It has long been a mystery of Aztec arithmetic: What is three arms plus five bones?
Now researchers know: Five hearts.
The odd symbols had been noted for centuries — thousands of them appear in Aztec property registries that were created around 1540. But no one knew the value of the symbols or how they were used to represent the size of land plots for tax assessment and other purposes.
After three decades of work, geographer Barbara Williams and mathematician Maria del Carmen Jorge y Jorge have found a solution that reveals a complex surveying system with a rudimentary ability to calculate the area of irregular shapes and manipulate fractional amounts.
“It cracks the code,” said Williams, a professor emeritus at the University of Wisconsin.
The researchers, who published their findings Friday in the journal Science, based their analysis on two books called the Codex Vergara and the Codice de Santa Maria Asuncion. The manuscripts were written on paper brought by the Spanish conquistadors, who had arrived in Mexico two decades earlier.
The pages of the books are filled with tiny property maps. For each plot, there are two drawings — one showing the lengths of the sides and another showing the area. The measurements are represented by seven symbols: lines, dots, arrows, hearts, hands, arms and bones. Each map also includes the name of the property owner and the soil type.
Researchers already knew what each map represented and the value of some of the measurements. A line, for example, was the standard unit of length, which was known as a “tlaquahuitl,” or rod, and in modern units would measure 2.5 meters.
When the researchers knew the values of the units in roughly rectangular plots, they could easily follow the logic of the Aztecs and reproduce their calculations by multiplying lengths and widths.
But they were stymied in calculating many plots because they didn’t know the value of the units. The breakthrough came when Jorge y Jorge, a professor at the National Autonomous University of Mexico, found that some areas were prime numbers.
That meant some of the unknown symbols had to represent fractions of a rod, she said.
By trial and error, she decoded the system.
A hand equaled three-fifths of a rod, an arrow was one-half, a heart was two-fifths, an arm was one-third, and a bone was one-fifth.
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