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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. Graphs in cultures. II. A study in ethnomathematics. Arch. Hist. Exact Sci. 39 (1988), no. 1, 75--95. (Reviewer: M. P. Closs.) SC: 01A10, MR: 90d:01003.
Discusses the cultural background and mathematical properties of the continuous graphs traced by the Booshong and Tshokwe, who live in the Angola/Zaire/Zambia region of Africa. The Bushoong are a subgroup in the Kuba chiefdom, and exchange their art for food and raw materials. They have interesting ways of classifying designs, which are touched on by the author. The problems in continuous tracing among the Bushoong are primarily the domain of children. Ascher discusses the tracing algorithms used. In the Tshokwe, continuously traced graphs play an important role in the story-telling tradition. The author gives examples of how some diagrams are used to discuss a rite of passage and in connection with the muyombo trees representing the village ancestors. In some cases, the notion of inside/outside is important (an aspect of the Jordan curve theorem). Ascher discusses geometric characteristics of the graphs (for example, many are regular of degree 4), and algorithms for drawing the curves. Closely related topics: Continuous Tracing Problems, The Bushoong, TheTshokwe, and Storytelling Traditions.
Ascher, Marcia. Graphs in cultures: a study in ethnomathematics. Historia Math. 15 (1988), no. 3, 201--227. (Reviewer: M. P. Closs.) SC: 01A10, MR: 90d:01002.
As the author observes, the philosopher Wittgenstein pointed to the problem of tracing graphs or figures as one that everyone can recognize as mathematical. Related problems have occurred in a variety of cultures. In western Europe, problems of tracing graphs or figures have occurred in Danish folk puzzles, where they were used as an alternative to dancing. Two patterns that are traced out are said to be similar to those on an artifact from Viking times, and are said to have mystical significance; and two others are said to be useful in witchcraft. Similar problems occur in other cultures as well. The article focuses on the context of the puzzles and the methods used to solve them in New Ireland and the Republic of Vanuatu, especially on the island of Malekula. A number of designs from Vanuatu have mythic significance. There is a tradition that one must complete a certain diagram to enter the Land of the Dead; failure results in being eaten. The methods used to draw the diagrams are also very interesting. In many cases, Ascher shows how individual drawing elements are transformed by processes such as reflection and rotation and are combined in systematic ways to draw the figure. Other types of mathematical ideas from Malekula include a drum signaling system with rhythms for each clan, rank, grade of pig, and special phrases, and a six-class marriage system which the elders explained with diagrams in the sand. Closely related topics: Continuous Tracing Problems, The Malekula of Vanuatu, New Ireland, Storytelling Traditions, The Philosophy of Mathematics, and Denmark Folk Tradition.
Ascher, Marcia and Ascher, Robert. Ethnomathematics. Hist. of Sci. 24 (1986), no. 64, part 2, 125--144. (Reviewer: Jens Høyrup.) SC: 01A10 (92A20), MR: 88a:01005.
Discusses the danger of identifying non-literate mathematics with "primitive" mathematics. Warns against assuming that because a group has two sets of number words (as in the Blackfoot Indians, who are said to use different sets of numbers for the living and the dead), the group therefore doesn't understand the underlying identity between the different words. Regarding logic, when asked the question "All Kpelle men are rice farmers. Mr Smith is not a rice farmer. Is he a Kpelle man?", one Kpelle respondent answered "If you know a person, if a question comes up about him you are able to answer. But if you do not know the person, if a question comes up about him, its hard for you to answer." The authors emphasize that a response like this doesn't show a lack of ability in logical reasoning, but just differences in views in talking about people you don't know and about 'playing along' with a questioner. The authors discuss how the Sioux viewed the circle as a more natural shape than the (western) line. Kinship systems of the Aranda of Australia, and in Ambrym in the New Hebrides. How elders in Ambrym used diagrams to elucidate the kinship systems, and explicitly explained the patricycles of degree 2 and the matricycles of degree 3. An interesting question for a student might be to investigate if the Aranda system (with six groups) is optimal in ruling out certain types of marriages that are too close. Closely related topics: Ethnomathematics General, Number Words, Logic, Kinship Systems, The Aranda, Ambrym, New Hebrides, The Blackfoot Indians, The Sioux, and The Kpelle of Guinea.
