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

Edited by Joyce Marcus, University of Michigan, Ann Arbor, MI, and approved July 29, 2011 (received for review May 14, 2011)

## 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.

Land 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 Acolhua–Aztec 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 straightforward if individual field boundaries could be identified in situ and resurveyed. Five centuries of landscape change recently culminating in massive urbanization precludes this method, except for one small codex locality that we have tentatively identified and reconstructed with on-the-ground observations. Overall, our area accuracy testing employs Acolhua data in mathematical evaluation tools specifically designed to interpret codex information.

## Vergara Quadrilateral Database

Data for all 408 codex quadrilateral fields consist of three variables: (*i*) field side lengths (*a*, *b*, *c*, *d*) in T, (*ii*) depicted field shapes (without linear scale, correctly drawn angles, or diagonals), and (*iii*) field areas in T^{2}. For inclusion in the working database, quadrilaterals had to satisfy a primary requirement that no side be longer than the sum of the other three, meaning that the field boundaries must close. This is fulfilled by all but one field that was deleted from the data universe along with 21 others because of incomplete data or illegibility, resulting in a working database of 386 quadrilaterals (Table S1 provides quadrilateral side lengths, areas, and computer-generated field shapes).

## Methods

The corpus consists of 122 right-angled quadrilaterals (90 squares and 32 rectangles) and 264 irregular fields. For the former, 121 areas computed with the geometric rule “length × width” exactly match the recorded codex areas. The Acolhua procedure of recording areas in T^{2} (T × T) clearly demonstrates their abstract, mathematical concept of area corresponding to the natural step of counting square units enclosed within a quadrilateral. In contrast, had these been Colonial Spanish surveys, areas would have been expressed by the amount of maize or wheat sown or harvested (4⇓–6).

Unlike right-angled quadrilaterals, which have unique shape, testing area accuracy of the 264 irregular quadrilaterals is more problematic because, without side-angle data, these can vary in shape, and therefore area. However, quadrilateral side lengths do determine minimum and maximum possible values for its area. Therefore, as a test of survey accuracy we can say that Acolhua areas are feasible if they fall within these values. Otherwise, they are unfeasible, by which we mean mathematically impossible on a flat surface. Also, irregular quadrilateral field shapes are recoverable, providing a second test for feasibility. Fields that fail the feasibility test are especially illuminating because these allow a quantitative measurement of error. Below we present results from two different mathematical tools used in our analysis: (*i*) construction of field shapes and (*ii*) computation of maximum possible area.

## Use of Coordinates

First we consider the question, Are the corpus fields mathematically feasible? That is, is it possible to construct a quadrilateral (or several quadrilaterals) given the registered side lengths *a*, *b*, *c*, *d* (taken clockwise) and codex recorded area *A*_{c}? We use the coordinate plane to answer this question and find the possible shapes. Assume side *a* (ordered to be the longest) lies on the horizontal axis so that two of the quadrilateral’s vertices are (0,0) and (*a*,0). Let (*u*,*v*) and (*w*,*z*) be the other two unknown vertices.

Then by construction, using the Pythagorean theorem, the following equations must be satisfied: [1a][1b][1c]

Eq. **1(c)** for the area *A*_{c} is found by subdividing the area into triangles and rectangles (Fig. 2). By simplifying the equations in Eq. **1**, we obtain the unknowns (*w*,*z*) as functions of (*u*,*v*): [2]where is a constant (see *SI Appendix* for the proof).

After substitution of the equations in Eq. **2** into Eq. **1(c)**, we can solve for (*u*,*v*) as the intersection of a mathematically determined line and a circle given respectively by [3]where *P* = *a*(*a*^{2} + *b*^{2} - *c*^{2} - *d*^{2}), *Q* = 4*aA*_{c}, and .

(See further discussion in *SI Appendix*.)

Every solution (*u*,*v*) of system **3** gives a value for (*w*,*z*) using Eq. **2** and therefore determines the quadrilateral. For system **3**, only one of three outcomes can occur: (*i*) The line is tangent to the circle; that is, the quadrilateral is feasible and has only one possible shape; (*ii*) the line intersects the circle twice; that is, there are two sets of possible vertices (*u*_{1},*v*_{1}) and (*u*_{2},*v*_{2}) for the given data producing a feasible quadrilateral with two possible shapes; (*iii*) the line does not intersect the circle; that is, the quadrilateral is unfeasible, and thus the survey data are inaccurate (Fig. 3). An animation of this can be seen in Movie S1.

