THE SPREAD OF THE NUMERALS IN EUROPE
Of all the medieval writers, probably the one most influential in
introducing the new numerals to the scholars of Europe was Leonardo
Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy
mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]
The traveler of to-day may cross the Via Fibonacci on his way to the Campo
Santo, and there he may see at the end of the long corridor, across the
quadrangle, the statue of Leonardo in scholars garb. Few towns have honored
a mathematician more, and few mathematicians have so distinctly honored
their birthplace. Leonardo was born in the golden age of this city, the
period of its commercial, religious, and intellectual prosperity.[517]
{129} Situated practically at the mouth of the Arno, Pisa formed with Genoa
and Venice the trio of the greatest commercial centers of Italy at the
opening of the thirteenth century. Even before Venice had captured the
Levantine trade, Pisa had close relations with the East. An old Latin
chronicle relates that in 1005 "Pisa was captured by the Saracens," that in
the following year "the Pisans overthrew the Saracens at Reggio," and that
in 1012 "the Saracens came to Pisa and destroyed it." The city soon
recovered, however, sending no fewer than a hundred and twenty ships to
Syria in 1099,[518] founding a merchant colony in Constantinople a few
years later,[519] and meanwhile carrying on an interurban warfare in Italy
that seemed to stimulate it to great activity.[520] A writer of 1114 tells
us that at that time there were many heathen people--Turks, Libyans,
Parthians, and Chaldeans--to be found in Pisa. It was in the midst of such
wars, in a cosmopolitan and commercial town, in a center where literary
work was not appreciated,[521] that the genius of Leonardo appears as one
of the surprises of history, warning us again that "we should draw no
horoscope; that we should expect little, for what we expect will not come
to pass."[522]
Leonardo's father was one William,[523] and he had a brother named
Bonaccingus,[524] but nothing further is {130} known of his family. As to
Fibonacci, most writers[525] have assumed that his father's name was
Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] believe
that the name, even in the Latin form of _filius Bonaccii_ as used in
Leonardo's work, was simply a general one, like our Johnson or Bronson
(Brown's son); and the only contemporary evidence that we have bears out
this view. As to the name Bigollo, used by Leonardo, some have thought it a
self-assumed one meaning blockhead, a term that had been applied to him by
the commercial world or possibly by the university circle, and taken by him
that he might prove what a blockhead could do. Milanesi,[528] however, has
shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a
traveler, and was naturally assumed by one who had studied, as Leonardo
had, in foreign lands.
Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529]
the ancient Saldae on the coast of Barbary,[530] a royal capital under the
Vandals and again, a century before Leonardo, under the Beni Hammad. It had
one of the best harbors on the coast, sheltered as it is by Mt. Lalla
Guraia,[531] and at the close of the twelfth century it was a center of
African commerce. It was here that Leonardo was taken as a child, and here
he went to school to a Moorish master. When he reached the years of young
manhood he started on a tour of the Mediterranean Sea, and visited Egypt,
Syria, Greece, Sicily, and Provence, meeting with scholars as well as with
{131} merchants, and imbibing a knowledge of the various systems of numbers
in use in the centers of trade. All these systems, however, he says he
counted almost as errors compared with that of the Hindus.[532] Returning
to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in
1228.[534] In this work the numerals are explained and are used in the
usual computations of business. Such a treatise was not destined to be
popular, however, because it was too advanced for the mercantile class, and
too novel for the conservative university circles. Indeed, at this time
mathematics had only slight place in the newly established universities, as
witness the oldest known statute of the Sorbonne at Paris, dated 1215,
where the subject is referred to only in an incidental way.[535] The period
was one of great commercial activity, and on this very {132} account such a
book would attract even less attention than usual.[536]
It would now be thought that the western world would at once adopt the new
numerals which Leonardo had made known, and which were so much superior to
anything that had been in use in Christian Europe. The antagonism of the
universities would avail but little, it would seem, against such an
improvement. It must be remembered, however, that there was great
difficulty in spreading knowledge at this time, some two hundred and fifty
years before printing was invented. "Popes and princes and even great
religious institutions possessed far fewer books than many farmers of the
present age. The library belonging to the Cathedral Church of San Martino
at Lucca in the ninth century contained only nineteen volumes of
abridgments from ecclesiastical commentaries."[537] Indeed, it was not
until the early part of the fifteenth century that Palla degli Strozzi took
steps to carry out the project that had been in the mind of Petrarch, the
founding of a public library. It was largely by word of mouth, therefore,
that this early knowledge had to be transmitted. Fortunately the presence
of foreign students in Italy at this time made this transmission feasible.
(If human nature was the same then as now, it is not impossible that the
very opposition of the faculties to the works of Leonardo led the students
to investigate {133} them the more zealously.) At Vicenza in 1209, for
example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and
Spaniards, not to speak of representatives of divers towns of Italy; and
what was true there was also true of other intellectual centers. The
knowledge could not fail to spread, therefore, and as a matter of fact we
find numerous bits of evidence that this was the case. Although the bankers
of Florence were forbidden to use these numerals in 1299, and the statutes
of the university of Padua required stationers to keep the price lists of
books "non per cifras, sed per literas claros,"[538] the numerals really
made much headway from about 1275 on.
It was, however, rather exceptional for the common people of Germany to use
the Arabic numerals before the sixteenth century, a good witness to this
fact being the popular almanacs. Calendars of 1457-1496[539] have generally
the Roman numerals, while Koebel's calendar of 1518 gives the Arabic forms
as subordinate to the Roman. In the register of the Kreuzschule at Dresden
the Roman forms were used even until 1539.
While not minimizing the importance of the scientific work of Leonardo of
Pisa, we may note that the more popular treatises by Alexander de Villa Dei
(c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much
more widely used, and doubtless contributed more to the spread of the
numerals among the common people.
{134}
The _Carmen de Algorismo_[540] of Alexander de Villa Dei was written in
verse, as indeed were many other textbooks of that time. That it was widely
used is evidenced by the large number of manuscripts[541] extant in
European libraries. Sacrobosco's _Algorismus_,[542] in which some lines
from the Carmen are quoted, enjoyed a wide popularity as a textbook for
university instruction.[543] The work was evidently written with this end
in view, as numerous commentaries by university lecturers are found.
Probably the most widely used of these was that of Petrus de Dacia[544]
written in 1291. These works throw an interesting light upon the method of
instruction in mathematics in use in the universities from the thirteenth
even to the sixteenth century. Evidently the text was first read and copied
by students.[545] Following this came line by line an exposition of the
text, such as is given in Petrus de Dacia's commentary.
Sacrobosco's work is of interest also because it was probably due to the
extended use of this work that the {135} term _Arabic numerals_ became
common. In two places there is mention of the inventors of this system. In
the introduction it is stated that this science of reckoning was due to a
philosopher named Algus, whence the name _algorismus_,[546] and in the
section on numeration reference is made to the Arabs as the inventors of
this science.[547] While some of the commentators, Petrus de Dacia[548]
among them, knew of the Hindu origin, most of them undoubtedly took the
text as it stood; and so the Arabs were credited with the invention of the
system.
The first definite trace that we have of an algorism in the French language
is found in a manuscript written about 1275.[549] This interesting leaf,
for the part on algorism consists of a single folio, was noticed by the
Abbe Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then
seems to have been lost in the multitude of Paris manuscripts; for although
Chasles[552] relates his vain search for it, it was not rediscovered until
1882. In that year M. Ch. Henry found it, and to his care we owe our
knowledge of the interesting manuscript. The work is anonymous and is
devoted almost entirely to geometry, only {136} two pages (one folio)
relating to arithmetic. In these the forms of the numerals are given, and a
very brief statement as to the operations, it being evident that the writer
himself had only the slightest understanding of the subject.
