“The Origin of the Latin Numerals 1 to 1000”, American Journal of Archaeology 92 (1988)...

The Origin of the Latin Numerals 1 to 1000Author(s): Paul KeyserSource: American Journal of Archaeology, Vol. 92, No. 4 (Oct., 1988), pp. 529-546Published by: Archaeological Institute of AmericaStable URL: http://www.jstor.org/stable/505248 .Accessed: 29/01/2011 13:02

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The Origin of the Latin Numerals 1 to 1000* PAUL KEYSER

Abstract The standard theory 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 pictographic principle, and L, C, and (I) = 1000 as forms of Greek letters (aspirates) unused in Latin. There are four other important theories: a long-rejected one proposed by the fifth- century A.C. grammarian Priscian, a tally-mark theory (dating back to 1546), various pictographic theories (dating back to 1655), and various acrophonic theories. The tally-mark theory, though held by numerous scholars through the end of the 19th century, has received little attention in the last 75 years. Mommsen's mixed theory and the other four important theories are critically reviewed. Available numismatic and epigraphical evidence, certain logical principles, and historical considerations are ad- duced to support the rejection of Mommsen's theory and the acceptance of a theory which holds that the Latin nu-

merals are Etruscan tally-mark numerals which have undergone some alteration and abbreviation of 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 ad-

ditively (or subtractively).1 I consider here only the es- sential features which were in evidence by the mid- first 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 whole- number numerals and suggest a revision of the ac-

cepted view.

* This paper would have suffered without the assistance of the Interlibrary Loan staff at the Norlin Library, Uni- versity of Colorado, Boulder (Virginia Boucher, April Pe- terson, 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 Introduc- tion (New York 1983).

Bortolotti P. Bortolotti, "Congetture intorno una numerale notazione prealfabe- tica in Italia," BdI 1875, 155-60.

BU L.G. and J.F. Michaud eds., Biogra- phie universelle (Paris 1843).

CII A. Fabretti, Corpus Inscriptionum Italicarum et Glossarium Italicum (Turin 1867); Supplementi I (1872), II (1874), III (1878); Ap- pendice (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 Stu- dien (Leipzig 1877), Italian original: CII Suppl. I, Pt. 2 (1874).

Friedlein G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Grie- chen und Romer (Erlangen 1869; repr. Wiesbaden 1968).

Gordon A.E. Gordon, Illustrated Introduction to Latin Epigraphy (Berkeley 1983).

Gordon-Gordon J.S. and A.E. Gordon, Contributions to the Palaeography of Latin In- scriptions (CPCA 3.3, Berkeley 1967; repr. Milan 1977).

Gundermann G. Gundermann, Die Zahlzeichen (Giessen 1899).

Ifrah (1981) G. Ifrah, Histoire universelle des chif- fres (Paris 1981); trans. L. Blair, From One to Zero (New York 1985).

Meninger K. Meninger, Zahlwort und Ziffer2 2 (G6ttingen 1958); trans. P. Bro- neer, Number Words and Number Symbols (Cambridge, Mass. and London 1969).

Mommsen (1850) T. Mommsen, Die unteritalischen Dialekte (Leipzig 1850).

Mommsen (1887) T. Mommsen, "Zahl- und Bruchzei- chen," Hermes 22 (1887) 596-614 = Gesammelte Schriften 7 (Berlin 1909) 765-83.

Mommsen (1888) T. Mommsen, "Zu den r6mischen Zahl- und Bruchzeichen," Hermes 23 (1888) 152-56 = Gesammelte Schriften 7 (Berlin 1909) 783-87.

NBU J.C.F. Hoefer ed., Nouvelle biographie universelle (Paris 1853).

Zangemeister K. Zangemeister, "Entstehung der r6mischen Zahlzeichen," SBBerl 49 (1887) 1011-28.

A recent discussion of the origin of Latin numerals is Gordon 44-49; for a discussion of the additive principle see Ifrah (1981) 139-59, 337-47. In the earlier numeral systems, there are symbols only for the decades. From this ob- servation and from the fact that the natural (and almost uni- versal) base is 10, I conclude that any symbols for 5, 50, 500, etc. in additive non-place-value systems are secondary.

529 American Journal of Archaeology 92 (1988)

530 PAUL KEYSER [AJA 92

Table 1 lists the classical forms of the Latin whole- number numerals from 1 to 1000 (higher values derived from these are not discussed herein):2

Table 1. Latin Numerals to 1000

I V X 1' C (X) or o0or (D or (1) 1 5 10 50 100 500 1000

There are two or three ancient discussions of 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 acrophonic theory (1818 and later), and the unused-letter theory of Mommsen (1850). The last is combined with a pictographic theory for 1, V, and X by Mommsen, and this is the generally accepted theory today.

ANCIENT THEORIES

The two or three extant ancient discussions of the problem are all from the pens of grammarians. Isidore

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), mis- led 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 acrophonically from a Homeric form ('os, i'a, 'ov) of "one." This is sensible as the Greek numeral system in

2 Derived from Gordon 44-49 and Gordon-Gordon 181- 82 and 224 n. 1. On Q see T. Mommsen, "Quingenta Milia," Hermes 3 (1869) 467-68 and Hermes 10 (1876) 472 = Gesammelte Schriften 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. Suffice to say here that Priscian (fl. early sixth century A.C.) did not know of it (Keil, Gramm. Lat. 3.406-407) but Dante (Para- diso 19.127-29) does seem to. I may also note here that 50 4 became first I, I, then L (see Gordon 44-49), already in 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 epigraphical or palaeographical investiga- tions of the forms of the characters).

