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The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Angell, Ian O. Megalithic mathematics, ancient almanacs or neolithic nonsense. Bull. Inst. Math. Appl. 14 (1978), no. 10, 253--258. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 80f:01002.
Discusses different explanations for the shapes of megalithic stone rings. The author briefly discusses some of the theories of Alexander Thom, which involve an astronomical calendar and an effort to make the circumference equal to 3 times the "diameter" rather than the irrational pi. He then discusses two new theories of his own. One explains the shapes of the stone rings as extensions of the ellipse, generated with three or four pegs and a string rather than with just the usual two. The other explains the shapes as an effort to store shadow lengths. Neither theory may be given entirely in earnest. A theme of the paper is how theories may start as intellectual games, go out of control, and be changed into pseudo-science. Closely related topics: The Stone Builders, Astronomy, The Calendar, The Ellipse, Pseudoscience, and Alexander Thom.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Ascher, Marcia. Models and maps from the Marshall Islands: a case in ethnomathematics. Historia Math. 22 (1995), no. 4, 347--370. SC: 01A07 (01A13), MR: 1 364 080.
The Marshall islanders used their understanding of swell interaction to navigate, rather than the astronomical methods more familiar to us. These methods had the advantage of being usable when the sky was not visible. In fact, the author notes "one navigator recounted that an early part of his training was begin made to float in water at various places in order to learn how to feel what would later be shown and explained to him." Ascher explains how wave refraction and reflection explain the swell interactions, and how the Marshall islands map called the mattang was used to explain these interaction. She explains how the rebbelith and meddo maps (large and smaller scale) are not just literal descriptions of distances, but are also abstract representations of some of the same principles. Closely related topics: The Marshall Islands, Navigation, and Cartography.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Aveni, A. F. Tropical archeoastronomy. Science 213 (1981), no. 4504, 161--171. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01006.
Cultures in the tropics appear in general to have adopted a horizon and zenith approach to the sky, as opposed to the approach with the celestial pole (now Polaris) and the ecliptic/celestial equator, which is more familiar to most of us. Arorae in the Gilbert Islands (Kiribati) is very close to the equator, and navigators used stars on the horizon instead of compass directions. To them, constellations were also long chains of stars. Apparently, the people of the Caroline Islands also used a kind of star compass. In Polynesia and apparently in much of Oceania, islands were associated with stars that have zenith appearances above them; this is also useful in navigation. The Maori used a similar system. Various cultures in central and south America have been particularly interested in horizon and zenith events. These include the Maya, the Inca, and the Aztec, and are discussed in detail. There was a similar interest in the Chalchihuites culture, apparently influenced by astronomers of the Teotihuacán empire. Less is known about astronomy in Africa, but the Mursi of Ethiopia appear to corroborate the author's thesis, as may the Bambara of Sudan as well. Closely related topics: Astronomy, Kiribati (The Gilbert Islands), The Hawaiians, The Caroline Islands, Navigation, The Maya, The Chalchihuites, The Teotihuacán Empire, The Inca, Java, The Aztec, Oceania, The Mursi of Ethiopia, The Bambara of Sudan, and The Maori.Modify notes on this entry Modify bibliography entry Make comment on this entry
Aveni, A. and Hartung, H. The observation of the Sun at the time of passage through the zenith in Mesoamerica. Archaeoastronomy No. 3 suppl. J. Hist Astronom. 12 (1981), S51--S70. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82k:01003.
