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According to a
popular tradition, still tough in Egypt and North Africa, the “Arab” figures
would be the invention of a glazier geometrician
originating in the Maghreb, which would have imagined to give to the
nine significant figures an evocative form depending on the number of the
angles contained in the drawing of each one of them: an angle for the graphics
of figure 1, two angles for figure 2, three angles for the 3, and so on:
Part 1: The “Arab numerals” appeared
in India!
Part 2: Genius of the Indian
positional system
Part 3: The introduction of the
Indian numbering system out of ground of Islam then in Occident
Foreword
Nowadays, almost all the populations in the world use a decimal basic
positional numbering system resting on the figures from 1 to 9 (that one names,
wrongly or rightly, “Arab numerals”). Seeming the only able
system today, thanks to its unlimited numerical resources, to adapt to the
development of calculation. But its use became if current, because
essential, that one omits to wonder on his true origin and the context
(geographical and temporal) of the invention of a as
clever and perfect numeration as ours. But still is necessary it for that to be
freed from tough, harmful prejudices with the true knowledge of this important
page of our history, in particular according to which would owe us our current
numeration with Arab civilization. This is why we here will interest we in the real founders, alas too little known, of such a
projection of the human intelligence: Indian populations. Still today, the
general thought, influenced by incorrect ideas, too often forgets to return to
these creators the merit which is to them. The History, however, undoubtedly
tells us that modern numeration is the fruit, not
Arab
scientists, but well of Indian civilization.
Introduction
One often said in the literature arabo-Persian that
there were two works whose was glorifiait mainly the
Indian nation:
- its decimal
notation of position and its methods of calculation
- Chaturanga, ancestor
of the set of failures, invented one day by a Brahman of the name of Sessa, whose legend celebrates it
will enable us to start this important study.
The part of Chaturanga
was played four on a square chess-board of 8 boxes out of 8, with 8 parts, which
one advances according to points' obtained while launching the dice.
When the play was presented at the king of the
- “Good Sovereign, I would like that you as many corn
grains than it makes me give would be necessary some to fill the 64 boxes with my
chess-board: 1 grain for the first box, 2 for the second, and so on by putting
in each box twice more grains of corn than in the preceding one”.
The king, wounded in his pride by such a modest
request, was indignant somewhat but ensured the Brahman that it would have his
corn bag before the night.
The evening, the king enquit near his minister to know if this “insane of Sessa” had taken possession of its thin reward well. Hesitating, this one him iwudsrépondit
that the mathematicians attached at his court had not arrived yet at the end of
their operations. The king wanted that the problem is solved with its alarm
clock, but the order remained without effect the following day; courroucé, it congédia
calculators.
- “O Sovereign”, known as then one
of his advisers, “you were right well to return these inefficient operators. They used too old methods! They were still to deploy
the numerical possibilities their fingers and to use the successive columns of
an abacus. I let myself say that the calculators of the central province of the
kingdom employ since some generations already a quite higher method and much
faster than theirs. It is, appears it, most expeditious and easiest to retain.
And of the operations which would require of your mathematicians several
difficult working days would not represent for those of which I speak only one
very short lapse of time to you.
On these councils, one thus made come one from these
clever arithmeticians who, after having solved the problem in record time,
arose at the king to announce to him that it was hardly in his capacity to
provide the quantity of corn which had been required of him:
- “This one is well beyond knowledge and of the use
which we have of the numbers. In fact, for such a quantity, it would be necessary
you to store the corn in an attic of
The calculator then revealed with the sovereign the
characteristics of the revolutionary numeration of the scientists of its native
area:
- “The manner of representing the numbers which one
uses traditionally in your kingdom is well too complicated, because it is
encumbered of a whole panoply of distinguishing marks representing the units,
higher or equal to ten; it moreover is very limited, because largest of these
figures the hundred of thousand does not exceed; and it is completely
inoperative, no arithmetic operation not being possible by this means. The
system that we use in our province is, on the other hand, of a great simplicity
and an effectiveness without equal: by means of nine “signs” 1,2,3,4,5,6,7,8,9
(which represent the nine simple units but which have a different value
according to the position that they occupy in the writing of the numbers), and
of a tenth noted 0 (which does not mean “anything” and is used to mark the
units absent), it makes it possible to represent without difficulty any number,
so large is it. And it is precisely this simplicity which makes its superiority
as much as elegance and the facility that it gets for the practice of all the
operations of arithmetic”.
On these words, it taught with the king the principal
methods of calculation in question by operations and concludes:
-
“You can now yourself make sure, Ô Souverain,
that the quantity of grains required is exactly 18.446.744
“Definitely”, answered the king
extremely impressed, the play that this Brahman invented is as clever as its
request is subtle.
Such is the legend of Sessa,
which thus allots to Indian civilization the honor of this fundamental
realization that one calls modern numeration. Moreover, in spite
of the mythical character of the tale, this fact is perfectly authentic.
But we initially should measure the importance of this
numbering system written of which the use became today so frequent, if
familiar, that we ended up forgetting of them the depth and the true merits.
Such is the goal that this TPE assigns.
The “Arab numerals” appeared in
Why can one say that the graphic signs employed by Indian civilization
to indicate the units constituted the essential base of the “Western figures”
current?
