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# Africa - Mathematics and the Liberal Arts

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The Mathematics and the Liberal Arts pages are intendedto be a resource for student research projects and for teachersinterested in using the history of mathematics in their courses.Many pages focus on ethnomathematics and in theconnections between mathematics and other disciplines.The notes in these pages are intended as much to evoke ideasas 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 Mathematical Society. When the mathematicalreview (MR) number and reviewer are known to the author of these pages,they are given as part of the bibliographic citation. Subscribinginstitutions can access the more recent MR reviews online throughMathSciNet.

Altshiller-Court, Nathan. The Dawn of Demonstrative Geometry.Mathematics Teacher **57** (1964), 163--66.

The author argues that it seems unlikely that the Greeks could haveinvented their notion of proof so rapidly and in isolation. Instead, hesuggests that the notion of geometric proof was a secret that wasjealously guarded from all but the "inner sanctum" of the Egyptianpriesthood. (Of course, since his argument implies by its very naturethat Egyptian proofs were unlikely to have been written down, this willbe a hard argument to either prove or disprove.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

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 thecontinuous graphs traced by the Booshong and Tshokwe, who live in theAngola/Zaire/Zambia region of Africa. The Bushoong are a subgroup in theKuba chiefdom, and exchange their art for food and raw materials. Theyhave interesting ways of classifying designs, which are touched on by theauthor. The problems in continuous tracing among the Bushoong areprimarily the domain of children. Ascher discusses the tracingalgorithms used. In the Tshokwe, continuously traced graphs play animportant role in the story-telling tradition. The author gives examplesof how some diagrams are used to discuss a rite of passage and inconnection with the muyombo trees representing the villageancestors. In some cases, the notion of inside/outside is important (anaspect of the Jordan curve theorem). Ascher discusses geometriccharacteristics of the graphs (for example, many are regular of degree4), and algorithms for drawing the curves.Closely related topics:

Ascher, Marcia and Ascher, Robert. Ethnomathematics. Hist. ofSci. ** 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 hastwo sets of number words (as in the Blackfoot Indians, who are said touse different sets of numbers for the living and the dead), the grouptherefore doesn't understand the underlying identity between thedifferent words. Regarding logic, when asked the question "All Kpellemen are rice farmers. Mr Smith is not a rice farmer. Is he a Kpelleman?", one Kpelle respondent answered "If you know a person, if aquestion comes up about him you are able to answer. But if you do notknow the person, if a question comes up about him, its hard for you toanswer." The authors emphasize that a response like this doesn't show alack of ability in logical reasoning, but just differences in views intalking about people you don't know and about 'playing along' with aquestioner. The authors discuss how the Sioux viewed the circle as amore natural shape than the (western) line. Kinship systems of theAranda of Australia, and in Ambrym in the New Hebrides. How elders inAmbrym used diagrams to elucidate the kinship systems, and explicitlyexplained the patricycles of degree 2 and the matricycles of degree 3. Aninteresting question for a student might be to investigate if the Arandasystem (with six groups) is optimal in ruling out certain types ofmarriages that are too close.Closely related topics:

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 andzenith approach to the sky, as opposed to the approach with the celestialpole (now Polaris) and the ecliptic/celestial equator, which is morefamiliar to most of us. Arorae in the Gilbert Islands (Kiribati) is veryclose to the equator, and navigators used stars on the horizon instead ofcompass directions. To them, constellations were also long chains ofstars. Apparently, the people of the Caroline Islands also used a kindof star compass. In Polynesia and apparently in much of Oceania, islandswere associated with stars that have zenith appearances above them; thisis also useful in navigation. The Maori used a similar system. Variouscultures in central and south America have been particularly interestedin horizon and zenith events. These include the Maya, the Inca, and theAztec, and are discussed in detail. There was a similar interest in theChalchihuites culture, apparently influenced by astronomers of theTeotihuacn empire. Less is known about astronomy in Africa, but theMursi of Ethiopia appear to corroborate the author's thesis, as may theBambara of Sudan as well.Closely related topics:

Bernal, Martin. Response to a paper by R. Palter: "Black Athena,Afro-centrism, and the history of science" [Hist. Sci.**31** (1993), no. 93, part 3, 227--287; MR: 94i:01001].With comments by Palter. Hist. Sci. ** 32 **(1994), no. 98, part 4, 445--468. (Reviewer: Donald Cook.) SC: 01A16(01A07 01A20 01A70 01A80), MR: 96c:01005.

An important question in the history of Greek mathematics is how muchGreek mathematics was influenced by the mathematics of the Egyptians. Bernal suggests in Black Athena that the influence may be muchgreater than previously thought. Palter's review article BlackAthena, Afro-centrism, and the history of science disagreed with anumber of Bernal's points. Bernal responds here to Palter's review, Palter, Robert, Black Athena, Afro-centrism, and the history ofscience, and then Palter comments on Bernal's response. The responseand comment provide an excellent introduction to some of the issuesinvolved in the question of Egyptian influence and also to some of theissues of modern scholarship. It might be useful to have a class readand comment about this article. It is interesting that questions in thehistory of medicine play a more important role in this controversy thanone might at first expect. If the Greeks borrowed heavily from theEgyptians in medicine, it seems more reasonable that they borrowed in theother sciences as well.Closely related topics:

