For material on related topics, see The Ellipse.
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.
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: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, and The Pyramid.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Euclid, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, and Religion.Modify notes on this entry Modify bibliography entry Make comment on this entry
Atkinson, R. J. C. Obituary: Alexander Thom. J. Hist. Astronom. 17 (1986), no. 1, 73--75. SC: 01A70 (01A10), MR: 87h:01062.
As the author explains, some of the work of Alexander Thom remains controversial. However, Thom is to be credited with the invention of the subject of archaeoastronomy and with a number of interesting observations and theories. One of his interesting observations is the repeated occurrence of certain types of non-circular arrangements of stones. An interesting theory is his notion of a megalithic yard and rod, supposedly fairly consistent in Britain and Brittany. His theories of apparent alignments with solar and lunar events have been among the most influential, though are not always necessarily correct in all detail. Closely related topics: Alexander Thom, The Stone Builders, The Measurement of Distance, and Astronomy.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Polygons, and Heron.Modify notes on this entry Modify bibliography entry Make comment on this entry
Engels, Hermann. Quadrature of the circle in ancient Egypt. Historia Math. 4 (1977), 137--140. (Reviewer: L. Guggenbuhl.) SC: 01A15, MR: 56 #5124.
Explains the Egyptian formula for the area of a circle in terms of the practices of Egyptian stone masons. In order to form a relief, the stone masons covered their designs with a grid. The hypothesized construction involves an error which would confirm the now commonly held view that the ancient Egyptians did not properly understand the Pythagorean theorem. Closely related topics: Ancient Egypt, Coordinates, and Pythagorean Triangles and Triples.Modify notes on this entry Modify bibliography entry Make comment on this entry
Gerdes, Paulus. Three alternate methods of obtaining the ancient Egyptian formula for the area of a circle. Historia Math. 12 (1985), no. 3, 261--268. (Reviewer: Richard L. Francis.) SC: 01A15, MR: 86k:01004.
Gerdes gives three possible methods that the Egyptians could have used in discovering their "value" of pi, which is in effect 4(8/9)2, or about 3.16. All methods are empirical. One is connected with how rope can be coiled, one is with how mats can be formed using concentric rings, and one with arranging small balls or cylinders in a circle (the Egyptians are known to have used such objects). In al cases, if it is desired that the size of the circle be chosen so as to obtain (in effect) a perfect square value for pi, the Egyptian value arises naturally. Closely related topics: Ancient Egypt and Basket Making.Modify notes on this entry Modify bibliography entry Make comment on this entry
Hildebrandt, Stefan and Tromba, Anthony. The parsimonious universe. Shape and form in the natural world. Copernicus, New York, 1996. xiv+330 pp. ISBN: 0-387-97991-3. SC: 00A05 (01A99 49Q15), MR: 97c:00001.
This book has many interesting examples of how problems in optimization have been important both historically and in the world around us. For our purposes, we focus on Chapter 2, The Heritage of Ancient Science. The authors start here with a survey the history of some of the mathematics and applied mathematics of the Babylonians, Egyptians, and Greeks. They consider aspects such as astronomy, burning mirrors, and the discovery of the irrationals (they include a modulo 10 proof that the square root of two is irrational). Of course, this part of the book is not intended to be authoritative; the reader should beware of comments about the Egyptians and the Pythagorean theorem. The book continues with discussions of the Ptolemaic system (which they said was once thought to have been handed down from above) and of the heliocentric system. One of the more appealing parts of Chapter 2 is a discussion of the problem where Queen Dido of Carthage obtained the largest possible area that can be enclosed by the hide of an ox. She supposedly cut the hide into strips and formed it into a semicircle bounded by the sea. Elsewhere in the book there is quite a bit of discussion on optical shortest path problems. There are many fine illustrations both here and elsewhere. Example from Chapter 2 include the music of the spheres as imagined by Kepler, an illustration of Dido's minimization problem from the 1630s, pictures of medieval towns built with an optimization principle à la Dido, and a fronticepiece of a treatise on optics from the 1200s where refraction and burning mirrors are clearly illustrated. This book can be a fine educational resource for teachers trying to motivate ideas such as minimization problems in Calculus. Closely related topics: Optimization, Optics, Astronomy, Irrationals, Carthage, and Education.Modify notes on this entry Modify bibliography entry Make comment on this entry
Huylebrouck, Dirk. The $\pi$-room in Paris. Math. Intelligencer 18 (1996), no. 2, 51--53. SC: 01A99, MR: 1 395 091.