Introduction
Our system of modern counting, used today
in the whole world, is based on the figures 0,1,2,3,4,5,6,7,8,9 which are known
under the name of “Arab numerals”. Practically all the dictionaries qualify
these signs as being originating in Moslem civilization. For example, the
Webster dictionary, of average size, gives this significance to the heading
“Arab numerals”: one of the numerical symbols 0,1,2,3,4,5,6,7,8,9.
But do these figures really come from the
The opinion of Professor Maulana Sayyad Suleman Nadvi of the Academy Shibli d' Azamgarh (Uttar Pradesh, Indian area) deserves our
attention for this reason. It writes: Arabic says clearly that they learned
the figures from 1 to 9 from the Hindus (Indian civilization); this is why the
latter call these figures the “figures Hindsa” and
their system of figures Hisab Hindi (at that time,
the Sindhi language was known like Hindi and Sindhis
like Hindis). Europeans then learned from Arabic this numbering system and thus
gave birth to the term from “Arab numerals”.
Also we will leave the preceding remark in this part to
bring the evidence of the Indian origin of our Western figures.
Incorrect
assumptions on the origin of the “Arab numerals”
In the field of the preconceived
idea, a tradition still long-lived (and even dominating) nowadays allots to the
Arabs the invention of our current system of numeration. But the figures known
as “Arab” surely did not have the Arabs for inventors. The historians indeed
acquired since several generations already the certainty, evidence with the
support, that this denomination comprised actually a serious historical error.
Moreover, it is to this end advisable to note that, curiously, no trace of this
tradition was detected in the writings of the Arabs themselves.
And in
fact, many Arab treaties relating to mathematics and with arithmetic reveal
that the arabo-Moslems authors always knew to recognize,
without the least complex, that it was about a discovery carried out by foreign
scientists with their own culture.
But for unsuitable that is the
assumption of an Arab origin of our figures, it is however not
incomprehensible. A historical error like this one, because it was spread on a
wide geographical surface (in Europe) and remained hung in the spirits during
centuries, until our days, finds obligatorily its true raison d'être some
share.
In fact, this theory of “Arab
numerals” has be conveyed only in country European, undoubtedly since time of
low Middle Ages, in particular by authors of works of arithmetic or of
mathematics, which, to be distinguished from the current of their time, had
wanted to fill what had seemed to them to constitute a vacuum, by formulating
arbitrary assumptions resting on preconceived ideas, and by thus delivering the
historical truth to the chances of their individual inspirations. As for the
cause even of the error, we include/understand it of as much better than we know
today than the figures in question arrived to Occident at the end of Xè century via the Arabs. And it is because the Arabs had
reached a level cultural and scientific comparatively higher than that of the
Western people than these figures had ended up being equipped with “the Arabic”
denomination.
In this extract of the mathematical
Institution published in 1636, Laurembergus
affirms as follows:
“These ordinary characters, cruel, survived, and today the almost
whole ground makes use of it. In all, there are nine of them: 1,2,3,4,5,6,7,8,9, to which figure 0 is added, in other words
the figure appearing “anything”, “no thing”, zero Arabic. Of aucuns think that they are the Arabs who are the first
inventors of these signs (whereas others prefer Phéniciens,
or Chaldéens); opinion which is certainly not foreign
with the truth. Because, just as the Arabs have one day be Masters of
almost all the ground, it is probable that they were also the propagators of
sciences.
This testimony shows well how, according to the same inspirations,
according to preconceived ideas' and in support of a very light argumentation,
the imagination of the European authors of the time appealed to allot to the
Arabs sometimes, sometimes in Phéniciens or Chaldéens the discovery of our modern figures,
civilizations which one abundantly proved that they were foreign there.
In addition, at the beginning of the century, of the historians of
sciences (G.R Kaye, N.Bubnov and B.Carra
de Vaux in particular, which had been made the keenest adversaries of the
thesis of the Indian origin of our current system) pled that we were indebted
of this numeration to the mathematicians of ancient
According to them, indeed, the system would have occurred in the neo-pythagorean mediums a little before the beginning of the
Christian era. Wearing of
Naturally, the funds of this assumption was
cancelled by the fact that no trace was detected to date of employment among
Greeks of the Antiquity of a standard system in the same way than ours. But
without to be stripped by the solidity of the against-arguments brought by the
reality of the things, these authors had clung at their pure sights of the
spirit and had been locked up there at the point to deploy all their
imagination to provide to their prejudices all that could constitute a pretence
of proof or confirmation.
Thus, in
But, the remainder, as pointed out it so precisely J-F.Montucla, “if these characters come from the Greek
letters, they curiously changed on the road. Indeed, it is only by truncating these letters and while turning over
them in a quite strange way, that one comes to end to make them resemble our
figures. Moreover, it acts here much less theirs forms that of this system
[positional] clever, which by means of ten characters only, expresses any
possible number. The Greeks had too much genius not to feel the merit of this
invention; and they would promptly have adopted it if it had occurred on their premises, or even if they had been informed of it only”.
Lastly, it is advisable to recall, them to once and for all eliminate,
the principal legends and theories, very contestable, which still circulate
about the origin of the figures known as “Arab”: these whimsical explanations
are described in additional the documents part: whimsical
explanations about the origin of the “Arab numerals” this TPE.