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 withcombinatorics may be the house-cat-mice-wheat problem of the RhindPapyrus (Problem 79), which occurs in a similar form in a problem ofFibonacci's Liber Abaci and in an English nursery rhyme. All areconcerned with successive powers of 7. (2) The first occurrence ofcombinatorics per se may be in the 64 hexagrams of the I Ching. (However, the more modern binary ordering of these is first seen in Chinain the 10th century.) A Chinese monk in the 700s may have had a rule forthe number of configurations of a board game similar to go. In Greece,one of the very few references to combinatorics is a statement byPlutarch about the number of compound statements from 10 simplepropositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates onthe subject, so all apparently had some interest in the subject. (Plutarch's statement is also discussed in a recent article in theMonthly.) Boethius apparently had a rule for the number ofcombinations of n things taken two at a time. The authordiscusses interest in combinatorics in the Hindu world, by the Jainas,Varahamihira, and Bhaskara (the latter in the Lilavati). Thework of Brahmagupta should be relevant, but is not currently available inEnglish. The Arabs seem to have adopted their combinatorics from theHindus. The author also briefly discusses some interest in combinatoricsin the Jewish mathematical tradition; two examples are Rabbi ben Ezra andLevi ben Gerson. (3) Magic squares may first occur in the lo shudiagram, which is often linked with the I Ching. The authordiscusses how the idea of magic squares may have entered the Islamicworld, was then improved, appeared in the work of Manuel Moschopoulos,and possibly through him entered the Western world. What happened inChina is less clear. As the author suggests, the the work of Yang Huisuggests that there had been a Chinese tradition of work in magicsquares, already dead 's time. For example, the squares YangHui gives are not of types found elsewhere. In addition, Yang Hui seemsunclear on the techniques for construction. It is interesting that De laLoubre learned of a simple method for constructing magic squares inSiam. The author also discusses: the possibility of a Hindu study ofmagic squares; the presumably Arab source of Western magic squaremysticism; and later developments, such as Euler's questions onorthogonal Latin squares. (4) The author discusses how questions inpartitions 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 DeVetula, which gives the number of ways you can obtain any giventotal from a throw of 3 dice. Cardano and Galileo examined the subjectin more depth. (5) Combinatorial thinking in games and puzzles. Discusses the wolf-goat-cabbage, attributed to Alcuin. [Similar puzzlesalso occur in a variety of other cultures, but are not discussed in thisarticle.] Also discusses the Josephus problem, based on aprocess similar to the childhood process of "counting-out". The Josephusproblem is named for the Jewish historian Josephus of the 1st century AD,who supposedly saved his life with a correct solution. This problemunexpectedly turned up in Japan. (6) The author discusses how "Pascal's"triangle was possibly known to Omar Khayyam in the context of takingroots. The Hindu scholar Pingala may have known a method, but the caseis more cryptic. At any rate, it was known by the time of Halayudha, whomay have lived in the 900s AD. A more clear-cut reference occurs in thework of Nasir al-Din al-Tusi in 1265. In China, the triangle appears inthe work of Chu Shih-Chieh (1303), but may have been very ancient bythen. The triangle was used by Pascal and Fermat to resolve the "problemof points". This problem had the goal of determining how to distributestakes when a game ends early. ... Excellent article.Closely related topics:

Bogoshi, Jonas; Naidoo, Kevin and Webb, John. The oldest mathematicalartefact. Math. Gaz. ** 71 ** (1987), no. 458, 294.(Reviewer: M. P. Closs.) SC: 01A10, MR: 89a:01003.

As the authors note, the oldest mathematical artifact known may be apiece of baboon fibula with 29 notches, dating from around 35,000 BC, anddiscovered in the mountains between South Africa and Swaziland. Bycomparison, the Ishango bone dates from about 9000 BC, and theCzechoslovakian wolf's bone with 57 notches dates from about 30,000 BC. Bushmen clans in Nambia apparently use similar bones for calendar stickstoday. Includes photo.Closely related topics:

Bruins, Evert M. Egyptian arithmetic. Janus ** 68** (1981), no. 1-3, 33--52. (Reviewer: Paul Ernest.) SC: 01A15,MR: 83a:01003.

Discusses the construction of the 2/n table in the Rhindpapyrus, using an extensive computer search. Fairly technical. Doesn'tgive a magical answer, but does apparently discredit some other theories.Might be a topic suitable for some independent study projects.Closely related topics:

Crowe, Donald W. The geometry of African art. III. The smoking pipes ofBegho. The geometric vein, pp. 177--189, Springer, NewYork-Berlin, 1981. (Reviewer: M. P. Closs.) SC: 01A10 (51M20), MR:84b:01004.

Introduces the strip and plane patterns. Gives a useful flowchart forrecognizing them (and some examples). Then classifies the patternsappearing in smoking pipes from the Krama quarter of Begho, in Ghana. The most common strip pattern is the one usually referred to aspmm2 (number 7 in the author's own system). The most commonplane patterns are pmm and p4m. As the author notes,both of these can be easily created as rows of pmm2 strips. Representatives of all 7 strip patterns were found, but only 7 of the 17possible plane patterns occurred. The author also considered questions onthe relative preponderance of the various strip types by four differentlevels in the dig; no noticeable differences were found.Closely related topics:

Crowe, Donald W. The geometry of African art. II. A catalog of Beninpatterns. Historia Math. ** 2 ** (1975), 253--271.(Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986b.

Discusses the strip patterns and plane patterns occurring in Benin art.All 7 strip patterns and 12 of the 17 frieze patterns occur, though aboutfive of the frieze patterns which do occur are rare: two may only occuronce, and one of these may be based on a European model. The authorcompares the Benin patterns with the Bakuba patterns. Glide reflectionsare more rare in Benin art than in Bakuba art, possibly because glidereflection symmetries may arise most naturally from weaving patterns. Benin art also tents to be more representational, Bakuba art moreabstract. The author also considers Benin patterns to be less variedthan Bakuba patterns. However, it appears that the bronzework itself isnearly unsurpassed. A catalog is given with most of the strip patternsthe author has found in Benin art, along with one example of each of the12 plan patterns that occur. The author does not discuss this, but somepatterns combine elements of different symmetries: the authors example ofa p1 symmetry would have been classified differently if eitherof its two motifs were removed. Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art.Closely related topics:

Crowe, Donald W. The geometry of African art. I. Bakuba art. J.Geometry ** 1 ** (1971), 169--182. (Reviewer: M. P.Closs.) SC: 01A15 (20H15), MR: 58 #9986a.

Discusses strip and plane patterns occurring in Bakuba art, particularlyin textiles and woodcarving. The inspiration for many of these patternsseems to be from weaving, but at least one pattern may originate in thetechnique of sewing together triangles to make bark cloth. All sevenstrip patterns occur, and 12 of the 17 possible plane patterns. Discusses the relative proportions of some of these patterns, and givesan example of each. In all but one of the strip patterns, the authorgives both cloth and carved examples (the other is given in cloth only,being rare in wood). The author includes an appealing claim about one ofthe patterns, made by an earlier researcher (too enthusiastic in the viewof the authors): "it is probably the most remarkable example of thiskind... its discovery is certainly a mathematical accomplishment of thefirst magnitude." Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art.Closely related topics:

Dilke, O. A. W. Mathematics and measurement. Reading the Past, 2.University of California Press, Berkeley, CA; British MuseumPublications, Ltd., London, 1987. 64 pp. ISBN: 0-520-06072-5.(Reviewer: Richard L. Francis.) SC: 01A05 (01A15 01A20), MR: 89f:01003.

This very interesting book discusses many aspects of mathematics in theRoman empire, Egypt, Babylonia, Greece, and sometimes other cultures. Thebook discusses systems of measurement of length, area, volume, andweight, mathematical or para-mathematical subjects such as surveying,cartography, interest rates, taxes, time keeping, games, and numerology.Also discusses number systems. Much of the discussion on number systemsmay be familiar, but here there is also a little that may be a littleless familiar, such as the use of Etruscan letters in the early Romannumerals. In a work of this scope, the author of the book is not to befaulted that there may be some disagreement with occasional facts. Thediscussions on the mathematics of the Romans are particularlyinteresting; there are few other studies touching on Roman mathematicalpractices at all.Closely related topics:

Eglash, Ron. Fractal geometry in African material culture. Symmetry:natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult.Sci. ** 6 ** (1995), no. 1, 174--177. SC: 01A13(01A07), MR: 1 371 629.

