To refine search, see subtopics The Maya, The Aztec, The Inca, The Chalchihuites, and The Teotihuacán Empire. To expand search, see Indigenous American Mathematics and Central and South America. Laterally related topics: Indigenous Mathematics of North America and Guatemala (and Cakchiquel Indians).
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.
Ascher, Marcia. Before the conquest. Math. Mag. 65 (1992), no. 4, 211--218. SC: 01A12, MR: 93g:01006.
Discusses the Inca and the Maya. With the Inca, focuses on the quipu. Most quipus were destroyed by the Spanish, who thought them to be the work of the Devil, but some 550 remain. Discusses their basic structure. A fascinating puzzle in the article is a pair of quipus which seem to represent data in a similar yet inexplicable way. With the Maya, focuses on their calendar. Again, much has been destroyed. For example, there only four codices remain, whereas thousands were burned by the Spanish. Fortunately, many stelae still exist. These show a calendar system with a variety of cycles. These cycles to us suggest Chinese Remainder problems. Examples of cycles are the 260 day ritual almanac composed of a cycle of 13 numbers and 20 named dieties, the vague year of 365 days composed of a cycle of 20 numbers within a cycle of 18 named dieties plus 5 unnamed days, their least common multiple (the calendar round of 18,980 days), the long count of days (in effect, multiples of 360 days plus a remainder), a 9 day cycle of Lords of the night associated with gods of the underworld, a lunar cycle of 29 and 30 day months, 13 levels in the heaven, a cycle of 4 cardinal directions (associated with different colors), sometimes used in conjunction with an 819 day cycle of the rain god. The Mayans appear to have had keen astronomical knowledge. The author notes that the error between real and tabulated times of the position of Venus would be off by just two hours in 500 years. Closely related topics: The Inca, The Quipu, The Maya, The Calendar, Astronomy, and Chinese Remainder Problems.
Ascher, Marcia. The logical-numerical system of Inca quipus. Ann. Hist. Comput. 5 (1983), no. 3, 268--278. (Reviewer: M. P. Closs.) SC: 01A12 (68-03), MR: 85b:01003.
Spanish chroniclers have claimed that the messages on quipus "were as varied as ballads, peace negotiations, laws, and state history." The approximately 550 quipus that still exist show us instead a variety of remarkable ways for organizing structured data. The meaning of this data, however, has been largely lost. Numbers in some quipus show relationships that are still hard to explain. Closely related topic: The Quipu.
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, The Aztec, 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, The Aztec, and The Zuñi.
Aveni, Anthony F. Archaeoastronomy in the Maya region: a review of the past decade. Archaeoastronomy No. 3 suppl. J. Hist Astronom. 12 (1981), S1--S16. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82k:01004.
A survey of what is understood and not understood about Maya astronomy. A theme running through the article is that Maya astronomy can only be understood in its religious or cultural context, because astronomical events had to be made to occur on certain days of non-astronomical cycles. Mayan astronomy also has a different character from Old World astronomy in that it seems to be concerned mainly with events on the horizon, whereas Old World astronomy is generally concerned more with events on the ecliptic. This phenomenon is discussed in more detail in Aveni, A. F., Tropical archeoastronomy. The author also discusses alignments of sides of buildings, whose non-perpendicular sides have otherwise been sometimes claimed to have "presumably resulted from the inability of the Maya to lay out the right angles". Closely related topics: The Maya, Astronomy, and Religion.
Aveni, Anthony F.; Morandi, Steven J. and Peterson, Polly A. The Maya number of time: intervalic time reckoning in the Maya codices. I. Archaeoastronomy No. 20, suppl. J. Hist. Astronom. 26 (1995), S1--S28. (Reviewer: M. P. Closs.) SC: 01A12, MR: 97a:01004.
Some almanacs in the surviving Mayan codices have surprising irregularities. The authors explain how these almanacs may have been formed from more regular tables by a variety of factors, including astronomical, political, ritual, and numerological. Although parts of the paper may seem a bit dry, there is quite a bit that would merit further investigation arithmetically, astronomically, statistically, or archaeologically. Closely related topics: The Maya, Astronomy, and Numerology.
