The Egyptian origin of the Greek alphabetic numerals.
Introduction
Numbers are represented in Greek classical inscriptions in two different ways. The first system, the acrophonic, is so named because the signs used to represent five and multiples of 10 are taken from the first letters of the appropriate Greek numeral words ([GAMMA] = five = [TEXT NOT REPRODUCIBLE IN ASCII]; [DELTA] = 10 = [TEXT NOT REPRODUCIBLE IN ASCII]; H = 100 = [TEXT NOT REPRODUCIBLE IN ASCII]; X = 1000 = [TEXT NOT REPRODUCIBLE IN ASCII]; M = 10 000 = [TEXT NOT REPRODUCIBLE IN ASCII]); 50, 500, 5000 and 50 000 are formed using multiplicative combinations of the sign for five and the other signs ([??], [??], [??], and [??], respectively). Like the Roman numerals, the acrophonic system is a mixed base 5 and base 10 system, and numbers are written using long strings of the above signs combined into a single additive phrase (e.g. 12784 = [TEXT NOT REPRODUCIBLE IN ASCII]). The other system, which is the primary focus of this paper, is the Greek alphabetic systemalso known as the 'Milesian' or 'Ionic' systemwhich was regularly used in the Greek script from the third century BC to the fifteenth century AD, and still occasionally used today. Throughout this paper I will use the locationneutral term 'alphabetic numerals', without denying that the numerals originated among Ionian traders. The Greek alphabetic numerals are extraordinarily important for understanding the history of numeration, but the debate regarding their origin has hardly progressed in a century. Early theories holding that the numerals were developed in the eighth century BC or earlier, or that they, like the Greek alphabet, had a Semitic origin, have now been refuted (Gow 1883; Larfeld 190207). The major study of the alphabetic numerals that Marcus Tod had hoped to present in the 1950s was never completed, leaving us with only a single brief paper on the subject from this pioneer (Tod 1950). As Johnston indicates, the study of the early history of the Greek numerals (both alphabetic and acrophonic) has generally been ignored in favour of limited studies of regional variations that developed much later (1979: 27). When they have considered the topic, classical epigraphers have assumed that the alphabetic numerals were independently invented, without considering the possibility that the system had an external origin. In this paper I contend, on the basis of structural similarities and historical indications, that the Greek alphabetic numerals developed from the Egyptian demotic numerals in the context of Ionian trade with Lower Egypt in the early sixth century BC. The Greek alphabetic numerals The Greek majuscule alphabetic numerals are shown in Figure 1. Whereas our own numerals require only 10 symbols, the Greek 'alphabetic numerals require nine symbols for each value of the base (units, tens, hundreds, etc.). There was no zero or similar sign, because none was needed; the system is an additive one rather than one based on position or placevalue. Numerals were usually written from left to right, though righttoleft and boustrophedon inscriptions are not unknown. Thus, 562 would be written X[XI]B. The numeralsigns are the 24 familiar letters of the Greek alphabet, plus three archaic or foreign signs called episemons: vau or digamma (6), qoppa (90) and san or sampi (900). The episemons are necessary to complete the full complement of 27 characters to occupy all the values from 1 through 900, enabling any natural number less than 1000 to be written. Vau and qoppa were occasionally used phonetically in some archaic Greek dialects, with the rough phonetic values of [v] and [k], while sampi was apparently borrowed from Phoenician sade [ts], though it may have occasionally been used phonetically in certain archaic Greek dialects (Swiggers 1996: 26566). When the alphabetic numerals were developed, the Ionic script had characters for vau and qoppa, but not sampi. In the Phoenician alphabet, sade, equivalent to the Greek sampi, was located between pe (equivalent to pi/[PI]) and qof (equivalent to qoppa/[??]). But in borrowing sade (as sampi) for the alphabetic numerals as the third episemon, the Ionians abandoned its order in the Phoenician script and placed it at the end of the system, with the value of 900. However, I am not convinced that the ordering of the episemons proves very much about the date of origin of the alphabetic numerals (cf Jeffery 1990: 327). In this instance, I think we would be much better off sticking to the material evidence. [FIGURE 1 OMITTED] To express fractions in Greek alphabetic numerals, the most common technique was to place a small mark above and to the right of a given numeralsign to indicate a fraction with a numerator of 1 and a denominator of the alphabetic numeralsign. These 'unitfractions' or '1/x fractions' were commonly used in early Greek mathematical texts and elsewhere. In addition to the standard unitfractions, special signs existed for 1/2 ([??]' and [??]') and 2/3 ([omega]') (Thomas 1962: 45). While later mathematicians such as Heron and Diophantus used common (x/y) fractions with alphabetic numerals for both the numerator and the denominator, this was