Chandrasekhar, S. Shakespeare, Newton and Beethoven or patterns of creativity. Current Sci. 70 (1996), no. 9, 810--822. SC: 01A99, MR: 1 387 202.
Discusses the creative lives of Shakespeare, Newton, and Beethoven. The example of Newton contrasts with the other two, particularly in how old they were when they did their most creative work. While the best work of poets is often later in life, G. H. Hardy tells us that the best work of mathematicians is generally when they are young. Chandrasekhar gives the additional examples of the mathematicians or scientists James Clerk Maxwell, George Gabriel Stokes, and Albert Einstein. Lord Rayleigh's example is different, and gives us a possible explanation of the differences we've seen. In the words of J. J. Thomson, "There are some great men of science whose charm consists in having said the first word on a subject, in having introduced some new idea which has proved fruitful; there are others whose charm consists perhaps in having said the last word on the subject, and who have reduced the subject to logical consistency and clearness. I think by temperament Lord Rayleigh belonged to the second group." Chandrasekhar then discusses the importance of beauty to mathematics and science, and concludes with statements of scientists and poets on one or the other of the two disciplines (some comments are more favorable than others). Closely related topics: Creativity, Shakespeare, Isaac Newton (1642-1727), and Beethoven.
Cordrey, William A. Ancient Mathematics and the Development of Primitive Culture. Mathematics Teacher 32 (1939), 51--60.
Discusses number words and systems of time reckoning for a wide variety of groups. Although many readers may be familiar with the Egyptian and Babylonian number systems, there are many interesting examples from the indigenous peoples of North and South America. The reader may want to ignore statements regarding the relative levels of different cultures. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Number Systems, Number Words, The Reckoning of Time, and Indigenous American Mathematics.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
Deakin, Michael A. B. The origins of our number-words. Austral. Math. Soc. Gaz. 23 (1996), no. 2, 50--66. SC: 01A07 (00A99), MR: 97c:01003.
Discusses how our number words evolved from Proto-Indo-European, and how English also has other traces of Proto-Indo-European number concepts. In the process, discusses a number of other languages, some Indo-European and some not. The author begins with the examples of the words eight and four and their derivation from the Proto-Indo-European *oktou and *kwetwores. He then discusses how languages with a one-two-many system also often have three forms of a noun, and how the form for 2 (the dual form) left remnants in English. Discusses how 1, 2, 3, 4 were adjectives in Proto-Indo-European, and 5, 6, 7, ... were nouns; other Indo-European languages have signs of this. In the case of 2, even English does. The author advances the interesting theory that the Proto-Indo-European word for five, *penkwe derives from a word for 4. As he notes, there is some evidence of an ancient base 4 number system in the Proto-Indo-European word for 9, *newn or *newm. See, for example, the Latin word novem, meaning new. Old Japanese may have some evidence of an ancient base 4 system as well; this is particularly interesting if, as is sometimes now believed, Japanese and Proto-Indo-European derive from a common source (Nostratic). The author gives examples of languages with coexisting bases. These lend plausibility to his argument about 5. Closely related topic: Number Words.
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.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
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.
Henle, Jim. Classical mathematics. Baroque mathematics. Romantic mathematics? Mathematics jazz! Also atonal, New Age, minimalist, and punk mathematics. Amer. Math. Monthly 103 (1996), no. 1, 18--29. SC: 01A99 (00A30 00A69), MR: 1 369 148.
Music is often broken into Renaissance, Baroque, Classical, and Romantic periods. This classification is not used so consistently in art and literature, and is rarely applied to mathematics, but the author finds reasonable ways to define these eras for the other disciplines as well. He finds that the periods correspond closely in art and literature, and that they correspond closely in music and mathematics, but that the periods in the latter lag significantly behind the periods in the former. This may suggest some linking between the two fields, the exact nature of which still remains to be determined. The author makes a few good-natured guesses about relationships between mathematics and other types of music as well. Atonalism is associated with formalism, jazz with topology, and, in essence, new age with dynamical systems. A very enjoyable article, and could be a good reading assignment for students in either a History of Mathematics or a Philosophy of Mathematics course. Closely related topics: Music, Art History, Literature, and Philosophy.