These theories are all the more doubtful as, to believe about it their holding, the shapes of our current figures would result each
time from the imagination of a isolated individual. An individual who would
have forged these signs of all parts so that the shape of each sign concerned
concealed the idea of the number represented following a process sometimes
resorting to a graphic notation founded on as many angles, of features or
points which the illustrated number comprises of units, sometimes with
geometrical representations like the triangle, the rectangle, the square or the
circle, from which one would have deduced the signs in question according to a
simple rule of a geometrical nature. These theories claim thus jointly to
provide a “explanation” giving our current figures like the fruit of a kind of
spontaneous generation allotting to them upon the departure a perfectly
rationalized form.
Such conjectures are in truth quite sterile, because none them can
provide explanation to the completely considerable variety of the written forms
that the nine figures took during centuries and in various areas of the world,
as we will see it later. Not considering as well as the ultimate shape of the
modern figures (used for printing works), those indeed take into account only
the result of a very long history and thus neglect all the turnings of a slow
evolution spread out over several millenia.
They are there thus explanations a posteriori, brought by imaginations
pseudo-scientists, taken with the traps of appearances and easy deductions.
Legend with reality: modern figures, a properly Indian invention
In fact, it is with very an other line of scientists and calculators,
the mathematicians and astronomers of the Indian civilization, which we owe the
fundamental discovery of these figures, due to the development by these same
scientists of a system of position - event not less important than the control
of fire, the development of agriculture, or than the invention of the wheel,
writing or steam engine -: scientists who had had the spirit resolutely turned
towards the applications and who had been animated by a kind of passion at the
same time for the great numbers and numerical calculation.
Many facts prove it and of innumerable testimonys
come from all the horizons confirm it.
Among testimonys in favour
of the Indian origin of modern numeration, one in particular finds, as of 976,
that of a monk of the name of Vigila, established in
the north of Spain, which, in its work, the Vigilanus
Codex, written:
“And the same in connection with the figures of
the arithmetic one. It should be known that the
Indians have an extremely subtle intelligence, and that the other concepts
yield the step with regard to the arithmetic one to them, liberal geometry and
other disciplines. It is what appears best in the nine figures by which they
indicate each degree of any level. Here the shape of these figures:
1,2,3,4,5,6,7,8,9
In the same way, during more than thousand years, the arabo-Moslems authors never ceased proclaiming, in a
remarkable spirit of opening which makes them honor, that the discovery of
figures stripped of any visual intuition, integrated in a decimal notation of
position, was due to the Indians.
Thus, as of 810, Abu Ja'
far Muhammad ibn Musa Al Khuwarizmi,
Arab scientist famous for his work of popularization (see part 3), indicates,
in its work entitled “Treated of the addition and the subtraction according to
the calculation of the Indians”:
“… we
decided to expose the manner of calculating Indians using the nine characters
and of showing how, thanks to their simplicity and their concision, these
characters can express all the numbers”.
Al Khuwarizmi then explains, in detail, the
principle of the decimal notation of position, by announcing the Indian origin
of the nine digits and “the tenth figure in the shape of circle” (the zero), of
which it recommends “not to neglect the use in order not to confuse the
positions”.
Preceding testimonys all are thus unanimous to
proclaim that our written numeration current A indeed be the product of the
creative dashes of Indian civilization. In any rigour
however the difficulty of the convincing value of these various testimonys arises: the fact describes in those is often
certified by a former statement made by an eyewitness. However, a considerable
importance must be attached to these various testimonys,
because “the Hindu event” was evoked several times during more than thousand
years. Indeed, these two authors were not the only ones to describe the Indian
origin of our Western figures (see additional documents: testimonys).
But, for solid that they are, these testimonys
can never constitute for the “historical truth” which one knows but of simple
confirmations. It is thus advisable to be harnessed with a thorough graphic
study of these figures and their evolution in time, in order to establish a
direct bond between the figures which we currently use in
For that, we will show that Indian civilization arrived in all autonomy
to basic figures stripped of any visual intuition, while establishing that the
graphics attached to the Indian figures as of the high time currently preceded
not only all the varieties of use in India, Central Asia and Southeast Asia,
but also the respective shapes of the figures of the Eastern Arabs and Western
Arabic, as well as the C-W communication of our current figures and their
various European predecessors of the same kind.
Numerical notations of Indian origin
The goal of this part being to establish the indianity
of the origin of our current figures, we hereafter will review the used
numerical notations to
The oldest known writing of the Indian sub-continent is that which appears on
seals and plates of the age of
But this writing not being deciphered yet, the corresponding language remains
unknown; one cannot thus fill the broad ditch which separates these
inscriptions from the first texts known in writing and language properly Indian,
if as well is as a filiation existed between the two
systems.
In fact, the history of the properly Indian writings starts only with the
inscriptions of Ashoka, third emperor of the dynasty
of Maurya of Magadha, which had reigned on
The kharoshthî drift directly of the old Aramean alphabet, and is written like him from right to
left. This is why one gives him also to the name of writing “araméo-Indian”. Probably introduced with the IV è
century before our era, it will remain of use in the North-West of India until
the end of the IV è century after J-C.
As for the writing brâhmî, she was written
from left to right and used to note the sounds of the Sanskrit, language spread
through all the Indian regions and considered as the “language of the gods” (samskrita means “complete, perfect, final”).
