Numerals inside the Great Pyramid

A couple of weeks ago all the news was about some new red ochre markings found in a shaft on the interior of the Great Pyramid at Giza (a.k.a. the Pyramid of Khufu), identified using an exploratory robot. That was pretty cool. But if you’re a professional numbers guy (as I am) you’ll be doubly excited to learn that it is probable that those marks are hieratic numerals. If this interpretation is correct, these are almost certainly mason’s marks used to indicate some quantity involved in the construction. Other than the fact that I would like all news outlets to stop calling them hieroglyphs (they aren’t – the hieratic script is a cursive Egyptian script that differs significantly from the hieroglyphs, and the numerals look nothing alike), this is really cool. I do want to urge caution, however: this does not imply that the Great Pyramid was designed along some sort of mystical pattern or using some numerological precepts. It actually doesn’t tell us even that the marks indicate the length of the shaft (as Luca Miatello suggests in the new article) – it could just as easily be 121 bricks in a pile used to make a portion of the pyramid. I am also not 100% convinced of the ’121′ interpretation – the 100 could be a 200, very easily, or even some other sign altogether, for instance. But the idea that numerical marks using hieratic script would be made by the pyramid-makers is entirely plausible and helps show the role of hieratic script in the Old Kingdom. Although it’s hardly going to revolutionize our understanding of Egyptian mathematics, it may well help outline the functional contexts of the use of numerals in Old Kingdom Egypt.

An anti-hiatus?

Apparently the English language has a lexical gap – it has no good term for ‘the end of a hiatus’ (‘resumption’ and ‘recommencement’ hardly suffice semantically), but, in any case, my apologies for having been largely absent here the past couple of months. Our department of nine full-time faculty has just finished three simultaneous job searches, which for those of you in academia, will give you a very good sense of what I’ve been up to.

And a very happy Pi Day (3/14) to all of you who celebrate! I didn’t do much special at 1:59 pm – perhaps I should have toasted Archimedes or something like that. I’m fonder of Pi Approximation Day anyway (July 22), since 3 1/7 is much closer to pi than 3.14.

This week’s World Wide Words (the e-magazine authored for 15 years by the inestimably talented lexicographer, Michael Quinion) featured one of my favourite numerical words, chronogram, meaning number-riddles in which a date is encoded in text using Roman numerals. Quinion mentions in passing the “three big books” of James Hilton from the 19th century, but this does little justice to the 1500+ pages of chronograms Hilton compiled over two decades. The first two volumes available for free download from Google Books (vol 1 – 1882; vol 2 – 1885; vol 3, 1895, is inexplicably only in ‘snippet view’).

Lastly, here is some good advice for those who are (rightly) considering charitable donations in support of victims of the Sendai earthquake.

Numeration and Numeracy in Cognition, Language, and Culture

Last month, at the Society for Anthropological Sciences annual meeting in Charleston, SC, I organized a panel of some really interesting material on the broad topic of numeration. I want to take this opportunity (again) to thank all the presenters for their attendance and hard work. The abstracts (as also published in the conference program) were as follows:

Toward a cognitive, historical, linguistic anthropology of numerals
Stephen Chrisomalis (Wayne State University)

For over a century, the study of numeration, number systems and allied topics has been a key part of the comparative study of thought, language, and culture. The anthropology of numbers and mathematics has traditionally been a locus for unilinear evolutionary thought linked to notions of primitivity. The papers in this panel constitute a call for a culturally-grounded cognitive science of numeration within four of the disciplines of cognitive science (anthropology, linguistics, philosophy, and psychology).

Recent research in language evolution, linguistic relativity, and cultural aspects of mathematical cognition draw attention to the need for anthropologists to re-engage with this new agenda. First, the cross-cultural study of numerals allows the investigation and evaluation of universal and particular aspects of numeration and their relationship with social organization. Because numerals have multiple modalities (e.g., verbal, graphic, gestural), examining patterns in number systems beyond linguistics allows us to evaluate to what extent number concepts can be separated from language, including universal grammar. Finally, just as the cognitive anthropology of plant and animal taxonomy contributes to ecological and environmental anthropology, the cognitive anthropology of numerals and mathematics underpins economic anthropology and the anthropology of science.