Once the new system was known in France, even thus superficially, it would
be passed across the Channel to England. Higden,[553] writing soon after
the opening of the fourteenth century, speaks of the French influence at
that time and for some generations preceding:[554] "For two hundred years
children in scole, agenst the usage and manir of all other nations beeth
compelled for to leave hire own language, and for to construe hir lessons
and hire thynges in Frensche.... Gentilmen children beeth taught to speke
Frensche from the tyme that they bith rokked in hir cradell; and
uplondissche men will likne himself to gentylmen, and fondeth with greet
besynesse for to speke Frensche."
The question is often asked, why did not these new numerals attract more
immediate attention? Why did they have to wait until the sixteenth century
to be generally used in business and in the schools? In reply it may be
said that in their elementary work the schools always wait upon the demands
of trade. That work which pretends to touch the life of the people must
come reasonably near doing so. Now the computations of business until about
1500 did not demand the new figures, for two reasons: First, cheap paper
was not known. Paper-making of any kind was not introduced into Europe
until {137} the twelfth century, and cheap paper is a product of the
nineteenth. Pencils, too, of the modern type, date only from the sixteenth
century. In the second place, modern methods of operating, particularly of
multiplying and dividing (operations of relatively greater importance when
all measures were in compound numbers requiring reductions at every step),
were not yet invented. The old plan required the erasing of figures after
they had served their purpose, an operation very simple with counters,
since they could be removed. The new plan did not as easily permit this.
Hence we find the new numerals very tardily admitted to the counting-house,
and not welcomed with any enthusiasm by teachers.[555]
Aside from their use in the early treatises on the new art of reckoning,
the numerals appeared from time to time in the dating of manuscripts and
upon monuments. The oldest definitely dated European document known {138}
to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus,
written in the Albelda Cloister not far from Logrono in Spain, in 976 A.D.
The nine characters (of [.g]ob[=a]r type), without the zero, are given as
an addition to the first chapters of the third book of the _Origines_ by
Isidorus of Seville, in which the Roman numerals are under discussion.
Another Spanish copy of the same work, of 992 A.D., contains the numerals
in the corresponding section. The writer ascribes an Indian origin to them
in the following words: "Item de figuris arithmetic[e,]. Scire debemus in
Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et
geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum
est in nobem figuris, quibus designant unumquemque gradum cuiuslibet
gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow.
Some of the abacus forms[557] previously given are doubtless also of the
tenth century. The earliest Arabic documents containing the numerals are
two manuscripts of 874 and 888 A.D.[558] They appear about a century later
in a work[559] written at Shiraz in 970 A.D. There is also an early trace
of their use on a pillar recently discovered in a church apparently
destroyed as early as the tenth century, not far from the Jeremias
Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date
349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin
documents the Arabic forms were used as early as the thirteenth
century.[561]
On the early use of these numerals in Europe the only scientific study
worthy the name is that made by Mr. G. F. Hill of the British Museum.[562]
From his investigations it appears that the earliest occurrence of a date
in these numerals on a coin is found in the reign of Roger of Sicily in
1138.[563] Until recently it was thought that the earliest such date was
1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the
seals and medals containing dates that were at one time thought to be very
early have been shown by Mr. Hill to be of relatively late workmanship.
There are, however, in European manuscripts, numerous instances of the use
of these numerals before the twelfth century. Besides the example in the
Codex Vigilanus, another of the tenth century has been found in the St.
Gall MS. now in the University Library at Zuerich, the forms differing
materially from those in the Spanish codex.
The third specimen in point of time in Mr. Hill's list is from a Vatican
MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of
Boethius, of the same {140} (eleventh) century, and the sixth and seventh
are also from an eleventh-century MS. of Boethius at Chartres. These and
other early forms are given by Mr. Hill in this table, which is reproduced
with his kind permission.
EARLIEST MANUSCRIPT FORMS
[Illustration]
This is one of more than fifty tables given in Mr. Hill's valuable paper,
and to this monograph students {141} are referred for details as to the
development of number-forms in Europe from the tenth to the sixteenth
century. It is of interest to add that he has found that among the earliest
dates of European coins or medals in these numerals, after the Sicilian one
already mentioned, are the following: Austria, 1484; Germany, 1489
(Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485;
Italy, 1390.[565]
The earliest English coin dated in these numerals was struck in 1551,[566]
although there is a Scotch piece of 1539.[567] In numbering pages of a
printed book these numerals were first used in a work of Petrarch's
published at Cologne in 1471.[568] The date is given in the following form
in the _Biblia Pauperum_,[569] a block-book of 1470,
[Illustration]
while in another block-book which possibly goes back to c. 1430[570] the
numerals appear in several illustrations, with forms as follows:
[Illustration]
Many printed works anterior to 1471 have pages or chapters numbered by
hand, but many of these numerals are {142} of date much later than the
printing of the work. Other works were probably numbered directly after
printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are
numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,...
v,... vi, and followed by Roman numerals. This appears in the body of the
text, in spaces left by the printer to be filled in by hand. Another
book[572] of 1470 has pages numbered by hand with a mixture of Roman and
Hindu numerals, thus,
[Illustration] for 125 [Illustration] for 150
[Illustration] for 147 [Illustration] for 202
As to monumental inscriptions,[573] there was once thought to be a
gravestone at Katharein, near Troppau, with the date 1007, and one at
Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371
and one at Ulm of 1388.[574] Certain numerals on Wells Cathedral have been
assigned to the thirteenth century, but they are undoubtedly considerably
later.[575]
The table on page 143 will serve to supplement that from Mr. Hill's
work.[576]
{143}
EARLY MANUSCRIPT FORMS
[577] [Illustration] Twelfth century A.D.
[578] [Illustration] 1197 A.D.
[579] [Illustration] 1275 A.D.
[580] [Illustration] c. 1294 A.D.
[581] [Illustration] c. 1303 A.D.
[582] [Illustration] c. 1360 A.D.
[583] [Illustration] c. 1442 A.D.
{144}
[Illustration]
For the sake of further comparison, three illustrations from works in Mr.
Plimpton's library, reproduced from the _Rara Arithmetica_, may be
considered. The first is from a Latin manuscript on arithmetic,[584] of
which the original was written at Paris in 1424 by Rollandus, a Portuguese
physician, who prepared the work at the command of John of Lancaster, Duke
of Bedford, at one time Protector of England and Regent of France, to whom
the work is dedicated. The figures show the successive powers of 2. The
second illustration is from Luca da Firenze's _Inprencipio darte
dabacho_,[585] c. 1475, and the third is from an anonymous manuscript[586]
of about 1500.
[Illustration]
As to the forms of the numerals, fashion played a leading part until
printing was invented. This tended to fix these forms, although in writing
there is still a great variation, as witness the French 5 and the German 7
and 9. Even in printing there is not complete uniformity, {145} and it is
often difficult for a foreigner to distinguish between the 3 and 5 of the
French types.
[Illustration]
As to the particular numerals, the following are some of the forms to be
found in the later manuscripts and in the early printed books.
1. In the early printed books "one" was often i, perhaps to save types,
just as some modern typewriters use the same character for l and 1.[587] In
the manuscripts the "one" appears in such forms as[588]
[Illustration]
2. "Two" often appears as z in the early printed books, 12 appearing as
iz.[589] In the medieval manuscripts the following forms are common:[590]
[Illustration]
{146}
It is evident, from the early traces, that it is merely a cursive form for
the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal
strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.
