Fractions: the key to Aztec civilisation

How many bones make a heart? And if you add two arrows and three arms, do you get a number that can be divided into an exact number of rods?

That may sound like an obscure schoolroom algebra quiz, but it is also a line of questioning at the heart of cracking one of the great mathematical mysteries of the pre-modern world – the complex code used by the Aztecs to calculate land measurements.

Now, almost five hundred years after the Spanish arrived in Mexico and brought the Aztec civilisation to a violent end, two researchers believe they have solved the puzzle.

Barbara Williams, a geographer from the University of Wisconsin, and Maria del Carmen Jorge y Jorge, a mathematician from the National Autonomous University of Mexico, pored over two ancient manuscripts detailing measurements of irregular parcels of land in Asuncion – just outside Mexico City – and used several trial-and-error methods to work out the values of all those hearts and bones (and arrows and arms) used by the Aztecs to denote, well, something.

As the pair described in the published version of their findings, which appeared in yesterday's edition of the journal Science, Dr Jorge y Jorge realised that some of the land area values were prime numbers and thus indivisible into smaller units.

That led to a flurry of calculations resulting in the following: A bone is one-fifth of a rod; a heart is two-fifths; an arm is a third and an arrow is half.

"We found these smaller units of measure that we call monads that have the role of a fraction," Dr Jorge y Jorge reported. "We don't like to call them fractions, though, because they were considered as unitary entities like inches, seconds or minutes."

Researchers have known for some time that the rod, or tlalquahuitl, was the Aztecs' key measurement of length – just over eight feet. All this has intrigued scholars for centuries because the Aztecs left the most extensive mathematical writings of any pre-Columbian people. Scientists have long realised the Aztecs used not a decimal system but a vigesimal one (with 20, rather than 10, as its base). They familiarised themselves with basic Aztec arithmetic, in which a dot equals one, a bar five, and various other symbols represent 20 or multiples of 20. The land-surveying symbols were much harder to decipher, though, because they related to irregular plots of land on sloping hillsides.

The source material is two documents dating from about 1540, or two decades after the arrival of the Spanish conquistadors. One of them, the Codice de Santa Maria Asuncion, lays out the terrain, while the other, the Codex Vergara, contains the mathematical symbols. Both books are filled with small property maps.

Previous generations of scholars have realised that the Codex Vergara reveals evidence of a sophisticated understanding of multiplication and division as well as geometric principles.

But not everything about the Codex is now clear.

Dr Williams and Dr Jorge y Jorge found that the Aztec surveyors used as many as five different formulas to determine the area of irregularly shaped pieces of land. In some cases, they tried to find a median line cutting through a wobbly property boundary. In other cases, they divided the fields into triangles.

The researchers said they couldn't entirely understand why the surveyors used a given method in a given circumstance. And they couldn't make their numbers add up in every case. They accurately reproduced the Aztec calculations in 287 out of 369 plots that they examined.