doi: 10.1073/pnas.1107737108. Epub 2011 Aug 29.
Mathematical accuracy of Aztec land surveys assessed from records in the Codex Vergara
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- PMID:21876138
- PMCID: PMC3174618
- DOI: 10.1073/pnas.1107737108
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Mathematical accuracy of Aztec land surveys assessed from records in the Codex Vergara
María del Carmen Jorge et al. Proc Natl Acad Sci U S A..
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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.
Conflict of interest statement
The authors declare no conflict of interest.
Figures
A representative household record of fields in theCodex Vergara.A shows fields with their side lengths. Standard linear measures intlalcuahuitl (T) are denoted by lines (1 T) and dots (20 T). Shorter-than-standard distances include a heart (here appearing in all fields) and a hand (third field).B shows the same fields with their corresponding areas. To read the areas (given in T2), multiply lines and dots in the center or bottom margin of the rectangle by 20 and add the units in the upper right-hand tab. For example, the area of the last field is 26 × 20 + 4 = 524 T2 (ff. 22 v, 24 v).
Graphic representation in the coordinate plane of Eqs. 1. The longest quadrilateral side lengtha lies on the horizontal axis; clockwise followb,c, andd. Eq. 1c is the sum of the areas of the rectangle and triangles.
Graphic illustration of Eqs. 3. (0,0) is the center of
, the circle of radius side lengthb;
is the linePu + Qv = R. The position and slope of
depend on side lengths and area. Here the area is varying and the side lengths are fixed and ordered as in Fig. 2. As the area increases we have first caseA, a position of
that gives two feasible quadrilateral shapes, then caseB, the tangential position of
with only one feasible quadrilateral shape, and finally caseC, the position without intersection and thus no feasible quadrilateral. SeeMovie S1 .
, the circle of radius side lengthb; is the linePu + Qv = R. The position and slope of depend on side lengths and area. Here the area is varying and the side lengths are fixed and ordered as in Fig. 2. As the area increases we have first caseA, a position of that gives two feasible quadrilateral shapes, then caseB, the tangential position of with only one feasible quadrilateral shape, and finally caseC, the position without intersection and thus no feasible quadrilateral. See
Boundary coordinates (solid circles) of modern El Topote, which corresponds to the Vergara locality of Topotitla. Compañía Mexicana de Aerofoto, circa 1965.
Distribution of the percentage discrepancy (Pd) of recorded areas (Ac) with respect to the maximum possible area (Am). The horizontal axis represents each of the 386 quadrilaterals, ordered by their correspondingPd values (vertical axis). The solid line indicates zero discrepancy betweenAc andAm, and dotted lines delimit a ± 10% discrepancy. NegativePd values correspond to the unfeasible fields.
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References
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