Spatial-numeric associations in literates and illiterates
Samar Zebian (Lebanese American University, Beirut Lebanon)

Several independent studies have reported a cognitive association between small numbers and the left side of space and larger numbers andthe right side of space among individuals who read and write from left-to-right (SNARC effect). These associations are reversed for individuals who read and write from right-to-left. The SNARC effect has widely been taken as evidence that numbers are conceptualized as points along a mental number line, however there is growing evidence that this systematic spatial performance bias related to writing directionality is an instance of strategic processing rather than a reflection of inherent spatial attributes of numbers. In an attempt to explain the “deeper” origins of these associations researchers are examining the linkages between number and finger counting. The current study examines whether finger counting practices reveal consistent spatial-numeric associations and whether there are any spillover effects to other tasks that involve object sorting and counting and other non-counting but quantitative tasks such as line bisection and speeded parity judgment. If, in fact, finger counting practices and not the directionality of writing set up spatial-numeric associations than we should be able to observe the same type of spatial biases in literates and illiterates. Preliminary evidence suggests that the finger counting practices of literates and illiterates are not same and furthermore that the spatial biases found in finger counting are not observed across tasks.

Zero’s beginnings: the Mayan case
John Justeson (SUNY, Albany)

This paper addresses linguistic and (Mayan) historical evidence concerning the origins of a numerical concept of zero. Comparative linguistic evidence suggests that zero is not part of basic numerical cognition; rather, it develops out of computing practices of mathematical specialists. Specifically, while zero is often assumed to be prerequisite to the invention of positional notation, it seems on the contrary to emerge as a notational device within such systems. This is clearly the case in Mesoamerica. A system of place-value notation arose in Guatemala and Mexico among Mayans and epi-Olmecs by 36 BCE, with no symbol corresponding to a zero coefficient. Although data is limited, circumstantial evidence is consistent with the following scenario for the emergence of a numerical zero: Mayan calendar specialists developed discourse practices, associated with calendrically-timed ritual events, that used the word “lacking”; the associated dates were represented in a new, non-positional system of notation, which replaced positional notation except in calculating tables; the sign for “lacking” was transferred from the new notation into these tabular positional notations; as a side effect of the algorithms that specialists used to add and subtract positional numerals, the “lacking” symbol was reinterpreted numerically.

Methodological reflections on typologies for numeral systems
Theodore R. Widom and Dirk Schlimm (McGill University)

Past and present societies worldwide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. In the present paper we provide a general framework
within which the efficacy of these typologies can be assessed relative to certain desiderata. Using this framework, we discuss the two influential typologies of Zhang & Norman and Chrisomalis, and present a new typology which takes as its starting point the principles by which numeral systems represent multipliers (the principles of cumulation and cipherization), and
bases (those of integration, parsing, and positionality). We argue with many different examples that this provides a more refined classification of numeral systems than the ones put forward previously. We also note that the framework can be used to assess typologies not only of numeral systems, but of many domains.

Social relationships as a lexical source for numeral terms in Amazonia
Cynthia Hansen and Patience Epps (The University of Texas at Austin)

Due to the relatively high degree of etymological transparency found in the numeral systems of Amazonia, it is possible to see the range of lexical sources from which the numeral terminology emerges. In this paper, we present the range of strategies used to create numeral terms below 5, based on an extensive survey of the numeral systems of close to 200 Amazonian languages conducted by the authors. More specifically, we discuss a strategy that is well-attested in Amazonia but that is not attested elsewhere in the world: a ‘relational’ strategy where terms for 4 (and sometimes 3-10) are built using a social relationship term, such as ‘sibling’ or ‘companion’. We propose that this strategy mirrors a gestural counting strategy found throughout the region where fingers are grouped in pairs.

Cultural variation in numeration systems and their mapping onto the mental number line
Andrea Bender and Sieghard Beller (University of Freiburg)

The ability to exactly assess large numbers hinges on cultural tools such as counting sequences and thus offers a great opportunity to study how culture interacts with cognition. To obtain a more comprehensive picture of the cultural variance in number representation, we argue for the inclusion of cross-linguistic analyses. In this talk, we will briefly depict the specific counting systems of Polynesian and Micronesian languages that were once derived from an abstract and regular system by extension in three dimensions. The linguistic origins, cognitive properties, and cultural context of these specific counting systems are analyzed, and their implications for the nature of a (putative) mental number line are discussed.

