To refine search, see subtopics Number Systems, Numerology, Magic Squares, Bookkeeping, Modular Arithmetic, Algorithms, Logarithms, The Number Concept, The Abacus, Exponentials, Interpolation, Zero, Fractions, Irrationals, The Extraction of Roots, Mental Arithmetic, The Negative Numbers, and Imaginary and Complex Numbers. For more material on this topic, see subtopic The Real Number System. Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Measurement, Mathematics and Mysticism, Geometry, Discrete Mathematics, Optimization, Philosophy, Calculus, Statistics, Social Science, Logic, Computation, Probability, Applied Mathematics (General), Education, Algebra, Number Theory, Optics, Archaeology, Medicine, Creativity, Business, Fractals, and Science.
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.
Aiyar, S. Balakrishna. The Ganita-S\=ara-Sangraha of Mah\=av\=\i r\=ac\=arya. Mathematics Teacher 47 (1954), 528--33.
An overview of Mahavira's Ganita-Sara-Sangraha. The author makes the interesting observation that in Jainism, Mahavira's religion, mathematics was very popular, and was "accorded the status of one of the four anuyog\=as, which were the auxiliary sciences, the study of which helped the aspirant to the attainment of soul-liberation." Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Mahaviracarya.
Anagnostakis, Christopher and Goldstein, Bernard R. On an error in the Babylonian table of Pythagorean triples. Centaurus 18 (1973/74), 64--66. (Reviewer: E. M. Bruins.) SC: 01A15, MR: 58 #20994.
The authors explain a well-known mistake in the Babylonian tablet Plimpton 322 (column I, entry 10) as a consequence of a certain method of computation and of the neglect of a medial zero. It is a very appealing theory, and could give us some insight the way Babylonians did their mathematics. Other solutions have also been proposed. A good example of how we can learn from mistakes! Closely related topics: Sumerians and Babylonians, Pythagorean Triangles and Triples, and Algorithms.
Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
Ascher, Marcia. Before the conquest. Math. Mag. 65 (1992), no. 4, 211--218. SC: 01A12, MR: 93g:01006.
Discusses the Inca and the Maya. With the Inca, focuses on the quipu. Most quipus were destroyed by the Spanish, who thought them to be the work of the Devil, but some 550 remain. Discusses their basic structure. A fascinating puzzle in the article is a pair of quipus which seem to represent data in a similar yet inexplicable way. With the Maya, focuses on their calendar. Again, much has been destroyed. For example, there only four codices remain, whereas thousands were burned by the Spanish. Fortunately, many stelae still exist. These show a calendar system with a variety of cycles. These cycles to us suggest Chinese Remainder problems. Examples of cycles are the 260 day ritual almanac composed of a cycle of 13 numbers and 20 named dieties, the vague year of 365 days composed of a cycle of 20 numbers within a cycle of 18 named dieties plus 5 unnamed days, their least common multiple (the calendar round of 18,980 days), the long count of days (in effect, multiples of 360 days plus a remainder), a 9 day cycle of Lords of the night associated with gods of the underworld, a lunar cycle of 29 and 30 day months, 13 levels in the heaven, a cycle of 4 cardinal directions (associated with different colors), sometimes used in conjunction with an 819 day cycle of the rain god. The Mayans appear to have had keen astronomical knowledge. The author notes that the error between real and tabulated times of the position of Venus would be off by just two hours in 500 years. Closely related topics: The Inca, The Quipu, The Maya, The Calendar, Astronomy, and Chinese Remainder Problems.
Ascher, Marcia. The logical-numerical system of Inca quipus. Ann. Hist. Comput. 5 (1983), no. 3, 268--278. (Reviewer: M. P. Closs.) SC: 01A12 (68-03), MR: 85b:01003.
Spanish chroniclers have claimed that the messages on quipus "were as varied as ballads, peace negotiations, laws, and state history." The approximately 550 quipus that still exist show us instead a variety of remarkable ways for organizing structured data. The meaning of this data, however, has been largely lost. Numbers in some quipus show relationships that are still hard to explain. Closely related topic: The Quipu.
Aveni, Anthony F.; Morandi, Steven J. and Peterson, Polly A. The Maya number of time: intervalic time reckoning in the Maya codices. I. Archaeoastronomy No. 20, suppl. J. Hist. Astronom. 26 (1995), S1--S28. (Reviewer: M. P. Closs.) SC: 01A12, MR: 97a:01004.
Some almanacs in the surviving Mayan codices have surprising irregularities. The authors explain how these almanacs may have been formed from more regular tables by a variety of factors, including astronomical, political, ritual, and numerological. Although parts of the paper may seem a bit dry, there is quite a bit that would merit further investigation arithmetically, astronomically, statistically, or archaeologically. Closely related topics: The Maya, Astronomy, and Numerology.
Barit, Julian. The Lore of Number. Mathematics Teacher 61 (1968), 779--83.
Number symbolism among the Greeks, Hebrews, and in cases also Egyptians, Druids, Hindus. Discusses numbers up through 13. For further reading, suggests a book by W. Wynn Westcott called Numbers, Their Occult Powers and Mystic Virtues by the Theosophical Publishing Society, London, 1911. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Numerology.
Bidwell, James K. Maya Arithmetic. Mathematics Teacher 74 (1967), 762--68.
A discussion of the base 20 Mayan number system. It will be especially useful to those teaching mathematics at the elementary level. It does not discuss the Mayan calendrical system in detail, which is uses a mixed base of 20 and 360. As the author points out, his versions of Maya arithmetic may not be historically accurate---The main source on the subject, Father Diego de Landa (1524--1579), burned many of the existing Mayan manuscripts because he considered them heretical. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya, Number Systems, and Education.
