To expand search, see The Middle East. Laterally related topic: The Islamic World.
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.
Anagnostakis, Christopher and Goldstein, Bernard R. On an error in the Babylonian table of Pythagorean triples. Centaurus 18 (1973/74), 64--66. (Reviewer: E. M. Bruins.) SC: 01A15, MR: 58 #20994.
The authors explain a well-known mistake in the Babylonian tablet Plimpton 322 (column I, entry 10) as a consequence of a certain method of computation and of the neglect of a medial zero. It is a very appealing theory, and could give us some insight the way Babylonians did their mathematics. Other solutions have also been proposed. A good example of how we can learn from mistakes! Closely related topics: Pythagorean Triangles and Triples and Algorithms.
Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
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, The Circle, Polygons, and Heron.
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, 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.
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, and China.
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: The Circle, Zero, Aristotle, The Measurement of Area and Volume, and Heron.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. II. The Earliest Known Problem Text. Mathematics Teacher 50 (1957), 442--44.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. I.. Discusses a very old Babylonian problem text (c. 2000 BC), that seems to show an understanding of the proportionality of sides in similar right triangles. Continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. III., which has a different character from both of its predecessors. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Similarity and The Triangle.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. III. Trapezoids and Quadratics. Mathematics Teacher 50 (1957), 570--71.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. II.. The author discusses a single Babylonian problem. The problem is interesting more as a representative of a "typical" Babylonian problem than as a discovery that gives new insights into Babylonian mathematics. The problem involves the solution to a quadratic. The scribe uses an incorrect "formula" for the area of a trapezoid. The author discusses the solution both using modern notation and in a translation of the scribes actual language. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Quadratic Formula and The Measurement of Area and Volume.
Katz, Victor J. Essay reviews of Ethnomathematics [Brooks/Cole, Pacific Grove, CA, 1991; MR: 92c:01006] by M. Ascher and The crest of the peacock [Tauris, London, 1991; MR: 92g:01004] by G. G. Joseph. Historia Math. 19 (1992), no. 3, 310--315. SC: 01A07 (00A30), MR: 1 177 496.
Katz reviews and contrasts Marcia Ascher's book Ethnomathematics: A Multicultural View of Mathematical Ideas and George Gheverghese Joseph's book The Crest of the Peacock: Non-European Roots of Mathematics. He finds that both correct serious omissions in the literature (and in particular, in Morris Kline's Mathematical Thought from Ancient to Modern Times). Joseph focuses on the history of mathematics in the large civilizations of ancient Egypt, Babylonia, China, India, and the Islamic World. He wanted to highlight "(1) the global nature of mathematical pursuits of one kind or another; (2) the possibility of independent mathematical development within each cultural tradition; and (3) the crucial importance of diverse transmissions of mathematics across cultures, culminating in the creation of the unified discipline of modern mathematics." Katz seems disappointed only in the third thesis, "because the documentary evidence for transmission of mathematical ideas is lacking." (For example, he notes that "whether Diophantus was directly influenced by the Babylonian tradition is a subject of scholarly debate." Joseph's treatment of Indian mathematics seems to be particularly good "especially since it is difficult to find this material in other sources." The focus of Ascher's book is completely different. She looks at traditional non-literate peoples. As Katz notes, "She has no intention of claiming that the mathematics developed in the cultures she discusses had any influence on developments elsewhere. Her main goal is simply to show that mathematical ideas, even if not developed by those called mathematicians, can be found in many societies if one only knows where to look." Katz reports examples as coming from the Inuit, Navajo, Iroquois, and Incas of the Americas, the Malekula, Warlpiri, Maori and Caroline Islanders of Oceania, and the Tshokwe, Bushoong, and Kpelle of Africa. This very useful review concludes by highly recommending both books. Closely related topics: Ancient Egypt, China, India, The Islamic World, The Inuit, The Navajo, The Iroquois, The Inca, The Malekula of Vanuatu, The Warlpiri, The Maori, The Caroline Islands, TheTshokwe, The Bushoong, and The Kpelle of Guinea.
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: Ancient Egypt, Greece, China, India, The Stone Builders, Alexander Thom, and Pythagorean Triangles and Triples.
Knuth, Donald E. Ancient Babylonian algorithms. Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (1972), no. 7, 671--677; errata, ibid. 19 (1976), no. 2, 108; MR: 52#13133. SC: 01A15, MR: 52 #13132.
Were there computer scientists among the ancient Babylonians? Probably not. However, some of the ideas in computer science occurred to the ancient Babylonians as well. The author here discusses Babylonian algorithms in particular. Most algorithms are of course given as examples, but Knuth notes one text that is an exception: "Length and width is to be equal to the area. You should proceed as follows. Make two copies of one parameter. Subtract 1. Form the reciprocal. Multiply by the parameter you copied. This gives the width." Knuth explains, "In other words, if x+y=xy, it is possible to compute y by the procedure y=(x-1)-1x. The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples. The above procedure reads surprisingly like a program for a 'stack' machine like the Burroughs B5500!". Knuth finds a table involving compound interest where he finds evidence of a "DO I = 1 TO N" loop and something like a "WHILE" clause. He also discusses how one tablet may have been obtained by sorting a large set of numbers. "Thus, Inakibit seems to have the distinction of being the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" [However, note that this statement was made in 1972.] Some tablets cited are available here in English for the first time (Knuth translated them using German and French translations, and at times Akkadian and Sumerian vocabularies as well). See errata in Knuth, Donald E., Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Closely related topics: Computation, Algorithms, and Logarithms.
Knuth, Donald E. Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Comm. ACM 19 (1976), no. 2, 108. SC: 01A15, MR: 52 #13133.
An errata to Knuth, Donald E., Ancient Babylonian algorithms. The table that was sorted was not as extensive as Knuth previously believed, and involved a "file" of about 500 instead of about 800. As Knuth notes "My italicized statement on p. 676 that 'this table contains every one' of the 231 regular sexagesimal numbers of six digits or less, is false; the table contains only 136 of those 231." The misunderstanding was due to a failure "to read the accompanying German commentary carefully enough, since [Neugebauer] departed from his usual custom in this particular case. Many of the lines in his rendition of the table were not on the original clay tablet at all, they were interpolated to show what the tablet would have looked like if it had been complete." Closely related topics: Computation, Algorithms, and Logarithms.
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, 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.
Llyod, Daniel B. Further Evidences of Primeval Mathematics. Mathematics Teacher 57 (1966), 668--70.
A tablet from a dig at Tel Dhibayi near Baghdad shows how to find the dimensions of a triangle from its diagonal and area. The solution requires a knowledge of the Pythagorean theorem, and artfully sidesteps the difficulty of solving a quadratic equation by solving a pair of simple linear equations. Many other articles discuss similar tablets and solutions, but few so concisely as this. However, note that in the context of other Babylonian sources, the method of solution may be less obscure than the author seems to suggest. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Quadratic Formula and Pythagorean Triangles and Triples.
Pingree, David. The Mesopotamian origin of early Indian mathematical astronomy. J. Hist. Astronom. 4 (1973), no. 1, 1--12. (Reviewer: A. I. Volodarskii.) SC: 01A15 (01A25), MR: 58 #25.
Some of the most important questions in the history of mathematics are on the interactions between various cultures. Here, the author makes a strong case that early Indian mathematical astronomy (in the Jyotisavedanga) was influenced by Mesopotamia science. His discussion is somewhat technical, and may be hard to follow for those not knowledgeable about the subjects involved. Near the end of the article, the author writes "But there is one further question that we must raise before accepting this hypothesis of transmission. Was this an isolated phenomenon, or part of a general Iranian influence on Indian culture in the fifth and fourth centuries B.C.?" Although, as he notes, "our answer to that question is rather clouded by the scarcity of literary or archaeological data from the period in question", he finds that he is able to conclude with the statement "It is reasonable then, or at least so I believe, to see the origins of mathematical astronomy in India as just one element in a general transmission of Mesopotamian-Iranian cultural forms to northern India during the two centuries that antedated Alexander's conquest of the Achaemenid empire." Closely related topics: India and Astronomy.
Powell, Marvin A., Jr. The antecedents of old Babylonian place notation and the early history of Babylonian mathematics. Historia Math. 3 (1976), 417--439. (Reviewer: Richard L. Francis.) SC: 01A15, MR: 58 #9990.
The Mesopotamian positional notation is generally thought to have originated in the Old Babylonian period (c. 2000--1600 BC), but the author argues that it actually dates back even further, before the end of the Third Dynasty of Ur (c. 2112--2004 BC) or even to the middle of the third millennium BC. The author looks at several texts, and finds evidence of a positional way of thinking in the way units of measurement were used and in the kinds of errors made by students. As is often the case, errors can be very useful in understanding the procedures that were used to do mathematics. In one example, the author compares the errors made by two different students: One tablet is "rather a text ... written by a bungler who did not know the front from the back of his tablet, did not know the difference between standard numerical notation and area notation, and succeeded in making half a dozen writing errors in as many lines, but nevertheless was not without a modicum of ability and probably finished school with a low passing grade, took a post with the government and became a bureaucrat. The writer of no. 50 [the other tablet] no doubt became a scholar and died penniless. However probable these postulated eventualities may be, the modern scholar may well be more grateful to our third millennium bungler than to his competent classmate." (p. 432) Closely related topics: Number Systems and Measurement.
Ritter, James. Prime Numbers. Unesco Courier (November 1989), 12--17.
The title is a bit misleading. Discusses the work of Babylonian and Egyptian scribes and how they fit into society. Although neither society had a word for a mathematician, the ability to do mathematics was highly valued. One Mesopotamian king boasted of his academic achievements by stating proudly "I am perfectly able to subtract and add, [clever in] counting and accounting", and another says "I can find the difficult reciprocals and products which are not in the tables." In Babylonia and Egypt, mathematics was taught by creating a "network of typical examples in which a new problem can be related---by a form of interpolation---to those already known." An edited version appears in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Education and Ancient Egypt.
Schmandt-Besserat, Denise. Oneness, Twoness, Threeness. The Sciences 27 (1987), 44--48.
Writing developed in Sumeria from attempts to represent numbers. Objects such as animals and bushels of grain were represented in a one-to-one correspondence with small clay tokens--animals with cylinders and bushels of grain with spheres. When Sumerian society became more complex, new complex tokens were invented. These represented finished items such as garments, metalworks, jars of oil, and loaves of bread. The complex tokens could have elaborate markings and a wide variety of shapes. What made things change was the habit of putting plain tokens in solid clay envelopes to record quantities in legal documents. Since breaking the envelopes symbolically "broke the deal", accountants began impressing the tokens on the surface. Later, they realized that the envelopes themselves were unnecessary. Soon, the Sumerians also copied the markings on complex tokens onto a two-dimensional surface. Writing had been invented. The symbols for small and large quantities of grain (a wedge and a circle) came to be used to represent the numbers 1 and 10 when used in conjunction with two-dimensional representations of complex tokens. Abstract numbers had been invented as well. Not long after, the pictographs came to represent sounds. This worked fairly well until the first fully phonetic alphabet was invented by the Phoenicians, perhaps 1400 years later. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Development of Writing and Number Systems.
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.