Our computer program that calculates the unknown quadrilateral coordinates exactly recovers the same 122 right-angled quadrilaterals when tangency occurs [case (*i*)], verifying the validity of the equations and program. For the other 264 irregular quadrilaterals, 128 fields could feasibly have two possible quadrilateral shapes [case *ii*)]. The remaining 136 fields are the unfeasible quadrilaterals [case (*iii*)].

To further check feasibility, another computer program was designed to plot the possible shapes determined by the sets of vertices (*u*,*v*) and (*w*,*z*); these are reproduced in *SI Appendix* (Table S1). The majority of these plots have shapes similar to actual agricultural fields in the region depicted in the codex (see below), which further supports the accuracy of the data [*SI Appendix* (Table S1)].

## Use of Maximum Area

The plane coordinate feasibility test suggests that, given their side lengths, the surveyors incorrectly determined areas for slightly over one-third (35.23%) of the quadrilaterals. This seemingly high error rate raises the question of error magnitude. Acolhua area approximation apparently relied entirely on side lengths; no indication of the use of trigonometry has been detected in the Vergara or other indigenous land documents (2). But precise mathematical determination of an exact area requires angular data. For a quadrilateral with side lengths *a*, *b*, *c*, and *d*, if *ϕ* is the angle between *a* and *b*, and *ψ* is the angle between *c* and *d*, the exact area is given by Bretschneider’s formula (7) as[4]where *S* = (*s*-*a*)(*s*-*b*)(*s*-*c*)(*s*-*d*) with *s* = (*a* + *b* + *c* + *d*)/2, the semiperimeter. Even if angles are unknown, Eq. **4** can be used to compute the maximum possible area of the given quadrilateral, which provides a test for feasibility.

From Eq. **4** the maximum possible value *A*_{m} is achieved when the negative term is zero, and hence . Using this maximum area, outcomes (*i*), (*ii*), and (*iii*) of the coordinates findings are equivalent in the maximum area test to (*I*) *A*_{c} = *A*_{m} , (*II*) *A*_{c} < *A*_{m} , and (*III*) *A*_{c} > *A*_{m}.

Our program that calculates the values of *A*_{m} and compares them with *A*_{c}, as expected, produced the 122 right-angled quadrilaterals in case (*I*), 128 feasible irregular quadrilaterals in case (*II*), and 136 unfeasible cases in case (*III*). Thus, the coordinates test and the maximum area test both reveal the same mathematically inaccurate unfeasible fields.

Unexpected results occurred, however, in computation of the percentage of relative discrepancy between *A*_{m} and *A*_{c} defined as *Pd* = 100 × (*A*_{m} - *A*_{c})/*A*_{m}. Of the 136 unfeasible quadrilaterals, nearly three-quarters (72.79%, 99 fields) exceed *A*_{m} by less than 10% [Table 1, based on data in *SI Appendix* (Table S2)]. If a 10% error resulting from survey procedures is allowed, then these 99 unfeasible fields fall into the feasible group. Only the remaining 37 cases (27.21%) greatly overestimate a feasible area. Of these, 18 were discarded a priori in ref. 2 because the areas were obviously too big, probably the result of recording errors in the sequence of perimeter and area drawings or faulty entry of numerical values.

Aside from recording errors, area overestimation is in part a product of Acolhua computation algorithms. After the length × width rule, the most frequently used Acolhua algorithm is the Surveyors’ Rule (SR), where area is the product of averaged opposite sides (2). Among the 99 unfeasible fields with |*Pd*| < 10%, the SR was applied in nearly half of these (45) to compute the *A*_{c}. In 28 of the 45 (62.22%), overestimation is nearly negligible: |*Pd*| < 1%. Because an approximation given by the SR is always equal to or larger than the *A*_{m} (see further discussion in *SI Appendix*), some mathematically unfeasible cases necessarily result from its application. Nevertheless, in many cultures and settings, the SR commonly is employed to approximate area and mathematically is a very good approximation of the *A*_{m} when angles are not overly acute. Acolhua surveyors, however, may not have been aware of this deficiency.

For the 250 feasible fields (*A*_{c} < *A*_{m}), percentage differences from the maximum possible area (which are not errors in this situation) vary from 0% for the right-angled quadrilaterals to 5% for 82.8% of all feasible quadrilaterals, whereas 92.5% fall within 10% discrepancy. Only areas of 19 quadrilaterals differ by more than 10% from the *A*_{m}. The surprisingly large number of quadrilaterals with similar *A*_{c} and *A*_{m} values indicates that surveyors chose algorithms to approximate the largest possible land area within given field boundaries. This observation belies a notion that to lessen tax levies native surveyors might have intentionally produced inaccurate surveys by systematically underestimating linear measurements and areas.

