old Brâhmî notation, acquire a positional significance in a decimal base, and they are completed by an additional symbol in the form of a small circle or a dot which represents zero; this therefore is the birth of the Indian positional decimal notation, which was the ancestor of our modern numerical notation (see Chapter 24).
4th - 6th centuries CE. During this period, the Indian arithmeticians radically transform their traditional methods of calculation. They do away with the columns of their ancient sand abacus, and attribute values to written digits according to their decimal position. This, therefore, is the beginning of the modern number-system. It is also, however, the beginning of modern written arithmetic.
To begin with, their techniques, albeit liberated from the columns of the abacus, were but written imitations of the abacus procedures. As formerly practised, on a medium as inconvenient as the sand abacus, with interme- diate results noted after erasing the previous ones, these constrained the role of human memory and made it difficult if not impossible to check the calculation and correct errors made along the way. The Indian and Arab scholars subsequently developed procedures which did not involve erasure, but involved writing intermediate results above the working. While certainly advantageous for checking purposes, since every intermediate error remained to be seen, nevertheless this resulted in a cluttered work- sheet from which it was difficult to get a clear view of the progression of the calculation.
Because of this kind of complication, even using the nine digits and the zero, the new methods long remained beyond the grasp of ordinary mortals. Writing their calculations on a board with chalk without worrying about how many figures there were or, better still, rubbing them out successively with a cloth: such was the convenient and relaxed method which, even before the advent of pen and paper, allowed the Indian arithmeticians and their Arab and European successors to work in their own way and with unfettered imagination to arrive at simplifications of the rules and methods and ultimately to create the techniques which would give rise to our modern methods of written arithmetic. (See Chapters 24 and 25, and also the entry "Indian Methods of Calculation" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
4th - 6th centuries CE [W] . The earliest forms of the Bhattiprolu and Pâlî scripts appear: from these will be derived, respectively, all the alphabetic writing systems of South India, and those of Southeast Asia (see Chapter 24).
4th - 6th centuries CE. The earliest forms of the Bhattiprolu and Pâlî numerals appear: from these will be derived, respectively, all the number-systems of South India, and those of Southeast Asia (see Chapter 24).
4th - 9th centuries CE. This is probably the period during which the positional notation, with base 20 and a zero, of the Mayan astronomer-priests emerged. However, as a result of its forced conformity to the peculiarities of the Mayan calendar, this number-system exhibited an irregular use of the base 20 beyond the third digit position which robbed it, along with its zero, of practical operational value (see Chapter 22).
5th century CE [W]. The earliest known specimens of Arab writing found in pre-Islamic inscriptions. This script was cursive in style, and in time diversified to give rise to the Kufic script and the Naskhî script during the early centuries of Islam (see Chapters 19 and 25).
5th century CE [W]. The priest Mesrop Machtots draws inspiration from Greek to invent the Armenian alphabet (see Chapter 17).
5th - 7th centuries CE [W]. Emergence of the Ogham script in Celtic inscriptions in Ireland and Wales.
510. The Indian astronomer Âryabhata invents a special numerical notation for which it is necessary to have a full awareness of the concepts of zero and the principle of position. He further makes use of a remarkable method for calculating square and cube roots which it is impossible to perform unless the numbers are written down using the principle of position, the nine digits, and a tenth sign which plays the role of zero. (See Chapter 24, and also the entries "Âryabhata," "Âryabhata's Number- System," "Indian Mathematics, The history of" and "Square Roots, How Âryabhata calculated his" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
628. The Indian mathematician and astronomer Brahmagupta publishes Brahmasphutasiddhânta , which displays total mastery of positional decimal notation, using the nine digits and a zero. (See Chapters 24 and 25, and also the entries "Brahmagupta" and "Indian Mathematics, The history of" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
629. The mathematician Bhâskara publishes a Commentary on the Âryabhatîya. This work not only reveals complete mastery of the use of zero and of the positional decimal number-system: it also shows that the author is quite at ease with the Rule of Three and with arithmetical fractions, which he writes in a way very similar to ours, though lacking the horizontal bar which will not be introduced until several centuries later, by Arab mathematicians. (See Chapters 24 and 27, and also the entry "Bhâskara" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
7th century CE [W]. The earliest distinct forms of the Indian Nâgarî writing appear, from which the scripts of North and Central India will be derived (Bengâlî, Gujarâtî, Oriyâ, Kaîthî, Maithilî, Manipurî, Marâthî, Mârwarî, etc.(See Chapter 24, and also the entries "Indian Styles of Writing" and "Indian Styles of Writing, The materials of," in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)
7th century CE. The earliest distinct forms of the Indian Nâgarî numerals appear, from which the numerals of North and Central India will be derived (Bengâlî, Gujarâtî, Oriyâ, Kaîthî, Maithilî, Manipurî, Marâthî, Mârwarî, etc., see Chapter 24).
