THE SYMBOL ZERO
What has been said of the improved Hindu system with a place value does not
touch directly the origin of a symbol for zero, although it assumes that
such a symbol exists. The importance of such a sign, the fact that it is a
prerequisite to a place-value system, and the further fact that without it
the Hindu-Arabic numerals would never have dominated the computation system
of the western world, make it proper to devote a chapter to its origin and
history.
It was some centuries after the primitive Br[=a]hm[=i] and
Kharo[s.][t.]h[=i] numerals had made their appearance in India that the
zero first appeared there, although such a character was used by the
Babylonians[185] in the centuries immediately preceding the Christian era.
The symbol is [Babylonian zero symbol] or [Babylonian zero symbol], and
apparently it was not used in calculation. Nor does it always occur when
units of any order are lacking; thus 180 is written [Babylonian numerals
180] with the meaning three sixties and no units, since 181 immediately
following is [Babylonian numerals 181], three sixties and one unit.[186]
The main {52} use of this Babylonian symbol seems to have been in the
fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of
[Greek: o], for [Greek: ouden], with the meaning _vacant_.
"The earliest undoubted occurrence of a zero in India is an inscription at
Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line
20), 50 is written [Gwalior numerals 50]. 270 (line 4) is written [Gwalior
numerals 270]."[187] The Bakh[s.][=a]l[=i] Manuscript[188] probably
antedates this, using the point or dot as a zero symbol. Bayley mentions a
grant of Jaika Rashtrakuta of Bharuj, found at Okamandel, of date 738 A.D.,
which contains a zero, and also a coin with indistinct Gupta date 707 (897
A.D.), but the reliability of Bayley's work is questioned. As has been
noted, the appearance of the numerals in inscriptions and on coins would be
of much later occurrence than the origin and written exposition of the
system. From the period mentioned the spread was rapid over all of India,
save the southern part, where the Tamil and Malayalam people retain the old
system even to the present day.[189]
Aside from its appearance in early inscriptions, there is still another
indication of the Hindu origin of the symbol in the special treatment of
the concept zero in the early works on arithmetic. Brahmagupta, who lived
in Ujjain, the center of Indian astronomy,[190] in the early part {53} of
the seventh century, gives in his arithmetic[191] a distinct treatment of
the properties of zero. He does not discuss a symbol, but he shows by his
treatment that in some way zero had acquired a special significance not
found in the Greek or other ancient arithmetics. A still more scientific
treatment is given by Bh[=a]skara,[192] although in one place he permits
himself an unallowed liberty in dividing by zero. The most recently
discovered work of ancient Indian mathematical lore, the
Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.),
while it does not use the numerals with place value, has a similar
discussion of the calculation with zero.
What suggested the form for the zero is, of course, purely a matter of
conjecture. The dot, which the Hindus used to fill up lacunae in their
manuscripts, much as we indicate a break in a sentence,[194] would have
been a more natural symbol; and this is the one which the Hindus first
used[195] and which most Arabs use to-day. There was also used for this
purpose a cross, like our X, and this is occasionally found as a zero
symbol.[196] In the Bakh[s.][=a]l[=i] manuscript above mentioned, the word
_['s][=u]nya_, with the dot as its symbol, is used to denote the unknown
quantity, as well as to denote zero. An analogous use of the {54} zero, for
the unknown quantity in a proportion, appears in a Latin manuscript of some
lectures by Gottfried Wolack in the University of Erfurt in 1467 and
1468.[197] The usage was noted even as early as the eighteenth
century.[198]
The small circle was possibly suggested by the spurred circle which was
used for ten.[199] It has also been thought that the omicron used by
Ptolemy in his _Almagest_, to mark accidental blanks in the sexagesimal
system which he employed, may have influenced the Indian writers.[200] This
symbol was used quite generally in Europe and Asia, and the Arabic
astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in
connection with the alphabetic system of numerals. The occasional use by
Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence
of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also
the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the
Kashmirian Atharva-Veda, the writing being at least 400 years old.