Review: von Mengden, Cardinal Numerals

This review appeared originally in the LINGUIST List at http://linguistlist.org/issues/21/21-5213.html

AUTHOR: von Mengden, Ferdinand
TITLE: Cardinal Numerals
SUBTITLE: Old English from a Cross-Linguistic Perspective
SERIES: Topics in English Linguistics [TiEL] 67
PUBLISHER: De Gruyter Mouton
YEAR: 2010

Stephen Chrisomalis, Department of Anthropology, Wayne State University

SUMMARY

This monograph is a systematic analysis of Old English numerals that goes far
beyond descriptive or historical aims to present a theory of the morphosyntax of
numerals, including both synchronic and diachronic perspectives, and to
contribute to the growing linguistic literature on number concepts and numerical
cognition.

The volume is organized into five chapters and numbered subsections throughout
and for the most part is organized in an exemplary fashion. Chapters II and
III, where the evidence for the structure of the Old English numerals is
presented, will be of greatest interest to specialists in numerals. Chapter IV
will be of greatest interest to specialists in Old English syntax. Chapter V is
a broader contribution to the theory of word classes and should be of interest
to all linguists.

The author begins with an extensive theoretical discussion of number concepts
and numerals, working along the lines suggested by Wiese (2003). Chapter I
distinguishes numerals (i.e., numerically specific quantifiers) from other
quantifiers, and distinguishes systemic cardinal numerals from non-systemic
expressions like ‘four score and seven’. As the book’s title suggests, cardinal
numerals are given theoretical priority over ordinal numerals, and nominal forms
like ‘Track 29′ or ’867-5309′ are largely ignored. Cardinal numerals exist in
an ordered sequence of well-distinguished elements of expandable but
non-infinite scope. Here the author builds upon the important work of Greenberg
(1978) and Hurford (1975, 1987), without presenting much information about Old
English numerals themselves.

Chapter II introduces the reader to the Old English numerals as a system of
simple forms joined through a set of morphosyntactic principles. It is
abundantly data-rich and relies on the full corpus of Old English to show how
apparent allomorphs (like HUND and HUNDTEONTIG for ’100′) in fact are almost
completely in complementary distribution, with the former almost always being
used for multiplicands, the latter almost never. This analysis allows the
author to maintain the principle that each numeral has only one systemic
representation, but at the cost of making a sometimes arbitrary distinction
between systemic and non-systemic expressions. This links to a fascinating but
all-too-brief comparative section on the higher numerals in the ancient Germanic
languages, which demonstrates the typological variability demonstrated even
within a closely related subfamily of numeral systems.

Chapter III deals with complex numerals, a sort of hybrid category encompassing
various kinds of complexities. The first sort of complexity, common in Old
English, involves the use of multiple noun phrases to quantify expressions that
use multiple bases (e.g. ‘nine hundred years and ten years’ for ’910 years’).
The second complexity is the typological complexity of Old English itself; the
author cuts through more than a century of confusion from Grimm onward in
demonstrating conclusively that there is no ‘duodecimal’ (base 12) element to
Old English (or present-day English) — that oddities like ‘twelve’ and
‘hundendleftig’ (= 11×10) can only be understood in relation to the decimal
base. The third is the set of idiosyncratic expressions ranging from the
not-uncommon use of subtractive numerals, to the overrunning of hundreds (as in
modern English ‘nineteen hundred’), to the multiplicative phrases used
sporadically to express numbers higher than one million. Where a traditional
grammar might simply list the common forms of the various numeral words, here we
are presented with numerals in context and in all their variety.

Chapter IV presents a typology of syntactic constructions in which Old English
numerals are found: Attributive, Predicative, Partitive, Measure, and Mass
Quantification. In setting out the range of morphosyntactic features
demonstrated within the Old English corpus, the aim is not simply descriptive,
but rather, assuming that numerals are a word class, to analyze that class in
terms of the variability that any word class exhibits, without making
unwarranted comparisons with other classes.

In Chapter V the author argues against the prevalent view that numerals are
hybrid combinations of nouns and adjectives. While there are similarities,
these ought not to be considered as definitional of the category, but as results
of the particular ways that cardinal numerals are used. Because it is
cross-linguistically true that higher numerals behave more like nouns than lower
ones, this patterned variability justifies our understanding the cardinal
numerals as a single, independent word class. It is regarded as the result of
higher numerals being later additions to the number sequence — rather than
being ‘more nounish’, they are still in the process of becoming full numerals.
They are transformed from other sorts of quantificational nouns (like
‘multitude’) into systemic numerals with specific values, but retain vestiges of
their non-numeral past.