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 Loubère 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.
Bogoshi, Jonas; Naidoo, Kevin and Webb, John. The oldest mathematical artefact. 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 a piece of baboon fibula with 29 notches, dating from around 35,000 BC, and discovered in the mountains between South Africa and Swaziland. By comparison, the Ishango bone dates from about 9000 BC, and the Czechoslovakian wolf's bone with 57 notches dates from about 30,000 BC. Bushmen clans in Nambia apparently use similar bones for calendar sticks today. Includes photo. Closely related topics: TallySystems, South Africa, The Bushmen (southern Africa), and Archaeology.
Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.
Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ptolemy (Claudius Ptolemaeus), Trigonometry, and Interpolation.
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 Rhind papyrus, using an extensive computer search. Fairly technical. Doesn't give a magical answer, but does apparently discredit some other theories. Might be a topic suitable for some independent study projects. Closely related topics: The Rhind/Ahmes Papyrus and Algorithms.
Closs, Michael P. Mayan Head Variant Numerals. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 78--79.
The Mayan days of the year were associated with gods, and the Mayans used representations of gods for the numbers 0 through 19. Closs shows the "head numerals", identifies the gods, and explains how to recognize them. Excerpted from . Closely related topics: The Maya and Number Systems.
Cordrey, William A. Ancient Mathematics and the Development of Primitive Culture. Mathematics Teacher 32 (1939), 51--60.
Discusses number words and systems of time reckoning for a wide variety of groups. Although many readers may be familiar with the Egyptian and Babylonian number systems, there are many interesting examples from the indigenous peoples of North and South America. The reader may want to ignore statements regarding the relative levels of different cultures. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Number Systems, Number Words, The Reckoning of Time, and Indigenous American Mathematics.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
Diana, Lind Mae. The Peruvian Quipu. Mathematics Teacher 60 (1967), 623--28.
An introduction to the Quipu. The author observes that the quipu was used not only in Peru but also in other areas of South America. These others have not been as well preserved as those found in dry graves in coastal Peru. Discusses Nordenskiöld's theory that the burial quipus contain numerological and astronomical secrets. Briefly discusses the unusual Incan abacus. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Maya, The Quipu, Numerology, Astronomy, and The Abacus.
Dilke, O. A. W. Mathematics and measurement. Reading the Past, 2. University of California Press, Berkeley, CA; British Museum Publications, 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 the Roman empire, Egypt, Babylonia, Greece, and sometimes other cultures. The book discusses systems of measurement of length, area, volume, and weight, 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 systems may be familiar, but here there is also a little that may be a little less familiar, such as the use of Etruscan letters in the early Roman numerals. In a work of this scope, the author of the book is not to be faulted that there may be some disagreement with occasional facts. The discussions on the mathematics of the Romans are particularly interesting; there are few other studies touching on Roman mathematical practices at all. Closely related topics: The Roman Empire, Ancient Egypt, Sumerians and Babylonians, Greece, The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Surveying, Cartography, Banking, Taxation, The Reckoning of Time, Games, Numerology, and Number Systems.
Dwornik, Henryk. A $2\sp{n}$-number system in the arithmetic of prehistoric cultures. Organon No. 16-17 (1980/81), 199--222 (1983). (Reviewer: Garry J. Tee.) SC: 01A10, MR: 85f:01006.
The author attempts to explain use of base 12 or base 60 in otherwise primarily base 10 cultures as an attempt to reconcile a base 10 and a base 2n system. As evidence of such a base 2n system, the author discusses the use of "base" 2 worldwide in systems for measuring distance, area, volume, and weight. He also discusses how Indo-European languages show evidence of an ancient base 4 or 8 system in the words for nine, such as in the well-known example of the Latin novem for both new and nine. The numbers 4+1, 4-1, 16+1, and 16-1 are all represented neatly in base 60. The author discusses some advantages of a number system where numbers are represented by bn...b2b1b0 as in base two, except where bi=1, 0, or -1. As the author admits, all of this is highly speculative. The author also makes the interesting observation that some of the numbers used in Mayan cosmology become very symmetric when expressed in base 2 on a 3x3 board. The suggestion seems to be that base 2 computation may have been a motivating force for the Mayans. As we still have little knowledge about Mayan arithmetic, it may be awhile before we have a definitive answer. Closely related topics: Number Systems, The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Number Words, and The Maya.
Ellerman, David P. The mathematics of double entry bookkeeping. Math. Mag. 58 (1985), no. 4, 226--233. (Reviewer: D. J. Struik.) SC: 90C99 (01A99 20G99), MR: 87a:90151.
The double entry bookkeeping system was first described by Luca Pacioli in 1494, though it had been developed in the 1300s. One feature of the system is that it in effect constructs the negative numbers Z from the natural numbers omega. This same construction is regularly done as well in courses in logic and set theory and may also be relevant to courses on the foundations of our number system (e.g., for those planning to teach elementary school students). Closely related topics: Bookkeeping, The Negative Numbers, Luca Pacioli, and Logic.