The article briefly discusses an episode in the calculation of pi, where the value given by Shanks in the 1800s was incorrect after 527 of its 607 digits. The error was unnoticed until 1945. The authors include a picture of how the incorrect value was (not surprisingly) once recorded in the pi-room in Paris, at the Palais de la Découverte.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, and The Measurement of Area and Volume.Modify notes on this entry Modify bibliography entry Make comment on this entry
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, Zero, Aristotle, The Measurement of Area and Volume, and Heron.Modify notes on this entry Modify bibliography entry Make comment on this entry
Riese, Tara A. and Chen, Yong Zhuo. Crop circles and Euclidean geometry. Internat. J. Math. Ed. Sci. Tech. 25 (1994), no. 3, 343--346. (Reviewer: E. J. F. Primrose.) SC: 51M04 (01A99), MR: 95b:51018.
This article can be viewed as a supplement to an article by I. Peterson (Science News 141 (1992), no. 5, 76--77) in which the author discusses "Gerald S. Hawkins, a retired astronomer, who was fascinated by the intriguing configuration of crop circles near Stonehenge in southern England. After a systematic study of the crop formations, he discovered five geometric theorems which cannot be found in any Euclidean geometry textbooks and references. Four of them were stated in that article. The fifth, he left to the reader to figure out." These theorems turn out to be quite elementary, but might still be of some interest to an introductory geometry class; when Riese and Yong-Zhou Chen used Peterson's article in their geometry class they had "an exciting discussion on Hawkins's theorems", and the class was able to develop its own version of the fifth theorem. The class's theorem is given in the paper, together with three other simple theorems describing the relationships between circles and n-gons. Closely related topics: The Stone Builders and Education.Modify notes on this entry Modify bibliography entry Make comment on this entry
Robins, Gay and Shute, Charles C. D. Mathematical bases of ancient Egyptian architecture and graphic art. Historia Math. 12 (1985), no. 2, 107--122. (Reviewer: Jens Høyrup.) SC: 01A15, MR: 87c:01002.
The authors discuss the slopes that occur in Egyptian pyramids and artwork. The discussion of Egyptian artwork is particularly interesting because of the Egyptian's conscious use of squared grids. The authors find no evidence of circles or the value of pi being used in to determine the overall dimensions of the pyramids, and similarly with the golden ratio. Similarly, the authors find no evidence of pi or the golden ratio being found in slopes of lines in Egyptian artwork. Nevertheless, the authors carefully discuss such claims rather than simply dismissing them out of hand. The authors do, however, find that certain "slopes" seem to have been preferred to others (as the authors note, the Egyptians seem to have preferred to measure slopes as run per unit rise rather than our rise per unit run). The authors buttress their arguments about the artwork through their use of new photographs; these carefully avoid distortion by means of a shift lens. The article is only moderately technical. Closely related topics: Ancient Egypt, The Egyptian Pyramids, Proportion and the Golden Ratio, and Coordinates.Modify notes on this entry Modify bibliography entry Make comment on this entry
Seidenberg, A. On the volume of a sphere. Arch. Hist. Exact Sci. 39 (1988), no. 2, 97--119. (Reviewer: K.-B. Gundlach.) SC: 01A20 (01A15 01A17 01A25 01A32), MR: 89j:01012.
Abraham Seidenberg argues that there is a common source for Pythagorean and Chinese (or Chinese-like) mathematics. He suggests that Old-Babylonian mathematics is a derivative of a more ancient mathematics having a much clearer geometric component (p. 104), and is "in some respects ... is derivative of a Chinese-like mathematics" (p. 109). Van der Waerden holds a similar view on this, and tells us that the mathematics of the Chiu Chang Suan Shu represents the common source more faithfully than the Babylonian does. Seidenberg believes that the common source is most similar to the Sulvasutras. He discusses how questions of the sphere and the circle were treated by the Greeks, Chinese, Egyptians, and to a lesser extent Indians. He discusses the some similarities and differences in the work on the sphere in Greece (Archimedes, with a very brief account of the application of his Method), and in Chinese (first in the Chiu Chang Suan Shu, improved by Liu Hui or perhaps Tsu Ch'ung-Chih, and then further improved by the Tsu Ch'ung-Chih's son Tsu Keng-Chih). He believes that the problem of the volume of a sphere goes back to the common source, to the first part of the second millennium B.C. or earlier. An interesting and related topic is the topic of the equality of the proportionality constants pi that occur in the formulas for the area and circumference of a circle. Seidenberg examines the Moscow Papyrus, Chinese sources, and an Old-Babylonian text and finds that this fact seemed to be recognized in all three groups. He argues that the Egyptian, Babylonian, and Chinese approaches to the volume of a truncated pyramid may have derived from the same common source. He believe that the common source also used infinitesimal, Cavalieri-type, arguments as well. It is interesting as well that Heron, who as Seidenberg notes is sometimes considered to be continuing the Babylonian tradition, gives the formula 1/2(s+p)p+1/14(1/2s)2 for the area of a segment of a circle with chord s and height (sagita, arrow) p (with an Archimedean value of 22/7 for pi), and "that the 'ancients' took [the area as] 1/2(s+p)p and even conjectured that they did so because they took pi = 3." The paper is also interesting in that he discusses the development of some of his ideas from his early papers in the 60s until much later (the paper was received soon before his death). Closely related topics: The Sphere, The Pythagoreans, China, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), Sumerians and Babylonians, The Sulvasutras, Archimedes, Archimedes' Method, The Moscow Mathematical Papyrus, Heron, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
Seidenberg, A. The ritual origin of the circle and square. Arch. Hist. Exact Sci. 25 (1981), no. 4, 269--327. (Reviewer: M. P. Closs.) SC: 01A10 (51-03), MR: 83h:01008.