This article is very brief, but mentions several tantalizing examples offractals and recursive similarity in Africa. He gives an example offractals in the layout of the settlement of Mokoulek in Cameroon. Thereare apparently also hints of fractal architecture in ancient Egypt. Theauthor tells us that recursive scaling (infinite self-similar structures)is also seen in Ethiopian crosses, Egyptian cosmological icons, andCameroon bronzeware. The author also tells us that "specific scalingtechniques are particularly evident in Ghana, where the use of logspirals to represent self-organizing systems (biological morphogenesisand fluid turbulence is common", and that "binary recursion is used inBambara sand divination" [in Mali].Closely related topics:

Engels, Hermann. Quadrature of the circle in ancient Egypt. HistoriaMath. ** 4 ** (1977), 137--140. (Reviewer: L.Guggenbuhl.) SC: 01A15, MR: 56 #5124.

Explains the Egyptian formula for the area of a circle in terms of thepractices of Egyptian stone masons. In order to form a relief, the stonemasons covered their designs with a grid. The hypothesized constructioninvolves an error which would confirm the now commonly held view that theancient Egyptians did not properly understand the Pythagorean theorem.Closely related topics:

Evans, Brian. Number and form and content: a composer's path of inquiry.The Visual Mind, 113--120, Leonardo Book Series, MIT Press,Cambridge, Mass., 1993.

The author shows how the golden ratio occurs in music and art. Hisexamples include Mozart's Symphony in G Minor, Grant Wood'sAmerican Gothic, Piet Mondrian's Composition with Blue,and some of his own musical and visual compositions. More controversialexamples include the Great Pyramid in Egypt and Stonehenge, where theauthor shows how approximate values of both pi and the goldenratio can be found. The author mentions Luca Pacioli's statements on thegolden ratio in De Divina Proportione and discusses otheraspects of the philosophy of number and art as well.Closely related topics:

Eves, Howard. On the Practicality of the Rule of False Position.Mathematics Teacher **51** (1958), 606--8.

Eves shows how the method of false position can be simpler than our ownmethods by giving one example from the Ahmes Papyrus, three from theGreek Anthology of c. 500 AD, and two of his own. One of hisexamples is from surveying, and Eves says that it is the method asurveyor would probably use. In the other example of his own, he likensthe rule of false position to the method of similitude in geometricconstructions. Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Fauvel, John and Gerdes, Paulus. African slave and calculating prodigy:bicentenary of the death of Thomas Fuller. Historia Math.** 17 ** (1990), no. 2, 141--151. SC: 01A70 (01A10), MR:91h:01051.

Thomas Fuller, who showed remarkable ability in mental computation, wasborn in Africa and was sold as a slave when he was 14. It would beinteresting to know more about where he came from and what theeducational practices of the area he came from were. His abilities werenot isolated, as there is for example evidence of highly developedability in mental computation among the African slave traders of the era.The article is at least as much about the way Thomas Fuller'saccomplishments were discussed and used by his contemporaries as aboutFuller himself. The article includes the text of two sourcescontemporary with Fuller, one (one of the signers of theDeclaration of Independence). The authors also mention Francis Williams,who achieved some fame as a poet and a mathematician. Little is knownabout Williams' mathematics, but Gerdes does include a sample ofWilliams' verse (the sample is in Latin).Closely related topics:

Fischer, Irene K. At the dawn of geodesy. Bull.Géodésique ** 55 ** (1981), no. 2,132--142. SC: 01A10 (01A17 01A20 01A25), MR: 83g:01002.

The cultures in ancient Egypt and in Greece, China, and Babylonia all didwork in surveying, geodesy, and astronomy. However, they all haddifferent approaches to the subjects. The author explains that "Thestriking difference between the abstract, geometric approach of Greeceand the concrete, algebraic approach of Babylonia and China represent nota difference in talents but a difference in culture-bound interests." The reader should probably have some prior knowledge of the subjectmatter (and of geodesy in particular) to fully appreciate this article.Closely related topics:

Fletcher, E. N. R. The area of the curved surface of a hemisphere inancient Egypt. Math. Gaz. ** 54 ** (1970), no. 389,227--229. SC: 01A15, MR: 58 #9987.

Problem 10 of the Moscow papyrus discusses the surface area of a basketand is thought by some to compute the surface area of a hemisphere. Theauthor analyzes which units may have been used in the problem, andadvances the theory that the basket in question was, in fact,hemispherical, and was designed to hold 100 Hekat of corn. He notes thatthe units used in ancient Egypt appear to have some interestinggeometrical properties. For example, a circle with a radius of 1pes (or "foot", equal to 16 digits) was approximately equal inarea to a square with sides measuring 1 royal cubit. These are allfascinating possibilities.Closely related topics:

Gerdes, Paulus. Fivefold symmetry and (basket) weaving in variouscultures. Fivefold symmetry, 245--261, World Sci.Publishing, River Edge, NJ, 1992. SC: 52B99 (01A07), MR: 1 178 750.

Gerdes suggests that five-fold symmetries arose from efforts to solveproblems in basketweaving rather than in observations of five-foldsymmetry in natural phenomena (such as starfish). One way five-foldsymmetries can arise is by modifying the more obvious six-fold symmetries(such as those used by peasants in Mozambique) to fit a curved surface. The author reports that "these pentagonal-hexagonal baskets are, forinstance, also woven by the Ticuna and Omagua Indians (northeasternBrazil), by the Huarani Indians, by the Kha-ko in Laos, and by the Mendain India. One sees them also in China, Japan, and Indonesia." TheMalaysian sepak tackraw ball is similar to the soccer ball andis woven in the same way. The author reports that the peasants of theisland Roti (Indonesia) may have discovered a way to fold a regularpentagon as a kind of a thimble. The author shows how a similarpentagonal weaving pattern is used in weaving brooms in Mozambique. (Anear pentagram then appears inside the knot.) The author notes that asimilar method is used in Angola to hold together the bars of a cage. The author in addition discusses how hat weaving techniques can leadnaturally to three- and five-fold symmetries. The author's main exampleis with the hats of the Belu of central Timor, but he notes that relatedtechniques are used in northern Mozambique, southern Tanzania, and by theKuva of Congo. The author also shows a Chinese hat with five-foldsymmetry. Two other particularly interesting examples are "a burdenbasket ... from the Papago Indians (Arizona) which combines beautifully aglobal sevenfold symmetry with local fivefold symmetry", and the "centerof a Japanese basket, which combines global ninefold symmetry with localfivefold symmetry."Closely related topics:

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 prehistoricmathematics (e.g., the Ishango and Border Cave bones), number systems andsymbolism (including algorithms and education), games and puzzles (forexample, 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 apossibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies onlanguage and mathematical concepts. A goal of the studies mentioned isapparently to better understand mathematics learning in Africa. Somestudies focus on logic. Questions on interaction with ancient Egypt arestill largely open. A better understanding of Islamic mathematics insub-Saharan Africa is desirable as well. The author also touches onfactors connected with the slave trade; e.g., the remarkable but notperhaps entirely atypical abilities of Thomas Fuller. Includes anextensive bibliography.Closely related topics:

Gerdes, Paulus. Three alternate methods of obtaining the ancient Egyptianformula for the area of a circle. Historia Math. ** 12** (1985), no. 3, 261--268. (Reviewer: Richard L. Francis.) SC:01A15, MR: 86k:01004.