The origin of this writing was not elucidated yet. One wanted to make it derive
from the writing kharoshthî, but the provided
explanation was hardly convincing. It is known however that the brahmî drift of old the writing alphabetical of the
Western Semitic world, undoubtedly via an other araméenne variety which one did not find the specimens.
Always it is that as of second half of the first millenium
before our era, India is already largely opened with the foreign influences, of
the contacts being established for a long time with the Persians and the
tradesmen of origin araméenne which took the roads
active of Syria and Mésopotamie to the valley of
Indus.
The appearance of the brâhmî and its numerical
notation is however probably former at the time of Ashoka,
where it was already perfectly elaborate and widespread through the various
regions of the Indian sub-continent.
In all the cases, it is well this writing which will survive all the others,
consequently becoming the single source of all the writings which will develop
thereafter in
After the edicts of Ashoka, the numerical notation brâhmî will appear, in a slightly modified form, in
the contemporary inscriptions of the dynasty of Shunga
(which will règnera from - approximately 185 with -
75 on Magadha, in current Bihâr, in the south of the
course of Gange), then in those of the dynasty of Kanva (which will succeed the preceding one from -
approximately 730 - 30)
It then will be seen, in a form even more advanced:
- in the inscriptions of the
time of the dynasty of Shaka (Scythians which will
reign, of the II è front century J-C in Ier
century a. J-C, on the
- on the currencies struck by
the sovereigns of the dynasty of Shaka origin who
will reign of the II è to the IV è century of our era on Mahârâshtra (by taking the name of “Satraps”)
It will evolve/move still a little more in the writings of the dynasty
of Ândhra or Shâtakarni
(which will reign during the first two centuries of our era on the North-West
of Dekkan).
The system will appear then, in a form more evolved/moved even, in the
inscriptions of the time of the sovereigns Kushâna
(of which the reign will be spread out of Ier to the
III è century a. J-C and who, initially fixed in Gandhâra
and in Transoxiane, will launch out in the conquest
of India of the North-West).
And thus at the end of several more or less significant successive
modifications, the brâhmî will lead finally to
the development of various types of writings (numerical) very definitely
individualized, of which in particular the writing of style nâgarî
(or “town” writing, whose splendid later regularity made him take the name of devanâgarî or “nâgarî of
the gods”) which acquired an extreme importance thereafter, while becoming not
the principal writing of the Sanskrit, but also that of the Hindi, the great
language of current central India.
Types from which the principal groups of following figures will be
currently constituted of use in the Indian world (flow chart 24.28):
1. The group of the notations numerical of central and septentrional
a. notations derived from the writing nâgari:
- figures mahârâshtrî and its derivatives: figures marâthî….
- figures kutilâ and its derivatives: figures bengâlî, oriyâ, gujarâti….
B. notations derived from the writing shâradâ:
- figures sindhî,
panjâbî….
C. notations of Nepâl:
- figures siddham
and its derivatives: modern figures nepâlî….
D. notations of the Tibetan type:
- figures Tibetans
(derivatives of the figures siddham) and its
derivatives:
Mongolian figures…
E. notations of Chinese Turkestan (derivative of
the figures siddham):
- figures agnéens,
khotanais….
2. The group of the notations of southernmost
a. figures will kannara
B. figures Tamil
C. figures malayâlam
D.
figures sinhala (singhalais)
3. The group of the “Eastern” notations known as, resulting from the
notation known as “pâlî”, deriving
itself from the same source as the bhattiprolu:
a. old figures Khmer
B. figures cham
C. Malayan old figures
D. figures kawi: old Javanese
and Balinese old man
E. modern figures thai-Khmer
F. Burmese figures
(see chart 24.53)
additional cf documents
The apparently considerable differences that the writings of these
various groups will present later on will hold in fact, is with the specific
character of the languages and traditions to which they will have been adapted,
is still with the regional practices scribales and
the nature of the materials scripturaires employed.
Indeed, in
Thus, being given the number of languages which rose from the brâhmî, one still employs in
But naturally, this diversity does not go back to today.
It is besides that to which had testified about 1030 the Moslem astronomer to
origin Persian Al Biruni in his Kitab
fi tahqiq I my Li' L hind,
work constituting a talk on India, which here will interest us in the
continuation of our reflexion on the Indian origin of
the Western modern figures. In consequence of a stay of almost thirty years in
“The Indians do not have the use to assign with their letters an unspecified
employment in calculation, as soft assign some with our letters by classifying
them according to the order of their numerical values.
“And just as the figures of the letters [of their writing]
are different in [various areas of] their country, in the same way also the
signs of calculation [vary].
[…]
“What us [Arabs] employ [in fact of figures] is selected among what
there is best [and of more regular] at the Indians.
“But it does not matter the forms, provided that one knows the
significances which they contain”
However among the figures which one employed formerly and which one
employs still today most usually in the various regions of India, most regular
are precisely those of the nâgarî kind (about
which one spoke previously), that one calls also figures devenâgarî,
of the name of the superb writing into which they are integrated (the word
Sanskrit means “writing of the gods literally”).
