# Aztec Math Decoded, Reveals Woes of Ancient Tax Time

Brian Handwerk
for National Geographic News
April 3, 2008

Today's tax codes are complicated, but the ancient Aztecs likely shared your pain.

To measure tracts of taxable land, Aztec mathematicians had to develop their own specialized arithmetic, which has only now been decoded.

By reading Aztec records from the city-state of Tepetlaoztoc, a pair of scientists recently figured out the complicated equations and fractions that officials once used to determine the size of land on which tributes were paid.

Two ancient codices, written from A.D. 1540 to 1544, survive from Tepetlaoztoc. They record each household and its number of members, the amount of land owned, and soil types such as stony, sandy, or "yellow earth."

"The ancient texts were extremely detailed and well organized, because landowners often had to pay tribute according to the value of their holdings," said co-author Maria del Carmen Jorge y Jorge at the National Autonomous University in Mexico City, Mexico.

The Aztecs recorded only the total area of each parcel and the length of the four sides of its perimeter, Jorge y Jorge explained.

Officials calculated the size of each parcel using a series of five algorithms—including one also employed by the ancient Sumerians—she added.

"Rule of Thumb" and Other Body Parts

The Aztec arithmetic included fractional symbols like hearts, hands, and arrows that seem unusual to modern eyes. But to the Aztecs they likely had a relation to the familiar—the human body.

"For example the heart," Jorge y Jorge said.

"If you stretch out your left arm, that would be the measure from your heart to the tip of your finger. If you stretch both arms, the measure of the hand would be the distance between the tips of your two fingers.

"It's just very natural. Your body you carry with you all the time and it's very easy to refer whatever you want to measure to your body."

The primary land unit was likely the distance from the ground to the tip of a finger on an adult's upraised right arm—about 8.2 feet (2.5 meters), she said.

Jorge y Jorge and co-author B.J. Williams of the University of Wisconsin-Rock County report their findings in this week's issue of the journal Science.

"I think [the study] is neat because it shows that this sort of math and science was pretty practical in orientation," said Michael Smith, an archaeologist and Aztec expert at Arizona State University.

(Read related story: "Inca Tax Records Were Tied Up in Knots, Study Says" [August 11, 2005].)

"We have the idea that ancient societies were dominated by religion. Yeah, religion was important, but they were also very practical people doing very practical things," Smith said.

"With this sort of rule-of-thumb surveyor's math, they figured out a way to get it done."