A careful analysis suggests that two near-vertical tubes in central America were used to mark the zenith sun, and possibly the June solstice. At about the time that the second tube was used, transits of the Pleiades could be observed as well. Arguing for the significance of this is the recent knowledge of the importance of the Pleiades in the Aztec calendar. In the structure containing the second tube is also a doorway making a place on the horizon. At the period in question, Capella underwent helical rise at that place on the same day as the first zenith sun. Further evidence of the importance of zenith sun watching is found in a clay model of a temple offering and in two clay figurines. The author gives an example of how observations of the sun through an aperture were also important to the Zuñi (in this case through the roof of a Zuñi chief or priest). Closely related topics: The Maya, Astronomy, The Aztec, and The Zuñi.Modify notes on this entry Modify bibliography entry Make comment on this entry
Aveni, Anthony F. Archaeoastronomy in the Maya region: a review of the past decade. Archaeoastronomy No. 3 suppl. J. Hist Astronom. 12 (1981), S1--S16. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82k:01004.
A survey of what is understood and not understood about Maya astronomy. A theme running through the article is that Maya astronomy can only be understood in its religious or cultural context, because astronomical events had to be made to occur on certain days of non-astronomical cycles. Mayan astronomy also has a different character from Old World astronomy in that it seems to be concerned mainly with events on the horizon, whereas Old World astronomy is generally concerned more with events on the ecliptic. This phenomenon is discussed in more detail in Aveni, A. F., Tropical archeoastronomy. The author also discusses alignments of sides of buildings, whose non-perpendicular sides have otherwise been sometimes claimed to have "presumably resulted from the inability of the Maya to lay out the right angles". Closely related topics: The Maya, Astronomy, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, 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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Dahlke, Richard; Fakler, Robert A. and Morash, Ronald P. A sketch of the history of probability theory. Math. Ed. 5 (1989), no. 4, 218--232. (Reviewer: William J. Adams.) SC: 01A99 (60-03), MR: 91i:01148.
Focuses on the history of probability theory, but also touches on the development of statistics. Considers one ancient root of probability theory to be the gambling with astrogali. Mentions the related ancient Egyptian game "Hounds and Hackals", of c. 3500 BC. Discusses the table of frequencies of tosses of 3 die in De Vetula, and Cardano's and Galileo's explanations of the probabilities of such events. Galileo's telescope led him to consider some of the theory of errors, and he arrived, in effect, at some of the features of the normal probability distribution. (It is interesting that later on, Gauss refined some of his own work in statistics to rediscover the planetoid Ceres.) Discusses the "division of stakes" problem and its solution by Pascal and Fermat. The first book actually published on games of chance was written by Huygens. In addition, as the author explains, "Huygens was the first to use probability in studying vital statistics of humans. He used John Graunt's (London) now famous book displaying vital statistics to construct a mortality curve and to define the notions of mean and probable duration of life. Shortly thereafter, probability theory was being applied to annuities." The article continues through the beginning of the 1900s. Much of this later material is of course beyond the scope of these pages. Closely related topics: Probability, Statistics, Gambling, De Vetula, Girolamo Cardano, Galileo Galilei, Astronomy, Blaise Pascal, Pierre de Fermat, Christiaan Huygens, and Insurance.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Eves, Howard. On the Practicality of the Rule of False Position. Mathematics Teacher 51 (1958), 606--8.
Eves shows how the method of false position can be simpler than our own methods by giving one example from the Ahmes Papyrus, three from the Greek Anthology of c. 500 AD, and two of his own. One of his examples is from surveying, and Eves says that it is the method a surveyor would probably use. In the other example of his own, he likens the rule of false position to the method of similitude in geometric constructions. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of False Position, Ancient Egypt, Medieval Europe, Surveying, and Geometry.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Fischer, Irene K. At the dawn of geodesy. Bull. Géodésique 55 (1981), no. 2, 132--142. SC: 01A10 (01A17 01A20 01A25), MR: 83g:01002.