never the practice during the period relevant to the present discussion (Fowler 1999: 227). For larger numbers, two different techniques were used. For multiples of 1000, a small slanting mark, or hasta, placed to the left of a unitsign, indicated that the following sign's value should be multiplied by 1000; thus, [GAMMA] means three but [??] means 3000 (Threatte 1980: 115). Values above 10 000 are rarely encountered except in mathematical works, and individual mathematicians used different methods of representing them. Most commonly, placing a small alphabetic numeralphrase (less than 10 000) above a large M character (=myriades) indicated multiplication by 10 000; thus, 3 000 000 would be expressed with only two signs (300 x 1000), as [??] (Heath 1921: 3941). Any number less than 100 million could thus be expressed. The first reliable epigraphic evidence for alphabetic numerals comes from a Middle Corinthian krater of the Detroit painter dating to around 575 BC, on which '[SIGMA]YMI' is inscribed on the foot in Ionic characters (Johnston 1973: 186). Johnston, noting that the first three signs are used elsewhere as an abbreviation for [SIGMA]YMMIKTA 'mixed batch', argues for the reading of the fourth character as an archaic zeta (I) representing the alphabetic numeral 7. There is substantial evidence from Attica and Corinth for the use of alphabetic numerals as capacity markers on mercantile vases, especially in the late sixth and early fifth centuries BC (Hackl 1909). While it is often impossible to distinguish an alphabetic numeral from an ordinary letter, such identifications are reliable where numerals are separated from text by punctuation, where tallymarks accompany the alphabetic signs and provide the correct sum, where alphabetic numerals are accompanied by another numerical notation system, or where a set of two or three letters is possible as a numerical value but not phonetically. Figure 2 shows a remarkable inscription on the base of an Athenian vase (Louvre F211) of the Leagros group (525500 BC) containing the number 29 both in alphabetic numerals (K[theta]) and in a system used in the Levant and Cyprus by which tens and ones are indicated by horizontal and vertical strokes, respectively (Johnston 1979:31). That an Attic vase would be inscribed both with alphabetic Ionic numerals and abstract LevantineCypriote numerals abundantly confirms the international context of the representational systems of this period. [FIGURE 2 OMITTED] However, it is extremely unlikely that the numerals actually originated either in Attica or Corinth. Rather, the numerals were probably developed by individuals with ties to western Asia Minor, especially the Ionian cities of Miletus and Halicarnassus, where some early instances of the numerals are found (Heath 1921: 3233). While Miletus (from whence the adjective 'Milesian'), was geographically situated in Carla, the region of Asia Minor immediately to the south of Ionia, the Milesians spoke an Ionian dialect and used the Ionic script. All the examples of the alphabetic numerals from the sixth century BC, even those found outside Asia Minor, are written using Ionic signs and, where accompanied by writing, the purely phonetic signs are also Ionic (Johnston 1979: 27). Moreover, because these examples are all graffiti and dipinti on the bases of mercantile vases, and are found in conjunction with trademarks and other identificatory letters, it is likely that the numerals were first used and diffused by Ionian traders throughout the Mediterranean. Unfortunately, there are relatively few examples of alphabetic numerals from before the late fourth century BC, and even fewer whose provenances can be determined with certainty. Moreover, because all of them express values under 1000, we do not know how higher alphabetic numerals were written at this early date, if at all. After a period of Ionian cultural dominance between 575 BC and 475 BC, alphabetic numerals were used only rarely in the period between roughly 475 BC and 325 BC. While the system did not disappear entirely, it fell out of common use in favour of the acrophonic numerals as Athens' influence waxed and Ionia's waned. Some of the probable alphabetic numerals from this interim period include those found on IG [I.sup.2] 760, an inscription from the Acropolis dating to the Periclean period (Tod 1950: 137), casket inscriptions from Halicarnassus dating to around 350 BC (Heath 1921: 3233), and various sherdmarks from the Athenian Agora (Lang 1976). This period, the height of Greek achievement, corresponds exactly with the rise of Athens as an Aegean power, while Ionia's strength decreased after the Milesianled Ionian revolt of 499 to 494 BC. This is further evidence that the alphabetic numerals were invented in Asia Minor rather than Greece proper, and that the traditional name 'Ionic' or 'Milesian' given to the system is correct. From the Alexandrine period onwards, the alphabetic numerals began to be preferred over the acrophonic numerals throughout most of the Greekspeaking world, with only Athens retaining the acrophonic system until around 50 BC (Threatte 1980:117). The numerals were invariant throughout the Greekspeaking world, in