History of Mathematics Project | Roman Stylus Tablet

History of
Mathematics

Project

around 70 CE

Roman Stylus Tablet

Sequential Roman numerals on a wax tablet

Wax tablets were wooden (or occasionally ivory) boards covered with a layer of wax. They were widely employed as portable and erasable writing surfaces in antiquity and the Middle Ages. Writing was done by etching grooves in the wax using a stylus, and erasure using the other end of the stylus, which often contained a putty-knife-like surface for smoothing the wax. Since wax tablets were used for recording everyday activities such as tracking business accounts, doing simple computations, and recording notes and students' exercises, no special care was taken to preserve them. As a result, surviving examples typically originate from sources such as dump deposits.

This wax stylus writing tablet made of silver fir was found between 2010–2014 during the excavations of the Bloomberg London site in Walbrook, London. The three fragments are unique in being inscribed with nothing but numerical symbols. The tablet originated from the top layer of two dump deposits. The text seems to be a writing exercise for numerical symbols, possibly by a trainee clerk, bookkeeper or schoolboy.

Artifact dimensions

134 mm × 57.5 mm × 8 mm (8.5 mm thick on front, 7.4 mm on reverse)

Original artifact location

Londinium (historical name), London, United Kingdom (current name)

Current artifact location

Museum of London, London

Catalog number

Tab.Lond.Bloomberg

Timeline

Interactive Content

Computational Explanation

Roman numerals originated in ancient Rome and were used across the Roman Empire. They even remained the common way of writing numbers throughout Europe well into the Late Middle Ages (1250 to 1500 CE).

Roman numerals were written in several different forms over the centuries. The modern "standard" version consists of the letters I, V, X, L, C, D and M, which stand for the following numerical values:

In[]:=

numbers=Most@Flatten[Table[i10^j,{j,0,3},{i,{1,5}}]];symbols=RomanNumeral[numbers];

In[]:=

TextGrid[Transpose[{symbols,numbers}]]

Out[]=

I	1
V	5
X	10
L	50
C	100
D	500
M	1000

This form of Roman numeral is usually written using additive notation as concatenations of multiple copies of symbols:

In[]:=

StringJoin@@Flatten[Table@@@Transpose[{Reverse[symbols],NumberDecompose[5678,Reverse[numbers]]}]]

Out[]=

MMMMMDCLXXVIII

However, subtractive notation consisting of moving a single symbol to the left is used for numbers that differ by exactly the next smaller symbol:

In[]:=

RomanNumeral[{4,9,14,19,44,49,90,94,99}]

Out[]=

{IV,IX,XIV,XIX,XLIV,XLIX,XC,XCIV,XCIX}

Various extensions were used to represent large numbers. A summary of various notations was compiled by Petrus Bungus (1583–1584) and reproduced in Smith (1908):

The type of extension appearing on Tablet Tab.Lond.Bloomberg <WT78>, which has been dated to around 70 CE, is known as the apostrophus system. In this form of Roman numerals, 500 was written as IↃ (instead of D) and 1000 was written as CIↃ, ↀ or ∞ (instead of M). This system of surrounding numbers with groups of C/Ↄ symbols to denote thousands had its origins in the Etruscan numeral system and was used during and for a period following the Roman Republican period (509–27 BCE). In this system, each bracketing pair of C…|…Ↄ symbols corresponded to a factor of 10 starting with 1000 and additional unmatched Ↄs were used to denote 500, 5000, 50,000 and so on. Furthermore, just as CIↃ was sometimes joined to form the single character ↀ for 1000, the character sequence IↃↃ for 5000 was sometimes written as ↁ and the sequence CCIↃↃ for 10,000 as ↂ.

To implement the version of the apostrophus system appearing on this tablet, write a utility function to create left and right runs of Cs and Ↄs:

Apostrophus[n_,sides_]:=With[left=StringRepeat["C",n],right=StringRepeat["Ↄ",n],Switch[sides,1,"|"<>right,2,left<>"|"<>right]]

Now build the representations for thousands, including replacement of single and doubled (paired or unpaired) Cs with their special character equivalents:

In[]:=

ApostrophusRepresentation[n_Integer?Positive,collapse_:True]:=Module[{digs=Reverse@IntegerDigits[n/10^3],res},Print[digs];res=Row[Flatten[Reverse@MapIndexed[{If[#<5,{},Apostrophus[#2[[1]]+1,1]],Table[Apostrophus[#2[[1]],2],Mod[#,5]]}&,digs]],Spacer[1]];If[collapse,res/."CC|ↃↃ""ↂ","|ↃↃ""ↁ","C|Ↄ""ↀ",res]]/;Mod[n,10^3]0

The markings on the tablet can then be seen to correspond to the following sequences of thousands:

In[]:=

Row[TextGrid/@{Table[{n,ApostrophusRepresentation[n]},{n,{8,9,10,20,30,40}1000}],Table[{n,ApostrophusRepresentation[n]},{n,{60,70,80,90,100}1000}]},Spacer[2],BaseStyle18]

Out[]=

8000	ↁ ↀ ↀ ↀ
9000	ↁ ↀ ↀ ↀ ↀ
10000	ↂ
20000	ↂ ↂ
30000	ↂ ↂ ↂ
40000	ↂ ↂ ↂ ↂ

60000	\|ↃↃↃ ↂ
70000	\|ↃↃↃ ↂ ↂ
80000	\|ↃↃↃ ↂ ↂ ↂ
90000	\|ↃↃↃ ↂ ↂ ↂ ↂ
100000	CCC\|ↃↃↃ

Additional Reading

Bungus, P. Mysticæ Numerorum Significationis. Bergomi, 1583–1584.
Cajori, F. "Romans." § 46–61 in A History of Mathematical Notations, Vol. 1. New York: Dover, pp. 30–37, 1993.
Chrisomalis, S. "Italic Systems: Roman." Numerical Notation: A Comparative History. Cambridge, England: Cambridge University Press, pp. 109–131, 2010.
Menninger, K. W. "The 'German' Roman Numerals." In Number Words and Number Symbols : A Cultural History of Numbers. New York: Dover, pp. 281–294, 2013.
Smith, D. E. Rara Arithmetica: A Catalogue of the Arithmetics Written Before the Year MDCI with a Description of Those in the Library of George Arthur Plimpton of New York. Boston and London: Ginn, pp. 380–383, 1908.
Tomlin, R. S. O. Figs. 128 and 129 in Roman London’s First Voices: Writing Tablets from the Bloomberg Excavations, 2010–14. London: Museum of London Archaeology, pp. 237–239, 2016.

Roman Stylus Tablet

Timeline

Interactive Content

Computational Explanation

Additional Reading

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