Hughes, Barnabas. Hawaiian Number Systems. Mathematics Teacher 75 (1982), 253--56.
Discusses the original mixed base (4 and 10) Hawaiian system and the introduction of a strict base 10 system after the arrival of missionaries. Gives many examples of both types of number words. (One theory, due to W. D. Alexander, 1864, is that groupings by 4 became popular from the the custom of counting fish and such by taking a couple in each hand or by tying them in bundles of four.) The transition between the two number systems was apparently not entirely smooth; younger Hawaiians understood only the decimal system had difficulty with older Hawaiians, who for example used different words for forty when speaking of forty canoes than speaking of forty fish. The author also discusses the introduction of some other words into Hawaiian. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Hawaiians, Number Words, Mathematics in Language, and Number Systems.
Hughes, Barnabas B. The earliest known record of California Indian numbers. Historia Math. 1 (1974), no. 1, 79--82. SC: 01A15, MR: 57 #15836.
The author discusses a document from 1775 which is now thought to be the earliest written record of the number system of a California Indian tribe. The document includes numbers for one through 14. One interesting feature is that some of the words suggest a base 4 number base. Also interesting is the fact that some of the number words that were recorded are different from the ones recorded by Dixon and Kroeber for related Costanoan Indians in 1907. These differences between this these number words (from Angel Island) and the others (from Mission Santa Clara) may indicate the influence of other tribes. The author notes that since Angel Island is nearly a centerpoint of various waterways, this influence is not surprising. The document was written by Fr. Vincente Maria, the chaplain of a Spanish expedition, and the author of the article seems to suggest that the confessional practices of the time may have encouraged Indians to use a decimal system for numbers larger than ten. This was because sins had to be identified at confession both by kind and by number, and because the Indians were not likely to be otherwise understood. Closely related topics: Number Words, Number Systems, California Indians, and Religion.
Mamedov, Kh. S. Crystallographic patterns. Symmetry: unifying human understanding, II. Comput. Math. Appl. Part B 12 (1986), no. 3-4, 511--529. SC: 00A69 (01A99 20H15 51F15), MR: 87e:00008.
This article discusses how crystallographic patterns "and their distribution and connection with natural phenomena and subjects of pure and applied art." It is written as an essay from a personal point of view. As the author tells us "I have made no effort to restrict the style of my meditations. I have presented a flow of free and sincere statements, and have not attempted to impose on them a style which might conceal their individuality. A great advantage of such statements is that one's 'falsehoods' are merely considered to be delusions, thus somehow mollifying the anger of those strict critics who feel obliged to adhere to absolute truths." The author himself is a chemist, so it is not surprising that there is some discussion of how crystallographic patterns in art are similar to those in chemistry. However, his observations on art from his own background in a nomadic family from Azerbaijan may be at least as valuable. The author notes that M. C. Escher is often identified with the applied art of crystallographic patterns, but these ideas are common in many cultures. Crystallographic patterns involving elements such as colored symmetry "are very characteristic of ancient and medieval decorations of Siberia, Kazakhstan, Central Asia, Azerbaijan, and Asia Minor." Quite a few examples of the art in this article use Islamic khufic script, and as he notes it is common to attribute the rise of patterned art rather than representational art to religious demands. The author does not seem entirely sympathetic with this idea, writing "The the problem was 'explained with God's help.' It is evident that in such cases it is much easier for the representatives of some other tradition to invent a new explaining theory than to examine the artwork using the language of its own traditions." The author gives some examples of crystallographic patterns in his own art and that of his associates. Interesting and enjoyable article. Closely related topics: Symmetry, Plane Patterns, Religion, and M. C. Escher.
Nagy, Dénes. The 2,500-year old term symmetry in science and art and its "missing link" between the antiquity and the modern age. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 18--28. SC: 01A99, MR: 1 371 622.