It is remainder with these figures which allusion Al Biruni
made (which had a perfect command of the language and the writing of the Sanskrit),
by saying that the Arabs, by borrowing from the Indians their decimal notation
of position, had taken to them, in fact of notation for the nine units, “what
there was best and of more regular on their premises”. Thus the confirmation
comes which the figures which we use nowadays reached us well from
But, definitively to give a credible range in this
established fact, it is advisable to bring a concrete interpretation of what
was established by the scientists and was specified a little higher in this
part, i.e. the origin brâhmî of the figures nâgarî. An analysis of the graphic evolution
of the numerical notations which allowed the passage of one writing the other
will help us there.
Figures of the
original notation brâhmî
Let us return then to
the notation brâhmî original itself. We saw that this notation appears for the
first time in the middle of IIIè century before our
era in the edicts in language ardha-mâgadhî and
writing brâhmî, that the Ashoka emperor had made
engrave on rocks, polished sandstone columns and temples dug in the rock, in
various regions of its empire.
But the numerical notation contained in these edicts is unfortunately
very fragmentary, delivering for the series of the nine units only the
representations of numbers 1,2,4 and 6. (figure 24.27) One can already recognize on the document
our 6 current.
Figures of the
intermediate notations
The same system
appearing in a more significant way in the documents of the following times,
which follows thus will enable us to have an idea much
more precise of it.
One
sees the corresponding figures indeed appearing at the beginning of the time of
the dynasty of Shunga of Magadha (- IIè century) in Buddhist inscriptions which decorate the
walls of the caves of Nânâ Ghât:
(figure 24.30) One can bring closer
figures 1,2,4 and 6 here this writing with those of the notation nâgarî. Thus, one finds in these two series of figures the
single feature to indicate the 1, the two features to account for the 2. One
also recognizes in the two notations the reason for cross indicating figure 4,
and the cursive form characteristic of figure 6.
One sees there already, in
addition, the prefiguration of our figures 4,6,7 and 9.<! --[endif]--> The same series reappears a little later,
but in a form much more complete, in Ier or IIè century of our era, in the inscriptions of the Buddhist
caves of Nâsik: (figure 24.31)
As previously, the graphic bonds with the notation nâgarî
are here also obvious to note. One
recognizes there in addition the prototypes of our figures 4,5,6,7,8
and 9. Let us notice that
one finds this system in an increasingly diversified form, in particular in the
inscriptions of Mathurâ, like in the inscriptions of
the dynasties Kushâna and Ândhra,
the currencies of the Western Satraps, the inscriptions of the dynasty of Pallava.
Being the numeral series
from 1 to 9 which derive thus from the notation brâhmî
and constitute consequently the intermediary with the later derived series
including/understanding the style nâgarî, these signs
are called the “figures of the intermediate notations”.
While being spread in the
various regions of India and the neighbouring areas,
these intermediate notations, like the letters of the corresponding writings,
have sudden during centuries of the more or less significant graphic
modifications, in the final analysis to acquire extremely varied cursive forms,
appropriate each one to a regional style.
The origin of the
notation nâgarî: the writing gupta
One of the first
individualized notations was the gupta, used at the
time of the dynasty of the same name (of which the sovereigns reigned on all
the valley of Gange and its affluents
from approximately +240 to +535): (figure 24.38) It is
easy to see, graphically, that this notation presents many similarities with
the style of the intermediate notations; there is then no doubt as for its
remote origin brâhmî.
And it is well this notation
which was the source of the writing nâgarî.
The development of
the notation nâgarî
While being
refined, the writing gupta indeed gave birth as of VIIè century of our era to the writing of the style nâgarî.
And like the
numerical notation followed a parallel evolution, the figures of the gupta kind also generated the figures of the nâgarî type, whose formal evolution led thereafter to the
modern figures of the same name: (figure 24.39) These are well
these forms that the Arabs borrowed when they adopted Indian numeration. One
recognizes there besides without difficulty of the forms if not identical, at
least similar to our current figures 1,2,3,4,6,7,9 and 0.
(Let us notice that
this notation includes/understands the use of the zero, contrary to the
preceding ones: it is thus already integrated into a decimal system of
position).
We thus indeed
showed here that the notation borrowed by the Arabs derives undoubtedly from
the Indian original writing properly Indian: the brâhmî.
The problem of the
origin of the figures brâhmî
It then returns to
us now to elucidate the delicate problem of the origin of the first nine digits
brâhmî themselves.
This notation, as we saw, a long time
preserved for the first three units a ideographic
representation consisting in reproducing horizontal features as many the value
of the figure.
On the other hand, as
of their appearance, the figures from 4 to 9 were signs independent, detached
of any sensory intuition, not seeking to visually evoke the numbers represented
(even before the introduction of the zero and thus of the numeration of
position).
Many assumptions
on the origin of the first figures brâhmî, according
to which these signs could derive from the numeration used by the old age of
Other assumptions on the
origin of the first nine digits brâhmî were put
forth, in particular on the possibility of one loan to the alphabet brâhmî, or even to the Egyptians, but they were hardly
convincing.
The origin of the
first nine Indian digits
Another
assumption appears on the other hand much more plausible, and that even in the
absence of any documentation.
