The Origin of the Latin Numerals 1 to 1000
Author(s): Paul Keyser
Source: American Journal of Archaeology, Vol. 92, No. 4 (Oct., 1988), pp. 529-546
Published by: Archaeological Institute of America
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The Origin of the Latin Numerals 1 to 1000*
PAUL KEYSER
Abstract
The standardtheory of the origin of Latin numerals,
found in all the best handbooks, is that of Mommsen
(1850). He explains I, V, and X on a pictographicprinciple, and L, C, and (I) = 1000 as forms of Greek letters
(aspirates) unused in Latin. There are four other important theories: a long-rejectedone proposed by the fifthcentury A.C. grammarian Priscian, a tally-mark theory
(dating back to 1546), various pictographictheories (dating back to 1655), and various acrophonictheories. The
tally-mark theory, though held by numerous scholars
through the end of the 19th century,has receivedlittle attention in the last 75 years. Mommsen'smixed theoryand
the other four importanttheories are criticallyreviewed.
Available numismaticand epigraphicalevidence,certain
logical principles, and historical considerationsare adduced to support the rejectionof Mommsen's theory and
the acceptanceof a theory which holds that the Latin nu-
merals are Etruscantally-marknumeralswhich have undergonesome alterationand abbreviationof their forms.
INTRODUCTION
The Latin whole-number numerals form an additive non-place-value system, with a decimal base, i.e.,
distinct symbols exist only for 1, 5, 10, 50, 100, 500,
and 1000, and intermediate values are constructed additively (or subtractively).1 I consider here only the essential features which were in evidence by the midfirst century B.C. (e.g., ignoring S = 6, M = 1000,
and Q = 500,000 as late or rare). I review and criticize
the principal theories of the origin of the wholenumber numerals and suggest a revision of the accepted view.
* This
paper would have suffered without the assistance
of the InterlibraryLoan staff at the Norlin Library, University of Colorado, Boulder (Virginia Boucher, April Peterson, and Eladia Rivera) in obtaining obscure sources. I
am indebted to Profs. W.M. Calder III, C.F. Konrad, and
Werner Krenkel (Rostock) for references, advice, discussions, and encouragement.I use the following abbreviations
within:
ADB
Allgemeine Deutsche Biographie, 56
vols. (Leipzig 1875-1912).
Bonfante-Bonfante G. Bonfante and L. Bonfante, The
Etruscan Language: An Introduction (New York 1983).
Bortolotti
P. Bortolotti, "Congetture intorno
una numeralenotazioneprealfabetica in Italia,"BdI 1875, 155-60.
BU
L.G. and J.F. Michaud eds., Biographie universelle(Paris 1843).
A. Fabretti, Corpus Inscriptionum
CII
Italicarum et GlossariumItalicum
(Turin 1867); Supplementi I
(1872), II (1874), III (1878); Appendice (Florence 1880).
DBI
Dizionario biografico degli Italiani
(Rome 1960-).
deMatthaeis
G. deMatthaeis, Sull'origine de'numeri romani. Dissertazione (Rome
1818).
Fabretti (1877)
A. Fabretti, Palaeographische Studien (Leipzig 1877), Italian original: CII Suppl. I, Pt. 2 (1874).
Friedlein
G. Friedlein, Die Zahlzeichen und
das elementareRechnen der Griechen und Romer (Erlangen 1869;
repr. Wiesbaden 1968).
Gordon
A.E. Gordon,IllustratedIntroduction
to Latin Epigraphy (Berkeley
1983).
AmericanJournal of Archaeology92 (1988)
Gordon-Gordon
J.S. and A.E. Gordon, Contributions
to the Palaeography of Latin Inscriptions (CPCA 3.3, Berkeley
1967; repr. Milan 1977).
Gundermann
G. Gundermann, Die Zahlzeichen
(Giessen 1899).
Ifrah (1981)
G. Ifrah,Histoire universelledes chiffres (Paris 1981); trans. L. Blair,
From One to Zero (New York
1985).
K. Meninger, Zahlwortund Ziffer22
Meninger
(G6ttingen 1958); trans. P. Broneer, Number Wordsand Number
Symbols (Cambridge, Mass. and
London 1969).
Mommsen (1850)
T. Mommsen, Die unteritalischen
Dialekte (Leipzig 1850).
Mommsen (1887) T. Mommsen, "Zahl- und Bruchzeichen,"Hermes 22 (1887) 596-614
= GesammelteSchriften7 (Berlin
1909) 765-83.
Mommsen (1888) T. Mommsen, "Zu den r6mischen
Zahl- und Bruchzeichen,"Hermes
23 (1888) 152-56 = Gesammelte
Schriften7 (Berlin 1909) 783-87.
NBU
J.C.F. Hoefer ed., Nouvelle biographie universelle(Paris 1853).
K. Zangemeister, "Entstehung der
Zangemeister
r6mischen Zahlzeichen," SBBerl
49 (1887) 1011-28.
A recent discussion of the origin of Latin numerals is
Gordon 44-49; for a discussionof the additiveprinciple see
Ifrah (1981) 139-59, 337-47. In the earlier numeral systems, there are symbols only for the decades. From this observationand from the fact that the natural (and almostuniversal) base is 10, I concludethat any symbolsfor 5, 50, 500,
etc. in additivenon-place-valuesystemsare secondary.
529
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530
PAUL KEYSER
Table 1 lists the classical forms of the Latin wholenumber numerals from 1 to 1000 (higher values derived from these are not discussedherein):2
Table 1. Latin Numerals to 1000
I
1
V
5
X
1' C
(X)oro0or (Dor (1)
10 50 100 500
1000
There are two or three ancient discussionsof these
numerals, and four principal Renaissance and modern theories: various forms of a tally-mark theory
(1546 and later), the pictographic theory (1655 and
later), the acrophonictheory (1818 and later), and the
unused-lettertheory of Mommsen (1850). The last is
combinedwith a pictographictheory for 1,V, and X by
Mommsen, and this is the generally acceptedtheory
today.
ANCIENT THEORIES
The two or three extant ancient discussions of the
problemare all from the pens of grammarians.Isidore
2 Derived from Gordon44-49 and Gordon-Gordon18182 and 224 n. 1. On Q see T. Mommsen, "Quingenta
Milia," Hermes 3 (1869) 467-68 and Hermes 10 (1876)
472 = GesammelteSchriften 7 (Berlin 1909) 788-91, RE
24 (1963) 622.38-47, s.v. Q (H. Chantraine) and Gordon
44-45, n. 120; and on S see Gordon 46. On the lateness of
M as a numeral see Gordon 45, ns. 122-23. I hope elsewhere to treat the origin of the numeral M = 1000. Sufficeto
say here that Priscian (fl. early sixth century A.C.) did not
know of it (Keil, Gramm.Lat. 3.406-407) but Dante (Paradiso 19.127-29) does seem to. I may also note here that 50
4 becamefirst I, I, then L (see Gordon44-49), alreadyin
the classical period (first century B.C. to second century
A.C.). Just as it is desirable to print IVLII(not JULIJ)or
even 1III(and not IV) so I would find it desirable to print
these classical forms of the numerals (and not L, D or worst
of all M, save in epigraphicalor palaeographicalinvestigations of the forms of the characters).
3 Suet. Gram. 1 = Funaioli, Gramm.Rom. Frag., p. 411
Fr = p. 101 T 1 distinguishesthe two Ennii and assigns De
Litteris Sillabisque (and two other works) to our Ennius.
(Migne, PL 82 [1850] ?98 = Isid., Orig. 1.22 "De Notis
Uulgaribus"1 = Funaioli, Gramm.Rom. Frag., p. 4 T 3.) I
cite the text of W.M. Lindsay (Oxford 1911).
4 See RE 5 (1905) 2627.60-2628.10, s.v. Ennius
(F.
Skutsch) and E. Meyer, Einfiihrung in die lateinischeEpigraphik (Darmstadt 1973) 33. If M = 1000 is post-classical
(supra n. 2), I wonder what Ennius was explaining.
5Probus on numerals at Migne, PL 130 (1880)
?1196A-B. In Keil, Gramm.Lat. 4 (1864) Mommsen omits
this sectionof Probus (see pp. 347-52). D.E. Smith, Scientia
40 (1926) 7, cites Probus from the edition of J. deC. deTridino (Venice 1525) p. xxiii, in which work Probus is assigned a later version of Priscian (see below). One can tell
that it is later because "Probus"uses M not (I) to explain
1000 and 500, and because all referencesto Greek origins
are omitted. DeFeis (in 1898, infra n. 19) 16 also assigns
[AJA 92
of Seville, at Origines 1.22 "De Notis Vulgaribus" 1,
reports: Vulgares notas Ennius primus mille et centum inuenit. He refers to Ennius Posterior the grammarian (ca. first century B.C.).3 Does Isidore mean
that Ennius discovered 1100 common abbreviations
(hardly likely) or that Ennius was the first to discover
the (presumably acrophonic) explanation of C = 100
and M = 1000?4 M. Valerius Probus (first century
A.C.) in De Notis Antiquis following the letter X includes a brief paragraph "De Numeris," which is only
a list and offers no explanation of the symbols, but one
modern historian of mathematics (D.E. Smith), misled by a false ascription of a version of Priscian's
theory (see below) to Probus, has claimed that Probus
did offer an explanation.'
The late and hence elaborate theory of Priscian (ca.
A.D. 485) has long been rejected as inadequate.6 He
claims that I = 1, from the Greek symbol itself derived
froma Homericform('os, i'a, 'ov)of
acrophonically
"one." This is sensible as the Greek numeral system in
Priscian'stheory to Probus. On Probus, see Suet. Gram.24,
Euseb. (Hier.) Chron. ad 01. 208.4 (i.e., fi. A.D. 56), RE
23.1 (1957) 59.47-64.48, s.v. Probus (26) (R. Helm) and
RE 8A (1955) 195.67-212.58, s.v. M. Valerius Probus
(315) (R Hanslik); the work De Notis is discussed
208.30-209.9. Probus does discuss numeralselsewhere, but
only in lists of abbreviations:see Keil, Gramm. Lat. 4.308
(letter I, nos. 37, 38) similar to 322 (letter I, nos. 68-71)
similar to 343 (letter I, no. 14) on which see the notes p. 304;
Gramm. Lat. 4.317 (letter C, no. 14), 318 (letter C, no. 40)
and 337 (letter C, no. 47). Charisius (fourth centuryA.C.,
C. Barwick ed., 1964) says almost nothing about numerals
(only Gramm.Lat. 1.10.2-3 = Barwick7.5 on C = 100, and
Gramm.Lat. 1.10.7 = Barwick 7.10-11 on D = 500); Diomedes (fourth centuryA.C., Keil, Gramm. Lat. 1.297-529)
notes the numerical use of C (Gramm. Lat. 1.424.8-9), D
(424.13), I (424.27), L (425.5-6), V (425.34) and X (426.3).
6 Priscian on Latin numerals:
Keil, Gramm. Lat. 3.406407. Doubted alreadyby A. Alciati in 1546 (infra n. 10) and
by nearly everyone since. Three exceptions:Julius Caesar
Scaliger, De causis linguae Latinae libri tredecim (Heidelberg 1534) c. 41 (pp. 96-97); A. Dragoni, Sul metodoaritmeticodegli antichi Romani (Cremona1811) 22-23; and R.
Bombelli, Studifilologico-criticisulla genesi forma e valore
delle lettere dell'alfabetoitaliano (Rome/Turin 1866) at
least for 1(98), V (143-44), X (146), L (107) and C (72-73).
Bombelliquotes from the Tridino editionof "Probus"(Venice, 1525, supra n. 5) for D (76) and follows it on M (110).
Bombelli (13, 21, 144) cites "Hugo Herrmann de prim.
scrib. orig. cap. 28" as also agreeing with Priscian, but I
have been unable to confirm the existence of this work.
Bombelli himself later wrote a work Dell'antica numerazione italica e dei relativi numeri simbolici (Rome 1876) (not
availablein this country,see A. Pagliaini, Catalogogenerale
1:267). On H.A. Herrmann (1820-post 1871), see F.A.
Eckstein, Nomenclator philologum (Leipzig 1871; repr.
Hildesheim 1966) 241.
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THE ORIGINOF THE LATINNUMERALS1 TO 1000
1988]
which the symbol I = 1 was acrophonic,7but it is noteworthy that most additive non-place-value systems
use a symbol I for 1.8 Priscian claims that V = 5 because it is the fifth Latin vowel. This is special pleading-why vowels (and not consonantsor simply letters)? Why switch to an alphabeticprinciple?Priscian
claims that X = 10 because X is the 10th Greek consonant (6) or because X follows V in the Latin alphabet.
