The Universal History of Computing: From the Abacus to the Quantum Computer: From the Abacus to the Quantum Computer - Brossura

Ifrah, Georges

Brossura

ISBN 10: 0471441473 ISBN 13: 9780471441472

Casa editrice: Wiley, 2001

Riassunto

A fascinating compendium of information about writing systems both for words and numbers -Publishers Weekly "A truly enlightening and fascinating study for the mathematically oriented reader" -Booklist "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 intelligence-and how much further we still have to go.

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Informazioni sull?autore

GEORGES IFRAH is an independent scholar and former math teacher.
E. F. HARDING, the primary translator, is a statistician and mathematician who has taught at Aberdeen, Edinburgh, and Cambridge Universities.
SOPHIE WOOD, cotranslator, is a specialist in technical translation from French.
IAN MONK, cotranslator, has translated the works of Georges Perec and Daniel Pennac.
ELIZABETH CLEGG, cotranslator, is also an interpreter who has worked on a number of government and international agency projects.
GUIDO WALDMAN, cotranslator, has translated several classic literary works.

Estratto. © Ristampato con autorizzazione. Tutti i diritti riservati.

Excerpt

HISTORICAL SUMMARY OF ARITHMETIC,

NUMERICAL NOTATION, AND WRITING SYSTEMS

The writing of words and the writing of numbers show many parallels in theirhistories.

In the first place, human life was profoundly changed by each system, whichallowed spoken language on the one hand, and number on the other, to be recordedin lasting form.

Further, each system answered marvellously to the universal need, felt by everymember 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 hadexpressed or communicated: of words which were otherwise long silent, or ofcalculations long since completed.

Finally, and most importantly, each system granted direct access to the world ofideas and thoughts across space and time. By encapsulating thought, and byinspiring it in others, the writing down of thought imposed on it bothdiscipline and organisation.

Number and letter have often worn the same clothes, especially at times whenletters were used to stand for numerals. But this is superficial: at a muchdeeper level there is still a close correspondence between the alphabet and thepositional 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 finalstage in the development of writing and the final stage in the development ofnumerical notation. They are among the most powerful intellectual attributes ofthe modern human race.

The analogy is not limited to this, however. Throughout its history, eachwritten number-system evolved in a very similar way to the verbal writing systemthat it grew up with. This can be seen in the way that both reflected the spokenlanguage or the cultural traditions to which the language was adapted; or,again, in the mannerisms of local scribes, and in the influence of the verymaterials used for writing.

The main purpose of the chapter is to present a recapitulation of the history ofnumerical notation and arithmetical calculation. But so close is the parallelwith writing just noted, that we shall incorporate notes on the history ofwriting at relevant points; for clarity, these notes will be flagged with [W].

For all that, the writing of language and the notation of numbers differradically in one respect. For something to be called writing, its signs must berelated to a spoken language; it must reflect a conscious effort to representspeech: "Writing is a system of human communication which uses conventionalsigns, which are well defined and which represent a language, which can be sentout 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. Themental process of counting is not linked to any particular act of speech; we cancount to any number without speaking or even without thinking a single word. Weneed only create a "sign language for numbers", and in fact humankind devisedmany "number languages" before inventing even the word "number" andgoing on to use the human voice itself to measure concrete or abstractquantities.

While written characters correspond to the articulations of a spoken language,the signs in a numerical notation reflect components of thought, of a method ofthinking much more structured than the sounds of speech. This method of thinkingis itself a language (the language of numbers, no less), but to acquire thislanguage we need first of all to have a concept of distinct units and thecapacity to aggregate them. The language of number organises numerical conceptsinto a fixed order according to an idea which, on reflexion, we recognise as ageneral principle of recurrence or recursion [a principleaccording to which the evaluation of a complex entity is resolved by evaluatingits component entities of a lower degree of complexity, which in turn . . . ,until the simplest level is reached, at which each entity can be evaluatedimmediately. Transl.] It also makes use of a scale of magnitude (orbase) according to which numbers can be distributed over successivelevels called first-order, second-order, . . . units.

For a system of signs to constitute a written number-system, therefore, theymust in the first place have a structure which its user can conceive, in hismind, as a hierarchical system of units nested each within the next. Then theremust be a predetermined fixed number which gives the number of units on onelevel which must be aggregated together so as to constitute a single unit at thenext level; this number is called the base of the number-system.

In summary: a notation for numbers is a very special human communication systemusing conventional signs called figures which have a well definedmeaning, which can be sent and received, which are equally well understood byboth communicating parties [in other words, a code. Transl.] ,and which are attached to the natural whole numbers according to a mentallyconceived structure which obeys both the recurrence principle and theprinciple of the base.

Over the five thousand years which have elapsed since the emergence of theearliest number-system, of course people have not merely devised one singlenumber-system. Nor have there been an indefinite quantity of them - as can beseen from our Classification of the Written Number-systems of History inChapter 23, which brings together systems so separated in space and time as tobe effectively isolated from each other.

At the end of this chapter, we shall present the main conclusions of thisClassification in a series of comparative systematic tableaux which will exhibitthe mathematical characteristics of each number-system.

The order of succession of the number-systems in this series of tables will notbe purely chronological, but will follow a path which traces their evolution inlogic, as well as in time, from the most primitive to the most advanced.