7th century CE [W]. The earliest distinct forms of the stylised scripts of Southeast Asia appear ( Khmer, Malaysian, Shan, Kawi, etc., see Chapter 24).
7th century CE. The earliest distinct forms of the Indian numerals which will give rise to the stylised numerals of Southeast Asia appear (Khmer, Malaysian, Shan, Kawi, etc., see Chapter 24).
7th - 8th centuries CE. During this period, the Indian decimal notation, with the zero, spreads to the Indianised civilisations of Southeast Asia (Cambodia, Shan, Java, Malaysia, Bali, Borneo, etc., see Chapter 24).
7th - 8th centuries CE [W]. The era of the oldest known manuscripts in which we find Latin writing of "Visigoth" and "Luxeuil" type.
7th - 10th centuries CE [W]. The earliest distinct forms which will give rise to the stylised scripts of South India (Tamil, Malayâlam, Tulu, Telugu, Kannara, etc., see Chapter 24).
8th century CE. Under the influence of Indian Buddhist monks, the zero, of Indian origin, takes its place in the Chinese positional decimal number-system of "bar numbers" (the suan zí system, see Chapter 24).
8th century CE [W]. The first appearance of the so-called "minuscule" Greek writing (which will replace the older system in books from the ninth century onwards).
End of 8th century CE. At this time, the positional decimal notation with a zero enters the world of Islam. In the hands of the Arab scribes, the figures will undergo changes of form, in some cases far enough from the original forms that they appear to be new (see Chapter 25).
8th - 11th centuries CE [W]. Runic inscriptions from Viking times (from Uppland province in Sweden).
8th - 11th centuries CE [W]. The earliest distinct forms of Carolingian writing (the Corbie studio, manuscripts of the Bible written under the direction of the monk Maurdramnus, the dedication of the Gospels by Charlemagne, etc.).
820 - 850. The period of the great Muslim astronomer and mathematician Al Khuwarizmi, whose works contributed greatly to the knowledge and dissemination of the numerals and arithmetical methods which originated in India (see Chapter 25).
9th century CE. The Ghubar numerals of the Maghreb and Andalusian Arabs now appear (they are of Indian origin, and their form anticipates that of the European numerals of the Middle Ages and the Renaissance, before giving rise to our modern numerals. See Chapter 25.)
9th century CE [W]. The earliest distinct forms of the Indian Shâradâ script appear (a southern variant of the Gupta script), which will give rise to the scripts of Northwest India (Dogrî, Tâkarî, Multânî, Sindhî, Punjabî, Gurûmukhî, etc.(See Chapter 24.)
9th century CE. The earliest forms of the Indian Shâradâ numerals appear, which will give rise to the numerals of Northwest India (Dogrî, Tâkarî, Multânî, Sindhî, Punjabî, Gurûmukhî, etc., see Chapter 24).
9th century CE [W]. With the aim of converting the Bulgars, the bishop Cyril draws inspiration from Greek to invent the Glagolitic alphabet.
9th century CE [W]. The appearance of Japanese writing, properly speaking.
10th century CE [W]. In order to record the sounds of the Slavic languages, the bishop Clement draws inspiration from Greek to invent the Cyrillic alphabet. The first simplification of this writing which will later give rise to the modern Russian alphabet will be brought about by Peter the Great in the eighteenth century.
972 - 982. In the course of a voyage to Spain, the monk Gerbert d'Aurillac from Auvergne (later to become Pope Sylvester II, in 999) learns the "Arab" numerals and introduces them to Western Europe (see Chapter 26).
976 - 992. Two manuscripts from non-Muslim Spain illustrate the forms of nine figures which are very similar to numerals of the Ghubar type. These are the oldest known evidence of the presence of "Arab" numerals in Western Europe (see Chapter 26).
10th - 12th centuries CE. Europeans are carrying out arithmetic operations using the abacus with columns, of Roman origins and perfected by Gerbert d'Aurillac and his pupils. They use counters made of horn ( called apices) marked with the Arab numerals from 1 to 9, or with the Greek alphabetic numerals from æ to Ø, or with the Roman numerals from I to IX (see Chapter 26).
11th century CE [W]. The master calligrapher Lanfranc, in the Carolingian tradition, creates the script which will become the most beautiful pontifical writing of the twelfth century.
Middle of 11th century CE [W]. Printing is invented by the Chinese. They make use of separate characters made of baked clay, which later will be made of lead and then in copper. This invention is related by Qin Guo in 1056; he attributes it to Bi Xing and dates it at 1041.
12th century CE. The Indian sign for zero is introduced to Europe. The European arithmeticians henceforth do their calculations with the zero and the nine "Indo-Arabic" digits. Also, the rules of arithmetic, of Indian origin, are now called algorisms.
12th - 13th centuries CE [W]. Gothic script gradually replaces Carolingian script.
12th - 13th centuries CE [W]. Aztec writing emerges (see Chapter 22).
