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

Abstract

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.and surveying played an integral role in the development of ancient states as growing economic demands and political complexity required increasingly precise records of distribution, amount, and quality of agricultural resources. In Eurasia such records begin several millennia ago. In the Americas, time depth of land surveying is unknown, but two extant native-style pictorial records attest that a sophisticated system was used by AcolhuaAztec people prior to European contact.Painted circa A.D. 1543-1544, the Codex Vergara* and the Códice de Santa María Asunción* provide unparalleled data to reconstruct Acolhua-Aztec (hereafter Acolhua) metrology and arithmetic and to evaluate the credibility of the records. The first comprehensive study of these codices (1) demonstrated that Acolhua land surveyors/scribes recorded side lengths of hundreds of agricultural fields using a standard linear measure, the tlalcuahuitl ("land rod"; T, equal to 2.5 m) and shorter-than-standard distances depicted by hearts, arrows, and hands (metrological monads, which are simple, indivisible units). Unexpectedly, it was also demonstrated that surveyors reported field areas in square tlalcuahuitl (T 2 ), which were depicted pictographically by a spatially distinctive form of numerical notation (Fig. 1).A follow-up study of 367 quadrilateral fields in the Codex Vergara (hereafter the Vergara) reconstructed both Acolhua survey metrology and area algorithms from an emic perspective (that of cultural insiders) (2, 3). The study validated the previously known Acolhua standard linear measure and amassed quantitative evidence establishing both the metric values of Acolhua monads and their use as fractions in area computation. Using Acolhua congruence arithmetic, five recurrent algorithms were detected that exactly reproduced 78% of the recorded areas. These results indicated that areas were indeed computed rather than measured by some physical means.Although Acolhua arithmetic was functionally accurate within their cultural context, in this work we analyze the accuracy of their recorded areas from the etic (cultural outsider) perspective of Western mathematics (3). Such assessment would be fairly st...

A Mathematical Description of the Agricultural Fields of the Acolhua Codices Vergara and Santa María Asunción

“…This article not only summarizes some of the mathematical results obtained over the years in the study of the Acolhua codices CV and CSMA (see Harvey and Williams 1980; Jorge et al 2011; Williams and Harvey 1997; Williams and Hicks 2011; Williams and Jorge y Jorge 2008) but also extends previous results from CV to CSMA and adds some refinements, as we analyze the accuracy of the area computations from both codices. We present for the first time an analysis of polygons with more than four sides from both codices that includes areas and shapes.…”

“…For example, a quadrilateral with side lengths equal to 10 units can have any area between 0 and 100 square units, but no (flat) quadrilateral exists with side lengths equal to 10 units and an area larger than 100 square units. For this corpus, 137 fields (35.49%) were impossible (Jorge et al 2011).…”

“…In real fields, one cannot expect perfect accuracy. As we did in an earlier article (Jorge et al 2011), we define a field as feasible if its percentage relative difference, Pd , is larger than or equal to −10%; that is, if it lies to the right of the −10% line in Figure 3 or if its percentage relative error is less than 10%. Our choice of 10% is arbitrary, but we believe it is reasonable.…”

“…Using elementary analytic geometry and a computer program to draw all the possible shapes of a quadrilateral, Jorge and colleagues (2011:File SI) proved that, given the four side lengths and the area of a possible quadrilateral, there are at most two different shapes (Garza-Hume 2016; Garza-Hume and Jorge 2018).…”

Measurement in Cacaxtla: A Multicultural and Symbolic Convergence

“…The use of a complementary system to obtain intermediate measures is likely. This would have been for adding several subunits, for example a half unit plus one-third of another, in the same way that the length of the mural measures “1 + 1/3.” Since the Mesoamerican cultures did not use fractions (Jorge et al 2011), each subunit, with its corresponding value in relation to the general one, was transformed into a unit of its own. Unfortunately, because of the number of possible combinations, this becomes somewhat difficult to verify from the archaeological data.…”