Aztec Math Used Hearts and Arrows

How big is a heart? Two-fifths of a land rod* of course, according to the Aztecs

AZTEC MATH: The Aztec used symbols such as arrows and hearts to denote fractional units of measurement in surveying records like the Oztoticpac Lands Map pictured here. Image: COURTESY OF LIBRARY OF CONGRESS

The Aztecs had more numbers than we do, or at least symbols denoting numerical concepts. When it came to measuring land—critical for levying the proper tax or tribute—these medieval Mesoamericans used arrows, hearts, hands and other units representing fractions, according to a new study in Science.

To figure this out, mathematician Maria del Carmen Jorge y Jorge of the National Autonomous University of Mexico (U.N.A.M) channeled the mind of an Aztec land surveyor. That meant retraining herself to use a different numerical system and combing through the Codex Vergara, one of two remaining books that record Aztec land surveying.

Working with geographer Barbara Williams and del Carmen Jorge y Jorge counted 367 fields in this book with both an overall area for the plot of land as well as the lengths of the sides. Roughly 60 percent of these fields had areas that matched the basic mathematical rule of length multiplied by width or other common surveying calculations.

But the rest were off, usually by a small amount. And 69 had areas that were prime numbers such as 211—numbers that cannot be created by multiplying two whole numbers together, such as 20 times 10. Instead, del Carmen Jorge y Jorge determined that the Aztecs were using the equivalent of fractions.

"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."

The researchers will next try to assess the accuracy of the Aztec surveyors. The neighborhood of Asuncion outside Mexico City still bears the markings of the ancient Aztec terraced fields on its hillsides that were recorded in the Codex Vergara. "We were there trying to measure those terraces," del Carmen Jorge y Jorge says. "This is complicated because this is sloping land."

It is no doubt easier to measure sloping land with modern devices like satellite global positioning systems and computers than it is to try to inhabit an alternative mathematical system and devise the meaning of mysterious symbols—as well as grasp the algorithms that can explain how they were used. "I can use my math, my computers and whatever I want," she says. "With this paper, I am only using hand calculations."

*This article erroneously referred to a heart as measuring 2.5 land rods. The correct measurement is two-fifths.

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1. 1. davidcemin 10:01 PM 4/4/08

I think they used to measure the things with the available "devices", such as arrow, heart, hand, bone and arm, that, maybe, it was common for them.

2. 2. jabailo 05:57 PM 4/5/08

Sounds like we're too dumb to understand Aztec math.

3. 3. Îž_Heather 01:02 AM 4/6/08

Interesting article! I think, however, that there is a typo in the subtitle. It currently reads "How big is a heart? Two and a half land rods of course, according to the Aztecs" but in the article itself it says "five hearts is two units". This implies that a heart is two-fifths of a land-rod, not five-halves.

4. 4. davidmm 06:31 PM 4/7/08

I'm currently teaching a course on Mesoamerican Writing Systems at UNC-Chapel Hill, and I'll mention this article to my students. I do object to the "medieval" characterization of Aztec society. The term is normally used in English a pejorative way, connoting a backward social organization and cultural development. This is hardly acceptable when writing about any society--especially one that the article's author does not have any links to--and certainly not applicable to the Aztecs.

5. 5. dbiello 08:37 PM 4/7/08

Thanks for the math correction. How embarrassing. Somehow two-fifths became 2.5. I'll blame the decimal system and fractions for striking back at their competitor : )

As for medieval, I in no way meant that as a pejorative term, merely as descriptive of the time period involved. There is no doubt that the Aztecs had a sophisticated and advanced civilization.

6. 6. David C. Hewitt 04:05 AM 4/9/08

The Tialquahuitl at 2.5 m makes the 'arm' equal 0.833 m. This is uncomfortably close to the Iberian vara at 0.840 m. which is also known in Texas, California, Mexico and Peru where it is thought to have been taken by the Spanish. Do the authors have any positive indication of the use of these units by the Aztecs before the Spanish conquest? If not there would seem to be a distinct possibility that knowledge of fractions and the calculation of areas may well have been transfered along with the vara.
David C. Hewitt

7. 7. oolatec 07:56 AM 4/9/08

Studied Latin American cultures(one of my "A" subjects)
Always like to stay up-to-date
iceman

8. 8. georgejmyersjr 02:06 AM 4/16/08

Confusion results in western terms between "rod" and "rood" and apparently at one time "rod" was a unit of brick wall with a method to "estimate the value of a rod of brickwork..." (p. 742) and the measures of "roods" of "superficies...as under" 144 sq. inches are equal to 1 square foot...9 square feet ...1 square yard...2 7/9 square yards...1 square pace...10-89 square paces ...1 square pole...40 square poles...1 square rood...4 square roods...1 square acre". Followed by very large surfaces, as of countries are expressed in square miles, which is shown in conversion multiples of units of "Square feet";"Square yards";"Square poles";"Square Roods";"Square Acres". This leads to the comment above that "Perhaps the only standard that can be safely referred to at the present day is that belonging to the Royal Astronomical Society."!
p. 1224 - From "The Encyclopedia of Architecture - Historical, Theoretical, and Practical -The Classic 1867 Edition. Joseph Gwilt c) 1982 Crown Publishers, Inc.

9. 9. Chaos21754 09:06 PM 7/6/08

2.5 hearts = 2 arrows = 1 land rod.
5 hearts = 4 arrows = 2 land rods. Where does 2/5ths enter? It doesn't. I agree with jabailo concerning Aztec math.

10. 10. Chaos21754 09:08 PM 7/6/08

5hearts = 4 arrows = 2 land rods.
2.5 hearts = 2 arrows = 1 land rod. Where does 2/5ths enter? It doesn't. I agree with jabailo.

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