EVALUATION

This is an extremely important volume, one that deserves a readership far beyond
historical linguists interested in Germanic languages. It is not the last word
on the category status of cardinal numerals, cross-linguistic generalizations
about number words, or the linguistic aspects of numerical cognition, but it
represents an exceedingly detailed and well-conceived contribution to all these
areas. While virtually any grammar can be relied upon to present a list of
numerals, virtually none deals with the morphosyntactic complexities and
historical dimensions of this particular domain that exist for almost any
language. Minimal knowledge of Old English is required to understand and
benefit from the volume.

The specialist in numerals will be struck by the richness and depth of the
author’s specific insights regarding numerical systems in general, using the Old
English evidence to great effect. Because it is one of very few monographs to
be devoted specifically to a single numeral system, and by far the lengthiest
and theoretically the most sophisticated (cf. Zide 1978, Olsson 1997, Leko
2009), there is time and space to deal with small complexities whose broader
relevance is enormous. The volume thus strikes that fine balance between
empiricism and theoretical breadth required of this sort of cross-linguistic
study rooted in a single language.

With regard to the prehistory of numerals, we are very much working from a
speculative framework, and where the author treads into this territory, of
necessity the argument is more tenuous. It may be true that for most languages,
the hands and fingers are the physical basis for the counting words, but
Hurford’s ritual hypothesis (1987), of which von Mengden does not think highly,
is at the very least plausible for some languages if not for all. These issues
are not key to the argument, which is all the more striking given that they are
presented conclusively in Chapter I.

A potential limitation of the volume is that, by restricting his definition of
numerals to cardinals (by far the most common form in the Old English corpus),
the author is forced into an exceedingly narrow position, so that, ultimately,
ordinals, nominals, frequentatives, and other forms are derived from numerals
but are not numerals as a word class, but something else. But the morphosyntax
of each of these forms has its own complexities — think of the nominal ’007′ or
the decimal ’6.042′ – that deserve attention from specialists on numerals.
Numerals may well be neither adjectives nor nouns, but omitting the clearly
numerical is not a useful way to show it. Similarly, the insistence that each
language possesses one and only one systemic set of cardinal numerals is
problematic in light of evidence such as that presented by Bender and Beller
(2006).

When comparing with other sorts of numerical expressions, e.g. numerical
notations, the author is on shakier grounds. It is certainly not the case, as
the author claims that the Inka khipus had a zero symbol, and it is equally the
case that the Babylonian sexagesimal notation and the Chinese rod-numerals did
(Chrisomalis 2010). Similarly, the author seems to suggest that in present-day
English, any number from ‘ten’ to ‘ninety-nine’ can be combined multiplicatively
with ‘hundred’, whereas in fact *ten hundred, *twenty hundred, … *ninety hundred
are well-formed in Old English but not in later varieties.

It is curious that von Mengden does not link the concept of numerical ‘base’ to
that of ‘power’, but rather to the patterned recurrence of sequences of
numerals. Rather than seeing ’10′, ’100′ and ’1000′ as powers of the same base
(10), they are conceptualized as representing a series of bases that combine
with the recurring sequence 1-9. But a system that is purely decimal, except
that numbers ending with 5 through 9 are constructed as ‘five’, ‘five plus one’
… ‘five plus four’, would by this definition have a base of 5 even though powers
of 5 have no special structural role and even though 5 never serves as a
multiplicand. This definition is theoretically useful in demonstrating that Old
English does not have a duodecimal (base-12) component, but as a
cross-linguistic definition will likely prove unsatisfactory.

Because the Old English numerals are all Germanic in origin, with no obvious
loanwords, it is perhaps unsurprising that language contact and numerical
borrowing play no major role in this account. Yet on theoretical grounds the
borrowing of numerals, including the wholesale replacement of structures and
atoms for higher powers, is of considerable importance cross-linguistically.
Comparative analysis will need to demonstrate whether morphosyntactically,
numerical loanwords are similar to or different from non-loanwords.

The author has incorporated the work of virtually every major recent theorist on
numerals, and the volume is meticulously referenced. There are a few irrelevant
typos, and a few somewhat more serious errors in tables and text that create
ambiguity or confusion, but no more than might be expected in any volume of this
size.