Emmer, Michele. Art and mathematics: the Platonic solids. The Visual Mind, 215--220, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author begins by mentioning some ancient representations of Platonic solids. These include a pair of Egyptian die from the Ptolemaic dynasty, an Etruscan dodecahedron (at least 2500 years old), two Celtic dodecahedra, and a West German dodecahedron from the 2nd century BC. The author continues with a discussion of the regular solids in Plato's Timaeus. The author notes that Dürer's Melancholia, which includes a truncated rhombohedron, is sometimes thought to show the influence of Luca Pacioli. The magic square in the painting gives some evidence for this; Dürer's engraving may be one of the earliest depictions of a magic squares in the West, but an earlier manuscript by Pacioli showed an interest in them. On the other hand, Luca Pacioli's De Divina Proportione relied heavily on, and perhaps even appropriated the work of Piero della Francesca. The book is also notable for its pictures of the regular solids, attributed to Leonardo da Vinci. Also discusses work on the regular solids due to Johannes Kepler, including Kepler's recognition of a duality and his idea of a combination of two tetrahedra called a stella octangula. The author notes that the notion of the stella octangula also appears in Pacioli's De Divina Proportione. In addition, Kepler's stellated dodecahedron occurs in mosaics in the San Macro Cathedral in Venice; this work is thought to have been done by Paolo Uccello. Regarding Uccello, the author quotes Donatello as saying to his close friend "Ah Paolo, this perspective of yours makes you neglect what we know for what we don't know. These things are no use except for marquetry." (The source is Vasari's Vita di Paolo Uccello.) The author, Michele Emmer, collaborated on the film Art and Mathematics. Closely related topics: The Regular Solids, Plato, Art, The Etruscans, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.
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, was born in Africa and was sold as a slave when he was 14. It would be interesting to know more about where he came from and what the educational practices of the area he came from were. His abilities were not isolated, as there is for example evidence of highly developed ability in mental computation among the African slave traders of the era. The article is at least as much about the way Thomas Fuller's accomplishments were discussed and used by his contemporaries as about Fuller himself. The article includes the text of two sources contemporary with Fuller, one by Benjamin Rush (one of the signers of the Declaration of Independence). The authors also mention Francis Williams, who achieved some fame as a poet and a mathematician. Little is known about Williams' mathematics, but Gerdes does include a sample of Williams' verse (the sample is in Latin). Closely related topics: Thomas Fuller (1710-1790) and Mental Arithmetic.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
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 Ahmes Papyrus), where 1, 2, 6, 7, 9, and finally 9 loaves of bread are divided among 10 men. The results are given in terms of unit fractions (if you include 2/3 as a unit fraction). Gillings gives pictures of each of the divisions, and argues convincingly that the division of bread would generally appear to be more fair to the typical (presumably uneducated) ancient Egyptian laborer than a more modern division would be. This is because each laborer would get pieces of both the 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, would increase the number of cuts.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt and Fractions.
Grattan-Guinness, I. Mozart 18, Beethoven 32: hidden shadows of integers in classical music. History of mathematics, 29--47, Academic Press, San Diego, CA, 1996. SC: 01A99 (00A69), MR: 97a:01075.
Discusses number symbolism in the works of Mozart and Beethoven. With Mozart, discusses in particular Die Zauberflöte and the last three symphonies (and particularly the Symphony in g of 1788). There is also some evidence that Mozart used gematria. Literary sources also attest to Mozart's interest in numerology. With Beethoven, focuses primarily on Piano Sonata op. 111 (no. 32), the Diabelli Variations, and the Missa Solemnis. The choice of opus numbers themselves appear to show an interest in numerology. The author suggests that some knowledge of the history and conventions of numerology would be useful before reading this article. The author's own article in the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences may be useful in this regard. The author also suggests some avenues for future research. Closely related topics: Music, Numerology, Gematria, Wolfgang Amadeus Mozart (1756-1791), and Beethoven.
Grattan-Guinness, I. Some numerological features of Beethoven's output. Ann. of Sci. 51 (1994), no. 2, 103--135. SC: 01A99 (00A69), MR: 1 278 119.
The author discusses possible occurrences of number symbolism in Beethoven's compositions. A large number of examples are used to buttress his arguments, and some prior familiarity with Beethoven's work might be useful. In some cases, numbers occur as the number of measures or notes of a them or motif, and in other cases in Beethoven's choice of opus numbers. (In contrast with the common practice of the time, Beethoven chose his opus numbers himself, and the numbers chosen could at times be seriously at variance with the order of composition.) The author's conclusions have been controversial, partly because Beethoven has often been regarded as being quite poor at arithmetic. The author discusses this objection and aspects of methodology in some detail. Closely related topics: Numerology, Music, and Beethoven.
Harvey, H. R. and Williams, B. J. Aztec arithmetic: positional notation and area calculation. Science 210 (1980), no. 4469, 499--505. (Reviewer: M. P. Closs.) SC: 01A10, MR: 81k:01002.
It has long been thought that the Mayans were the only Mesoamerican people to have developed a positional number system. However, as the authors have noted, the Aztecs also had such a system (using lines and dots). The treatment of zero may be less consistent than it was with the Mayans. The authors discuss Aztec calculations of area as well. The Aztecs clearly used some sort of algorithm to compute these areas. (It's difficult to assess the calculations perfectly since areas of quadrilaterals are only determined by the lengths of the sides in the special case of triangles.) The authors discuss why the mathematics discussed in this article was unlikely to have come from the Spanish. The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked. Closely related topics: The Aztec, Number Systems, The Measurement of Area and Volume, Number Words, and The Maya.