The number-systems which are found throughout history fall into three maintypes, each divided into several kinds (Fig. 1.40):

A. the additive type of number-system. They are based on the additiveprinciple and each of their figures has a particular value which is always thesame regardless of its position in the representation of the number (Fig. 1.14to 16). Basically, they are simply written versions of more ancient methods ofcounting with objects (Fig. 1.1 to 13);

B. the hybrid type of number-system. These use a kind of mixedmultiplicative and additive principle ( Fig. 1. 28 to 32) , and are essentiallytranscriptions of oral number-systems of varying degrees of organisation;

C. the positional type of number-system. These are based on theprinciple that the value of a particular figure depends on its position in therepresentation 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 thereforeindicate the above types of additive, hybrid, and positional number-systemsrespectively. The kinds within these types will be indicated by numbers attachedto 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 truenature of our modern system of numerical notation, and therefore to understandwhy 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 inIndia. Its birth, then and there, came about through the remote chance thatthree great ideas came together, namely: well-conceived figures representing thebase digits (1, . . . , 9) ; their use according to a principle of position; andtheir completion by a sign 0 for zero which not only served to mark the absenceof a base digit in a given position but also - and above all - served to denotethe null number.

When we talk of "our modern number-system", by the way, we do not only mean theway we now write numbers worldwide, but also any of the other number-systemsalso 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 theDictionary of Indian Numerical Symbols (Chapter 24, Part II).

This "temporal logic" alone could demonstrate the deep unity of all humanculture; but it is not merely a question of order and system revealed solely onlines of time - this would be to ignore both transmission within each cultureand also, above all, transmission from one culture to another. It would alsogloss over the true chronology of events in which, in our story, we see culturesboth overlapping each other, and leapfrogging past each other. At certain times,some cultures have been far in advance of others. Some other peoples have clungto inadequate number-systems throughout their history, whether by failure tobreak out of the prison of an inadequate system, or through a conservatism whichattached them to a poor tradition.

For these reasons, we now embark on a systematic chronological résumé which willtrace out this logic of time. Fig. 1.41, to be found below, will show how fromthe different civilisations emerged our different classifications ofnumber-systems (defined in Fig. 1. 14 to 16, 1. 28 to 32, and 1.38 to 39); andFig. 1.41 is the inverse of Fig. 1.42, showing how the different number-systemsemerged 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 ofcalculations or counts previously carried out, in order to compensate for thedefects of human memory.

We shall also follow the principal stages of development of arithmeticalcalculation, which evolved in parallel with the writing of numbers. It beganwith counting on the fingers, and with pebbles; continued through many- colouredstrings, the abacus, checkerboards and abacuses traced in wax or dust or sand,and finally made contact with written numerical notation when our modernpositional numerical notation, and the zero on which it depends, werediscovered.

The dates of "first appearance" given in the following Figures are the resultsof archaeological, epigraphical and palaeographical research, to mention a fewof the domains of study from which the information has been gleaned. They do notcorrespond to the definite date of an invention or a discovery. A date which wegive below merely means the established date, according to these researches, ofthe 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 ordiscovery of something, and the date by which it came into common use; and thelatter must in turn also be distinguished from the dates of the earliestinstances of which we are currently aware.

Quite possibly, a discovery occurred many generations prior to itspopularisation, and there may well be a delay between this and the date of theearliest evidence we possess today. There may be many reasons for this. Often, adiscovery long remained the esoteric property of a closed sect, or of aspecialised elite who jealously guarded their monopoly of an arcane art. In manycases, no doubt, documents which had existed prior to those of which we knowhave perished; or perhaps they have yet to be discovered. Archaeological ordocumentary discoveries which remain to be made may yet cause changes in theconclusions which we present below.

Chronological summary

At undetermined dates, beginning in prehistory, the following sequence ofdevelopments occurred. Entries flagged with [W] refer specifically todevelopments in writing, rather than numbers.

* The human race was, in the earliest stages of its evolution, at the mostprimitive stage of the notion of number, which was confined to such number (upto four or five) as could be assimilated at a glance. This never- theless awokein the human mind a realisation of the concrete aspects of objects which itdirectly perceived.

* By force of necessity, aided by native intelligence and by the capacity forthought, human beings little by little learned to solve an increasing range ofproblems. For numerical magnitudes greater than four, people devised proceduresbased on the manipulation of concrete objects which enabled them to achieve, upto a point, results which met their needs of the moment. These were simply basedon a principle of counting by one-to-one correspondence, and amongst them may befound 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 beexpressed in articulated speech, accompanied by corresponding gestures.

* By force of habit, counting according to these parts of the body (once adoptedaccording to an invariable routine) slowly became part abstract, part concrete,thereby suggesting less and less the specific part of the body and more and morethe concept of a certain corresponding number which increasingly tended tobecome detached from the notion of the body and to become applicable to objectsof any kind. (For a detailed account of the above, see Chapter 1.)

* The resulting necessity to make a distinction between the numerical symbolitself and the name of the concrete object or image led people finally to make aclean break between the two, and the relationship between them disappeared fromtheir minds. Thenceforth, people progressively learned to count and to conceiveof numbers in an abstract sense, not related to specific concrete countingtokens. 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 beencreated. Day by day, the notion of successor became established in thehuman mind, and what had been a motley collection of concrete objects became astructured abstract system, at first based on gesture before assuming verbal orwritten forms. It became a spoken system when the names of the numbers wereinvented as abstractions out of custom, usage and memory. Much later, in asimilar way written systems came about when all kinds of graphic symbols werebrought into use - scratched or drawn or painted lines, marks hollowed out ofclay or carved in stone, various figurative symbols, and so on (see Chapter 2).