contrast with the highly variable acrophonic system. As such, the alphabetic numerals could easily be used as an effective instrument of communication and trade among diverse regions of Greece (Dow 1952: 23). Whereas Greece before Alexander was highly fragmented, rendering the development of a universal Greek numerical notation system unlikely, Alexandrine and especially Roman Greece provided a suitable environment for the development of a single panHellenic notation. It was used throughout the Greekspeaking world until the fall of the Byzantine Empire in the fifteenth century, and, like Roman numerals in Western Europe, is still occasionally used today for page numbering. Demotic numerals: a forgotten system Three distinct numerical notation systems were used in ancient Egypt: the hieroglyphic, the hieratic and the demotic. The earliest, the hieroglyphic numerals, was developed as early as 3250 BC (Dreyer 1998). It had numeralsigns for each exponent of 10, which could be repeated up to nine times each as necessary to write a desired number, it was thus structurally similar to the Roman numerals and the Greek acrophonic numerals, save that it lacked the quinary (base5) subbase of these two systems. Starting in the twentysixth century BC, the hieratic script emerged as a cursive shorthand of hieroglyphic, and was designed for writing in ink on papyrus documents and clay ostraca. The Old Kingdom hieratic numerals were no more than cursive variants of their hieroglyphic counterparts, and were based on the same principle. However, by the Middle Kingdom, groups of signs became condensed into single ciphers, so that the hieroglyphic [??][??][??] for 300 was 999 in Old Kingdom hieratic but [??] in Middle Kingdom and later hieratic. Thus, where there is only one hieroglyphic numeralsign for each exponent of 10, each of which can be repeated up to nine times in a single number, there are nine hieratic numeralsigns for each exponent of 10, which cannot be repeated. Hieratic numerals are those used in all of the Middle Kingdom Egyptian mathematical texts and most of the surviving economic records from the Middle and New Kingdoms. Nevertheless, the hieroglyphic numerals continued to be used for nearly all monumental writings, thus producing a rather stark functional division of the two scripts and their accompanying numerical notation systems. The hieroglyphic script and numerals did not disappear completely until the fourth century AD. Yet another division emerged in the late eighth century BC (TwentyFifth Dynasty), when the 'business hand' hieratic script used in the Nile Delta diverged from that used in Upper Egypt and became what we now know as the demotic script (Rimer 1996: 82). While the hieratic and demotic scripts were not mutually legible, the two sets of numerals operated on the same principle, and the early demotic numeralsigns were very similar to their hieratic counterparts. For a short time, the hieratic numerals continued to be used extensively in Upper Egypt, while the new demotic system was used in Lower Egypt. Early in the TwentySixth Dynasty, however, the demotic script and numerals were granted royal preference for commercial and administrative functions throughout Egypt, after which time the hieratic script was used only in a limited set of religious texts. The last hieratic texts date to around 200 AD. Demotic, on the other hand, was widely used in the Late and Ptolemaic periods, and, though it became less common after the Roman conquest of Egypt, survived until around 450 AD before being replaced by the Greekderived Coptic alphabet. The demotic numeralsigns shown in Figure 3 are typical of those found in papyri of the Late and Ptolemaic periods (Sethe 1916: Tables I and III). While there was considerable paleographic variation in the forms of the numerals used by different scribes, the structure of the demotic numerals remained constant at all times and locations. The system has a decimal base, with signs for each multiple of each exponent of 10, and is always written from right to left. Thus, 4637 would be written [TEXT NOT REPRODUCIBLE IN ASCII]. Some of the signs for the thousands may be multiplicative in nature, as there is a general resemblance between the signs for the hundreds and the corresponding signs for the thousands, although the paleography of the demotic numerals is too incomplete to warrant a firm conclusion. Sethe (1916: Table I) suggests that for the missing 5000 and 7000 signs, additive phrases incorporating two lower signs (3000+2000, 4000+3000) were used. Above 10 000, the demotic numerals are multiplicative combinations of two lower signs (though such high numbers are fairly rare); see, for instance, the multiplicative expressions for 90 000 ([??] = 9 x 10 000) and 100 000 ([??] = 10 x 10 000) in the demotic mathematical papyri (Parker 1972: 86). To express fractions, a small stroke placed above a given integer x indicated a unitfractional value of 1/x. This procedure was not always followed for 1/2, 1/3, 2/3, 1/4 and 5/6, for which special signs unrelated to the integer signs existed (Sethe 1916: Table III; Parker 1972: 86). [FIGURE 3 OMITTED] The demotic numerals served a wide variety