Hildebrandt, Stefan and Tromba, Anthony. The parsimonious universe. Shape and 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 optimization have been important both historically and in the world around us. For our purposes, we focus on Chapter 2, The Heritage of Ancient Science. The authors start here with a survey the history of some of the mathematics and applied mathematics of the Babylonians, Egyptians, and Greeks. They consider aspects such as astronomy, burning mirrors, and the discovery of the irrationals (they include a modulo 10 proof that the square root of two is irrational). Of course, this part of the book is not intended to be authoritative; the reader should beware of comments about the Egyptians and the Pythagorean theorem. The book continues with discussions of the Ptolemaic system (which they said was once thought to have been handed down from above) and of the heliocentric system. One of the more appealing parts of Chapter 2 is a discussion of the problem where Queen Dido of Carthage obtained the largest possible area that can be enclosed by the hide of an ox. She supposedly cut the hide into strips and formed it into a semicircle bounded by the sea. Elsewhere in the book there is quite a bit of discussion on optical shortest path problems. There are many fine illustrations both here and elsewhere. Example from Chapter 2 include the music of the spheres as imagined by Kepler, an illustration of Dido's minimization problem from the 1630s, pictures of medieval towns built with an optimization principle à la Dido, and a fronticepiece of a treatise on optics from the 1200s where refraction and burning mirrors are clearly illustrated. This book can be a fine educational resource for teachers trying to motivate ideas such as minimization problems in Calculus. Closely related topics: Optimization, Optics, Astronomy, Irrationals, The Circle, Carthage, and Education.
Hughes, Barnabas. Hawaiian Number Systems. Mathematics Teacher 75 (1982), 253--56.
Discusses the original mixed base (4 and 10) Hawaiian system and the introduction of a strict base 10 system after the arrival of missionaries. Gives many examples of both types of number words. (One theory, due to W. D. Alexander, 1864, is that groupings by 4 became popular from the the custom of counting fish and such by taking a couple in each hand or by tying them in bundles of four.) The transition between the two number systems was apparently not entirely smooth; younger Hawaiians understood only the decimal system had difficulty with older Hawaiians, who for example used different words for forty when speaking of forty canoes than speaking of forty fish. The author also discusses the introduction of some other words into Hawaiian. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Hawaiians, Number Words, Mathematics in Language, and Number Systems.
Hughes, Barnabas B. The earliest known record of California Indian numbers. Historia Math. 1 (1974), no. 1, 79--82. SC: 01A15, MR: 57 #15836.
The author discusses a document from 1775 which is now thought to be the earliest written record of the number system of a California Indian tribe. The document includes numbers for one through 14. One interesting feature is that some of the words suggest a base 4 number base. Also interesting is the fact that some of the number words that were recorded are different from the ones recorded by Dixon and Kroeber for related Costanoan Indians in 1907. These differences between this these number words (from Angel Island) and the others (from Mission Santa Clara) may indicate the influence of other tribes. The author notes that since Angel Island is nearly a centerpoint of various waterways, this influence is not surprising. The document was written by Fr. Vincente Maria, the chaplain of a Spanish expedition, and the author of the article seems to suggest that the confessional practices of the time may have encouraged Indians to use a decimal system for numbers larger than ten. This was because sins had to be identified at confession both by kind and by number, and because the Indians were not likely to be otherwise understood. Closely related topics: Number Words, Number Systems, California Indians, and Religion.
Jones, Phillip S. Irrationals or Incommensurables. I. Their discovery, and a "Logical Scandal". Mathematics Teacher 49 (1956), 123--27.
The discovery of irrationals. Discusses an appealing theory, due to Kurt von Fritz, that the discovery of irrationals grew out of a study of the pentagram. Von Fritz is in support of the traditional theory that discovery or irrationals was due to Hippasus of Metapontum. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Irrationals, The Pentagram, and Hippasus of Metapontum.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. I. Zero, Pi, and Polygons. Mathematics Teacher 50 (1957), 162--65.
Supplements Archibald, Raymond Clare, Babylonian Mathematics, discussing some work by Neugebauer and others 1936 and 1957. Discusses the invention of the zero in (later) Babylonia and its appearance in Greece. (Zero was apparently first regarded as a true number by Aristotle.) Also discusses a value of 3 1/8 for pi (reported by M.E.M. Bruins, anticipated by Neugebauer), a problem to determine the radius of a circle circumscribing an isosceles triangle with two sides of 50 and one of 60 (an often discussed example, originally discovered by Bruins, that is still a good algebra problem, using only the Pythagorean theorem), and a table giving areas of pentagons, hexagons, and heptagons from the square of a side. Not all are accurate, but agree with analogous values given later by Heron (c. 75 AD). Heron's table included the regular nonagon as well. The article is continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II., which however, has a somewhat smaller scope. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Circle, Zero, Aristotle, The Measurement of Area and Volume, and Heron.
Knuth, Donald E. Ancient Babylonian algorithms. Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (1972), no. 7, 671--677; errata, ibid. 19 (1976), no. 2, 108; MR: 52#13133. SC: 01A15, MR: 52 #13132.
Were there computer scientists among the ancient Babylonians? Probably not. However, some of the ideas in computer science occurred to the ancient Babylonians as well. The author here discusses Babylonian algorithms in particular. Most algorithms are of course given as examples, but Knuth notes one text that is an exception: "Length and width is to be equal to the area. You should proceed as follows. Make two copies of one parameter. Subtract 1. Form the reciprocal. Multiply by the parameter you copied. This gives the width." Knuth explains, "In other words, if x+y=xy, it is possible to compute y by the procedure y=(x-1)-1x. The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples. The above procedure reads surprisingly like a program for a 'stack' machine like the Burroughs B5500!". Knuth finds a table involving compound interest where he finds evidence of a "DO I = 1 TO N" loop and something like a "WHILE" clause. He also discusses how one tablet may have been obtained by sorting a large set of numbers. "Thus, Inakibit seems to have the distinction of being the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" [However, note that this statement was made in 1972.] Some tablets cited are available here in English for the first time (Knuth translated them using German and French translations, and at times Akkadian and Sumerian vocabularies as well). See errata in Knuth, Donald E., Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Closely related topics: Sumerians and Babylonians, Computation, Algorithms, and Logarithms.