This dual suggestionshows that he has no idea how X
= 10 originated, and is even more egregious special
pleading than his suggestionfor V-why switch to the
Greekalphabet? Priscian holds that L = 50 because v
= 50 in the Greek alphabeticsystem (itself true') and
v becomes A (i.e., L) in some Latin words. This is
again ingenious special pleading, refuted by the fact
that the symbol for 50 was originally T (see Table 1
above). Priscian holds that C = 100 because it is the
initial letter of centum, but why only here is the system acrophonic?Priscian claims that D = 500 because
D follows C in the Latin alphabet, which is unconvincing, but is at least consistentwith his second suggestion for X. But C does not follow L, nor L X, and the
symbol for 500 was D not D (see Table 1). Priscian
holds that (X) = 1000 from the Greek acrophonicX =
1000 with ( ) added to distinguish it from X. Why
now return to a Greek acrophonic principle? Priscian's theory is a farragoof special pleadingsupported
only by assertion and useful only to show that neither
he nor any contemporaryknew the origin of the Latin
whole-numbernumeral system.
RENAISSANCE
AND MODERN
TALLY-MARK
THEORIES
The legal scholar Andrea Alciati (1492-1550), the
mathematician Nicolo Tartaglia (1499/1500-1557),
and the great Protestant scholar Pierre de la Ram&e
(Petrus Ramus, 1515-1572) are the first writers
known to have questioned Priscian's theory (or Ennius's theory as preservedin Isidore's remark) or to
have proposeda theory of their own. It is notable that
both Alciati and Ramus present in essence the same
theory and neither present it as their own but rather
as if it were a fairly well-known alternative (Ramus:
Romani . . . utuntur...
ad omnemrnurnerum, Alciati:
credendumitaque). This coincidenceand the scholars'
BSA
7See M.N. Tod,"TheGreekNumericalNotation,"
18 (1911/1912) 98-132. The Greek symbols for 50, 500,
5000, and 50000 are derivative(cf. supra n. 1).
8 SeeIfrah(1981) 139-265passimandMeninger26-59,
73-85. I am indebtedto E.L. Bennett(Athens)fordrawing
my attentionto Meninger's work.
9 See M.N. Tod, "The Alphabetic Numeral System in
Attica,"BSA 45 (1950) 126-39.
531
apparentindependenceof each other (as confirmedby
numerous differences in detail), as well as Alciati's
passing referenceto (the historian) BenedettoGiovio
(1471-1554), raise the (unconfirmed)suggestionthat
this theory is pre-Renaissance (but it may be due to
Zeitgeist).
Alciati (1546) explicitly derivesthe Latin numerals
ab agricolarumtesseris.10He claims that V is simply
two lines joined, and that X is the same two totidem
transversas.In fact V ought to be derivedfrom X, not
X fromV by some development.The symbolL for 50 is
painfully elicitedfrom a symbol X, itself derivedfrom
X = 10. Alciati scientifically cites epigraphical evidence:in antiquis marmoribusnon uno in loco reperitur eafigura esse descripta,to an erroneousconclusion
(that X became L for 50 whereas we now know that
the ancient symbol for 50 was \t or the like). The
symbol for 100 is alleged to be [X, an ancient symbol
for C: tertium. . . quodin alphabetoest elementumsic
[X antiquieffigaverunt.For this he cites palaeographical evidence: antiquissimi libri vel ante tempora
Langobardorumscripti. I am unaware of any such
evidenceor of any acceptanceof this theory for C. The
symbolfor 500 he derivesfrom the symbol X for 50 by
the addition of a line to form the symbol X. At this
point it becomes clear that he is guessing-a symbol
A might well become D (but scarcely D, the correct
form of the symbolfor 500), but neither he nor anyone
has ever cited any evidencethat Z = 500. Moreover,if
A = 500 derivesby the additionof a line from X = 50,
why did X not deriveby the additionof a line from V
= five?Finally the symbolfor 1000 is derivedlogically
from the symbol [X for 100 by the additionof another
line to form [X]. Then alii . .. celeritatiscausa sic depingebant (1). The essence of his suggestion is sensible, as is his attemptto deriveall the numeralssystematically, but his theory is seriouslyif not fatally weakened by his inability to cite evidencefor Z , by the elusivenessof his evidencefor [X = C, by his ignoranceof
= 50, and by inconsistentlyderiving X = 50 from X
= 10, but A = 500 from X = 50.
Tartaglia (1556) restrictedhis theoryto 1,V, X, and
a symbol for 50, arguing from an extant use of tallymarks.' The first characterafter I was X ("la prima
fu vna croce obliqua"),and the second was V ("la se-
10 A. Alciati, Parergon:Iuris Libri VII Posteriores(Lyons
1546) Book X, c. XXV, p. 107. On Alciati (1492-1550) the
juris consult see R. Abbondanzain DBI 2 (1960) 69-77.
11N. Tartaglia, General Trattatodi Numeri, et Misure
(Venice 1556) Pt. I, pp. 3v-5. On Tartaglia the mathematician, see Lessicouniversaleitaliano22 (1979) 451 or E. Bortolotti in Enciclopediaitaliana 33 (1937) 286.
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532
PAUL KEYSER
[AJA 92
conda abreuiatura ... "). For 50 Tartaglia claims that
a symbol \ was used ("la terza abbreuiatura fu vna
sola linea obliquamente intagliata"), which does not
easily derive from or yield a symbol \Vor L. Tartaglia
does not explain why he stops at 50.
Ramus (1569) is the first scholar known to have
correctly insisted on the derivative nature of the symbols for 5, 50, and 500, as well as being the first known
to have correctly insisted on the systematic nature of
the symbols for 1, 10, 100, 1000.12 His theory begins
with the supposition that I = 1 represents a single
stroke or tally. Then X = 10 is held to represent two
crossed strokes for the second rank (or place) of our
base-ten numbers. Ramus states that C = 100 because
E is three strokes (for the third rank). But C is a twostroke character,'3 and the three-stroke form E is not
the natural successor to the two-stroke form X. Finally
M = 1000 is alleged to arise from a four-stroke form E
or m. While E is a natural successor to E, it is not
clear why this symbol alone was tipped (E -> m)
nor is this theory consistentwith the earlier forms of
1000 (see Table 1). Given these four as the primary
symbols, those for 5, 50, and 500 were each derived
from half the symbol for 10, 100, and 1000 respectively. Thus half X gives V for 5, half E gives L for 50,
and half the form ( for 1000 gives D for 500. While
the idea is sensible and systematic,the origins of L and
D cannot be correct as they stand, for 50 was first
\j and not L, and 500 was first D and not D. It is also
troubling that D must be derivedfrom(Dand not E or
m, theoretically the earlier forms. Ramus was followed by Matthaeus Hostus later in the same century,
by G.J. Voss and many others in the next two centuries,14 and could gain acceptance up through the
mid-19th century."'Ramus's perceptionon the basis
of literary evidencealone of the essentially systematic
nature of the Latin numeral system, and his recognition of the derivative nature of the symbols for the
12P. Ramus (Pierre de la Ram&e),Scholarummathematicarum libri unus et triginta (Basel 1569) 117. I have not
examined the later reprints (Frankfort1599, 1627). On this
work and the reprintssee W.J. Ong, S.J., Ramus and Talon
Inventory (Cambridge,Mass. 1958) nos. 703, 705, 706 respectively.I am indebtedto Fr. Ong for bringing the copy of
the 1569 edition containing Ramus's autographnotes to my
attention (24 November 1986): see J.F. Daly, "Ramus:Recently Discovered Unpublished Edition of His Mathematical Works," Manuscripta 17 (1973) 80-90. Daly notes
(p. 83) that the changesto this work are slight, and there are
none to p. 117, which I have been able to confirmthanks to
the kindnessof Miss C.E. Weidle, Rare Books Librarianof
the Pius XII Memorial Library, in providing xerographic
copies of pp. 116-18 (9 February 1987).
13Gordon-Gordon100, and fig. 7.
14 M.
Hostus, De numerationeemendataueteribusLatinis
et Graecis usitata (Antwerp 1582) 11-17. Hostus (p. 16)
cites Alciati, but his system is closer to that of Ramus: V is
half X, X e primae notaegeminatae decussationenata videri
potest, C is three units E, and D is either acrophonicfor dimidium [mille] or from M becomingAA. On Hostus (15091587) see Schimmelpfennig in ADB 13 (1881) 191 and
M. Cantor, MathematischeBeitrdge zum Kulturlebender
Vdlker (Halle 1863) 159. Those who adopt this theory
(none cite Ramus or Alciati) are the following: 1) G.J. Voss,
De universaeMatheseosnaturaet constitutione(Amsterdam
1650) c. 8, ?4; on Voss see the monographby C.S.M. Rademaker, Life and Workof GerardusJoannes Vossius (15771649) (Respublica Literaria Neerlandica 5, Assen 1981),
who does not mention this work by Voss although it is listed
in the National Union Catalogue 642:618 NV 0241831-5.
2) P. Borel, Trisor de rechercheset antiquitez gauloises et
franqoises, riduites en ordre alphabgtique(Paris 1655) 95
(as cited in deMatthaeis, p. IX, n. 5, p. XIX, n. 27: I have
been unable to obtain a copy of this) at least for L and C (he
adoptsa pictographictheory for 1,V and X-see n. 26 infra):
"Les anciens faisoient leur C cent, comme un long E qui
n'avait pas de barre au milieu de sorte que le coupant en
deux, la moitie forme un L qui vaut 50.";on Borel (?16201689) see BU 5 (1843) 76 and NBU 6 (1853) 697-99.
3) C.F. Milliet de Chales, Cursusseu mundusmathematicus
'(Lyons 1674) 2(Lyons 1690) Vol. 1, p. 28 who revealingly
remarksPaulo difficiliorerit origocharacterumnumeralium
a Romanis usurpatorum;on Chales (erroneously spelled
Challes in some works), the Jesuit mathematician(16211678), see C.M. Pillet in BU 7 (1843) 410 and see s.v.
"Challes"in NBU 9 (1855) 569-70. 4) F. Bianchini, La
istoria universale provata con monumenti (Rome 1697;
repr. 1747) c. 3, p. 112 (citing Voss); on Bianchini (16621729), the antiquarian and polymath, see S. Rotta in DBI
10 (1968) 187-94. 5) P.D. Huet, Huetiana ou pensdes diverses '(Paris 1722) 2(Amsterdam1723) c. 47, p. 112 only
for L and C, D and (I);on Huet the scholar and bishop, see
BU 20 (1858) 101-105 and C. Hippeau in NBU 25 (1858)
380-90. 6) J.C. Heilbronner,Historia Matheseosuniversae
(Leipzig 1742) Librum IV, caput 1 (pp. 732-35); on Heilbronner the mathematician (ca. 1706-ca. 1747), see M.
Cantorin ADB 11 (1880) 313. 7) E. Corsini,Notae Graecorum; sive vocumet numerorumcompendiumquae in aereis
atque marmoreisGraecorumtabulis observantur(Florence
1749) cap. 3 and Prolegom.(as cited in deMatthaeisp. X, n.
10 and p. XX, n. 29: I have been unable to obtain a copy of
this work); on Corsini see U. Baldini in DBI 29 (1983)
620-25. It is to deMatthaeis'screditthat he cites all sevenof
these works plus Alciati (though not Ramus or Tartaglia).
1' Most importantly G. deMatthaeis (in 1818) VIII, X,
XVIII-XXII; see also J. Leslie, Philosophyof Arithmetic2
(Edinburgh/London 1820) 8-10 without citation, G.F.
Grotefend, Lateinische Grammatik32 (Frankfurt 1820)
163 without citation and G.H.F. Nesselmann, Die Algebra
der Griechen (Berlin 1842; repr. Frankfurt 1969) 88-89.
On Nesselmann (1811-1881), a mathematicianand semiticist, see M. Cantor in ADB 23 (1886) 445-46. The only
citation of Alciati known to me after deMatthaeisis L. Gerschel, "Commentcomptaient les anciens Romains,"Hommages &L'on Hermann (CollLatomus44, Brussels 1960)
386-97.
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THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
1988]
533
Fig. 1. Etruscanfuneraryinscriptionusing * = 100. TLE 890. (After M. Torelli, StEtr 33 [1965] pl. CIVa)
multiples of five, are great advances over Priscian's
puzzle-piece theory, but Ramus's explanations of L,
C, D, and M are less than convincing when confronted
with the evidence available now.