3 Suet. Gram. 1 = Funaioli, Gramm. Rom. Frag., p. 411 Fr = p. 101 T 1 distinguishes the 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 lateinische Epi- graphik (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 section of Probus (see pp. 347-52). D.E. Smith, Scientia 40 (1926) 7, cites Probus from the edition of J. deC. deTri- dino (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 references to Greek origins are omitted. DeFeis (in 1898, infra n. 19) 16 also assigns

Priscian's theory 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 numerals elsewhere, 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 century A.C., C. Barwick ed., 1964) says almost nothing about numerals (only Gramm. Lat. 1.10.2-3 = Barwick 7.5 on C = 100, and Gramm. Lat. 1.10.7 = Barwick 7.10-11 on D = 500); Dio- medes (fourth century A.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. 3.406- 407. Doubted already by A. Alciati in 1546 (infra n. 10) and by nearly everyone since. Three exceptions: Julius Caesar Scaliger, De causis linguae Latinae libri tredecim (Heidel- berg 1534) c. 41 (pp. 96-97); A. Dragoni, Sul metodo arit- metico degli antichi Romani (Cremona 1811) 22-23; and R. Bombelli, Studi filologico-critici sulla genesi forma e valore delle lettere dell'alfabeto italiano (Rome/Turin 1866) at least for 1 (98), V (143-44), X (146), L (107) and C (72-73). Bombelli quotes from the Tridino edition of "Probus" (Ven- ice, 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 available in this country, see A. Pagliaini, Catalogo generale 1:267). On H.A. Herrmann (1820-post 1871), see F.A. Eckstein, Nomenclator philologum (Leipzig 1871; repr. Hildesheim 1966) 241.

1988] THE ORIGIN OF THE LATIN NUMERALS 1 TO 1000 531

which the symbol I = 1 was acrophonic,7 but 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 consonants or simply letters)? Why switch to an alphabetic principle? Priscian claims that X = 10 because X is the 10th Greek conso- nant (6) or because X follows V in the Latin alphabet. This dual suggestion shows that he has no idea how X = 10 originated, and is even more egregious special pleading than his suggestion for V-why switch to the Greek alphabet? Priscian holds that L = 50 because v = 50 in the Greek alphabetic system (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 consistent with 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 acrophonic X = 1000 with ( ) added to distinguish it from X. Why now return to a Greek acrophonic principle? Pris- cian's theory is a farrago of special pleading supported only by assertion and useful only to show that neither he nor any contemporary knew the origin of the Latin whole-number numeral 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 En- nius's theory as preserved in Isidore's remark) or to have proposed a 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 omnemr nurnerum, Alciati:

credendum itaque). This coincidence and the scholars'

apparent independence of each other (as confirmed by numerous differences in detail), as well as Alciati's passing reference to (the historian) Benedetto Giovio (1471-1554), raise the (unconfirmed) suggestion that this theory is pre-Renaissance (but it may be due to Zeitgeist).

Alciati (1546) explicitly derives the Latin numerals ab agricolarum tesseris.10 He claims that V is simply two lines joined, and that X is the same two totidem transversas. In fact V ought to be derived from X, not X from V by some development. The symbol L for 50 is painfully elicited from a symbol X, itself derived from X = 10. Alciati scientifically cites epigraphical evidence: in antiquis marmoribus non uno in loco reperi- tur ea figura esse descripta, to an erroneous conclusion (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 . . . quod in alphabeto est elementum sic [X antiqui effigaverunt. For this he cites palaeographical evidence: antiquissimi libri vel ante tempora Langobardorum scripti. I am unaware of any such evidence or of any acceptance of this theory for C. The symbol for 500 he derives from 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 symbol for 500), but neither he nor anyone has ever cited any evidence that Z = 500. Moreover, if A = 500 derives by the addition of a line from X = 50, why did X not derive by the addition of a line from V = five? Finally the symbol for 1000 is derived logically from the symbol [X for 100 by the addition of another line to form [X]. Then alii . .. celeritatis causa sic de- pingebant (1). The essence of his suggestion is sensible, as is his attempt to derive all the numerals system- atically, but his theory is seriously if not fatally weak- ened by his inability to cite evidence for Z , by the elu- siveness of his evidence for [X = C, by his ignorance of

= 50, and by inconsistently deriving X = 50 from X = 10, but A = 500 from X = 50.

Tartaglia (1556) restricted his theory to 1, V, X, and a symbol for 50, arguing from an extant use of tally- marks.' The first character after I was X ("la prima fu vna croce obliqua"), and the second was V ("la se-

7See M.N. Tod, "The Greek Numerical Notation," BSA 18 (1911/1912) 98-132. The Greek symbols for 50, 500, 5000, and 50000 are derivative (cf. supra n. 1).

8 See Ifrah (1981) 139-265 passim and Meninger 26-59, 73-85. I am indebted to E.L. Bennett (Athens) for drawing my attention to Meninger's work.

9 See M.N. Tod, "The Alphabetic Numeral System in Attica," BSA 45 (1950) 126-39.

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. Abbondanza in DBI 2 (1960) 69-77.

11 N. Tartaglia, General Trattato di Numeri, et Misure (Venice 1556) Pt. I, pp. 3v-5. On Tartaglia the mathematician, see Lessico universale italiano 22 (1979) 451 or E. Bor- tolotti in Enciclopedia italiana 33 (1937) 286.