This monograph is a major contribution to the literature on numerals and
numerical cognition. Its value will be in its rekindling of debates long left
dormant, and its integration of Germanic historical linguistics, syntax,
semantics, and cognitive linguistics within a fascinating study of this
neglected lexical domain.

REFERENCES:

Bender, A., and S. Beller. 2006. Numeral classifiers and counting systems in
Polynesian and Micronesian languages: Common roots and cultural adaptations.
Oceanic Linguistics 45, no. 2: 380-403.

Chrisomalis, Stephen. 2010. Numerical Notation: A Comparative History. New York:
Cambridge University Press.

Greenberg, Joseph H. 1978. Generalizations about numeral systems. In Universals
of Human Language, edited by J. H. Greenberg. Stanford: Stanford University Press.

Hurford, James R. 1975. The Linguistic Theory of Numerals. Cambridge: Cambridge
University Press.

Hurford, James R. 1987. Language and Number. Oxford: Basil Blackwell.

Leko, Nedžad. 2009. The syntax of numerals in Bosnian. Lincom Europa.

Olsson, Magnus. 1997. Swedish numerals: in an international perspective. Lund
University Press.

Wiese, Heike. 2003. Numbers, Language, and the Human Mind. Cambridge: Cambridge
University Press.

Zide, Norman H. 1978. Studies in the Munda numerals. Central Institute of Indian
Languages.

As I was going through the Times…

Recently, there has been a “Puzzle Moment” in the science section of the New York Times, with an eclectic mix of articles combining scientific pursuits with cognitive and linguistic play of various sorts. One that caught my eye is ‘Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus’ by Pam Belluck [1], which is an account of the well-known Rhind Mathematical Papyrus. The RMP is an Egyptian mathematical text dating to around 1650 BCE, and is one of the most complete and systematic known accounts of ancient Egyptian mathematics. It’s a fascinating text, written in the Egyptian hieratic script rather than the more famous hieroglyphs, and it gives us considerable insight into the economy, social organization, and technical practices of the Second Intermediate Period.

The central conceit of the Times article is that the well-known “As I was going to St. Ives” poem-puzzle has its earliest ancestor in the RMP. This is vaguely true in that the RMP has a section involving repeated multiplication by seven, resulting in an addition problem. But Ahmes the scribe, despite his insistence that his text would reveal “obscurities and all secrets”, was not writing a mystery, but an exercise that formed part of scribal training, in an era where the literacy rate was at most 1-2%. While one can argue fairly that this is not a ‘real’ problem, and that the structure of it is meant to hold the learner’s attention through its repetitions, to call it a puzzle is only true in the broadest possible sense.

I’m a professional numbers guy, not an Egyptologist, but the article we are presented with not only tells us nothing new about the Rhind. I was very pleased, on the one hand, to see Marcel Danesi, whose work may be familiar to many readers of this blog, commenting on the widespread cross-cultural and cross-historical interest in puzzles (not only numerical puzzles, but including them). It’s not often enough that linguistic anthropologists get quoted in the Times. And like Danesi, I have broadly universalist sympathies. But I disagree with Danesi, who has made this claim about the RMP elsewhere, in his The Puzzle Instinct (Indiana, 2004, pp. 6-7) that it was “shrouded in mystery” or that “mystery, wisdom, and puzzle-solving were intrinsically entwined in the ancient world.”

The better example of numerical play in Egyptian scribal traditions mentioned in the Times article is the Horus eye, or wedjat, a combination of six symbols whose constituent parts signify the fractional series {1/2, 1/4, 1/8, 1/16, 1/32, 1/64} which when summed totals 63/64, or nearly one (see below). As the Egyptologist Sir Alan Gardiner reckoned it, the remaining 1/64 would be provided by Thoth who would heal the Eye and thus produce unity. It’s a nice story, and at least at some periods or for some writers, this narrative may have been relevant.