This assumption rests
above all, indeed, on the fact that civilizations which were subjected to the
same needs according to same initial conditions', social, psychological,
intellectual and material, generally independently, borrowed from/to each other
the same ways to arrive to results if not identical, at least similar. However, it is precisely what explains the
reason of the existence of certain of the same figures invoices and often of
the same numerical value as the figures brâhmî, than
one finds attested in other civilizations and whose date generally goes back to
several centuries before the time of the Ashoka
emperor. (figure 24.57)
By consulting figures 24.57 and 24.27 to
24.31, one will thus recognize nonIndian signs
completely similar to the various alternatives of figures 1,2
and 3 of Indian civilization, just as the obvious analogy between the 5 nabatéen or palmyrénien and old
the 5 Indian, as well as the similarity which present to the hieratic or
demotic figures 7 and 9 Egyptians with their respective Indian
correspondents. In fact, these formal
analogies are explained, not by the not very probable thesis of a possible
transmission of the system by one of civilizations concerned, but rather with
universal constants released by the fundamental rules of the history of
paleography. They come owing to the fact that civilizations in question wrote
on supports similar to those of the former Indians and used tracer tools of the
same type, for example the calame (kind of reed which
one soaked the point crushed in a dye), which was used to them to write on
papyrus or parchment.
However, one knows up to which
point the nature of this instrument influenced the handwritten writing of all
these people.
Thus the superposition of two
or three horizontal features, joined together initially in only one sign by a
binding, gave birth, at the ones as at the others, with of the same graphics
invoices than the 2 and the 3 Indians, whose paleographic alternatives
thereafter diversified considerably according to the times, the areas and
practices' of the scribes (figure
24.58), for finally leading to the prototypes of the signs which we use
nowadays.
This explanation supposes of
course that the consecutive features constituting the old Indian ideographic
notation of the first three numbers were laid out horizontally. It is in any
case what the posterior inscriptions brâhmî in IIIè century reveal before J-C, like those of the time of
the dynasty of Gupta (+IIIè/+IVè century); this
figurative représention using “laid down” features,
visually inspired, will persist even by places until VIIIè
century after J-C. (figures 24.30, 24.31 and 24.38).
And yet, if one examines the
edicts “brâhmî” of the emperor Ashoka
(260 approximately front J-C), one notices that, from one end to another of the
empire of Maurya, numbers 1,2 and 3 were represented
not by superimposed horizontal features, but by one, two or three vertical bars
(figure 24.27).
This change of orientation was
it due to reasons of an aesthetic nature? It is as not very probable as the
explanation which would give for reason the convenience of this new notation.
Because to repeat a feature, one, two or even three times, that it is
vertically or horizontally, does not have anything esthetics and practically
raises of the same gesture, of which only the practice can establish the
difference.
In fact, this phenomenon has
another probable explanation. The Indians used for a long time in their texts Sanskrits in worms and prose a punctuation mark (called danda) in the shape of small vertical feature (½), to mark
the end of worms or part of sentence, and which they doubled (½ ½) to indicate the end of a sentence, a verse or a stanza.
However, the danda having constituted an innovation
of IIè a. J-C, one understands that the vertical
notations of the first three units had to lie down as from this time to avoid
any confusion. However, it is only one simple conjecture without proof nor confirmation.
Another question: why the
Indians did a long time preserve at the first three units such a ideographic notation, whereas, as of their appearance on
the preceding documents, the figures from 4 to 9 are already signs graphically
advanced, corresponding to figures independent, detached of any visual
intuition?
In fact, this ambiguity is
explained simply: whereas it was necessary to proceed to a radical
transformation of the groupings from 4 to 9 features to avoid a tiresome
writing milks by feature, it was not inevitably useful, indeed, to operate any
modification on the assemblies of the units lower or equal to 4; and that, not
only because of the fast character of one notation using features up to three
units, but more especially because the eye always manages to easily
distinguish, without counting, all the units aligned until the fourth, row
beyond whose the artifice of counting becomes essential. Let us note that the
Chinese and the Egyptians of Antiquity found themselves in a similar situation.
But then, which is the idea
which governed the formation of the six other digits brâhmî?
The preceding considerations on the universality of the rules of the evolution
of paleography in all the cultures let think that these graphic signs were
probably not create artificially for the needs for the cause, with a purely
conventional aim, but rather were the fruit of a graphic advance on the basis
of prototypes made up of primitive groupings of as many features representing
the unit. And as the bars representing the numbers from 1 to 3 were vertical
before even being horizontal, one can thus suppose rightly that the first nine
figures brâhmî constituted the vestiges of an old
indigenous numerical notation, undoubtedly older than the brâhmi
itself, where the nine units were represented per as many vertical features
necessary (see figure 24.59).
To give a need for fast
notation and the required to save time, these groupings of features
evolved/moved graphically taking into account the possibilities and of the
requirements of materials of writing put at contribution in India during
centuries, and also of the constraints even of the tracer tool (calame). These prototypes of figures became complicated
little by little by the use of many bindings (figure 24.60), to finally undergo a deep
modification of layout not having more anything to see with the initial forms,
outcome with signs distinct detached from any sensory intuition: the figures brâhmî the first three centuries before our era. Such is
the most plausible explanation which one can give of the origin of the first
nine Indian digits. (see figure 24.61 to 24.69) to as much say under
these conditions that the numerical notation brâhmî
indigenous and was deprived of any foreign influence. In other words, according
to any probability, the nine Indian figures were born well in
The Arabs thus
borrowed a typically Indian numeration well before transmitting it to Western
civilization.