Abraham Seidenberg advances a theory that the circle first arose in the context of the ritual enactment of a creation myth. In many cases, stars seem to play an important role in these myths. Seidenberg's research suggests that participants in these myths generally moved in a circle in imitation of the stars in the heavens. It is interesting that individuals in these societies often move in the same direction as the stars, and movement in the opposite direction is often considered unlucky. The fact that the Aztec god Tezcatlipoca is missing is right foot, forcing him to walk clockwise in a circle may be related. Seidenberg suggests that the creation myth is the origin for the dance around the may pole, which is for example observed near the summer solstice in northern Scandinavia today. Analogous rituals may play (or have played) a role in a wide variety of other cultures as well; examples are found in the Aztecs, ancient Indians, American Indians, and Greeks. (Spinning tops may have a ritual significance as well.) Special support is given to Seidenberg's these through the fact that in some cases, a pole may have been set up at an angle so as to point towards the pole star. Seidenberg notes that the moon might have motivated the circle rather than the stars, but the sun is unlikely to. His investigations tend to confirm this, and also suggest that lunar culture is older than solar culture. Seidenberg believes that the square arose from the circle, through the process of dividing a group into a dual organization, where for example members of one group marry someone in the other group and also (as he notes) play complementary roles in ritual. If a society divides a second time, one can think of it dividing the tribal circle into four parts. He finds some evidence of this as well. The four parts naturally define a square. His theory therefore implies that the circle arose first and that the square arose as a dual form of the circle; there is some other evidence (e.g., architectural) that may tend to confirm this. Seidenberg mentions several interesting dualities involving the circle and the square. The Altar of Heaven in Peking, for example, exhibits the equations Heaven : Earth = circle : square = three : two = South : North = White : Yellow. In Sinhalese art he finds the equation circle : square = standing : sitting. In the Omaha tribe he finds the equations that Sky : Earth = superior : inferior = one : two. He also notes the equations Heaven : Earth = Male : Female and Male : Female = one : two. The former is well known, and the latter is extensively discussed in Seidenberg, A., The ritual origin of counting The ancient Egyptians appear to be an exception as they associated the square with the earth and the circle with the sky. A fascinating paper. Closely related topics: Myth and Ritual, Religion, Anthropology, General, Kinship Systems, The Square, and Abraham Seidenberg.Modify notes on this entry Modify bibliography entry Make comment on this entry
Stern, M. D. A remarkable approximation to $\pi$. Math. Gaz. 69 (1985), no. 449, 218--219. (Reviewer: H. W. Guggenheimer.) SC: 01A15, MR: 86m:01006.
I Kings 7:23 states "And he (Solomon) made a molten sea, ten cubits from one one brim to the other: it was round all about, and its height was five cubits; and a line of thirty cubits did compass it round about." This appears to give a value of 3 for pi. The author's theory (which has recently been making the rounds) that a more accurate value (3)(111/106) is coded in the text by the use of gematria. The written form has numerical value 111 and the spoken form has numerical value 106. The corresponding value of pi is 3.141509, accurate to better than one part in ten thousand. Stern's theory has come under attack, and not least for this reason, this short article makes a good subject for classroom discussion. Closely related topic: The Jewish Tradition.Modify notes on this entry Modify bibliography entry Make comment on this entry
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