The modern version of this hypothesis is that successive decade-symbols (for 10, 100, etc.) of the Etruscan numerals are made by successive crossings or circlings of the single tally-mark. Then the symbols for
half-decade values (5, 50, 500, etc.) are made by halving the symbol for the succeeding decade. The modern
version has a number of merits: it is systematic, in-
16 Ifrah
(1981) 139-59 cites numerous parallels. A.P.
Ninni, "Sui segni prealfabetici,"AttiVen Ser. 6:7 (1888/
1889) 679-86, pls. 13-16, describeda survival(?) of the use
of tally-mark numerals in Italy.
17 Alciati (supra n. 10), Ramus (supra n. 12), Friedlein
(who may have derived his theory from Hostus whom he
cites in P1. 1 or from Nesselmann whom he includes in his
annotatedbibliographyon p. 3-but he cites no authorities
for his theory and his suggestion for C = 100 is very different from that of Ramus in Hostus/Nesselmann), and
Bortolotti. It is perhaps significant that when I examined
the Etruscannumeralswithout having seen this theorymentioned or advocatedI came to the same conclusion.
18So Alciati (supra n. 10), Tartaglia (supra n. 11), Ramus
(supra n. 12), Hostus, Voss, Chales, Bianchini, Heilbronner, and Corsini (supra n. 14), deMatthaeis, Leslie, Grotefend, and Nesselmann (supra n. 15). In addition, Friedlein
27-28 (?40), Bortolotti 158, Fabretti (1877) 155-57,
Zangemeister 1014-16, E. L6ffler, Ziffern und Ziffernsysteme der Kulturvblkerin alter und neuer Zeit (Mathematische Bibliothek 1, Leipzig/Berlin 1912) 50-52, B. Lefebvre, Notes d'histoiredes mathematiques(Louvain 1920)
30 for I and X only (for \/, C, and (I)he follows Mommsen,
as noted infra n. 54), Meninger 47-52 and Ifrah (1981)
139-59. (Fabretti cites deMatthaeis, Zangemeister cites
Friedlein, Fabretti, and Bortolotti,Liffler cites Zangemeister, Meninger does not cite but follows Zangemeister [see
pp. 52, 293, no. 31] and Ifrah cites Meninger.) Zangemeis-
volves only one principle, can be paralleled,16 and has
been discovered, apparently independently, by a number of scholars.17
The Etruscan numerals (Appendix I) are more immediately tractable. One = I is the single tally, and X
= 10 is the second decade, made by crossing the single
tally.'" Then A = 5 (or V in the Latin case) is the half
of X.'9 The third decade symbol (see figs. 1-4) is now
known to have three crossed lines: * = 100 (see Appendix I), of which half is 4 = 50, though earlier
workers on this theory derived / = 50 in a variety of
Zahlzeiter has also been followedby L. Saalschtitz,"UOber
chen der alten Vilker," Schriftenderphysikalisch-bkonomischen Gesellschaft zu Kdnigsberg 33 (1892) [4]-[9];
M. Cantor, VorlesungenfiberGeschichteder Mathematik1
(Leipzig 1894) 487-88; E.M. Thompson, A Handbookof
Greekand Latin Palaeography3(London1906;repr. Chicago 1975) 105; and V. Gardthausen,"Die r6mischeZahlzeichen," Germanisch-romanischeMonatsschrift Ser. 2:1
(1909) 401-405. The warm criticismsof Zangemeisterin
Mommsen (1888) amountmostlyto forcefulrestatementsof
Mommsen'sown theory,but he correctlynotedthat "decussare" probably never meant "multiplyby ten by crossing."
This scarcely invalidates Zangemeister's work. See also
n. 86 infra for criticismsof Zangemeisterin detail.
19Supra n. 18. L. deFeis, "I dadi scritti di Toscanella ed i
numeri etruschi,"Giornalelinguisticodi archeologia,storia
e letteratura 10 (1883) 241-55 and "Origine dei numeri
etruschi,"Dissertazionidella PontificiaAccademiaRomana
di ArcheologiaSer. 2:7 (1898) 1-19, pl. 1 holds that A preceded X, so that X = twice A. He has similar remarks on
4\ and A = 500. See the remarkssupra ns. 1, 7, and infra n.
32 on the probablepriority of X, *, etc. to A, 4, etc. I do
not cite deFeis (save infra ns. 98, 99 and 103 for his citation
of evidencefor the formsof Etruscannumerals)as he represents a retreatas comparedto Bortolotti(i.e., as he advocates
A as prior to X). On deFeis the priest (1844-19??), see Dizionario biograficouniversale1 (1907) 645.
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534
PAUL KEYSER
[AJA 92
Fig. 2. Etruscanfuneraryinscriptionusing * = 100. CIE 5757. (After M. Cristofani,StEtr 34 [1966] pl. LXXIIa)
other ways.20 At this point a difficulty arises-how
to get C from * ? Many ingenious methods have
been tried, none entirely convincing21 (but see infra
pp. 541-43).
Explanations of the symbol for 1000 in this system
tend to be arbitrary. A few examples will suffice.
Friedlein holds that "fir 1000 war es am natfirlichsten auf 1 zuriickzugehen und durch Einfassung diese
neue Einheit ebenso von dem gew6hnlichen 1 zu unterscheiden, wie das Zeichen fiir 100 von dem fMr
10."22 Thus Friedlein has 0= 100 and 0= 1000. It
seems unlikely that the simpler symbol had the higher
value. Further, why should 100 be X = 10 circled and
1000 be I = 1 circled? Zangemeister claims that (X) =
1000 derives from a fourth "crossing" of * = 100.23
This may be, but no reason is given for the use of
20
See supra ns. 18 and 19. Neither Alciati nor Ramus
knew of Etruscan * = 100, and so tried to make C a threestrokecharacterand L its half (see ns. 10, 12, 14, 15). Friedlein knows C is not the original symbol for 100 but believes
that ? is. From this he derivesuwas its half for 50. DeMatthaeis derived 4 = 50 from an alternate version of X (two
crossedlines) and is followedby Fabretti 155-57. Bortolotti
vacillates and comes to no conclusion on C. Zangemeister,
Meninger and Ifrah (1981) hold the theoryexpressedin the
text.
21 See Zangemeister 1017-18 who providesvarious unattested graphical modifications. Friedlein simply gives up
and returns to Priscian'sallegation of acrophonicstatus for
C. Meninger 49 and Ifrah (1981) 158 providevariationson
Zangemeister's theme, Ifrah possibly rightly (see infra
n. 86).
22 Friedlein 28. Voss (and Chales, either
independentlyor
without citation) suggestthatfour strokesmade ]-1which became I) and that five strokes made MEwhich became (I).
This is not an advance on Ramus or Alciati. Bianchini,
Heilbronner,Corsini,deMatthaeis, Leslie, Grotefend,Nesselmann, Fabretti,and L6ffierall follow Voss at one remove
or another (supra n. 18).
23
24
Zangemeister 1018.
And is paralleled-see Ninni (supra n. 16).
25 Meninger 47-52 and Ifrah (1981) 139-59. Nor do deMatthaeis, Fabretti 155-57 and Bortolotti provide an ex-
curved lines (* would be possible and natural as the
fourth crossing24). Meninger and Ifrah do not explain
@0.25
RENAISSANCE AND MODERN PICTOGRAPHIC
THEORIES
Already in antiquity the origins of writing were
sometimes ascribed to the use of pictographs (e.g.,
Tac. Ann. 11.14). I have been unable to trace the origins of the pictographic theory of Latin numerals before 1655, but it probably arose in the same or similar
speculations.26 It was adopted by Mommsen in the
first edition of his Rimische Geschichte (1856) for I,
V, and X, and has found its way thence into some
handbooks (Sandys and Roby in particular).27 As was
realized already in 1818, however, there is no agreeplanation of (I).
26
I1.Taylor, The Alphabet(London 1883) Vol. 1, p. 6 cites
Grotefend as an advocate of the pictographictheory, but
gives no reference;G.F. Grotefend (1775-1853) in 1820
(supra n. 15) was an advocateof the tally-marktheory:what
does Taylor mean? Advocates of the pictographictheory
known to deMatthaeisare 1) Borel (supra n. 14) 95 as cited
in deMatthaeisp. IX, n. 5 (I have not yet been able to obtain
a copy of this work): "On met ... IIIIpour 4, parcequecela
representeles quatre doigtsde la main.... Et I'Vqui vaut 5
est marque par le cinquieme doigt qui est le pouce, lequel
etant ouvertforme un V avec le doigt index, et deux Vjoints
par le pointe font un X qui vaut 10." On Borel see supra
n. 14. 2) Huet (supra n. 14) c. 47, p. 112 only for I, V and X;
on Huet see supra n. 14. 3) "Berbyde Mailly" in an article,
"Sur le clou, que les payens attachoientsolemnelmentdans
leurs temples,"Mercure de France (March 1728) 479 and
in Varietieshist.physiq. et litter.ou Recherchesd'un savant
2 (Paris 1752) 320 (cited in deMatthaeis p. IX, n. 6 and
p. XIV, n. 18), but I have been unable to confirmthe existence of this writer. Add to deMatthaeis's list Charles
deBrosses, Traite'de la formation mechaniquedes langues
(Paris 1765; repr. 1801) 1.468-72. On deBrosses (17091777) the historian and politician, see Michaud et Foisset
s.v. "Brosses,"BU 5 (1843) 616-18.
27 T. Mommsen, Rimische Geschichte' 1 (Berlin 1856)
1.14.191, 196, 201: in all later editionshe retainedthe theo-
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1988]
THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
Fig. 3. Etruscan funerary inscription using
364 bis 1. (Museo Guarnacci,Volterra 175)
= 100. CII
ment among advocates of this theory as to the pictures
from which the larger-valued symbols were derived,
and little agreement even about X.28
According to all forms of this theory, I = 1 represents an extended finger and V = 5 represents the
whole hand, thumb away from fingers.29 But in the
finger-counting system used by the Latins, five was
represented on the left hand by raising together the
thumb and all fingers save the middle finger, which
was bent into the palm30-a gesture which only
slightly resembles V. For X = 10, disagreement is rife.
Borel (1655), deBrosses (1675), Huet (1722), and
Mommsen derive it from the doubled hand,31 which
implies that the symbol V is prior to the symbol X (unlikely, as noted above, and elsewhere Mommsen
agrees32). Villicus provides a picture of crossed hands
with interleaved fingers, and Barrett claims that X derived from crossed arms.33 Such variety can indicate
only uncertainty, and the gesture for 10 in the fingercounting system was none of these but was made on
the left hand with the three outstretched fingers together and the index finger bent to touch the outstretched thumb at the thumb's outer joint-not an X.
As Mommsen advocates another theory for the
larger-valued symbols, and as there is wild disagreement among advocates of the pictographic theory about
the picture to be connected with the larger-valued
symbols, we consider separately the three theories.
ry, but pagination varied. See also: H.J. Roby, A Grammar
of the Latin Language (London 1876) pt. 1, p. 441 with hesitation; Taylor (supra n. 26) Vol. 1, pp. 6-7, Vol. 2, p. 139;
Mommsen (1887) 598 (767-68); Mommsen (1888) 153-54;
F. Villicus, Die Geschichte der Rechenkunst3 (Vienna
1897) 13-14; J.A.S. Barrett, "A Note on the Roman Numerals," Proceedingsof the Royal Society of Edinburgh 28
(1907/1908) 161-82, esp. 173-74, 177; W.W. Rouse Ball,
A ShortAccountof the History of Mathematics4(Cambridge
1908; repr. New York 1960) 126; D.E. Smith, "The Roman
Numerals," Scientia 40 (1926) 1-8, 69-78, esp. 1-2, 4-5;
J.E. Sandys, Latin Epigraphy2, rev. by S.G. Campbell
(Cambridge 1927) 54; Encyclopedia Britannica 16 (1958)
612, s.v. Numerals (D.E. Smith) and in later editions.
28 See deMatthaeis p. IX "mostransiassai confusi ed incerti nello spiegare come sieno indi nate le figure V, X, L, etc."
29 See the references supra ns. 26, 27. DeBrosses (supra
n. 26) has only the thumb and index finger extended.
30 See RE S.14 (1974) 112.26-113.32, s.v. digitorumcomputus (H. Hommel); Reallexikonfiir Antike und Christen-
535
DeBrosses makes L on the left hand with index finger and thumb at right angles and C "pourroitetre la
meme figure en courbant les deux memes doigts."
