To expand search, see Indigenous Mathematics of Central and South America. Laterally related topics: The Maya, The Inca, The Chalchihuites, and The Teotihuacán Empire.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Aveni, A. F. Tropical archeoastronomy. Science 213 (1981), no. 4504, 161--171. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01006.
Cultures in the tropics appear in general to have adopted a horizon and zenith approach to the sky, as opposed to the approach with the celestial pole (now Polaris) and the ecliptic/celestial equator, which is more familiar to most of us. Arorae in the Gilbert Islands (Kiribati) is very close to the equator, and navigators used stars on the horizon instead of compass directions. To them, constellations were also long chains of stars. Apparently, the people of the Caroline Islands also used a kind of star compass. In Polynesia and apparently in much of Oceania, islands were associated with stars that have zenith appearances above them; this is also useful in navigation. The Maori used a similar system. Various cultures in central and south America have been particularly interested in horizon and zenith events. These include the Maya, the Inca, and the Aztec, and are discussed in detail. There was a similar interest in the Chalchihuites culture, apparently influenced by astronomers of the Teotihuacán empire. Less is known about astronomy in Africa, but the Mursi of Ethiopia appear to corroborate the author's thesis, as may the Bambara of Sudan as well. Closely related topics: Astronomy, Kiribati (The Gilbert Islands), The Hawaiians, The Caroline Islands, Navigation, The Maya, The Chalchihuites, The Teotihuacán Empire, The Inca, Java, Oceania, The Mursi of Ethiopia, The Bambara of Sudan, and The Maori.
Aveni, A. and Hartung, H. The observation of the Sun at the time of passage through the zenith in Mesoamerica. Archaeoastronomy No. 3 suppl. J. Hist Astronom. 12 (1981), S51--S70. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82k:01003.
A careful analysis suggests that two near-vertical tubes in central America were used to mark the zenith sun, and possibly the June solstice. At about the time that the second tube was used, transits of the Pleiades could be observed as well. Arguing for the significance of this is the recent knowledge of the importance of the Pleiades in the Aztec calendar. In the structure containing the second tube is also a doorway making a place on the horizon. At the period in question, Capella underwent helical rise at that place on the same day as the first zenith sun. Further evidence of the importance of zenith sun watching is found in a clay model of a temple offering and in two clay figurines. The author gives an example of how observations of the sun through an aperture were also important to the Zuñi (in this case through the roof of a Zuñi chief or priest). Closely related topics: The Maya, Astronomy, and The Zuñi.
Harvey, H. R. and Williams, B. J. Aztec arithmetic: positional notation and area calculation. Science 210 (1980), no. 4469, 499--505. (Reviewer: M. P. Closs.) SC: 01A10, MR: 81k:01002.
It has long been thought that the Mayans were the only Mesoamerican people to have developed a positional number system. However, as the authors have noted, the Aztecs also had such a system (using lines and dots). The treatment of zero may be less consistent than it was with the Mayans. The authors discuss Aztec calculations of area as well. The Aztecs clearly used some sort of algorithm to compute these areas. (It's difficult to assess the calculations perfectly since areas of quadrilaterals are only determined by the lengths of the sides in the special case of triangles.) The authors discuss why the mathematics discussed in this article was unlikely to have come from the Spanish. The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked. Closely related topics: Number Systems, The Measurement of Area and Volume, Number Words, and The Maya.