To refine search, see subtopics Hungary, Greece, The Roman Empire, The Celts, Medieval Europe, Germany, England, Denmark, Switzerland, Russia, Italy, France, Spain, The Etruscans, Holland /The Netherlands, and Austria. Laterally related topics: Africa, The Americas, Oceania, Asia, and The Middle East.
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.
Altshiller-Court, Nathan. The Dawn of Demonstrative Geometry. Mathematics Teacher 57 (1964), 163--66.
The author argues that it seems unlikely that the Greeks could have invented their notion of proof so rapidly and in isolation. Instead, he suggests that the notion of geometric proof was a secret that was jealously guarded from all but the "inner sanctum" of the Egyptian priesthood. (Of course, since his argument implies by its very nature that Egyptian proofs were unlikely to have been written down, this will be a hard argument to either prove or disprove.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Geometry, Proof, Ancient Egypt, and Greece.
Andersen, Kirsti. The mathematical treatment of anamorphoses from Piero della Francesca to Niceron. History of mathematics: states of the art, 3--28, Academic Press, San Diego, CA, 1996.
Discusses the mathematics of anamorphoses and the history of the subject from the mathematical point of view. Begins with a short discussion of problems stemming from the well-known fact that cylindrical columns seem smaller towards the top. Dürer discussed how one can use letters of different size on such a column so that rows of print will all appear the same size. His student Erhard Schön did some work using anamorphoses proper. (This was about the same time as Hans Holbein's Ambassadors.) Piero della Francesca's De Prospectiva Pingendi includes a discussion of how to construct a particular anamorphic drawing, but little further progress was made until the 1600s. The author notes that artists didn't seem to use the same mathematical techniques when using more extreme perspectives as they used with more normal perspectives. In fact, written works from the time suggest that orthogonal projections were used. The author gives examples from the work of of Daniele Barbaro [Italy 1500s], Paolo Giovanni Lomazzo [Italy 1500s], Egnazio Danti [Italy 1500s], Guidobaldo del Monte [France 1600s], Samuel Marolois [France 1600s], and Salomon de Caus [France 1600s]. (The case of Lomazzo is unclear: he suggested using threads for the construction, but didn't state clearly how they were to be used.) After Niceron, more mathematically accurate techniques were used; the author gives an example of a work by Emmanuel Maignan [France 1600s], who was influenced by Niceron. The problems of mirror anamorphoses apparently originated in China by about 1600. Artists apparently either worked intuitively (as in China), or by using approximate constructions. Approximate constructions still appear today in the work of the 20th century Swedish artist Hans Hamngren. A mathematically precise treatment of the problem (and of a problem using a conical mirror) was given by Jean-Louis Vaulezard in the 1600s, but even Niceron gave only an approximate method. The author suggests that Vaulezard's students were perhaps the only ones who constructed curved-mirror anamorphoses using mathematically accurate methods. (Computer analyses might be useful to verify this.) Using a computer algebra system, the author has derived the equations for the curves which will project to a coordinate grid. The curve is not given in the text, but the author tells us that it is not one of the familiar curves, has degree 6, and has rather complicated coefficients. Closely related topics: Anamorphoses, The Column, Albrecht Dürer, Erhard Schön, Piero della Francesca, China, Jean-Louis Vaulezard, and Jean-François Niceron.
Artmann, Benno. The cloisters of Hauterive. Math. Intelligencer 13 (1991), no. 2, 44--49. SC: 00A69 (01A99), MR: 1 098 219.
The author discusses geometric principles behind Gothic tracery. The Gothic style developed in France about 1150, but spread widely in the next few centuries. Examples are taken from Reims, Haina, Strasbourg, and Esslingen. The geometric principles are by no means trivial; some make rather challenging exercises. The author discusses the windows of the cloisters of Hauterive in some detail. Hauterive is a Cistercian monastery near Fribourg in Switzerland, and the cloister dates from 1320-1328. The windows there are unusually geometric, and the author advances the theory that the windows amount to a kind of commentary on Book IV of Euclid's Elements. One window, however, can not be constructed with straightedge and compass: it involves the construction of a regular 9-gon. The author notes that a regular 15-gon may have originally been envisioned, but that "esthetic considerations overwhelmed mathematics." Interesting article. A number of illustrations, a few of which appear in Artmann, Benno; Swetz, Frank J., The Geometry of Gothic Church Windows. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, Euclid, and Religion.
Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.
Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Religion.
Ascher, Marcia. Graphs in cultures: a study in ethnomathematics. Historia Math. 15 (1988), no. 3, 201--227. (Reviewer: M. P. Closs.) SC: 01A10, MR: 90d:01002.
As the author observes, the philosopher Wittgenstein pointed to the problem of tracing graphs or figures as one that everyone can recognize as mathematical. Related problems have occurred in a variety of cultures. In western Europe, problems of tracing graphs or figures have occurred in Danish folk puzzles, where they were used as an alternative to dancing. Two patterns that are traced out are said to be similar to those on an artifact from Viking times, and are said to have mystical significance; and two others are said to be useful in witchcraft. Similar problems occur in other cultures as well. The article focuses on the context of the puzzles and the methods used to solve them in New Ireland and the Republic of Vanuatu, especially on the island of Malekula. A number of designs from Vanuatu have mythic significance. There is a tradition that one must complete a certain diagram to enter the Land of the Dead; failure results in being eaten. The methods used to draw the diagrams are also very interesting. In many cases, Ascher shows how individual drawing elements are transformed by processes such as reflection and rotation and are combined in systematic ways to draw the figure. Other types of mathematical ideas from Malekula include a drum signaling system with rhythms for each clan, rank, grade of pig, and special phrases, and a six-class marriage system which the elders explained with diagrams in the sand. Closely related topics: Continuous Tracing Problems, The Malekula of Vanuatu, New Ireland, Storytelling Traditions, The Philosophy of Mathematics, and Denmark Folk Tradition.
