To refine search, see subtopics Germany in the 1800s, Germany in the 1600s, Germany in the 1500s, Germany in Ancient Times, Germany in the 1700s, and Germany in the 1900s. To expand search, see Europe. Laterally related topics: Hungary, Greece, The Roman Empire, The Celts, Medieval Europe, England, Denmark, Switzerland, Russia, Italy, France, Spain, The Etruscans, Holland /The Netherlands, and Austria.
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.
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.
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.
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.
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.
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.
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).
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.