3. "Three" usually had a special type in the first printed books, although
occasionally it appears as [Symbol].[591] In the medieval manuscripts it
varied rather less than most of the others. The following are common
forms:[592]
[Illustration]
4. "Four" has changed greatly; and one of the first tests as to the age of
a manuscript on arithmetic, and the place where it was written, is the
examination of this numeral. Until the time of printing the most common
form was [Symbol], although the Florentine manuscript of Leonard of Pisa's
work has the form [Symbol];[593] but the manuscripts show that the
Florentine arithmeticians and astronomers rather early began to straighten
the first of these forms up to forms like [Symbol][594] and [Symbol][594]
or [Symbol],[595] more closely resembling our own. The first printed books
generally used our present form[596] with the closed top [Symbol], the open
top used in writing ( [Symbol]) being {147} purely modern. The following
are other forms of the four, from various manuscripts:[597]
[Illustration]
5. "Five" also varied greatly before the time of printing. The following
are some of the forms:[598]
[Illustration]
6. "Six" has changed rather less than most of the others. The chief
variation has been in the slope of the top, as will be seen in the
following:[599]
[Illustration]
7. "Seven," like "four," has assumed its present erect form only since the
fifteenth century. In medieval times it appeared as follows:[600]
[Illustration]
{148}
8. "Eight," like "six," has changed but little. In medieval times there are
a few variants of interest as follows:[601]
[Illustration]
In the sixteenth century, however, there was manifested a tendency to write
it [Symbol].[602]
9. "Nine" has not varied as much as most of the others. Among the medieval
forms are the following:[603]
[Illustration]
0. The shape of the zero also had a varied history. The following are
common medieval forms:[604]
[Illustration]
The explanation of the place value was a serious matter to most of the
early writers. If they had been using an abacus constructed like the
Russian chotue, and had placed this before all learners of the positional
system, there would have been little trouble. But the medieval {149}
line-reckoning, where the lines stood for powers of 10 and the spaces for
half of such powers, did not lend itself to this comparison. Accordingly we
find such labored explanations as the following, from _The Crafte of
Nombrynge_:
"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the
first place of the rewele....
"If it stonde in the secunde place of the rewle, he betokens ten tymes hym
selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty,
for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he
stondis on the lyft side & in the secunde place, he betokens ten tyme hym
selfe. And so go forth....
"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre
tokens no[gh]t, bot he makes the figure to betoken that comes after hym
more than he shuld & he were away, as thus 10. here the figure of one
tokens ten, & yf the cifre were away & no figure byfore hym he schuld token
bot one, for than he schuld stonde in the first place...."[605]
It would seem that a system that was thus used for dating documents, coins,
and monuments, would have been generally adopted much earlier than it was,
particularly in those countries north of Italy where it did not come into
general use until the sixteenth century. This, however, has been the fate
of many inventions, as witness our neglect of logarithms and of contracted
processes to-day.
As to Germany, the fifteenth century saw the rise of the new symbolism; the
sixteenth century saw it slowly {150} gain the mastery; the seventeenth
century saw it finally conquer the system that for two thousand years had
dominated the arithmetic of business. Not a little of the success of the
new plan was due to Luther's demand that all learning should go into the
vernacular.[606]
During the transition period from the Roman to the Arabic numerals, various
anomalous forms found place. For example, we have in the fourteenth century
c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a
manuscript of the fifteenth century 12901 for 1291.[609] In the same
century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and
MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To
the next century belongs the form 1vojj for 1502. Even in Sfortunati's
_Nuovo lume_[612] the use of ordinals is quite confused, the propositions
on a single page being numbered "tertia," "4," and "V."
Although not connected with the Arabic numerals in any direct way, the
medieval astrological numerals may here be mentioned. These are given by
several early writers, but notably by Noviomagus (1539),[613] as
follows[614]:
[Illustration]
{151}
Thus we find the numerals gradually replacing the Roman forms all over
Europe, from the time of Leonardo of Pisa until the seventeenth century.
But in the Far East to-day they are quite unknown in many countries, and
they still have their way to make. In many parts of India, among the common
people of Japan and China, in Siam and generally about the Malay Peninsula,
in Tibet, and among the East India islands, the natives still adhere to
their own numeral forms. Only as Western civilization is making its way
into the commercial life of the East do the numerals as used by us find
place, save as the Sanskrit forms appear in parts of India. It is therefore
with surprise that the student of mathematics comes to realize how modern
are these forms so common in the West, how limited is their use even at the
present time, and how slow the world has been and is in adopting such a
simple device as the Hindu-Arabic numerals.
* * * * *
{153}
INDEX
_Transcriber's note: many of the entries refer to footnotes linked from the
page numbers given._
Abbo of Fleury, 122
`Abdall[=a]h ibn al-[H.]asan, 92
`Abdallat[=i]f ibn Y[=u]suf, 93
`Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6
Abenragel, 34
Abraham ibn Meir ibn Ezra, _see_ Rabbi ben Ezra
Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74
Ab[=u] 'l-[H.]asan, 93, 100
Ab[=u] 'l-Q[=a]sim, 92
Ab[=u] 'l-[T.]eiyib, 97
Ab[=u] Na[s.]r, 92
Ab[=u] Roshd, 113
Abu Sahl Dunash ibn Tamim, 65, 67
Adelhard of Bath, 5, 55, 97, 119, 123, 126
Adhemar of Chabanois, 111
A[h.]med al-Nasaw[=i], 98
A[h.]med ibn `Abdall[=a]h, 9, 92
A[h.]med ibn Mo[h.]ammed, 94
A[h.]med ibn `Omar, 93
Ak[s.]aras, 32
Alanus ab Insulis, 124
Al-Ba[.g]d[=a]d[=i], 93
Al-Batt[=a]n[=i], 54
Albelda (Albaida) MS., 116