Knuth, Donald E. Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Comm. ACM 19 (1976), no. 2, 108. SC: 01A15, MR: 52 #13133.
An errata to Knuth, Donald E., Ancient Babylonian algorithms. The table that was sorted was not as extensive as Knuth previously believed, and involved a "file" of about 500 instead of about 800. As Knuth notes "My italicized statement on p. 676 that 'this table contains every one' of the 231 regular sexagesimal numbers of six digits or less, is false; the table contains only 136 of those 231." The misunderstanding was due to a failure "to read the accompanying German commentary carefully enough, since [Neugebauer] departed from his usual custom in this particular case. Many of the lines in his rendition of the table were not on the original clay tablet at all, they were interpolated to show what the tablet would have looked like if it had been complete." Closely related topics: Sumerians and Babylonians, Computation, Algorithms, and Logarithms.
Lambert, Joseph B.; Ownbey-McLaughlin, Barbara and McLaughlin, Charles D. Maya arithmetic. Amer. Sci. 68 (1980), no. 3, 249--255. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82f:01002.
As the authors explain, there is no real evidence that the Mayas used either multiplication or division. However, the authors discuss how the Mayan notation would suit itself naturally to both of these operations. The authors suggest that in the Mayan calendrical system, where multiples of 360 are used in the third place instead of multiples of 202=400, it might be easier to "multiply by converting calendrical figures to vigesimal, perform the operations, and reconvert the answer to calendrical notation." The authors don't discuss a simple way to do this conversion. Closely related topics: The Maya and Algorithms.
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 as a kind of abacus. The word sungka is from the Philippines, but the author tells us that a similar board is "known over a wide area of the Malayo-Polynesian world from Madagascar to Polynesia, and also through Southeast Asia, India, and even mainland Africa." As the author notes, "documentation for this usage is very hard to come by". The arithmetical algorithms that the author advances for the sungka board have few surprises to someone familiar with abacus systems, but the article has some interesting remarks about other uses of the sungka board and about some number systems from India, the Philippines, and elsewhere in Asia that used mixed number bases. The author is particularly interested in eight-based counting systems, and believes that the Sungka board is particularly relevant in this regard: "The board has two large wells at each end, with each large well having a corresponding row of seven smaller wells. These two rows of seven are parallel and thus the board has a total of 16 wells divided into two groups of eight." The wells were apparently once filled with various numbers of things such as cowrie shells. In the examples given, the wells are used for powers of 10. Apparently the sungka board is now used at least as much for divination. As the author explains, "Its main purpose in modern times is to serve as a sedentary game. In the Philippines, and probably elsewhere, the Sungka Board is also still occasionally used for popular divination, especially by elders enquiring on whether travel by youths is auspicious on a certain day, or by girls interested in finding out whether and when they will get married." Closely related topics: The Philippines, The Abacus, Divination, Indo-Malay Archipelago, Polynesia, and Africa.
Miller, G. A. Gerbert's Letter to Adelbold. School Science and Mathematics 21 (1921), 649--53.
Gerbert puts circles and squares inside an equilateral triangle, and attempts to explain why they give different answers for the area. We think of these answers as estimates, but Gerbert's letter contains no hint of a limiting process. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Abacus, Gerbert, Pope Sylvester II, The Measurement of Area and Volume, and Limit.
Patel, D. M. Symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9 & 0 in Sanskrit and English languages. Math. Ed. (Siwan) 15 (1981), no. 1, B1--B3. (Reviewer: Brij Mohan.) SC: 01A99 (01A32), MR: 82h:01080.
There have been many theories on the origins of the numerals 1 through 9. The numerals for 1, 2, and 3 are frequently thought to based on one two or three tally marks or fingers, drawn in the case of 2 and 3 so that the number is written in one stroke. There have been many theories for the origins of the other numerals. Patel suggests that the Hindu-Arabic numerals 4, 6, 7, 8, and 9 were derived from shapes made with the fingers (perhaps some kind of finger numerals?). It's likely that the last word has not yet been said. He also notes similarities between the Sanskrit and English words for the numbers one through nine; these similarities are however already very well known. Closely related topics: The Hindu-Arabic Numerals, India, and Finger Numerals.
Powell, Marvin A., Jr. The antecedents of old Babylonian place notation and the early history of Babylonian mathematics. Historia Math. 3 (1976), 417--439. (Reviewer: Richard L. Francis.) SC: 01A15, MR: 58 #9990.
The Mesopotamian positional notation is generally thought to have originated in the Old Babylonian period (c. 2000--1600 BC), but the author argues that it actually dates back even further, before the end of the Third Dynasty of Ur (c. 2112--2004 BC) or even to the middle of the third millennium BC. The author looks at several texts, and finds evidence of a positional way of thinking in the way units of measurement were used and in the kinds of errors made by students. As is often the case, errors can be very useful in understanding the procedures that were used to do mathematics. In one example, the author compares the errors made by two different students: One tablet is "rather a text ... written by a bungler who did not know the front from the back of his tablet, did not know the difference between standard numerical notation and area notation, and succeeded in making half a dozen writing errors in as many lines, but nevertheless was not without a modicum of ability and probably finished school with a low passing grade, took a post with the government and became a bureaucrat. The writer of no. 50 [the other tablet] no doubt became a scholar and died penniless. However probable these postulated eventualities may be, the modern scholar may well be more grateful to our third millennium bungler than to his competent classmate." (p. 432) Closely related topics: Sumerians and Babylonians, Number Systems, and Measurement.