Source: wikimedia.org

But the Horus-eye illustrates one of the central problems in the transliteration of Egyptian texts, namely that while the vast majority of Egyptian mathematically-relevant texts are written in the cursive hieratic script, they are transcribed, and all-too-frequently theorized, as if they were hieroglyphs. This transcriptional practice leads us to think of the Rhind as a hieroglyphic text that just happens to be in hieratic in the original, but in the case of the Horus eye it couldn’t be more misleading. The Horus symbols in the Rhind don’t look like the above image, and more generally, the hieratic numerals look nothing like, and behave nothing like, the hieroglyphic numerals. We now call of these six Horus eye components by the less evocative name of ‘capacity system submultiples’ in recognition of the fact that these components were originally nonpictographic, part of a metrological system of grain measurement, and only at a much later date were they composed into the wedjat-eye. This isn’t to say that the Egyptians weren’t numerically playful, but they weren’t especially playful in the Rhind.

In short, the RMP is not an especially good example of numerical play in Egypt, and certainly not an especially relevant example from a cross-cultural perspective. It illustrates, to be sure, that mathematical texts are not purely functional or economic documents, but include semiotic and linguistic elements far beyond their pragmatic use. But this is not new knowledge about the Rhind or about mathematics. And it runs a grave risk of othering a document whose function was largely pedagogical, and is thus not so different than, for instance, the ‘ready-reckoners’ of early-capitalist sixteenth-century England.

I am thrilled to see numerical texts treated as objects of inquiry beyond the facile ‘Did they get the answer right?’ I am sympathetic to Danesi’s claim that puzzles and riddles have universal salience. Yet I worry that, at least in the case of the Rhind, the link to puzzle-like behavior is so far-fetched that it turns our best glimpse into Egyptian sociomathematical practice into an inappropriately arcane and obscurantist account. This ‘mysteries of lost Egypt’ nonsense should have been set aside decades ago.

If you wanted to pull out some cross-cultural examples of numerical play, you could easily find lots of better examples, from well-covered territory such as Hebrew gematria practices, to the richly evocative varnasankhya systems of number-word associations in premodern South Asian texts, to the complex cluster of quasi-cryptographic numerical systems used by Ottoman administrators and military officers. Or if you were really stuck on Egypt, you could investigate the cryptic numerals used on late Egyptian votive rods and Ptolemaic inscriptions, richly infused with homophony. (For a more extensive discussion of these and others, see my Numerical Notation: A Comparative History (Cambridge, 2010). There is a rich, although disciplinarily diverse, comparative body of material on numerical practices including puzzles, but the Rhind just isn’t part of it.

(Crossposted to the Society for Linguistic Anthropology blog)

Just clouding around

This evening I was playing around with Wordle (which makes gorgeous word clouds from user-provided text) in preparation for an in-class exercise in my intro linguistic anthropology class tomorrow. And of course I ended up throwing other chunks of text in there, including this rather fetching visual representation of chapter 13 (the conclusion) of Numerical Notation:

Counting change: the anthropology of numerical notation

(Not sure if any of my readership is in the Detroit area and might be able to attend, but just in case…)

Counting Change

The anthropology of numerical notation

Stephen Chrisomalis
Assistant Professor, Anthropology
Monday, April 5, 12:30 – 2:30 pm
McGregor Conference Center, Wayne State University, Rooms B & C

In his new book, Numerical Notation: A Comparative History (Cambridge, 2010), Stephen Chrisomalis presents a linguistic, cognitive, and anthropological history of written numeral systems, comparing over 100 different systems used over the past 5,500 years. He invites members of the community of scholars and the public to join him for this book launch and luncheon.

In this lecture, Dr. Chrisomalis aims his work on numerals at the heart of anthropological theory, seeking to refigure thinking about historical processes in the social sciences. As the integrative core of the social sciences, anthropology has an obligation to compare processes of change across time and space. The richness and time-depth of the evidence in Numerical Notation stand in opposition to narrower visions of anthropology as the study of contemporary life.

Free and open to the public
Lunch will be provided to all guests
Discount flyers for Numerical Notation will be available

Download flyer
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What word has the highest Roman numeral value?

Today at Futility Closet comes a report on the Mesembryanthemum:

The South African flower Mesembryanthemum draws its name from the Greek roots for middle, embryo, and flower. It’s believed to be the English word containing the highest “score” in Roman numerals — four Ms.

I have no quarrel with the first sentence but the second struck me as immediately improbable, because there are no other letters in Mesembryanthemum with a Roman numeral value so its total value is only 4000. I was immediately able to think of a couple that equal it – words that should be familiar to virtually anyone – and with a little searching was able to find a word that I knew and could define with a total of 4502 (Correction: 4602 – I should learn to add better). Can you find my word? More importantly, can you beat my word’s score?