Bérczi, Sz. Symmetry and technology in ornamental art of old Hungarians and Avar-Onogurians from the archaeological finds of the Carpathian Basin, seventh to tenth century A.D. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 715--730. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058b.
Analysis of symmetries can be very helpful in better understanding archaeological art and artifacts. The types of symmetries not only show what the author describes as "intuitive mathematical development in ornamental art" but can also help trace relationships between different communities. Such studies are now relatively new, but with time should become "an accepted, standard part of the description of archaeological finds". In this article, the author discusses how all 7 types of strip/frieze patterns occur in Old Hungarian ornamental art, and develops a notion of a double frieze pattern, which is intermediary between frieze patterns and plane patterns. A number of these patterns occur (sometimes individualized) in Avar-Onogurian artifacts. The author's classification of double frieze patterns focuses on how the patterns are generated horizontally and vertically, and may be more useful for archaeological purposes than classification by the related plane patterns. The author gives examples of some plane patterns that came up somewhat naturally, including patterns from weaving, chained ring structures, and the optimal fitting of furs (a pmg plane pattern). The author compares the frequencies of certain symmetry patterns in collections from several cultures. Closely related topics: Hungary in the Middle Ages, Frieze Patterns, Plane Patterns, Double Frieze Patterns, Archaeology, and Metal Work.
Bernal, Martin. Response to a paper by R. Palter: "Black Athena, Afro-centrism, and the history of science" [Hist. Sci. 31 (1993), no. 93, part 3, 227--287; MR: 94i:01001]. With comments by Palter. Hist. Sci. 32 (1994), no. 98, part 4, 445--468. (Reviewer: Donald Cook.) SC: 01A16 (01A07 01A20 01A70 01A80), MR: 96c:01005.
An important question in the history of Greek mathematics is how much Greek mathematics was influenced by the mathematics of the Egyptians. Bernal suggests in Black Athena that the influence may be much greater than previously thought. Palter's review article Black Athena, Afro-centrism, and the history of science disagreed with a number of Bernal's points. Bernal responds here to Palter's review, Palter, Robert, Black Athena, Afro-centrism, and the history of science, and then Palter comments on Bernal's response. The response and comment provide an excellent introduction to some of the issues involved in the question of Egyptian influence and also to some of the issues of modern scholarship. It might be useful to have a class read and comment about this article. It is interesting that questions in the history of medicine play a more important role in this controversy than one might at first expect. If the Greeks borrowed heavily from the Egyptians in medicine, it seems more reasonable that they borrowed in the other sciences as well. Closely related topics: Greece, Ancient Egypt, and Medicine.
Biggs, N. L. The roots of combinatorics. Historia Math. 6 (1979), no. 2, 109--136. (Reviewer: J. Dieudonné.) SC: 05-03 (01A15 01A20 01A25 01A30 01A32 01A40 01A45), MR: 80h:05003.
(1) As the author explains, the most ancient problem connected with combinatorics may be the house-cat-mice-wheat problem of the Rhind Papyrus (Problem 79), which occurs in a similar form in a problem of Fibonacci's Liber Abaci and in an English nursery rhyme. All are concerned with successive powers of 7. (2) The first occurrence of combinatorics per se may be in the 64 hexagrams of the I Ching. (However, the more modern binary ordering of these is first seen in China in the 10th century.) A Chinese monk in the 700s may have had a rule for the number of configurations of a board game similar to go. In Greece, one of the very few references to combinatorics is a statement by Plutarch about the number of compound statements from 10 simple propositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates on the subject, so all apparently had some interest in the subject. (Plutarch's statement is also discussed in a recent article in the Monthly.) Boethius apparently had a rule for the number of combinations of n things taken two at a time. The author discusses interest in combinatorics in the Hindu world, by the Jainas, Varahamihira, and Bhaskara (the latter in the Lilavati). The work of Brahmagupta should be relevant, but is not currently available in English. The Arabs seem to have adopted their combinatorics from the Hindus. The author also briefly discusses some interest in combinatorics in the Jewish mathematical tradition; two examples are Rabbi ben Ezra and Levi ben Gerson. (3) Magic squares may first occur in the lo shu diagram, which is often linked with the I Ching. The author discusses how the idea of magic squares may have entered the Islamic world, was then improved, appeared in the work of Manuel Moschopoulos, and possibly through him entered the Western world. What happened in China is less clear. As the author suggests, the the work of Yang Hui suggests that there had been a Chinese tradition of work in magic squares, already dead by Yang Hui's time. For example, the squares Yang Hui gives are not of types found elsewhere. In addition, Yang Hui seems unclear on the techniques for construction. It is interesting that De la Loubère learned of a simple method for constructing magic squares in Siam. The author also discusses: the possibility of a Hindu study of magic squares; the presumably Arab source of Western magic square mysticism; and later developments, such as Euler's questions on orthogonal Latin squares. (4) The author discusses how questions in partitions arose in gambling, such as the throwing of astrogali (huckle bones, which can land 4 ways) or dice (which can land in 6 ways). An early systematic study is in the late Medieval Latin poem De Vetula, which gives the number of ways you can obtain any given total from a throw of 3 dice. Cardano and Galileo examined the subject in more depth. (5) Combinatorial thinking in games and puzzles. Discusses the wolf-goat-cabbage, attributed to Alcuin. [Similar puzzles also occur in a variety of other cultures, but are not discussed in this article.] Also discusses the Josephus problem, based on a process similar to the childhood process of "counting-out". The Josephus problem is named for the Jewish historian Josephus of the 1st century AD, who supposedly saved his life with a correct solution. This problem unexpectedly turned up in Japan. (6) The author discusses how "Pascal's" triangle was possibly known to Omar Khayyam in the context of taking roots. The Hindu scholar Pingala may have known a method, but the case is more cryptic. At any rate, it was known by the time of Halayudha, who may have lived in the 900s AD. A more clear-cut reference occurs in the work of Nasir al-Din al-Tusi in 1265. In China, the triangle appears in the work of Chu Shih-Chieh (1303), but may have been very ancient by then. The triangle was used by Pascal and Fermat to resolve the "problem of points". This problem had the goal of determining how to distribute stakes when a game ends early. ... Excellent article. Closely related topics: Combinatorics, The Rhind/Ahmes Papyrus, Leonardo of Pisa (Fibonacci), The I Ching, Logic, Plutarch, Chrysippus, Hipparchus, Xenocrates, Boethius (Ancius Manlius Torquatus Severinus Boetius), Jainism, Varahamihira, Brahmagupta, Bhaskara, The Islamic World, The Jewish Tradition, Rabbi ben Ezra, Levi ben Gerson, Magic Squares, Manuel Moschopoulos, Yang Hui, Siam, Mathematics and Mysticism, Leonhard Euler, Gambling, De Vetula, Girolamo Cardano, Galileo Galilei, Puzzles, Alcuin, The Josephus Problem, Japan, Pascal's Triangle, Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Pingala, Halayudha, Nasir al-Din al-Tusi, Chu Shih-chieh, Blaise Pascal, and Pierre de Fermat.
Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.
Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ptolemy (Claudius Ptolemaeus), Trigonometry, and Interpolation.
Bruins, Evert M. The division of the circle and ancient arts and sciences. Janus 63 (1976), no. 1--3, 61--84. (Reviewer: J. L. Berggren.) SC: 01A15 (01A20), MR: 57 #12015.
One Etruscan cup, made in Caere about 500 BC, and now in the Museum of Fine Arts in Budapest, has both an 11-gon and a 14-gon inscribed on it. As the author notes, one possible reason why both were given together could be that the sum of the sides of an 11-gon and of a 14-gon imperceptibly deviates from the radius of a circle inscribing them. Moreover, methods known in the old Babylonian period could be used to provide excellent approximations to the lengths of the sides. All this raises questions about the level of Etruscan mathematical development, about which little is still known (their language still being poorly understood). The author also discusses Heron's rather accurate method for approximating the area of a circle. The article is very interesting, but the reader should be forewarned that it is a bit technical. Closely related topics: The Etruscans, Sumerians and Babylonians, The Circle, Polygons, and Heron.
Byrne, Catriona. The left-handed Pythagoras. Math. Intelligencer 12 (1990), no. 3, 52--53. SC: 01A99, MR: 1 059 227.
The author notes an relief at Notre Dame de Chartres (dating from the 1100s) where Pythagoras is depicted as being left-handed. The author suggests that left-handedness is distinctly higher among mathematicians than in a random population. It would be interesting to know if any such association were perceived in the middle ages. Closely related topics: Medieval Europe and Biology.
Chandrasekhar, S. Shakespeare, Newton and Beethoven or patterns of creativity. Current Sci. 70 (1996), no. 9, 810--822. SC: 01A99, MR: 1 387 202.
Discusses the creative lives of Shakespeare, Newton, and Beethoven. The example of Newton contrasts with the other two, particularly in how old they were when they did their most creative work. While the best work of poets is often later in life, G. H. Hardy tells us that the best work of mathematicians is generally when they are young. Chandrasekhar gives the additional examples of the mathematicians or scientists James Clerk Maxwell, George Gabriel Stokes, and Albert Einstein. Lord Rayleigh's example is different, and gives us a possible explanation of the differences we've seen. In the words of J. J. Thomson, "There are some great men of science whose charm consists in having said the first word on a subject, in having introduced some new idea which has proved fruitful; there are others whose charm consists perhaps in having said the last word on the subject, and who have reduced the subject to logical consistency and clearness. I think by temperament Lord Rayleigh belonged to the second group." Chandrasekhar then discusses the importance of beauty to mathematics and science, and concludes with statements of scientists and poets on one or the other of the two disciplines (some comments are more favorable than others). Closely related topics: Creativity, Shakespeare, Isaac Newton (1642-1727), and Beethoven.
Cox, Steven J. The shape of the ideal column. Math. Intelligencer 14 (1992), no. 1, 16--24. (Reviewer: Peeter Müürsepp.) SC: 01A99 (00A69), MR: 93a:01072.
Discusses the shape of the "ideal" column. Shows how the aesthetic and perceptual ideals of Greek and Roman times were relayed by Vitruvius and later by Alberti and others. Then shows how later scientists considered the problem from the point of view of structural strength instead. A key player in this new point of view was Lagrange. The author discusses mistakes in Lagrange's work and in the work of some later scientists and mathematicians. It is interesting that the author himself has made investigations in this area (together with M. L. Overton). The article Kirmser, Philip G. and Hu, Kuo-Kuang, The shape of the ideal column reconsidered is critical of these investigations, and includes a response by Cox. Closely related topics: The Column, Vitruvius, Leone Battista Alberti (1404?--1472), Statics, and Joseph Louis Lagrange.
Cromwell, Peter R. Celtic knotwork: mathematical art. Math. Intelligencer 15 (1993), no. 1, 36--47. SC: 01A07 (00A69), MR: 1 199 275.