## Measurements on the Ground

A preferred method to test Acolhua area accuracy would be to resurvey the codex fields on the ground to compare with codex data. Although known to have been located in the village of Tepetlaoztoc (8) some 20 miles northeast of Mexico City and now undergoing rapid urbanization, place names of the five Vergara communities no longer are recognized by the population. Nevertheless, colonial period texts (9, 10) provide descriptions by which to equate the Vergara hamlet of Topotitla [locality 2 in *SI Appendix* (Table S3)] with a modern land tract called El Topote in the barrio (ward) of Asunción. Landscape change does not allow identification of individual properties, but for accuracy assessment the measured area of El Topote may be compared with the sum of Topotitla recorded areas.

Modern El Topote is a triangular-shaped, gently sloping tract of agricultural land bounded on the north by a centuries-old stone wall, on the east by a ravine, and on the west by a road located a short distance from a second ravine, which, in the sixteenth century and today, is known as Tzila (Fig. 4). To determine the tract area, we took 18 boundary Global Positioning System coordinates ground-truthed with a Google Earth image and plotted 42 points on an air photo^{†}. To calculate the area we applied the Surveyor’s Formula to a Lambert plane projection of the surface^{‡} (11, 12). Our resultant approximation of El Topote is 12.41 ha (124,071.52 m^{2}) (13). An estimated error for our area approximation of El Topote is unavailable without a professional survey. Error uncertainties arise from the lack of tract boundary markers except for the east-west stone wall. The boundary ravine on the east has clearly backwasted into the level surface of El Topote above it; to compensate for erosion we took coordinates from the ravine center. Also, it is not entirely clear that the road on the west exactly follows the sixteenth-century boundary.

For Topotitla, Acolhua surveyors recorded 38 fields: 24 quadrilaterals and 14 polygons with more than four sides. Area data are complete except for three quadrilaterals and one six-sided field [see *SI Appendix* (Table S4)]. We estimated quadrilateral areas by the *A*_{m} computed from recorded side lengths. A different method was required for computing the *A*_{m} for the six-sided field. The codex drawings of irregular polygons clearly reflect additions to, or subtractions from, a quadrilateral form, to accommodate neighboring properties, small structures, and other nonagricultural features (Fig. 1). Mathematically, a polygon inscribed in a circle has the maximum area among all the possible configurations for the given sides (13). A program written to find the enclosing circles of Topotitla polygons allowed determination of the vertices’ coordinates, and thereby calculation of maximum areas with the Surveyor’s Formula. This method necessarily introduces area overestimation because all vertices lie on the enclosing circle thus ignoring indentations. The resulting field shapes are mathematical abstractions rather than typical field forms.

With the four substitutions of maximum area for recorded area, the adjusted sum of Topotitla recorded field areas is taken to be 13.5 ha (134,997 m^{2}), to which we add 580 m^{2} for dwelling areas of nine households listed in the Vergara population census^{§}. Adjusted agricultural land and house plots total 135,577 m^{2}, whereas our measurement of El Topote is 124,071.52 m^{2}, a difference of about one hectare (11,505.48 m^{2}). The adjusted Acolhua area is 9.15% greater than that of the ground measurement of El Topote and thus by definition is unfeasible. Notably, this finding fits the pattern of a 10% discrepancy margin with respect to the *A*_{m} exhibited by a large majority of the unfeasible quadrilaterals.

## Summary, Discussion, and Conclusions

Of the 386 Vergara quadrilateral fields, 65% are feasible; that is, quadrilaterals mathematically exist with the Acolhua recorded side lengths and areas. Although feasibility does not prove such quadrilaterals existed with these areas, 92% of the feasible fields differ from the maximum possible area by less than 10%, suggesting that the recorded areas were good approximations of the actual ones. Our computer-generated shapes of the feasible fields support this conclusion. For the unfeasible fields (*A*_{c} > *A*_{m}), nearly three-quarters overestimate the maximum possible area by no more than 10%. The on-the-ground results of the El Topote/Topotitla study also pertain to this group. For the entire quadrilateral corpus, 86% of the recorded areas fall within 10% of the maximum possible area, but more striking, nearly 75% have a discrepancy of less than 5%. Graphic distribution of percentage discrepancies clearly shows concentration of quadrilateral areas within these values with relatively few outliers, which suggests systematic rather than random Acolhua computation procedures (Fig. 5).

