Roman mathematics
Roman mathematics encompasses the mathematical practices and contributions during the Roman Empire, which began around 212 BCE and extended through several centuries. Although often perceived as merely adopting Greek mathematical methods, the Romans made significant practical advancements. One enduring legacy is the Roman numeral system, characterized by letters representing specific values, such as I (1), V (5), and X (10), which remains in use today. To facilitate calculations, Romans developed the portable abacus, aiding merchants and military surveyors in their work.
Roman surveyors utilized geometry and tools like the groma to create grid patterns for cities, influencing urban planning across Europe and into modern America. Additionally, the Roman calendar introduced by Julius Caesar marked a notable shift in timekeeping, laying the groundwork for the Julian and later Gregorian calendars. This fusion of mathematics with cosmology garnered the interest of philosophers, although tensions arose between traditional pagan views and emerging Christian thought, leading to significant historical events, including the tragic fate of the mathematician Hypatia. Overall, Roman mathematics reflects a complex interplay of cultural influences and practical applications that shaped the mathematical landscape of the ancient world and beyond.
Roman mathematics
Summary: The ancient Romans, who are often remembered for their applied mathematics, made important contributions to surveying, time-keeping, and astronomy.
The Roman period for mathematics could be said to have started when a Roman soldier was sent to seize Archimedes during the capture of Syracuse. Told by Archimedes to wait as he finished his diagrams, the soldier lost patience with the old man and slew him. The popular stereotype of the Romans is that they did little to advance Greek discoveries in mathematics, instead merely applying Greek methods to practical problems. This conception is not entirely fair. The Roman Empire was not one homogenous zone, but was rather a collection of culturally diverse provinces. For this reason, many works produced during the time of Roman rule, like the books of Ptolemy, writing in Alexandria, Egypt, are written in ancient Greek rather than Latin. Therefore, these books could be considered Greek, Roman, or Greco-Roman depending on the context. However, despite this diversity, the Roman period led to the dominance of some mathematical practices that still have an influence in the twenty-first century.
Roman Numerals
One of the most distinctive remnants of Roman mathematics is the use of Roman numerals, which are letters that stand for specific values and usually work as additive values. The numerals are
I = 1 V = 5 X = 10
L = 50 C = 100 D = 500
M = 1000.
So: LXXVII = 50 + 2(10) + 5 + 2(1) = 77.
The numerals are written with the largest values at the left, proceeding to the smaller values. They can also have subtractive constructions. I preceding subtracts one from a 10 to make nine. X before an L or C produces 40 or 90, and C before D or M produces 400 or 900. So
MCMXLVIII =
1000 + (1000 − 100) + (50 − 10) + 5 + 3(1) = 1948.
The origins of the system are unknown. It has been proposed that they were based on tally marks, with I being a notch, V being a double notch to mark five, and 10 as crossed-notches (though it could also be that X was formed from two V symbols). The number IV to represent 4 is a later addition based on medieval Latin and does not seem to have been used by the Romans, who instead used IIII.
This system is not very helpful for arithmetic, and so it is little surprise to find that the Romans developed the portable abacus to ease mathematical operations. This device was a tray with a number of columns etched into it that could hold pebbles. A pebble (in Latin, the word “calculus”) had a value depending on the column that held it. Moving a pebble a column to the left increased its value by a factor of 10. Such an abacus could be used by merchants in the city or by surveyors working for the military.
Survey
Roman surveyors employed geometry to divide the landscape and lay out cities with effects that can still be seen in the twenty-first century. The key to Roman survey was a tool called a groma, which was a tall staff with a beam, known as a rostro, at right-angles to the staff at the top. The rostro supported a wooden cross, and at each end of the cross-beams was hung a plumb line. Sighting across these lines allowed Roman surveyors to lay out grids of perpendicular lines in the landscape. Surveyors could then divide land for agricultural purposes, and some field systems in Europe are based on these ancient surveys. The groma also left an impression on modern cities. The Romans frequently built new cities in conquered territories, for either native inhabitants or new settlements of veteran soldiers. At the heart of a Roman settlement lay the forum, the central civic space, which usually lay at the intersection of the Cardo maximus (the main north-south street) and the Decumanus maximus (the main east-west street). This system created new cities with grid-plans in which the main intersection was laid out by a groma. These perpendicular grids were the origins of many European settlements and was adopted in the planning of many U.S. cities in the nineteenth century.
The Roman Calendar
The Roman calendar instituted by Julius Caesar made a radical change to time-reckoning in Europe. Before this development, European calendars outside Rome were usually luni-solar calendars. As such, each month was related to the lunar cycle, which is not commensurate with the solar year, and so periodically whole months, known as “inter-calary months” would be inserted into the year to keep the months in step with the seasons. Insertions would usually have to be done every two or three years. Even ancient authors recognized that this system was inefficient, including Herodotus, who wrote in the late fifth century b.c.e. that the Egyptians had a much more accurate solar calendar. In 45 b.c.e., Julius Caesar adapted the Egyptian method of time-keeping for Roman use.
Each month was counted as a period of days, usually 30 or 31 but with 28 or 29 in February. In addition, Julius Caesar laid down rules for when an inter-calary day would be added to February. The Egyptians corrected the calendar by adding a day every fourth year. Unfortunately, the Romans counted inclusively, meaning that the leap year was in the fourth year, rather than after the fourth year. For example, 2020 is a leap year. For the ancient Romans, the second year in the cycle is 2021 and the third is 2022. Therefore, 2023 is the fourth and the Romans of Julius Caesar’s time would have made this a leap year, rather than 2024. Augustus Caesar corrected this error in the early years of the first century c.e.
This method of keeping the years remained until the reforms of Pope Gregory XIII in 1582, though Britain and the American colonies did not implement the Gregorian calendar until 1752. The difference between the two calendars is that years divisible by 100 are not leap years, unless the year is divisible by 400. Otherwise, years are marked by the same cycle of months as the ancient Romans did.
Mathematics and the Cosmos
Even though ancient mathematicians had a relatively small set of tools based in geometry and arithmetic, these could be used to create incredibly intricate models. Ptolemy proposed a model of the universe that contained circles rotating upon circles to reproduce the movement of the planets. The connections between mathematics and cosmology made mathematics attractive to philosophers of the Roman period. The assertion that mathematics could reveal truth became increasingly contentious in late antiquity. Pagan philosophers came into conflict with a new religious sect, Christianity, which was increasingly powerful. One notorious incident was the killing of Hypatia, a female mathematician philosopher, in the city of Alexandria by a Christian mob. For some ancient historians, her death marks the end of the period known as classical antiquity.
Bibliography
Cuomo, Serafina. Ancient Mathematics. London: Routledge, 2001.
Dilke, Oswald. Roman Land Surveyors: Introduction to the Agrimensores. Newton Abbot, England: David and Charles, 1971.
Hannah, R. Time in Antiquity. London: Routledge, 2009.
Jaeger, Mary. Archimedes and the Roman Imagination. Ann Arbor: University of Michigan Press, 2008.