Gerdes gives three possible methods that the Egyptians could have used indiscovering their "value" of pi, which is in effect4(^{8}/_{9})^{2}, or about 3.16. All methods areempirical. One is connected with how rope can be coiled, one is with howmats can be formed using concentric rings, and one with arranging smallballs or cylinders in a circle (the Egyptians are known to have used suchobjects). In al cases, if it is desired that the size of the circle bechosen so as to obtain (in effect) a perfect square value forpi, the Egyptian value arises naturally.Closely related topics:

Gerdes, Paulus P. J. On ethnomathematical research and symmetry. Symmetryin a kaleidoscope, 2. Symmetry Cult. Sci. ** 1 **(1990), no. 2, 154--170. SC: 01A07, MR: 1 188 949.

Gerdes begins with a discussion of why symmetry is such a commonphenomenon in human culture. He notes that some symmetries which arerare in nature (e.g., rotational symmetries of order 2) are commonamongst us. Gerdes gives the example of rotational symmetry being used inthe tattoos of the Makonde of northern Mozambique. Gerdes explains howsymmetries such as the rotational symmetry of order 2 can arise naturallyin solving problems in such areas as weaving. Gerdes then turns to thegeometry of the line drawings made by the Tamil women in South India(during harvest month) and those made by the Tshokwe. These drawingshave some strong similarities, and in both cases show an interest intracing out a figure with a single continuous line. They also show astrong interest in symmetry, and Gerdes gives examples of how designswhich fail to follow the one-line cultural norm may also fail to displaythe expected symmetries, suggesting that such drawings are degradationsof more symmetric ones drawn with one line. The author advances aconstruction principle that can be used to construct both the Tamil andTshokwe patterns. (Although the author doesn't note this, it isinteresting that this principle is very similar to another principle thathas been advanced for Celtic knot friezes!) Gerdes then discusses somemathematical properties of curves made using his construction principle. He also discusses some other interesting topics in his ethnomathematicalresearch. For example, the author mentions that he has a found a newhypothesis on the origin of the Egyptian formula for the volume of atruncated pyramid, and has also found an infinite series proof for thePythagorean theorem.Closely related topics:.Also possibly relevant:

Gerdes, Paulus and Bulafo, Gildo. Sipatsi. Technology, art and geometryin Inhambane. Translated from the Portuguese andGerdes. Instituto Superior Pedagógico, EthnomathematicsResearch Project, Maputo, 1994. 102 pp. (Reviewer: J. S. Joel.) SC:01A07 (00A08 00A69 01A13 51M20), MR: 95f:01002.

The authors discuss the construction and mathematical properties of theMozambican sipatsi, which are essentially woven handbags. Theyare generally decorated with strip or frieze patterns, and in fact all 7possible types of strip patterns occur in the sipatsi from Inhambaneprovince in Mozambique. This book includes a description of theprocesses used to create the sipatsi, a catalog of the strippatterns found, and a chapter designed for people using thesipatsi to teach mathematics. The authors also give just a fewexamples of strip patterns on wooden spoons (also from Inhambaneprovince) and on vases and pots (from Maputo).Closely related topics:

Gillings, R. J. Problems 1 to 6 of the Rhind Mathematical Papyrus.Mathematics Teacher **56** (1962), 61--69.

Discusses problems 1-6 of the Rhind Mathematical Papyrus (or AhmesPapyrus), where 1, 2, 6, 7, 9, and finally 9 loaves of bread are dividedamong 10 men. The results are given in terms of unit fractions (if youinclude^{2}/_{3}as a unit fraction). Gillings givespictures of each of the divisions, and argues convincingly that thedivision of bread would generally appear to be more fair to the typical(presumably uneducated) ancient Egyptian laborer than a more moderndivision would be. This is because each laborer would get pieces of boththe same number and size, at least if you consider two^{1}/_{3}pieces as being the same number and size as one^{1}/_{3}piece. (Although Gillings doesn't discuss this,this latter problem could be resolved by replacing^{2}/_{3}with^{1}/_{2}+^{1}/_{6}. This, however, wouldincrease the number of cuts.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Gillings, R. J. The Volume of a Truncated Pyramid in Ancient EgytianPapryi. Mathematics Teacher **57** (1964), 552--55.

Gillings gives a clever way to derive the formulaV=^{1}/_{3}(a^{2}+ab+b^{2})for the volume of a truncated pyramid, using only the formula for thevolume of a complete pyramid and other methods that the Egyptians had attheir disposal. As he shows, fairly simple arguments suffice whenb=a/2,a/3,..., and also whenb=^{2}/_{3}a. Since to the Egyptians,every number could be represented as a finite sum of unit fractions, thedemonstration is now complete. Of course we (or the Greeks) would requiresomething like the method of exhaustion. (Even without it, the jump to ageneral number is a difficult step, and not trivial geometrically.) (Since in the Moscow papyrus, b=a/2, one might wonderif perhaps the Egyptians did not know the general case after all.)Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Grünbaum, Branko. The emperor's new clothes: full regalia, G-string,or nothing? With comments and Jean Pedersen. Math.Intelligencer ** 6 ** (1984), no. 4, 47--56. (Reviewer:H. S. M. Coxeter.) SC: 01A15 (01A60 05B45 20F32 52A45), MR: 86d:01004.

