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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.
Artmann, Benno. The cloisters of Hauterive. Math. Intelligencer 13 (1991), no. 2, 44--49. SC: 00A69 (01A99), MR: 1 098 219.
The author discusses geometric principles behind Gothic tracery. The Gothic style developed in France about 1150, but spread widely in the next few centuries. Examples are taken from Reims, Haina, Strasbourg, and Esslingen. The geometric principles are by no means trivial; some make rather challenging exercises. The author discusses the windows of the cloisters of Hauterive in some detail. Hauterive is a Cistercian monastery near Fribourg in Switzerland, and the cloister dates from 1320-1328. The windows there are unusually geometric, and the author advances the theory that the windows amount to a kind of commentary on Book IV of Euclid's Elements. One window, however, can not be constructed with straightedge and compass: it involves the construction of a regular 9-gon. The author notes that a regular 15-gon may have originally been envisioned, but that "esthetic considerations overwhelmed mathematics." Interesting article. A number of illustrations, a few of which appear in Artmann, Benno; Swetz, Frank J., The Geometry of Gothic Church Windows. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Euclid.Modify notes on this entry Modify bibliography entry Make comment on this entry
Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.
Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, and The Circle.Modify notes on this entry Modify bibliography entry Make comment on this entry
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 and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
Biggs, N. L. The roots of combinatorics. Historia Math. 6 (1979), no. 2, 109--136. (Reviewer: J. Dieudonné.) SC: 05-03 (01A15 01A20 01A25 01A30 01A32 01A40 01A45), MR: 80h:05003.
(1) As the author explains, the most ancient problem connected with combinatorics may be the house-cat-mice-wheat problem of the Rhind Papyrus (Problem 79), which occurs in a similar form in a problem of Fibonacci's Liber Abaci and in an English nursery rhyme. All are concerned with successive powers of 7. (2) The first occurrence of combinatorics per se may be in the 64 hexagrams of the I Ching. (However, the more modern binary ordering of these is first seen in China in the 10th century.) A Chinese monk in the 700s may have had a rule for the number of configurations of a board game similar to go. In Greece, one of the very few references to combinatorics is a statement by Plutarch about the number of compound statements from 10 simple propositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates on the subject, so all apparently had some interest in the subject. (Plutarch's statement is also discussed in a recent article in the Monthly.) Boethius apparently had a rule for the number of combinations of n things taken two at a time. The author discusses interest in combinatorics in the Hindu world, by the Jainas, Varahamihira, and Bhaskara (the latter in the Lilavati). The work of Brahmagupta should be relevant, but is not currently available in English. The Arabs seem to have adopted their combinatorics from the Hindus. The author also briefly discusses some interest in combinatorics in the Jewish mathematical tradition; two examples are Rabbi ben Ezra and Levi ben Gerson. (3) Magic squares may first occur in the lo shu diagram, which is often linked with the I Ching. The author discusses how the idea of magic squares may have entered the Islamic world, was then improved, appeared in the work of Manuel Moschopoulos, and possibly through him entered the Western world. What happened in China is less clear. As the author suggests, the the work of Yang Hui suggests that there had been a Chinese tradition of work in magic squares, already dead by Yang Hui's time. For example, the squares Yang Hui gives are not of types found elsewhere. In addition, Yang Hui seems unclear on the techniques for construction. It is interesting that De la Loubčre learned of a simple method for constructing magic squares in Siam. The author also discusses: the possibility of a Hindu study of magic squares; the presumably Arab source of Western magic square mysticism; and later developments, such as Euler's questions on orthogonal Latin squares. (4) The author discusses how questions in partitions arose in gambling, such as the throwing of astrogali (huckle bones, which can land 4 ways) or dice (which can land in 6 ways). An early systematic study is in the late Medieval Latin poem De Vetula, which gives the number of ways you can obtain any given total from a throw of 3 dice. Cardano and Galileo examined the subject in more depth. (5) Combinatorial thinking in games and puzzles. Discusses the wolf-goat-cabbage, attributed to Alcuin. [Similar puzzles also occur in a variety of other cultures, but are not discussed in this article.] Also discusses the Josephus problem, based on a process similar to the childhood process of "counting-out". The Josephus problem is named for the Jewish historian Josephus of the 1st century AD, who supposedly saved his life with a correct solution. This problem unexpectedly turned up in Japan. (6) The author discusses how "Pascal's" triangle was possibly known to Omar Khayyam in the context of taking roots. The Hindu scholar Pingala may have known a method, but the case is more cryptic. At any rate, it was known by the time of Halayudha, who may have lived in the 900s AD. A more clear-cut reference occurs in the work of Nasir al-Din al-Tusi in 1265. In China, the triangle appears in the work of Chu Shih-Chieh (1303), but may have been very ancient by then. The triangle was used by Pascal and Fermat to resolve the "problem of points". This problem had the goal of determining how to distribute stakes when a game ends early. ... Excellent article. Closely related topics: Combinatorics, The Rhind/Ahmes Papyrus, Leonardo of Pisa (Fibonacci), The I Ching, Logic, Plutarch, Chrysippus, Hipparchus, Xenocrates, Boethius (Ancius Manlius Torquatus Severinus Boetius), Jainism, Varahamihira, Brahmagupta, Bhaskara, The Islamic World, The Jewish Tradition, Rabbi ben Ezra, Levi ben Gerson, Magic Squares, Manuel Moschopoulos, Yang Hui, Siam, Mathematics and Mysticism, Leonhard Euler, Gambling, De Vetula, Girolamo Cardano, Galileo Galilei, Puzzles, Alcuin, The Josephus Problem, Japan, Pascal's Triangle, Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Pingala, Halayudha, Nasir al-Din al-Tusi, Chu Shih-chieh, Blaise Pascal, and Pierre de Fermat.Modify notes on this entry Modify bibliography entry Make comment on this entry
Chorbachi, W. K. In the tower of Babel: beyond symmetry in Islamic design. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 751--789. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A30 92K99), MR: 91a:01058c.