* This proliferation of representations created problems, which were solved bythe invention of the principle of the base of a number-system (the base10 being the most commonly used throughout history). Using every kind of objectand device (the fingers, pebbles, strings of pearls, little rods, . . .) peoplegradually arrived at the abstractions embodied in the procedures of calculationand in the operations of arithmetic (see Chapter 2).

From this point in the history, we are able to assign approximate dates to thesuccessive stages of development.

35000 - 20000 BCE. The earliest notched bones of prehistory are themost ancient known archaeological objects which had been used for numericalends. They are in fact graphical representations of numbers, though we do notknow what precise purposes they served (see Chapter 4).

20000 BCE [W]. The first pictures on rock appeared in Europe: these, among theearliest known visual representations of human thought, were made by engravedor 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 claytokens of various sizes and shapes (cones, discs, spheres, small balls,little rods, tetrahedra etc.). Some of them bear parallel lines, some crossesand other motifs, while others are decorated with carved figurines representingevery kind of object (jugs, animal heads, etc.). These relief drawings (whichsurely had significance for their creators and users) are clear evidence of thedevelopment of symbolic thought. We do not know, however, whether these are theelements of some system nor whether this corresponds to one of the intermediatestages between a systematic purely symbolic expression of human thought and itsformal expression in a spoken language (see Chapter 10).

9th - 2nd millennium BCE. The peoples of the Middle East (from Anatolia andPalestine to Iran and Mesopotamia, from Syria to Sudan) made their calculationsusing cones, spheres, rods and other clay objects which stood for the differentunit magnitudes of a number-system. Such systems can be found, from the fourthmillennium BCE onwards, in Elam and in Sumer, in a much elaborated form whichwill give rise, not only to written counts, but also to some extent to thegraphical forms of the Sumerian and proto-Elamite figures (see Chapters 10 and12).

6th - 5th millennia BCE [W]. The earliest ceramic artefacts, on whichmotifs have been painted, engraved, cut out or impressed on the raw clay, orengraved after firing, appear in the Middle East. These are evidently graphicalrepresentations emanating from some symbolic system, but we do not know theirmeaning or purpose.

At the same time, in Asia Minor (Çatal Hüyük) and later in Mesopotamia, thereappear the earliest seals (carved objects which can be used to impressa relief design on soft material such as clay).

4th millennium BCE (?). The people of Sumer have an oral number- system, to base60 (see Chapters 8 and 9) . This base 60 has come down to us via theBabylonians, the Greeks and the Arabs, and we use it yet for the minutes andseconds of time, and for the measure of angle in minutes and seconds of adegree.

3500 BCE [W]. The first cylindrical seals appear in Elam and inMesopotamia. They are small cylinders of stone, precious or semi-precious,bearing an engraved symbolic design. Every man of a certain standing had one ofthese: it represented the very person of its bearer and therefore was associatedwith all economic or judicial aspects of his life. By rolling the cylinder ontoany object of clay, the proprietor of the seal thereby impressed his"signature", or his right of property, upon it. The different designs did notconstitute "writing" in the strict sense of the word; rather, they had asymbolic significance subject to every kind of interpretation.

3300 - 3200 BCE. The figures of the Sumerian number-system and the figures ofthe proto-Elamite number-system make a simultaneous appearance at this time.These are the most ancient written number-systems at present known (see Chapters8 and 10).

3200 - 3100 BCE [W]. The writing signs of Sumer, the most ancient writing systemknown, make their appearance. These are pictograms which represent every kind ofobject, and they are found on clay tablets which seem to have been used for someeconomic purpose. However, this is still not a true writing system, since thesigns 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 thevarious parts of speech, and will have become linked to spoken language, whichis the most highly developed way of analysing and communicating reality (seeChapter 8).

3000 BCE [W]. In ancient Persia, the proto-Elamite writing signs appear (seeChapter 10).

3000 - 2900 BCE [W]. The signs of Egyptian hieroglyphic writing appear (seeChapter 14).

3000 - 2900 BCE. The figures of the Egyptian number-system appear (see Chapter14).

2700 BCE [W]. The cuneiform characters (in the form of angles and wedges) of theSumerian writing system appear on their clay tablets (see Chapter 8).

2700 BCE. The cuneiform figures of the Sumerian number-system appear (seeChapter 8).

2700 - 2300 BCE. For doing arithmetic, the people of Sumer now abandon their oldcalculi and invent their abacus, a kind of table of successivecolumns, ruled beforehand, which delimit the successive orders of magnitude oftheir sexagesimal number-system. By clever manipulation of small balls or rodson the abacus, they are able to perform all sorts of calculations (see Chapter12).

2600 - 2500 BCE [W]. Egyptian hieratic writing appears, a cursive abbreviationof hieroglyphic writing and used alongside the latter for the sake of rapidwriting 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 ofSumer to write down their own speech. This is the beginning of the Akkadianscript from which will emerge the Babylonian and Assyrian writing systems.

2350 BCE [W]. Appearance of the writing of Ebla (the capital of the Semitekingdom situated at Tell Mardikh, to the South of Aleppo in Syria), a cuneiformscript cut into clay tablets, for their Western Semitic dialect which was closeto Ugaritic, Phoenician and Hebrew.

