To refine search, see subtopics The Balance and the Measurement of Weight, The Measurement of Area and Volume, The Measurement of Distance, Leveling, Angular Measure, and The Astrolabe and Related Instruments. Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Arithmetic, 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.
Ammarell, Gene. Sky calendars of the Indo-Malay archipelago. History of oriental astronomy (New Delhi, 1985), 241--247, Cambridge Univ. Press, Cambridge, 1987. SC: 01A13 (01A07), MR: 1 160 818.
The people of the Indo-Malay archipelago used astronomical events such as the heliacal risings or culminations of stars, the solstices, and the zenith sun to make calendars or otherwise determine the most favorable time for rice planting. There is sometimes a need to measure or mark angles in this context, and methods used include shadow methods (marking the lengths of the tangents on some sticks), an ingenious method of tilting a bamboo stick filled with water, and a method of noting when kernels of rice rolled off an open palm when raised to Orion at dusk. (In the case of one tribe, someone observed that "the time was right for planting when a man looked up to see the Pleiades and his fat fell off!") Closely related topics: Indo-Malay Archipelago, The Calendar, Astronomy, Angular Measure, Agriculture, The Kenyah, The Kayan, Java, The Dyak, The Maloh, and The Iban.
Atkinson, R. J. C. Obituary: Alexander Thom. J. Hist. Astronom. 17 (1986), no. 1, 73--75. SC: 01A70 (01A10), MR: 87h:01062.
As the author explains, some of the work of Alexander Thom remains controversial. However, Thom is to be credited with the invention of the subject of archaeoastronomy and with a number of interesting observations and theories. One of his interesting observations is the repeated occurrence of certain types of non-circular arrangements of stones. An interesting theory is his notion of a megalithic yard and rod, supposedly fairly consistent in Britain and Brittany. His theories of apparent alignments with solar and lunar events have been among the most influential, though are not always necessarily correct in all detail. Closely related topics: Alexander Thom, The Stone Builders, The Measurement of Distance, The Circle, and Astronomy.
Deshpande, M. N. Archaeological sources for the reconstruction of the history of sciences of India. Indian J. History Sci. 6 (1971), 1--22. (Reviewer: A. I. Volodarskii.) SC: 01A25 (01A10), MR: 58 #15813.
A broad review of the archaeology of ancient India, focusing on the sciences. Perhaps a third of the article is devoted to a discussion of the Harappan civilization, and particularly Harappa and Mohenjo-Daro. Little is directly known about Harappan mathematics, but there are strong suggestions that there would have been some significant knowledge of surveying and possibly astronomy. The author also discusses the Harappan system of weights and measures. A good area for future research, particularly if some progress is made in reading the Harappan script. Closely related topics: The Harappan Civilization, Surveying, Astronomy, The Balance and the Measurement of Weight, The Measurement of Distance, and Archaeology.
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.
Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.
Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Vitruvius, Architecture, Symmetry, Proportion and the Golden Ratio, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
Fletcher, E. N. R. The area of the curved surface of a hemisphere in ancient Egypt. Math. Gaz. 54 (1970), no. 389, 227--229. SC: 01A15, MR: 58 #9987.
Problem 10 of the Moscow papyrus discusses the surface area of a basket and is thought by some to compute the surface area of a hemisphere. The author analyzes which units may have been used in the problem, and advances the theory that the basket in question was, in fact, hemispherical, and was designed to hold 100 Hekat of corn. He notes that the units used in ancient Egypt appear to have some interesting geometrical properties. For example, a circle with a radius of 1 pes (or "foot", equal to 16 digits) was approximately equal in area to a square with sides measuring 1 royal cubit. These are all fascinating possibilities. Closely related topics: Ancient Egypt, Surface Area, The Sphere, and The Measurement of Area and Volume.
Gillings, R. J. The Volume of a Truncated Pyramid in Ancient Egytian Papryi. Mathematics Teacher 57 (1964), 552--55.
