Mathematical accuracy of Aztec land surveysassessed from records in the Codex VergaraMaría del Carmen Jorgea, Barbara J. Williamsb, C. E. Garza-Humea,1, and Arturo Olveraa

aInstituto de Investigaciones en Matemáticas Aplicadas y en Sistemas and Proyecto Universitario de Fenómenos No Lineales y Mecánica, UniversidadNacional Autónoma de México, Ciudad Universitaria, 04510 Mexico D.F., Mexico; and bDepartment of Geography and Geology,University of Wisconsin–Rock County, Janesville, WI 53546

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

Land surveying in ancient states is documented not only for Eurasiabut also for the Americas, amply attested by two Acolhua–Aztecpictorial manuscripts from the Valley of Mexico. The Codex Vergaraand the Códice de Santa María Asunción consist of hundreds ofdrawings of agricultural fields that uniquely record surface areasas well as perimeter measurements. A previous study of the CodexVergara examines how Acolhua–Aztecs determined field area byreconstructing their calculation procedures. Here we evaluate theaccuracy of their area values using modern mathematics. The find-ings verify the overall mathematical validity of the codex records.Three-quarters of the areas are within 5% of the maximum possi-ble value, and 85% are within 10%, which compares well with re-ported errors by Western surveyors that postdate Aztec–Acolhuawork by several centuries.

Aztec culture ∣ ethnomathematics ∣ Mexican codex ∣ Bretschneider’sFormula ∣ Surveyors’ Rule

Land surveying played an integral role in the development ofancient states as growing economic demands and political

complexity required increasingly precise records of distribution,amount, and quality of agricultural resources. In Eurasia suchrecords begin several millennia ago. In the Americas, time depthof land surveying is unknown, but two extant native-style pictorialrecords 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 theCódice de Santa María Asunción* provide unparalleled data toreconstruct Acolhua–Aztec (hereafter Acolhua) metrology andarithmetic and to evaluate the credibility of the records. The firstcomprehensive study of these codices (1) demonstrated thatAcolhua land surveyors/scribes recorded side lengths of hundredsof agricultural fields using a standard linear measure, the tlalcua-huitl (“land rod”; T, equal to 2.5 m) and shorter-than-standarddistances depicted by hearts, arrows, and hands (metrologicalmonads, which are simple, indivisible units). Unexpectedly, it wasalso demonstrated that surveyors reported field areas in squaretlalcuahuitl (T2), which were depicted pictographically by aspatially distinctive form of numerical notation (Fig. 1).

A follow-up study of 367 quadrilateral fields in the CodexVergara (hereafter the Vergara) reconstructed both Acolhuasurvey metrology and area algorithms from an emic perspective(that of cultural insiders) (2, 3). The study validated the pre-viously known Acolhua standard linear measure and amassedquantitative evidence establishing both the metric values ofAcolhua monads and their use as fractions in area computation.Using Acolhua congruence arithmetic, five recurrent algorithmswere detected that exactly reproduced 78% of the recorded areas.These results indicated that areas were indeed computed ratherthan measured by some physical means.

Although Acolhua arithmetic was functionally accurate withintheir cultural context, in this work we analyze the accuracy oftheir recorded areas from the etic (cultural outsider) perspectiveof Western mathematics (3). Such assessment would be fairlystraightforward if individual field boundaries could be identified

in situ and resurveyed. Five centuries of landscape change recentlyculminating in massive urbanization precludes this method, exceptfor one small codex locality that we have tentatively identified andreconstructed with on-the-ground observations. Overall, our areaaccuracy testing employs Acolhua data in mathematical evaluationtools specifically designed to interpret codex information.

Vergara Quadrilateral DatabaseData for all 408 codex quadrilateral fields consist of three vari-ables: (i) field side lengths (a, b, c, d) in T, (ii) depicted fieldshapes (without linear scale, correctly drawn angles, or diago-

Fig. 1. A representative household record of fields in the Codex Vergara. Ashows fields with their side lengths. Standard linear measures in tlalcuahuitl(T) are denoted by lines (1 T) and dots (20 T). Shorter-than-standard distancesinclude a heart (here appearing in all fields) and a hand (third field). B showsthe 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 20and add the units in the upper right-hand tab. For example, the area of thelast field is 26 × 20þ 4 ¼ 524 T2 (ff. 22 v, 24 v).

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

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.

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

1To whom correspondence should be addressed. E-mail: clara@mym.iimas.unam.mx.