2300 BCE [W]. Proto-Indian writing appears in the Indus valley at Mohenjodaroand Harappâ (in what is now Pakistan). This writing of the ancient Induscivilisation (25th - 18th centuries BCE) is separated by a hiatus of over twothousand years from the earliest written texts in any true Indian language andin true Indian writing. It is not known how to bridge this gap, nor, indeed, ifit ever was bridged.

End of 3rd millennium BCE. The Semites of Mesopotamia are now slowly adopting acuneiform decimal notation which has come down to them from their predecessors.In everyday use, this system will come to supplant the Sumerian sexagesimalsystem (of which, however, the base 60 will survive in the positional notationof the Babylonian scholars). At the same time, the ancient Sumerian abacusundergoes a radical transformation: instead of using beads or rods, they tracetheir cuneiform figures inside the ruled columns of a large clay tablet; in thecourse of calculation, these figures are successively erased according as thesuccessive partial results are obtained (see Chapter 13).

2000 - 1660 BCE [W]. The hieroglyphic writing of the Minoan civilisation appearsin Crete, found at Knossos and Mallia on bars and tablets of clay which appearto have been accountancy documents (see Chapter 15).

2000 - 1660 BCE. At the same time appear the hieroglyphic figures which theyused 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 wereundoubtedly inventories of some kind. Somewhat casual in style, this scriptoccurs not only in administrative quarters but also in sanctuaries and probablyin private houses too (see Chapter 15).

1900 BCE. At the same time appear the "Linear A" figures which they used fornumbers (see Chapter 15).

1900 - 1600 BCE [W]. The cuneiform script of the Semites of Mesopotamiagradually supplants the Sumerian script and spreads across the Near East, whereit will even become the official script of the chancelleries.

1900 - 1200 BCE. The decimal cuneiform number-system of the Semites ofMesopotamia 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 isnot yet in possession of a zero (see Chapter 13).

17th century BCE [W]. The first known venture into an alphabetic script - theSemites who were in the service of the Egyptians in the Sinai made use of simplephonetic symbols derived from Egyptian hieroglyphics (the so-called"proto-Sinaitic inscriptions" of Serabit al Khadim).

17th century BCE. Notwithstanding the very rudimentary nature of theirhieroglyphic and hieratic numerals, the Egyptians are able to make use of themfor arithmetical calculations (see Chapter 14). These methods relieve theburden on the memory (since it is sufficient simply to know how to multiply anddivide by 2), but they are not unified and they lack flexibility; they aretime-consuming, and are very complicated in comparison with the procedures ofour own day.

16th century BCE. By now, the Egyptian hieratic number-system has come to theend of its graphical evolution (see Chapter 14).

15th century BCE [W]. Desiring abbreviation, and keen to break away from thecomplicated Egyptian and Assyro-Babylonian writing systems then in use in theNear East, the Semites of the Northwest who were settled along the Syrian andPalestinian coasts develop the very first purely alphabetical writing system inhistory, thereby inventing the alphabet. This superior method oftranscribing words, capable of being adapted to any spoken language, henceforthallows all the words of any language to be written by means of a small numberof simple phonetic symbols called letters(see Chapter 17).

15th century BCE [W]. The hieroglyphic writing of the Hittite civilisationappears. This script will not only be used for religious and dedicatorypurposes, but also - and above all - for secular purposes (see Chapter 15).

15th century BCE. At the same time, the Hittite hieroglyphic number-systemappears (see Chapter 15).

1350 - 1200 BCE [W]. The Creto-Mycenaean script called "Linear B" appears inCrete (found at Knossos) and in Greece (Pylos, Mycenae, etc.); it is amodification of "Linear A", used to write an archaic Greek dialect (see Chapter15).

1350 - 1200 BCE. At the same time, the "Linear B" numerals appear (see Chapter15).

14th century BCE [W]. The oldest known entirely alphabetic script appears, foundon tablets from Ugarit (Ras Shamra, near Aleppo in Syria). This is a cuneiformscript whose alphabet has only thirty letters; it was used to write a Semiticlanguage related to Phoenician and Hebrew (see Chapter 17).

End of the 14th century BCE [W]. One of the oldest known specimens of archaicChinese writing appears at Xiao. It is found on inscriptions made on bones or ontortoise shells, and its main purpose was to enable communication between theworld of the living and the world of the spirits by means of various divinatoryand religious practices (see Chapter 21).

End of the 14th century BCE. At the same time, we find the oldest known Chinesenumerals (archaic Chinese number-system, in inscriptions on the bones andtortoise shells at Xiao dun; see Chapter 21).

End of the 12th century BCE [W]. The earliest known specimens of the WesternSemitic "linear" alphabet, a precursor of all modern alphabets (see Chapter 17),are used by the Phoenicians. Since they had dealings with a great variety ofpeoples, these notable merchants and bold navigators diffused their alphabet farand wide. In the East, they will pass it first to their immediate neighbours(Moabites, Edomites, Ammonites, Hebrews, etc.), including the Aramaeans who inturn will spread it from Syria to Egypt and to Arabia, and from Mesopotamia tothe Indian sub-continent. From the ninth century BCE, it will spread also roundthe whole Mediterranean seaboard and be progressively adopted by the Westernpeoples who will adapt it to their own languages, and modify it by the additionof some further symbols.

9th century BCE [W]. Ancient I raelite inscriptions in palaeo-Hebraiccharacters, derived directly from the 22 Phoenician characters, appear (seeChapter 17).

9th century BCE. The Hebrews adopt the Egyptian hieratic numerals, which theyespecially make use of in correspondence (see Chapter 18).