Bidwell, James K. Maya Arithmetic. Mathematics Teacher 74 (1967), 762--68.
A discussion of the base 20 Mayan number system. It will be especially useful to those teaching mathematics at the elementary level. It does not discuss the Mayan calendrical system in detail, which is uses a mixed base of 20 and 360. As the author points out, his versions of Maya arithmetic may not be historically accurate---The main source on the subject, Father Diego de Landa (1524--1579), burned many of the existing Mayan manuscripts because he considered them heretical. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya, Number Systems, and Education.
Closs, Michael P. Mayan Head Variant Numerals. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 78--79.
The Mayan days of the year were associated with gods, and the Mayans used representations of gods for the numbers 0 through 19. Closs shows the "head numerals", identifies the gods, and explains how to recognize them. Excerpted from . Closely related topics: The Maya and Number Systems.
Diana, Lind Mae. The Peruvian Quipu. Mathematics Teacher 60 (1967), 623--28.
An introduction to the Quipu. The author observes that the quipu was used not only in Peru but also in other areas of South America. These others have not been as well preserved as those found in dry graves in coastal Peru. Discusses Nordenskiöld's theory that the burial quipus contain numerological and astronomical secrets. Briefly discusses the unusual Incan abacus. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya, The Quipu, Numerology, Astronomy, and The Abacus.
Dwornik, Henryk. A $2\sp{n}$-number system in the arithmetic of prehistoric cultures. Organon No. 16-17 (1980/81), 199--222 (1983). (Reviewer: Garry J. Tee.) SC: 01A10, MR: 85f:01006.
The author attempts to explain use of base 12 or base 60 in otherwise primarily base 10 cultures as an attempt to reconcile a base 10 and a base 2n system. As evidence of such a base 2n system, the author discusses the use of "base" 2 worldwide in systems for measuring distance, area, volume, and weight. He also discusses how Indo-European languages show evidence of an ancient base 4 or 8 system in the words for nine, such as in the well-known example of the Latin novem for both new and nine. The numbers 4+1, 4-1, 16+1, and 16-1 are all represented neatly in base 60. The author discusses some advantages of a number system where numbers are represented by bn...b2b1b0 as in base two, except where bi=1, 0, or -1. As the author admits, all of this is highly speculative. The author also makes the interesting observation that some of the numbers used in Mayan cosmology become very symmetric when expressed in base 2 on a 3x3 board. The suggestion seems to be that base 2 computation may have been a motivating force for the Mayans. As we still have little knowledge about Mayan arithmetic, it may be awhile before we have a definitive answer. Closely related topics: Number Systems, The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Number Words, and The Maya.
Groemer, H. The symmetries of frieze ornaments in Maya architecture. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 203 (1994), 101--116 (1995). (Reviewer: J. S. Joel.) SC: 01A12 (00A69 01A07 51M20 52C20), MR: 96b:01006.
The author discusses the frieze patterns that occur in Mayan architecture, with occasional references to the frieze patterns found on Mayan pottery. All seven basic types of frieze patterns occur, though one (the one with only glide reflections) is rather rare. The author notes that many of the symmetries appear to be derived from the symmetries of the same base motif, which is merely translated; it is acknowledged that this distinction is not a mathematical one. The author also distinguishes between discrete and continuous patterns. One interesting pattern is classified as having only translations and vertical reflections, but as the author notes, the "negative space" has an upside-down version of the same ornament This particular pattern could be classified as a bichromatic strip pattern, but apparently the "negative space" symmetry is lost in some other examples of the motif. The author finds, to his surprise, that there seems to be little Toltec or Zapotek-Mixtec influence in the Mayan frieze patterns. Closely related topics: The Maya, Architecture, Frieze Patterns, and Pottery.
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: The Aztec, Number Systems, The Measurement of Area and Volume, Number Words, and The Maya.