The ultimate
evolution of the Indian figures Aujourd'hui, our
modern figures 1,2,3,4,5,6,7,8,9,0 are widespread everywhere in the world and
thus constitute a kind of universal language being able to be
included/understood as well by an Indian, an Arab, a Burmese, Kampuchean, a
Korean, an Chinese or a Japanese as by Australian, an European, an American or
an African.
This form is however not only in which is
expressed the current decimal notation. Particular C-Ws communication
representing the same numbers coexist still indeed beside this series in
a certain number of Eastern countries. From the
the graphic evolution
of the Indian figures in the Islamic countries of the East
This divergence is in fact due to the use which one makes the Arabs of
the East of Indian numeration, in particular via their scribes.
When this
numeration arrived to the Arabs, the nine Indian figures, at the beginning,
purely and were simply recopied. In the middle of IXè
century, the figures of the Eastern Arabs still resembled their prototype
Indians of the style nâgarî of the same time:
But once passed between the hands of the scribes arabo-Moslems,
the Indian figures underwent relatively important graphic modifications, moving
away then little by little from their initial prototypes. In other words, while coming to fit among
the elements of this writing and while approaching the various corresponding graphic
styles, the numerical notation of Indian origin underwent variations of layout
to lead finally to apparently original series.
But this stylization of the
Indian figures does not explain all. If one attentively examines the Arab
manuscripts of the first centuries of Islam, one notes indeed that a change of
orientation took place at the time on the Indian notation.
And thus in the Moslem
countries of the
With what this change of orientation was it due? With
practical, primarily material reasons. Thus, during the first centuries of the
“Hégire”, the Arab scribes of the East were indeed
accustomed to tracing the characters of their cursive, not from right to left
as authorizes it the Arab writing, but from top to bottom, the lines following
one another from left to right.
And for reading,
they did not have any more that to turn over their manuscripts of 90°, in the
direction of the needles of a watch, so that the lines were laid out normally
and that the reading was made line well towards the left.
This way of
proceeding drew in fact its origin of considerations primarily related to the
handwritten writing on sheets from papyrus.
As for the zero, it was initially represented by a “small circle similar
to the letter O”, as Al Khuwarizmi said it which thus
referred to the Arab letter ha, whose form is precisely that of a small round.
But with long, this
round became so small that it was finally reduced at a simple point. And it is under this stylized
and a little modified C-W communication that the nine figures of Indian origin
were spread through the Eastern provinces of the Moslem world, after having
been fixed in a series which was not to know during centuries but completely
unimportant modifications any more, carrying mainly on the shape of figures 5
and 0 (figure 25.3).
And it is what the
Arabs always called under the name of figures Hindi (“Indian figures”).
Figures known as “ghubar” of the Western Arabs
But the
preceding figures are not completely at the origin of our “Arab” figures. We
hold the current figures of the Arabs, it is true, but Western Arabs (those
which populated
These Arab numerals
Western, known as “ghubar”, have a C-W communication
of appearance completely different from the figures Hindi of the Eastern
provinces of the countries of Islam (figure 25.5).
The differences
between these two types of notation are actually due only to the practices of
the scribes and the copyists of each area concerned.
But it is especially
the history even styles of the Arab writing which will enable us to better
seize the phenomenon.
As of the appearance
of Islam, this writing evolved to two quite distinct types:
- a concise cursive style,
drifting of that of the pre-Islamic inscriptions, from where left the writing
known as “coufic”, style of a monumental penmanship,
characterized by a horizontal base line on which rigid and angular C-Ws
communication come to be established vertically. Being useful for the
inscriptions on stone, wood or metal, it was used in the legal and religious
texts.
- a style even more cursive,
directly resulting from the first Arab handwritten writings, which gave rise to
the writing naskhi, “the writing of the copyists” whose
derivatives are most widespread at the present time, which replaced the “coufic” writing little by little. This style, employed in
the current texts on papyrus or parchment, is characterized by flexible and
round forms.
However, if one refers now to the layout of the Eastern and Western Arab
numerals (figures 25.3
and 25.5), one notes that cursive the figures known as Hindi are
graphically of a form much rounder than those of the
On the other hand,
the figures “ghubar” present indisputably, even if
they are cursive signs, a more angular, stiffer and more rigid character. Arab
the writing known as “Maghrebian” was thus at the
bottom only one “coufic” manuscript, the faithful Maghrebians and Andalusians being
always remained, announce it, with the old traditions of Islam (in writing
lying thanks to the old coufic writing).
Always it is
that, in spite of the variations existing between the two graphic series, the
Indian influence appears there clearly.
Thus, while
proceeding to a comparison, even summary, with the Indian figures of nâgarî type, one of course finds in the series “ghubar” the 1 Indian, but also the 2, the 3, the 4 (with
for Arabic, a light modification of orientation compared to its precursor), the
6, the 7, the 9 and the 0, as well as the 5 and the 8 (figure 25.7).
From a paleographic
point of view, it there thus no difference between the figures “Hindi” of the
Middle East and figures “ghubar” of the Maghreb, two
series coming from the same source; the Indian origin of those as of these is
thus from now on obvious.
<! --[endif]-->Et it is well under the style “ghubar” of the Western Arabs that the Indian figures, on
the basis of Spain, will reach the Christian people of Western Europe, before
even taking the form of the figures which we currently know…
Note
In order to recapitulate
all the evolution of the Indian figures, of the first figures brâhmî to nonmodern figures, a
flow chart is present in the section additional documents: evolution of the
Indian figures.
