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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.
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.
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.
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.
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.
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.
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).
Pressman, Ian and Singmaster, David. The jealous husbands and the missionaries and cannibals. Math. Gaz. 73 (1989), no. 464, 73--81. (Reviewer: E. Keith Lloyd.) SC: 01A99 (05A99), MR: 92b:01086.
There are three river crossing problems in the Propositiones ad Acuendos, which is generally attributed to Alcuin: the problem of three jealous husbands (each of whom won't let another man be alone with his wife), the problem of the wolf, goat, and cabbage, and the problem of "the two adults and two children where the children weigh half as much as the adults." The authors discusses modifications of these problems and attempted solutions by Luca Pacioli, Tartaglia, and others. Modifications include the addition of more people, an island in the center, and a bigger boat. A more recent version is the problem of the Missionaries and the Cannibals, where the cannibals must never outnumber the missionaries. The authors give some solutions and theorems on minimality, although they leave their discovery of a 16 move solution to the four-couples-with-an-island problem as "a nice exercise for the reader". The authors don't discuss this, but problems similar to the wolf-goat-cabbage problem have appeared in a variety of cultures. Closely related topics: Alcuin, Discrete Mathematics, Luca Pacioli, Niccolò Fontana (Tartaglia) (1499?-1557), and Mathematics in Recreation.
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.
Scriba, Christoph J. Mathematics and music. (Danish) Normat 38 (1990), no. 1, 3--17, 52. SC: 01A99 (00A69), MR: 91i:01154.
The author discusses the relationship between mathematics and music from Pythagorean through modern times. His story begins in in Pythagorean times, and as he explains, the notes of the musical scale were then determined by the ratio of a perfect fifth, i.e. 3:2. Twelve intervals of a fifth are roughly equal to seven octaves, but are in reality slightly more than seven octaves, the discrepancy being the "Pythagorean comma" of 312:219, or roughly 74:73. Whole steps in the scale were in the ratio 9:8, and half steps were in the ratio 256:243. Thus two half steps were slightly less than one whole step. In fact, Philolaus noted that one whole note is equal to two half notes plus a Pythagorean comma. Archytas showed that intervals like the octave 2:1, fifth 3:2, fourth 4:3, and whole tone 9:8, or any other interval in the ratio (n+1):n cannot in fact be divided with rational numbers into two equal intervals. However, he noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5. 5:4 can be thought of as a major third, and 6:5 can be though of as a minor third. So the ratio 3:2 is divided as 6:5:4. Similarly, the fourth of 4:3 can be divided into the product of 7:6 and 8:7, so the ratio 4:3 is divided as 8:7:6. The interval 7:6 can be though of as a shrunken minor third and 8:7 can be though of as an enlarged whole tone. Scriba suggests that the germs of the idea of making this division lie with the Babylonians.In the Renaissance, the musical scale was modified to take some of these ideas into account through the work of theoreticians like Ludovico Fogliano and Giusseppe Zarlino. For example, the ratio for the notes E:C and A:F were changed from the Pythagorean 81:64 (two whole tones) to the ratio 5:4. B moved to stay a whole tone of 9:8 above A. Thus the half tones F:E and c:B were now in the ratio 16:15 rather than the Pythagorean 256:243. The whole tones C:D, F:G, and A:B remained in the ratio 9:8, but the whole tones D:E and G:A were now in the ratio 10:9. (It was roughly in the same time interval that intervals of a third began to be considered consonant.) Sharps and flats did not coincide: C sharp and D flat were for example different notes. However, it wasn't long before there were efforts to make a scale of 12 uniform steps. The first to attempt to do so was Galileo Galilei's father, Vincenso Galilei. He tried to make each step of size 18:17, though that of course led to problems. It was Simon Stevin who first had the idea of making uniform steps of size 21/12.
Later on, some mathematicians even began to question the division of the scale into 12 tones, with the idea that a division into a different number of notes might lead to a more perfect representation of the intervals. For example Christiaan Huygens defined a 31-tone system of temperament in his Lettre touchant le cycle harmonique. One source even suggests that this has "led indirectly to a tradition of 31-tone music in the Netherlands in this century". Leonhard Euler's efforts involved an attempt to reconcile the ideal "octave" 2:1 with the ideal "fifth" 3:2. He analyzed the problem by using a continued fraction representation of the ratio log 2:log 3/2. The convergent 12/7 corresponds to the popular division of 7 octaves into a circle of 12 fifths. Other convergents include 17/12, 29/17, 41/24, and 53/31. In the last case, for example, 31 octaves would be divided into 53 fifths. These didn't answer the question of what kind of equally tempered scale best reconciles the intervals of an octave, fifth, and third (2:1, 3:2, and 5:4) simultaneously. This may or may not influence the course of music, but Scriba shows how an algorithm by the Norwegian mathematician Viggo Brun (1885-1978) gives an answer. If the best answers are written in terms of the number of steps in the three intervals, the best approximations are (2,1,1), (3,2,1), (5,3,2), (7,4,2), (12,7,4), (19,11,6), (31,18,10), (34,20,11), (53,31,17), (87,51,28), .... The triple (12,7,4) is the common case with 12 semitones in an octave, 7 in a "major fifth", and 4 in a "major third". As Scriba explains, the case of the 31 tone scale has been especially important historically. In fact, Scriba tells us that it was back in the middle of the 1600s that Nicolas Vicentino described a "archicembalo" with six manuals with the octave divided into 31 parts; as mentioned above, Huygens clarified this. Moreover, Scriba tells us that Zarlino and Salinas shortly thereafter discussed the division of the octave into 19 equal parts. There is apparently an organ built according to the principles of the Dutch physicist D. Fokker (1887-1972) that also divides the octave into 31 parts (it is now in the Teylers Museum in Haarlem). Along a different line, Euler tried to design a mathematical system to quantify the dissonance of chords, but it apparently did not work very well.
The next part of the article discusses some of the work of Wolfgang Graesers (1906-1928), who tried to do a mathematical study of Bach's Art of the Fugue (this was published under the name Bachs "Kunst der Fuge" (German) in the Bach-Jahrbuch 1924, pages 1-104). Here, group theoretic notions reflect the kinds of transformations, such as inversion, that can be used in a fugue. A background in music theory may be useful in understanding Graesers's work.