An interesting and personal account of how the author discovered geometric manuscripts written for Islamic artisans. With this discover, the author gives a new historical and scientific basis to the study of certain kinds of Islamic art. Much work preceding the author's had focused on religious, mystical, or perceptual interpretations of the work. Many ideas were primarily hypothetical, such as the (incorrect) idea that all Islamic art derives from the circle. The author suggests that many religious and mystical interpretations of Islamic geometric art should not be regarded as being historically based. Instead, the author shows how some Islamic art is highly mathematical, showing concerns with such topics as Pythagorean triangles and the notion of similarity (he gives an example where a shape appears in three different scales, each similar shape being derived from the last by a clever process). Much of the article discusses these in the context of a cyclic quadrilateral appearing in Islamic art with sides 1, 2, 2, 71/2. The author even noted an Islamic anticipation of a shape used to produce Penrose tilings. The author suggests that symmetry groups, while useful, can not alone give a full understanding of Islamic art. Closely related topics: The Islamic World, Art, Plane Patterns, Pythagorean Triangles and Triples, Penrose Tilings, and Mathematics and Mysticism.Modify notes on this entry Modify bibliography entry Make comment on this entry
Gardner, Arthur O. The History of Mathematics as a Part of the History of Mankind. Mathematics Teacher 61 (1968), 524--26.
Briefly discusses how factors such as religion and warfare have influenced the development of mathematics. Attributes the success of Leonardo of Pisa (Fibonacci) to the unconventional ideas of his sovereign, Emperor Frederick II of the house of Hanover. Martin Luther is an example of an important theologian who supported mathematics: "If I had children, they should not only study language and history, but they should also learn singing and music, together with the whole of mathematics." Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Warfare and Leonardo of Pisa (Fibonacci).Modify notes on this entry Modify bibliography entry Make comment on this entry
Gerdes, P. Reconstruction and extension of lost symmetries: examples from the Tamil of South India. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 791--813. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058d.
Gerdes discusses the designs drawn (or formerly drawn) by Tamil women in South India during the harvest month Margali. The author shows that some of the diagrams may be degradations of earlier patterns that display more symmetry and/or are constructed according to the cultural ideal of having only one line. Gerdes also discusses drawing algorithms; many algorithms work by applying a series of simple transformation rules to a simpler motif. The function of these diagrams appears to be religious. As the author explains, "Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali." Closely related topics: The Tamil of South India, Continuous Tracing Problems, and Symmetry.Modify notes on this entry Modify bibliography entry Make comment on this entry
Hansen, David W. The Dependence of Mathematics on Reality. Mathematics Teacher 64 (1971), 715--19.
Discusses how the greatest mathematicians have been vitally concerned with the real world. Uses Archimedes, Newton, and Gauss as examples. Archimedes did so much applied work that it is hard to see how he fits Plutarch's description of considering mechanical work ignoble and inferior. The case of Newton is of course well known. An interesting example is Gauss, who used the motto "Thou, nature art my goddess;to thy laws/My services are bound" from Shakespeare's King Lear. Newton and Gauss were also very interested in religion. Philosophy was very important to Gauss. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Applied Mathematics (General), Archimedes, Isaac Newton (1642-1727), Karl Friedrich Gauss (1777-1855), and Philosophy. Also possibly relevant: Literature.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, and California Indians.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, and The Jewish Tradition.Modify notes on this entry Modify bibliography entry Make comment on this entry
MacDonnell, Joseph. Jesuit geometers. A study of fifty-six prominent Jesuit geometers during the first two centuries of Jesuit history. With a foreword by John W. Padberg. Studies in Jesuit Topics. Series IV, 11. Institute of Jesuit Sources, St. Louis, MO; Vatican Observatory Publications, Vatican City, 1989. iv+106 pp. ISBN: 0-912422-94-7. (Reviewer: J. V. Field.) SC: 01A99 (01A40 01A45 01A50 51-03), MR: 92e:01082.