Gillings gives a clever way to derive the formula V=1/3(a2+ab+b2) for the volume of a truncated pyramid, using only the formula for the volume of a complete pyramid and other methods that the Egyptians had at their disposal. As he shows, fairly simple arguments suffice when b=a/2,a/3,..., and also when b=2/3a. Since to the Egyptians, every number could be represented as a finite sum of unit fractions, the demonstration is now complete. Of course we (or the Greeks) would require something like the method of exhaustion. (Even without it, the jump to a general number is a difficult step, and not trivial geometrically.) (Since in the Moscow papyrus, b=a/2, one might wonder if perhaps the Egyptians did not know the general case after all.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt, The Pyramid, The Measurement of Area and Volume, and The Method of Exhaustion.
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.
Jones, Phillip S. Irrationals or Incommensurables. III. The Greek solution. Mathematics Teacher 49 (1956), 282--85.
Shows how Eudoxus' Method of Exhaustion is used to prove that circles are to one another as the squares on their diameters. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of Exhaustion, Eudoxus, The Measurement of Area and Volume, and The Circle.
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.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. III. Trapezoids and Quadratics. Mathematics Teacher 50 (1957), 570--71.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II.. The author discusses a single Babylonian problem. The problem is interesting more as a representative of a "typical" Babylonian problem than as a discovery that gives new insights into Babylonian mathematics. The problem involves the solution to a quadratic. The scribe uses an incorrect "formula" for the area of a trapezoid. The author discusses the solution both using modern notation and in a translation of the scribes actual language. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and The Measurement of Area and Volume.
Lorch, Richard. The sphera solida and related instruments. Special issue dedicated to Olaf Pedersen on his sixtieth birthday. Centaurus 24 (1980), 153--161. (Reviewer: K.-B. Gundlach.) SC: 01A99 (85-03), MR: 82a:01057.
The sphera solida or "solid sphere" is "essentially a globe, on which the stars and principal celestial circles are depicted, and a frame of horizon and meridian circles." Related instruments include the astrolabe, and particularly the spherical astrolabe. On the other hand, the sphera solida should not be confused with the armillary sphere. As an example how the sphera solida was used, the author explains that "To align the sphere with the Heavens in the daytime, and so obtain the configuratio celi, a pin is stuck into the degree of the sun in the ecliptic and the sphere is turned until the pin has no shadow. At night the same can be achieved by the less spectacular method of taking the altitude of a known star and shifting the sphere till the representation of the star has the same altitude--just as in a plane astrolabe." (p. 157) Much of the article focuses on the literary sources on the sphera solida, which are "at least as old as the fourteenth century." The author concludes that the ultimate source may be Arabic, and mentions a related Islamic globe made in 1279. "But unfortunately there is no clear Arabic exemplar for the text of the Sphera solida." This article has a rather scholarly tone, was doubtless difficult to research; it ends with the unusual note "Finit tractatus. Deo gratias." Closely related topics: Medieval Europe, The Islamic World, and The Astrolabe and Related Instruments.
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.
North, J. D. Astrolabes and the hour-line ritual. J. Hist. Arabic Sci. 5 (1981), no. 1-2, 113--114. SC: 01A99, MR: 84h:01102.
The author examined 132 astrolabes in the Museum of the History of Science in Oxford, and concluded that they were of less value than one might expect for timekeeping: "Our of 132 astrolabes examined, 41 instruments have the unequal-hour lines, and yet only four could have been used in at best a rough and ready way to find unaided the unequal hour." Equally interesting, the author observes that "not a single medieval instrument has survived in a form which would suggest that the unequal-hour lines were used meaningfully." All this is in spite of the fact that the author observed that "it seems to be commonly believed that a standard part of the engraving of the back of an astrolabe is a set of hour-lines forming, as it were, a double horary quadrant." Closely related topics: The Astrolabe and Related Instruments and The Reckoning of Time.
Petruso, Karl M. Additive progression in prehistoric mathematics: a conjecture. Historia Math. 12 (1985), no. 2, 101--106. (Reviewer: Garry J. Tee.) SC: 01A10 (01A15), MR: 86m:01005.
A collection of stone balance weights was recovered from a Late Bronze Age ship (c. 1200 BC) that sank off the coast of southern Turkey (near Cape Gelidonya, modern Finike). Some of these weights are sphendonoid in shape ("approximately the shape of an olive pit"), and appear to be multiples 1, 3, 5, 7, 12, 31, 50, and 54 of a hypothetical unit weight of 9.3 grams (the error is within about 2 percent). There are five weights of 7, and one weight of each of the others. Initially, these balance weights defied analysis, but the author (Petruso) realized that they nearly form a Fibonacci series; he posits the existence of missing weight of 2 and 19. Two problems with this interpretation are the fact that a weight of 7 occurs instead of a weight of 8, and the fact that the weight of 54 does not fit into his system. He suggests that the weight of 8 is a "purposeful and quite utilitarian shift in the basic Fibonacci series .... [to] allow the generation of a 50-unit (rather than 55-unit) mass further along the series." He also notes that the units of 19+31+50 would conveniently add up to 100. As for the 54 unit weight, "it might well have had a specific, idiosyncratic (industrial) purpose which is now lost to us." The author notes that one particular advantage of the Fibonacci-like system is that the accuracy of the individual weights could be quickly checked: for example, one can weigh the 12 against the 5 and the 7. Altogether a fascinating theory, readily readable. Closely related topics: The Balance and the Measurement of Weight, Leonardo of Pisa (Fibonacci), Archaeology, and The Late Bronze Age.
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 Number Systems.
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, Diophantus, India, The Middle East, The Islamic World, Analytic Geometry, and Arithmetic.
Seidenberg, A. and Casey, J. The ritual origin of the balance. Arch. Hist. Exact Sci. 23 (1980/81), no. 3, 179--226. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01008.
The author's trace the beginnings of the balance back to a rituals where principals contended against each other on a kind of see-saw (somewhat similar sports are of course known from medieval times). The grain-crusher and water-lifter are similar, and perhaps derived from, the see-saw; the fact that one stands on these suggested to the authors that the contestants may have been standing on the see-saw. The authors note that in ancient Egypt, one's heart was believed to be weighed against a feather in order to decide whether one would be able to enter the afterlife. Other parts of the body, such as hair, can be used to represent an individual, and in other instances these may have been weighed instead; the authors give examples of rites where hair is weighed. An interesting use of the balance in Greece is from the Iliad where Zeus weighs Achilles and Hector on pans of a balance. "That of Hector sinks toward Hades and Hector falls, slain by Achilles." An even more interesting weighing ritual was once common in the far east, where a ruler was balanced against a quantity of a precious substance such as gold, and gave that substance (and thereby symbolically himself) to his people. The authors found many other interesting examples in a wide variety of cultures and world religions. The authors believe that only items of ritual significance were weighed at first, and that widespread commercial use came much later. Although the authors don't focus greatly on this, they also briefly discuss the different kinds of balances (and the balance-like instrument used to carry loads on the shoulders) and the weight multiples that were used on balances. Closely related topics: The Balance and the Measurement of Weight, Myth and Ritual, Religion, Sports, Ancient Egypt, Greece, The Islamic World, and Abraham Seidenberg.
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 Number Systems.
Swetz, Frank J. The Method of Archimedes. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 180--181.
Shows how Archimedes used his Method to discover the formula for the volume of a sphere. (Of course Archimedes also gave a rigorous proof using Eudoxus' Method of Exhaustion.) Closely related topics: Archimedes' Method, Archimedes, The Measurement of Area and Volume, and The Sphere.
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, Games, Continuous Tracing Problems, Architecture, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.