From the Los Angeles Times

Aztec math finally adds up: A three-decade study 'cracks the code' of  
symbols in registries indicating a complex land survey system
By Alan Zarembo
Los Angeles Times Staff Writer

April 4, 2008

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  

"It cracks the code," said Williams, a professor emerita at the  
University of Wisconsin.

The researchers, who published their findings today 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 Spanish conquistadors, who had arrived in Mexico  
two decades earlier.

The researchers said the property drawings in the books were probably  
transcribed from even older documents written on tree bark or cotton  

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 tlalquahuitl, or rod, and in modern  
units would measure a little more than 8 feet.

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 the values of some areas were prime numbers.

That meant that 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 3/5 of a rod, an arrow was 1/2 , a heart was 2/5 , an  
arm was 1/3 , and a bone was 1/5 .

A set of at least five formulas emerged showing how the Aztec  
surveyors determined the areas of irregular shapes. In some cases, the  
Aztecs averaged opposite sides and then multiplied. In others, they  
bisected the fields into triangles.

Of the 369 plots the researchers examined, they could accurately  
reproduce the Aztec math in 287 cases, according to the study.

Still, Williams and Jorge y Jorge don't understand how the Aztec  
surveyors decided which formula to use for each area calculation.

In addition, it is unclear whether the same system was used in other  
city-states and if it applied to measurements besides land dimensions.
[log in to unmask]

s. e. anderson is author of "The Black Holocaust for Beginners"
Social Activism is not a hobby: it's a Lifestyle lasting a Lifetime