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 interesting problems in number theory. There are several proofs that every fraction can be represented as a sum of unit fractions, and these vary in the number of fractions produced and the maximum size of the denominators (these proofs are given as Fibonacci-Sylvester, Erdös (1950), Golomb (1962), Bleicher (1968, using Farey series), and Bleicher (1972, using continued fractions)). He also discusses various conjectures about unit fractions. For example, Erdös and Strauss conjectured that 4/n can always be written as the sum of three or less Egyptian fractions, and Sierpinski 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 the denominators of the unit fractions are required to be squares, or to be cubes, or to be square free. Closely related topics: Ancient Egypt and Number Theory.
Sanford, Vera. Counters: Computing if You Can Count to Five. Mathematics Teacher 43 (1950), 368--70.
As the author points out, the words calculator and calculus come from the Latin calculus (a small stone). Small stones were used in early counting boards, which were something like loose abacuses. Similar counting boards were used into the 1700s. The author explains how to use one to add, subtract, and multiply using a Roman-numeral type system (so, for example, the counting board has rows for 1, 5, 10, 50, 100, 500, and 1000). Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: The Abacus.
Schaaf, William L. Mathematics and World History. Mathematics Teacher 23 (1930), 496--503.
Concerned with the idea the different cultures have different ways of thinking about mathematical concepts. Schaaf takes the number concept as an example. The idea of number and magnitude was concrete and geometric to the Greeks, and was closely tied with the idea of measurement. This notion was changed by Diophantus, who may have been influenced by the mathematics of India and the Middle East. Similar ideas in the Islamic world may have reached Europe in the middle ages. A new concept of number was introduced with Descartes in Analytic Geometry. Since then, mathematics has become still more abstract. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Number Concept, Geometry, Greece, Measurement, Diophantus, India, The Middle East, The Islamic World, and Analytic Geometry.
Schaaf, William L. Mathematics as a Cultural Heritage. Arithmetic Teacher 8 (1961), 5--9.
Briefly discusses some of the key characteristics of the mathematics of the Babylonians, Egyptians, Greeks, and of Medieval Europe. Then discusses adoption of the Hindu-Arabic numerals, the development of computation, and more abstract mathematics. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt, Greece, Medieval Europe, and The Hindu-Arabic Numerals.
Schmandt-Besserat, Denise. Oneness, Twoness, Threeness. The Sciences 27 (1987), 44--48.
Writing developed in Sumeria from attempts to represent numbers. Objects such as animals and bushels of grain were represented in a one-to-one correspondence with small clay tokens--animals with cylinders and bushels of grain with spheres. When Sumerian society became more complex, new complex tokens were invented. These represented finished items such as garments, metalworks, jars of oil, and loaves of bread. The complex tokens could have elaborate markings and a wide variety of shapes. What made things change was the habit of putting plain tokens in solid clay envelopes to record quantities in legal documents. Since breaking the envelopes symbolically "broke the deal", accountants began impressing the tokens on the surface. Later, they realized that the envelopes themselves were unnecessary. Soon, the Sumerians also copied the markings on complex tokens onto a two-dimensional surface. Writing had been invented. The symbols for small and large quantities of grain (a wedge and a circle) came to be used to represent the numbers 1 and 10 when used in conjunction with two-dimensional representations of complex tokens. Abstract numbers had been invented as well. Not long after, the pictographs came to represent sounds. This worked fairly well until the first fully phonetic alphabet was invented by the Phoenicians, perhaps 1400 years later. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Development of Writing, Sumerians and Babylonians, and Number Systems.
Schrader, Dorothy V. De arithmetica, Book I, of Boethius. Mathematics Teacher 61 (1968), 615--28.
Paraphrases Book I of Boethius' De arithmetica, which is in turn based on the Arithmetica of Nichomachus. This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today. Boethius begins what we might think of as modular arithmetic (even and odd, and later evenly-even, evenly-odd, oddly-even), but the classification of numbers and parts of numbers soon acquires an unexpected complexity. The article gives an excellent introduction to the character of Medieval arithmetic/number theory. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Boethius (Ancius Manlius Torquatus Severinus Boetius), Number Theory, and Nichomachus of Gerasa.
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, Number Theory, Gerbert, Pope Sylvester II, Religion, Medieval Europe, and The Islamic World.
Schroeder, Manfred R. Number theory and the real world. Math. Intelligencer 7 (1985), no. 4, 18--26. (Reviewer: M. Mendès France.) SC: 11-02 (00A69 01A99), MR: 87b:11001.