Bh[=a]skara (c. 1150) used a small circle above a number to indicate
subtraction, and in the Tartar writing a redundant word is removed by
drawing an oval around it. It would be interesting to know whether our
score mark [score mark], read "four in the hole," could trace its pedigree
to the same sources. O'Creat[203] (c. 1130), in a letter to his teacher,
Adelhard of Bath, uses [Greek: t] for zero, being an abbreviation for the
word _teca_ which we shall see was one of the names used for zero, although
it could quite as well be from [Greek: tziphra]. More rarely O'Creat uses
[circle with bar], applying the name _cyfra_ to both forms. Frater
Sigsboto[204] (c. 1150) uses the same symbol. Other peculiar forms are
noted by Heiberg[205] as being in use among the Byzantine Greeks in the
fifteenth century. It is evident from the text that some of these writers
did not understand the import of the new system.[206]
Although the dot was used at first in India, as noted above, the small
circle later replaced it and continues in use to this day. The Arabs,
however, did not adopt the {56} circle, since it bore some resemblance to
the letter which expressed the number five in the alphabet system.[207] The
earliest Arabic zero known is the dot, used in a manuscript of 873
A.D.[208] Sometimes both the dot and the circle are used in the same work,
having the same meaning, which is the case in an Arabic MS., an abridged
arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the
numerals are [symbols]. The form for 5 varies, in some works becoming
[symbol] or [symbol]; [symbol] is found in Egypt and [symbol] appears in
some fonts of type. To-day the Arabs use the 0 only when, under European
influence, they adopt the ordinary system. Among the Chinese the first
definite trace of zero is in the work of Tsin[210] of 1247 A.D. The form is
the circular one of the Hindus, and undoubtedly was brought to China by
some traveler.
The name of this all-important symbol also demands some attention,
especially as we are even yet quite undecided as to what to call it. We
speak of it to-day as _zero, naught_, and even _cipher_; the telephone
operator often calls it _O_, and the illiterate or careless person calls it
_aught_. In view of all this uncertainty we may well inquire what it has
been called in the past.[211]
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As already stated, the Hindus called it _['s][=u]nya_, "void."[212] This
passed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[213] When
Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this
character as _zephirum_.[214] Maximus Planudes (1330), writing under both
the Greek and the Arabic influence, called it _tziphra_.[215] In a treatise
on arithmetic written in the Italian language by Jacob of Florence[216]
{58} (1307) it is called _zeuero_,[217] while in an arithmetic of Giovanni
di Danti of Arezzo (1370) the word appears as _ceuero_.[218] Another form
is _zepiro_,[219] which was also a step from _zephirum_ to zero.[220]
Of course the English _cipher_, French _chiffre_, is derived from the same
Arabic word, _a[s.]-[s.]ifr_, but in several languages it has come to mean
the numeral figures in general. A trace of this appears in our word
_ciphering_, meaning figuring or computing.[221] Johann Huswirt[222] uses
the word with both meanings; he gives for the tenth character the four
names _theca, circulus, cifra_, and _figura nihili_. In this statement
Huswirt probably follows, as did many writers of that period, the
_Algorismus_ of Johannes de Sacrobosco (c. 1250 A.D.), who was also known
as John of Halifax or John of Holywood. The commentary of {59} Petrus de
Dacia[223] (c. 1291 A.D.) on the _Algorismus vulgaris_ of Sacrobosco was
also widely used. The widespread use of this Englishman's work on
arithmetic in the universities of that time is attested by the large
number[224] of MSS. from the thirteenth to the seventeenth century still
extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in
England given by Halliwell,[225] ten listed in Coxe's _Catalogue of the
Oxford College Library_, one in the Plimpton collection,[226] one in the
Columbia University Library, and, of course, many others.
From _a[s.]-[s.]ifr _has come _zephyr, cipher,_ and finally the abridged
form _zero_. The earliest printed work in which is found this final form
appears to be Calandri's arithmetic of 1491,[227] while in manuscript it
appears at least as early as the middle of the fourteenth century.[228] It
also appears in a work, _Le Kadran des marchans_, by Jehan {60}
Certain,[229] written in 1485. This word soon became fairly well known in
Spain[230] and France.[231] The medieval writers also spoke of it as the
_sipos_,[232] and occasionally as the _wheel_,[233] _circulus_[234] (in
German _das Ringlein_[235]), _circular {61} note_,[236] _theca_,[237] long
supposed to be from its resemblance to the Greek theta, but explained by
Petrus de Dacia as being derived from the name of the iron[238] used to
brand thieves and robbers with a circular mark placed on the forehead or on
the cheek. It was also called _omicron_[239] (the Greek _o_), being
sometimes written o or [Greek: ph] to distinguish it from the letter _o_.
It also went by the name _null_[240] (in the Latin books {62} _nihil_[241]
or _nulla_,[242] and in the French _rien_[243]), and very commonly by the
name _cipher_.[244] Wallis[245] gives one of the earliest extended
discussions of the various forms of the word, giving certain other
variations worthy of note, as _ziphra_, _zifera_, _siphra_, _ciphra_,
_tsiphra_, _tziphra,_ and the Greek [Greek: tziphra].[246]
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