Since its inception, land surveying has involved error attributable to technology, procedures, and blunders. One error source in Acolhua surveys stems from measuring instruments of rope or rod used over long side-length distances, many over 100 m, which introduces problems such as maintaining straight line of sight. Also, because distance monads were not consistently recorded, rounding apparently was an accepted measurement protocol, which could introduce an error of +/-0.5 to 2.5 m on a side.

In computations, algorithms such as the SR inherently may produce error. Choice of algorithm also contributes, as illustrated by Id 5-21-02 (Table S1) with side lengths 26, 32, 30, and 10. The *A*_{c} of 588 T^{2} is given exactly by the SR, yet the *A*_{m} is only 538.27 T^{2}. The surveyor perhaps did not have (or chose not to use) a more appropriate area algorithm. Calculations with or without monads likewise produce varying results in accuracy. Greater precision by using the most appropriate algorithm and the arrow monad is exemplified in Id 3-11-09 with side lengths 31 + arrow, 8, 33, 9, and *A*_{c} = 273 T^{2}. The codex figure approximates a trapezoid, but in dimensions closely resembles a rectangle. A logical algorithm of choice would be the SR, giving 274 T^{2}, but instead the surveyor chose one of two variants of the Triangle Rule [*A*_{c} = (*a* × *d*)/2 + (*b* × *c*)/2], giving an area of 273.75 T^{2}, compared to the *A*_{m} = 273.02 T^{2}, both of which when rounded down yield the *A*_{c}. In this case, rounding down to the integer value of T^{2} introduces a negligible error, but the Acolhua practice of recording all areas in integers could have increased the rounding error to +/-1 T^{2}. Also, blunders in recording numbers are nearly inevitable. As expected working in base-20, some errors detected in the codex are multiples of 20. For example, for Id 01-18-01 with side lengths 34, 20, 34, and 20, length × width yields 680 T^{2}, whereas the *A*_{c} = 660 T^{2} that quite likely derived from a missing line in the area record.

Given these and other error sources, and working without angles, one might expect Acolhua survey records to be grossly inaccurate by Western standards. To the contrary, we find the work of the Acolhua surveyors/scribes to be reasonably to very accurate. For perspective, nearly a century later, errors by English surveyors were of similar magnitude, despite employment of more sophisticated instruments. Saxton’s 1613 map of Manningham, for example, registers an approximate 10% discrepancy between land parcel areas recorded in his written survey and those on his map. Even by 1658, area error could be as much as 25% (16, 17). Land surveys in the southern American colonies were notoriously inaccurate. For example, a meticulous analysis of early seventeenth-century land grants that were resurveyed in the eighteenth century revealed area errors ranging from 150% overestimation to 42% underestimation (18, 19). A century later in Mexico a property area measured in 1757 was found in an 1872 resurvey to have been overestimated by 8.9% (20). Survey inaccuracy still remained an issue at the close of the nineteenth century.

Quantification of the mathematical integrity of the Codex Vergara serves to underscore accomplishments in rural land surveying, applied mathematics, and record keeping in the Aztec world. Spanish administrators did not avail themselves of the land-survey competency of their native subjects. Instead, they attempted to impose order upon a chaotic system imported from Peninsular Spain. In 1536 their confusion was denounced by the first Viceroy of New Spain: “in this city [Mexico City] there exists no measure for measuring land” (5). Although Acolhua surveying succumbed in the Conquest aftermath, the *Codex Vergara* provides testimony of their thoroughly modern concept of area, the functionality of their geometry for area determination, and the mathematical accuracy of their work when assessed by Western standards.

## Acknowledgments

We gratefully acknowledge the Department of Mathematics and Mechanics of the Institute for Applied Mathematics and Systems Research (IIMAS) of the National Autonomous University of Mexico (UNAM) for research and technical support, especially the expertise of A. C. Pérez Arteaga and R. Chávez for design of databases and figures.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: clara{at}mym.iimas.unam.mx.

Author contributions: M.C.J., B.J.W., C.E.G.-H., and A.O. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1107737108/-/DCSupplemental.

↵

^{*}National Library of France, Ms. Mexicain 37–39, Paris, and National Library of Mexico, National Autonomous University of Mexico, Mexico City, Ms. 1497bis, respectively.↵

^{†}Boundary coordinates of El Topote listed in Table S3 were taken with a nonprofessional receiver that has a random error of up to 5 m per coordinate.↵

^{‡}The Surveyor’s Formula, not to be confused with the Surveyors’ Rule, gives the exact area of a polygon with known coordinate vertices.↵

^{§}Dwelling area estimates are based on archaeological data adapted from Evans (14) and Parsons (15). We do not adjust the*A*_{c}for other possible nonagricultural landscape features, such as walls or paths.

Freely available online through the PNAS open access option.

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