How did the ancient Romans really use maths in their everyday lives?
Here, Dr Richard Marshall, Lecturer in Classics, writes about how our assumptions about ancient mathematical thinking could be hiding some interesting secrets.
Contents:
- The fall of Archimedes and historical assumptions
- Who used maths in the ancient world?
- Real-world and theory
- Who were the agrimensores?
- Agrimensores and their geometrical problem texts
- A long-forgotten mathematical method
- Unlocking ancient knowledge: The Agrimensores and Roman Mathematics’ (AgRoMa) project
- Bringing together ancient maths and modern education
- Test yourself
The fall of Archimedes and historical assumptions
When we think of maths in ancient Rome, one story tends to stand out, and it’s not necessarily a flattering one.
The Romans have never really lived down their most (in)famous mathematical feat: the murder of one of the greatest mathematicians of all time — Archimedes — during the siege of Syracuse in 212 BC.
Discovered by a Roman soldier, Archimedes is said to have made the famous but futile plea ‘noli, obsecro, istum disturbare’ (‘Do not, I pray, disturb those’), gesturing to geometrical figures he had drawn in the sand. As the soldier advanced for the killing stroke, Greek mathematics was literally trampled under the hobnails of the Roman conqueror.
This image has long coloured perceptions of Rome’s relationship with mathematics: brutal indifference.
But is this the full picture?
The death of Archimedes, by Felice Giani, died 1823.
Who used maths in the ancient world?
Just like today, people in ancient Greece and Rome studied and practised mathematics for a variety of reasons – some practical, some intellectual, and some playful.
Some were taught geometry and arithmetic at school, though evidence suggests that beyond basic numeracy and simple mental arithmetic, more complex calculations and geometric constructions formed no regular part of the school programme.[1]
Some people, like merchants, engaged with mathematics to run their businesses. Others, such as architects, surveyors, or military engineers, to build new structures or plan fortifications. A few people studied mathematics for pleasure and recreation. More advanced geometrical and arithmetical problems had no practical purpose in antiquity, but were investigated in a spirit of intellectual curiosity and cordial competition.
Real-world and theory
It was the ancient Greeks who championed theoretical maths as a distinct field of study. And while only a small number of ancient mathematicians actually worked in this way, the axiomatic-deductive method — where arguments are built step-by-step from basic assumptions (axioms) using logical reasoning — has come to capture most academic attention.
But we also have collections of mathematical exercises written in the language of ‘real world’ scenarios. Today these collections are generally anonymous and the solutions usually lack any formal proofs. This non-deductive mathematics is the mathematics that actually kept the ancient world running. It enabled architects to calculate the quantity of material required for a new building, or a surveyor to equitably assign portions of land to colonists in a newly settled or conquered territory.
One profession that needed a wide range of mathematical expertise in antiquity was that of the Roman agrimensores. The work wasn’t glamorous, but it was essential, and they needed maths — not just in theory, but in practice.
Tombstone of Lucius Aebutius Faustus, an agrimensor and freedman (former slave). The cross-shaped object in the relief below is a groma (surveying instrument). Museo della civiltà romana. CC-BY-SA-2.0.
Who were the agrimensores?
The agrimensores were the land surveyors of ancient Rome — professionals tasked with measuring, dividing, and recording land across the empire.
Their work underpinned everything from agricultural planning to urban development, and they played a vital role in Rome’s expansion by laying out new colonies and settling land disputes.
Far from being simple technicians, the agrimensores combined practical geometry with a working knowledge of Roman law and measurement systems, leaving behind a body of written work that offers a rare window into the applied mathematics of the ancient world.
Roman mathematical problems and diagrams from the sixth-century Codex Arcerianus. Wolfenbüttel, Herzog August Bibliothek, Guelferb. 36.23, f. 10v. CC BY-SA 3.0 DE.
Agrimensores and their geometrical problem texts
Many of the problems the agrimensores trained with look remarkably similar to those encountered in a modern mathematical education, though the way in which solutions are presented look alien to us. This is because all mathematical operations are fully verbalised as step-by-step instructions, rather than presented in the symbolic notation of modern mathematics.
An example:
Question:
A field that tapers along its length (i.e. a trapezium) is 200 feet long, on one side 130 feet wide, on the other 70 feet. I want to know the area in iugera (a iugerum is 28,800 square Roman feet).
Solution:
Let us find it like this. I add the smaller and larger side together, this comes to 200 feet; I take off a half, this comes to 100 feet; I multiply this by the length, that is 100 by 200, this comes to 20,000; I divide by the hundredth part, this comes to 200, I divide by the twenty-fourth and then by the twelfth part of the same sum: this comes to 3/4 plus 1/36; there are this many iugera.[2]
The approach is rather like following a cooking recipe, in that the reader is expected to apply a sequence of simple arithmetical operations to find the unknown quantity. Other algorithms are provided for triangular, rectangular, polygonal, and circular and semi-circular fields: exactly what one would expect to find amongst a collection of surveying manuals.