Grnbaum's article: The author discusses the commonmisconceptions that the Egyptians and the artists of the Alhambra hadused all 17 types of plane patterns. In fact, the Egyptians appear tohave missed the five symmetry groups which have three-fold rotations. The sources for these misconceptions are discussed as well. The authorhas done fairly extensive research on the subject, and has concluded thattwo of the four plane patterns missing from the Alhambra seem not toappear at all in Islamic art (these are pg and pgg; thetwo missing at the Alhambra but present elsewhere are p2 andp3m1). A final theme of the author's is that the language ofsymmetry groups may at times be inadequate to discuss patterns, and canalso be misleading in connection with the intentions of the artiststhemselves.The response and Jean Pedersen:The author's acknowledge Grnbaum's correction about the Egyptians. Theauthors note that the Egyptians and Moore's between them only missed onesymmetry group, p3m1. They comment briefly on Chinese andJapanese designs, and quote Schattschneider, who notes that Chinese andJapanese artwork features rotations and glide reflections much morestrongly than Islamic art does. Schattschneider also cites anillustration from a Japanese book that seems to suggest that underlyinglattices of squares, equilateral triangles, rhombuses, and parallelogramswere consciously used in developing symmetry patterns. The authorsacknowledge the limitations of group theory in discussing symmetry, butalso emphasize its usefulness.Closely related topics:

Høyrup, Jens. Sub-scientific mathematics: observations on apre-modern phenomenon. Hist. of Sci. ** 28 **(1990), no. 79, part 1, 63--87. (Reviewer: David Singmaster.) SC: 01A10(01A05 01A12 01A80), MR: 91j:01007.

Hyrup makes a distinction between scientific andsubscientific mathematics. These fields correspond somewhat topure and applied mathematics. However, by using thisnew terminology, the author hopes to avoid suggesting that"subscientific" mathematics is always derived from "scientific"mathematics in the way that "applied" mathematics is derived from "pure"mathematics. Hyrup discusses the distinction between scientific andsubscientific mathematics and also their various kinds of relationships. His examples are drawn from Greece, Egypt, India, the Islamic World (withreferences to the Silk route), and from the Carolingian Propositionesad acuendos jevenes. (The latter is traditionally associated withAlcuin.) Hyrup touches on relevant work by the mathematicians Hero,Diophantus, and al Khwarizmi. Surveying is discussed as a particularlyimportant type of subscientific mathematics.Closely related topics:

Hildebrandt, Stefan and Tromba, Anthony. The parsimonious universe. Shapeand form in the natural world. Copernicus, New York, 1996.xiv+330 pp. ISBN: 0-387-97991-3. SC: 00A05 (01A99 49Q15), MR: 97c:00001.

This book has many interesting examples of how problems in optimizationhave been important both historically and in the world around us. Forour purposes, we focus on Chapter 2, The Heritage of AncientScience. The authors start here with a survey the history of someof the mathematics and applied mathematics of the Babylonians, Egyptians,and Greeks. They consider aspects such as astronomy, burning mirrors, andthe discovery of the irrationals (they include a modulo 10 proof that thesquare root of two is irrational). Of course, this part of the book isnot intended to be authoritative; the reader should beware of commentsabout the Egyptians and the Pythagorean theorem. The book continues withdiscussions of the Ptolemaic system (which they said was once thought tohave been handed down from above) and of the heliocentric system. One ofthe more appealing parts of Chapter 2 is a discussion of the problemwhere Queen Dido of Carthage obtained the largest possible area that canbe enclosed by the hide of an ox. She supposedly cut the hide intostrips and formed it into a semicircle bounded by the sea. Elsewhere inthe book there is quite a bit of discussion on optical shortest pathproblems. There are many fine illustrations both here and elsewhere. Example from Chapter 2 include the music of the spheres as imagined byKepler, an illustration of Dido's minimization problem from the 1630s,pictures of medieval towns built with an optimization principle laDido, and a fronticepiece of a treatise on optics from the 1200s whererefraction and burning mirrors are clearly illustrated. This book can bea fine educational resource for teachers trying to motivate ideas such asminimization problems in Calculus.Closely related topics:

Jones, Phillip S. The history of mathematics---new sources and uses.Southeast Asian Bull. Math. ** 4 ** (1980), no. 1,1--5. (Reviewer: C. R. Fletcher.) SC: 01A15, MR: 83m:01002.

The author gives a few brief examples of how problems in the Ahmespapyrus could be used for pedagogical purposes.Closely related topics:

Katz, Victor J. Essay reviews of Ethnomathematics [Brooks/Cole,Pacific Grove, CA, 1991; MR: 92c:01006] by M. Ascher and The crest ofthe 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: AMulticultural View of Mathematical Ideas and George GhevergheseJoseph's book The Crest of the Peacock: Non-European Roots ofMathematics. He finds that both correct serious omissions in theliterature (and in particular, in Morris Kline's Mathematical Thoughtfrom Ancient to Modern Times). Joseph focuses on the history ofmathematics in the large civilizations of ancient Egypt, Babylonia,China, India, and the Islamic World. He wanted to highlight "(1) theglobal nature of mathematical pursuits of one kind or another; (2) thepossibility of independent mathematical development within each culturaltradition; and (3) the crucial importance of diverse transmissions ofmathematics across cultures, culminating in the creation of the unifieddiscipline of modern mathematics." Katz seems disappointed only in thethird thesis, "because the documentary evidence for transmission ofmathematical ideas is lacking." (For example, he notes that "whetherDiophantus was directly influenced by the Babylonian tradition is asubject of scholarly debate." Joseph's treatment of Indian mathematicsseems to be particularly good "especially since it is difficult to findthis material in other sources." The focus of Ascher's book iscompletely different. She looks at traditional non-literate peoples. AsKatz notes, "She has no intention of claiming that the mathematicsdeveloped in the cultures she discusses had any influence on developmentselsewhere. Her main goal is simply to show that mathematical ideas, evenif not developed by those called mathematicians, can be found in manysocieties if one only knows where to look." Katz reports examples ascoming from the Inuit, Navajo, Iroquois, and Incas of the Americas, theMalekula, Warlpiri, Maori and Caroline Islanders of Oceania, and theTshokwe, Bushoong, and Kpelle of Africa. This very useful reviewconcludes by highly recommending both books.Closely related topics:

Knorr, W. R. The geometer and the archaeoastronomers: on the prehistoricorigins of mathematics. Review of: Geometry and algebra in ancientcivilizations [Springer, Berlin, 1983; MR: 85b:01001] by B. L. vander Waerden. British J. Hist. Sci. ** 18 ** (1985),no. 59, part 2, 197--212. SC: 01A10, MR: 87k:01003.