Then D is the index of the right hand curved and
joined to the thumb of the same hand held out straight,
while (I) is D doubled.34His own hesitationon C and
the lack of correspondencebetween these gesturesand
the known finger-countinggestures (as noted below)
do not inspire confidence.
Villicus explains L as a hand gesture similar to that
for V,35but 50 was \V(not easily obtainedfrom a hand
gesture), and in the finger-countingsystemwas repre-
Fig. 4. Etruscan numerical graffiti on terracotta, showing
* = 100. AppCII 114, pl. IV.
tum 7 (1968) 915-20, s.v. Finger (K. Gross); Encyclopedia
Britannica 9 (1954) 249, s.v. Finger Numerals (D.E.
Smith); J.H. Turner, "Roman Elementary Mathematics:
The Operations,"CJ 47 (1951) 63-74, 106-108 (a reference I owe to E.A. Fredricksmeyer);E.M. Sanford, "De
Loquela Digitorum," CJ 23 (1928) 588-93; D.E. Smith,
History of Mathematics 2 (New York 1925) 196-202 (and
the important referencesthere); L.J. Richardson, "Digital
Reckoning among the Ancients," American Mathematical
Monthly 23 (1916) 7-13 (and the important references
there) or Villicus (supra n. 27) 10-11. All further remarks
on this finger-countingsystem refer to these authorities.
31See supra n. 26 for Borel and Huet, and Mommsen
(1887) 598.
32 Mommsen (1888) 155, 153 (sections 10 and 2).
33Villicus (supra n. 27) 14, and Barrett(supra n. 27) 174,
177.
34DeBrosses (supra n. 26) 469-70.
35 Villicus (supra n. 27) 14 for L, C, D, and (D.
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536
PAUL KEYSER
sented on the left hand by the fingers outstretched
with the thumb folded into the palm. Villicus suggests
that C was a hand gesture with all digits outstretched
to form a C as seen from the edge, but 100 was represented in the finger-counting system on the left hand
by bending the little finger with the other digits extended. Villicus suggests a similar form for D, with the
fingers bent sharply to join the thumb at its end, but
this gesture when made resembles O not D, and 500 in
the finger-counting system was represented on the
right hand by the same gesture that represented five
on the left: middle finger only bent. Finally T is derived from a doubled D, which falls afoul of the remarks on X and V above. It is to be noted that Villicus
was well aware of the finger-counting system.
J.A.S. Barrett claims that a new gesture was needed
every time five of the previous gesture were used.36
But this should lead to a quinary system (i.e., symbols
for 1, 5, 25, 125, etc.), rather than to a decimal system,
and this fails to explain the absence of such forms as
or DDDD. For T = 50, Barrett claims that
VV,
W''V,
a picture of raised arms, represented by T, was used.
Yet 50 is not T but \. Barrett holds that the system
involved a "teapot handle" gesture for C = 100 and the
"teapot handle" position for D: what is one to make of
this? Finally (1) represents both arms forming the
"teapot handle" simultaneously. Moreover, he claims
that no further gestures are possible, thus "explaining" the "fact" that (D is the largest symbol used. But
we can imagine many more gestures, e.g., arms joined
over head T, arms lowered +, and we can also
imagine positions or gestures of the feet.
The pictographic theory is initially attractive but finally invalidated by an inability to generate consistent
"pictures" of 4, C, D, and (D. Pictographic theories
36Barrett (supra n. 27) 175-76 for the allegationabout the
five-fold repetition; pp. 177-82 for the "derivation"of the
signs for L, C, D, and (D.
37Ifrah (1981) cc. 10-14 discusses numeration systems
and finds none whose origin was pictographic.Neither Villicus (supra n. 27) nor Mommsen (1887), (1888) are able to
offer parallels. Barrett (supra n. 27) 170-75 adduces the
"parallel"that some numeral words and many short distance-measuresare derivedfrom parts of the body. But this
proves either too much or nothing: the generality of such
numeral words should result in a numeral system identical
to the Roman in most parts of the world, and this we do not
see. D. Schmandt-Besserat, "An Ancient Token System:
The Precursorof Numerals and Writing,"Archaeology39.6
(1986) 32 claims that the Sumerian symbols for 10 and 60
were in origin pictographsof standardcontainersor heaps
of those quantities: i.e., if true, the Sumerian system would
involve something similar to writing a pictographof an egg
carton for 12, and reading "dozen-eggs"or some such. This
parallel is rather remote in resemblance,space and time-
[AJA 92
assume 500 = D, but 500 = D. We may ask if there is
any numeral system which is known to be pictographic, and if so why do the advocates of this theory not cite
it?37 Given no evidence for it, and the existence (in the
finger-counting system) of evidence against it, we
must pronounce this theory no improvement over
Priscian or the acrophonic theories.
MODERN ACROPHONIC THEORIES
Acrophonic theories of the origin of the Latin
whole-number numerals, like bad pennies, turn up
again and again. The earliest advocate, Francesco
Orioli, provided explanations of the symbols for 5, 10,
and 50.38 He holds that A (the Etruscan form for five,
see Appendix I) is derived from the Greek acrophonic
LI for w'vrEc. He claims that X = 10 arises from + (an
old form of T), which is acrophonic from a (hypothetical) early form *tesen = 10. But we now know that the
symbol X is a sibilant, not a dental, in Etruscan,39 and
T was always T, not + , in form.40 I query whether an
acrophonic numeral system, presumably originally
Etruscan on Orioli's hypothesis, would adopt a Greek
symbol for five and a Latin symbol for 10. Finally,
Orioli derives 4 (the Etruscan form for 50, see Appendix I) from the Etruscan letter Y (chi), allegedly
acrophonic for quinquaginta. But it is hardly likely
that 5 and 50 in an acrophonic system would have
different initials (cf. the Greek system41), or that an
Etruscan system would use a Latin word, and Orioli
is unable to parallel the use of aspiro-velar chi for the
labio-velar qu. It is to his credit that Orioli seems to
have been the first to recognize that the original form
of L was \.
Karl Otfried Mtiller (1797-1840), the father of
scientific Etruscology, not long after Orioli's first at-
but it is the closest available.
38 F. Orioli in three works: 1)
"Sull'originedei numeri
etruschi e romani e sull'infissione solenne del chiodo annuale," Opuscoli letterari 1 (1818) 208-26, esp. 219-24;
2) Spiegazioni d'unagemma etruscadel Museo reale di Parigi (Bologna 1825); and 3) "Nuovo commento sopra una
gemma etrusca del Museo di Parigi," Spighe e Paglie 4
(Corfu 1844) 137-41. In 1818 he acceptedC from centum,
though he believedthat the Etruscansthemselveshad no numerals larger than 50 (infra n. 58). In 1825 he changed his
mind basedon the symbol ? in the abacus-gem(see App. I).
On Francesco Orioli (1783-1856) see A.E. Morandi, La
figura e l'opera di F. Orioli l'archeologo(Viterbo 1984), a
referenceI owe to an anonymousrefereeof AJA.
39 E. Fiesel, "X Representsa Sibilant in Early Etruscan,"
AJP 57 (1936) 261-70.
40 A.E. Gordon, "On the Origins of the Latin Alphabet:
Modern Views," CSCA2 (1969) 157-70.
41 See Tod (supra n. 7).
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1988]
THE ORIGINOF THE LATIN NUMERALS1 TO 1000
tempt, ventured a more systematic but wholly speculative acrophonic theory of the Etruscan numerals.42
Miiller suggested that V was acrophonic for a (hypothetical) Etruscan word *u- for 5, but it is now generally accepted that Etruscan five was "max."43 Miler
suggested that X or + was acrophonic for a (hypothetical) Etruscan word *t- for 10. But we now know
that neither X nor + is a form of T.44 Miller suggested that /4 was acrophonic for a (hypothetical)
Etruscan word *X- for 50, but it is likely that even in
Etruscan the words for 5 and 50 began with the same
letter.45 It is to his credit that Muller seems to have
been the first to realize that the original formof C was
not C.
Camillo Tarquini, Jesuit and later Cardinal, advocated a Semitic acrophonic system (1864) at least for
A, X, 4, and 9= 1000 ( * = 100 is "una vera cifra
somigliante alla cifra Fenicia").46 He claims that A is
an Etruscan letter with the value of "heth" (the aspirate from which Greek 7/ derived) for Hebrew "hImesh" (nmn) = 5, but we now know that the value of
the late epichoric Etruscan letter A was m.47 By various unconvincing special pleas, X (6 or X) is related
(via the aspirate T, chi) to the guttural 'ayin for Hebrew "'eser" ('it) = 10. To avoid the difficulty that
50 and 5, even in Semitic languages, begin with the
same phoneme, Tarquini claims that "era necessario
di adoperaretal lettera, il cui sono fosse al Hheth [sic]
aspro il piuiaffine ... talle era certamenteil Koph ...
la lettera \/, la quale e certamente un
Koph." Finally ? is related to the letter 'aleph for Hebrew "'elef" ('XK) = 1000. The special pleading, the
cioe a dire...
error of fact in the case of A, the inability to fit * (or
D' , not even mentioned by Tarquini) into the system
would each refute this theory-together they are fatal.
Nor is confidence increased when Tarquini proceeds
to decipher the Etruscan language as Phoenician.
Tarquini does correctlyemphasize the connectionof
the Etruscan numerals to the Latin.
42 K.O. Miuller, Die Etrusker' 2 (Breslau 1828) 317-21.
Muller connects Etruscan + = 100 with 0 and ? = 1000
with f (Etruscan 8) in the same acrophonicsystem. Mtiller
himself is aware of the defect that (at his time) not one
Etruscan numeral-wordwas known (p. 320).
43Bonfante-Bonfante79.
44 See Fiesel
(supra n. 39) on X as a sibilant.
45Bonfante-Bonfante79.
46 Camillo Tarquini, S.J. "Dichiarazionedell'epigrafedel
lampadario di Cortona, della lettera A, e delle note numeriche degli Etruschi,"Dissertazioni della PontificiaAccademia Romana di Archeologia15 (1864) 68-93. We find A =
5 explained on pp. 70, 77-78, X = 10 on pp. 80-84, \ = 50
on pp. 78-79, * = 100 on pp. 84-85 and ? = 1000 on p. 85.
On Tarquini see C. Testore in Enciclopedia cattolica 11
537
Recently, two scholarshave vainly attemptedto revive an acrophonic theory. Pisani48 correctly notes
that Latin I = 1 is also the symbol for 1 in the Greek
acrophonic system, and that both the Greek and Latin
systems have symbols for the same set of numbers (i.e.,
for 1, 5, 10, 50, etc.). From this he invalidly concludes
that the two systems must embody the same principle-but the symbol I = 1 is widely attested (as noted
above)and the existenceof symbolsfor the same numbers is evidence of a common need, not a common origin. The remainder of Pisani's theory is as speculative
as Muiller's, but less systematic. He claims that in an
older alphabet V = k (as in the Runic futhark, "Ru-
nenalphabet") for *kuinkue. Similarly, X = t, for
*tecem, in Etruscan (which had no d); but when Pisani wrote, X (variant + ) had been known to be a sibilant in Etruscan, as noted above, for over 15 years.
Pisani claimed that \ = qu, for quinquaginta, in some
older alphabet. But on the analogy of the Greek system invoked by Pisani, the symbol for 50 ought to be
A. Pisani acceptsC as acrophonicfor centum,but explains D as the half of (D (1000). This admission in
itself invalidates his analogy to the Greek acrophonic
principle: the symbol for 500 ought to be A. In the
case of 0, Pisani states: "m6chte ich . . . den Wert von
h annehmen," and supports this with a derivation of
Latin mille from an archaic form *heili from IE
*ghesl- = 1000.49
Pisani's student Rix in his teacher's Festschriftso
accepts Pisani's idea that the system is acrophonic, but
rejectseverythingelse. He flirts with A as a variantof
Etruscan m, hence acrophonic for "maX" = five, but
concludes that A = half X, "das Wahrscheinlichste
ist." Rix holds X =- to be acrophonic for Etruscan
*San = 10, which is possible.5' After explaining 'b as
half *, the Etruscan 100 (see Appendix I), Rix
claims that Etruscan ? = 100 is acrophonic for *0- =
100, while Etruscan *= 100 is acrophonic for *s- =
100. Finally Rix suggests that 0 = 1000 derives from
(1953) 1765.