9th - 8th centuries BCE [W]. The alphabetic script of Phoenician origin spreadsacross the near East and the Eastern Mediterranean (Aramaeans, Hebrews, Greeks,etc.).

End of 9th century BCE [W]. The Greeks perfect the principle of the modernalphabetical system by adding symbols for the vowels to the consonants of theoriginal Phoenician alphabet. This is the first alphabet to have a strict andintegrated notation for the vowels (see Chapter 17). In turn, this alphabet willinspire the Italic alphabets (Oscan, Umbrian, Etruscan, etc.) and then Latin,later giving rise to the Coptic, Gothic, Armenian, Georgian and Cyrillicalphabets.

8th century BCE [W]. Egyptian demotic script appears, a cursive script, arisingfrom a local branch of Egyptian hieratic script but more abbreviated, which itwill later supplant in everyday use (see Chapter 14).

8th century BCE [W]. Appearance of the Italic alphabets, in particular theEtruscan (see Chapter 17).

8th century BCE. The earliest clearly differentiated forms of Egyptian demoticnumerals 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 Semiticnumerals; 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 inpre-Columbian Central America.

5th century BCE. The Zapotecs use an additive number-system with base 20, whichcan be found in use amongst all the pre-Columbian peoples of Central America(Mayas, Mixtecs, Aztecs, etc.) to within minor graphical variations (see Chapter22). For purposes of calculation, these peoples certainly do not make use ofthis ill-adapted number-system; on the contrary, they make use of calculatinginstruments. Although Central-American archaeology has yielded up nothingrelevant to this subject, we may nevertheless get an idea of it by appealing toethnology and to history which provide us with numerous analogies. We cantherefore suppose that, on the example of certain African societies, they madeuse of rods, each corresponding to an order of magnitude, along which they slidpierced pebbles. They possibly proceeded in the same way as the Apache, theMaidu and the Havasupai of North America who threaded pearls and shells ontocoloured threads; or, perhaps more plausibly, like the Incas of South Americawho distributed pebbles, beans or grains of maize onto the squares of acheckerboard on a kind of tray made of stone, pottery or wood, or drawn on thefloor (see Chapters 12 and 22).

5th century BCE [W]. Aramaean script becomes generally adopted for internationalcorrespondence in the Middle East, henceforth supplanting the Assyro-Babylonianscript for this purpose.

5th century BCE. The Aramaean numerals, which have already reached their finalform, spread over the Middle East (Mesopotamia, Syria, Palestine, Egypt,Northern Arabia, etc.).

5th century BCE. Earliest archaeological evidence for the use of the Greekabacus: tables made out of wood or marble, pre-set with small counters in woodor metal suitable for the mathematical calculations (see Chapter 16). ThePersians in the time of Darius were to use this type of abacus, and after themthe Etruscans and Romans. The Western Christian world was to inherit the use ofthis abacus, which they were to continue until the French Revolution (seeChapters 16, 25 and 26).

5th century BCE. The Greek acrophonic number-system spreads across the Hellenicworld (see Chapter 16).

5th century BCE. The acrophonic numerals of Southern Arabia appear in theinscriptions of the kingdom of Sheba (see Chapter 16).

End of 4th century BCE. The earliest known records of the Greek alphabeticnumber-system appear in Egypt, showing that the Greek letter numerals are ingeneral 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, asused by the Babylonian scholars. This was a cuneiform character. Although usedin the positional Babylonian number-system to signify the absence of asexagesimal unit, it is nevertheless still not perceived as a number in its ownright (see Chapter 13).

3rd century BCE. The Greek alphabetical number-system spreads across the MiddleEast and the Eastern Mediterranean (see Chapter 17).

3rd century BCE [W]. The Aramaeo-Indian Kharoshthî writing appears in the edictsof 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 nowPakistan and Afghanistan (see Chapter 24).

3rd century BCE. At the same time, Kharoshthî numerals appear in Northwest Indiaand 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 theWestern Semitic world, no doubt with an intermediate Aramaean form of which nospecimens have been found. This will become the earliest truly Indian script,and will be the origin of all of the alphabetic scripts of the Indiansub-continent and of Southeast Asia. Over the centuries, it will undergo manychanges which culminate in many distinctly different types of writing, such asGupta, Bhattiprolu and Pâlî . Gupta in turn will splitinto Nâgarî , Siddham, and ´ Shâradâ from which willdescend all the current scripts to be found in Central and Northern India, inNepal, in Tibet, and in Chinese Turkestan. Bhattiprolu will give rise to thescripts of Southern India and Ceylon, while Pâlî will give rise to the scriptsof Southeast Asia. The apparently considerable differences between all thesescripts are due either to the natures of the languages and traditions to whichthey have been adapted, or to regional differences between scribes anddifferences between writing materials. (See "Indian Styles Of Writing" and"Indian Styles of Writing (The materials of)" in the Dictionary of IndianNumerical 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 trulyIndian numerals, occur more and more frequently in later inscriptions (Shunga,Ândhra, Shaka, Kshatrapa etc.); and they are the prototypes of all the numericalnotations 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 ofthe digits 1 to 9 in our own number-system and in the modern Arabicnumber-system (see Chapter 24).

3rd century BCE to 4th century CE [W]. Greek manuscript writing splits intothree types: "book" script, official script, and the script used for privatedocuments.