The reviewer discusses van der Waerden's book Geometry and Algebra inAncient Civilizations. Although the reviewer clearly admires vander Waerden for his work in algebra and in the history of mathematics ingeneral, he is highly critical of the conclusions reached in van derWaerden's book. A basic theme of the book is that there is apre-Babylonian ancestor to mathematics in Babylonia, ancient Egypt,Greece, China and India; thus the book can therefore be thought of inpart as a further development of Abraham Seidenberg's theories on theritual origins of ancient mathematics. The reviewer takes issue withseveral facts cited in the book, and in addition with three assumptionsthat he sees van der Waerden using explicitly or implicitly in the book:"(1) independent discovery is so rare that it may effectively bediscounted as a working hypothesis for relating technical traditions; (2)derivative traditions are inferior to their source traditions; (3)borrowing from one tradition to another is not selective, but entails theadoption of whole bodies of technique." (The phrase "inferior to" in (2)could just as well be replaced by "degraded in".) The reviewer suggestsin addition that van der Waerden has not been sufficiently critical inaccepting claims and others about advanced mathematicsin megalithic monuments, and sees these claims as forming "the veritablelinchpin of van der Waerden's thesis". The author briefly discusses someof Thom's work in megalithic mathematics, and concludes that he finds noreal evidence of the Pythagorean theorem, the ellipse, or a standard unitof distance in neolithic times. The review concludes with the statement"I fear even more the regrettable impact on credulous nonspecialists whomay not know to distinguish between the general enterprise of scientificresearch and the reckless notions of some scientists."Closely related topics:

Kudlek, Manfred. Calendar systems. Mathematische Wissenschaften gesternund 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 varietyof cultures. Explains some of the astronomical issues involved. Theauthor discusses calendars of Egypt, Babylonia, the Roman Empire, Greece(Athens), the Islamic World (especially Persia), India, China (only givesa taste, since more than 50 official calendars were used), Japan andVietnam (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 ofEaster. (The computation of Easter was of course one of the primary goalsof mathematics instruction in the middle ages.) There is information onhow to correlate these calendars as well (in terms of Julian dates).Closely related topics:

Lumpkin, Beatrice. From Egypt to Benjamin Banneker: African origins offalse position solutions. Vita mathematica (Toronto, ON, 1992;Quebec City, PQ, 1992), 279--289, MAA Notes, 40, Math. Assoc.America, Washington, DC, 1996. SC: 01A05 (01A13), MR: 1 391 748.

Discusses the work of the Benjamin Banneker, who is perhaps the mostinteresting early American mathematician. The author gives a fineintroduction to Banneker's life; this is necessarily brief, because asthe author observes, his house burned down on the day of his funeral,destroying almost all his papers. She notes that there were hints of hisgenius starting with his building of a wood clock at the age of 22 (heused a borrowed pocket watch as a model; unfortunately, the clock wasdestroyed in the fire); he thereafter became famous for his ability tosolve and create mathematical puzzles. "People sent him puzzles from allover the colonies and later from the new republic." His work became moreserious when he was 57 and borrowed some books and astronomy instrumentsfrom a neighbor. He taught himself the mathematics he needed to becomean astronomer, and published local almanacs including things such as theplanetary positions and the times of sunrise, sunset, moonrise, moonset,eclipses, and tides. "Based on Banneker's work on his almanac, he wasappointed an astronomer on the team of surveyors that drew up the outlinefor the new nation's capital, Washington, DC. Banneker was appointedbecause he was one of the few in the country capable of doing such work.Charles Leadbetter, author of an astronomy book that Banneker studied,wrote that knowledge of astronomy in London was 'so rare, ... not one of20,000 hath attained to it.' Knowledge of astronomer", Lumpkincontinues, "was even rarer in the new United States. Banneker's work soimpressed Thomas Jefferson, then Secretary of State, that he wroteBanneker that he was sending a copy of the almanac to the Paris Academyof Sciences." Most amazing of all is that Banneker accomplished all thisas an African American who had spent most of his life thus far hardphysical labor. After this introduction, the author focuses on howBanneker and other mathematicians used the rule of false position. Shenotes, the rule of false position was used by the Egyptians in the timeof the Rhind Papyrus and in a variety of other Egyptian sources (e.g.,the Kahun and Berlin papyri), in the work of Alexandrian Greeks likeDiophantus (c. 250 AD), in the work of Islamic mathematicians such as AbuKamil (b. 850 AD), and in the work of the mathematician Leonardo of Pisa(Fibonacci) (who was also influenced by the work in Northern Africa). Theauthor then discusses some interesting false position problems fromBanneker's own work.Closely related topics:

Lumpkin, Beatrice. Note: the Egyptians and Pythagorean triples.Historia Math. ** 7 ** (1980), no. 2, 186--187. SC:01A15, MR: 81c:01004.

The author notes that some ancient Egyptian problems suggest a knowledgeof certain Pythagorean triangles. For example, in the Berlin Papyrusthere are problems where a given square is to be written as the sum oftwo squares in a given ratio. The solutions involve the fact that6^{2}+8^{2}=10^{2}and12^{2}+16^{2}=20^{2}; these facts are familiar tous from our knowledge of the (3,4,5) right triangle. She also notes thatthe Egyptian units of measurement suggest a knowledge of the Pythagoreantheorem in the special case of an isosceles right triangle. "Thedouble remen is the diagonal of a square whose side was onecubit. By changing the units of measurement from cubits todouble remens, the area of a figure would be doubled."Closely related topics:

Manansala, Paul. Sungka mathematics of the Philippines. Indian J.Hist. Sci. ** 30 ** (1995), no. 1, 13--29. (Reviewer:J. S. Joel.) SC: 01A29 (01A13), MR: 96g:01009.

The author discusses the Sungka Board, which may once have been used asa kind of abacus. The word sungka is from the Philippines, butthe author tells us that a similar board is "known over a wide area ofthe Malayo-Polynesian world from Madagascar to Polynesia, and alsothrough Southeast Asia, India, and even mainland Africa." As the authornotes, "documentation for this usage is very hard to come by". Thearithmetical algorithms that the author advances for the sungka boardhave few surprises to someone familiar with abacus systems, but thearticle has some interesting remarks about other uses of the sungka boardand about some number systems from India, the Philippines, and elsewherein Asia that used mixed number bases. The author is particularlyinterested in eight-based counting systems, and believes that the Sungkaboard is particularly relevant in this regard: "The board has two largewells at each end, with each large well having a corresponding row ofseven smaller wells. These two rows of seven are parallel and thus theboard has a total of 16 wells divided into two groups of eight." Thewells were apparently once filled with various numbers of things such ascowrie shells. In the examples given, the wells are used for powers of10. Apparently the sungka board is now used at least as much fordivination. As the author explains, "Its main purpose in modern times isto serve as a sedentary game. In the Philippines, and probablyelsewhere, the Sungka Board is also still occasionally used for populardivination, especially by elders enquiring on whether travel by youths isauspicious on a certain day, or by girls interested in finding outwhether and when they will get married."Closely related topics:

Neugebauer, O. On the orientation of pyramids. Special issue dedicated toOlaf Pedersen on his sixtieth birthday. Centaurus ** 24** (1980), 1--3. (Reviewer: H. W. Guggenheimer.) SC: 01A15, MR:81k:01004.