Cromwell discusses a theory for the construction of Celtic knot friezes. These knot patterns may have been inspired by basketry (or maybe by textiles). He then analyzes the patterns in the knot friezes using a notion of a two-sided frieze pattern. There turn out to be 31 such patterns; 7 of these are the standard monochromatic strip patterns; 17 are exactly analogous to the bichromatic strip patterns; and 7 are like the monochromatic strip patterns but require the two sides to be identical. These last 7 "grey" patterns can't occur in knotwork, since the two sides of a crossing are not identical. Of the 24 monochromatic and bichromatic patterns, 12 cannot occur in Celtic knotwork because they would require strings that don't tie up, and 2 require a string straight through the centerline (and also don't occur). The other 10 can theoretically appear. Of these 10, two do not seem to occur at all, and one occurs but with an apparently different constriction technique (an example of this type is thought to be Scandinavian). The author is able to explain the rareness of these symmetry types in terms of the theory for their construction and from the fact that Celtic know friezes were generally finite and had their ends knotted together; these constraints require construction with an even grid, and the three problematic patterns require construction with an odd grid. This explains the type which does occur appears to use a different construction technique. In fact, the author found only one Celtic pattern that uses an odd grid. (And of course it can't be used in a bounded way, though it can be used in a kind of border.) All 7 of the monochromatic frieze patterns were apparently used in generating the existing know patterns, assuming the theory of construction is true (the author makes no claims that it is). The author includes examples of his own for the 3 problematic odd-grid know patterns. Excellent article. The author includes a good bibliography of related topics. It goes as far as Norwegian peasant art, for example. Not inordinately technical, in spite of the way it might sound. Closely related topics: The Celts, Knots and Knotwork, Two Sided Frieze Patterns, Frieze Patterns, Bichromatic Strip Patterns, Weaving, and Basket Making.
Crowe, D. W. and Washburn, D. K. Groups and geometry in the ceramic art of San Ildefonso. Proceedings of the conference on groups and geometry, Part A (Madison, Wis., 1985). Algebras Groups Geom. 2 (1985), no. 3, 263--277. (Reviewer: H. S. M. Coxeter.) SC: 05B45 (00A05 01A12 20F32 52A45), MR: 87k:05055.
Discusses the types of frieze patterns and bichromatic strip patterns occurring in the pottery of the pueblo of San Ildefonso in New Mexico. The people of San Ildefonso are Tewa speaking and are thought to be of Anasazi descent. However, it should be noted that the pottery has apparently been influenced by the Spanish and by attempts to make it more readily salable. All 7 of the strip patterns and 14 of the 17 possible bichromatic strip patterns are exhibited. (The authors supply the missing 3 bichromatic strip patterns in a similar style. The authors supplement their discussion with an explanation of the appealing Coxeter notation for classifying the bichromatic patterns (the standard classification system is cumbersome) and give a table of the correspondences between various systems. A historical aside briefly discusses the study of plane patterns in the context of the Alhambra, where there is still some disagreement on which patterns are represented. Closely related topics: The Pueblo of San Ildefonso, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
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.
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.
Ellerman, David P. The mathematics of double entry bookkeeping. Math. Mag. 58 (1985), no. 4, 226--233. (Reviewer: D. J. Struik.) SC: 90C99 (01A99 20G99), MR: 87a:90151.
The double entry bookkeeping system was first described by Luca Pacioli in 1494, though it had been developed in the 1300s. One feature of the system is that it in effect constructs the negative numbers Z from the natural numbers omega. This same construction is regularly done as well in courses in logic and set theory and may also be relevant to courses on the foundations of our number system (e.g., for those planning to teach elementary school students). Closely related topics: Bookkeeping, The Negative Numbers, Luca Pacioli, and Logic.
Emmer, Michele. Art and mathematics: the Platonic solids. The Visual Mind, 215--220, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author begins by mentioning some ancient representations of Platonic solids. These include a pair of Egyptian die from the Ptolemaic dynasty, an Etruscan dodecahedron (at least 2500 years old), two Celtic dodecahedra, and a West German dodecahedron from the 2nd century BC. The author continues with a discussion of the regular solids in Plato's Timaeus. The author notes that Dürer's Melancholia, which includes a truncated rhombohedron, is sometimes thought to show the influence of Luca Pacioli. The magic square in the painting gives some evidence for this; Dürer's engraving may be one of the earliest depictions of a magic squares in the West, but an earlier manuscript by Pacioli showed an interest in them. On the other hand, Luca Pacioli's De Divina Proportione relied heavily on, and perhaps even appropriated the work of Piero della Francesca. The book is also notable for its pictures of the regular solids, attributed to Leonardo da Vinci. Also discusses work on the regular solids due to Johannes Kepler, including Kepler's recognition of a duality and his idea of a combination of two tetrahedra called a stella octangula. The author notes that the notion of the stella octangula also appears in Pacioli's De Divina Proportione. In addition, Kepler's stellated dodecahedron occurs in mosaics in the San Macro Cathedral in Venice; this work is thought to have been done by Paolo Uccello. Regarding Uccello, the author quotes Donatello as saying to his close friend "Ah Paolo, this perspective of yours makes you neglect what we know for what we don't know. These things are no use except for marquetry." (The source is Vasari's Vita di Paolo Uccello.) The author, Michele Emmer, collaborated on the film Art and Mathematics. Closely related topics: The Regular Solids, Plato, Art, The Etruscans, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.
Evans, Brian. Number and form and content: a composer's path of inquiry. The Visual Mind, 113--120, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author shows how the golden ratio occurs in music and art. His examples include Mozart's Symphony in G Minor, Grant Wood's American Gothic, Piet Mondrian's Composition with Blue, and some of his own musical and visual compositions. More controversial examples include the Great Pyramid in Egypt and Stonehenge, where the author shows how approximate values of both pi and the golden ratio can be found. The author mentions Luca Pacioli's statements on the golden ratio in De Divina Proportione and discusses other aspects of the philosophy of number and art as well. Closely related topics: Proportion and the Golden Ratio, Music, Art, Wolfgang Amadeus Mozart (1756-1791), Luca Pacioli, The Egyptian Pyramids, and The Stone Builders.
