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A multidisciplinary and duo of scientists have analyzed the Codex Vergara. The Codex Vergara is a 660 year old document made by Aztec surveyors used to measure the surface of a field. It contains schematic drawings and measurements of individual fields. They were able to figure out that that Aztec surveyors probably used several types of algorithms to calculate area. There’s no doubt that they calculated areas by multiplying length by width. But when these people were confronted with irregular four-sided lots, they came up a different approach from what we’re used too, such as multiplying the average of two opposite sides by an adjacent side.

The research has been published in yesterday’s issue of Science under the title, “Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic.” The authors of this study were also able to figure out that when a measurement did not match their standard units for distance, a “land rod” or about 2.5 meters, they added symbols, such as an arrow, a heart, a hand, or a bone, to indicate the deviation. Each of these symbols corresponded to different fractions of a land rod.

I thought the Aztec number system was decoded a while ago. I knew they Aztec used a vigesimal system, where the number 20 was its base. That differs from modern day decimal based systems. Furthermore, in Aztec arithmetic, a dot equals 1, a bar represents 5, and there are other symbols for 20 and various multiples thereof…. but we didn’t know if the Aztec figured out fractions of a whole number. I guess now we do.

Scientific American took this very astute observation and made a rather demeaning ethnocentric headline out of it, but for those of us without access to Science, they were able to summarize the amount each of these symbols represent,

“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.”

    Williams, B.J., Jorge y Jorge, M.d. (2008). Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic. Science, 320(5872), 72-77. DOI: 10.1126/science.1153976