Neugebauer gives a theory that explains how the Egyptians could haveoriented their pyramids without using the advanced astronomical knowledgesometimes attributed to them. The theory relies on the construction ofan accurately shaped pyramidal model (for example the capstone of thefuture pyramid), and on watching the shadow of the model in the course ofthe day. The biggest question about this procedure may be the questionof how the model can be made accurately enough. Nevertheless, this theoryrepresents a great simplification over many other theories.Closely related topics:

Palter, Robert. Black Athena, Afro-centrism, and the history ofscience. Hist. Sci. ** 31 ** (1993), no. 93, part3, 227--287. (Reviewer: Donald Cook.) SC: 01A16 (01A07 01A20 01A70), MR:94i:01001.

Martin Bernal's Black Athena created a bit of a sensation whenit first came out. Robert Palter discusses aspects of Bernal's articleand also other arguments of afro-centrists. Palter particularly focuseson the question of whether Egyptian mathematics and science influencedthe Greeks. Bernal suggests that the influence may be quite large, andPalter argues that all existing evidence points to the influence beingquite small. An important area in Palter's discussions is ancientastronomy, where Palter discusses the general character of Egyptianastronomy, and argues that some claims about it have been vastlyexaggerated; much of this discussion focuses on discrediting claims made Palter then notes that Peter Tompkins, author ofSecrets of the Great Pyramid, seems to suggest that Newton wasled by Egyptian science to discover his law of gravitation. AboutTompkins, Bernal writes that "it it a tragedy that Tompkins's brilliantand scholarly book has been stripped of its scholarly apparatus". Palterwrites "It seems never to have occurred to Bernal that the absence ofscholarly apparatus in Tompkins's account of Newton has a very simpleexplanation: no scholarly evidence exists to support that account." Whendiscussing Egyptian mathematics proper, Palter focuses discusses thegeneral character, and then square roots (or a relative lack of them),the value of pi, the controversial problem in the Moscow papyruson the surface area of a basket, the Pythagorean theorem (or the relativelack of it, arguments on the special case of involving the diagonal ofthe square), and the notion (or absence of notion) of an irrationalnumber. Palter attacks claims Diop (see Civilizationor barbarism: An authentic anthropology) that Archimedes stole someof his most famous mathematics from the Egyptians. Palter then discussespyramidology, and some of the claims cited by Bernal that "one can findsuch relations as pi, phi, the 'golden number' andPythagoras' triangle from them." The final section, discusses thesimilarities and differences between Egyptian and Greek medicine. Although Mathematics is not so directly involved here, strong Egyptianinfluence in Greek medicine could argue for the plausibility of influenceof other Egyptian science on Greek science as well. A very interestingpaper. Apart from the fact that Palter's article serves as a kind ofreview of Bernal's book, it is worth reading for its discussions on thenature of Egyptian mathematics and science. Bernal responds to Palter'sarticle in Bernal, Martin, Response to a paper by R. Palter: "Black Athena,Afro-centrism, and the history of science" [Hist. Sci.31(1993), no. 93, part 3, 227--287; MR: 94i:01001].Closely related topics:

Rees, Charles S. Egyptian fractions. Math. Chronicle ** 10** (1981), no. 1-2, 13--30. (Reviewer: Bruno Poizat.) SC: 10A30(01A15), MR: 82m:10016.

This article uses the Egyptian preference for dealing with unit fractions(except in the case of 2/3) as a starting point for some interestingproblems in number theory. There are several proofs that every fractioncan be represented as a sum of unit fractions, and these vary in thenumber of fractions produced and the maximum size of the denominators(these proofs are given as Fibonacci-Sylvester, Erds (1950), Golomb(1962), Bleicher (1968, using Farey series), and Bleicher (1972, usingcontinued fractions)). He also discusses various conjectures about unitfractions. For example, Erds and Strauss conjectured that 4/ncan always be written as the sum of three or less Egyptian fractions, andSierpinski made the same conjecture for numbers of the form 5/n.The author also discusses some interesting results by R. L. Graham(1963). As an example, Graham proves some interesting theorems where thedenominators of the unit fractions are required to be squares, or to becubes, or to be square free.Closely related topics:

Ritter, James. Prime Numbers. Unesco Courier (November 1989),12--17.

The title is a bit misleading. Discusses the work of Babylonian andEgyptian scribes and how they fit into society. Although neither societyhad a word for a mathematician, the ability to do mathematics was highlyvalued. One Mesopotamian king boasted of his academic achievements bystating proudly "I am perfectly able to subtract and add, [clever in]counting and accounting", and another says "I can find the difficultreciprocals and products which are not in the tables." In Babylonia andEgypt, mathematics was taught by creating a "network of typical examplesin which a new problem can be related---by a form of interpolation---tothose already known." An edited version appears in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Robins, Gay and Shute, Charles. The Rhind mathematical papyrus. Anancient Egyptian text. With a note by T. G. H. James. British MuseumPublications, Ltd., London, 1987. 88 pp. ISBN: 0-7141-0944-4.(Reviewer: K.-B. Gundlach.) SC: 01A15, MR: 89a:01005.

This interesting book discusses some of the main features of theAhmes/Rhind Papyrus in a way that is accessible to a broad audience. Thisbook differs from the classic book on the Rind Mathematical Papyrus byChance, Manning, and Archibald (The Rhind mathematical papyrus, MAA, twovolumes, 1927 and 1929) in several respects. The present book is moretopically oriented, generally less comprehensive, and probably moresuitable for a general audience. The present authors make little attemptto discuss the source directly; Chance helps the reader go directly intothe Egyptian hieroglyphics by giving phonetic transcriptions and fairlydirect translations. The present book, on the other hand, gives avirtually complete color set of photographs for the papyrus. Thesephotographs are clearer and more comprehensive than the ones in Chance. The reader should be forewarned that some statements in the book may bemisleading; see the review by K.-B. Gundlach for more details.Closely related topics:

Robins, Gay and Shute, Charles C. D. Mathematical bases of ancientEgyptian architecture and graphic art. Historia Math. **12 ** (1985), no. 2, 107--122. (Reviewer: Jens Høyrup.) SC:01A15, MR: 87c:01002.