Albert, J., 62
Albert of York, 103
Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93
Alcuin, 103
Alexander the Great, 76
Alexander de Villa Dei, 11, 133
Alexandria, 64, 82
Al-Faz[=a]r[=i], 92
Alfred, 103
Algebra, etymology, 5
Algerian numerals, 68
Algorism, 97
Algorismus, 124, 126, 135
Algorismus cifra, 120
Al-[H.]a[s.][s.][=a]r, 65
`Al[=i] ibn Ab[=i] Bekr, 6
`Al[=i] ibn A[h.]med, 93, 98
Al-Kar[=a]b[=i]s[=i], 93
Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126
Al-Kind[=i], 10, 92
Almagest, 54
Al-Ma[.g]reb[=i], 93
Al-Ma[h.]all[=i], 6
Al-M[=a]m[=u]n, 10, 97
Al-Man[s.][=u]r, 96, 97
Al-Mas`[=u]d[=i], 7, 92
Al-Nad[=i]m, 9
Al-Nasaw[=i], 93, 98
Alphabetic numerals, 39, 40, 43
Al-Q[=a]sim, 92
Al-Qass, 94
Al-Sakh[=a]w[=i], 6
Al-[S.]ardaf[=i], 93
Al-Sijz[=i], 94
Al-S[=u]f[=i], 10, 92
Ambrosoli, 118
A[.n]kapalli, 43
Apices, 87, 117, 118
Arabs, 91-98
Arbuthnot, 141
{154}
Archimedes, 15, 16
Arcus Pictagore, 122
Arjuna, 15
Arnold, E., 15, 102
Ars memorandi, 141
[=A]ryabha[t.]a, 39, 43, 44
Aryan numerals, 19
Aschbach, 134
Ashmole, 134
A['s]oka, 19, 20, 22, 81
A[s.]-[s.]ifr, 57, 58
Astrological numerals, 150
Atharva-Veda, 48, 49, 55
Augustus, 80
Averroes, 113
Avicenna, 58, 74, 113
Babylonian numerals, 28
Babylonian zero, 51
Bacon, R., 131
Bactrian numerals, 19, 30
Baeda, 2, 72
Bagdad, 4, 96
Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53
Ball, C. J., 35
Ball, W. W. R., 36, 131
B[=a][n.]a, 44
Barth, A., 39
Bayang inscriptions, 39
Bayer, 33
Bayley, E. C., 19, 23, 30, 32, 52, 89
Beazley, 75
Bede, _see_ Baeda
Beldomandi, 137
Beloch, J., 77
Bendall, 25, 52
Benfey, T., 26
Bernelinus, 88, 112, 117, 121
Besagne, 128
Besant, W., 109
Bettino, 36
Bhandarkar, 18, 47, 49
Bh[=a]skara, 53, 55
Biernatzki, 32
Biot, 32
Bjoernbo, A. A., 125, 126
Blassiere, 119
Bloomfield, 48
Blume, 85
Boeckh, 62
Boehmer, 143
Boeschenstein, 119
Boethius, 63, 70-73, 83-90
Boissiere, 63
Bombelli, 81
Bonaini, 128
Boncompagni, 5, 6, 10, 48, 49, 123, 125
Borghi, 59
Borgo, 119
Bougie, 130
Bowring, J., 56
Brahmagupta, 52
Br[=a]hma[n.]as, 12, 13
Br[=a]hm[=i], 19, 20, 31, 83
Brandis, J., 54
B[r.]hat-Sa[m.]hita, 39, 44, 78
Brockhaus, 43
Bubnov, 65, 84, 110, 116
Buddha, education of, 15, 16
Buedinger, 110
Bugia, 130
Buehler, G., 15, 19, 22, 31, 44, 49
Burgess, 25
Buerk, 13
Burmese numerals, 36
Burnell, A. C., 18, 40
Buteo, 61
Calandri, 59, 81
Caldwell, R., 19
Calendars, 133
Calmet, 34
Cantor, M., 5, 13, 30, 43, 84
{155}
Capella, 86
Cappelli, 143
Caracteres, 87, 113, 117, 119
Cardan, 119
Carmen de Algorismo, 11, 134
Casagrandi, 132
Casiri, 8, 10
Cassiodorus, 72
Cataldi, 62
Cataneo, 3
Caxton, 143, 146
Ceretti, 32
Ceylon numerals, 36
Chalfont, F. H., 28
Champenois, 60
Characters, _see_ Caracteres
Charlemagne, 103
Chasles, 54, 60, 85, 116, 122, 135
Chassant, L. A., 142
Chaucer, 121
Chiarini, 145, 146
Chiffre, 58
Chinese numerals, 28, 56
Chinese zero, 56
Cifra, 120, 124
Cipher, 58
Circulus, 58, 60
Clichtoveus, 61, 119, 145
Codex Vigilanus, 138
Codrington, O., 139
Coins dated, 141
Colebrooke, 8, 26, 46, 53
Constantine, 104, 105
Cosmas, 82
Cossali, 5
Counters, 117
Courteille, 8
Coxe, 59
Crafte of Nombrynge, 11, 87, 149
Crusades, 109
Cunningham, A., 30, 75
Curtze, 55, 59, 126, 134
Cyfra, 55
Dagomari, 146
D'Alviella, 15
Dante, 72
Dasypodius, 33, 67, 63
Daunou, 135
Delambre, 54
Devan[=a]gar[=i], 7
Devoulx, A., 68
Dhruva, 49
Dicaearchus of Messana, 77
Digits, 119
Diodorus Siculus, 76
Du Cange, 62
Dumesnil, 36
Dutt, R. C., 12, 15, 18, 75
Dvived[=i], 44
East and West, relations, 73-81, 100-109
Egyptian numerals, 27
Eisenlohr, 28
Elia Misrachi, 57
Enchiridion Algorismi, 58
Enestroem, 5, 48, 59, 97, 125, 128
Europe, numerals in, 63, 99, 128, 136
Eusebius Caesariensis, 142
Euting, 21
Ewald, P., 116
Fazzari, 53, 54
Fibonacci, _see_ Leonardo of Pisa
Figura nihili, 58
Figures, 119. _See_ numerals.
Fihrist, 67, 68, 93
Finaeus, 57
Firdus[=i], 81
Fitz Stephen, W., 109
Fleet, J. C., 19, 20, 49
{156}
Florus, 80
Fluegel, G., 68
Francisco de Retza, 142
Francois, 58
Friedlein, G., 84, 113, 116, 122
Froude, J. A., 129
Gandh[=a]ra, 19
Garbe, 48
Gasbarri, 58
Gautier de Coincy, 120, 124
Gemma Frisius, 2, 3, 119
Gerber, 113
Gerbert, 108, 110-120, 122
Gerhardt, C. I., 43, 56, 93, 118
Gerland, 88, 123
Gherard of Cremona, 125
Gibbon, 72
Giles, H. A., 79
Ginanni, 81
Giovanni di Danti, 58
Glareanus, 4, 119
Gnecchi, 71, 117
[.G]ob[=a]r numerals, 65, 100, 112, 124, 138
Gow, J., 81
Grammateus, 61
Greek origin, 33
Green, J. R., 109
Greenwood, I., 62, 119
Guglielmini, 128
Gulist[=a]n, 102
Guenther, S., 131
Guyard, S., 82
[H.]abash, 9, 92
Hager, J. (G.), 28, 32
Halliwell, 59, 85
Hankel, 93
H[=a]r[=u]n al-Rash[=i]d, 97, 106
Havet, 110
Heath, T. L., 125
Hebrew numerals, 127
Hecataeus, 75
Heiberg, J. L., 55, 85, 148
Heilbronner, 5
Henry, C., 5, 31, 55, 87, 120, 135
Heriger, 122
Hermannus Contractus, 123
Herodotus, 76, 78
Heyd, 75
Higden, 136
Hill, G. F., 52, 139, 142
Hillebrandt, A., 15, 74
Hilprecht, H. V., 28
Hindu forms, early, 12
Hindu number names, 42
Hodder, 62
Hoernle, 43, 49
Holywood, _see_ Sacrobosco
Hopkins, E. W., 12
Horace, 79, 80
[H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6
Hostus, M., 56
Howard, H. H., 29
Hrabanus Maurus, 72
Huart, 7
Huet, 33
Hugo, H., 57
Humboldt, A. von, 62
Huswirt, 58
Iamblichus, 81
Ibn Ab[=i] Ya`q[=u]b, 9
Ibn al-Adam[=i], 92
Ibn al-Bann[=a], 93
Ibn Khord[=a][d.]beh, 101, 106
Ibn Wahab, 103
India, history of, 14
writing in, 18
Indicopleustes, 83
Indo-Bactrian numerals, 19
{157}
Indr[=a]j[=i], 23
Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93
Jacob of Florence, 57
Jacquet, E., 38
Jamshid, 56
Jehan Certain, 59
Jetons, 58, 117
Jevons, F. B., 76
Johannes Hispalensis, 48, 88, 124
John of Halifax, _see_ Sacrobosco
John of Luna, _see_ Johannes Hispalensis
Jordan, L., 58, 124
Joseph Ispanus (Joseph Sapiens), 115
Justinian, 104
Kale, M. R., 26
Karabacek, 56
Karpinski, L. C., 126, 134, 138
K[=a]ty[=a]yana, 39
Kaye, C. R., 6, 16, 43, 46, 121
Keane, J., 75, 82
Keene, H. G., 15
Kern, 44
Kharo[s.][t.]h[=i], 19, 20
Khosr[=u], 82, 91
Kielhorn, F., 46, 47
Kircher, A., 34
Kit[=a]b al-Fihrist, _see_ Fihrist
Kleinwaechter, 32
K[=l]os, 62
Koebel, 4, 58, 60, 119, 123
Krumbacher, K., 57
Kuckuck, 62, 133
Kugler, F. X., 51
Lachmann, 85
Lacouperie, 33, 35
Lalitavistara, 15, 17
Lami, G., 57
La Roche, 61
Lassen, 39
L[=a][t.]y[=a]yana, 39
Leboeuf, 135
Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133
Lethaby, W. R., 142
Levi, B., 13
Levias, 3
Libri, 73, 85, 95
Light of Asia, 16
Luca da Firenze, 144
Lucas, 128
Mah[=a]bh[=a]rata, 18
Mah[=a]v[=i]r[=a]c[=a]rya, 53
Malabar numerals, 36