You can to also refer
you on figures 24.61
to 24.69.
Additional documents
You can find the documents here (tables of graphic analysis, charts, flow
charts, testimonys…) and the written passages of
which references indicated throughout the TPE.
Written
passages
- Whimsical
Explanations about the origin of the “Arab numerals” (cf
page of the TPE)
- Incorrect Assumptions on the origin of the figures brâhmî
(cf page of the TPE)
Documents:
- Graphic Analysis
- Charts: chart of India…
- Flow charts: evolution of the Indian figures…
- Testimonys (in favour of
the Indian origin of our figures)
- Classification of the zeros of the history
You will find at the end of this section a table of
agreement of principal numerations which have existed throughout the world for
more than 5000 years, in order to better replace the Indian numeral system in
its at the same time historical and mathematical context.
....................................................................
Whimsical
explanations about the origin of the “Arab numerals”
According to a
popular tradition, still tough in Egypt and North Africa, the “Arab” figures
would be the invention of a glazier geometrician
originating in the Maghreb, which would have imagined to give to the
nine significant figures an evocative form depending on the number of the
angles contained in the drawing of each one of them: an angle for the graphics
of figure 1, two angles for figure 2, three angles for the 3, and so on:
A French author of the end of XIXè century, P.Voizot, also gives like “probable” the assumption of the
formation of these numerary figures by assemblies of
features:
Another assumption of the same kind was put forth in 1642 by the Italian Jesuit
Mario Bettini, then taken again in 1651 by the German
George Philip Harsdörffer. The explanation
relates to this time the number of points which would have initially been used
as ideographic representation with the nine first order units decimal, and
which one would have then connected between them to form the nine signs that
one knows:
Another similar theory was emitted by Weidler in
1737, according to which the invention of the modern figures would have been
the result of a partition of the figure formed by a circle and two of its
diameters. Thus, the vertical diameter would have given the form of the 1; the
same diameter, supplemented on both sides by two arcs of circle opposed, that
of figure 2; a half-circle provided with the median horizontal ray that of
figure 3; and so on until zero the, resulting one according to the theory, of
the figure formed by the entire circle:
In addition, en1778, the Spaniard Carlos Maur
establishes theories according to which the signs in question would have drawn
their form, that is to say of a particular provision of stones being used to
count (see figure
Let us announce finally an eccentric explanation given by Jacob Leupold in 1727, known under the name of legend of the ring
of Solomon, according to whom the figures which we currently use would have
been formed successively starting from this ring registering a square and its
diagonals:
Incorrect
assumptions on the origin of the figures brâhmî<>
Several
assumptions were put forth on this subject:
- The assumption
of an origin indusienne, according to which, the
Indian writings deriving from that of the old age of Indus (- XXVè/-XVIIIè century), the Indian figures brâhmî would originate in the notation indusienne. The objection with this thesis relates to the
alleged bond between the Indian letters and the characters picto-ideographic
of the writing proto-Indian. It is established indeed perfectly that the
writing brâhmî drift in fact of the old
alphabets of the Western Semitic world via a araméenne variety. However a broad hiatus of more than two
thousand years separates the documents from this civilization from the first
texts in writing brâhmî and properly Indian
language. And like the writing indusienne was not
deciphered yet, one is unaware of how to fill this ditch. As much to say that
this assumption does not rest on no base, since it is known if a filiation existed or not between the figures indusiens and Indian figures themselves (more especially as
the documentation delivered by the age of Indus is very lacunar):
it is thus to reject.
- The assumption
of one loan to numeration “araméo-Indian”, according
to whom, since the Indian letters derive from the Aramean
alphabet, one could suppose that the figures brâhmî
kids of the one of the old numerical notations of the Western Semitic world.
But this assumption is cancelled by differences too much considerable between
the two writings. Thus, the notation araméo-Indian is
done from right to left, whereas the brâhmî is
written from left to right. In addition, in the system kharoshtî,
the numbers from 4 to 9 are generally illustrated by repetitions of as many
vertical bars representing the unit, while the system brâhmî
gives them independent signs stripped of any direct visual intuition. This
assumption can thus hardly be retained.
- The assumption of one loan to the alphabet
kharoshtî, according
to which the figures
brâhmî
would have rather constituted a loan of the letters of
the alphabet
kharoshtî, taken as initial of the
names corresponding Sanskrits (figure 24.55). However, the signs given for the supposed phonetic
values are very resembling (not to say identical) known letters carrying of
other values, from where an objection. Moreover, it is extremely probable that
the
brâhmî already
had a long history before Ashoka, since at the time
it was already not only perfectly elaborate, but more especially widespread
through all the regions of the Indian sub-continent. And if the
kharoshtî, introduced
into the North-West of India at the time of Alexandre
the Large one (towards - 326) did not penetrate ahead than the areas of Panjâb and of Gândharâ (the
extreme North-West of India), they is most probably because it had run up
against the strong competition of a preexistent properly Indian writing, namely
the
brâhmî itself,
which one can thus make go up the use in Vè century
approximately before our era. The assumption of an influence of the
kharoshtî on
the formation of the writing and numeration
brâhmî thus
appears improbable.
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