Barbara Jean Harris Williams

Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic

Acolhua-Aztec land records depicting areas and side dimensions of agricultural fields provide insight into Aztec arithmetic. Hypothesizing that recorded areas resulted from indigenous calculation, in a study of sample quadrilateral fields we found that 60% of the area values could be reproduced exactly by computation. In remaining cases, discrepancies between computed and recorded areas were consistently small, suggesting use of an unknown indigenous arithmetic. In revisiting the research, 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.” The findings also clarify variance between Acolhua and Tenochca linear units, long an issue in understanding Aztec metrology.

Mathematical accuracy of Aztec land surveys assessed from records in the Codex Vergara.

Land surveying in ancient states is documented not only for Eurasia but also for the Americas, amply attested by two Acolhua–Aztec pictorial manuscripts from the Valley of Mexico. The Codex Vergara and the Códice de Santa María Asunción consist of hundreds of drawings of agricultural fields that uniquely record surface areas as well as perimeter measurements. A previous study of the Codex Vergara examines how Acolhua–Aztecs determined field area by reconstructing their calculation procedures. Here we evaluate the accuracy of their area values using modern mathematics. The findings verify the overall mathematical validity of the codex records. Three-quarters of the areas are within 5% of the maximum possible value, and 85% are within 10%, which compares well with reported errors by Western surveyors that postdate Aztec–Acolhua work by several centuries.