We learn in this interesting article that number theory has applications to, or at least connections, with the real world. The author begins with a discussion of the division of the scale into twelve equal semitones, and how this appears natural from the continued fraction representation of log23. Next, he discusses the acoustics of concert halls, and how ceilings designed with a knowledge of quadratic residues can better convert sound waves traveling longitudinally into lateral waves, and thereby produce a more accurate stereophonic effect. Another suggestion of the author on wave diffraction involves primitive roots. (If the reader wants to really understand this part of the article, some knowledge of physics will be necessary.) The author then discusses of applications of finite fields to error correcting codes and even a verification of Einstein's General Theory of Relativity (the slowing of electromagnetic radiation in a gravitational field, observed with radar echos of the planets Venus and Mercury). The applications of modular arithmetic to cryptography and fast methods of multiplication are more widely known, but will come as a pleasant surprise to the uninitiated. Many other applications are also briefly mentioned. The author has written a book Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information and Computing (Springer-Verlag, Berlin, 1984) that discusses these and other applications in more detail. Closely related topics: Number Theory, Music, Acoustics, Astronomy, and Information Theory.
Seidenberg, A. km, a widespread root for ten. Arch. History Exact Sci. 16 (1976/77), no. 1, 1--16. (Reviewer: Richard L. Francis.) SC: 01A10, MR: 58 #4778.
Seidenberg studies number words in a wide variety of languages and finds some surprising similarities. He argues from these similarities that these number words, and therefore the corresponding number concepts, arose one place and spread throughout the world by a diffusion process. Here, and also in his article Seidenberg, A., The ritual origin of counting, he notes several similarities in the construction of number words in three languages that are built on the number words for one and two (Gumulgul in Australia, Bakairi in South America, and of the Bushmen in South Africa). These include the fact that in building odd numbers, the word one comes at the end, and also the fact that there is on connective. Similarities like these seem particularly natural under the diffusion hypothesis. However, the strongest evidence would come from number words themselves; for as Seidenberg notes, "If the number-vocabularies of the Gumulgul, the Bakairi, and the Bushman had been the same, and not merely the same in structure, probably everybody, or nearly everybody, would concede that the words derive from a single source." In fact, Seidenberg does find that one root, km, seems to appear in many number words world-wide. After looking at a wide variety of languages, Seidenberg concludes (p. 11) that the original word km meant "one", and thereafter began to be used for "one" larger unit, and particularly for the unit ten. He also finds at least one example of the root km meaning four; one wonders if perhaps this use might even have predated its use for ten. There have been attempts to explain the proto Indo-European root for ten in a way that conflicts with Seidenberg's theory; the notion of ten as "two hands" seems popular. Seidenberg discusses these attempts, but feels that they are rather ad hoc. Certainly from the point of a mathematician, Seidenberg's theory is very appealing. Closely related topics: Number Words, Number Systems, and Abraham Seidenberg.
Seidenberg, A. The ritual origin of counting. Arch. Hist. Exact Sci. 2 (1962b), 1-40.
It is common to argue that counting and other elementary mathematics arose spontaneously throughout the world in response to a practical, or perhaps psychological, need. Abraham Seidenberg argues instead for a diffusion theory, that counting arose only once, and then spread throughout the world. In fact, many common associations with numbers suggest such a common origin. One such association that Seidenberg is the idea that odd numbers are male and even numbers are female; this is certainly well known from the Pythagoreans, but turns out to be nearly universal. Seidenberg proposes that counting in fact originally arose in a ritual context. Seidenberg draws from a wide variety of anthropological sources for rituals and myths that hint at what this common origin might have been. He finds that counting "was frequently the central feature of a rite, and that participants in ritual were numbered." He focuses more specifically on creation rituals. He suggests that in the enaction of creation myths, men and women may have come onto the scene alternately, easily explaining the odd/male even/female association. He finds that his ideas clarify "pure 2-counting, which is the oldest stratum of counting we can detect." In pure-2 counting, there are separate words for one and two and these are used to form all other number words. He illustrates this with number words from diverse languages such as the Gumulgal of Australia, the Bakairi of South America, and the Bushmen of South Africa. He sheds additional light on his hypothesis with discussions of the possible origin of counting taboos (and connections with ritual sacrifice), of ancient one-one-correspondence "tally" systems (e.g., counting people with stones), of taxation systems, of money, and of gematria. Seidenberg also gives us some fascinating examples of counting in world religions. These include the analogy The Lord : His people = the shepherd : his sheep, the analogy The shepherd : his sheep = the moon : the stars. These two lead one to expect the moon to count the stars; and Seidenberg in fact finds evidence of this in ancient Babylonia. He argues from the equation The Lord's people = the stars of the heaven to The Lord's people = the sand upon the seashore that one would expect to find a ritual counting of sand. In fact, he finds the notion of Counter of the Sands both in Buddhism and among the Ancient Greeks. The equation The Lord = The Counter seems to be confirmed in two of the ninety-nine beautiful names of Allah, namely The Counter and the Reckoner; and there is further confirmation in Chapter's XV and XIX of the Qu'ran. This is a fascinating article, connecting mathematics with a wide variety of disciplines. Closely related topics: Myth and Ritual, Storytelling Traditions, Anthropology, General, Counting, TallySystems, Taxation, Number Words, The Pythagoreans, Gematria, Religion, The Islamic World, and Abraham Seidenberg.
Sizer, Walter S. Mathematical notions in preliterate societies. Math. Intelligencer 13 (1991), no. 4, 53--60. (Reviewer: U. D'Ambrosio.) SC: 01A07 (01A12 01A13), MR: 93a:01002.
The author discusses the ethnomathematics of nonliterate societies. There is little detail, as the article is rather brief, but the author does mention the number concept and counting, fractions (very briefly), elementary geometric notions (e.g., that of a line), symmetry, string figures, and games of strategy. One note on the article: there are strong similarities behind the mathematics in different parts of the world. There is a theory that this similarity is due to a common origin. The author credits Cantor for this idea. It was first fully developed, however, by Abraham Seidenberg. Closely related topics: Ethnomathematics General, The Number Concept, Fractions, Geometry, Symmetry, Games, and String Figures. Also possibly relevant: Abraham Seidenberg.