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

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nals), and (iii) field areas in T2. For inclusion in the workingdatabase, quadrilaterals had to satisfy a primary requirement thatno side be longer than the sum of the other three, meaning thatthe field boundaries must close. This is fulfilled by all but onefield that was deleted from the data universe along with 21 othersbecause of incomplete data or illegibility, resulting in a workingdatabase of 386 quadrilaterals (Table S1 provides quadrilateralside lengths, areas, and computer-generated field shapes).

MethodsThe corpus consists of 122 right-angled quadrilaterals (90 squares and 32rectangles) and 264 irregular fields. For the former, 121 areas computed withthe geometric rule “length ×width” exactly match the recorded codex areas.The Acolhua procedure of recording areas in T2 (T × T) clearly demonstratestheir abstract, mathematical concept of area corresponding to the naturalstep of counting square units enclosed within a quadrilateral. In contrast,had these been Colonial Spanish surveys, areas would have been expressedby the amount of maize or wheat sown or harvested (4–6).

Unlike right-angled quadrilaterals, which have unique shape, testing areaaccuracy 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 maximumpossible values for its area. Therefore, as a test of survey accuracy we cansay that Acolhua areas are feasible if they fall within these values. Otherwise,they are unfeasible, by which we mean mathematically impossible on a flatsurface. Also, irregular quadrilateral field shapes are recoverable, providing asecond test for feasibility. Fields that fail the feasibility test are especiallyilluminating because these allow a quantitative measurement of error.Below we present results from two different mathematical tools used inour analysis: (i) construction of field shapes and (ii) computation of maximumpossible area.

Use of CoordinatesFirst we consider the question, Are the corpus fields mathema-tically 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 Ac? We use thecoordinate plane to answer this question and find the possibleshapes. Assume side a (ordered to be the longest) lies on the hor-izontal axis so that two of the quadrilateral’s vertices are ð0;0Þ andða;0Þ. Let ðu;vÞ and ðw;zÞ be the other two unknown vertices.

Then by construction, using the Pythagorean theorem, the fol-lowing equations must be satisfied:

u2 þ v2 ¼ b2; ðw − uÞ2 þ ðv − zÞ2 ¼ c2; [1a]

ða − wÞ2 þ z2 ¼ d2; [1b]

2Ac ¼ vwþ zða − uÞ: [1c]

Eq. 1(c) for the area Ac is found by subdividing the area intotriangles and rectangles (Fig. 2). By simplifying the equations inEq. 1, we obtain the unknowns ðw;zÞ as functions of ðu;vÞ:

w ¼ Mða − uÞ þ 2Acva2 þ b2 − 2au

; z ¼ 2Acða − uÞ −Mva2 þ b2 − 2au

; [2]

whereM ¼ 12ða2 þ c2 − b2 − d2Þ 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 ðu;vÞ as the intersection of a mathematically deter-mined line and a circle given respectively by

L:PuþQv ¼ R; C: u2 þ v2 ¼ b2; [3]

where P ¼ aða2 þ b2 − c2 − d2Þ, Q ¼ 4aAc, and R ¼ ðM − a2Þ2þ4A2

c þ a2b2 − d2ða2 þ b2Þ.(See further discussion in SI Appendix.)

Every solution ðu;vÞ of system 3 gives a value for ðw;zÞ usingEq. 2 and therefore determines the quadrilateral. For system 3,only one of three outcomes can occur: (i) The line is tangent tothe circle; that is, the quadrilateral is feasible and has only onepossible shape; (ii) the line intersects the circle twice; that is,there are two sets of possible vertices ðu1;v1Þ and ðu2;v2Þ for thegiven data producing a feasible quadrilateral with two possibleshapes; (iii) the line does not intersect the circle; that is, the quad-rilateral 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 quadrilat-eral coordinates exactly recovers the same 122 right-angled quad-rilaterals when tangency occurs [case (i)], verifying the validity ofthe equations and program. For the other 264 irregular quadri-

Fig. 2. Graphic representation in the coordinate plane of Eqs. 1. The longestquadrilateral side length a lies on the horizontal axis; clockwise follow b, c,and d. Eq. 1c is the sum of the areas of the rectangle and triangles.

Fig. 3. Graphic illustration of Eqs. 3. ð0;0Þ is the center of C, the circle ofradius side length b; L is the line PuþQv ¼ R. The position and slope ofL depend on side lengths and area. Here the area is varying and the sidelengths are fixed and ordered as in Fig. 2. As the area increases we have firstcase A, a position ofL that gives two feasible quadrilateral shapes, then caseB, the tangential position of L with only one feasible quadrilateral shape,and finally case C, the position without intersection and thus no feasiblequadrilateral. See Movie S1.