The authors discuss the slopes that occur in Egyptian pyramids andartwork. The discussion of Egyptian artwork is particularly interestingbecause of the Egyptian's conscious use of squared grids. The authorsfind no evidence of circles or the value of pi being used in todetermine the overall dimensions of the pyramids, and similarly with thegolden ratio. Similarly, the authors find no evidence of pi orthe golden ratio being found in slopes of lines in Egyptian artwork. Nevertheless, the authors carefully discuss such claims rather thansimply dismissing them out of hand. The authors do, however, find thatcertain "slopes" seem to have been preferred to others (as the authorsnote, the Egyptians seem to have preferred to measure slopes as run perunit rise rather than our rise per unit run). The authors buttress theirarguments about the artwork through their use of new photographs; thesecarefully avoid distortion by means of a shift lens. The article is onlymoderately technical.Closely related topics:

Schaaf, William L. Mathematics as a Cultural Heritage. ArithmeticTeacher **8** (1961), 5--9.

Briefly discusses some of the key characteristics of the mathematics ofthe Babylonians, Egyptians, Greeks, and of Medieval Europe. Thendiscusses adoption of the Hindu-Arabic numerals, the development ofcomputation, and more abstract mathematics. Reprinted in Swetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Seidenberg, A. On the volume of a sphere. Arch. Hist. Exact Sci.** 39 ** (1988), no. 2, 97--119. (Reviewer: K.-B. Gundlach.)SC: 01A20 (01A15 01A17 01A25 01A32), MR: 89j:01012.

Abraham Seidenberg argues that there is a common source for Pythagoreanand Chinese (or Chinese-like) mathematics. He suggests thatOld-Babylonian mathematics is a derivative of a more ancient mathematicshaving a much clearer geometric component (p. 104), and is "in somerespects ... is derivative of a Chinese-like mathematics" (p. 109). Vander Waerden holds a similar view on this, and tells us that themathematics of the Chiu Chang Suan Shu represents the commonsource more faithfully than the Babylonian does. Seidenberg believesthat the common source is most similar to the Sulvasutras. Hediscusses how questions of the sphere and the circle were treated by theGreeks, Chinese, Egyptians, and to a lesser extent Indians. Hediscusses the some similarities and differences in the work on the spherein Greece (Archimedes, with a very brief account of the application ofhis Method), and in Chinese (first in the Chiu Chang SuanShu, improved or perhaps Tsu Ch'ung-Chih, and thenfurther improved by the Tsu Ch'ung-Chih's son Tsu Keng-Chih). Hebelieves that the problem of the volume of a sphere goes back to thecommon source, to the first part of the second millennium B.C. orearlier. An interesting and related topic is the topic of the equality ofthe proportionality constants pi that occur in the formulas forthe area and circumference of a circle. Seidenberg examines the MoscowPapyrus, Chinese sources, and an Old-Babylonian text and finds that thisfact seemed to be recognized in all three groups. He argues that theEgyptian, Babylonian, and Chinese approaches to the volume of a truncatedpyramid may have derived from the same common source. He believe that thecommon source also used infinitesimal, Cavalieri-type, arguments as well.It is interesting as well that Heron, who as Seidenberg notes issometimes considered to be continuing the Babylonian tradition, gives theformula^{1}/_{2}(s+p)p+^{1}/_{14}(^{1}/_{2}s)^{2}for the area of a segment of a circle with chord s and height(sagita, arrow) p (with an Archimedean value of^{22}/_{7}for pi), and "that the 'ancients'took [the area as]^{1}/_{2}(s+p)p and evenconjectured that they did so because they took pi = 3." Thepaper is also interesting in that he discusses the development of some ofhis ideas from his early papers in the 60s until much later (the paperwas received soon before his death).Closely related topics:

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 whereprincipals contended against each other on a kind of see-saw (somewhatsimilar sports are of course known from medieval times). Thegrain-crusher and water-lifter are similar, and perhaps derived from, thesee-saw; the fact that one stands on these suggested to the authors thatthe contestants may have been standing on the see-saw. The authors notethat in ancient Egypt, one's heart was believed to be weighed against afeather in order to decide whether one would be able to enter theafterlife. Other parts of the body, such as hair, can be used torepresent an individual, and in other instances these may have beenweighed instead; the authors give examples of rites where hair isweighed. An interesting use of the balance in Greece is from theIliad where Zeus weighs Achilles and Hector on pans of abalance. "That of Hector sinks toward Hades and Hector falls, slain byAchilles." An even more interesting weighing ritual was once common inthe far east, where a ruler was balanced against a quantity of a precioussubstance such as gold, and gave that substance (and thereby symbolicallyhimself) to his people. The authors found many other interestingexamples in a wide variety of cultures and world religions. The authorsbelieve that only items of ritual significance were weighed at first, andthat widespread commercial use came much later. Although the authorsdon't focus greatly on this, they also briefly discuss the differentkinds of balances (and the balance-like instrument used to carry loads onthe shoulders) and the weight multiples that were used on balances.Closely related topics:

Swetz, Frank J. Seeking Relevance? Try the History of Mathematics.Mathematics Teacher **77** (1984), 54--62.

Focuses on how the history of mathematics can be used to improvemathematics education. It can not only breath new life into the subject,but also allow students to better understand mathematics as a mode ofinquiry. If students see mathematical ideas in other times [and in othercultures], they can appreciate the ideas better in our own. Swetz givesexamples from the development of algorithms for arithmetic (includingsquare roots). Ancient demonstrations of mathematical ideas, such as the"husan-thu" proof of the Pythagorean theorem from China can beconceptually more suitable for students than more synthetic modern ones.Ancient "homework problems" from Babylonia, China, and Medieval Italy canbe more interesting than the more dry and formulaic modern equivalents.(See Swetz, Was Pythagoras Chinese? for many interestingexamples from China.) Although the author doesn't discuss this, theChinese problems in surveying led to interesting questions in algebra,with fourth and higher degree equations. Swetz discusses how Descartes'idea of a coordinate grid was earlier used by Renaissance artists,ancient Egyptian tomb painters, and various cartographers. Reprinted inSwetz, Frank J., From Five Fingers to Infinity.Closely related topics:

Zaslavsky, Claudia. Africa counts. Number and pattern in African culture.Prindle, Weber & Schmidt, Inc., Boston, Mass., 1973. x+328pp. SC: 01A10, MR: 58 #20993.

This book is an excellent introduction to the mathematics of (primarilysub-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 andPattern in African Culture ..., Claudia Zaslavsky presented anoverview of the available literature on mathematics in the history ofsub-Saharan Africa. She discussed written, spoken, and gesture counting,number symbolism, concepts of time, numbers and money, weights andmeasures, record-keeping (sticks and strings), mathematical games, magicsquares, graphs, and geometric forms, while Donald Crowe contributed achapter on geometric symmetries in African art." Regarding geometricsymmetries, it is primarily the frieze patterns and plane patterns thatare discussed; there is surely more work to be done on the bichromaticfrieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensivebibliography. Another useful resource is the newsletter distributed bythe African Mathematical Union's Commission on the History of Mathematicsin Africa (AMUCHMA).Closely related topics: