Fondo Reservado de la Biblioteca Nacional de México, UNAM

Aztec census. Snippets from the Códice de Santa María Asunción, very similar to the Codex Vergara, record the border lengths of Mesoamerican farms (right) and calculate their areas (left).

Aztec Maps Put Cortés to Shame

As far as tax collectors in colonial Mexico went, Gonzalo de Salazar, often dubbed "El Gordo," was a pinchpenny. The conquistador-turned-regional-chief demanded steep tributes from his charges living in an area called Tepetlaoxtoc just north of what is now Mexico City. To expose El Gordo's greed, census takers from the Acolhua-Aztecs, a subset of the larger Aztec group, set out to count their own numbers in the mid-1500s and tally the extent of their farmland and hence their tax burden. They did a remarkably good job, a new study suggests. The early surveyors calculated the sizes of their farms with a degree of accuracy likely beyond the means of El Gordo or his cronies.

The Tepetlaoxtoc census, also known as the Codex Vergara, was much more than a simple survey. This paint-on-paper record incorporated icons for every adult and child in the region, as well as detailed maps for at least 386 farms. The surveyors measured the borders around each of these fields and then calculated their areas in square tlalcuahuitls, units equal to roughly 2.5 meters.

Using these records, Clara Garza-Hume, a mathematician at the National Autonomous University of Mexico in Mexico City, and colleagues went back to the codex to check the Aztecs' math. That was easy for the rectangular-shaped farms but much harder for the more than 200 recorded plots that, although still four-sided, lacked that uniform shape. The Aztecs hadn't yet stumbled upon trigonometry, so their maps failed to record at what angles the farms' borders joined up. Since the exact angles weren't clear from the maps, such odd four-sided shapes could have taken on a number of different forms, Garza-Hume says. "The side lengths remain constant, but you can still wiggle the figure and obtain many different areas."

What the team could do, however, was calculate the wiggle range of possible shapes for each of the fields. And the surveyors "did quite well" in matching those shapes, she says. The Aztecs calculated the sizes of their farms within a 10% error range about 85% of the time, Garza-Hume and her colleagues report online today in the Proceedings of the National Academy of Sciences. The few high guesses likely stemmed from reliance on the "surveyor's rule" to compute areas, she adds. This old trick, in which surveyors average out the lengths of a quadrilateral's opposing sides and then multiply them together, is notorious for giving high numbers.

But the Aztecs could just have easily have fudged their measurements, trying to trick their governor out of a few spools of cloth. Luckily, a field near the modern town of Texcoco still vouches for their honesty; this sloping lot contains the remnants of 38 old farms censused in the codex. The boundaries between the individual plots here had long ago eroded away, but Garza-Hume and her colleagues could still make out the larger borders of the region. Using GPS markers, they reckoned that the 38 farms had once taken up about 135,577 square meters, not too far off from the Aztecs' estimate of 124,072 square meters. El Gordo can keep his place in infamy.

Although El Gordo fought against lower taxes, the Codex Vergara did put the Spanish in their place, at least mathematically. Early colonialists were largely clueless when it came to land surveys, rarely knowing for sure where their expansive cattle ranches started or stopped, says Andrew Sluyter, a geographer at Louisiana State University in Baton Rouge. "Really, you don't get to that kind of map in Mexico ... until the Enlightenment, the 1700s."

Studies like these are important because they show the Mesoamericans' prowess in fields outside of astronomy, says Michael Smith, an archaeologist at Arizona State University, Tempe. Hernán Cortés and his countryman burned library after library in the old Aztec kingdom, leaving few records of day-to-day achievements behind. Still, indigenous Mexicans didn't always use their record-keeping acumen for good, Smith adds. The Aztecs, conquerors themselves, would have needed meticulous notes to squeeze every penny out of their squashed foes.

Correction: This article has been amended to correct the following details: The Codex Vergara was printed on paper, not cloth; it included depictions of 386, not 367, plots; it also included icons representing every adult and child, not strict “likenesses.” The field containing the 38 farms is not “empty” and is near, not in, Texcoco.

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