2nd century BCE. In Plutarch, we find mention of the sand abacus alongside theabacus with tokens. A board, with a raised border, is filled with fine sand onwhich 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 theChristian population of the mediaeval West, and they carry out theircalculations using either Roman numerals or the Greek alphabetic numerals (seeChapter 16).

2nd century BCE [W]. This epoch sees the Chinese invention of paper. (Accordingto some, it was invented by Cai-Lun who achieved it by boiling up unravelledtissues and old fishing nets.) They also invent xylography: the text tobe reproduced is written on the polished surface of a wooden board, and then thewood surrounding the writing is cut away so as to leave it in relief; ink isapplied, and a sheet of paper is pressed on. (See further information aboutpaper in the course of the entry "Pâtîganita" in the Dictionary of IndianNumerical Symbols, Chapter 24, Part II.)

2nd century BCE. The earliest known documents which refer to use of the Chineseabacus, and to "calculation with rods" (suan zí) in which small bamboosticks are placed in successive squares of a checkerboard (see Chapter 21).

2nd century BCE. The earliest known documents which affirm the use of apositional decimal notation by the Chinese. This system does not however have azero (see Chapter 21), and in fact is simply a written counterpart of the methodof "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 arederived from the cursive Aramaean script (see Chapter 17).

2nd century BCE. The earliest documents in which we can see the use of modernHebrew alphabetic numerals (see Chapter 17).

2nd century BCE to 3rd century CE. Indian arithmeticians perform theircalculations by tracing their nine digits in Brâhmî notation on the floor,within consecutive columns already delineated, with a pointed rod. A similarprocedure will later be used by the Arabs, especially those of the Maghreb andof Andalusia. Boards covered with fine sand, with flour or with some otherpowder are also used, with a stylus whose point is used to trace the figures andits flat end to erase them. This board might be placed on the floor, on a stoolor on a table, or might be furnished with legs like those used much later in theArab, Turkish and Persian administrations. The board might be made in a smallversion which could be kept in a case. (See Chapters 24 and 25, and also theentry "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 theemergence of the lì shu graphics which will evolve towards the modernsystem of Chinese characters (see Chapter 21).

1st century BCE. Horace notes the use of the wax abacus, as well as the abacuswith rods, by the Romans: a real "portable calculator" which could be hung overthe shoulder, this consists of a board made of bone or wood, covered with a thinlayer of black wax, on which the lines for the columns, and the figures, aredrawn with an iron stylus (see Chapter 16).

Start of the Common Era [W]. A cursive branch of the ancient Aramaean evolves togive rise to Arabic script.

1st century CE. At this period we find the earliest archaeological evidence ofthe Roman "pocket abacus". This is a small metal plate with parallel slots alongwhich mobile beads can be slid; each is associated with a numerical order ofmagnitude. It is therefore very similar to the bead abacus which in modern timesstill 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 giverise 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 inpre-Columbian Central America (see Chapter 22).

End of 3rd century CE. The earliest examples of use of the "Long Count" fordates by Mayan astronomers (see Chapter 22).

3rd - 4th centuries CE [W]. The earliest known cases of runic script used byGermanic peoples (Futhark alphabet).

Beginning of 4th century CE [W]. Pharnavaz, first king of the country which laybetween Armenia and the Caucasus, is inspired by Greek to invent theMkhedrouli alphabet, ancestor of the Georgian alphabet (see Chapter 17).

4th century CE. Appearance of Ethiopian numerals in the inscriptions from Aksumin 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 theGothic 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 alphabetappear, from which all the alphabetical scripts of central India, Nepal, Tibetand 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 andChinese Turkestan will be derived (see Chapter 24, and also the entry "GuptaNumerals" in the Dictionary of Indian Numerical Symbols, Chapter 24, Part II).

4th - 5th centuries CE. The first nine digits of the Indian system, derived fromthe 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 ora dot which represents zero; this therefore is the birth of the Indianpositional decimal notation, which was the ancestor of our modern numericalnotation (see Chapter 24).

4th - 6th centuries CE. During this period, the Indian arithmeticians radicallytransform their traditional methods of calculation. They do away with thecolumns of their ancient sand abacus, and attribute values to written digitsaccording to their decimal position. This, therefore, is the beginning of themodern number-system. It is also, however, the beginning of modern writtenarithmetic.

To begin with, their techniques, albeit liberated from the columns of theabacus, were but written imitations of the abacus procedures. As formerlypractised, on a medium as inconvenient as the sand abacus, with interme- diateresults noted after erasing the previous ones, these constrained the role ofhuman memory and made it difficult if not impossible to check the calculationand correct errors made along the way. The Indian and Arab scholars subsequentlydeveloped procedures which did not involve erasure, but involved writingintermediate results above the working. While certainly advantageous forchecking purposes, since every intermediate error remained to be seen,nevertheless this resulted in a cluttered work- sheet from which it wasdifficult 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. Writingtheir calculations on a board with chalk without worrying about how many figuresthere were or, better still, rubbing them out successively with a cloth: suchwas the convenient and relaxed method which, even before the advent of pen andpaper, allowed the Indian arithmeticians and their Arab and European successorsto work in their own way and with unfettered imagination to arrive atsimplifications of the rules and methods and ultimately to create the techniqueswhich would give rise to our modern methods of written arithmetic. (See Chapters24 and 25, and also the entry "Indian Methods of Calculation" in the Dictionaryof Indian Numerical Symbols, Chapter 24, Part II.)