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.
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.
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.
Gardner, Arthur O. The History of Mathematics as a Part of the History of Mankind. Mathematics Teacher 61 (1968), 524--26.
Briefly discusses how factors such as religion and warfare have influenced the development of mathematics. Attributes the success of Leonardo of Pisa (Fibonacci) to the unconventional ideas of his sovereign, Emperor Frederick II of the house of Hanover. Martin Luther is an example of an important theologian who supported mathematics: "If I had children, they should not only study language and history, but they should also learn singing and music, together with the whole of mathematics." Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Religion, Warfare, and Leonardo of Pisa (Fibonacci).
Gerdes, Paulus P. J. On ethnomathematical research and symmetry. Symmetry in a kaleidoscope, 2. Symmetry Cult. Sci. 1 (1990), no. 2, 154--170. SC: 01A07, MR: 1 188 949.
Gerdes begins with a discussion of why symmetry is such a common phenomenon in human culture. He notes that some symmetries which are rare in nature (e.g., rotational symmetries of order 2) are common amongst us. Gerdes gives the example of rotational symmetry being used in the tattoos of the Makonde of northern Mozambique. Gerdes explains how symmetries such as the rotational symmetry of order 2 can arise naturally in solving problems in such areas as weaving. Gerdes then turns to the geometry of the line drawings made by the Tamil women in South India (during harvest month) and those made by the Tshokwe. These drawings have some strong similarities, and in both cases show an interest in tracing out a figure with a single continuous line. They also show a strong interest in symmetry, and Gerdes gives examples of how designs which fail to follow the one-line cultural norm may also fail to display the expected symmetries, suggesting that such drawings are degradations of more symmetric ones drawn with one line. The author advances a construction principle that can be used to construct both the Tamil and Tshokwe patterns. (Although the author doesn't note this, it is interesting that this principle is very similar to another principle that has been advanced for Celtic knot friezes!) Gerdes then discusses some mathematical properties of curves made using his construction principle. He also discusses some other interesting topics in his ethnomathematical research. For example, the author mentions that he has a found a new hypothesis on the origin of the Egyptian formula for the volume of a truncated pyramid, and has also found an infinite series proof for the Pythagorean theorem. Closely related topics: Symmetry, The Tamil of South India, TheTshokwe, Continuous Tracing Problems, The Celts, Ancient Egypt, and Pythagorean Triangles and Triples. Also possibly relevant: Mozambique, Tattoos, and Weaving.
Grattan-Guinness, I. Mozart 18, Beethoven 32: hidden shadows of integers in classical music. History of mathematics, 29--47, Academic Press, San Diego, CA, 1996. SC: 01A99 (00A69), MR: 97a:01075.
Discusses number symbolism in the works of Mozart and Beethoven. With Mozart, discusses in particular Die Zauberflöte and the last three symphonies (and particularly the Symphony in g of 1788). There is also some evidence that Mozart used gematria. Literary sources also attest to Mozart's interest in numerology. With Beethoven, focuses primarily on Piano Sonata op. 111 (no. 32), the Diabelli Variations, and the Missa Solemnis. The choice of opus numbers themselves appear to show an interest in numerology. The author suggests that some knowledge of the history and conventions of numerology would be useful before reading this article. The author's own article in the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences may be useful in this regard. The author also suggests some avenues for future research. Closely related topics: Music, Numerology, Gematria, Wolfgang Amadeus Mozart (1756-1791), and Beethoven.
Grattan-Guinness, I. Some numerological features of Beethoven's output. Ann. of Sci. 51 (1994), no. 2, 103--135. SC: 01A99 (00A69), MR: 1 278 119.
The author discusses possible occurrences of number symbolism in Beethoven's compositions. A large number of examples are used to buttress his arguments, and some prior familiarity with Beethoven's work might be useful. In some cases, numbers occur as the number of measures or notes of a them or motif, and in other cases in Beethoven's choice of opus numbers. (In contrast with the common practice of the time, Beethoven chose his opus numbers himself, and the numbers chosen could at times be seriously at variance with the order of composition.) The author's conclusions have been controversial, partly because Beethoven has often been regarded as being quite poor at arithmetic. The author discusses this objection and aspects of methodology in some detail. Closely related topics: Numerology, Music, and Beethoven.
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.
Hĝyrup 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. Hĝyrup 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.) Hĝyrup 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.
Hansen, David W. The Dependence of Mathematics on Reality. Mathematics Teacher 64 (1971), 715--19.
Discusses how the greatest mathematicians have been vitally concerned with the real world. Uses Archimedes, Newton, and Gauss as examples. Archimedes did so much applied work that it is hard to see how he fits Plutarch's description of considering mechanical work ignoble and inferior. The case of Newton is of course well known. An interesting example is Gauss, who used the motto "Thou, nature art my goddess;to thy laws/My services are bound" from Shakespeare's King Lear. Newton and Gauss were also very interested in religion. Philosophy was very important to Gauss. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Applied Mathematics (General), Archimedes, Isaac Newton (1642-1727), Karl Friedrich Gauss (1777-1855), Religion, and Philosophy. Also possibly relevant: Literature.
Hargittal, István and Lengyel Györgi. The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Journal of Chemical Education 61 (1984), 1033.
All 7 frieze patterns can be found in Hungarian needlework. The authors give an example of each pattern. A related article by the same authors is Hargittal, István and Lengyel Györgi, The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary, and Needlework.
Hargittal, István and Lengyel Györgi. The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Journal of Chemical Education 62 (1985), 35--36.
