On 4/4/08 the journal "Science" published:
The following points may be pertinent to discuss aspects of the Aztec Surveying article:
1. Cortes, nor any Conquistadore, formally used a formal base 10 decimal system prior to 1521. Base 10 decimals had not been invented yet! Base 10 decimals replaced Egyptian fractions around 1585 AD when the decimal system was formally approved by the Paris Academy. So I have little doubt that Spanish officials did not significantly use algorithms or decimals in ways that Aztecs or any Mesoamerican would have learned by 1521, the latest date possible connected to the Aztec 'surveying' documents, as Luis apts mentions.
2. Barbara Williams, et al, misunderstood congruences used in the Dresden Codex. She reported Aztec surveying congruences, in the context of Mayan congruences, to mean proportion, related to geometry congruences, two very different subjects.
Modern residue math uses a closely related term, as Carl F. Gauss mentions in "Discussion on Arithmetic", 1801, when he published the system in Europe. Any good history of number theory textbook (like Oystein Ore's, " Number Theory and its History") sheds appropriate Mayan, and Chinese light on the theoretical congruence subject.
That is, the Chinese Remainder Theorem, per Ore, would have been found in Mesoamerican calendars, and cosmological data, and not Aztec surveying's geometrical proportions, applied by algorithms.
I suspect one reason that two Chinese news services picked up on this story in April 2008 was that Chinese culture aptly considers Mayan and Chinese planetary and lunar cycles as closely related. For example, the Chinese Remainder Theorem uses the East's circular metaphor, rather than the West's arrow of time metaphor as built into its base 10 decimal system.
3. Cortes and Conquistadores, likely had not rounded off in a decimal system, or orientated maps to North, as we do today (related to the diameter of the earth measured by major and minor axis to compute latitudes and longitudes). Columbus had used ocean navigation methods (learned from the Portuguese) to approximate latitude and longitude by reading the "Armed Guards" of Ursa Major's solar and sidereal star clocks, viewing an eclipse on Jamaica, and one other eclipse.
In other words, as many may recall, Bishop de Landa edited Columbus' log book(s), and burnt Mesoamerican codicies, destroying critical math records on both sides of the Atlantic Ocean.
It is unknown if Mayans or Aztecs included a version of latitude or longitude on their maps. But some location information had to be attached.
4. What we do know is that Aztecs, like all Mesoamericans, loved and acted upon the four-direction metaphor. Mesoamericans in 1492, the direction East meant good, south meant pretty good, west meant fairly bad, and north meant totally evil.
Aztecs governed and taxed from the north, the evil position. As I recall reading, based on the Aztec calendar, and other traditions included Mesoamericans being given four names at birth, denoting the four directions. This was done, as the story goes, so that everyone acts in the four moral modes at various times of their life. Hence, today Mexican tradition continues the Aztec (Mesoamerican) tradition, per my wife and her 12 siblings, quickly attaching nicknames in at least four ways: good, not so good, pretty bad, and very bad situations.
As many know, Cortes, learned of the Aztec use of the four-direction metaphor, and failed to create an image of governing from the Eastern direction. Cortes, and the Spanish ruling class, governed the New World from the north, literally using evil policies. Cervantes wrote the most famous book in Spanish literature, Don Quixote, on a related topic, adding the Inquisition and other themes, proposing to solve aspects of the several evil Spanish governance policies. Of course, Cervantes failed to gain respect for his New World governance suggestions, the central issue pertinent to this post.
Spain did alter its 'converting by the sword' polices during the early 1600's to a policy that 'only' enslaved native peoples. Native people were attached for life, as well as their children, grandchildren, ... (as long as natives did not speak Spanish, and did not live in a Spanish manner) to estates, or Fincas, and thereafter earning no voting rights ... a policy that continues today-- in Guatemala, and many native areas of Latin America.
That is to ask, did Aztecs include direction orientations, and well defined locations to their taxing land area based maps? None have been deduced, per Barbara Williams, et al. But, is the lack of evidence proof that Aztecs did not use one or more of the four directions as major or minor aids to their surveying projects? I look forward to investigating this aspect of Aztec Surveying documents.
5. Barbara Williams, et al, justified the use of decimal round-offs, and algorithms to better understand map integer and fractional data by saying: ".... Anecdotal evidence suggests such practices (algorithmic rounding off) are quite natural, and they are readily observable in everyday measurement procedures".
I find the use of algorithm and rounded off decimal arguments unpersuasive. A wider perspective should be taken. Raw survey map data should be read as written by the originator, without losing additional Mesoamerican data in the translation.
In summary, the above five points show that Barbara Williams, et al, had taken excellent steps in certain respects to bring to life Aztec integers and fractions used in surveying land for taxation/tribute purposes. On the downside their paper did not fairly report Mayan astronomy, Mayan congruences, modern congruences, ancient Mesoamerican arithmetic, and Conquistadore arithmetic used from 1492 to 1521.