One can find in this book that Jesuit mathematicians and scientists made interesting and important contributions not only in geometry, but in a number of other areas, some closely related and some not. One particularly interesting figure in the book is Anthanatius Kircher (1602-1680). He, for example, did such diverse creative things as design a water organ, observe tiny animals under a microscope that he connected with the bubonic plague, and showed the Egyptian Coptic language "was a vestige of early Egyptian. ... In fact it was because of Kircher's work that scientists knew what to look for when interpreting the Rosetta stone." Not every person discussed made discoveries that are quite so sensational, of course, but those who didn't still help give a more full view of mathematics in the periods the the author is interested in. Naturally, in a book of this length it is impossible to go into all Jesuit contributions in detail (and the reviewer was perhaps a little too harsh in his criticisms). Closely related topic: Geometry.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Language and Literature, and M. C. Escher.Modify notes on this entry Modify bibliography entry Make comment on this entry
Martzloff, Jean-Claude. Pi in the Sky. Unesco Courier (Nov., 1989), 22--28.
Very brief. Includes a bit on the influence of divination, astronomy/astrology, Confucianism, and Taoism on the development of Chinese mathematics. The emphasis on the answer rather than the proof shows a Taoist influence, "on the grounds that the fallacious arguments of the sophists showed its limits". Also a bit on how mathematics and mathematicians fit into Chinese society. Appears in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Divination, Astronomy, Astrology, Confucianism, and Taoism.Modify notes on this entry Modify bibliography entry Make comment on this entry
Mathews, Jerold. A Neolithic oral tradition for the van der Waerden/Seidenberg origin of mathematics. Arch. Hist. Exact Sci. 34 (1985), no. 3, 193--220. (Reviewer: M. Folkerts.) SC: 01A10 (01A25), MR: 88b:01005.
Abraham Seidenberg advanced a theory that mathematics arose from a common origin, and that some the mathematics was preserved by an oral tradition, and very likely a religious tradition, perhaps one like the one seen in the Indian Sulvasutras. Van der Waerden's book Geometry and Algebra in Ancient Civilizations takes a similar views, and in fact van der Waerden credits Seidenberg for making him look at the history of mathematics a new way. As Mathews notes, the Chinese Chiu Chang Suan Shu is very important in van der Waerden's work. Mathews relies heavily on this work as well to "give a small, coherent, and basic core of geometry concerning rectangles and their parts, ..., which may serve as what van der Waerden has called an 'oral tradition current in the Neolithic age.'" He states the he hoped "to give this hypothesized ancient core some credence through its relation to the Chiu Chang and its explanatory power. After giving a thorough discussion of this geometry, he then carefully analyzes the ninth chapter of the Chiu Chang in terms of this core. He is able to find a strong match, though his conclusions on one of the problems (Problem 20) are not consistent with those of some other researchers, who find in problem 20 instead suggestions of something like Horner's method. A very interesting article. Hopefully future papers will discuss how well the author's geometry agrees with the ancient geometry of other cultures. As he notes, "Until I can thoroughly test his conjecture on, say, the Babylonian corpus, I can argue for the merits of my conjecture only on such grounds as the simplicity of explanation it allows, or its congruence with received results or figures." Closely related topics: The Neolithic Era, Geometry, The Sulvasutras, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
Schrader, Dorothy V. The Arithmetic of the Medieval Universities. Mathematics Teacher 60 (1967), 264--75.
The history of the notion of the liberal arts, particularly in the middle ages. The role of arithmetic (computational and theoretical). The abacus of Gerbert. The computation of Easter. The influence of the Arabic texts. Different attitudes towards arithmetic at different times and in different places. An excellent introduction to the mathematics of the middle ages, though of course it omits much on topics such as geometry and astronomy. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Liberal Arts, Arithmetic, Number Theory, Gerbert, Pope Sylvester II, Medieval Europe, and The Islamic World.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, The Islamic World, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Anthropology, General, The Circle, Kinship Systems, The Square, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Sports, Ancient Egypt, Greece, The Islamic World, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
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