The Hungarians of the late 1800s may be among the earliest people known to have "discovered" all 17plane patterns. The authors give an example of each pattern from Hungarian needlework. For the related article on frieze patterns, see Hargittal, István and Lengyel Györgi, The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary in the 1800s, and Needlework.
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.
Jablan, Slavik. Ornament today. Geometry in the pre-scientific period; ornament today, 33--65, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 92g:01008.
The author discusses how a wide variety of mathematical notions can be used to help describe and understand the patterns occurring in art. One of the most important is, of course, the notion of symmetry, including those in the rotational symmetry patterns, frieze patterns, plane patterns, and their bichromatic (or antisymmetry) variants. More complex types of patterns also occur in art, and as Grünbaum, Grünbaum, and Shephard observed in their article Symmetry in Moorish and other ornaments, many of the problems originating from these are still unsolved. Examples are given from the Paleolithic to the 20th century. The author touches on (to give a few examples) interlace patterns (often considered to be connected with weaving), similarity symmetry, symmetries in higher dimensional spaces, and on some of the ideas of the theory of tilings, including Penrose tilings and hyperbolic tilings. The author also gives examples from the work of artists including M. C. Escher, B. Riley, and R. Neal. A fine article. A fine article. It could easily take a class an entire semester to examine in detail all the ideas presented. Closely related topics: Art, Pattern, Symmetry, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Penrose Tilings, Weaving, Similarity, and M. C. Escher.
Jones, Phillip S. Irrationals or Incommensurables. I. Their discovery, and a "Logical Scandal". Mathematics Teacher 49 (1956), 123--27.
The discovery of irrationals. Discusses an appealing theory, due to Kurt von Fritz, that the discovery of irrationals grew out of a study of the pentagram. Von Fritz is in support of the traditional theory that discovery or irrationals was due to Hippasus of Metapontum. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Irrationals, The Pentagram, and Hippasus of Metapontum.
Jones, Phillip S. 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.
Kemp, Martin. Spirals of life: D'Arcy Thompson and Theodore Cook, with Leonardo and Dürer in retrospect. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 37--54. SC: 01A99 (92-03), MR: 96j:01047.
Discusses theories of how art appears in biology. The author starts with St. Augustine, who concluded "If, then, we argue from the facts, first, that as everyone admits, not a single visible organ of the body serving a definite function is lacking in beauty, and, second, that there are some parts which have beauty and no apparent function, it follows, I think, that in the creation of the human body God put form before function." The author then discusses and compares the investigations of D'Arcy Thompson and Theodore Cook into the mathematical/biological manifestations of the spiral. Thompson and Cook agreed on many issues, though Thompson didn't approve of the "mystical conceptions" that he found in Cook's work. Specific topics discussed include the appearance of the golden ratio in biological systems (often in the guise of the Fibonacci series), turbulence, and transformations that take one biological object into a related one (one of Thompson's examples compares the skulls of Hyrachyus agrarius and Aceratherium tridactylum). In the process, the author touches on the work of Albrecht Dürer and Leonardo da Vinci (as the title suggests). Obviously, this article can not to be comprehensive, and the author himself tells us that the article is itself intended as a preface; it serves this function well. Both Thompson and Cook were well aware of the mathematical difficulties involved in thoroughly understanding the phenomena they wrote of. Cook wrote "It would only be possible to imagine life or beauty as being 'strictly' mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them." And Thompson wrote "And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies beyond our powers of analysis." The author writes that Thompson ended his book with "a plea for biological mathematicians and mathematical biologists to cultivate 'a field which few have entered and no man has explored'". He continues "Thompson's plea did not fall upon deaf ears, but it is only recently that new techniques of computer modeling have begun to realize something of the potential of some of his techniques." Closely related topics: Art, Biology, Spirals, Topology, Proportion and the Golden Ratio, Albrecht Dürer, and Leonardo da Vinci (1452-1519).
Kilmister, C. W. Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number. Math. Gaz. 64 (1980), no. 429, 149--158. (Reviewer: K. E. Hirst.) SC: 01A99 (00A05 03A05), MR: 82i:01075.
Ever since Zeno's paradoxes, mathematicians, philosophers, and logicians have been discussing the nature of the infinite. The author starts by discussing one of Zeno's four paradoxes, the Dichotomy. This leads to a discussion of Aristotle's views of the infinite. Needless to say, philosophical problems remained, and Hermann Weyl made one attempt to rectify them. Weyl advised caution in dealing with impredicative definitions, which he believed could lead to a vicious circle. Unfortunately, as Weyl notes "This vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis." And in fact, if impredicative definitions are abandoned entirely, we must also abandon the notion that a bounded infinite set has a least upper bound and of course the related theorem (Bolzano-Wierestrass) that a bounded infinite set has a limit point. As the author notes, "On 9 February 1918, Polya and Weyl made a bet in Zürich, with twelve witnesses (all mathematicians). About [the least upper bound property], Weyl prophesied 'A. Within twenty years, Polya, or a majority of leading mathematicians, will admit that the concepts of number, set and countability involved are completely vague; and that there is no more point in asking about the truth of [the least upper bound property] than of the main assertions of Hegel's physics. B. It will be recognized by Polya, or a majority of leading mathematicians, that in any wording [the least upper bound property] is false...'" When the bet was called, everyone agreed that Polya had won with the single exception of Kurt Gödel. The author notes "if the construction of the real numbers contains subtleties that troubled such an acute intellect as Weyl's as recently as 1917, and still worried Gödel in 1940, it is not to be wondered at that some of our first-year undergraduates find it hard to stomach. Perhaps they are wiser than we are." Closely related topics: Zeno, Aristotle, Hermann Weyl, Infinity, Paradox, and Philosophy.
King, Charles. Leonardo Fibonacci. Fibonacci Quarterly 1 (1963), 15--19.