47 See Bonfante-Bonfante
64 following M. Pallottino,
Thesauruslinguae Etruscae I (Rome 1978) 421, A.J. Pfiffig, Die etruskische Sprache (Graz 1969) 20-21 and
J. Heurgon, "Note sur la lettre A dans les inscriptions
etrusques," Studi in honore di Luisa Banti (Rome 1965)
177-89.
48 V. Pisani, "Die rdmischen Zahlzeichen, ein alteres
r6mischesAlphabetund Lat. mille,"RhM 96 (1953) 89-93.
Pisani(supran. 48) 92.
ZahlwortundZifferin altenMitsoH. Rix, "Buchstabe,
41
telitalien," Studi linguistici in onore di Vittore Pisani 2
(Brescia 1969) 845-46.
51 Bonfante-Bonfante79 or Pfiffig (supra n. 47) 123-30
for numeral words.
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538
PAUL KEYSER
Etruscan *5- = 1000. It is unlikely that there were
two words for 100, there is no evidence that * was
ever a letter in Etruscan, and we have no idea what
the Etruscan words for 100 or 1000 were.52
Acrophonic theories are a Procrustean bed into
which Latin (or Etruscan) numerals can be fitted only
by speculation or special pleading. It is noteworthy
that none of these theories has won acceptance in detail by any scholar other than its originator (Pisani
accepts only Rix's idea that the system is acrophonic,
and the derivation of the symbol X from *t- is found in
three of five versions of the theory).
MOMMSEN S UNUSED-LETTER
THEORY
For the origin of \V= 50, C = 100, and ( = 1000,
Mommsen firmly adopts the unparalleled explanation
that they are derived "ohne Zweifel" from letters of
the (Chalcidic) model alphabet which were useless for
Latin and so used for the continuation of the numeral
series.53 He explains I, V, and X on the (entirely different) pictographic system (as noted above). This
dual theory is the best known and most widely accepted of all the theories. Mommsen has been fol52 Supra n. 51 and see L.H. Jeffery, Local Scripts of Archaic Greece (Cambridge 1961) for Arcado-Locrian +k=
psi: 104-105, 206-207, 213-14, 259. This letter has nothing
to do with Etruscan *1= 100.
53 Mommsen (1850) 19-20, 33-34, Mommsen (1887) 589601 and Mommsen (1888) 152-56. He refers to the theory
in Geschichte des rbmischen Miinzwesens (Berlin 1860)
188-89 (trans. L.C.P. Blacas, Histoire de la monnaie romaine 1 [Paris 1865] 201). The theory is unparalleled,and
Miller (supra n. 42) 320 remarks that there are only two
ways to use letters as numerals:in an acrophonicsystemand
in an alphabetic system (cf. Tod, supra ns. 7 and 9). See
supra n. 5 and infra n. 83 on the arbitrary systems of an
eighth-centurymanuscript.
54 See (in chronologicalorder) E. Goebel (in a review of
A. Vanitek, LateinischeGrammatik)ZOstG 10 (1856) 764;
F. Ritschl, "Zur Geschichte des lateinischen Alphabets,"
RhM 24 (1869) 1-32 (esp. 12-13, 14 n. 27, 18, 28-31) =
OpPh (Leipzig 1878; repr. Hildesheim 1978) 691-726;
Roby (supra n. 27); W. Deecke and K.O. Miller, Die
Etrusker22 (Leipzig 1877; repr. Graz 1965) 534 and plate
facing p. 560; Taylor (supra n. 26) 6-7; E. Hiibner, Exempla scripturae epigraphicae Latinae (Berlin 1885) LXXLXXI; Hubner, Handbuch der klassischenAltertumswissenschaft21 (Munich 1892) 651; A. Kirchoff, Studien zur
Geschichte des griechischen Alphabets4 (Gutersloh 1887;
repr. Amsterdam 1970) 132-33; F. Bucheler, "AltesLatein
XVIII," RhM 46 (1891) 238-41 = Kleine Schriften23 (Osnabruick1965) 203-205 (the latter referenceI owe to W.M.
Calder III); R. Cagnat, Cours d'?pigraphie latine 2(Paris
1890) 3(Paris 1898) 4(Paris 1914) 30-32; R. Kohner and
F. Holzweissig, Ausfiihrlichegrammatik der lateinischen
Sprache2 1 (Hannover 1912) 5, 630-31; Lefebvre (supra
n. 18) 30; Smith (supra n. 27) 1-5; M. Leumann, Lateinische Grammatik: Handbuch der Altertumswissenschaft
[AJA 92
lowed by a galaxy of greats, and his theory is found in
all the best handbooks.54
Mommsen claims that I, V, and X are pre-alphabetic for two reasons: "das verschiedene in ihnen obwaltende graphische Princip," and their resemblance to
the Etruscan symbols.55 The first tends to prove their
non-alphabetic character. The second point might
rather indicate that the numerals are contemporary
with the alphabet, as the Latin alphabet is generally
conceded to be Etruscan in origin.16 Mommsen's realization that the Latin and Etruscan symbols are related is admirable, but his second point as stated unfairly represents the case: not only I, V, and X, but
also Latin \V = 50 strongly resemble the corresponding Etruscan numerals (see Appendix I). Finally,
Mommsen's own hesitation on the origin of X shows
that his argument (in favor of a pre-alphabetic origin)
is not persuasive: he at first derived X = 10 from X =
"ks," which he believed to be unused in Etruscan."
Mommsen's theory has been criticized on the grounds
that it implies that the Latins originally had no symbolic numerals larger than 10.58 This implication
Mommsen accepts, citing as an unconvincing parallel
2.2.1 (Munich 1926) 47; Sandys (supra n. 27) 54; F. Cajori,
A History of MathematicalNotations 1 (Lasalle, Ill. 1928)
30-31; D.E. Smith, History of Mathematics2 (New York
1925) 55-56 (hesitantly); D.E. Smith and J. Ginsburg,
Numbers and Numerals (Contributionsof Mathematics to
Civilization 1, New York 1937) 14 (at least for V and X as
pictographs), repr. in J.R. Newman ed., The World of
Mathematics1 (New York 1956) 448; D. Diringer, The Alphabet2 (New York 1948) 536 (for L, C, D, and 0D);Encyclopedia Britannica 16 (1958) 612, s.v. Numerals (D.E.
Smith); M. Guarducci, L'epigrafia greca 1 (Rome 1967)
220; H. Jensen, Die Schrift in Vergangenheitund Gegenwart3 (Berlin 1969) 512-23 (who also takes note of the
"tally-mark"theory; I owe this referenceto Werner Krenkel, Rostock);A.J. Vaccaro,La numeracidnlatina:aspectos
y problemas (La Plata, Argentina 1969) 17-19; Meyer
(supra n. 4) 30-33; Der kleine Pauly 5 (1975) 1451.30-37,
43-46, s.v. Zahlensystem,Zahlw6rter(D. Najock);Gordon
44-49; and O.A.W. Dilke, Mathematicsand Measurement
(Berkeley/London 1987) 15. I do not know what to make of
J. Walter Graham, "X= 10,"Phoenix 23 (1969) 347-58, in
which the numeral system in use at Olynthus (X, 8, Wt= 10,
100, 1000) is first explained as alphabetic by positing (an
unattested)V = 5 in a system lacking any other half-decade
symbols and a (speculative)substitution -> 8, and then
(entirely without evidence)is linked to Etruscannumerals.
11Mommsen
(1887) 598.
56Gordon (supra n. 40).
17
Mommsen (1850) 20.
Zangemeister 1012-13. For examples of primitive incompetencein countingabove some small number,see Strabo 11.4.4 (the Caucasian Albanians count no higher than
100) cited by Hostus (supra n. 14) 14 and Heilbronner
(supra n. 14) 732; Arist. [Pr.] 911a.1-4 (some Thracians
count no higher than four) on which see T.L. Heath, Math58
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1988]
THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
539
the Duilius-inscription (260 B.C., CIL I2, 2.25 = ILS
65) where (irf = 100,000 was repeated at least 21
(probably 33 or 34) times.59
When we consider the unused-letter theory per se,
we find further faults. That \Vresembles the Chalcidic
chi, and (D the Chalcidic phi, is undeniable, but there
are no extant examples of (Chalcidic) ? = 100 in
Latin inscriptions.60 Mommsen cites the Etruscan
abacus-cameo (CII2578 ter)6' for ? = 100, but there
it is more likely that ? = 1000 (see Appendix I).
Mommsen claims that 0 became C by a graphic process, a possibility which he elsewhere denies.62 Moreover, Mommsen claims that the Etruscans, who did
use 0, distinguished ? = 100 from ? = 0 by removing the cross in theta,63 but Etruscan theta retained
the cross into the fifth century,64 and the removal
probably occurred because they no longer needed to
distinguish it from the unused letter O.65 (Numerals
are more likely to be differentiated from letters, than
letters from numerals: cf. our 5 and S.)
It must also be noted that Mommsen's theory must
be specially modified to account for D = 500. D is not
an unused letter, but rather resembles half of 0D, and
this is how Mommsen explains it66 (i.e., he retains
Ramus's theory). This admission tends to invalidate
his theory-he now has three principlesfromwhich to
constructseven symbols.
Even Mommsen's fundamentalnotion is questionable. A necessary condition for the validity of his
theory is that the three letters ( y chi, 0 theta, and
P phi) needed for \V, C, and (Dbe unused and that
they be the only letters of the model alphabet which
were not used. Neither half of this dual condition is
met (see Appendix II). Both
(?) and M (san) in the
model alphabet are unattested in Latin. There is
(slight) evidence that Y , 0, and 4) were known.
Finally, Y, 0, and 4, or even W and M, may have
been includedin some formalLatin abecedaria(cf. the
retention of the unused letters B, D, and O in early
Etruscan abecedaria67),and the replacementof zeta
by G suggests that the unused zeta, at least, was retained (see Appendix II). In fact, we have only one
early Latin abecedarium before the first century
A.C.,68CIL 12, 2.2903, which is dated to ca. 350-300
B.C. and contains Z ( I ) between F and H (but no
other unused letters): here at least the (presumably
unused) I was retained.
Followers of Mommsen have modifiedhis theoryin
ways which have not won general acceptance.Ritschl
hesitates at a pictographicorigin for V = 5 and X =
ematics in Aristotle (Oxford 1949) 259; and thirdly Ifrah
(1981) 5-19 for some modern examples. Such people are
preliterateand give no grounds for separatingthe Latin numeral system into two parts. In 1818 Orioli (supra n. 38)
221-22 had already claimed that the Etruscanshad no numerals greater than 50, arguing from the absence of evidence. In 1825 he changed his mind (supra n. 38).
"9Mommsen (1888) 153.
60 Mommsen (1850) 33 cited CIL 1', 1156 = CIL 10,
6514 = CIL 12, 2.1510 = ILS 3819, but the numeral is not
8 = 100 but (0 = 1000, as Mommsen (1887) 599 n. 1 concedes. Mommsen was in the first case only following Muiller
(supra n. 42) 319, and was himself followed by Hiibner
1885 (supra n. 54) LXXI. Meyer (supra n. 4) 30 is still (in
1973) unable to produceany examples ("lisst sich diese Annahme nicht durch belegte Beispiele sichern"). R.S. Conway, The Italic Dialects 2 (Cambridge 1897) inscr. 168 =
CIL 2873 quater reads 8e9 not as lert (i.e., hef retrograde) but as eee = 300, though he admits that this is
doubtful,and Gundermann35 remarksthat it is "leereVermutung."The plate in CII seems clear (hef not eee ) and I
am unaware of any scholarly acceptanceof Conway's reading. Note that CII 2873 quater is an undated Central Oscan inscription and Conway has C = 100 in another undated Central Oscan inscription (59 = CII 2806). In Umbrian the only relevant inscriptionsare Conway 354, of the
Gracchanperiod, in which C = 100, and the TabulaEugubina VIIb.4 (cf. CIL I2, 2.366 + 2872) CCC = 300. See
n. 101 infra.
6' Mommsen (1887) 599 n. 2. The reading of this
gem is
doubtful;see Appendix I.
62
Allegation: Mommsen (1887) 599; denial: Mommsen
(1888) 155.
63 Mommsen (1887) 599.
64Bonfante-Bonfante 64, following Pallottino (supra
n. 47) 421, and Pfiffig (supra n. 47) 20.
65 Some Greeks also "removed"
the cross from 9 at about
the same time, even though they retained O. The resultant
epigraphic 0 is called "dotted0" (from the center-punch
markof the cuttingcompass,presentalso in the ordinaryO).