Edit to add: The word is ‘IMMunoCoMproMIseD’, for a total of 4602, as one clever reader has discerned. This is just a game, of course, but there is a long tradition of Roman numeral chronograms in Europe, passages in which the sum of the Roman numerals gives a significant date. A multi-volume corpus of thousands of these was published in the 19th century and is practically begging for reanalysis – the inscriptions can be located and dated securely (Hilton 1882, 1885, 1895).

Hilton, James. 1882. Chronograms. London: Elliot Stock.
———. 1885. Chronograms Continued. London: Elliot Stock.
———. 1895. Chronograms Collected. London: Elliot Stock.

Why paleography matters: solution

Well, we haven’t had an answer, so I’ll give it to you:

Solution: Plate adapted from Jenkinson 1926: Figure 10.

All of the numeral-phrases except the bottom right read CXLVII (= 147). I’ve highlighted every other character in red to emphasize the distinct characters, making this solution comprehensible if not obvious. The fact that you, my readership (several of whom have training in paleography) couldn’t find the solution even after two clues demonstrates the ongoing importance of paleography as a scholarly profession, and also demonstrates the highly aberrant character of the hand.

This plate is taken from a 1926 article ‘The use of Arabic and Roman numerals in English archives’ by Sir Hilary Jenkinson, the doyen of British archivists (and who, shockingly, lacks a Wikipedia page!), who was also one of the major figures in early-twentieth-century British paleography (Jenkinson 1915, 1926). Jenkinson writes of this plate:

First there is the unimportant but highly curious development in one or two Courts of a special Roman numeration for the membranes of their Rolls, which in its later stages is practically unreadable. Our illustration (fig. 10) shows the number 147 as it was written in the years 1421, 1436, and 1466 and finally the number 47 as it appears in 1583: it does not show the worst that might be selected and is only half the size of the original (Jenkinson 1926: 274).

Although I have been familiar with this plate for over a decade, for the life of me I can’t find a way to read 47 (XLVII) out of the bottom-right phrase. I’m sure that Jenkinson had some grounds (probably contextual) for believing this to be the numeral, but I just can’t get 47 out of it. Weirdly enough, I can find 147 in it, but that may just be a product of its association with the other three phrases. Note that the other three are chronologically close while the fourth is nearly a century later. Anyway, I’m really stumped on this one.

But I actually wish to disagree with Jenkinson on one word: ‘unimportant’. The cursive transformation of cumulative numeral phrases is important in the history of numeration because it is the most common means by which cumulative systems, which rely on the repetition of like symbols whose values are added (e.g. XXX = 30), turn into ciphered systems, which do not do so. The figure below (borrowed from my book) shows how the Egyptian hieroglyphic numerals for 6, 9, and 300 became paleographically reduced over time into hieratic forms that do not show any evidence of their original cumulative structure. The ligaturing of individual signs eventually leads to the reconceptualization of the entire set of signs as a single unit. In fact, this process occurred (slightly differently) with the Brahmi numerals 1 through 3 of ancient India, which eventually became the ciphered figures of the Indian, Arabic, and Western numerals used by virtually everyone today (see this chart, for instance).

Cursive reduction of Egyptian numerals (after Chrisomalis 2010: 47)

The Roman numerals ceased to be used in manuscript-writing throughout early modern Europe, so there was no opportunity for the form of this particular court hand to lead from extreme cursivization to something structurally distinct in the Roman numerals. But it could have happened, just as it happened before several times. And that (along with so many other things) is why paleography matters.

Chrisomalis, Stephen. 2010. Numerical Notation: A Comparative History. New York: Cambridge University Press.
Jenkinson, Hilary. 1915. Palaeography and the practical study of court hand. Cambridge: Cambridge University Press.
Jenkinson, Hilary. 1926. The use of Arabic and Roman numerals in English archives. The Antiquaries Journal 6:263-275.

Juvenile ethnopaleography

Ms. 1 (APTC 1)

Arthur Chrisomalis, The Lines
Construction paper. ff. 2. Unfoliated. 11×8.5 in. Bound at left with four staples. Dated 2010 (?) in hybrid numerical notation (see below).