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 \V or 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 two- stroke 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 consistent with 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 respec- tively. 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 derived from (D and 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 cen- turies,14 and could gain acceptance up through the mid-19th century."' Ramus's perception on the basis of literary evidence alone of the essentially systematic nature of the Latin numeral system, and his recogni- tion of the derivative nature of the symbols for the

12 P. Ramus (Pierre de la Ram&e), Scholarum mathemati- carum libri unus et triginta (Basel 1569) 117. I have not examined the later reprints (Frankfort 1599, 1627). On this work and the reprints see W.J. Ong, S.J., Ramus and Talon Inventory (Cambridge, Mass. 1958) nos. 703, 705, 706 re- spectively. I am indebted to Fr. Ong for bringing the copy of the 1569 edition containing Ramus's autograph notes to my attention (24 November 1986): see J.F. Daly, "Ramus: Re- cently Discovered Unpublished Edition of His Mathemati- cal Works," Manuscripta 17 (1973) 80-90. Daly notes (p. 83) that the changes to this work are slight, and there are none to p. 117, which I have been able to confirm thanks to the kindness of Miss C.E. Weidle, Rare Books Librarian of the Pius XII Memorial Library, in providing xerographic copies of pp. 116-18 (9 February 1987).

13 Gordon-Gordon 100, and fig. 7. 14 M. Hostus, De numeratione emendata ueteribus Latinis

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 notae geminatae decussatione nata videri potest, C is three units E, and D is either acrophonic for di- midium [mille] or from M becoming AA. On Hostus (1509- 1587) see Schimmelpfennig in ADB 13 (1881) 191 and M. Cantor, Mathematische Beitrdge zum Kulturleben der Vdlker (Halle 1863) 159. Those who adopt this theory (none cite Ramus or Alciati) are the following: 1) G.J. Voss, De universae Matheseos natura et constitutione (Amsterdam 1650) c. 8, ?4; on Voss see the monograph by C.S.M. Rade- maker, Life and Work of Gerardus Joannes Vossius (1577- 1649) (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 recherches et 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 adopts a pictographic theory 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 (?1620- 1689) see BU 5 (1843) 76 and NBU 6 (1853) 697-99. 3) C.F. Milliet de Chales, Cursus seu mundus mathematicus '(Lyons 1674) 2(Lyons 1690) Vol. 1, p. 28 who revealingly remarks Paulo difficilior erit origo characterum numeralium a Romanis usurpatorum; on Chales (erroneously spelled Challes in some works), the Jesuit mathematician (1621- 1678), 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 (1662- 1729), the antiquarian and polymath, see S. Rotta in DBI 10 (1968) 187-94. 5) P.D. Huet, Huetiana ou pensdes di- verses '(Paris 1722) 2(Amsterdam 1723) 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 Matheseos universae (Leipzig 1742) Librum IV, caput 1 (pp. 732-35); on Heil- bronner the mathematician (ca. 1706-ca. 1747), see M. Cantor in ADB 11 (1880) 313. 7) E. Corsini, Notae Graeco- rum; sive vocum et numerorum compendium quae in aereis atque marmoreis Graecorum tabulis observantur (Florence 1749) cap. 3 and Prolegom. (as cited in deMatthaeis p. 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's credit that he cites all seven of these works plus Alciati (though not Ramus or Tartaglia).

1' Most importantly G. deMatthaeis (in 1818) VIII, X, XVIII-XXII; see also J. Leslie, Philosophy of Arithmetic2 (Edinburgh/London 1820) 8-10 without citation, G.F. Grotefend, Lateinische Grammatik3 2 (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 mathematician and semiti- cist, see M. Cantor in ADB 23 (1886) 445-46. The only citation of Alciati known to me after deMatthaeis is L. Ger- schel, "Comment comptaient les anciens Romains," Hom- mages & L'on Hermann (CollLatomus 44, Brussels 1960) 386-97.


Fig. 1. Etruscan funerary inscription using * = 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 Etrus- can numerals are made by successive crossings or cir-

clings of the single tally-mark. Then the symbols for half-decade values (5, 50, 500, etc.) are made by halv-

ing the symbol for the succeeding decade. The modern version has a number of merits: it is systematic, in-

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 im-

mediately 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 Ap- pendix I), of which half is 4 = 50, though earlier workers on this theory derived / = 50 in a variety of

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, described a 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 annotated bibliography on 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 Etruscan numerals without having seen this theory mentioned or advocated I came to the same conclusion.

18 So Alciati (supra n. 10), Tartaglia (supra n. 11), Ramus (supra n. 12), Hostus, Voss, Chales, Bianchini, Heilbron- ner, and Corsini (supra n. 14), deMatthaeis, Leslie, Grote- fend, and Nesselmann (supra n. 15). In addition, Friedlein 27-28 (?40), Bortolotti 158, Fabretti (1877) 155-57, Zangemeister 1014-16, E. L6ffler, Ziffern und Ziffernsys- teme der Kulturvblker in alter und neuer Zeit (Mathema- tische Bibliothek 1, Leipzig/Berlin 1912) 50-52, B. Le- febvre, Notes d'histoire des 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 Zangemeis- ter, Meninger does not cite but follows Zangemeister [see pp. 52, 293, no. 31] and Ifrah cites Meninger.) Zangemeis-

ter has also been followed by L. Saalschtitz, "UOber Zahlzei- chen der alten Vilker," Schriften der physikalisch-bkonomi- schen Gesellschaft zu Kdnigsberg 33 (1892) [4]-[9]; M. Cantor, Vorlesungen fiber Geschichte der Mathematik 1 (Leipzig 1894) 487-88; E.M. Thompson, A Handbook of Greek and Latin Palaeography3 (London 1906; repr. Chica- go 1975) 105; and V. Gardthausen, "Die r6mische Zahlzei- chen," Germanisch-romanische Monatsschrift Ser. 2:1 (1909) 401-405. The warm criticisms of Zangemeister in Mommsen (1888) amount mostly to forceful restatements of Mommsen's own theory, but he correctly noted that "decus- sare" probably never meant "multiply by ten by crossing." This scarcely invalidates Zangemeister's work. See also n. 86 infra for criticisms of Zangemeister in detail.