Malayalam numerals, 36
Mannert, 81
Margarita Philosophica, 146
Marie, 78
Marquardt, J., 85
Marshman, J. C., 17
Martin, T. H., 30, 62, 85, 113
Martines, D. C., 58
M[=a]sh[=a]ll[=a]h, 3
Maspero, 28
Mauch, 142
Maximus Planudes, 2, 57, 66, 93, 120
Megasthenes, 77
Merchants, 114
Meynard, 8
Migne, 87
Mikami, Y., 56
Milanesi, 128
Mo[h.]ammed ibn `Abdall[=a]h, 92
Mo[h.]ammed ibn A[h.]med, 6
Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8
Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i]
Molinier, 123
Monier-Williams, 17
{158}
Morley, D., 126
Moroccan numerals, 68, 119
Mortet, V., 11
Moseley, C. B., 33
Mo[t.]ahhar ibn [T.][=a]hir, 7
Mueller, A., 68
Mumford, J. K., 109
Muwaffaq al-D[=i]n, 93
Nabatean forms, 21
Nallino, 4, 54, 55
Nagl, A., 55, 110, 113, 126
N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40
Narducci, 123
Nasik cave inscriptions, 24
Na[z.][=i]f ibn Yumn, 94
Neander, A., 75
Neophytos, 57, 62
Neo-Pythagoreans, 64
Nesselmann, 58
Newman, Cardinal, 96
Newman, F. W., 131
Noeldeke, Th., 91
Notation, 61
Note, 61, 119
Noviomagus, 45, 61, 119, 150
Null, 61
Numerals,
Algerian, 68
astrological, 150
Br[=a]hm[=i], 19-22, 83
early ideas of origin, 1
Hindu, 26
Hindu, classified, 19, 38
Kharo[s.][t.]h[=i], 19-22
Moroccan, 68
Nabatean, 21
origin, 27, 30, 31, 37
supposed Arabic origin, 2
supposed Babylonian origin, 28
supposed Chaldean and Jewish origin, 3
supposed Chinese origin, 28, 32
supposed Egyptian origin, 27, 30, 69, 70
supposed Greek origin, 33
supposed Phoenician origin, 32
tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148
O'Creat, 5, 55, 119, 120
Olleris, 110, 113
Oppert, G., 14, 75
Pali, 22
Pancasiddh[=a]ntik[=a], 44
Paravey, 32, 57
P[=a]tal[=i]pu[t.]ra, 77
Patna, 77
Patrick, R., 119
Payne, E. J., 106
Pegolotti, 107
Peletier, 2, 62
Perrot, 80
Persia, 66, 91, 107
Pertz, 115
Petrus de Dacia, 59, 61, 62
Pez, P. B., 117
"Philalethes," 75
Phillips, G., 107
Picavet, 105
Pichler, F., 141
Pihan, A. P., 36
Pisa, 128
Place value, 26, 42, 46, 48
Planudes, _see_ Maximus Planudes
Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148
Pliny, 76
Polo, N. and M., 107
{159}
Praendel, J. G., 54
Prinsep, J., 20, 31
Propertius, 80
Prosdocimo de' Beldomandi, 137
Prou, 143
Ptolemy, 54, 78
Putnam, 103
Pythagoras, 63
Pythagorean numbers, 13
Pytheas of Massilia, 76
Rabbi ben Ezra, 60, 127
Radulph of Laon, 60, 113, 118, 124
Raets, 62
Rainer, _see_ Gemma Frisius
R[=a]m[=a]yana, 18
Ramus, 2, 41, 60, 61
Raoul Glaber, 123
Rapson, 77
Rauhfuss, _see_ Dasypodius
Raumer, K. von, 111
Reclus, E., 14, 96, 130
Recorde, 3, 58
Reinaud, 67, 74, 80
Reveillaud, 36
Richer, 110, 112, 115
Riese, A., 119
Robertson, 81
Robertus Cestrensis, 97, 126
Rodet, 5, 44
Roediger, J., 68
Rollandus, 144
Romagnosi, 81
Rosen, F., 5
Rotula, 60
Rudolff, 85
Rudolph, 62, 67
Ruffi, 150
Sachau, 6
Sacrobosco, 3, 58, 133
Sacy, S. de, 66, 70
Sa`d[=i], 102
['S]aka inscriptions, 20
Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93
['S][=a]rad[=a] characters, 55
Savonne, 60
Scaliger, J. C., 73
Scheubel, 62
Schlegel, 12
Schmidt, 133
Schonerus, 87, 119
Schroeder, L. von, 13
Scylax, 75
Sedillot, 8, 34
Senart, 20, 24, 25
Sened ibn `Al[=i], 10, 98
Sfortunati, 62, 150
Shelley, W., 126
Siamese numerals, 36
Siddh[=a]nta, 8, 18
[S.]ifr, 57
Sigsboto, 55
Sih[=a]b al-D[=i]n, 67
Silberberg, 60
Simon, 13
Sin[=a]n ibn al-Fat[h.], 93
Sindbad, 100
Sindhind, 97
Sipos, 60
Sirr, H. C., 75
Skeel, C. A., 74
Smith, D. E., 11, 17, 53, 86, 141, 143
Smith, V. A., 20, 35, 46, 47
Smith, Wm., 75
Sm[r.]ti, 17
Spain, 64, 65, 100
Spitta-Bey, 5
Sprenger, 94
['S]rautas[=u]tra, 39
Steffens, F., 116
Steinschneider, 5, 57, 65, 66, 98, 126
Stifel, 62
{160}
Subandhus, 44
Suetonius, 80
Suleim[=a]n, 100
['S][=u]nya, 43, 53, 57
Suter, 5, 9, 68, 69, 93, 116, 131
S[=u]tras, 13
Sykes, P. M., 75
Sylvester II, _see_ Gerbert
Symonds, J. A., 129
Tannery, P., 62, 84, 85
Tartaglia, 4, 61
Taylor, I., 19, 30
Teca, 55, 61
Tennent, J. E., 75
Texada, 60
Theca, 58, 61
Theophanes, 64
Thibaut, G., 12, 13, 16, 44, 47
Tibetan numerals, 36
Timotheus, 103
Tonstall, C., 3, 61
Trenchant, 60
Treutlein, 5, 63, 123
Trevisa, 136
Treviso arithmetic, 145
Trivium and quadrivium, 73
Tsin, 56
Tunis, 65
Turchill, 88, 118, 123
Turnour, G., 75
Tziphra, 57, 62
[Greek: tziphra], 55, 57, 62
Tzwivel, 61, 118, 145
Ujjain, 32
Unger, 133
Upanishads, 12
Usk, 121
Valla, G., 61
Van der Schuere, 62
Var[=a]ha-Mihira, 39, 44, 78
V[=a]savadatt[=a], 44
Vaux, Carra de, 9, 74
Vaux, W. S. W., 91
Ved[=a][.n]gas, 17
Vedas, 12, 15, 17
Vergil, 80
Vincent, A. J. H., 57
Vogt, 13
Voizot, P., 36
Vossius, 4, 76, 81, 84
Wallis, 3, 62, 84, 116
Wappler, E., 54, 126
Waeschke, H., 2, 93
Wattenbach, 143
Weber, A., 31
Weidler, I. F., 34, 66
Weidler, I. F. and G. I., 63, 66
Weissenborn, 85, 110
Wertheim, G., 57, 61
Whitney, W. D., 13
Wilford, F., 75
Wilkens, 62
Wilkinson, J. G., 70
Willichius, 3
Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138
Wolack, G., 54
Woodruff, C. E., 32
Word and letter numerals, 38, 44
Wuestenfeld, 74
Yule, H., 107
Zephirum, 57, 58
Zephyr, 59
Zepiro, 58
Zero, 26, 38, 40, 43, 45, 49, 51-62, 67
Zeuero, 58
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Notes
al-Mekk[=i] on a treatise on [.g]ob[=a]r arithmetic (explained later)
called _Al-murshidah_, found by Woepcke in Paris (_Propagation_, p. 66),
there is mentioned the fact that there are "nine Indian figures" and "a
second kind of Indian figures ... although these are the figures of the
[.g]ob[=a]r writing." So in a commentary by [H.]osein ibn Mo[h.]ammed
al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]`ilm
el-[h.]is[=a]b_ (Extract from Arithmetic) by `Abdalq[=a]dir ibn `Al[=i]
al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of
the forms of the figures of Hindu signs, such as were established by the
Hindu nation." [Woepcke, _Propagation_, p. 63.]]
which, of course, are interpolations. An interesting example of a forgery
in ecclesiastical matters is in the charter said to have been given by St.
Patrick, granting indulgences to the benefactors of Glastonbury, dated "In
nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei
anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in
saying that Dionysius Exiguus, a Scythian monk, first arranged the
Christian chronology c. 532 A.D., this can hardly be other than spurious.
See Arbuthnot, loc. cit., p. 38.
[1] "_Discipulus._ Quis primus invenit numerum apud Hebraeos et AEgyptios?
_Magister._ Abraham primus invenit numerum apud Hebraeos, deinde Moses; et
Abraham tradidit istam scientiam numeri ad AEgyptios, et docuit eos: deinde