4th - 6th centuries CE [W] . The earliest forms of the Bhattiprolu and Pâlîscripts appear: from these will be derived, respectively, all the alphabeticwriting systems of South India, and those of Southeast Asia (see Chapter 24).

4th - 6th centuries CE. The earliest forms of the Bhattiprolu and Pâlî numeralsappear: from these will be derived, respectively, all the number-systems ofSouth India, and those of Southeast Asia (see Chapter 24).

4th - 9th centuries CE. This is probably the period during which the positionalnotation, 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 Mayancalendar, this number-system exhibited an irregular use of the base 20 beyondthe third digit position which robbed it, along with its zero, of practicaloperational value (see Chapter 22).

5th century CE [W]. The earliest known specimens of Arab writing found inpre-Islamic inscriptions. This script was cursive in style, and in timediversified to give rise to the Kufic script and the Naskhî script during theearly centuries of Islam (see Chapters 19 and 25).

5th century CE [W]. The priest Mesrop Machtots draws inspiration from Greek toinvent the Armenian alphabet (see Chapter 17).

5th - 7th centuries CE [W]. Emergence of the Ogham script in Celtic inscriptionsin Ireland and Wales.

510. The Indian astronomer Âryabhata invents a special numerical notation forwhich it is necessary to have a full awareness of the concepts of zero and theprinciple of position. He further makes use of a remarkable method forcalculating square and cube roots which it is impossible to perform unless thenumbers are written down using the principle of position, the nine digits, and atenth 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 IndianNumerical Symbols, Chapter 24, Part II.)

628. The Indian mathematician and astronomer Brahmagupta publishesBrahmasphutasiddhânta , which displays total mastery of positionaldecimal notation, using the nine digits and a zero. (See Chapters 24 and 25,and also the entries "Brahmagupta" and "Indian Mathematics, The history of" inthe Dictionary of Indian Numerical Symbols, Chapter 24, Part II.)

629. The mathematician Bhâskara publishes a Commentary on theÂryabhatîya. This work not only reveals complete mastery of the use ofzero and of the positional decimal number-system: it also shows that the authoris quite at ease with the Rule of Three and with arithmetical fractions, whichhe writes in a way very similar to ours, though lacking the horizontal bar whichwill not be introduced until several centuries later, by Arab mathematicians.(See Chapters 24 and 27, and also the entry "Bhâskara" in the Dictionary ofIndian Numerical Symbols, Chapter 24, Part II.)

7th century CE [W]. The earliest distinct forms of the Indian Nâgarîwriting appear, from which the scripts of North and Central India will bederived (Bengâlî, Gujarâtî, Oriyâ, Kaîthî, Maithilî, Manipurî, Marâthî, Mârwarî,etc.(See Chapter 24, and also the entries "Indian Styles of Writing" and "IndianStyles of Writing, The materials of," in the Dictionary of Indian NumericalSymbols, 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 bederived (Bengâlî, Gujarâtî, Oriyâ, Kaîthî, Maithilî, Manipurî, Marâthî, Mârwarî,etc., see Chapter 24).

7th century CE [W]. The earliest distinct forms of the stylised scripts ofSoutheast Asia appear ( Khmer, Malaysian, Shan, Kawi, etc., see Chapter 24).

7th century CE. The earliest distinct forms of the Indian numerals which willgive 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, withthe 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 wefind Latin writing of "Visigoth" and "Luxeuil" type.

7th - 10th centuries CE [W]. The earliest distinct forms which will give rise tothe 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, ofIndian origin, takes its place in the Chinese positional decimal number-systemof "bar numbers" (the suan zí system, see Chapter 24).

8th century CE [W]. The first appearance of the so-called "minuscule" Greekwriting (which will replace the older system in books from the ninth centuryonwards).

End of 8th century CE. At this time, the positional decimal notation with a zeroenters the world of Islam. In the hands of the Arab scribes, the figures willundergo changes of form, in some cases far enough from the original forms thatthey appear to be new (see Chapter 25).

8th - 11th centuries CE [W]. Runic inscriptions from Viking times (from Upplandprovince 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 themonk Maurdramnus, the dedication of the Gospels by Charlemagne, etc.).

820 - 850. The period of the great Muslim astronomer and mathematician AlKhuwarizmi, whose works contributed greatly to the knowledge and disseminationof the numerals and arithmetical methods which originated in India (see Chapter25).

9th century CE. The Ghubar numerals of the Maghreb and Andalusian Arabs nowappear (they are of Indian origin, and their form anticipates that of theEuropean numerals of the Middle Ages and the Renaissance, before giving rise toour modern numerals. See Chapter 25.)

9th century CE [W]. The earliest distinct forms of the Indian Shâradâ scriptappear (a southern variant of the Gupta script), which will give rise to thescripts 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, whichwill 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 Cyrildraws 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, thebishop Clement draws inspiration from Greek to invent the Cyrillic alphabet. Thefirst simplification of this writing which will later give rise to the modernRussian alphabet will be brought about by Peter the Great in the eighteenthcentury.

972 - 982. In the course of a voyage to Spain, the monk Gerbert d'Aurillac fromAuvergne (later to become Pope Sylvester II, in 999) learns the "Arab" numeralsand introduces them to Western Europe (see Chapter 26).

976 - 992. Two manuscripts from non-Muslim Spain illustrate the forms of ninefigures which are very similar to numerals of the Ghubar type. These are theoldest known evidence of the presence of "Arab" numerals in Western Europe (seeChapter 26).