See Jeffery (supra n. 52) 29 and passim. The conclusion
may be drawn that the Etruscans "removed"the cross
without concernfor a distinctionbetween 9 and any other
character.
66 Mommsen
(1887) 599-600.
67 See the list of abecedariain Bonfante-Bonfante106-109
and the longer list in Pallottino (supra n. 47) 409-10 and
the table in Pfiffig (supra n. 47) 19. From Pallottino's list
(partly dated) it would seem that B, D, and O were dropped
in the sixth century B.C.
68 On CIL I2, 2.2903, cf. infra n. 107. Gordon (supra
n. 40) 167 cited only Pompeiian (i.e., first century A.C.)
abecedaria, as did Ariodante Fabretti (from Garrucci and
Zangemeister)7-8. See also Meyer (supra n. 4) 26-27. We
know that none of these six unused letters were included in
the formal abecedariumby the time of Cicero,as he (Nat.D.
2.93) indicates that the alphabet contained only 21 letters.
Other extant abecedariaare even later, e.g., the two fifthsixth-century abecedaria in P. Antinoi fr. IV:see H.J.M.
Milne, Greek Shorthand Manuals (London 1934) 70,
pl. IX; B.L. Ullman, "Two Latin Abecedariafrom Egypt,"
AJP 56 (1935) 147-48; and R. Cavenaile, CorpusPapyrorum Latinarum(Wiesbaden 1958) 136-37, nos. 58-59.
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540
PAULKEYSER
10, and would like to see X derived from the cross in
9= theta and V as half X (rather than X as V doubled).69Thus Ritschl reduces the number of principles needed from three to two at the expense of introducing further special pleading. Deecke, in revising
Miiller's Die Etrusker, related X = 10, / = 50, and
9 = 1000(?) to the aspirates X, 0, 0 in use in Etruscan, used here in reversealphabeticalorder.70 He explains Etruscan * = 100 as a Cypro-Lycian import,
and A as the half of X. Deecke too requiresthree principles, as well as special pleading. Mentz adopts a
similar theory, much more systematic but less in accord with the evidence.71 He claims that the Latins
took the last three letters of the Greek alphabet (in
arbitrary order) for the decade values, and halved
them for the half-decadevalues. Thus (D= 1000,)\ =
100, X = 10, then D = 500, ? = 50, V = 5. (Note that
this gives \ an incorrect value and gives the wrong
forms for 500 and 50.)
It is perhaps not without significance that Manu
Leumann, who had acceptedthis theory in 1926, has
recently(1977) expresseddoubts.72In summary,I suggest that, though authorizedby Mommsen and accepted by many handbooks,this theory is but a small advanceon Priscian,and is not supportedby the evidence.
A FEW RENAISSANCE
))
"NEBENTHEORIEN
AND MODERN
Under this heading I considera few theories which
do not fall into any of the preceding classifications.
None of these theories deservesor has attractedmuch
acceptance.They are includedfor completeness.
Hostus's (1582) theory that the numeralsowe their
origin to the abacus73is dependentonly on the observation that the abacushas decimalcolumns,often with
a "fives"row, which the Greek abacus had also. This
insight hints at a relation between the abacus and additive non-place-value numeral systems, but cannot
69 Ritschl(supran. 54) 13, 18. Followed
by Cajori(supra
n. 54) andRoby(supran. 27). X wasusedas a symbolfor0:
see AppendixII.
70 DeeckeandMtiller(supran. 54) 534.
71
A. Mentz, Geschichteder griechisch-r6imischenSchrift
(Leipzig1920)41.
72
Leumann(supran. 54) in 1926;it is in the new edition
(1977)of the sameworkthathe hesitates:p. 5.
73Hostus(supran. 14) 20-22, refutedby Cantor(supra
n. 14) 160. Gundermann
42 approvesCantor'srefutation.
Eitherindependently
orwithoutcitation,Hostusis followed
by RE S.3 (1918) 11.9-64, s.v. Abacus (9) (A. Nagl). Nagl
is himselffollowedby C.M. Taisbak,"RomanNumerals
and the Abacus,"ClMed 26 (1965) 147-60. An apparently
independentsuggestionis due to P.A. Dapre,"TheOrigin
[AJA92
yield the forms of the numerals, as is amply demonstrated by the very different forms of the Greek acrophonic numerals.74
Lanzi (1824) assumes that the Etruscan numerals
(to 50) were an alphabeticsystem.75He alleges that X
= 6 was the 10th letter and that T was the fifth letter
following (counting inclusively) so that T (or T ) had
the value 50. The numeralA = 5 was the lower half of
X. But his explanation of the values of X and T contradictsthe facts:X = S was at the end of the Etruscan
alphabet, EB= ksi was the 15th letter of alphabets
which includedit, and T was the seventhletter following (counting inclusively). Furthermore, as Fabretti
points out, there is no evidence that any other letters
were used for alphabeticnumerals;we find XX, XXX,
XXXXand not n, P, Z (as would be requiredby Lanzi's theory).76
Faulmann (1880) advocates a muddled Semitic
acrophonic system, confused with a native Latin
alphabeticsystem for L, and a naive Latin acrophonic
system for C = centum and M = mille.77 Five quinque
derives from *que-que = and-and (so he alleges) and
in Hebrew this would be 11(vav-vav), "dessendilteste
Form Y war," which meant 2 (how 2 became 5 he
does not explain). X derives from Greek K from Hebrew kaph 2, standing for "Alles, die Ganze," (bZ)
(presumably "all"became "10"from the 10 fingers).
Then K, the 10th letter of the Latin alphabet, is followed by L, and as 2 became 5, so 20 became 50. I do
not understandhis explanationof D = 500.
In such a context even Gundermann's(1899) Semitic alphabetic system will seem partially plausible.78While wholly speculativeand supportedonly by
special pleading, his theory has at least the merit of
being systematic. All symbols V, X, \/, C, D, and
(Dare explained as derivative forms of Semitic alphabetic numerals, though the forms are often not
close and he must resort to several different Semitic
alphabets.
of the Roman Numerals," Didaskalos 5 (1976) 359-60:
Cantor'srefutationholds.
74
SeeNagl (supran. 73) ontheGreekabacusanditsasso-
ciation with the Greek acrophonicnumerals.
75 L.A. Lanzi, Saggio di lingua etrusca e di altre antiche
d'Italia2(Florence 1824/1825) 385-86 (?XIV).
and T, see the abecedariain Pallot76 For the positionof EB
tino (supra n. 47) 409-10, and see Fabretti (1877) 154-55
on the absenceof fl, P, Z.
K. Faulmann, IllustrierteGeschichteder Schrift (Vien77
na and Leipzig 1880) 546-47.
Gundermann45-49. The special pleading of his theory
78
does not invalidate his learned and useful survey of Mediterraneannumeral systems.
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THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
1988]
Only brief mention can be accorded to the theory of
Minshall (1976) that I is a mark, and X is used every
10 marks (so far a tally-mark theory), and that "there
are really only three more symbols which" would be
used to write on "stubborn materials[:] the acute angle
V, the right angle I [for 50], and the obtuse angle"
(for 100).79 According to Minshall, "after [100] a
third mark [sc., in addition to the two in <] was introso that < = 500... and two triangles toduced..
soon
had their lines changed to curves .. to
gether
produce co." Thus he advocates a geometrical theory
for 5 as well as 50 and up. The lack of a parallel, the
essentially arbitrary order in which the angles are assigned, and the hypothesized disconnection of 50 and
100 as well as the hypothetical priority of 500 to 1000
all render the theory unconvincing.
Although Faulmann, Gundermann, and Tarquini
find (apparently independently) a Semitic origin for
all or part of the Latin whole-number numeral system, the wide divergence of their views reveals the essentially arbitrary nature of their speculations (which
owes more to Zeitgeist than to Wissenschaft) and
gives no ground for assuming a Semitic origin.
PRINCIPLES
OF EXPLICATION
Certain principles must be borne in mind in evaluating theories about the origin of the Roman numerals. One: it is far more likely that one system underlay
the seven numeral symbols than that several did.80
Two: the Etruscans dominated Rome at a formative
period of its history and provided an alphabet, and so
one ought to seek to derive the Latin numerals from
the Etruscan."8 Three: a system or principle which
can be paralleled is to be preferred to one which can7"B.W. Minshall, "The Roman Numerals,"Didaskalos5
(1976) 262-65.
80 Cf. Zangemeister 1012, who seeks a solution which has
"ein einheitliches Entstehungsprinzipftir die ganze Reihe
bis 1000 incl.";Nesselmann (supra n. 15) 90 (with reference
to the tally-mark theory): "Diese Deutung ist so ungeheuer
einfach und ungezwungen und mit einer solchen Consequenz in sich abgeschlossen,dass man kaum begreift, wie
irgend eine andere sich hat heraus bilden k6nnen." Most
theorists have realized the importance of this principle of
economy, but the realization is noticeably lacking from
Mommsen'stheory.
81 On the Etruscan origin of the Latin alphabet, see esp.
Gordon (supra n. 40) with the literature there cited. The
Etruscan origin of the Latin numerals was first explicitly
realized by G.R. Carli, Count Rubbi, Delle antichith italiche 1 (Milan 1788) 22, apparentlyfollowing up a remark
by Maffei who thought that "gli Etruschigli avesseroappresi dei Romani."On G.R. Carli (1720-1795) see E. Apih in
DBI 20 (1977) 161-67. On the Etruscan origin of Latin
numerals see also Orioli 1818 (supra n. 38) 225; G. Micali,
L'Italia avanti il dominio dei romani 1 (Milan 1826) 230;
541
not (omnibus paribus). These first three are principles
of economy of hypothesis. Four: symbols for decadevalues (1, 10, 100, etc.) are prior to those for half-decade values (5, 50, 500, etc.), whether the half-decade
values are derived from the following decade symbol,
from the preceding decade symbol (as in Greek), or
from no decade symbol.82 Five: letters are only used as
numerals either acrophonically or alphabetically.83
Six: acrophonic systems must use compound symbols
(e.g., l for rITEvTby •'Ka = 50) for the half-decade
symbols.84
Few, if any, of these principles are satisfied by
Mommsen's theory, or by Priscian's. Since the Latin
symbols for the half-decade values are not compound,
the system is not acrophonic. Since the Latin symbols
are only seven in number, the system is not alphabetic.
Consideration of these principles and the theories presented above suggests that the most likely theory is the
tally-mark theory: Etruscan numerals, based on a single simple system in which the half-decade symbols
are graphically half the succeeding decade-value symbols, were adopted by the Romans (and have parallels
elsewhere). Yet there are a number of flaws in the
theory as it stands.
A SUGGESTION
I suggest that the Latin whole-number numeral
system (in Table 1) is a modified version of the Etruscan tally-mark system presented above (see supra
pp. 531-34). The symbol I for 1 is (as always) a single
tally-mark. The symbol X for 10 is Etruscan, conceived of as a "second-rank" symbol, and formed from
two lines crossing. The Etruscan symbol for 100 (*)
was conceived of as a "third-rank" symbol and was
Cantor (supra n. 14) 161; Tarquini (supra n. 46) 70, 77;
Friedlein 27; Fabretti (1877) 155-56; Mommsen (1887)
598 and Zangemeister1013. On G. Micali (1776-1844) see
Eckstein(supra n. 6) 373.
82 See supra ns. 1, 7 and 32 on the priorityof decadesymbols. On the Greek acrophonicsystem, see Tod (supra n. 7).
On other examples of additivenon-place-valuesystemsembodying symbols for half-decade values, see Ifrah (1981)
148, 152-57 (Sabaean Arabs, Lycians, Mayas, Palmyrean
Arameans,and varioustally-marksystems).
83 That the use of letters for numerals must be either
alphabeticor acrophonicwas first explicitly noted by Milller (supra n. 42) 320, who was followed by Cantor (supra
n. 14) 161-62. Cf. Tod (supra ns. 7 and 9) and Friedlein27.
Ifrah (1981) gives examples only of alphabeticor acrophonic uses of letters as numerals. Only the Notae Papianae et
Einsidlenses (ca. eighth century A.C.) provide evidence of
randomletter-use for numerals:Keil, Gramm.Lat. 4.330.
84 Cf. Tod
(supra n. 7). They must because the name for
such half-decadevalues is always derivedfrom the previous
decade(except in the case of five and one): see Meninger v. 1
passim.
