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The subtractive method, in which a smaller number placed to the left of a larger one was subtracted from it, is nowadays a dominant way of writing values not covered by the seven basic Roman numeral symbols. At times, however, we can encounter values such us 9 written using the additive method. Thus, instead of the usual IX symbol, the value is marked as VIIII – Roman five symbol with four Roman one symbols added.
A common view holds that the subtractive method was popularized by medieval scholars, while the Romans themselves preferred the additive system. Indeed, researchers of the early twentieth century tended to support this theory, and some of them – such as Florian Cajori – even claimed that the first examples of subtractive notation appeared only in the Middle Ages.
In his A History of Mathematical Notations, Cajori, referring among others to earlier analyses by Adriano Cappelli published in Lexicon Abbreviaturarum, argued that all recorded instances of the number 9 written as IX date no earlier than the mid-fifteenth century CE. At the same time, he maintained that there were no historical examples of the subtractive form for the number 4. The only subtractive notation he identified from around the second century BCE was the number 83 written as XXCIII instead of the standard LXXXIII. According to him, this form resulted from a lack of space at the end of a tablet, which forced a non-standard solution.
Interestingly, Cajori also cited cases of phonetic spelling (reflecting pronunciation) of numbers that contradicted both the additive and subtractive systems. The forms IIIX and VIX, mentioned in the Lexicon diplomaticum by W.J. Ludolph, were intended to render the Latin terms tertio decimo and sexto decimo, referring to the numbers 13 and 16 respectively.
From scriptorium to printing press
If subtractive notation was known to the Romans, why is its wider use so often associated with the Middle Ages?
The main problem with the additive system was legibility. A notation consisting of many repeated symbols is difficult to read and prone to error. When copying manuscripts in the scriptorium, it was easy to confuse VIIII with VIII or IIII with III. Subtractive notation – shorter and more unambiguous – provided a natural remedy for this weakness, especially under the intense working conditions of medieval copyists.
The material context of writing was also significant. In the Middle Ages, scribes and stonemasons, particularly in regions distant from Rome, introduced new variants of notation, adapted to their needs. Paradoxically, this loosening of classical standards created space for the broader adoption of subtractive notation as a more economical solution. Ultimately, the invention of printing in the fifteenth century CE enforced a certain degree of standardisation. Typesetters required consistent rules, and subtractive notation, being more compact, proved more practical in their work.
Legio XVIII or XIIX?
Modern research provides concrete examples of subtractive notation in ancient Rome, thereby challenging previous claims of its medieval origin. One of the best-documented cases is the inscription on the cenotaph of Marcus Caelius, a Roman centurion who fell in the Battle of the Teutoburg Forest in 9 CE. On this monument, the Eighteenth Legion is written as XIIX rather than XVIII. Although this form appears non-standard from the perspective of later rules, it demonstrates that the principle of placing a smaller value before a larger one to indicate subtraction was known to Roman stonemasons at the beginning of the first millennium CE.
Importantly, the form XIIX is neither a mason’s error nor an isolated anomaly. The same notation appears in other ancient sources referring to this legion, suggesting that it was a deliberate and established convention – at least in relation to this particular unit. It is also worth noting that XIIX represents a double subtraction: two units are subtracted from twenty, which goes beyond the rules later codified in the Middle Ages and commonly used today.
An even earlier example is the calendar known as the Fasti Antiates Maiores, created between 84 and 55 BCE. In this pre-Julian, the oldest surviving, Roman calendar we find forms such as XXIIX for the 28 days of February. As in the case of the cenotaph of Marcus Caelius, this notation indicates the practical use of subtraction long before medieval scholars supposedly popularized the method.
The examples cited above demonstrate that subtractive notation was not a medieval invention but a practice present in ancient Rome long before the Christian era. Its use, however, was inconsistent and never standardised. Although Cajori was mistaken in some of his assumptions, subtractive notation indeed became dominant in written texts only with the invention of printing press, more than a thousand years after the fall of the western Roman Empire.
