Having tax woes? The Aztecs, who lived in Central Mexico and whose empire was particularly strong between the 12^{th} and 15^{th} centuries, shared your pain. They might not have had to fill out 1040 forms, but they did have to pay taxes, and the calculations were not simple, according to a paper published April 4 in *Science Magazine* which deciphered the mathematics of two codices from around 1540-1544. (“Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic” by Barbara J. Williams and María del Carmen Jorge y Jorge; see the abstract here.)

LiveScience records del Carmen Jorge y Jorge as saying, “These are the only prehispanic documents that contain perimeter and area data, as far as I know. Most of the documents from this time were lost.” You can see a big picture here or a detailed picture here. (Isn’t it neat? Don’t you love looking at old maps, especially with little perimeter and area notations?) The codices also had household information (how many people there were) and soil quality (stony, sandy, or yellow earth), but it’s the area and perimeter calculations that were most important from a mathematics perspective.

To determine taxes, the Aztecs had to be able to measure the size of their land tracts. The standard unit of measurement was a *tlalquahuitl* (a land rod or a land stick), which was a little over 8 feet long. In order to get more accurate measurements (and we all know that the governerment wants us to be accurate in measuring how much money we’ll owe!), this had to be broken into smaller units of measurement. For example, an arrow represented half of the *tlalquahuitl*, a heart represented two-fifth of the *tlalquahuitl*; and a hand represented three-fifths, according to an article Friday in Scientific American.

Once these fractional measurements were in place, the calculations still took some deciphering. The Aztecs had a mathematics system that (at least on the surface) has some similarities with the earlier Mayan system: both used a base 20 system with dots and lines (although the Mayans used lines for 5s and dots for 1s, and the base 20 aspect was shown through position; the Aztecs used dots for 20s and lines for 1s according to LiveScience), both used mathematics largely for calendar computations, and both had a symbol for zero (see this abstract from a 1980 article). The mathematics for determining these land areas, however, apparently used some computations that were previously unknown in modern times.

What those computations actually were, I’m not sure. National Geographic records del Carmen Jorge y Jorge as explaining that there were five different algorithms that were used by officials (“including one also employed by the ancient Sumerians”). The abstract from the published paper explains, “we discovered^{ }evidence for the use of congruence principles, based on proportions^{ }between the standard linear Acolhua measure and their units^{ }of shorter length. This procedure substitutes for computation^{ }with fractions and is labeled ‘Acolhua congruence arithmetic.’^{ }”

As a sidenote, this site has directions for making codices as a craft project, suitable for kids or adults.

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