Josephus." [Bede, _De computo dialogus_ (doubtfully assigned to him),
_Opera omnia_, Paris, 1862, Vol. I, p. 650.]
"Alii referunt ad Phoenices inventores arithmeticae, propter eandem
commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus
sepulchrum est Lutetiae in comitio Maturinensi, refert ad Arabes." [Ramus,
_Arithmeticae libri dvo_, Basel, 1569, p. 112.]
Similar notes are given by Peletarius in his commentary on the arithmetic
of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed.,
p. 14): "La valeur des Figures commence au coste dextre tirant vers le
coste senestre: au rebours de notre maniere d'escrire par ce que la
premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont ete
les premiers traffiquers de marchandise."
[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the
Indians." [Waeschke's German translation, Halle, 1878, p. 3.] Willichius
speaks of the "Zyphrae Indicae," in his _Arithmeticae libri tres_ (Strasburg,
1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le
pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is
not correct, therefore, in saying ("Memoire sur la propagation des chiffres
indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol.
I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical
and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin
edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to
give the Hindu origin.
[3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly
Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in
omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui
dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte
supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental
rival of Recorde, had the same idea: "Primum autem appellamus dexterum
locum, eo quod haec ars vel a Chaldaeis, vel ab Hebraeis ortum habere
credatur, qui etiam eo ordine scribunt"; but this refers more evidently to
the Arabic numerals. [_Arithmeticae practicae methodvs facilis_, Antwerp,
1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.
Even the modern Jewish writers claim that one of their scholars,
M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C.
Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.]
[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La
prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol.
9 of the 1592 edition.]
[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv
geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The
printer used the letters _rv_ for _w_ in "new" in the first edition, as he
had no _w_ of the proper font.]
[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui,
ab Indis usque, siue Chaldaeis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0
circulus, qui nihil significat." [_De VI. Arithmeticae practicae
speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.]
[7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins cum
notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos
natura et constitutione liber_, Amsterdam, 1650, p. 34) calls them
"Barbaras numeri notas." The word at that time was possibly synonymous with
Arabic.
[8] His full name was `Ab[=u] `Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a]
al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the
country about the present Khiva and bordering on the Oxus, and lived at
Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and
230 of the Mohammedan era, that is, between 835 and 845 A.D., although some
put the date as early as 812. The best account of this great scholar may be
found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti
della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5.
Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey
in the _Zeitschrift der deutschen Morgenlaend. Gesellschaft_, Vol. XXXIII,
p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenlaend.
Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur
Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und
Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der
Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachtraege," in Vol. XIV, p.
158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.;
F. Woepcke in _Propagation_, p. 489. So recently has he become known that
Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter
Arabes celebris Geometra, scripsit de figuris planis & sphericis."
[_Historia matheseos universae_, Leipzig, 1742, p. 438.]
In this work most of the Arabic names will be transliterated substantially
as laid down by Suter in his work _Die Mathematiker_ etc., except where
this violates English pronunciation. The scheme of pronunciation of
oriental names is set forth in the preface.
[9] Our word _algebra_ is from the title of one of his works, Al-jabr
wa'l-muq[=a]balah, Completion and Comparison. The work was translated into
English by F. Rosen, London, 1831, and treated in _L'Algebre
d'al-Kh[=a]rizmi et les methodes indienne et grecque_, Leon Rodet, Paris,
1878, extract from the _Journal Asiatique_. For the derivation of the word
_algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857;
Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B.
Boncompagni. "Almuchabala" also was used as a name for algebra.
[10] This learned scholar, teacher of O'Creat who wrote the _Helceph_
("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"),
studied in Toledo, learned Arabic, traveled as far east as Egypt, and
brought from the Levant numerous manuscripts for study and translation. See
Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p.
131; Woepcke in _Propagation_, p. 518.
[11] The title is _Algoritmi de numero Indorum_. That he did not make this
translation is asserted by Enestroem in the _Bibliotheca Mathematica_, Vol.
I (3), p. 520.
[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds:
"Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso
numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de
opera quod fit per eas aliquid quod esset leuius discentibus, si deus
uoluerit." [Boncompagni, _Trattati d'Aritmetica_, Rome, 1857.] Discussed
by F. Woepcke, _Sur l'introduction de l'arithmetique indienne en Occident_,
Rome, 1859.
[13] Thus in a commentary by `Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l
al-An[s.][=a]r[=i
[14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at
much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical
Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III,
1907, p. 489.
[15] _Alberuni's India_, Arabic version, London, 1887; English translation,
ibid., 1888.