The cultures in ancient Egypt and in Greece, China, and Babylonia all did work in surveying, geodesy, and astronomy. However, they all had different approaches to the subjects. The author explains that "The striking difference between the abstract, geometric approach of Greece and the concrete, algebraic approach of Babylonia and China represent not a difference in talents but a difference in culture-bound interests." The reader should probably have some prior knowledge of the subject matter (and of geodesy in particular) to fully appreciate this article. Closely related topics: Surveying, Astronomy, Ancient Egypt, Greece, China, and Sumerians and Babylonians.Modify notes on this entry Modify bibliography entry Make comment on this entry
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).Modify notes on this entry Modify bibliography entry Make comment on this entry
Høyrup, Jens. Sub-scientific mathematics: observations on a pre-modern phenomenon. Hist. of Sci. 28 (1990), no. 79, part 1, 63--87. (Reviewer: David Singmaster.) SC: 01A10 (01A05 01A12 01A80), MR: 91j:01007.
Høyrup makes a distinction between scientific and subscientific mathematics. These fields correspond somewhat to pure and applied mathematics. However, by using this new terminology, the author hopes to avoid suggesting that "subscientific" mathematics is always derived from "scientific" mathematics in the way that "applied" mathematics is derived from "pure" mathematics. Høyrup discusses the distinction between scientific and subscientific mathematics and also their various kinds of relationships. His examples are drawn from Greece, Egypt, India, the Islamic World (with references to the Silk route), and from the Carolingian Propositiones ad acuendos jevenes. (The latter is traditionally associated with Alcuin.) Høyrup touches on relevant work by the mathematicians Hero, Diophantus, and al Khwarizmi. Surveying is discussed as a particularly important type of subscientific mathematics. Closely related topics: Applied Mathematics (General), Greece, Ancient Egypt, India, The Islamic World, Alcuin, Heron, Diophantus, Surveying, and Abu Abdullah Muhammed ibn Musa al Khwarizmi.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Hively, Ray and Horn, Robert. Geometry and astronomy in prehistoric Ohio. Archaeoastronomy No. 4 (1982), S1--S20. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 84f:01002.
The geometrically designed earth-works near Newark, Ohio have been the subject of curiosity for centuries. They are Hopewellian, and are now dated at approximately 0-250 AD. From a purely geometric point of view the site is interesting because of its use of a circle and an almost equilateral octagon. The authors have carefully analyzed the available survey data. They first determined that the site was constructed using a standardized unit of length, and then considered possible astronomical alignments in the site. They found no convincing evidence of solar or planetary alignments, but they did find quite a bit of evidence for lunar alignments. Important lunar points include the minimum and maximum north and south extremes for the Moon's rise and set points, and there is in fact the possibility that all 8 of these points are recorded, though the evidence for some is stronger than the evidence for others. It appears that some deviation from symmetry in the octagon may have resulted from efforts to incorporate the given alignments. This study suggests that the builders may have been interested in the 18.61 year lunar cycle. The authors do not consider stellar alignments, since uncertainties in the date of the site make effects of precession unacceptably large. A related Hopewellian earth-works construction is discussed in Hively, Ray and Horn, Robert, Hopewellian geometry and astronomy at High Bank. Closely related topics: Hopewellian Indians, Astronomy, and Polygons.Modify notes on this entry Modify bibliography entry Make comment on this entry
Hively, Ray and Horn, Robert. Hopewellian geometry and astronomy at High Bank. Archaeoastronomy No. 7 Suppl. J. Hist. Astronom. 15 (1984), S85--S100. (Reviewer: M. P. Closs.) SC: 01A12, MR: 86f:01005.
This paper continues the investigations that the authors started with Hively, Ray and Horn, Robert, Geometry and astronomy in prehistoric Ohio. In the present article, the authors discuss the Hopewellian earthworks construction at High Bank in Ohio. Like the Newark construction, this includes a circle and an equilateral octagon. This site is oriented roughly 90o differently, however, and the octagon is on a different scale than at Newark. Nevertheless, both sites were apparently constructed using the same standard of length. [The octagon may have been constructed using a different procedure than the octagon at Newark.] There are possible alignments to the same lunar events as at Newark, and there are also possible alignments to sunrise and sunset on both the summer and the winter solstice. All may differ, of course, in their likelihood of being intentional. Like its predecessor, a very interesting article. Some suggestions for future research are given. Closely related topics: Hopewellian Indians, Astronomy, and Polygons.Modify notes on this entry Modify bibliography entry Make comment on this entry
Hunt, J. N. House numbering in revolutionary Paris. Bull. Inst. Math. Appl. 31 (1995), no. 9-10, 145--145. SC: 01A99 (01A50), MR: 1 352 301.