10th - 12th centuries CE. Europeans are carrying out arithmetic operations usingthe abacus with columns, of Roman origins and perfected by Gerbert d'Aurillacand his pupils. They use counters made of horn ( called apices) markedwith the Arab numerals from 1 to 9, or with the Greek alphabetic numerals fromæ to Ø, or with the Roman numerals from I to IX (see Chapter 26).

11th century CE [W]. The master calligrapher Lanfranc, in the Carolingiantradition, creates the script which will become the most beautifulpontifical writing of the twelfth century.

Middle of 11th century CE [W]. Printing is invented by the Chinese. They makeuse of separate characters made of baked clay, which later will be made of leadand then in copper. This invention is related by Qin Guo in 1056; he attributesit to Bi Xing and dates it at 1041.

12th century CE. The Indian sign for zero is introduced to Europe. The Europeanarithmeticians henceforth do their calculations with the zero and the nine"Indo-Arabic" digits. Also, the rules of arithmetic, of Indian origin, are nowcalled algorisms.

12th - 13th centuries CE [W]. Gothic script gradually replaces Carolingianscript.

12th - 13th centuries CE [W]. Aztec writing emerges (see Chapter 22).

12th - 16th centuries CE. A ferocious dispute takes place between theAbacists(adherents of methods of calculating by counters on the abacus,and prisoners of a system seamed with ancient number-systems such as the Romannumerals and the Greek alphabetic numerals) and the Algorists,proponents of methods of written arithmetic using the Indian numerals and thezero (see Chapter 26).

12th - 15th centuries CE. The period where the forms of the "Arabic" numeralsbecome established in Europe, where they will eventually evolve into theirmodern forms (see Chapter 26).

1202. Following his travels in North Africa and the Middle East, the Italianmathematician Leonard of Pisa, better known as Fibonacci, publishes LiberAbaci("A Treatise on the Abacus"). Over the ensuing three centuries, thisbook will prove to be a most fruitful source of development of arithmetic andalgebra in Western Europe (see Chapter 26).

13th century CE. In this period we find the earliest documents which illustratethe use of the Chinese abacus (suan pan). This is a rectangular frame ofwood 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 (thelower) of which there are five balls on each rod, and on the other side (theupper) 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 oneto provide a simple means to carry out all the operations of arithmetic; Westernobservers are usually astonished at the speed and dexterity with which even themost complicated arithmetic can be done. The same kind of instrument is stillemployed in modern times, and not only in Japan: it may be found also in Russia(the stchoty), in Iran, in Afghanistan (the choreb), in Armeniaand in Turkey, but in these cases it has a different structure, and is of morebasic design than the suan pan. The Japanese soroban, on theother hand, will later benefit from a considerable refinement. In the nineteenthcentury, it lost one of the upper pair of balls; and during the Second World Warit lost one of the lower five. These balls are in fact superfluous to the strictneeds of the Chinese instrument, so it might seem that the Japanese instrument,having been reduced to its necessary essentials, represents the ultimateperfection of design. However, a skilled operator can use the extra balls torepresent an intermediate result whose value exceeds nine and thereby gain speedand facility; learning to use the Japanese abacus well requires a longer andmore difficult training, and the acquisition of a more elaborate and precisefinger technique (see Chapter 21).

14th - 15th centuries CE [W]. The Italians develop the Humanist script, ascholarly style based on the Carolingian script of the ninth, tenth, andeleventh centuries.

c. 1440 [W]. In Holland, the first attempts at typographic printing are made.The printer Laurens Janszoon first printed playing cards from woodcuts, thenwhole pages of text, and was led on to make single characters out of wood whichhe then used to print a small eight-page book called Horarium.

c. 1540 [W]. Printing is reinvented, this time in the West by JohannesGensfleisch, known as Gutenberg, in Mainz. Recognising the inconvenience ofmobile characters made of wood, and that they are ill suited to a goodimpression, 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 willbe that the rapid spread of typographic procedures will replace the ancientGothic and Humanistic calligraphies, and will accelerate the evolution ofhandwriting, which it will cause to settle into a more and more standardisedform; the second, the more important, will be that it becomes possible toproduce as many copies as one wants of any literary, scientific, orphilosophical work, which will lead to a wider and wider dissemination ofknowledge within Western Europe. This will lead to a radical transformation ofsociety 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 ofprinting in Europe (see Chapter 26).

15th - 16th centuries CE. A progressive generalisation of calculation methodsoccurs, due to the use of "Arabic" numerals and the zero. The Algoristshave triumphed and the Abacists are in retreat. Calculation on theabacus will continue to be done by tradesmen, financiers and other businessmen,and only with the French Revolution will these archaic methods disappear (seeChapter 26).

1478. The publication of the Treviso Arithmetic , a manual of practicalarithmetic by an anonymous author, evinces the diffusion of "Arabic" numeralsand 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 definitionof a number-system to base m where m is an arbitrary integergreater than or equal to 2 (see Fig. 1.39).

Continues...

Excerpted from The Universal History of Computingby Georges Ifrah Copyright © 2002 by Georges Ifrah. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Copyright © 2002 Georges Ifrah
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Dettagli dell?edizione

• Editore: Wiley
• Data di pubblicazione: 2001
• ISBN 10: 0471441473
• ISBN 13: 9780471441472
• Rilegatura: Copertina flessibile
• Lingua: Inglese
• Numero di pagine: 419
• Contatto del produttore: non disponibile