Sleight, E. R. The Art of Nombryng. Mathematics Teacher 35 (1942), 112--16.
The Art of Nombryng is from England in the 1400s, and is a translation of de Arte Numerandi, which was in turn written in the 1200s and is attributed to Sacrobosco. It explains how to do the basic operations of arithmetic, including mediation and duplication, and going as far as the extraction of square and cube roots. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sacrobosco (John of Holywood) and England in the 1400s.
Sleight, E. R. The Craft of Nombrynge. Mathematics Teacher 32 (1939), 243--48.
As we are told, The Craft of Nombrynge is based on the Canto de Algorismo by Alexander de Villa Dei (1220). It explains how to add, subtract, double, and divide by two, but does not discuss general division or the extraction of roots. (The method of multiplication is essentially the galley method.) Topics are introduced from the Latin Canto, and the remaining text is given in English. Arithmetic (algorism) is attributed to a supposed King Algor of India. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Extraction of Roots, Alexander de Villa Dei, and England in the 1400s.
Swetz, Frank. The Evolution of Mathematics in Ancient China. Mathematics Teacher 52 (1979), 10--19.
An overview of Chinese mathematics, including the discovery of the lo shu magic square (thought to have a plan of universal harmony), square roots, the Chinese remainder theorem, and polynomials of high degree (including a quintic in x2). Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Algebra, and Magic Squares.
Swetz, Frank J. Bodily Mathematics. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . P. 52.
Many people have used parts of the body to represent numbers. "Hand" is a common source of the word for "five" [consider the English words "five" and "fist"]. An extreme example is in the Kewa people of Papua New Guinea, who count from 1 to 68 on different parts of the body. An illustration is given. The body is often used to represent lengths and volumes. Closely related topics: The Kewa People, Number Systems, and Measurement.
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 improve mathematics education. It can not only breath new life into the subject, but also allow students to better understand mathematics as a mode of inquiry. If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own. Swetz gives examples from the development of algorithms for arithmetic (including square roots). Ancient demonstrations of mathematical ideas, such as the "husan-thu" proof of the Pythagorean theorem from China can be conceptually more suitable for students than more synthetic modern ones. Ancient "homework problems" from Babylonia, China, and Medieval Italy can be more interesting than the more dry and formulaic modern equivalents. (See Swetz, Was Pythagoras Chinese? for many interesting examples from China.) Although the author doesn't discuss this, the Chinese 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 in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Education, Computation, China, Algebra, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.
Taussky, Olga. From Pythagoras' theorem via sums of squares to celestial mechanics. Math. Intelligencer 10 (1988), no. 1, 52--55. (Reviewer: \v Stefan Porubsk\'y.) SC: 01-01 (01A99), MR: 89e:01002.
The author discusses parameterization of Pythagorean triangles, the law of quadratic reciprocity, representation of numbers in a fixed finite number of sums of squares numbers, quadratic forms, and connections with the complex numbers, quaternions, and Cayley numbers. The author tells that H. Levy and E. Isaacson observed the law of quadratic reciprocity in the study of water waves on a sloping beach (if sound waves behaved in an analogous way, would there be an applications in acoustics?). We see a surprising application of the parameterization of Pythagorean triangles in astronomy: E. Stiefel found observed that a straight line u1=c in the parameter plane (u1,u2) gives us triples (x,y,r) corresponding to a parabola, and if one moves along this line at a constant rate, one moves in a parabolic path according to Kepler's second law. Closely related topics: Pythagorean Triangles and Triples, Imaginary and Complex Numbers, Number Theory, Algebra, Acoustics, and Astronomy.
Woodruff, Charles E. The Evolution of Modern Numerals from Ancient Tally Marks. American Mathematical Monthly 16 (1909), 125--33.
A theory that the Hindu-Arabic numerals actually started out in China. Gives a possible evolution of each of the digits 1--9. There are many other theories as well, so it would be valuable to find evidence of some of these "missing links". Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Hindu-Arabic Numerals and China.
Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians North of Mexico. School Science and Mathematics 33 (1933), 363--72.
Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Basket Making, Pottery, and Tattoos.
Zaslavsky, Claudia. Africa counts. Number and pattern in African culture. Prindle, Weber & Schmidt, Inc., Boston, Mass., 1973. x+328 pp. SC: 01A10, MR: 58 #20993.
This book is an excellent introduction to the mathematics of (primarily sub-Saharan) Africa. The best tribute to its importance may be in Gerdes, Paulus, On mathematics in the history of sub-Saharan Africa. Gerdes writes "In her classical study Africa Counts: Number and Pattern in African Culture ..., Claudia Zaslavsky presented an overview of the available literature on mathematics in the history of sub-Saharan Africa. She discussed written, spoken, and gesture counting, number symbolism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric forms, while Donald Crowe contributed a chapter on geometric symmetries in African art." Regarding geometric symmetries, it is primarily the frieze patterns and plane patterns that are discussed; there is surely more work to be done on the bichromatic frieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensive bibliography. Another useful resource is the newsletter distributed by the African Mathematical Union's Commission on the History of Mathematics in Africa (AMUCHMA). Closely related topics: Sub-Saharan Africa, TallySystems, Finger Numerals, Counting, Numerology, The Reckoning of Time, Money, Measurement, Games, Continuous Tracing Problems, Architecture, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.