19 Supra n. 18. L. deFeis, "I dadi scritti di Toscanella ed i numeri etruschi," Giornale linguistico di archeologia, storia e letteratura 10 (1883) 241-55 and "Origine dei numeri etruschi," Dissertazioni della Pontificia Accademia Romana di Archeologia Ser. 2:7 (1898) 1-19, pl. 1 holds that A pre- ceded X, so that X = twice A. He has similar remarks on 4\ and A = 500. See the remarks supra ns. 1, 7, and infra n. 32 on the probable priority of X, *, etc. to A, 4, etc. I do not cite deFeis (save infra ns. 98, 99 and 103 for his citation of evidence for the forms of Etruscan numerals) as he represents a retreat as compared to Bortolotti (i.e., as he advocates A as prior to X). On deFeis the priest (1844-19??), see Di- zionario biografico universale 1 (1907) 645.


Fig. 2. Etruscan funerary inscription using * = 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 natfirlich- sten auf 1 zuriickzugehen und durch Einfassung diese neue Einheit ebenso von dem gew6hnlichen 1 zu un-

terscheiden, 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

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 ori-

gins 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 agree-

20 See supra ns. 18 and 19. Neither Alciati nor Ramus knew of Etruscan * = 100, and so tried to make C a three- stroke character and L its half (see ns. 10, 12, 14, 15). Fried- lein knows C is not the original symbol for 100 but believes that ? is. From this he derives uw as its half for 50. DeMat- thaeis derived 4 = 50 from an alternate version of X (two crossed lines) and is followed by Fabretti 155-57. Bortolotti vacillates and comes to no conclusion on C. Zangemeister, Meninger and Ifrah (1981) hold the theory expressed in the text.

21 See Zangemeister 1017-18 who provides various unattested graphical modifications. Friedlein simply gives up and returns to Priscian's allegation of acrophonic status for C. Meninger 49 and Ifrah (1981) 158 provide variations on Zangemeister's theme, Ifrah possibly rightly (see infra n. 86).

22 Friedlein 28. Voss (and Chales, either independently or without citation) suggest thatfour strokes made ]-1 which became I) and that five strokes made ME which became (I). This is not an advance on Ramus or Alciati. Bianchini, Heilbronner, Corsini, deMatthaeis, Leslie, Grotefend, Nes- selmann, Fabretti, and L6ffier all follow Voss at one remove or another (supra n. 18).

23 Zangemeister 1018. 24 And is paralleled-see Ninni (supra n. 16). 25 Meninger 47-52 and Ifrah (1981) 139-59. Nor do de-

Matthaeis, Fabretti 155-57 and Bortolotti provide an ex-

planation of (I). 26 I1. Taylor, The Alphabet (London 1883) Vol. 1, p. 6 cites

Grotefend as an advocate of the pictographic theory, but gives no reference; G.F. Grotefend (1775-1853) in 1820 (supra n. 15) was an advocate of the tally-mark theory: what does Taylor mean? Advocates of the pictographic theory known to deMatthaeis are 1) Borel (supra n. 14) 95 as cited in deMatthaeis p. IX, n. 5 (I have not yet been able to obtain a copy of this work): "On met ... IIII pour 4, parceque cela represente les quatre doigts de la main.... Et I'V qui vaut 5 est marque par le cinquieme doigt qui est le pouce, lequel etant ouvert forme un V avec le doigt index, et deux V joints 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) "Berby de Mailly" in an article, "Sur le clou, que les payens attachoient solemnelment dans leurs temples," Mercure de France (March 1728) 479 and in Varieties hist. physiq. et litter. ou Recherches d'un savant 2 (Paris 1752) 320 (cited in deMatthaeis p. IX, n. 6 and p. XIV, n. 18), but I have been unable to confirm the existence of this writer. Add to deMatthaeis's list Charles deBrosses, Traite' de la formation mechanique des langues (Paris 1765; repr. 1801) 1.468-72. On deBrosses (1709- 1777) 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 editions he retained the theo-


Fig. 3. Etruscan funerary inscription using = 100. CII 364 bis 1. (Museo Guarnacci, Volterra 175)

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 finger- counting system was none of these but was made on the left hand with the three outstretched fingers to-

gether 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.

DeBrosses makes L on the left hand with index finger and thumb at right angles and C "pourroit etre 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.34 His own hesitation on C and the lack of correspondence between these gestures and the known finger-counting gestures (as noted below) do not inspire confidence.

Villicus explains L as a hand gesture similar to that for V,35 but 50 was \V (not easily obtained from a hand gesture), and in the finger-counting system was repre-

Fig. 4. Etruscan numerical graffiti on terracotta, showing * = 100. AppCII 114, pl. IV.

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 Nu- merals," Proceedings of the Royal Society of Edinburgh 28 (1907/1908) 161-82, esp. 173-74, 177; W.W. Rouse Ball, A Short Account of 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 "mostransi assai confusi ed incer- ti 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. digitorum com- putus (H. Hommel); Reallexikon fiir Antike und Christen-

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 references there); 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-counting system refer to these authorities.

31 See supra n. 26 for Borel and Huet, and Mommsen (1887) 598.

32 Mommsen (1888) 155, 153 (sections 10 and 2). 33 Villicus (supra n. 27) 14, and Barrett (supra n. 27) 174,

177. 34 DeBrosses (supra n. 26) 469-70. 35 Villicus (supra n. 27) 14 for L, C, D, and (D .


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 VV, W''V,

or DDDD. For T = 50, Barrett claims that 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

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 Ap- pendix 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-

36 Barrett (supra n. 27) 175-76 for the allegation about the five-fold repetition; pp. 177-82 for the "derivation" of the signs for L, C, D, and (D.

37 Ifrah (1981) cc. 10-14 discusses numeration systems and finds none whose origin was pictographic. Neither Vil- licus (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 dis- tance-measures are derived from 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 Precursor of Numerals and Writing," Archaeology 39.6 (1986) 32 claims that the Sumerian symbols for 10 and 60 were in origin pictographs of standard containers or heaps of those quantities: i.e., if true, the Sumerian system would involve something similar to writing a pictograph of an egg carton for 12, and reading "dozen-eggs" or some such. This parallel is rather remote in resemblance, space and time-

but it is the closest available. 38 F. Orioli in three works: 1) "Sull'origine dei numeri

etruschi e romani e sull'infissione solenne del chiodo an- nuale," Opuscoli letterari 1 (1818) 208-26, esp. 219-24; 2) Spiegazioni d'una gemma etrusca del Museo reale di Pa- rigi (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 accepted C from centum, though he believed that the Etruscans themselves had no numerals larger than 50 (infra n. 58). In 1825 he changed his mind based on 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 reference I owe to an anonymous referee of AJA.

39 E. Fiesel, "X Represents a Sibilant in Early Etruscan," AJP 57 (1936) 261-70.

40 A.E. Gordon, "On the Origins of the Latin Alphabet: Modern Views," CSCA 2 (1969) 157-70.

41 See Tod (supra n. 7).


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 form of 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 "hI- mesh" (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 He- brew "'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 adoperare tal lettera, il cui sono fosse al Hheth [sic] aspro il piui affine ... talle era certamente il Koph ... cioe a dire... la lettera \/, la quale e certamente un

Koph." Finally ? is related to the letter 'aleph for He- brew "'elef" ('XK) = 1000. The special pleading, the 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 correctly emphasize the connection of the Etruscan numerals to the Latin.

Recently, two scholars have vainly attempted to re- vive 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 existence of symbols for the same numbers is evidence of a common need, not a common ori-

gin. 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 Pisa- ni 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 accepts C as acrophonic for 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

rejects everything else. He flirts with A as a variant of 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

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 acrophonic system. Mtiller himself is aware of the defect that (at his time) not one Etruscan numeral-word was known (p. 320).

43 Bonfante-Bonfante 79. 44 See Fiesel (supra n. 39) on X as a sibilant. 45 Bonfante-Bonfante 79. 46 Camillo Tarquini, S.J. "Dichiarazione dell'epigrafe del

lampadario di Cortona, della lettera A, e delle note nume- riche degli Etruschi," Dissertazioni della Pontificia Accade- mia Romana di Archeologia 15 (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

(1953) 1765. 47 See Bonfante-Bonfante 64 following M. Pallottino,

Thesaurus linguae Etruscae I (Rome 1978) 421, A.J. Pfif- fig, 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 r6misches Alphabet und Lat. mille," RhM 96 (1953) 89-93.

41 Pisani (supra n. 48) 92. so H. Rix, "Buchstabe, Zahlwort und Ziffer in alten Mit-

telitalien," Studi linguistici in onore di Vittore Pisani 2 (Brescia 1969) 845-46.

51 Bonfante-Bonfante 79 or Pfiffig (supra n. 47) 123-30 for numeral words.


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 ac-

cepted of all the theories. Mommsen has been fol-

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 obwal- tende 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 un- fairly 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 sym- bolic numerals larger than 10.58 This implication Mommsen accepts, citing as an unconvincing parallel

52 Supra n. 51 and see L.H. Jeffery, Local Scripts of Ar- chaic 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) 589- 601 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 ro- maine 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 acrophonic system and 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-century manuscript.

54 See (in chronological order) E. Goebel (in a review of A. Vanitek, Lateinische Grammatik) 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 Etrusker2 2 (Leipzig 1877; repr. Graz 1965) 534 and plate facing p. 560; Taylor (supra n. 26) 6-7; E. Hiibner, Exem- pla scripturae epigraphicae Latinae (Berlin 1885) LXX- LXXI; Hubner, Handbuch der klassischen Altertumswis- senschaft2 1 (Munich 1892) 651; A. Kirchoff, Studien zur Geschichte des griechischen Alphabets4 (Gutersloh 1887; repr. Amsterdam 1970) 132-33; F. Bucheler, "Altes Latein XVIII," RhM 46 (1891) 238-41 = Kleine Schriften2 3 (Os- nabruick 1965) 203-205 (the latter reference I 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, Ausfiihrliche grammatik der lateinischen Sprache2 1 (Hannover 1912) 5, 630-31; Lefebvre (supra n. 18) 30; Smith (supra n. 27) 1-5; M. Leumann, Lateini- sche Grammatik: Handbuch der Altertumswissenschaft

2.2.1 (Munich 1926) 47; Sandys (supra n. 27) 54; F. Cajori, A History of Mathematical Notations 1 (Lasalle, Ill. 1928) 30-31; D.E. Smith, History of Mathematics 2 (New York 1925) 55-56 (hesitantly); D.E. Smith and J. Ginsburg, Numbers and Numerals (Contributions of 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 Mathematics 1 (New York 1956) 448; D. Diringer, The Al- phabet2 (New York 1948) 536 (for L, C, D, and 0D); Ency- clopedia Britannica 16 (1958) 612, s.v. Numerals (D.E. Smith); M. Guarducci, L'epigrafia greca 1 (Rome 1967) 220; H. Jensen, Die Schrift in Vergangenheit und Gegen- wart3 (Berlin 1969) 512-23 (who also takes note of the "tally-mark" theory; I owe this reference to Werner Kren- kel, Rostock); A.J. Vaccaro, La numeracidn latina: 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, Mathematics and 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 Etruscan numerals.

11 Mommsen (1887) 598. 56 Gordon (supra n. 40). 17 Mommsen (1850) 20. 58 Zangemeister 1012-13. For examples of primitive in-

competence in counting above some small number, see Stra- bo 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, Math-


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 \V resembles 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 More- over, Mommsen claims that the Etruscans, who did use 0, distinguished ? = 100 from ? = 0 by remov-

ing 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 principles from which to construct seven symbols.

Even Mommsen's fundamental notion is question- able. 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 (D be 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 included in some formal Latin abecedaria (cf. the retention of the unused letters B, D, and O in early Etruscan abecedaria67), and the replacement of 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.,68 CIL 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 modified his theory in ways which have not won general acceptance. Ritschl hesitates at a pictographic origin for V = 5 and X =

ematics in Aristotle (Oxford 1949) 259; and thirdly Ifrah (1981) 5-19 for some modern examples. Such people are preliterate and give no grounds for separating the Latin numeral system into two parts. In 1818 Orioli (supra n. 38) 221-22 had already claimed that the Etruscans had no numerals greater than 50, arguing from the absence of evidence. In 1825 he changed his mind (supra n. 38).

"9 Mommsen (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 con- cedes. 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 produce any examples ("lisst sich diese An- nahme nicht durch belegte Beispiele sichern"). R.S. Con- way, 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 Gundermann 35 remarks that it is "leere Ver- mutung." The plate in CII seems clear (hef not eee ) and I am unaware of any scholarly acceptance of Conway's reading. Note that CII 2873 quater is an undated Central Os- can inscription and Conway has C = 100 in another undated Central Oscan inscription (59 = CII 2806). In Um- brian the only relevant inscriptions are Conway 354, of the Gracchan period, in which C = 100, and the Tabula Eugu- bina 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. 64 Bonfante-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 "dotted 0" (from the center-punch mark of the cutting compass, present also in the ordinary O). See Jeffery (supra n. 52) 29 and passim. The conclusion may be drawn that the Etruscans "removed" the cross without concern for a distinction between 9 and any other character.

66 Mommsen (1887) 599-600. 67 See the list of abecedaria in Bonfante-Bonfante 106-109

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 abecedarium by the time of Cicero, as he (Nat.D. 2.93) indicates that the alphabet contained only 21 letters. Other extant abecedaria are even later, e.g., the two fifth- sixth-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 Abecedaria from Egypt," AJP 56 (1935) 147-48; and R. Cavenaile, Corpus Papyro- rum Latinarum (Wiesbaden 1958) 136-37, nos. 58-59.


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).69 Thus Ritschl reduces the number of principles needed from three to two at the expense of intro- ducing 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 Etrus- can, used here in reverse alphabetical order.70 He explains Etruscan * = 100 as a Cypro-Lycian import, and A as the half of X. Deecke too requires three principles, as well as special pleading. Mentz adopts a similar theory, much more systematic but less in ac- cord 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-decade values. 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 accepted this theory in 1926, has recently (1977) expressed doubts.72 In summary, I suggest that, though authorized by Mommsen and accepted by many handbooks, this theory is but a small advance on Priscian, and is not supported by the evidence.

A FEW RENAISSANCE AND MODERN

"NEBENTHEORIEN ))

Under this heading I consider a few theories which do not fall into any of the preceding classifications. None of these theories deserves or has attracted much acceptance. They are included for completeness.

Hostus's (1582) theory that the numerals owe their origin to the abacus73 is dependent only on the obser- vation that the abacus has decimal columns, 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

yield the forms of the numerals, as is amply demon- strated by the very different forms of the Greek acrophonic numerals.74

Lanzi (1824) assumes that the Etruscan numerals (to 50) were an alphabetic system.75 He 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 numeral A = 5 was the lower half of X. But his explanation of the values of X and T con- tradicts the facts: X = S was at the end of the Etruscan alphabet, EB = ksi was the 15th letter of alphabets which included it, and T was the seventh letter following (counting inclusively). Furthermore, as Fabretti points out, there is no evidence that any other letters were used for alphabetic numerals; we find XX, XXX, XXXX and not n, P, Z (as would be required by Lan- zi's theory).76

Faulmann (1880) advocates a muddled Semitic acrophonic system, confused with a native Latin alphabetic system 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), "dessen dilteste Form Y war," which meant 2 (how 2 became 5 he does not explain). X derives from Greek K from He- brew 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 understand his explanation of D = 500.

In such a context even Gundermann's (1899) Se- mitic alphabetic system will seem partially plausi- ble.78 While wholly speculative and supported only by special pleading, his theory has at least the merit of being systematic. All symbols V, X, \/, C, D, and (D are explained as derivative forms of Semitic alphabetic numerals, though the forms are often not close and he must resort to several different Semitic alphabets.

69 Ritschl (supra n. 54) 13, 18. Followed by Cajori (supra n. 54) and Roby (supra n. 27). X was used as a symbol for 0: see Appendix II.

70 Deecke and Mtiller (supra n. 54) 534. 71 A. Mentz, Geschichte der griechisch-r6imischen Schrift

(Leipzig 1920) 41. 72 Leumann (supra n. 54) in 1926; it is in the new edition

(1977) of the same work that he hesitates: p. 5. 73 Hostus (supra n. 14) 20-22, refuted by Cantor (supra

n. 14) 160. Gundermann 42 approves Cantor's refutation. Either independently or without citation, Hostus is followed by RE S.3 (1918) 11.9-64, s.v. Abacus (9) (A. Nagl). Nagl is himself followed by C.M. Taisbak, "Roman Numerals and the Abacus," ClMed 26 (1965) 147-60. An apparently independent suggestion is due to P.A. Dapre, "The Origin

of the Roman Numerals," Didaskalos 5 (1976) 359-60: Cantor's refutation holds.

74 See Nagl (supra n. 73) on the Greek abacus and its asso- ciation with the Greek acrophonic numerals.

75 L.A. Lanzi, Saggio di lingua etrusca e di altre antiche d'Italia2 (Florence 1824/1825) 385-86 (?XIV).

76 For the position of EB and T, see the abecedaria in Pallot- tino (supra n. 47) 409-10, and see Fabretti (1877) 154-55 on the absence of fl, P, Z.

77 K. Faulmann, Illustrierte Geschichte der Schrift (Vien- na and Leipzig 1880) 546-47.

78 Gundermann 45-49. The special pleading of his theory does not invalidate his learned and useful survey of Medi- terranean numeral systems.


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 intro-

duced.. so that < = 500... and two triangles to-

gether soon had their lines changed to curves .. to

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 as-

signed, 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 es-

sentially 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 eval-

uating 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 can-

not (omnibus paribus). These first three are principles of economy of hypothesis. Four: symbols for decade- values (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 rITEvT by •'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 sin-

gle 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 Etrus- can 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

7" B.W. Minshall, "The Roman Numerals," Didaskalos 5 (1976) 262-65.

80 Cf. Zangemeister 1012, who seeks a solution which has "ein einheitliches Entstehungsprinzip ftir 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 Conse- quenz 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's theory.

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 ita- liche 1 (Milan 1788) 22, apparently following up a remark by Maffei who thought that "gli Etruschi gli avessero appre- si 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;

Cantor (supra n. 14) 161; Tarquini (supra n. 46) 70, 77; Friedlein 27; Fabretti (1877) 155-56; Mommsen (1887) 598 and Zangemeister 1013. On G. Micali (1776-1844) see Eckstein (supra n. 6) 373.

82 See supra ns. 1, 7 and 32 on the priority of decade symbols. On the Greek acrophonic system, see Tod (supra n. 7). On other examples of additive non-place-value systems em- bodying symbols for half-decade values, see Ifrah (1981) 148, 152-57 (Sabaean Arabs, Lycians, Mayas, Palmyrean Arameans, and various tally-mark systems).

83 That the use of letters for numerals must be either alphabetic or acrophonic was first explicitly noted by Mill- ler (supra n. 42) 320, who was followed by Cantor (supra n. 14) 161-62. Cf. Tod (supra ns. 7 and 9) and Friedlein 27. Ifrah (1981) gives examples only of alphabetic or acrophonic uses of letters as numerals. Only the Notae Papianae et Einsidlenses (ca. eighth century A.C.) provide evidence of random letter-use for numerals: Keil, Gramm. Lat. 4.330.

84 Cf. Tod (supra n. 7). They must because the name for such half-decade values is always derived from the previous decade (except in the case of five and one): see Meninger v. 1 passim.


Fig. 5. Etruscan bronze coin with X mark of value. Vatican, Medagliere 22e. (After A. Sambon, Les monnaies antiques 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 succeed-

ing 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 con-

ception, 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 in- structive 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 Etrus-

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. Fragmentary Etruscan inscription with CC = 200. (After M. Cristofani, StEtr 38 [1970] pl. XXVIII)

85 See Jeffery (supra n. 52) 5-6 in general and 23-25 for examples.

86 The theory of Zangemeister 1016-17 that *-> XK-> C lacks foundation, as noted by Mommsen (1888) 155. Gundermann 38-39 explains X as "rechtsliufiges C und linkslaufiges D ; der strich I zwischen den beiden Formen bedeutet die... Einheit" and compares the Cyprian [IF for "one drachma." This explanation is not convincing. After developing the idea in the text, I was gratified to note that Ifrah (1981) 158 suggests a similar evolution, as does Gardthausen (supra n. 18) 403 ("die Lateiner vereinfachten dieses Zeichen [ ] zu C").

87 Pliny HN 33.44-45 implies a date of 269 B.C. for the introduction of 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 (Copenhagen 1957) 187, and 2 (Nationalmuseets

Skrifter, Arkaeologisk-historisk Raekke 9, Copenhagen 1961) passim (esp. 384, 391) who advocated the date 212 1 B.C. See also M.H. Crawford, Roman Republican Coinage 1 (Cambridge 1974) 3-35 and T.V. Buttrey, "The Morgan- tina Excavations and the Date of the Roman Denarius," Congresso Internazionale di Numismatica 2: Atti (Rome 1965) 261-67 (with reply by Mattingly and discussion 269-73). The earliest dated attestation of the denarius ligature 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 XVI is not attested before 141 B.C.: Crawford 1.260 (issue no. 224), 70 years after the introduction of the denarius (after 136 B.C. the attested marks of value vary in form fre- quently). Absence of evidence is not evidence of absence- we do not know when the denarius ligature N was introduced, but it is not attested before 136 B.C. See the undated inscriptions infra n. 99.


Fig. 7. Italian inscriptions with C = 100 (left and right) and possible Etruscan 500 symbol (center). Left to right: CII2806, CII 2229, CII 2883.

strokes combined with a circle for the upper decades.88 It is impossible to know why this change occurred, but its occurrence supports the likelihood of such a change in the Etruscan system. The shift in the Etruscan system occurs between 100 and 1000, from crossed strokes to crossed strokes in 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 circle of @ would no doubt be 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 naturally to the well-attested formal Lat- in numeral D .93 Although the development of Etrus- can 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 attested in Etruscan, is the half of the original symbol GI for 1000.94 1 am unaware of any theory (besides this modified tally-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 D might have been

KV, the lower half of G, rather than D, the right half of E.D

Fig. 8. Etruscan abacus gem. Paris, Bibliotheque Nationale. CII 2578 ter.

88 The similarity has been noted also 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 hieroglyphic numerals); S. Dow, "Minoan Writing," AJA 58 (1954) 77-129, esp. 123-25; W.F. Anderson, "Arithmetical Proce- dure in Minoan Linear A and in Minoan-Greek Linear B," AJA 62 (1958) 363-68; and D.J. Struik, "Minoan and My- cenaean Numerals," Historia Mathematica 9 (1982) 54-58. Another numeral system with some similarity is the Lycian: see R. Shafer, "Lycian Numerals," AO 18.4 (1950) = Sym- bolae ad Studia Orientis Pertinentes Frederico Hroznj Dedi- catae, Pars Quinta 251-61: I = 1, V = 5, O = 10. Shafer notes the similarity (p. 251).

89 Gordon-Gordon 109 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; Gundermann 30-32 (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 Gundermann 30-32 (no. 17); Gordon no. 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-Gordon 182; Meyer (supra n. 4) 31-32; and Gordon 44-49.

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-Gordon p. 241, n. 1. Cf. the various derivations of D presented above. Gordon astutely notes that

D may have been derived from "a (D -type symbol with horizontal as well as vertical stroke inside the circle" (p. 46). See Appendix I on the apparent failure of the Latins to adopt the Etruscan $ = 10,000.


Fig. 9. Etruscan lead tablet with • = 1000 and • = 10,000. CIE 6310. (After M. Torelli, ArchCl 18 [1966] pl. XCIVa)

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 Renais- sance 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 Ra- mus, modified as in this paper, best explains the origin of the Latin numerals.

Appendix I: Etruscan Numerals

Table 2 lists the known forms of the Etruscan numerals (supported in the text following).

Table 2. Etruscan Numerals

> I A X 4,

*,)I(,C[w] e,? [P[] $ ?2 1 5 10 50 100 500 1000 5000 10000

The numerals >, I, A, X, and /\ are well attested and agreed upon: > to A occur on coins, I to A\ in funerary inscriptions (and ' is well attested in Latin).95 The numerals [ w] and [ v] are unattested96 and are tenta- tive suggestions for the forms, based on the principle observed in the cases of A and 4 of using the lower half of the succeeding decade symbol 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),98 and

Fig. 10. Possible example of Etruscan 500 symbol, from amphora. (After L.I.F. Janssen, Inscriptiones Etruscae [Lei- den 1840] pl. IV.48)

95 See 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 convenience I 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 provides a photo- graph, 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

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 second in 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) terracotta fragment: G.F. Gamurrini and A. Fabretti, Appendice al CII (Florence 1880) no. 114 (pl. IV) (tris) =


in two Latin inscriptions."99 The rounded form AK is 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 "Pa-

risian" 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 Commenta- tors 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 in-

scriptions 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-

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

deFeis 1898 (supra n. 19) 14, pl. I.V = Gundermann 39-40 with figure = Ifrah (1981) 144 with figure (and Zange- meister 1016) = my fig. 4; 2) on the base of a vase: deFeis 1898 (supra n. 19) 14, pl. I.III = Gundermann 39 = Zange- meister 1016, n. 2.

99 DeFeis 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's error: "]CETERIS-MEIS/]-

VSTRINV.LICERET/]AGR P" *XX" (read XXX?);

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.

0oo See 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. Mar- chetti, "La metrologie des monnaies etrusques les plus an- ciennes," Contributti introduttivi allo studio della moneta- zione etrusca: Atti del V convegno del Centro Internazionale 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). '01 On 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 campagna di scavi a Punta della Vi- pera e scoperto di 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. 103 Miiller (supra n. 42) 318, 320 and pl. IV.2 (at end),

Mommsen (1850) 19, Friedlein 27-28, Fabretti (1877) 156, Mommsen (1887) 599, Ninni (supra n. 16) 683-84, Gun- dermann 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. 105 See Buonamici (supra n. 103) 244 and Zangemeister 1021. 106 See supra n. 87. 107 For the Etruscan alphabet see Bonfante-Bonfante 79,

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 Buchstabenfolge des

546 PAUL KEYSER

A Hz0H f I ) IMwPr O PM9 "rY X

A B C D E F H - K L N -0 P - Q RS T V X

Fig. 11. Etruscan alphabet (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 "'VPETI'" are examples. The letter Y (chi) is attested in an un-

dated funerary inscription (CIL I2, 2.476.6): "A.

S[I]RPIOS ES \ ," but the -IOS ending of the gentili- tial 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 ac- knowledged. 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

DEPARTMENT OF CLASSICS

UNIVERSITY OF COLORADO AT BOULDER

BOULDER, COLORADO 80309

lateinischen Alphabets," RhM 15 (1860) 463-67, 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 century A.C. is cited supra n. 68.

108 I ignore here the later reintroduction of Z. 109

Sandys (supra n. 27) 35. 110 New letters are generally added to the end of the alpha-

bet: cf. v, , X, X,, co in Greek or the later reintroduction of 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 reference I owe to C.F. Konrad) or Vetter (supra n. 101) 13-28 (a reference I owe to an anonymous AJA referee); for the dating see Dizionario epigrafico di antichitac romane 4 (1957) 715-17, s.v. Lex (G. Tibiletti) (a reference I owe to an anonymous AJA referee).

112 On " received as "ss" see Diomedes in Gramm. Lat.

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. 113 Jeffery (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, Odys- seus -> Ulixes (which also shows the familiar d -> 1 of Latin, cf. Lewis and Short s. "L" 11.3, A. Walde, Latei- nisches etymologisches Wbrterbuch3 [Heidelberg 1938] xi and C.D. Buck, Comparative Grammar of Greek and Latin [Chicago 1933; repr. 1952] 123), sestius -> sextius, and Aias -> Aiax. See CIL 5.1880folex (for foles), 5.5583 ses- tum (for sextum), 5.6726 conius (for coniux), and 5.893, 900 and 8280 milex (for miles). 117 Note added in proof: Nancy de Grummond has kindly

directed my attention to Livia Giacardi, "L'origine della numerazione romana," Centro Studi "Cristiano Mancini" per la Storia del Pensiero Matematico 1 (Foligno 1987), not yet available in this country.

“The Origin of the Latin Numerals 1 to 1000”, American Journal of Archaeology 92 (1988)...

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