A variety of systems for numbering houses were used in Paris, both before and after the Revolution. The author discusses several of these systems, each of which had at least one fatal flaw. For example, in one system, the same number could be used several times on one street, so that if you were dropped in the middle of a street and wanted to find a given address, it could be impossible to know what direction to proceed. After many unsuccessful attempts to develop a workable system, an "ordinary citizen by the name of Garros [proposed] the eminently reasonable system in which numbers were to be attached to successive doorways, odd numbers on the left and even numbers on the right, beginning from the end nearest to the centre of Pairs. Although Initially rejected for flimsy reasons such as 'It needed equal numbers of houses on each side,' or 'What about the banks of the Seine?,' it was generally well received." An earlier suggestion had also been kept, "to number houses in the direction of river flow for streets that were more or less parallel to the Seine, and away from the river for the remainder." As the author observes discussing one of the systems, "a Graph Theorist might devise a more convenient system", and indeed some of the issues involved could lead to interesting problems in graph theory. Closely related topics: Cartography, France in the 1700s, and Graph Theory.Modify notes on this entry Modify bibliography entry Make comment on this entry
Kudlek, Manfred. Calendar systems. Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 395--428. (Reviewer: J. S. Joel.) SC: 01A99 (00A69), MR: 92j:01079.
A rare and unusually wide ranging look at calendar systems in a variety of cultures. Explains some of the astronomical issues involved. The author discusses calendars of Egypt, Babylonia, the Roman Empire, Greece (Athens), the Islamic World (especially Persia), India, China (only gives a taste, since more than 50 official calendars were used), Japan and Vietnam (their calendars were connected with China), Java, Bali, Guatamala (by the Cakchiquel Indians), revolutionary France, the Mayas, and in the Jewish tradition. Discusses the computation of the date of Easter. (The computation of Easter was of course one of the primary goals of mathematics instruction in the middle ages.) There is information on how to correlate these calendars as well (in terms of Julian dates). Closely related topics: The Calendar, Ancient Egypt, Sumerians and Babylonians, The Roman Empire, Greece, The Islamic World, India, China, Japan, Vietnam, Java, Bali, The Maya, Guatemala (and Cakchiquel Indians), France in the 1700s, The Jewish Tradition, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Martzloff, Jean-Claude. Pi in the Sky. Unesco Courier (Nov., 1989), 22--28.
Very brief. Includes a bit on the influence of divination, astronomy/astrology, Confucianism, and Taoism on the development of Chinese mathematics. The emphasis on the answer rather than the proof shows a Taoist influence, "on the grounds that the fallacious arguments of the sophists showed its limits". Also a bit on how mathematics and mathematicians fit into Chinese society. Appears in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: China, Divination, Astronomy, Astrology, Confucianism, and Taoism.Modify notes on this entry Modify bibliography entry Make comment on this entry
Neugebauer, O. On the orientation of pyramids. Special issue dedicated to Olaf Pedersen on his sixtieth birthday. Centaurus 24 (1980), 1--3. (Reviewer: H. W. Guggenheimer.) SC: 01A15, MR: 81k:01004.
Neugebauer gives a theory that explains how the Egyptians could have oriented their pyramids without using the advanced astronomical knowledge sometimes attributed to them. The theory relies on the construction of an accurately shaped pyramidal model (for example the capstone of the future pyramid), and on watching the shadow of the model in the course of the day. The biggest question about this procedure may be the question of how the model can be made accurately enough. Nevertheless, this theory represents a great simplification over many other theories. Closely related topics: The Egyptian Pyramids, The Pyramid, and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
Ollerenshaw, Kathleen. Some personal delights in geometry---from earliest days to fractals. Bull. Inst. Math. Appl. 27 (1991), no. 4, 65--75. SC: 01A99 (51-03 58-03), MR: 1 110 875.