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laterals, 128 fields could feasibly have two possible quadrilateralshapes [case ii)]. The remaining 136 fields are the unfeasiblequadrilaterals [case (iii)].

To further check feasibility, another computer program wasdesigned to plot the possible shapes determined by the sets ofvertices ðu;vÞ and ðw;zÞ; these are reproduced in SI Appendix(Table S1). The majority of these plots have shapes similar toactual 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 AreaThe plane coordinate feasibility test suggests that, given their sidelengths, the surveyors incorrectly determined areas for slightlyover one-third (35.23%) of the quadrilaterals. This seeminglyhigh error rate raises the question of error magnitude. Acolhuaarea approximation apparently relied entirely on side lengths; noindication of the use of trigonometry has been detected in theVergara or other indigenous land documents (2). But precisemathematical determination of an exact area requires angulardata. For a quadrilateral with side lengths a, b, c, and d, if ϕ isthe 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

A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS − abcd cos2

�ϕþ ψ

2

�s; [4]

where S ¼ ðs-aÞðs-bÞðs-cÞðs-dÞ with s ¼ ðaþ bþ cþ dÞ∕2, thesemiperimeter. Even if angles are unknown, Eq. 4 can be usedto compute the maximum possible area of the given quadrilateral,which provides a test for feasibility.

From Eq. 4 the maximum possible value Am is achieved whenthe negative term is zero, and hence Am ¼ ffiffiffi

Sp

. Using this max-imum area, outcomes (i), (ii), and (iii) of the coordinates findingsare equivalent in the maximum area test to (I) Ac ¼ Am , (II)Ac < Am , and (III) Ac > Am.

Our program that calculates the values of Am and comparesthem with Ac, as expected, produced the 122 right-angled quad-rilaterals in case (I), 128 feasible irregular quadrilaterals in case(II), and 136 unfeasible cases in case (III). Thus, the coordinatestest and the maximum area test both reveal the same mathema-tically inaccurate unfeasible fields.

Unexpected results occurred, however, in computation of thepercentage of relative discrepancy between Am and Ac defined asPd ¼ 100 × ðAm − AcÞ∕Am. Of the 136 unfeasible quadrilaterals,nearly three-quarters (72.79%, 99 fields) exceed Am by less than10% [Table 1, based on data in SI Appendix (Table S2)]. If a10% error resulting from survey procedures is allowed, then these99 unfeasible fields fall into the feasible group. Only the remain-ing 37 cases (27.21%) greatly overestimate a feasible area. Of

these, 18 were discarded a priori in ref. 2 because the areas wereobviously too big, probably the result of recording errors in thesequence of perimeter and area drawings or faulty entry ofnumerical values.

Aside from recording errors, area overestimation is in part aproduct of Acolhua computation algorithms. After the length ×width rule, the most frequently used Acolhua algorithm is theSurveyors’ Rule (SR), where area is the product of averaged op-posite sides (2). Among the 99 unfeasible fields with jPdj < 10%,the SR was applied in nearly half of these (45) to compute theAc. In 28 of the 45 (62.22%), overestimation is nearly negligible:jPdj < 1%. Because an approximation given by the SR is alwaysequal to or larger than the Am (see further discussion in SIAppendix), some mathematically unfeasible cases necessarilyresult from its application. Nevertheless, in many cultures andsettings, the SR commonly is employed to approximate area andmathematically is a very good approximation of the Am whenangles are not overly acute. Acolhua surveyors, however, may nothave been aware of this deficiency.

For the 250 feasible fields (Ac < Am), percentage differencesfrom the maximum possible area (which are not errors in thissituation) vary from 0% for the right-angled quadrilaterals to5% for 82.8% of all feasible quadrilaterals, whereas 92.5% fallwithin 10% discrepancy. Only areas of 19 quadrilaterals differby more than 10% from the Am. The surprisingly large numberof quadrilaterals with similar Ac and Am values indicates thatsurveyors chose algorithms to approximate the largest possibleland area within given field boundaries. This observation beliesa notion that to lessen tax levies native surveyors might haveintentionally produced inaccurate surveys by systematically un-derestimating linear measurements and areas.

Measurements on the GroundA preferred method to test Acolhua area accuracy would be toresurvey the codex fields on the ground to compare with codexdata. Although known to have been located in the village ofTepetlaoztoc (8) some 20 miles northeast of Mexico City and nowundergoing rapid urbanization, place names of the five Vergaracommunities no longer are recognized by the population. Never-theless, colonial period texts (9, 10) provide descriptions by whichto equate the Vergara hamlet of Topotitla [locality 2 in SIAppendix (Table S3)] with a modern land tract called El Topotein the barrio (ward) of Asunción. Landscape change does notallow identification of individual properties, but for accuracyassessment the measured area of El Topote may be comparedwith the sum of Topotitla recorded areas.