12th - 16th centuries CE. A ferocious dispute takes place between the Abacists(adherents of methods of calculating by counters on the abacus, and prisoners of a system seamed with ancient number-systems such as the Roman numerals and the Greek alphabetic numerals) and the Algorists, proponents of methods of written arithmetic using the Indian numerals and the zero (see Chapter 26).
12th - 15th centuries CE. The period where the forms of the "Arabic" numerals become established in Europe, where they will eventually evolve into their modern forms (see Chapter 26).
1202. Following his travels in North Africa and the Middle East, the Italian mathematician Leonard of Pisa, better known as Fibonacci, publishes Liber Abaci("A Treatise on the Abacus"). Over the ensuing three centuries, this book will prove to be a most fruitful source of development of arithmetic and algebra in Western Europe (see Chapter 26).
13th century CE. In this period we find the earliest documents which illustrate the use of the Chinese abacus (suan pan). This is a rectangular frame of wood traversed by a certain number of rods along which slide seven wooden balls. A longitudinal wooden slat divides the interior into two parts, on one side (the lower) of which there are five balls on each rod, and on the other side (the upper) two. Each rod corresponds to a power of 10, increasing to the left.
Of all the ancient calculating instruments, the Chinese abacus is the only one to provide a simple means to carry out all the operations of arithmetic; Western observers are usually astonished at the speed and dexterity with which even the most complicated arithmetic can be done. The same kind of instrument is still employed in modern times, and not only in Japan: it may be found also in Russia (the stchoty), in Iran, in Afghanistan (the choreb), in Armenia and in Turkey, but in these cases it has a different structure, and is of more basic design than the suan pan. The Japanese soroban, on the other hand, will later benefit from a considerable refinement. In the nineteenth century, it lost one of the upper pair of balls; and during the Second World War it lost one of the lower five. These balls are in fact superfluous to the strict needs of the Chinese instrument, so it might seem that the Japanese instrument, having been reduced to its necessary essentials, represents the ultimate perfection of design. However, a skilled operator can use the extra balls to represent an intermediate result whose value exceeds nine and thereby gain speed and facility; learning to use the Japanese abacus well requires a longer and more difficult training, and the acquisition of a more elaborate and precise finger technique (see Chapter 21).
14th - 15th centuries CE [W]. The Italians develop the Humanist script, a scholarly style based on the Carolingian script of the ninth, tenth, and eleventh centuries.
c. 1440 [W]. In Holland, the first attempts at typographic printing are made. The printer Laurens Janszoon first printed playing cards from woodcuts, then whole pages of text, and was led on to make single characters out of wood which he then used to print a small eight-page book called Horarium.
c. 1540 [W]. Printing is reinvented, this time in the West by Johannes Gensfleisch, known as Gutenberg, in Mainz. Recognising the inconvenience of mobile characters made of wood, and that they are ill suited to a good impression, he develops metallic characters, completely regular and adjustable, together with the requisite typographic techniques.
This achievement will later have at least two important consequences. One will be that the rapid spread of typographic procedures will replace the ancient Gothic and Humanistic calligraphies, and will accelerate the evolution of handwriting, which it will cause to settle into a more and more standardised form; the second, the more important, will be that it becomes possible to produce as many copies as one wants of any literary, scientific, or philosophical work, which will lead to a wider and wider dissemination of knowledge within Western Europe. This will lead to a radical transformation of society and the inauguration of a new era in Europe.
15th - 16th centuries CE. After undergoing various apparently major changes, which are however simply due to the natural tendencies of handwriting, the "Arab" numerals take on a fixed form once and for all, thanks to the upsurge of printing in Europe (see Chapter 26).
15th - 16th centuries CE. A progressive generalisation of calculation methods occurs, due to the use of "Arabic" numerals and the zero. The Algorists have triumphed and the Abacists are in retreat. Calculation on the abacus will continue to be done by tradesmen, financiers and other businessmen, and only with the French Revolution will these archaic methods disappear (see Chapter 26).
1478. The publication of the Treviso Arithmetic , a manual of practical arithmetic by an anonymous author, evinces the diffusion of "Arabic" numerals and the increasing favour which the new methods are finding in Western Europe (see Chapter 25).
1654. The French mathematician Blaise Pascal gives the first general definition of a number-system to base m where m is an arbitrary integer greater than or equal to 2 (see Fig. 1.39).
Chapter 1: Historical Summary of Arithmetic, Numerical Notation, and Writing Systems.
Chapter 2: From the Particular to the General Arithmetic Leads to Algebra.
Chapter 3: From Calculation to Calculus.
Chapter 4: Binary Arithmetic and Other Non-Decimal Systems.
Chapter 5: From Clockwork Calculator to Computer: The History of Automatic Calculation.
Chapter 6: What is a Computer?
Chapter 7: Information, the New Universal Dimension.
Conclusion: Intelligence, Science, and the Future of Mankind.
List of Abbreviations.