AZTEC MATH: The Aztec used symbols such as arrows and hearts to denote fractional units of measurement in surveying records like the Oztoticpac Lands Map pictured here.
The Aztecs had more numbers than we do, or at least symbols denoting numerical concepts. When it came to measuring land—critical for levying the proper tax or tribute—these medieval Mesoamericans used arrows, hearts, hands and other units representing fractions, according to a new study in Science.
To figure this out, mathematician Maria del Carmen Jorge y Jorge of the National Autonomous University of Mexico (U.N.A.M) channeled the mind of an Aztec land surveyor. That meant retraining herself to use a different numerical system and combing through the Codex Vergara, one of two remaining books that record Aztec land surveying.
Working with geographer Barbara Williams and del Carmen Jorge y Jorge counted 367 fields in this book with both an overall area for the plot of land as well as the lengths of the sides. Roughly 60 percent of these fields had areas that matched the basic mathematical rule of length multiplied by width or other common surveying calculations.
But the rest were off, usually by a small amount. And 69 had areas that were prime numbers such as 211—numbers that cannot be created by multiplying two whole numbers together, such as 20 times 10. Instead, del Carmen Jorge y Jorge determined that the Aztecs were using the equivalent of fractions.
"We found these smaller units of measure that we call monads that have the role of a fraction," she says. "We don't like to call them fractions, though, because they were considered as unitary entities like inches, seconds or minutes."
To denote half the Aztec basic unit of measure—known by Aztec experts as tlalquahuitl or land rods—the surveyors used an arrow symbol. So for a field that measured 20 land rods by 10 land rods plus an arrow (or 20 multiplied by 10.5), the correct area was 210. "Two arrows is one unit, five hearts is two units, five hands is three units," del Carmen Jorge y Jorge notes.
These extra units—arrow, heart, hand, bone and arm—cannot be subdivided further, standing alone as essentially extra numbers. It is unclear what exactly these measurements equal, but the team speculates that an arrow is the measure of the length from the shoulder to the hand (like an archer with a taut bow), a heart is a measure of the length from that organ to the tip of the hand and a hand as the measure from outstretched hand to outstretched hand—just as an English foot is the measure of a man's foot. "That could be an interpretation," del Carmen Jorge y Jorge says. "We cannot prove it."
The researchers will next try to assess the accuracy of the Aztec surveyors. The neighborhood of Asuncion outside Mexico City still bears the markings of the ancient Aztec terraced fields on its hillsides that were recorded in the Codex Vergara. "We were there trying to measure those terraces," del Carmen Jorge y Jorge says. "This is complicated because this is sloping land."
It is no doubt easier to measure sloping land with modern devices like satellite global positioning systems and computers than it is to try to inhabit an alternative mathematical system and devise the meaning of mysterious symbols—as well as grasp the algorithms that can explain how they were used. "I can use my math, my computers and whatever I want," she says. "With this paper, I am only using hand calculations."