of commercial, legal and other administrative functions during the Late period and the Ptolemaic era, and were written not only on clay and papyrus but also on stone. A number of demotic mathematical papyri have survived from the Ptolemaic period, confirming the suitability of the system for arithmetical and mathematical purposes (Parker 1972; Gillings 1978). Unfortunately, the demotic numerals are probably the most neglected among numerical notation systems worldwide, having been largely ignored by historians of numeration (e.g. Menninger 1969; Guitel 1975; Ifrah 1998). As a result, the historical importance of this system for economic and cultural contacts in the eastern Mediterranean region has been grossly underestimated. The Egyptian connection: demotic to alphabetic Most classicists interested in the topic now accept the alphabetic numerals as an early sixth century BC invention in western Asia Minor (Johnston 1979, Jeffery 1990). The question that remains unasked, however, is whether this development was stimulated, directly or indirectly, by some other numerical notation system. The unstated assumption of most classicists, Egyptologists, and scholars of numeration appears to be that the Greek numerals were independently invented. Instead, I propose that the structure of the Greek alphabetic numerals was borrowed directly from the Egyptian demotic numerals, except with the use of Greek alphabetic signs instead of abstract demotic signs. Almost sixty years ago, Boyer, recognizing the similarity between the two systems, thought this to be indicative of a historical connection; however his paper was not primarily oriented towards the historical demonstration of this argument, and in any case this insight has since been forgotten or ignored (Boyer 1944:159). New evidence uncovered in the interim has strengthened the case for the Egyptian origin of the Greek alphabetic numerals. One of the difficulties in tracing the origin of the alphabetic numerals is that, because they are the first to use phonetic signs as numeral signs, it is impossible to use paleographic evidence of their similarity to any earlier system. As the Greek system uses the alphabetic signs of its own script, there will be no similarity with the numeralsigns of any earlier script, since all earlier systems use abstract rather than phonetic signs. But in almost every important structural feature, the Greek alphabetic and Egyptian demotic systems are identical or very similar. They are both base 10 systems with nine signs for each exponent of the base (19, 1090, etc.), and both lack a zero or similar placeholder. The Greeks did eventually adopt a zero with the alphabetic numerals for astronomical calculations based on a Babylonian stimulus, but this was a much later development and need not be considered here. Such alphabetic systems obviously differ from our own, but more importantly, they are quite distinct from systems used at the time of the invention of the alphabetic numerals. The Roman numerals, Greek acrophonic numerals and Egyptian hieroglyphic numerals have limited sets of numeralsigns that are repeated to indicate their addition (e.g. CCCLXXXVIII = 388). So, too, with slight structural differences for the higher exponents, are the Phoenician, Aramaic, and AssyroBabylonian 'common' systems. The positional Babylonian system used in astronomy, in which the value oft sign depends on its position in a numeralphrase, is very different in principle from both alphabetic and demotic numerals, and in any case was not in use in the sixth century BC. Thus, no numerals in use in the Near East in the sixth century BC, except the demotic and hieratic numerals, are remotely similar in structure to the Greek alphabetic numerals. It is quite unlikely that the hieratic numerals were ancestral to the alphabetic numerals, since they were becoming rather restricted in the context of their use by this period. Only the demotic numerals remain as a plausible ancestor for the alphabetic numerals. Moreover, while it has yet to be established if the alphabetic numerals used multiplicative notation at an early date, surely it is suggestive that both alphabetic and demotic numerals eventually use multiplication with 10 000 as a multiplicand. The alphabetic system is also multiplicative for the thousands, which is not the case for the demotic numerals. It is interesting to speculate why the Greeks, for 10 000 and higher, would not simply have continued the series using 1090 and 100900 preceded by the hasta (I = 10 000; K = 20 000; [LAMBDA] = 30 000, etc.). It is possible that these two separate levels represent progressive steps in the system's development, with the higher (myriads) series being a later development. However, I think it more likely that this feature is a clue to the alphabetic numerals' history. If the Greek numerals were derived from the demotic numerals, it would be reasonable for the Greeks to adopt the multiplicative principle at the same level as in the demotic numerals, namely 10 000. However, because the Greek alphabet only has 24 letters, and requires three episemons to reach the 27 signs needed to get as high as 900, it would not have been feasible