[16] _Chronology of Ancient Nations_, London, 1879. Arabic and English
versions, by C. E. Sachau.
[17] _India_, Vol. I, chap. xvi.
[18] The Hindu name for the symbols of the decimal place system.
[19] Sachau's English edition of the _Chronology_, p. 64.
[20] _Litterature arabe_, Cl. Huart, Paris, 1902.
[21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p.
182 seq.
[22] Al-Mas`[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys
Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de
Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.
[23] _Les prairies d'or_, Vol. VIII, p. 289 seq.
[24] _Essays_, Vol. II, p. 428.
[25] Loc. cit., p. 504.
[26] _Materiaux pour servir a l'histoire comparee des sciences
mathematiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849,
pp. 438-439.
[27] He made an exception, however, in favor of the numerals, loc. cit.,
Vol. II, p. 503.
[28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp.
426-427.
[29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc.
cit., note Vol. II, p. 510].
[30] "Liber Artis Logisticae a Mohamado Ben Musa _Alkhuarezmita_ exornatus,
qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in
praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit.,
p. 427.]
[31] Macoudi, _Le livre de l'avertissement et de la revision_. Translation
by B. Carra de Vaux, Paris, 1896.
[32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind
included a treatment of arithmetic.
[33] A[h.]med ibn `Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12.
[34] _India_, Vol. II, p. 15.
[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist,"
_Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For
further references to early Arabic writers the reader is referred to H.
Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also
"Nachtraege und Berichtigungen" to the same (_Abhandlungen_, Vol. XIV,
1902, pp. 155-186).
[36] Suter, loc. cit., note 165, pp. 62-63.
[37] "Send Ben Ali,... tum arithmetica scripta maxime celebrata, quae
publici juris fecit." [Loc. cit., p. 440.]
[38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II,
_Scritti_ (1862); published by Baldassarre Boncompagni, Rome. Also _Tre
Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854.
[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum
introductus" etc. In another place, as a heading to a separate division, he
writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum
he sunt 9 8 7 6 5 4 3 2 1."
[40] See _An Ancient English Algorism_, by David Eugene Smith, in
_Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le
plus ancien traite francais d'algorisme," _Bibliotheca Mathematica_, Vol.
IX (3), pp. 55-64.
[41] These are the two opening lines of the _Carmen de Algorismo_ that the
anonymous author is explaining. They should read as follows:
Haec algorismus ars praesens dicitur, in qua
Talibus Indorum fruimur bis quinque figuris.
What follows is the translation.
[42] Thibaut, _Astronomie, Astrologie und Mathematik_, Strassburg, 1899.
[43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que
l'astronomie primitive est originaire de la Chine, et qu'elle a ete
empruntee par les anciens peuples occidentaux a la sphere chinoise; ouvrage
accompagne d'un atlas celeste chinois et grec_, The Hague and Leyden, 1875.
[44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7.
[45] R. C. Dutt, _History of India_, London, 1906.
[46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896.
[47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen
Morgenlaendischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327.
[48] _Geschichte der Math._, Vol. I, 2d ed., p. 595.
[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt,
"Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale
gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Buerk, loc.
cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp.
137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the
Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875;
Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani,"
_Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105.
[50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887.
[51] It is generally agreed that the name of the river Sindhu, corrupted by
western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of
India. Reclus, _Asia_, English ed., Vol. III, p. 14.
[52] See the comments of Oppert, _On the Original Inhabitants of
Bharatavar[s.]a or India_, London, 1893, p. 1.
[53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary
records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy
_lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic
applied to monetary affairs and mensuration), probably in the 5th century
B.C. [Buehler, _Indische Palaeographie_, Strassburg, 1896, p. 5.]
[54] R. C. Dutt, _A History of Civilization in Ancient India_, London,
1893, Vol. I, p. 174.
[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put
it c. 620 B.C.
[56] I.e. 100.10^7.
[57] There is some uncertainty about this limit.
[58] This problem deserves more study than has yet been given it. A
beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit a
la Grece_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in
the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke,
_Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source
of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910,
pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50
and 76-79. It will be discussed more fully in Chapter VI.
[59] I.e. to 100,000. The lakh is still the common large unit in India,
like the myriad in ancient Greece and the million in the West.
[60] This again suggests the _Psammites_, or _De harenae numero_ as it is
called in the 1544 edition of the _Opera_ of Archimedes, a work in which
the great Syracusan proposes to show to the king "by geometric proofs which
you can follow, that the numbers which have been named by us ... are
sufficient to exceed not only the number of a sand-heap as large as the
whole earth, but one as large as the universe." For a list of early
editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p.
227.
[61] I.e. the Wise.
[62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893,
pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of
India_, London, 1893, p. 2.
[63] For a list and for some description of these works see R. C. Dutt, _A
History of Civilization in Ancient India_, Vol. II, p. 121.
[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth
century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the
_Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p.
2.].
[65] Marshman, loc. cit., p. 2.
[66] A. C. Burnell, _South Indian Palaeography_, 2d ed., London, 1878, p. 1,
seq.
[67] This extensive subject of palpable arithmetic, essentially the history
of the abacus, deserves to be treated in a work by itself.
[68] The following are the leading sources of information upon this
subject: G. Buehler, _Indische Palaeographie_, particularly chap. vi; A. C.
Burnell, _South Indian Palaeography_, 2d ed., London, 1878, where tables of
the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the
Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol.
XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in
_The Academy_, January 28, 1882, with a repetition of his argument in his
work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R.
Kaye, loc. cit., in some respects one of the most critical articles thus
far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888,
Vol. III, with facsimiles of many Indian inscriptions, and _Indian
Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_,
Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R.
Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856,
p. 262 seq.; and _Epigraphia Indica_ (official publication of the
government of India), Vols. I-IX. Another work of Buehler's, _On the Origin
of the Indian Br[=a]hma Alphabet_, is also of value.
[69] The earliest work on the subject was by James Prinsep, "On the
Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic
Society of Bengal_, 1838, following a preliminary suggestion in the same
journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian
Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June,
1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F.
Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic
Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de
Piyadasi_, 2 vols., Paris, 1887.
[70] For a discussion of the minor details of this system, see Buehler, loc.
cit., p. 73.
[71] Julius Euting, _Nabataeische Inschriften aus Arabien_, Berlin, 1885,
pp. 96-97, with a table of numerals.
[72] For the five principal theories see Buehler, loc. cit., p. 10.
[73] Bayley, loc. cit., reprint p. 3.
[74] Buehler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian
Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.
[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri
Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay
Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.
[76] Ib., p. 405. He gives also a plate and an interpretation of each
numeral.
[77] These may be compared with Buehler's drawings, loc. cit.; with Bayley,
loc. cit., p. 337 and plates; and with Bayley's article in the
_Encyclopaedia Britannica_, 9th ed., art. "Numerals."
[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia
Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"
_Epigraphia Indica_, Vol. VII, pp. 47-74; Buehler, _Palaeographie_, Tafel
IX.
[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London,
1863, p. 217; M. R. Kale, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898,
p. 110, and other authorities as cited.
[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C.
Senart, _Notes d'epigraphie indienne_. Given by Buehler, loc. cit., Tafel I.
[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart,
loc. cit.; Buehler, loc. cit.
[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian
Antiquary_, Vol. VI, p. 155 seq.
[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l
Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from
a squeeze of the original.
[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey
Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79,
and Vol. VIII, pp. 59-96.
[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_,
1890, p. 639.
[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p.
381, and Vol. II, p. 201.