Documents the evolution of the word symmetry from its beginnings in ancient Greece. As the author explains, the word originally had a somewhat different meaning: symmetry = syn together + metron measure, suggesting the notion of commensurability. The word was adopted into Latin but was apparently rare in the middle ages. It's reappearance can probably be credited to the importance to the Renaissance of the De architectura libri decem of Vitruvius (1st century BC). The author discusses the Hebrew, Indian, and Chinese words for symmetry as well. At the end of the article the author enumerates some modern generalizations and uses of symmetry. For example, the author mentions "Noether's theorems connecting symmetry transformations (invariances) and conservation laws", Gell-Mann and Ne'eman's classification of elementary particles, and "Graeser's reconstruction of Bach's Kunst der Fuge". Closely related topics: Symmetry, Greece, Vitruvius, Physics, and Music.
Rav, Yehuda. On the interplay between logic and philosophy: a historical perspective. Theoria (San Sebastián) (2) 8 (1993), no. 19, 1--21. (Reviewer: Pierre Kerszberg.) SC: 03A05 (01A99 03-03), MR: 95c:03014.
The author discusses some of the connections between philosophy, logic, mathematics, and language. He focuses mainly on the West but also touches slightly on China. The reader should probably have a relatively strong background in philosophy before attempting this article. There is a long bibliography that should be useful for students making further investigations in these areas. Closely related topics: Philosophy, Logic, Language and Linguistics, and China.
Schmandt-Besserat, Denise. Oneness, Twoness, Threeness. The Sciences 27 (1987), 44--48.
Writing developed in Sumeria from attempts to represent numbers. Objects such as animals and bushels of grain were represented in a one-to-one correspondence with small clay tokens--animals with cylinders and bushels of grain with spheres. When Sumerian society became more complex, new complex tokens were invented. These represented finished items such as garments, metalworks, jars of oil, and loaves of bread. The complex tokens could have elaborate markings and a wide variety of shapes. What made things change was the habit of putting plain tokens in solid clay envelopes to record quantities in legal documents. Since breaking the envelopes symbolically "broke the deal", accountants began impressing the tokens on the surface. Later, they realized that the envelopes themselves were unnecessary. Soon, the Sumerians also copied the markings on complex tokens onto a two-dimensional surface. Writing had been invented. The symbols for small and large quantities of grain (a wedge and a circle) came to be used to represent the numbers 1 and 10 when used in conjunction with two-dimensional representations of complex tokens. Abstract numbers had been invented as well. Not long after, the pictographs came to represent sounds. This worked fairly well until the first fully phonetic alphabet was invented by the Phoenicians, perhaps 1400 years later. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Development of Writing, Sumerians and Babylonians, and Number Systems.
Seidenberg, A. km, a widespread root for ten. Arch. History Exact Sci. 16 (1976/77), no. 1, 1--16. (Reviewer: Richard L. Francis.) SC: 01A10, MR: 58 #4778.
Seidenberg studies number words in a wide variety of languages and finds some surprising similarities. He argues from these similarities that these number words, and therefore the corresponding number concepts, arose one place and spread throughout the world by a diffusion process. Here, and also in his article Seidenberg, A., The ritual origin of counting, he notes several similarities in the construction of number words in three languages that are built on the number words for one and two (Gumulgul in Australia, Bakairi in South America, and of the Bushmen in South Africa). These include the fact that in building odd numbers, the word one comes at the end, and also the fact that there is on connective. Similarities like these seem particularly natural under the diffusion hypothesis. However, the strongest evidence would come from number words themselves; for as Seidenberg notes, "If the number-vocabularies of the Gumulgul, the Bakairi, and the Bushman had been the same, and not merely the same in structure, probably everybody, or nearly everybody, would concede that the words derive from a single source." In fact, Seidenberg does find that one root, km, seems to appear in many number words world-wide. After looking at a wide variety of languages, Seidenberg concludes (p. 11) that the original word km meant "one", and thereafter began to be used for "one" larger unit, and particularly for the unit ten. He also finds at least one example of the root km meaning four; one wonders if perhaps this use might even have predated its use for ten. There have been attempts to explain the proto Indo-European root for ten in a way that conflicts with Seidenberg's theory; the notion of ten as "two hands" seems popular. Seidenberg discusses these attempts, but feels that they are rather ad hoc. Certainly from the point of a mathematician, Seidenberg's theory is very appealing. Closely related topics: Number Words, Number Systems, and Abraham Seidenberg.