Modern El Topote is a triangular-shaped, gently sloping tractof agricultural land bounded on the north by a centuries-old stonewall, on the east by a ravine, and on the west by a road located ashort distance from a second ravine, which, in the sixteenthcentury and today, is known as Tzila (Fig. 4). To determine thetract area, we took 18 boundary Global Positioning System coor-dinates ground-truthed with a Google Earth image and plotted42 points on an air photo†. To calculate the area we appliedthe Surveyor’s Formula to a Lambert plane projection of thesurface‡ (11, 12). Our resultant approximation of El Topote is12.41 ha (124;071.52 m2) (13). An estimated error for our areaapproximation of El Topote is unavailable without a professionalsurvey. Error uncertainties arise from the lack of tract boundarymarkers except for the east-west stone wall. The boundary ravineon the east has clearly backwasted into the level surface of ElTopote above it; to compensate for erosion we took coordinates

Table 1. Percentage discrepancy (Pd) between maximum area(Am) and codex area (Ac) for unfeasible and feasible fields

Pd for unfeasiblefields, %

No. offields, n

Pd for feasiblefields, %

No. offields, n

−1 to 0 35 0 122−3 to −1 30 0 to +5 85−3 to −5 16 +5 to +10 24−5 to −10 18 above +10 19−20 to −10 16− 50 to −20 11−100 to −50 7below −100 3Total 136 Total 250

Unfeasible fields (35% of the corpus) have registered areas above Am

and a negative % sign. Feasible fields (65% of the corpus) have registeredareas below Am and a positive % sign.

†Boundary coordinates of El Topote listed in Table S3 were taken with a nonprofessionalreceiver 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 areaof a polygon with known coordinate vertices.

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from the ravine center. Also, it is not entirely clear that the roadon the west exactly follows the sixteenth-century boundary.

For Topotitla, Acolhua surveyors recorded 38 fields: 24 quad-rilaterals and 14 polygons with more than four sides. Area dataare complete except for three quadrilaterals and one six-sidedfield [see SI Appendix (Table S4)]. We estimated quadrilateralareas by theAm computed from recorded side lengths. A differentmethod was required for computing the Am for the six-sided field.The codex drawings of irregular polygons clearly reflect additionsto, or subtractions from, a quadrilateral form, to accommodateneighboring properties, small structures, and other nonagricul-tural features (Fig. 1). Mathematically, a polygon inscribed ina circle has the maximum area among all the possible configura-tions for the given sides (13). A program written to find the en-closing circles of Topotitla polygons allowed determination of thevertices’ coordinates, and thereby calculation of maximum areaswith the Surveyor’s Formula. This method necessarily introducesarea overestimation because all vertices lie on the enclosing circlethus ignoring indentations. The resulting field shapes are math-ematical abstractions rather than typical field forms.

With the four substitutions of maximum area for recordedarea, the adjusted sum of Topotitla recorded field areas is takento be 13.5 ha (134;997 m2), to which we add 580 m2 for dwellingareas of nine households listed in the Vergara population census§.Adjusted agricultural land and house plots total 135;577 m2,whereas our measurement of El Topote is 124;071.52 m2, a dif-ference of about one hectare (11;505.48 m2). The adjusted Acol-hua area is 9.15% greater than that of the ground measurementof El Topote and thus by definition is unfeasible. Notably, thisfinding fits the pattern of a 10% discrepancy margin with respectto the Am exhibited by a large majority of the unfeasible quad-rilaterals.

Summary, Discussion, and ConclusionsOf the 386 Vergara quadrilateral fields, 65% are feasible; that is,quadrilaterals mathematically exist with the Acolhua recordedside lengths and areas. Although feasibility does not prove suchquadrilaterals existed with these areas, 92% of the feasible fieldsdiffer from the maximum possible area by less than 10%, suggest-ing that the recorded areas were good approximations of theactual ones. Our computer-generated shapes of the feasible fieldssupport this conclusion. For the unfeasible fields (Ac > Am),nearly three-quarters overestimate the maximum possible area byno more than 10%. The on-the-ground results of the El Topote/Topotitla study also pertain to this group. For the entire quadri-lateral corpus, 86% of the recorded areas fall within 10% of themaximum possible area, but more striking, nearly 75% have adiscrepancy of less than 5%. Graphic distribution of percentagediscrepancies clearly shows concentration of quadrilateral areaswithin these values with relatively few outliers, which suggestssystematic rather than random Acolhua computation procedures(Fig. 5).