However, the collections also include more contrived problems that suggest the agrimensores’ interest in mathematics was not simply instrumental. For instance:
There is a field, which has a length of 900 feet, but I have raised however many iugera it covers to the sixth power and found the width. I want to know the width or area in iugera.[3]
Today our first impulse would be to turn this question into symbolic notation. Let A = the area in iugera, b = the length in iugera, c = the width. We’re assuming the field is rectangular in shape, a hunch confirmed by the handy accompanying diagram. So to find the area, we’d start by using the equation for the area of a rectangle, A = b x c. We’re told that the width is equal to the area (measured in iugera) raised to the power six, so we can substitute this information for c in our equation: A = b x A6. We also know (because again we’ve been told this) that b = 900 feet. There are 28,800 Roman feet in a iugerum, so…
Needless to say, this problem is hardly likely to be encountered in the day-to-day work of a surveyor! This is an arithmetical exercise disguised in the language of surveying.
Other problems from the same collections involve finding the areas of fields with highly improbable shapes (one of these is our project logo), investigate the properties of a class of mathematical object known as polygonal numbers (notionally based on the ways in which pebbles can be arranged to form triangles, squares, pentagons, etc. of increasing size), or tacitly rely on the binomial theorem (an understanding of how to solve equations like (a + b)2.
Though reliant on Greek mathematics, these are profoundly Roman problems: their whole number solutions require the use of Roman units of measurement.
The existence of such problems, completely devoid of any practical application, leads to a surprising conclusion. We know that as a group, the Roman land surveyors did not belong to the most elevated strata of society, but it is clear from these texts that they cultivated a knowledge of mathematics for its own sake, as an intellectual exercise and marker of professional status, engaging creatively with Greek mathematics to create an independent tradition grounded in their own socio-cultural and linguistic reality.
A long-forgotten mathematical method
Practical mathematics is a key element of how the ancient world functioned, but the history of this tradition in the Latin-speaking world has been largely ignored. This is because most of our evidence for this kind of mathematics is found within collections of manuals assembled by the agrimensores.
Despite their significance, the contributions of the agrimensores have been largely overlooked so far — not least because historians in the nineteenth century were only interested in what these texts could tell us about Roman land use and imperialism, rather than intellectual or social history.[4] As a result, much of the evidence for Roman mathematics is still locked away in ancient and medieval manuscripts.
This is where the AgRoMa project comes in.
Unlocking ancient knowledge: The Agrimensores and Roman Mathematics’ (AgRoMa) project
The Agrimensores and Roman Mathematics’ project (AgRoMa), generously supported by a grant from the AHRC, aims to transform our access to this evidence by making modern editions, translations, and studies of this material available for specialists and the general public.
In the first stage of our project, just completed, we have reviewed the evidence of over one hundred manuscripts and are beginning the work of reconstructing the Latin texts of the individual problem collections.
Mathematical problems from the collection attributed to Epaphroditus and Vitruvius Rufus, preserved in a tenth century manuscript from Northern France, brought to England by William de Clare in 1277 and gifted to St Augustine’s Abbey, Canterbury. Cambridge, Trinity College Library, R.15.14, ff. 52v-53r
Bringing together ancient maths and modern education
We are also interested in investigating how these materials might be used in a modern educational environment, comparing assumptions underlying ancient and modern approaches to basic numeracy and mathematical education.
In the next stage of the project, we plan to create and trial activities in local primary schools and museums based on the problems encountered in our ancient evidence. One of the powerful characteristics of ancient mathematics is that it works by following fully verbalised algorithms, so students can arrive at the correct answers without understanding the theoretical basis of results or mastering mathematical notation.
We hope such characteristics may be useful for opening up the subject for students with mathematical anxiety or mathemaphobia.
Our activities will also support the teaching of the National Curriculum’s Key Stage 2 Roman numeral component. A recent study suggest that knowledge of Roman numerals among the general population is very patchy. Although dismissed as increasingly irrelevant in a world of digital clocks, Roman numerals are also good for developing number sense and the skill of subitising, that is, the ability to recognize small quantities instantly without counting them (think of the difference between III and 98746129).
Before we can bring Roman practical mathematics to the public, however, one final hurdle needs to be cleared. Though Roman numerals are easily typed on a modern computer, no support currently exists for encoding Roman fraction symbols. The project is thus working with the Medieval Unicode Font Initiative (MUFI) to develop new Unicode font standards so that Roman mathematics can be correctly printed.
Test yourself
Before leaving, why not test your own knowledge of Roman mathematics? Answers can be found at the bottom of this blog.
Can you correctly interpret the following Roman numerals?
- III
- XVIII
- XIX
- LXXVI
- MCMXVII
- MCDXLIV
What about the following question?
- In a right-angle triangle, whose hypotenuse measured in feet is 25, the surface 150 feet, can you state separately its height and base?
You might also like
- read the press release: Researchers bring ancient Roman maths up to date
- visit the project website for educational resources and guides for further reseach
- find out more about the author, Dr Richard Marshall, Lecturer in Classics at Newcastle University
- explore Newcastle University’s School of History, Classics and Archaeology
- discover our world of research into human history, led by our expert professors, lecturers, and research staff
References
[1] For an overview of ancient mathematical education, see Nathan Sidoli, ‘Mathematics Education’, in W.M. Bloomer, A Companion to Ancient Education, Chichester 2015, pp. 387-400.
[2] [Varro] F1.4 Bubnov (trans. Marshall)
[3] [Varro] F1.22 Bubnov (trans. Marshall)
[4] The first true scholarly edition of the collection set a precedent by simply omitting most of the mathematical material (F. Blume, K. Lachmann, and A. Rudorff, Gromatici Veteres = Die Schriften der römischen Feldmesser, 2 vols, Berlin, 1848-1852).
Test answers:
- 3
- 18
- 19
- 76
- 1917
- 1444
- 20 feet and 15 feet