Seidenberg, A. The ritual origin of counting. Arch. Hist. Exact Sci. 2 (1962b), 1-40.
It is common to argue that counting and other elementary mathematics arose spontaneously throughout the world in response to a practical, or perhaps psychological, need. Abraham Seidenberg argues instead for a diffusion theory, that counting arose only once, and then spread throughout the world. In fact, many common associations with numbers suggest such a common origin. One such association that Seidenberg is the idea that odd numbers are male and even numbers are female; this is certainly well known from the Pythagoreans, but turns out to be nearly universal. Seidenberg proposes that counting in fact originally arose in a ritual context. Seidenberg draws from a wide variety of anthropological sources for rituals and myths that hint at what this common origin might have been. He finds that counting "was frequently the central feature of a rite, and that participants in ritual were numbered." He focuses more specifically on creation rituals. He suggests that in the enaction of creation myths, men and women may have come onto the scene alternately, easily explaining the odd/male even/female association. He finds that his ideas clarify "pure 2-counting, which is the oldest stratum of counting we can detect." In pure-2 counting, there are separate words for one and two and these are used to form all other number words. He illustrates this with number words from diverse languages such as the Gumulgal of Australia, the Bakairi of South America, and the Bushmen of South Africa. He sheds additional light on his hypothesis with discussions of the possible origin of counting taboos (and connections with ritual sacrifice), of ancient one-one-correspondence "tally" systems (e.g., counting people with stones), of taxation systems, of money, and of gematria. Seidenberg also gives us some fascinating examples of counting in world religions. These include the analogy The Lord : His people = the shepherd : his sheep, the analogy The shepherd : his sheep = the moon : the stars. These two lead one to expect the moon to count the stars; and Seidenberg in fact finds evidence of this in ancient Babylonia. He argues from the equation The Lord's people = the stars of the heaven to The Lord's people = the sand upon the seashore that one would expect to find a ritual counting of sand. In fact, he finds the notion of Counter of the Sands both in Buddhism and among the Ancient Greeks. The equation The Lord = The Counter seems to be confirmed in two of the ninety-nine beautiful names of Allah, namely The Counter and the Reckoner; and there is further confirmation in Chapter's XV and XIX of the Qu'ran. This is a fascinating article, connecting mathematics with a wide variety of disciplines. Closely related topics: Myth and Ritual, Storytelling Traditions, Anthropology, General, Counting, TallySystems, Taxation, Number Words, The Pythagoreans, Gematria, Religion, The Islamic World, and Abraham Seidenberg.
Seidenberg, A. The ritual origin of the circle and square. Arch. Hist. Exact Sci. 25 (1981), no. 4, 269--327. (Reviewer: M. P. Closs.) SC: 01A10 (51-03), MR: 83h:01008.