[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol.
III.
[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.
[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British
Museum.
[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127
ff.
[91] Fleet, loc. cit.
[92] Bayley, loc. cit., p. 335.
[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i],
Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however,
that many of these copper plates, being deeds of property, have forged
dates so as to give the appearance of antiquity of title. On the other
hand, as Colebrooke long ago pointed out, a successful forgery has to
imitate the writing of the period in question, so that it becomes evidence
well worth considering, as shown in Chapter III.
[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central
India. [Fleet, loc. cit., Plate XIV.]
[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal
Presidency. [Fleet, loc. cit., Plate XXIV.]
[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay
Presidency. [Fleet, loc. cit., Plate XXIV.]
[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit.,
Plate XXXVI, C.]
[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]
[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]
[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.
[101] It seems evident that the Chinese four, curiously enough called
"eight in the mouth," is only a cursive [4 vertical strokes].
[102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J.
Hager, _An Explanation of the Elementary Characters of the Chinese_,
London, 1801.
[103] H. V. Hilprecht, _Mathematical, Metrological and Chronological
Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A,
Cuneiform Texts Published by the Babylonian Expedition of the University of
Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_,
Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773.
[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea,"
_Proceedings of the Society of Biblical Archaeology_, XXI, p. 301, London,
1899.
[105] For a bibliography of the principal hypotheses of this nature see
Buehler, loc. cit., p. 77. Buehler (p. 78) feels that of all these hypotheses
that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most
plausible, although he does not adduce any convincing proof. Th. Henri
Martin, "Les signes numeraux et l'arithmetique chez les peuples de
l'antiquite et du moyen age" (being an examination of Cantor's
_Mathematische Beitraege zum Culturleben der Voelker_), _Annali di matematica
pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author,
"Recherches nouvelles sur l'origine de notre systeme de numeration ecrite,"
_Revue Archeologique_, 1857, pp. 36, 55. See also the tables given later in
this work.
[106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII.
[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are
generally regarded as unwarranted.
[108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The
Academy_ of Jan. 28, 1882.
[109] Taylor, _The Alphabet_, loc. cit., table on p. 266.
[110] Buehler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg,
1898, footnote, pp. 52, 53.
[111] Albrecht Weber, _History of Indian Literature_, English ed., Boston,
1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the
initial letters of the numerals themselves...: the zero, too, has arisen
out of the first letter of the word _[s.]unya_ (empty) (it occurs even in
Pingala). It is the decimal place value of these figures which gives them
significance." C. Henry, "Sur l'origine de quelques notations
mathematiques," _Revue Archeologique_, June and July, 1879, attempts to
derive the Boethian forms from the initials of Latin words. See also J.
Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and
Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838,
especially Plate XX, p. 348; this was the first work on the subject.
[112] Buehler, _Palaeographie_, p. 75, gives the list, with the list of
letters (p. 76) corresponding to the number symbols.
[113] For a general discussion of the connection between the numerals and
the different kinds of alphabets, see the articles by U. Ceretti, "Sulla
origine delle cifre numerali moderne," _Rivista di fisica, matematica e
scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and
120, and continuation in 1910.
[114] This is one of Buehler's hypotheses. See Bayley, loc. cit., reprint p.
4; a good bibliography of original sources is given in this work, p. 38.
[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit.,
p. 64, and tables in plate XXIII.
[116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan,
1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in
_Bibliotheque Britannique_, Geneva, 1812). See also the recent article by
Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally
Marks," _American Mathematical Monthly_, August-September, 1909.
Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal fuer die reine
und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the
priority of the Chinese claim for a place system and the zero, but upon the
flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et
hieroglyphique des chiffres et des lettres de tous les peuples_, Paris,
1826; G. Kleinwaechter, "The Origin of the Arabic Numerals," _China Review_,
Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la
connaissance que les Chinois ont eue de la valeur de position des
chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de
Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in
China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.
Moseley, "Numeral Characters: Theory of Origin and Development," _American
Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals
from Chinese characters, in much the same way as is done by Major Woodruff,
in the article above cited.
[117] The Greeks, probably following the Semitic custom, used nine letters
of the alphabet for the numerals from 1 to 9, then nine others for 10 to
90, and further letters to represent 100 to 900. As the ordinary Greek
alphabet was insufficient, containing only twenty-four letters, an alphabet
of twenty-seven letters was used.
[118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a
somewhat rare work from which the following quotation is taken:
"_Quis est harum Cyphrarum autor?_
"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum
fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs
(quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci
tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet
videre.
"Graecorum Literae corruptae.
[Illustration]
_"Sed qua ratione graecorum literae ita fuerunt corruptae?_
"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in
nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in
septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola
differunt elegantia, vt apparet."
See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788,
pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit.
[119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on
p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum
modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id
esse, merosque esse Graecorum characteres aio; a librariis Graecae linguae
ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam
primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta]
extremis suis truncatum. [gamma], si in sinistram partem inclinaveris &
cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa
loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek:
kata katheton], & infra basim descendit; basis vero ipsa ultra crus
producta eminet. Vides quam 5 simile sit [Greek: toi] [epsilon]; infimo
tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek:
episemon bau] quod ita notabatur [digamma], rotundato ventre, pede
detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est
[Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem &
commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est
[alt theta]."
I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_,
Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin
Calmet, "Recherches sur l'origine des chiffres d'arithmetique," _Memoires
pour l'histoire des sciences et des beaux arts_, Trevoux, 1707 (pp.
1620-1635, with two plates), derives the current symbols from the Romans,
stating that they are relics of the ancient "Notae Tironianae." These
"notes" were part of a system of shorthand invented, or at least perfected,
by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de
nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol.
XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman
numerals.
[120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum,
mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665.
[121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100.
[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex
figura circuli secti inuenti."
[123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p.
333.
[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara,"
_Proceedings of the Society of Biblical Archaeology_, Vol. XX, p. 25
(London, 1898). Terrien de Lacouperie states that the Chinese used the
circle for 10 before the beginning of the Christian era. [_Catalogue of
Chinese Coins_, London, 1892, p. xl.]
[125] For a purely fanciful derivation from the corresponding number of
strokes, see W. W. R. Ball, _A Short Account of the History of
Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud,
_Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres
arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la
forme des chiffres usuels," _Annales de l'universite de Grenoble_, 1907,
Vol. XIX, pp. 657-674, also a note in _Revue Archeologique_, 1890, Vol. XVI
(3), pp. 342-348; one of the earliest references to a possible derivation
from points is in a work by Bettino entitled _Apiaria universae
philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus
eximios traducta, et facillimis demonstrationibus confirmata_, Bologna,
1545, Vol. II, Apiarium XI, p. 5.
[126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently
Sanskrit, and the 4, 7, and possibly 9 are from India.
[127] _Alphabetum Grandonico-Malabaricum_, Romae, MDCCLXXII, p. 90. The
zero is not used, but the symbols for 10, 100, and so on, are joined to the
units to make the higher numbers.
[128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS.
in the library of Professor Smith, probably of the eighteenth century,
substantially these forms are given.
[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and
numerous other forms found in India, as well as those of other oriental
countries, are given by A. P. Pihan, _Expose des signes de numeration
usites chez les peuples orientaux anciens et modernes_, Paris, 1860.
[130] Buehler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum
Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by
Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E.
Jacquet, "Mode d'expression symbolique des nombres employe par les Indiens,
les Tibetains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835.
[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the
earliest epigraphical instance of this usage in India proper.
[132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq.
[133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407.