to find nine extra signs for the values 10009000. Consequently, the inventor(s) of the alphabetic numerals may have had the idea of using multiplication for the thousands values as well as the ten thousands. Furthermore, Greek arithmetical techniques for dealing with fractions show a remarkable continuity with the Egyptian unitfraction (1/x) tradition of computation (Knorr 1982). Both the demotic numerals and the early Greek alphabetic numerals used unit fractions by placing a small mark above a given integer to indicate the appropriate unit fraction. Furthermore, both use alternative nonunit fractions for specific fractional values, although this is more prevalent in the demotic numerals than in the Greek alphabetic numerals, which only do so for 1/2 and 2/3. Historians of mathematics are unanimous that the Greeks borrowed the unitfraction technique from the Egyptians, and I see no reason to doubt that the Greek use of special signs for 1/2 and 2/3 is also a result of Egyptian influence. It is thus remarkable that the similar representation of Greek and Egyptian fractions has never led historians of mathematics to see the similarity of the entire alphabetic numeral system with the demotic numerals. Historical context Let us turn now to the historical connections between Egypt and Greece. The demotic numerals were the predominant ones in use in Egypt (especially Lower Egypt) in the early sixth century BC, having ousted the hieratic numerals a century earlier. This was just the time when Greeks were starting to encounter Egyptians in large numbers for the purposes of international trade. Most notable among the earliest Greeks in Egypt were the Ionian traders who set up an important emporion at Naukratis in the western Nile delta around 625 BC (during the reign of Psammetichos I) to facilitate exchange with Egypt. Naukratis acted as a 'portoftrade' by means of which the Sake kings of Egypt could administer and control foreign trade with Greece, and was thus the central locus for trade and cultural contact between the two regions (Moller 2000). Inscriptions in the Ionic Greek alphabet have been found at Naukratis dating from the seventh century BC (Heath 1921: 33). The rise of Naukratis and international trade in Greece coincides precisely with the first use of commercial inscriptions (rather than simple owners' marks) under the feet of Greek vases (Johnston 1979: 2). Moreover, it is probable that the practice of placing trademarks on vases originated in Naukratis and similar communities (Johnston 1979: 51). To my knowledge, however, no alphabetic numerals are attested among the inscriptions found at Naukratis, while many inscriptions contain acrophonic numerals (Gardner 1888). While most of the acrophonic numerals from Naukratis date from later periods than are relevant to the present discussion, the sherd London 1965.930.747 (ca. 500 BC) contains an acrophonic pricemark (Johnston 1979: 61). However, acrophonic and alphabetic numerals were used contemporaneously at many places at this period, so this piece of evidence is not significantly detrimental to my theory. Regardless of whether the alphabetic numerals were actually used at Naukratis, that the period shortly after 600 BC marked the beginning of enormous trade between the Aegean and Egypt can hardly be disputed. It is entirely probable, then, that Ionian merchants in Egypt developed the alphabetic numerals after seeing the demotic numerals in use, and then transmitted their invention to the cities of Ionia and Ionian Carla, from whence their use became more generalised. Given the context of the earliest alphabetic numerals on mercantile containers, we need not consider the role of the Carian and Ionian mercenaries used by the Saite kings of Egypt in the transmission I am proposing. It seems astonishing that the alphabetic numerals, having been found extensively throughout Greece by the late sixth and early fifth centuries BC, would almost entirely cease to be used for over a century, only to reappear in large quantities in the late fourth century BC. While their disappearance can be attributed to the decline in importance of Ionia in the early fifth century BC, their reappearance around 325 BC does not correspond to any reassertion of power by the Ionian cities. Instead, their reappearance corresponds almost exactly in time and space with the rise of the Ptolemies in Egypt. Alphabetic numerals are used in a Greek marriage contract from Elephantine dating to 317/316 BC indicating a dowry payment of 1000 drachmas (Ifrah 1998: 233). Figure 4 shows a column from the Hibeh Papyrus i 27, a GrecoEgyptian astronomical document dating to around 300 BC, which provides the earliest attested example of the use of the numerals in a scientific context (Fowler and Turner 1983). The signs for whole numbers are indistinguishable from ordinary letters, while the signs with long oblique lines above them indicate 'unitfractions'. The numbers marked with arrows in the text indicate 12 8/15 (10 + 2 + 1/2 + 1/30) and 11 7/15 (10 + 1 + 1/3 + 1/10 + 1/30), respectively. The complexity and frequency of the numerals in this text suggest that the system may have been used on papyrus somewhat