Since its inception, land surveying has involved error attribu-table to technology, procedures, and blunders. One error sourcein Acolhua surveys stems from measuring instruments of rope orrod used over long side-length distances, many over 100 m, whichintroduces 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 þ∕ − 0.5 to 2.5 m on a side.

In computations, algorithms such as the SR inherently mayproduce error. Choice of algorithm also contributes, as illustratedby Id 5-21-02 (Table S1) with side lengths 26, 32, 30, and 10. TheAc of 588 T2 is given exactly by the SR, yet the Am is only538.27 T2. The surveyor perhaps did not have (or chose not touse) a more appropriate area algorithm. Calculations with orwithout monads likewise produce varying results in accuracy.Greater precision by using the most appropriate algorithm andthe arrow monad is exemplified in Id 3-11-09 with side lengths31 + arrow, 8, 33, 9, and Ac ¼ 273 T2. The codex figure approx-imates a trapezoid, but in dimensions closely resembles a rectan-gle. A logical algorithm of choice would be the SR, giving 274 T2,but instead the surveyor chose one of two variants of the TriangleRule [Ac ¼ ða × dÞ∕2þ ðb × cÞ∕2], giving an area of 273.75 T2,compared to the Am ¼ 273.02 T2, both of which when roundeddown yield the Ac. In this case, rounding down to the integervalue of T2 introduces a negligible error, but the Acolhua practiceof recording all areas in integers could have increased the round-ing error to þ∕ − 1 T2. Also, blunders in recording numbersare nearly inevitable. As expected working in base-20, some er-rors detected in the codex are multiples of 20. For example, for Id01-18-01 with side lengths 34, 20, 34, and 20, length × width yields680 T2, whereas the Ac ¼ 660 T2 that quite likely derived from amissing line in the area record.

Fig. 4. Boundary coordinates (solid circles) of modern El Topote, whichcorresponds to the Vergara locality of Topotitla. Compañía Mexicana deAerofoto, circa 1965.

Fig. 5. Distribution of the percentage discrepancy (Pd) of recorded areas(Ac) with respect to the maximum possible area (Am). The horizontal axisrepresents each of the 386 quadrilaterals, ordered by their correspondingPd values (vertical axis). The solid line indicates zero discrepancy betweenAc and Am, and dotted lines delimit a � 10% discrepancy. Negative Pd valuescorrespond to the unfeasible fields.

§Dwelling area estimates are based on archaeological data adapted from Evans (14) andParsons (15). We do not adjust the Ac for other possible nonagricultural landscapefeatures, such as walls or paths.

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Given these and other error sources, and working withoutangles, one might expect Acolhua survey records to be grosslyinaccurate by Western standards. To the contrary, we find thework of the Acolhua surveyors/scribes to be reasonably to veryaccurate. For perspective, nearly a century later, errors by Englishsurveyors were of similar magnitude, despite employment ofmore sophisticated instruments. Saxton’s 1613 map of Manning-ham, for example, registers an approximate 10% discrepancybetween land parcel areas recorded in his written survey andthose on his map. Even by 1658, area error could be as much as25% (16, 17). Land surveys in the southern American colonieswere notoriously inaccurate. For example, a meticulous analysisof early seventeenth-century land grants that were resurveyedin the eighteenth century revealed area errors ranging from 150%overestimation to 42% underestimation (18, 19). A century laterin Mexico a property area measured in 1757 was found in an 1872resurvey to have been overestimated by 8.9% (20). Survey inac-curacy still remained an issue at the close of the nineteenthcentury.

Quantification of the mathematical integrity of the CodexVergara serves to underscore accomplishments in rural landsurveying, applied mathematics, and record keeping in the Aztecworld. Spanish administrators did not avail themselves of theland-survey competency of their native subjects. Instead, theyattempted to impose order upon a chaotic system imported fromPeninsular Spain. In 1536 their confusion was denounced by thefirst Viceroy of New Spain: “in this city [Mexico City] there existsno measure for measuring land” (5). Although Acolhua surveyingsuccumbed in the Conquest aftermath, the Codex Vergara pro-vides testimony of their thoroughly modern concept of area, thefunctionality of their geometry for area determination, and themathematical accuracy of their work when assessed by Westernstandards.

ACKNOWLEDGMENTS. We gratefully acknowledge the Department ofMathematics and Mechanics of the Institute for Applied Mathematics andSystems 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.

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