This manuscript is a juvenile work probably composed on 02/06/2010. There is textual and ethnographic evidence to suggest that the scribe (age 4.5) was aided by a more competent master. Currently the MS is magnetically affixed to the archivum refrigeratum in the scribe’s home.

fol. 1r: Text in orange ink, 3 lines. (Plate 1)
1v: Vertical lines in pencil, crossed with horizontal and diagonal lines in green, pink, red, grey, black, purple, and orange ink. Apparently nonrepresentational.
2r: Vertical lines in pencil, crossed with horizontal and vertical lines in red ink. Apparently nonrepresentational.
2v: Stylized depiction of a smiling human figure in orange ink. Elongated fingers and toes, highly enlarged ears. May be a scribal self-portrait.

Text (fol. 1r)

1. THE LINES
2.    ArTHu
3. ||0|0     r

Plate 1: APTC 1, fol. 1r.

Notes

l. 1: Written in a bold majuscule hand in orange ink. Dotted pencil marks underlying the ink suggest that this line was prepared by the scribe with the aid of a master, an observation later confirmed ethnographically.
l. 2: Written in a hybrid minuscule/majuscule hand; H dips well below the line. Absence of pencil marks and unusual features of the hand suggest that it was composed unaided (confirmed ethnographically).
l. 3: The final character ‘r’ at the right margin seems to be properly attached to the end of l. 2; it is spaced significantly apart from the unusual characters at left. The reading ’11010′ as an Arabic numeral is improbable given limitations on scribe’s counting abilities. Ethnographic interview with scribe’s assistant/instructor/mother confirms that this is a hybrid notation for the date of composition, 2010. Upon learning of the custom of dating the frontispieces of books from his mother, the author inquired how to do so, and was told, “Put 2-0-1-0″, at which point he wrote the characters ||, then paused and examined the characters. At this point his mother informed him that he could just write the number 2, to which he replied, “That is a 2, in lines.” Thereupon he continued to write the final three characters, noting, however, that the last 0 “is more of an oval.”

One is reminded of the unusual notation developed by Ocreatus in his 1130 Helcep Sarracenicum ‘Saracen Calculation’ which combines the ordinary Roman numerals I, II, III … IX with a circle for O, thus producing a mixed system of the additive Roman numerals and positional Western numerals (Burnett 2006; Chrisomalis 2010: 120). Thus, Ocreatus wrote 1089 as I.O.VIII.IX. Attested in only one MS (Cashel, G.P.A. Bolton Library, Medieval MS 1), this notation represents an effort to incorporate positionality into existing systems of notation, and is related to the debate between the abacists and algorithmists over the proper contexts of use for the newer Western numerals.

Despite the temptation to see in this line a re-invention or re-discovery of Ocreatus’ notation, the ethnographic evidence suggests that this is unlikely. Although the author is familiar with Roman numerals in the context of clocks, the description of the characters || as ‘lines’ rather than ‘numbers’ or ‘Roman numerals’ suggests instead an effort to incorporate tallying principles. It is probable that the Roman numerals derive ultimately from an as-yet unattested Italic practice of tallying used in the 6th century BCE or earlier (Keyser 1988; Chrisomalis 2010: 95-6). Yet tallying is distinct from cumulative-additive numeration like the Roman numerals in that it is produced sequentially as an open-ended count; one cannot simply add signs to the Roman XIII – the equivalent tally might be IIIIVIIIIXIII (Chrisomalis 2010: 15). It is therefore probable that this notation represents the scribe’s attempt to combine the elementary tallying principle of one-to-one correspondence with the familiar Western numerals.

The ethnographic evidence that the scribe paused (as if bemused) upon the production of || might suggest that this form is a scribal error; given, however, that he is conversant with the Western numerals and is capable of producing them unaided, the hypothesis cannot be discounted that the use of || for 2 represents an aesthetically motivated decision. While the title of the work, ‘The Lines’, may refer to the vertical and horizontal lines in fol. 1v and 2r, it may equally be a reference to the lines in the date in fol. 1r.

Works cited

Burnett, Charles. 2006. The semantics of Indian numerals in Arabic, Greek and Latin. Journal of Indian Philosophy 34:15-30.
Chrisomalis, Stephen. 2010. Numerical Notation: A Comparative History. New York: Cambridge University Press.
Keyser, Paul. 1988. The origin of the Latin numerals 1 to 1000. American Journal of Archaeology 92:529-546.