By the numbers.
This map from 1540 has notations in Aztec numbers depicting dimensions of lands near Texcoco, the ancient capital of the Acolhua Aztecs.

Library of Congress, Geography and Map Division

How Aztecs Did the Math

The Aztecs, who ruled central Mexico for several hundred years before the Spanish arrived in 1519, left the most extensive mathematical writings of any pre-Columbian people. Two manuscripts in particular have intrigued scholars because they portray land holdings in the Valley of Mexico along with their measurements, using the Aztec numbering system, for purposes of taxation. Now a geographer and a mathematician have zeroed in on just what methods Aztec surveyors used to measure the surface of a field in one of these documents, the Codex Vergara.

Scientists long ago deciphered the Aztec number system, a vigesimal system (using 20 as its base) as opposed to our decimal system. In Aztec arithmetic, a dot equals 1, a bar represents 5, and there are other symbols for 20 and various multiples thereof. The Codex Vergara, painted about 1540, contains schematic drawings and measurements of individual fields. Previous research on it has revealed an understanding of multiplication and division as well as certain principles of geometry.

Now, Barbara Williams of the University of Wisconsin-Rock County in Janesville, with Maria del Carmen Jorge y Jorge of the National Autonomous University of Mexico, has analyzed the Codex Vergara to discover just how Aztec surveyors estimated land area in parcels that were often irregularly shaped. The analysis reveals a "very practical kind of arithmetic and record keeping," says Michael Smith, an archaeologist at Arizona State University, Tempe.

In a paper published in tomorrow's issue of Science (4 April, p. 72) the authors show that Aztec surveyors probably used several types of algorithms to calculate area. Some parcels involved simply multiplying length by width. But in other, irregular four-sided lots, they had to come up with different approaches, such as multiplying the average of two opposite sides by an adjacent side.

Furthermore, when a measurement did not match a precise number of "land rods"--their standard unit of linear measurement, which corresponded to about 2.5 meters--the Aztecs added symbols, such as an arrow, a heart, a hand, or a bone, to indicate remaining length that was less than one rod. Working back from the recorded land areas, the authors determined that these corresponded to different fractions of a land rod.

Although the Aztecs are the only early Americans to have left these kinds of technical documents, it's reasonable to assume that other groups such as the numerically sophisticated Maya used similar systems, Smith says. "There's a view that ancient peoples were obsessed with religion and that science and knowledge were all directed at religious ends," he adds. But the paper shows that the Aztecs apparently liked to get their measurements right--and certainly when it came to taxation.

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