Katz, Victor J. Essay reviews of Ethnomathematics [Brooks/Cole, Pacific Grove, CA, 1991; MR: 92c:01006] by M. Ascher and The crest of the peacock [Tauris, London, 1991; MR: 92g:01004] by G. G. Joseph. Historia Math. 19 (1992), no. 3, 310--315. SC: 01A07 (00A30), MR: 1 177 496.
Katz reviews and contrasts Marcia Ascher's book Ethnomathematics: A Multicultural View of Mathematical Ideas and George Gheverghese Joseph's book The Crest of the Peacock: Non-European Roots of Mathematics. He finds that both correct serious omissions in the literature (and in particular, in Morris Kline's Mathematical Thought from Ancient to Modern Times). Joseph focuses on the history of mathematics in the large civilizations of ancient Egypt, Babylonia, China, India, and the Islamic World. He wanted to highlight "(1) the global nature of mathematical pursuits of one kind or another; (2) the possibility of independent mathematical development within each cultural tradition; and (3) the crucial importance of diverse transmissions of mathematics across cultures, culminating in the creation of the unified discipline of modern mathematics." Katz seems disappointed only in the third thesis, "because the documentary evidence for transmission of mathematical ideas is lacking." (For example, he notes that "whether Diophantus was directly influenced by the Babylonian tradition is a subject of scholarly debate." Joseph's treatment of Indian mathematics seems to be particularly good "especially since it is difficult to find this material in other sources." The focus of Ascher's book is completely different. She looks at traditional non-literate peoples. As Katz notes, "She has no intention of claiming that the mathematics developed in the cultures she discusses had any influence on developments elsewhere. Her main goal is simply to show that mathematical ideas, even if not developed by those called mathematicians, can be found in many societies if one only knows where to look." Katz reports examples as coming from the Inuit, Navajo, Iroquois, and Incas of the Americas, the Malekula, Warlpiri, Maori and Caroline Islanders of Oceania, and the Tshokwe, Bushoong, and Kpelle of Africa. This very useful review concludes by highly recommending both books. Closely related topics: Ancient Egypt, Sumerians and Babylonians, China, India, The Islamic World, The Inuit, The Navajo, The Iroquois, The Inca, The Malekula of Vanuatu, The Warlpiri, The Maori, The Caroline Islands, TheTshokwe, The Bushoong, and The Kpelle of Guinea.
Kudlek, Manfred. Calendar systems. Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 395--428. (Reviewer: J. S. Joel.) SC: 01A99 (00A69), MR: 92j:01079.
A rare and unusually wide ranging look at calendar systems in a variety of cultures. Explains some of the astronomical issues involved. The author discusses calendars of Egypt, Babylonia, the Roman Empire, Greece (Athens), the Islamic World (especially Persia), India, China (only gives a taste, since more than 50 official calendars were used), Japan and Vietnam (their calendars were connected with China), Java, Bali, Guatamala (by the Cakchiquel Indians), revolutionary France, the Mayas, and in the Jewish tradition. Discusses the computation of the date of Easter. (The computation of Easter was of course one of the primary goals of mathematics instruction in the middle ages.) There is information on how to correlate these calendars as well (in terms of Julian dates). Closely related topics: The Calendar, Ancient Egypt, Sumerians and Babylonians, The Roman Empire, Greece, The Islamic World, India, China, Japan, Vietnam, Java, Bali, The Maya, Guatemala (and Cakchiquel Indians), France in the 1700s, The Jewish Tradition, and Religion.
Lambert, Joseph B.; Ownbey-McLaughlin, Barbara and McLaughlin, Charles D. Maya arithmetic. Amer. Sci. 68 (1980), no. 3, 249--255. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82f:01002.
As the authors explain, there is no real evidence that the Mayas used either multiplication or division. However, the authors discuss how the Mayan notation would suit itself naturally to both of these operations. The authors suggest that in the Mayan calendrical system, where multiples of 360 are used in the third place instead of multiples of 202=400, it might be easier to "multiply by converting calendrical figures to vigesimal, perform the operations, and reconvert the answer to calendrical notation." The authors don't discuss a simple way to do this conversion. Closely related topics: The Maya and Algorithms.