A brief survey of the work of Fibonacci, Leonardo of Pisa. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topic: Leonardo of Pisa (Fibonacci).
Knight, Gordon. The geometry of Maori art---spirals. New Zealand Math. Mag. 22 (1985), no. 1, 4--7. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51060.
The Maoris frequently use spirals in their tattoos and wood carvings. These appear very much like the spirals of Archimedes, but often interlace two or more such spirals. Although the easiest way to construct a spiral similar to the spiral of Archimedes may be to use sets of concentric semicircles (or other segments of circles) offset with respect to one another, the author believes that the Maoris didn't use this technique. "In Spirals of Archimedes, and, it seems, in Maori spirals, there is a gradual, rather than an abrupt, change in curvature." The author gives several examples from Maori artwork; there are examples with 2, 3, and 4 interlaced spirals. The author notes that the 3 spiral form is more common in tattooing patterns than in carving. Apparently there was once a 6 spiral pattern on one of the figures guarding the gateway of Papawai Pa. The center of the spiral can be varied somewhat; for example, two spirals can come together in an S-curve. In one case, "the plain ridges, which form an S-curve, are made to cross over the notched spirals, giving a woven effect. According to Phillips this was chiefly an Arawa modification." The author concludes with a note that the spiral of Archimedes should perhaps have a Maori name instead. He suggests that an investigation of these spirals might be useful in mathematics education (when polar coordinates are studied). Closely related topics: Spirals, The Maori, Tattoos, Wood Carving, Archimedes, and Education.
Knorr, W. R. The geometer and the archaeoastronomers: on the prehistoric origins of mathematics. Review of: Geometry and algebra in ancient civilizations [Springer, Berlin, 1983; MR: 85b:01001] by B. L. van der Waerden. British J. Hist. Sci. 18 (1985), no. 59, part 2, 197--212. SC: 01A10, MR: 87k:01003.
The reviewer discusses van der Waerden's book Geometry and Algebra in Ancient Civilizations. Although the reviewer clearly admires van der Waerden for his work in algebra and in the history of mathematics in general, he is highly critical of the conclusions reached in van der Waerden's book. A basic theme of the book is that there is a pre-Babylonian ancestor to mathematics in Babylonia, ancient Egypt, Greece, China and India; thus the book can therefore be thought of in part as a further development of Abraham Seidenberg's theories on the ritual origins of ancient mathematics. The reviewer takes issue with several facts cited in the book, and in addition with three assumptions that he sees van der Waerden using explicitly or implicitly in the book: "(1) independent discovery is so rare that it may effectively be discounted as a working hypothesis for relating technical traditions; (2) derivative traditions are inferior to their source traditions; (3) borrowing from one tradition to another is not selective, but entails the adoption of whole bodies of technique." (The phrase "inferior to" in (2) could just as well be replaced by "degraded in".) The reviewer suggests in addition that van der Waerden has not been sufficiently critical in accepting claims by Alexander Thom and others about advanced mathematics in megalithic monuments, and sees these claims as forming "the veritable linchpin of van der Waerden's thesis". The author briefly discusses some of Thom's work in megalithic mathematics, and concludes that he finds no real evidence of the Pythagorean theorem, the ellipse, or a standard unit of distance in neolithic times. The review concludes with the statement "I fear even more the regrettable impact on credulous nonspecialists who may not know to distinguish between the general enterprise of scientific research and the reckless notions of some scientists." Closely related topics: Sumerians and Babylonians, Ancient Egypt, Greece, China, India, The Stone Builders, Alexander Thom, and Pythagorean Triangles and Triples.
Kokomoor, F. W. The Status of Mathmatics in India and Arabia during the "Dark Ages" of Europe. Mathematics Teacher 29 (1936), 224--31.
A survey of some of the work in mathematics during the middle ages. The focus is on the Islamic world. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Islamic World, India, China, Medieval Europe, and The Middle Ages.
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.
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Proportion and the Golden Ratio, Leonardo of Pisa (Fibonacci), The Pentagram, and Education.
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.
Lumpkin, Beatrice. From Egypt to Benjamin Banneker: African origins of false position solutions. Vita mathematica (Toronto, ON, 1992; Quebec City, PQ, 1992), 279--289, MAA Notes, 40, Math. Assoc. America, Washington, DC, 1996. SC: 01A05 (01A13), MR: 1 391 748.
Discusses the work of the Benjamin Banneker, who is perhaps the most interesting early American mathematician. The author gives a fine introduction to Banneker's life; this is necessarily brief, because as the author observes, his house burned down on the day of his funeral, destroying almost all his papers. She notes that there were hints of his genius starting with his building of a wood clock at the age of 22 (he used a borrowed pocket watch as a model; unfortunately, the clock was destroyed in the fire); he thereafter became famous for his ability to solve and create mathematical puzzles. "People sent him puzzles from all over the colonies and later from the new republic." His work became more serious when he was 57 and borrowed some books and astronomy instruments from a neighbor. He taught himself the mathematics he needed to become an astronomer, and published local almanacs including things such as the planetary positions and the times of sunrise, sunset, moonrise, moonset, eclipses, and tides. "Based on Banneker's work on his almanac, he was appointed an astronomer on the team of surveyors that drew up the outline for the new nation's capital, Washington, DC. Banneker was appointed because he was one of the few in the country capable of doing such work. Charles Leadbetter, author of an astronomy book that Banneker studied, wrote that knowledge of astronomy in London was 'so rare, ... not one of 20,000 hath attained to it.' Knowledge of astronomer", Lumpkin continues, "was even rarer in the new United States. Banneker's work so impressed Thomas Jefferson, then Secretary of State, that he wrote Banneker that he was sending a copy of the almanac to the Paris Academy of Sciences." Most amazing of all is that Banneker accomplished all this as an African American who had spent most of his life thus far hard physical labor. After this introduction, the author focuses on how Banneker and other mathematicians used the rule of false position. She notes, the rule of false position was used by the Egyptians in the time of the Rhind Papyrus and in a variety of other Egyptian sources (e.g., the Kahun and Berlin papyri), in the work of Alexandrian Greeks like Diophantus (c. 250 AD), in the work of Islamic mathematicians such as Abu Kamil (b. 850 AD), and in the work of the mathematician Leonardo of Pisa (Fibonacci) (who was also influenced by the work in Northern Africa). The author then discusses some interesting false position problems from Banneker's own work. Closely related topics: Benjamin Banneker, The Method of False Position, The Rhind/Ahmes Papyrus, Ancient Egypt, Diophantus, Abu Kamil (b. 850), and Leonardo of Pisa (Fibonacci).