Dame Kathleen Ollerenshaw discusses some of her favorite results and ideas of geometry. The examples range from Euclid to the present, and include illustrations of projective geometry, a fixed point principal (two superimposed identical maps on different scales will share a point in common), the nine-point circle (with proof), Pascal's mystic hexagram theorem and its generalization to general conics, and Briachon's theorem, obtained as the dual of Pascal's theorem. She briefly discusses the attempt to represent astronomy in geometrical terms, mentioning a frantic search for a "Clock in the Sky" for navigational purposes, achieved to some extent by observations of the moons of the planet Jupiter. She closes with some illustrations and a brief discussion of fractals. One of her examples is her own (apparently new) observation that if one has three circles intersecting in pairs, the three chords joining the points of intersection meet in a point; a proof is given in the article The Ollerenshaw point. Closely related topics: Geometry, Projective Geometry, Geometric Fixed Point Principles, Line-Point Duality, Astronomy, The Reckoning of Time, and Fractals.Modify notes on this entry Modify bibliography entry Make comment on this entry
Palter, Robert. Black Athena, Afro-centrism, and the history of science. Hist. Sci. 31 (1993), no. 93, part 3, 227--287. (Reviewer: Donald Cook.) SC: 01A16 (01A07 01A20 01A70), MR: 94i:01001.
Martin Bernal's Black Athena created a bit of a sensation when it first came out. Robert Palter discusses aspects of Bernal's article and also other arguments of afro-centrists. Palter particularly focuses on the question of whether Egyptian mathematics and science influenced the Greeks. Bernal suggests that the influence may be quite large, and Palter argues that all existing evidence points to the influence being quite small. An important area in Palter's discussions is ancient astronomy, where Palter discusses the general character of Egyptian astronomy, and argues that some claims about it have been vastly exaggerated; much of this discussion focuses on discrediting claims made by John Pappademos. Palter then notes that Peter Tompkins, author of Secrets of the Great Pyramid, seems to suggest that Newton was led by Egyptian science to discover his law of gravitation. About Tompkins, Bernal writes that "it it a tragedy that Tompkins's brilliant and scholarly book has been stripped of its scholarly apparatus". Palter writes "It seems never to have occurred to Bernal that the absence of scholarly apparatus in Tompkins's account of Newton has a very simple explanation: no scholarly evidence exists to support that account." When discussing Egyptian mathematics proper, Palter focuses discusses the general character, and then square roots (or a relative lack of them), the value of pi, the controversial problem in the Moscow papyrus on the surface area of a basket, the Pythagorean theorem (or the relative lack of it, arguments on the special case of involving the diagonal of the square), and the notion (or absence of notion) of an irrational number. Palter attacks claims by Cheikh Anta Diop (see Civilization or barbarism: An authentic anthropology) that Archimedes stole some of his most famous mathematics from the Egyptians. Palter then discusses pyramidology, and some of the claims cited by Bernal that "one can find such relations as pi, phi, the 'golden number' and Pythagoras' triangle from them." The final section, discusses the similarities and differences between Egyptian and Greek medicine. Although Mathematics is not so directly involved here, strong Egyptian influence in Greek medicine could argue for the plausibility of influence of other Egyptian science on Greek science as well. A very interesting paper. Apart from the fact that Palter's article serves as a kind of review of Bernal's book, it is worth reading for its discussions on the nature of Egyptian mathematics and science. Bernal responds to Palter's article in Bernal, Martin, Response to a paper by R. Palter: "Black Athena, Afro-centrism, and the history of science" [Hist. Sci. 31 (1993), no. 93, part 3, 227--287; MR: 94i:01001]. Closely related topics: Ancient Egypt, Greece, Astronomy, Archimedes, The Egyptian Pyramids, Pythagorean Triangles and Triples, and Medicine.Modify notes on this entry Modify bibliography entry Make comment on this entry
Pingree, David. The Mesopotamian origin of early Indian mathematical astronomy. J. Hist. Astronom. 4 (1973), no. 1, 1--12. (Reviewer: A. I. Volodarskii.) SC: 01A15 (01A25), MR: 58 #25.
