To refine search, see subtopics The Hindu-Arabic Numerals, The Quipu, TallySystems, and Finger Numerals. To expand search, see Arithmetic. Laterally related topics: Numerology, Magic Squares, Bookkeeping, Modular Arithmetic, Algorithms, Logarithms, The Number Concept, The Abacus, Exponentials, Interpolation, Zero, Fractions, The Real Number System, Irrationals, The Extraction of Roots, Mental Arithmetic, The Negative Numbers, and Imaginary and Complex Numbers.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Ascher, Marcia. Before the conquest. Math. Mag. 65 (1992), no. 4, 211--218. SC: 01A12, MR: 93g:01006.
Discusses the Inca and the Maya. With the Inca, focuses on the quipu. Most quipus were destroyed by the Spanish, who thought them to be the work of the Devil, but some 550 remain. Discusses their basic structure. A fascinating puzzle in the article is a pair of quipus which seem to represent data in a similar yet inexplicable way. With the Maya, focuses on their calendar. Again, much has been destroyed. For example, there only four codices remain, whereas thousands were burned by the Spanish. Fortunately, many stelae still exist. These show a calendar system with a variety of cycles. These cycles to us suggest Chinese Remainder problems. Examples of cycles are the 260 day ritual almanac composed of a cycle of 13 numbers and 20 named dieties, the vague year of 365 days composed of a cycle of 20 numbers within a cycle of 18 named dieties plus 5 unnamed days, their least common multiple (the calendar round of 18,980 days), the long count of days (in effect, multiples of 360 days plus a remainder), a 9 day cycle of Lords of the night associated with gods of the underworld, a lunar cycle of 29 and 30 day months, 13 levels in the heaven, a cycle of 4 cardinal directions (associated with different colors), sometimes used in conjunction with an 819 day cycle of the rain god. The Mayans appear to have had keen astronomical knowledge. The author notes that the error between real and tabulated times of the position of Venus would be off by just two hours in 500 years. Closely related topics: The Inca, The Quipu, The Maya, The Calendar, Astronomy, and Chinese Remainder Problems.
Ascher, Marcia. The logical-numerical system of Inca quipus. Ann. Hist. Comput. 5 (1983), no. 3, 268--278. (Reviewer: M. P. Closs.) SC: 01A12 (68-03), MR: 85b:01003.
Spanish chroniclers have claimed that the messages on quipus "were as varied as ballads, peace negotiations, laws, and state history." The approximately 550 quipus that still exist show us instead a variety of remarkable ways for organizing structured data. The meaning of this data, however, has been largely lost. Numbers in some quipus show relationships that are still hard to explain. Closely related topic: The Quipu.
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 and Education.
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.
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 topic: The Maya.
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 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, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
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, and Numerology.
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: The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Number Words, and The Maya.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
Harvey, H. R. and Williams, B. J. Aztec arithmetic: positional notation and area calculation. Science 210 (1980), no. 4469, 499--505. (Reviewer: M. P. Closs.) SC: 01A10, MR: 81k:01002.
It has long been thought that the Mayans were the only Mesoamerican people to have developed a positional number system. However, as the authors have noted, the Aztecs also had such a system (using lines and dots). The treatment of zero may be less consistent than it was with the Mayans. The authors discuss Aztec calculations of area as well. The Aztecs clearly used some sort of algorithm to compute these areas. (It's difficult to assess the calculations perfectly since areas of quadrilaterals are only determined by the lengths of the sides in the special case of triangles.) The authors discuss why the mathematics discussed in this article was unlikely to have come from the Spanish. The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked. Closely related topics: The Aztec, The Measurement of Area and Volume, Number Words, and The Maya.
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, and Mathematics in Language.
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, California Indians, and Religion.
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 and Measurement.
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 and Sumerians and Babylonians.
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 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.
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 and Measurement.
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.
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.