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542
PAUL KEYSER
Fig. 5. Etruscanbronzecoin with X markof value. Vatican,
Medagliere 22e. (After A. Sambon, Les monnaiesantiques
de l'Italie 1 [Paris 1906] 76-77, no. 132)
formed from three lines. The Etruscan symbols for 5
and 50 are (logically) the lower half of their succeeding decade symbols: thus A is (the lower) half of the X
and 4\ is (the lower) half of *. The Latins inverted
these half-decade symbols to obtain V and \/ (only
later did \/ become I, and then L). We do not know
why the Latins inverted the symbols (note that, in conception, I and X may have been inverted as well), but
the numerous dyslexic alterations of Phoenician letters made to obtain the Greek alphabet form an instructive parallel." Note also that Etruscan numerals
(written retrograde) may be read prograde and inverted: IAX * becomes 4 XVI (66 in both cases).
So much for I through \/; what of Etruscan * =
100 and Latin C = 100? C is not acrophonic for
centum, as the Etruscans used C (see Appendix I).
There is evidence that * came to be written(
in the
course of time (see Appendix I). I suggest that I became C by a process of abbreviation.86 This may have
been helped by distraction from Etruscan > = /2. The
Latin denarius symbol (ligature) X probably would
not have been introduced were * or even ( still in
use, which suggests that we may be able (tentatively)
to date the abbreviation X to C (whether by Etrus85See Jeffery (supra n. 52) 5-6 in general and 23-25 for
examples.
86 The theoryof Zangemeister1016-17 that *->
XK->
C lacks foundation, as noted by Mommsen (1888) 155.
Gundermann 38-39 explains X as "rechtsliufigesC und
linkslaufiges D ; der strich I zwischen den beiden Formen
bedeutet die... Einheit" and compares the Cyprian
[IF for "onedrachma."This explanationis not convincing.
After developingthe idea in the text, I was gratifiedto note
that Ifrah (1981) 158 suggests a similar evolution, as does
Gardthausen(supra n. 18) 403 ("dieLateinervereinfachten
dieses Zeichen [ ] zu C").
87 Pliny HN 33.44-45 implies a date of 269 B.C. for the
introductionof the denarius, but the modern viewpoint, as
discussed in RE 24 (1963) 880.36-881.6, s.v. quinarius
(H. Chantraine), is based on R. Thomsen, Early Roman
Coinage 1 (Copenhagen1957) 187, and 2 (Nationalmuseets
[AJA 92
cans or Latins) to before the introduction of the denarius (ca. 211 B.C.).87 The earliest dated Latin C =
100 (186 B.C.) is in the S.C. de Bacchanalibus (CIL
12, 2.581 = ILS 18, lines 9, 18), while the Etruscan C
= 100 is first found in the second century B.C. (see
Appendix I). Thus if the denarius ligature X was
introduced shortly after the denarius was, we may
tentatively date the abbreviation of I to ca. 250-200
B.C. A Latin symbol for 100 of the third century B.C.
is a desideratum (note that the Duilius-inscription
CIL I2, 2.25 = ILS 65 is unreliable as it is a first century A.C. copy and we cannot be certain that the
symbol forms were preserved). There are two undated
examples of a Latin symbol * which seem numerical
and probably stand for 100 (see Appendix I).
The Minoan-Mycenaean whole-number numeral
system is rather similar to the Etruscan system: in this
system, strokes are used for the lower decades, and
Fig. 6. FragmentaryEtruscan inscriptionwith CC = 200.
(After M. Cristofani,StEtr 38 [1970] pl. XXVIII)
Skrifter, Arkaeologisk-historiskRaekke 9, Copenhagen
1961) passim (esp. 384, 391) who advocatedthe date 212 1
B.C. See also M.H. Crawford,Roman RepublicanCoinage
1 (Cambridge1974) 3-35 and T.V. Buttrey,"The Morgantina Excavations and the Date of the Roman Denarius,"
CongressoInternazionale di Numismatica 2: Atti (Rome
1965) 261-67 (with reply by Mattingly and discussion
269-73). The earliest dated attestationof the denariusligature is a mark of value on a coin of 136 B.C.: Crawford,
1.269 (issue no. 238). On the other hand, the mark of value
XVIis not attested before 141 B.C.: Crawford 1.260 (issue
no. 224), 70 years after the introductionof the denarius
(after 136 B.C. the attestedmarksof value vary in form frequently). Absence of evidence is not evidenceof absencewe do not know when the denarius ligature N was introduced, but it is not attestedbefore 136 B.C. See the undated
inscriptionsinfra n. 99.
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1988]
THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
543
Fig. 7. Italian inscriptionswith C = 100 (left and right) and possible Etruscan500 symbol (center).Left to right:CII2806, CII
2229, CII 2883.
strokescombinedwith a circlefor the upper decades.88
It is impossibleto know why this changeoccurred,but
its occurrencesupportsthe likelihoodof such a change
in the Etruscansystem. The shift in the Etruscansystem occurs between 100 and 1000, from crossed
strokesto crossedstrokesin a circle (see Appendix I).
The Etruscan 0 or 9 = 1000 became (X) when written quickly (note that O is a two-stroke character
( )89 and the circleof @would no doubtbe written as
O was written). The form (x) (attested in Latin as
00 90) written cursively would lead naturally to the
well-attested Latin form o.91 This "horizontal-8"
figure can also appear in a "compressed"form 00 92
which leads naturallyto the well-attestedformal Latin numeral D.93 Although the developmentof Etruscan 0 to Latin ( is not obvious at first glance, every
step is attested in Latin. (A systematic diachronic
study of these forms is needed.)
Finally, the Latin symbol D for 500, probably not
attestedin Etruscan,is the half of the original symbol
GIfor 1000.94 1am unawareof any theory (besidesthis
modifiedtally-mark theory) which explains the horizontal bar of D. The Etruscan symbols for half-decade values (A, 4\ ? are the lower halves (logically) of
the corresponding decade-value symbols. I suggest
that the original Etruscan form of Dmight have been
KV,the lower half of G, rather than D, the right half
of E.D
Fig. 8. Etruscanabacusgem. Paris, BibliothequeNationale.
CII 2578 ter.
88 The similarityhas been notedalso by M. Torelli, ArchCl
18 (1966) 288, n. 13: "Curiose,ma solo casuali, a mio avviso,
le somiglianze tra questo sistema di numerazione e quello
miceneo."On the Minoan-Mycenaean numeral system see:
E.L. Bennett,Jr., Minoan Linear B Index (1953) 107; A.J.
Evans and J.L. Myres, Scripta Minoa 2 (Oxford 1909) 51
(and see v. 1, pp. 256-59 for the Minoan hieroglyphicnumerals); S. Dow, "Minoan Writing," AJA 58 (1954)
77-129, esp. 123-25; W.F. Anderson,"ArithmeticalProcedure in Minoan Linear A and in Minoan-GreekLinear B,"
AJA 62 (1958) 363-68; and D.J. Struik, "Minoanand Mycenaean Numerals,"Historia Mathematica9 (1982) 54-58.
Another numeral system with some similarityis the Lycian:
see R. Shafer, "LycianNumerals,"AO 18.4 (1950) = Symbolaead StudiaOrientisPertinentesFredericoHroznj Dedicatae,Pars Quinta251-61: I= 1, V = 5, O = 10. Shafernotes
the similarity (p. 251).
89Gordon-Gordon109 and 94-95, fig. 7.
9oSee Gordon-Gordon 181-82, Gundermann 30-32
(form no. 15), and Ifrah (1981) 140-41. The last two cite
CIL X, 1019.
91 See Friedlein pls. 9, 11, 12; Gundermann30-32 (no.
16); Meninger 51, plate; Meyer (supra n. 4) 31-32; and
Ifrah (1981) 140-41. Meyer cites CIL X, 1273 = ILS 6344.
92 See Gundermann30-32 (no. 17); Gordonno. 48 = CIL
12, 2.25 = ILS 65 (and see pp. 45-46); and Ifrah (1981)
140-41.
93 See Friedlein pls. 7, 9; Fabretti 162-63; Gundermann
30-32 (no. 18); Gordon-Gordon182; Meyer (supra n. 4)
31-32; and Gordon44-49.
94 See infra n. 96 on possible examples of D in Etruscan
inscriptions. Note that the symbol is D , half of (, not D
half of (D:see Gordon-Gordonp. 241, n. 1. Cf. the various
derivationsof D presentedabove.Gordonastutelynotes that
D may have been derivedfrom "a (D-type symbolwith horizontal as well as verticalstrokeinside the circle"(p. 46). See
Appendix I on the apparent failure of the Latins to adopt
the Etruscan$ = 10,000.
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544
PAUL KEYSER
Fig. 9. Etruscanlead tablet with
•
= 1000 and • = 10,000. CIE 6310. (After M. Torelli, ArchCl 18 [1966]
CONCLUSION
The various modern theories can only explain the
Etrusco-Latin numerals by invoking special pleading
and are little better than Priscian's ancient theory.
The well-nigh universally accepted theory found in all
the handbooks (Mommsen's unused letter plus pictographic theory) should be rejected. In light of our increased knowledge of Etruscan numerals, the Renaissance theory of Ramus can be simplified to explain the
Etrusco-Latin numerals as a tally-mark system which
has undergone some development and abbreviation of
its forms. Of extant theories only this one is simple,
systematic, recognizes the influence of the Etruscans
on the Romans, and provides parallels to the forms.
Based on the evidence, the tally-mark theory of Ramus, modified as in this paper, best explains the origin
of the Latin numerals.
[AJA 92
pl. XCIVa)
The numerals>, I, A, X, and /\ are well attestedand
agreedupon:> to A occuron coins, Ito A\in funerary
inscriptions (and ' is well attested in Latin).95The
numerals[ w] and [v] are unattested96and are tentative suggestionsfor the forms, based on the principle
observed in the cases of A and 4 of using the lower
half of the succeedingdecadesymbol for the half-decade symbol.
The Etruscan numeral * = 100 (later X or C) is
well attested but not well known. The original form
* occurs in funerary inscriptions of long-lived men
(figs. 1-3),97 in numerical"graffiti"(?) (fig. 4),98and
Appendix I: EtruscanNumerals
Table 2 lists the known forms of the Etruscan numerals (supported in the text following).
Table 2. Etruscan Numerals
> I
?2
1
A X
5
4,
10 50
*,)I(,C[w]
100
e,?
[P[]
$
500 1000 5000 10000
95See Bonfante-Bonfante 64; Pallottino (supra n. 47)
373-75, 422; Pfiffig (supra n. 47) 130; Deecke and Miiller
(supra n. 54) Vol. 1, Suppl. 1 (379-434) passim. Examples
of Etruscan > to X on coins may be found in nearly every
sale or auction catalogue;for convenienceI list a few recent
examples of A = 50: Miinzen und Medallien 68 (15 April
1986) no. 1 and Miinzen und Medallien 64 (30 January
1984) no. 1. For Latin examples of / = 50, see CIL 12,
2.2871 (= CII 2276), 2877, 2931 (the CIL providesa photograph, pl. 20), 2977 or 2978 (and the form I is found CIL
12, 2.585.E28, 638, 675, 676 and 677 for examples).
96 Unless deFeis (1883, supra n. 19) 250 is right in citing
Fig. 10. Possibleexampleof Etruscan500 symbol,fromamphora. (After L.I.F. Janssen, InscriptionesEtruscae [Leiden 1840] pl. IV.48)
CII 2229, or unless the example in L.I.F. Janssen, Musei
Lugduno-Batavi: Inscriptiones Etruscae (Leiden 1840)
29-30 (no. 48) pl. IV, is truly a numeral.The first is in my
fig. 7, center,the secondin my fig. 10.
97 See 1) M. Torelli, StEtr 33 (1965) 472-73 and pl. CIVa
+ M. Pallottino, StEtr 34 (1966) 355-56 = TLE 890 (second century B.C.) = my fig. 1; 2) M. Cristofani,StEtr 34
(1966) 363 and pl. LXXIIa = CIE 5757 = my fig. 2; 3) CII
364 bis 1 (pl. XXVI), Mus. Guarnacci,Volterra(175) = my
fig. 3.