Some of the most important questions in the history of mathematics are on the interactions between various cultures. Here, the author makes a strong case that early Indian mathematical astronomy (in the Jyotisavedanga) was influenced by Mesopotamia science. His discussion is somewhat technical, and may be hard to follow for those not knowledgeable about the subjects involved. Near the end of the article, the author writes "But there is one further question that we must raise before accepting this hypothesis of transmission. Was this an isolated phenomenon, or part of a general Iranian influence on Indian culture in the fifth and fourth centuries B.C.?" Although, as he notes, "our answer to that question is rather clouded by the scarcity of literary or archaeological data from the period in question", he finds that he is able to conclude with the statement "It is reasonable then, or at least so I believe, to see the origins of mathematical astronomy in India as just one element in a general transmission of Mesopotamian-Iranian cultural forms to northern India during the two centuries that antedated Alexander's conquest of the Achaemenid empire." Closely related topics: India, Sumerians and Babylonians, and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
Proverbio, Edoardo. The contribution of the mechanical clock to the improvement of navigation. Longitude zero 1884--1984 (Greenwich, 1984). Vistas Astronom. 28 (1985), no. 1-2, 95--103. SC: 01A99, MR: 809 625.
It is a relatively simple matter to measure latitude with simple instruments; your latitude is for example nearly equal to the altitude of the pole star above your horizon. Longitude can in theory be determined by what amounts to determining your time zone; this can be determined by noting the time of sunrise. If you note that the sun rises three hours later than it did at home, you would expect to be about 3 time zones, or 45 degrees to the west of home. However, until the mid 1700s, there was no accurate way to keep track of time at sea; traditional methods such as water clocks were hopeless on a moving ship. A solution was proposed by the mathematician and astronomer Galileo, who discovered the moons of Jupiter. These moons occasionally eclipse each other, and if one could predict when that would happen, one would in effect have a clock in the sky. Other mathematical/astronomical methods were proposed as well; in theory if you have accurate enough predictions of the orbit of the moon, you can predict time by observations of the moon as well. Unfortunately, mathematical methods were not yet adequate to predict the positions of astronomical objects with enough accuracy, and the computations could have been difficult for the average sailor in any case. So attention began to focus again on finding a more accurate clock. Some of the problems in clock design involved mathematics as well. For example, it was known that a pendulum will swing in roughly equal time regardless of the size of the swing. (A famous story tells of how Galileo discovered this in church one day, by comparing with his pulse.) "Roughly equal" wasn't good enough, and a mathematically very interesting solution was suggested by the mathematician and scientist Christiaan Huygens. His suggestion involved improving the accuracy of the pendulum by using the tautochrone property of the cycloid. Huygens tried a number of other things as well. Of course, there is much more that doesn't involve mathematics so directly. A fascinating article. Closely related topics: Navigation, The Reckoning of Time, The Clock, Astronomy, Galileo Galilei, Christiaan Huygens, and The Cycloid.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Information Theory, and Arithmetic.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Arithmetic, Computation, China, Algebra, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.Modify notes on this entry Modify bibliography entry Make comment on this entry
Swetz, Frank J. The Platonic Solids. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . P. 171.
Very brief. Discusses the platonic solids, their symbolism to the Pythagoreans, and their use in Kepler's astronomical theories. Closely related topics: The Regular Solids, The Pythagoreans, Johannes Kepler (1571-1630), and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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.Modify notes on this entry Modify bibliography entry Make comment on this entry
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