Abraham Seidenberg advances a theory that the circle first arose in the context of the ritual enactment of a creation myth. In many cases, stars seem to play an important role in these myths. Seidenberg's research suggests that participants in these myths generally moved in a circle in imitation of the stars in the heavens. It is interesting that individuals in these societies often move in the same direction as the stars, and movement in the opposite direction is often considered unlucky. The fact that the Aztec god Tezcatlipoca is missing is right foot, forcing him to walk clockwise in a circle may be related. Seidenberg suggests that the creation myth is the origin for the dance around the may pole, which is for example observed near the summer solstice in northern Scandinavia today. Analogous rituals may play (or have played) a role in a wide variety of other cultures as well; examples are found in the Aztecs, ancient Indians, American Indians, and Greeks. (Spinning tops may have a ritual significance as well.) Special support is given to Seidenberg's these through the fact that in some cases, a pole may have been set up at an angle so as to point towards the pole star. Seidenberg notes that the moon might have motivated the circle rather than the stars, but the sun is unlikely to. His investigations tend to confirm this, and also suggest that lunar culture is older than solar culture. Seidenberg believes that the square arose from the circle, through the process of dividing a group into a dual organization, where for example members of one group marry someone in the other group and also (as he notes) play complementary roles in ritual. If a society divides a second time, one can think of it dividing the tribal circle into four parts. He finds some evidence of this as well. The four parts naturally define a square. His theory therefore implies that the circle arose first and that the square arose as a dual form of the circle; there is some other evidence (e.g., architectural) that may tend to confirm this. Seidenberg mentions several interesting dualities involving the circle and the square. The Altar of Heaven in Peking, for example, exhibits the equations Heaven : Earth = circle : square = three : two = South : North = White : Yellow. In Sinhalese art he finds the equation circle : square = standing : sitting. In the Omaha tribe he finds the equations that Sky : Earth = superior : inferior = one : two. He also notes the equations Heaven : Earth = Male : Female and Male : Female = one : two. The former is well known, and the latter is extensively discussed in Seidenberg, A., The ritual origin of counting The ancient Egyptians appear to be an exception as they associated the square with the earth and the circle with the sky. A fascinating paper. Closely related topics: Myth and Ritual, Religion, Anthropology, General, The Circle, Kinship Systems, The Square, and Abraham Seidenberg.
Seidenberg, A. and Casey, J. The ritual origin of the balance. Arch. Hist. Exact Sci. 23 (1980/81), no. 3, 179--226. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01008.
The author's trace the beginnings of the balance back to a rituals where principals contended against each other on a kind of see-saw (somewhat similar sports are of course known from medieval times). The grain-crusher and water-lifter are similar, and perhaps derived from, the see-saw; the fact that one stands on these suggested to the authors that the contestants may have been standing on the see-saw. The authors note that in ancient Egypt, one's heart was believed to be weighed against a feather in order to decide whether one would be able to enter the afterlife. Other parts of the body, such as hair, can be used to represent an individual, and in other instances these may have been weighed instead; the authors give examples of rites where hair is weighed. An interesting use of the balance in Greece is from the Iliad where Zeus weighs Achilles and Hector on pans of a balance. "That of Hector sinks toward Hades and Hector falls, slain by Achilles." An even more interesting weighing ritual was once common in the far east, where a ruler was balanced against a quantity of a precious substance such as gold, and gave that substance (and thereby symbolically himself) to his people. The authors found many other interesting examples in a wide variety of cultures and world religions. The authors believe that only items of ritual significance were weighed at first, and that widespread commercial use came much later. Although the authors don't focus greatly on this, they also briefly discuss the different kinds of balances (and the balance-like instrument used to carry loads on the shoulders) and the weight multiples that were used on balances. Closely related topics: The Balance and the Measurement of Weight, Myth and Ritual, Religion, Sports, Ancient Egypt, Greece, The Islamic World, and Abraham Seidenberg.
Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians North of Mexico. School Science and Mathematics 33 (1933), 363--72.
Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Basket Making, Pottery, and Tattoos.
Zaslavsky, Claudia. Africa counts. Number and pattern in African culture. Prindle, Weber & Schmidt, Inc., Boston, Mass., 1973. x+328 pp. SC: 01A10, MR: 58 #20993.
This book is an excellent introduction to the mathematics of (primarily sub-Saharan) Africa. The best tribute to its importance may be in Gerdes, Paulus, On mathematics in the history of sub-Saharan Africa. Gerdes writes "In her classical study Africa Counts: Number and Pattern in African Culture ..., Claudia Zaslavsky presented an overview of the available literature on mathematics in the history of sub-Saharan Africa. She discussed written, spoken, and gesture counting, number symbolism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric forms, while Donald Crowe contributed a chapter on geometric symmetries in African art." Regarding geometric symmetries, it is primarily the frieze patterns and plane patterns that are discussed; there is surely more work to be done on the bichromatic frieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensive bibliography. Another useful resource is the newsletter distributed by the African Mathematical Union's Commission on the History of Mathematics in Africa (AMUCHMA). Closely related topics: Sub-Saharan Africa, TallySystems, Finger Numerals, Counting, Numerology, The Reckoning of Time, Money, Measurement, Games, Continuous Tracing Problems, Architecture, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.