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Symmetry, Philosophy, Greece, Physics, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
Mamedov, Kh. S. Crystallographic patterns. Symmetry: unifying human understanding, II. Comput. Math. Appl. Part B 12 (1986), no. 3-4, 511--529. SC: 00A69 (01A99 20H15 51F15), MR: 87e:00008.
This article discusses how crystallographic patterns "and their distribution and connection with natural phenomena and subjects of pure and applied art." It is written as an essay from a personal point of view. As the author tells us "I have made no effort to restrict the style of my meditations. I have presented a flow of free and sincere statements, and have not attempted to impose on them a style which might conceal their individuality. A great advantage of such statements is that one's 'falsehoods' are merely considered to be delusions, thus somehow mollifying the anger of those strict critics who feel obliged to adhere to absolute truths." The author himself is a chemist, so it is not surprising that there is some discussion of how crystallographic patterns in art are similar to those in chemistry. However, his observations on art from his own background in a nomadic family from Azerbaijan may be at least as valuable. The author notes that M. C. Escher is often identified with the applied art of crystallographic patterns, but these ideas are common in many cultures. Crystallographic patterns involving elements such as colored symmetry "are very characteristic of ancient and medieval decorations of Siberia, Kazakhstan, Central Asia, Azerbaijan, and Asia Minor." Quite a few examples of the art in this article use Islamic khufic script, and as he notes it is common to attribute the rise of patterned art rather than representational art to religious demands. The author does not seem entirely sympathetic with this idea, writing "The the problem was 'explained with God's help.' It is evident that in such cases it is much easier for the representatives of some other tradition to invent a new explaining theory than to examine the artwork using the language of its own traditions." The author gives some examples of crystallographic patterns in his own art and that of his associates. Interesting and enjoyable article. Closely related topics: Symmetry, Plane Patterns, Religion, Language and Literature, and M. C. Escher.
McClendon, R. B. Leonardo of Pisa and His Liber quadratorum. American Mathematical Monthly 26 (1919), 1--8.
The author discusses some of the most important work in Fibonacci's Liber quadratorum, and convincingly makes the case that Leonardo was the greatest genius in number theory between Diophantus and Fermat. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Leonardo of Pisa (Fibonacci) and Number Theory.
Meserve, Bruce E. The Evolution of Geometry. Mathematics Teacher 49 (1956), 372--82.
Discusses the history of geometry starting with the Egyptians and Babylonians and continuing into modern times. The rise and decline of Greek geometry, the logical structure of Greek proofs. Contributions by the Islamic world on the parallel postulate. Contributions of Renaissance artists studying perspective. Analytic geometry. More on the parallel postulate. Non-Euclidean geometry. The development of projective geometry and algebraic geometry. The article concludes with a discussion of how computational technology might change the nature of mathematics. Reprinted in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Geometry, Analytic Geometry, Projective Geometry, Algebraic Geometry, Greece, The Islamic World, The Parallel Postulate, and Perspective.
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.
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.
Discusses the history and philosophy of origami and then (in a little more depth) discusses some of its applications. The author discusses applications in math and science education, and also in art, design, and technology. A particularly interesting application of paper-folding and the theory of polyhedra is in music education, where one researcher devised "a 'tower' of five octahedra, to illustrate some basic concepts in musicology. His inspiration was from a work by Möbius written in 1861. Ganter's compound polyhedron illustrates geometrically the following concepts and their connections: the vertices correspond to the notes of the chromatic scale, the edges corresponds to the thirds and fifths, and the triangular faces correspond to the triads." He mentions that M. C. Escher was interesting in construction paper models (though it is not really clear how deep that interest lay). It is interesting that the well-known book by T. Sundara Row entitled Geometric Exercises in Paper Folding seems to be independent from the Japanese traditions. Closely related topics: Origami, Symmetry, Japan, Education, Music, M. C. Escher, and August Ferdinand Möbius (1790-1868).
Nagy, Dénes. The 2,500-year old term symmetry in science and art and its "missing link" between the antiquity and the modern age. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 18--28. SC: 01A99, MR: 1 371 622.
Documents the evolution of the word symmetry from its beginnings in ancient Greece. As the author explains, the word originally had a somewhat different meaning: symmetry = syn together + metron measure, suggesting the notion of commensurability. The word was adopted into Latin but was apparently rare in the middle ages. It's reappearance can probably be credited to the importance to the Renaissance of the De architectura libri decem of Vitruvius (1st century BC). The author discusses the Hebrew, Indian, and Chinese words for symmetry as well. At the end of the article the author enumerates some modern generalizations and uses of symmetry. For example, the author mentions "Noether's theorems connecting symmetry transformations (invariances) and conservation laws", Gell-Mann and Ne'eman's classification of elementary particles, and "Graeser's reconstruction of Bach's Kunst der Fuge". Closely related topics: Symmetry, Language and Literature, Greece, Vitruvius, Physics, and Music.
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.
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,