98 1) terracottafragment:G.F. Gamurriniand A. Fabretti,
Appendiceal CII (Florence 1880) no. 114 (pl. IV) (tris) =
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1988]
THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000
545
in two Latin inscriptions."99The rounded form AKis
found on coins (fig. 5).100 The Etruscan use of C,
though rare, is attested (figs. 6 and 7).?1'
The Etruscan numeral e, 0 is attested on the "Parisian" abacus-gem (CII2578 ter), bis (fig. 8a-b), and
on a lead (defixio? letter? oracular inquiry?) tablet
(CIE 6310 = TLE 878), tris (fig. 9).102 Commentators have assigned it the value 100 or 1000.103 If it is
100, then we have the unparalleled case of two different numerals for the same number in the same system
( E and $). If ? = 100, then based on the two inscriptions cited, the next sign $, not dissimilar to E,
is 1000. Thus, whatever the value of (, the symbol
for 1000 resembles a sign e. (This similarity may
explain the Latin failure to adopt the Etruscan $=
10,000 symbol.) The Etruscan 500 was probably
, and Figures 7 and 10 show , symbols which are
possibly numerical.
The Etruscan numeral $ is attested on the lead
tablet (CIE 6310) tris. The abacus-gem has a pair of
numerals in the next row higher than the row of ?,
but they are damaged,104 as attested by the variety in
the drawings;15 their form is similar to $ in having
bars extending beyond a circle. (Note also the Mi-
deFeis 1898 (supra n. 19) 14, pl. I.V = Gundermann39-40
with figure = Ifrah (1981) 144 with figure (and Zangemeister 1016) = my fig. 4; 2) on the base of a vase: deFeis
1898 (supra n. 19) 14, pl. I.III = Gundermann39 = Zangemeister 1016, n. 2.
99DeFeis 1898 (supra n. 19) 14, pl. I.IV lists several
Etruscan inscriptions without references.There is also the
difficult inscription published by B. Nogara, "Iscrizioni
etrusche di Bieda (Biera)," RM 30 (1915) 299, which appears to read "vt 100 a" ( 4+T1 ). The Latin examples
are 1) Gamurrini and Fabretti (supra n. 98) 16 no. 114,
n. 1, unless it is a stone-cutter'serror:"]CETERIS-MEIS/]*XX"
(read
XXX?);
P"
VSTRINV.LICERET/]AGR
2) CIL I2, 2.468b, in capite, *, which is clearly numerical
as 468a has X (decem);is the extra stroke the work of the
idle hand? I have been unable to obtain photographs of
either inscription.
0ooSee Deecke and Miiller (supra n. 54) Vol. 1, Suppl. 1,
pp. 424-26 (third-second century B.C.) = Fabretti (1877)
159, two examples; Thomsen (supra n. 87) 1.207, four examples, all with weights given; and more recently, P. Marchetti, "La metrologie des monnaies etrusques les plus anciennes," Contributtiintroduttiviallo studio della monetazione etrusca:Atti del V convegnodel CentroInternazionale
di Studi Numismatici 1975 (Naples 1976) 221-72, pls.
xxxiv-vi: see 253-60 and pl. xxxvi.1 (my fig. 5). The value
100 of the symbol C is secured by the weights of the coins
(in Thomsen: 40 to 30 gm, compare coins with 4\ mark of
value, weights 31 to 19 gm; X is not 75).
'01On Etruscan C = 100, see M. Cristofani, StEtr 38
(1970) 288 and pl. XXVIII ( DD): second century B.C.
(my fig. 6): Cristofani suspects Roman influence. Fabretti
(1877) 161, cites CII Suppl. 2, 122 ( IIIIDX ) and p. 163
cites CIH2806,pl. LII ( IIXX ), but the first seems Latin in
CII Suppl. 2, 122 and the second is listed as Central Oscan
by Conway (supra n. 60). In addition, Oscan and Umbrian
inscriptions attest C = 100: Conway no. 354 (supra n. 60),
Tabula Eugubina VIIb.4 (supra n. 60), CII 2883, pl. LV
( A)D), E. Vetter, Handbuch der italischen Dialekte 1
(Heidelberg 1953) no. 357 = CIE 8452 (Faliscan): CV (?),
Vetter, no. 60 (Oscan): XC, Vetter, no. 70 (Oscan): CXII,
Vetter, no. 233 (Umbrian, second half of second century
B.C.): CL[sic]VIIII,and note Vetter, no. 354 (amphora
marks) which may include numerals.In all these cases photographs would be helpful; CII 2806 (left) and 2883 (right)
are in my fig. 7. (I doubt that Vetter, no. 233 has "L"for 50.)
102 The Parisian abacus-gem CII 2578 ter is pictured in
Meninger 111 and in Morandi (supra n. 38) pl. XIV: my
fig. 8. The first edition of CIE 6310 is M. Torelli and
M. Pallottino, "Terza campagnadi scavi a Punta della Vipera e scopertodi una laminetta plumbea iscritti,"ArchCl
18 (1966) 283-99, pl. XCIVa (my fig. 9). DeFeis 1898 (supra n. 19) 14 cites the vase (supra n. 98) for ? = 1000 also.
103Miiller (supra n. 42) 318, 320 and pl. IV.2 (at end),
Mommsen (1850) 19, Friedlein27-28, Fabretti (1877) 156,
Mommsen (1887) 599, Ninni (supra n. 16) 683-84, Gundermann 35-37, and Rix (supra n. 50) 850 all incorrectly
take ? = 100, while Bortolotti 158, Deecke and Miiller
(supra n. 54) Vol. 2, 533, deFeis 1898 (supra n. 19) 248 and
pl., and Zangemeister 1020-23 correctly take ? = 1000.
G. Buonamici, Epigrafia etrusca (Florence 1932) 244-47
cites Orioli, MUller, and Mommsen versus Bortolotti and
deFeis, but comes to no conclusion; similarly Pallottino
(supra n. 47) 422.
104 The photographs, Meninger 111 and Morandi (supra
n. 38) pl. XIV, are unclear.
105See Buonamici (supra n. 103) 244 and Zangemeister
1021.
noan-Mycenaean parallel of 0 = 100, O= 1000.)106
We can be fairly sure that this Etruscan numeral is
not simply a variant of (. First, it appears in a different row of the abacus-gem. Second, if it were a
variant, the same numeral would be repeated six times
on the lead tablet: but in the Etrusco-Latin system no
decade symbol is repeated more than four times (unless there is no succeeding decade symbol).
Appendix II:
Unused Letters in the Latin Alphabet
I do not intend to give a detailed history of the
Etruscan or Latin alphabets; rather I here discuss
only the question of "unused letters." The Etruscan
alphabet, written left-to-right for ready comparison
with the Latin alphabet below it, is given in Figure
11.107
106See supra n. 87.
107 For
the Etruscan alphabet see Bonfante-Bonfante79,
Pallottino (supra n. 47) 421-22, Pfiffig (supra n. 47) 17-23
and Jeffery (supra n. 52) 236, pls. 47-48. For the Latin
alphabet see T. Mommsen "Ober die Buchstabenfolgedes
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546
PAUL KEYSER
A
A
Hz0H I
B C D E
F
H -
)
O fPM9
IMwPr
K L
N -0
P
-
"rY
Q RS
X
T V X
Fig. 11. Etruscanalphabet (upper, left to right) and Latin alphabet (lower)
The dashes indicate letters in the original Etruscan
alphabet not attested in Latin. The unattested letters
* (ksi) and M (san) require no discussion. The letter
0 (theta, but see below on X as a form of theta) was
not used, unless CIL 3.6010.142 "ME eILLVS" is an
example. The letter D (phi) was not used unless CIL
I2, 2.544c: "CP/j .", or CIL I2, 2.2658
are examples. The letter Y (chi) is attested"'VPETI'"
in an undated funerary inscription (CIL I2, 2.476.6): "A.
S[I]RPIOS ES \ ," but the -IOS ending of the gentilitial name may suggest Greek influence if it is not an
archaism for -IUS. All of this is not inconsistent with
the retention of ?, V, and Y in some model abecedaria. The letters I (zeta) and X (varies) require more
discussion.
The letter Z, or I , seems to have been used.10s If
Martianus Capella (fl. A.D. 425), 3.261, is to be believed, Appius Claudius Caecus (censor 312 B.C.)
hated Z due to its sound, which implies that Z was
used then. Plutarch (Quaest. Rom. 54.277D) states
that Sp. Carvilius (cos. 234, 228 B.C.) introduced G.
But from epigraphic evidence we know that G was in
use by the early third century B.C., whatever position
it held.10' That it found its way into the place of zeta
suggests strongly that zeta remained in the abecedarium until it was replaced by G,110 as is confirmed in
the one early (350-300 B.C.) Latin abecedarium we
have (CIL I2, 2.2903) which has I between F and H
(and no G). In addition to this evidence, we have the
attested examples of Z in Varro Ling. 7.26 quoting
from the Carmen Saliare (one Z), and the Oscan law
on the reverse of the Tabula Bantina CIL I2, 2.582 =
CIL 11, 197 (24 Zs).111 Note that the Latins received C
(the Greek cognate z) as "ss."112 Thus, we may conclude that the lack of attestation of Z fails to prove its
nonexistence in the Latin abecedarium.
The letter X is more of a puzzle than usually acknowledged. The form X is used in Euboean and
West Greek alphabets for ksi in that position (between N and O),113 but in seventh-sixth century B.C.
(south) Etruscan it has its "Latin" position and the
value of a sibilant,114 while in sixth-fifth century B.C.
(northeast) Etruscan it has the value and position (between H and I) of theta.115 Lewis and Short note that a
number of words show a tendency to change s -> x in
Latin.116 All this somewhat muddles the question of
unused letters: some (X anyway) were evidently used
in rather ad hoc ways.117
lateinischen Alphabets," RhM 15 (1860) 463-67,
1.422.31-423.1, 426.8-11, Isid. Orig. 1.4.15, and Petrus
Diaconus in Gramm. Lat. 4.334. More recently: Kiihner
and Holzweissig (supra n. 54) 8-9.
113Jeffery (supra n. 52) 79, 235, 248; Fiesel (supra n. 39).
114
Fiesel (supra n. 39).
115 Pallottino (supra n. 47) 421-22, not noted in
Pfiffig
(supra n. 47) 17-23.
"6 C.T. Lewis and C. Short, A Latin Dictionary (Oxford
1879) s. "X",citing assis -> axis, lassus -> laxus, Odysseus -> Ulixes (which also shows the familiar d -> 1 of
Latin, cf. Lewis and Short s. "L" 11.3, A. Walde, Lateinisches etymologisches Wbrterbuch3[Heidelberg 1938] xi
and C.D. Buck, ComparativeGrammarof Greekand Latin
[Chicago 1933; repr. 1952] 123), sestius -> sextius, and
Aias -> Aiax. See CIL 5.1880folex (forfoles), 5.5583 sestum (forsextum), 5.6726 conius (for coniux), and 5.893, 900
and 8280 milex (for miles).
117 Note added in proof: Nancy de Grummondhas
kindly
directedmy attentionto Livia Giacardi,"L'originedella numerazioneromana,"CentroStudi "CristianoMancini"per
la Storiadel Pensiero Matematico1 (Foligno 1987), not yet
available in this country.
Ritschl
(supra n. 54), RE 1 (1894) 1621.38-1626.12, s.v. Alphabet
(J. Schmidt), Leumann (supra n. 54) 44-49 and 2nd ed.
(supra n. 72) 1-3, M. Lejeune, "Notes de linguistique
italique: XIII. Sur les adaptations de l'alphabet etrusque
aux langues indo-europ~ennes d'Italie," REL 35 (1957)
88-105 and Gordon (supra n. 40). The one Latin abecedarium earlier than the first centuryA.C. is cited supra n. 68.
108
I ignore here the later reintroductionof Z.
109
Sandys (supra n. 27) 35.
110New letters are
generally added to the end of the alphabet: cf. v, , X,,
X, coin Greek or the later reintroductionof
Y, Z in Latin. Noting that the original form of Z was I , I
venture to suggest that whenever the letter I acquired its
serifs, which render its form similar to the original form of
Z, the letter Z had long fallen out of the formal abecedaria.
II For the Oscan text see C.G.
Bruns, Fontes luris Romani
Antiqui4 (Tubingen 1879) 45-50 (a referenceI owe to C.F.
Konrad) or Vetter (supra n. 101) 13-28 (a referenceI owe
to an anonymousAJA referee);for the dating see Dizionario
epigrafico di antichitac romane 4 (1957) 715-17, s.v. Lex (G.
Tibiletti) (a referenceI owe to an anonymousAJA referee).
112
On "received as "ss" see Diomedes in Gramm. Lat.
DEPARTMENT OF CLASSICS
UNIVERSITY OF COLORADO AT BOULDER
BOULDER, COLORADO 80309
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