prior to 300 BC. Coins dating to 266 BC indicating the regnal year of Ptolemy II Soter are the first coins anywhere in the world to bear alphabetic numerals (Tod 1950:138). The influence of demotic mathematics, reckoning, and mensuration on Greek practices, and vice versa, remains understudied, but it is clear that there was some influence in both directions (cf. Neugebauer 1957: 90; Fowler 1999: 231234). [FIGURE 4 OMITTED] I suggest that the rejuvenation of the alphabetic numerals in early Ptolemaic Egypt is further evidence of the connection between the demotic and alphabetic systems. We do not know exactly why the alphabetic numerals reemerged in Egypt in particular, though several possibilities suggest themselves. Perhaps the early Ptolemies recognised that the similarities between the alphabetic and demotic numerals (as compared to the acrophonic system) facilitated easy translation between the two systems. The recognition of the demotic numerals' brevity and usefulness for mathematics might have prompted the renewed use of the similar alphabetic system in GrecoEgyptian documents. It may even be that the alphabetic numerals were used in Egypt by Ionian Greeks in the interim period, and that the Ptolemaic resurrection of the system was a continuation of the older tradition of numeration. For instance, the alphabetic numerals could have been used on nowlost papyri in the fifth and early fourth centuries BC, meaning that in part, the 'disappearance' of the alphabetic system may have been an artifice of the differential survival of the materials on which each system was used. The resurgence of alphabetic numerals does not have anything to do with the adoption of the Ionian alphabet at Athens in 403 BC, because the Athenians continued to use acrophonic numerals for several centuries (Tod 1950: 138). In suggesting that the Greek alphabetic numerals were not independently invented, I do not mean to deny the Greeks' inventiveness or to downplay the interesting properties of the alphabetic numerals. As mentioned above, the alphabetic numerals are multiplicative for 10009000, obviating the need for nine more signs for those values. More importantly, by virtue of the fact that most of its numeralsigns would already be understood (and their order known) by literate Greeks, the alphabetic numerals, in contrast to the demotic, did not require the learning of an enormous new set of abstract signs. Rather, one needed only to learn the numerical values attached to the signs, and anyone who already knew the order of the alphabet could determine the signs' values as long as the episemons were taken into account. Diffusion from Egypt does not imply that there is nothing special or interesting about these local Greek developments. Conclusion Many questions remain to be answered before this argument can be fully settled. Evidence of the early use of higher (presumably multiplicative) alphabetic numerals would surely strengthen the case, providing an additional structural parallel between the two systems. The historical connection could be strengthened if it could be shown that alphabetic numerals were used at Naukratis at an early date, or in Egypt at any time in the Late period. However, this evidence is not essential for such a connection to be postulated, since there is so much circumstantial evidence for the transfer. The preponderance of evidence suggests that the Greek alphabetic numeral system was inspired by the demotic numerals of the early sixth century BC. This theory is a further contribution towards delineating the economic and intellectual transfers between Egypt and Greece in antiquity, though of course at a later date and with a more secure contextual foundation than that found in Bernal's (1987) analysis of the subject. The alternative to the hypothesis of Egyptian borrowing is that a nearly identical numerical notation system was developed independently by the Ionians within a few decades after they came into contact with Egyptians in large numbers and founded a colony at Naukratis, where traders were no doubt exposed to the demotic numerals widely used for administration and commerce throughout Lower Egypt. The structural and historical evidence all suggests a close relationship between the Greek and demotic systems. While the lack of paleographic evidence from the numeralsigns means that it is difficult to prove the case one way or the other, at the very least, the presumption that the Greek alphabetic numerals were independently invented can no longer be sustained, and ought to be replaced by a working hypothesis of direct diffusion from the demotic numerals. 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THOMAS, I. 1962. Selections illustrating the history of Greek mathematics, vol. 1. Cambridge, MA: Harvard University Press. THREATTE, L. 1980. The grammar of Attic inscriptions, vol. 1 (Phonology). Berlin: de Gruyter. TOD, M.N. 1950. The alphabetic numeral system in Attica, Annual of the British School at Athens 45: 126139. Stephen Chrisomalis, Department of Anthropology, McGill University, Quebec, Canada Received: 18 November 2001 Accepted: 14 October 2002 

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