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.
Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994.
A collection of articles aimed at a general audience. Most can be read independently. Here is a partial list.I. Why the History of Mathematics? Broad perspectives on the history of mathematics.1. Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.2. Schaaf, William L. Mathematics as a Cultural Heritage. Arithmetic Teacher 8 (1961), 5--9.Briefly discusses some of the key characteristics of the mathematics of the Babylonians, Egyptians, Greeks, and of Medieval Europe. Then discusses adoption of the Hindu-Arabic numerals, the development of computation, and more abstract mathematics. Closely related topics: Ancient Egypt, Greece, Medieval Europe, and The Hindu-Arabic Numerals.3. Schaaf, William L. Mathematics and World History. Mathematics Teacher 23 (1930), 496--503.Concerned with the idea the different cultures have different ways of thinking about mathematical concepts. Schaaf takes the number concept as an example. The idea of number and magnitude was concrete and geometric to the Greeks, and was closely tied with the idea of measurement. This notion was changed by Diophantus, who may have been influenced by the mathematics of India and the Middle East. Similar ideas in the Islamic world may have reached Europe in the middle ages. A new concept of number was introduced with Descartes in Analytic Geometry. Since then, mathematics has become still more abstract. Closely related topics: The Number Concept, Geometry, Greece, Measurement, Diophantus, India, The Middle East, The Islamic World, Analytic Geometry, and Arithmetic.4. Gardner, Arthur O. The History of Mathematics as a Part of the History of Mankind. Mathematics Teacher 61 (1968), 524--26.Briefly discusses how factors such as religion and warfare have influenced the development of mathematics. Attributes the success of Leonardo of Pisa (Fibonacci) to the unconventional ideas of his sovereign, Emperor Frederick II of the house of Hanover. Martin Luther is an example of an important theologian who supported mathematics: "If I had children, they should not only study language and history, but they should also learn singing and music, together with the whole of mathematics." Closely related topics: Religion, Warfare, and Leonardo of Pisa (Fibonacci).5. 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. Closely related topics: Applied Mathematics (General), Archimedes, Isaac Newton (1642-1727), Karl Friedrich Gauss (1777-1855), Religion, and Philosophy. Also possibly relevant: Literature.6. Swetz, Frank J. Seeking Relevance? Try the History of Mathematics. Mathematics Teacher 77 (1984), 54--62.Focuses on how the history of mathematics can be used to improve mathematics education. It can not only breath new life into the subject, but also allow students to better understand mathematics as a mode of inquiry. If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own. Swetz gives examples from the development of algorithms for arithmetic (including square roots). Ancient demonstrations of mathematical ideas, such as the "husan-thu" proof of the Pythagorean theorem from China can be conceptually more suitable for students than more synthetic modern ones. Ancient "homework problems" from Babylonia, China, and Medieval Italy can be more interesting than the more dry and formulaic modern equivalents. (See Swetz, Was Pythagoras Chinese? for many interesting examples from China.) Although the author doesn't discuss this, the Chinese problems in surveying led to interesting questions in algebra, with fourth and higher degree equations. Swetz discusses how Descartes' idea of a coordinate grid was earlier used by Renaissance artists, ancient Egyptian tomb painters, and various cartographers. Closely related topics: Education, Arithmetic, Computation, China, Algebra, Analytic Geometry, Renaissance Art, Ancient Egypt, and Cartography.II. In the Beginning... Early mathematics and some ethnomathematics.7. 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. Closely related topics: The Development of Writing, Sumerians and Babylonians, and Number Systems.8. Woodruff, Charles E. The Evolution of Modern Numerals from Ancient Tally Marks. American Mathematical Monthly 16 (1909), 125--33.A theory that the Hindu-Arabic numerals actually started out in China. Gives a possible evolution of each of the digits 1--9. There are many other theories as well, so it would be valuable to find evidence of some of these "missing links". Closely related topics: The Hindu-Arabic Numerals and China.9. Cordrey, William A. Ancient Mathematics and the Development of Primitive Culture. Mathematics Teacher 32 (1939), 51--60.Discusses number words and systems of time reckoning for a wide variety of groups. Although many readers may be familiar with the Egyptian and Babylonian number systems, there are many interesting examples from the indigenous peoples of North and South America. The reader may want to ignore statements regarding the relative levels of different cultures. Closely related topics: Number Systems, Number Words, The Reckoning of Time, and Indigenous American Mathematics.10. Bidwell, James K. Maya Arithmetic. Mathematics Teacher 74 (1967), 762--68.A discussion of the base 20 Mayan number system. It will be especially useful to those teaching mathematics at the elementary level. It does not discuss the Mayan calendrical system in detail, which is uses a mixed base of 20 and 360. As the author points out, his versions of Maya arithmetic may not be historically accurate---The main source on the subject, Father Diego de Landa (1524--1579), burned many of the existing Mayan manuscripts because he considered them heretical. Closely related topics: The Maya, Number Systems, and Education.11. Diana, Lind Mae. The Peruvian Quipu. Mathematics Teacher 60 (1967), 623--28.An introduction to the Quipu. The author observes that the quipu was used not only in Peru but also in other areas of South America. These others have not been as well preserved as those found in dry graves in coastal Peru. Discusses Nordenskiöld's theory that the burial quipus contain numerological and astronomical secrets. Briefly discusses the unusual Incan abacus. Closely related topics: The Maya, The Quipu, Numerology, Astronomy, and The Abacus.12. Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians North of Mexico. School Science and Mathematics 33 (1933), 363--72.Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Basket Making, Pottery, and Tattoos.13. Hughes, Barnabas. Hawaiian Number Systems. Mathematics Teacher 75 (1982), 253--56.Discusses the original mixed base (4 and 10) Hawaiian system and the introduction of a strict base 10 system after the arrival of missionaries. Gives many examples of both types of number words. (One theory, due to W. D. Alexander, 1864, is that groupings by 4 became popular from the the custom of counting fish and such by taking a couple in each hand or by tying them in bundles of four.) The transition between the two number systems was apparently not entirely smooth; younger Hawaiians understood only the decimal system had difficulty with older Hawaiians, who for example used different words for forty when speaking of forty canoes than speaking of forty fish. The author also discusses the introduction of some other words into Hawaiian. Closely related topics: The Hawaiians, Number Words, Mathematics in Language, and Number Systems.14. Barit, Julian. The Lore of Number. Mathematics Teacher 61 (1968), 779--83.Number symbolism among the Greeks, Hebrews, and in cases also Egyptians, Druids, Hindus. Discusses numbers up through 13. For further reading, suggests a book by W. Wynn Westcott called Numbers, Their Occult Powers and Mystic Virtues by the Theosophical Publishing Society, London, 1911. Closely related topic: Numerology.Historical Exhibit 2.1. Swetz, Frank J. Bodily Mathematics. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . P. 52.Many people have used parts of the body to represent numbers. "Hand" is a common source of the word for "five" [consider the English words "five" and "fist"]. An extreme example is in the Kewa people of Papua New Guinea, who count from 1 to 68 on different parts of the body. An illustration is given. The body is often used to represent lengths and volumes. Closely related topics: The Kewa People, Number Systems, and Measurement.Historical Exhibit 2.2. Closs, Michael P. Mayan Head Variant Numerals. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 78--79.The Mayan days of the year were associated with gods, and the Mayans used representations of gods for the numbers 0 through 19. Closs shows the "head numerals", identifies the gods, and explains how to recognize them. Excerpted from . Closely related topics: The Maya and Number Systems.III. Human Impact and the Societal Structuring of Mathematics Has articles focusing on the Egyptians, Babylonians, and Greeks.15. 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." Closely related topics: Education, Sumerians and Babylonians, and Ancient Egypt.16. 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. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and Pythagorean Triangles and Triples.17. 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. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.18. 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. Closely related topics: Sumerians and Babylonians, The Circle, Zero, Aristotle, The Measurement of Area and Volume, and Heron.19. 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. Closely related topics: Sumerians and Babylonians, Similarity, and The Triangle.20. 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. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and The Measurement of Area and Volume.21. Gillings, R. J. Problems 1 to 6 of the Rhind Mathematical Papyrus. Mathematics Teacher 56 (1962), 61--69.Discusses problems 1-6 of the Rhind Mathematical Papyrus (or Ahmes Papyrus), where 1, 2, 6, 7, 9, and finally 9 loaves of bread are divided among 10 men. The results are given in terms of unit fractions (if you include 2/3 as a unit fraction). Gillings gives pictures of each of the divisions, and argues convincingly that the division of bread would generally appear to be more fair to the typical (presumably uneducated) ancient Egyptian laborer than a more modern division would be. This is because each laborer would get pieces of both the same number and size, at least if you consider two 1/3 pieces as being the same number and size as one 1/3 piece. (Although Gillings doesn't discuss this, this latter problem could be resolved by replacing 2/3 with 1/2+1/6. This, however, would increase the number of cuts.) Closely related topics: Ancient Egypt and Fractions.22. Gillings, R. J. The Volume of a Truncated Pyramid in Ancient Egytian Papryi. Mathematics Teacher 57 (1964), 552--55.Gillings gives a clever way to derive the formula V=1/3(a2+ab+b2) for the volume of a truncated pyramid, using only the formula for the volume of a complete pyramid and other methods that the Egyptians had at their disposal. As he shows, fairly simple arguments suffice when b=a/2,a/3,..., and also when b=2/3a. Since to the Egyptians, every number could be represented as a finite sum of unit fractions, the demonstration is now complete. Of course we (or the Greeks) would require something like the method of exhaustion. (Even without it, the jump to a general number is a difficult step, and not trivial geometrically.) (Since in the Moscow papyrus, b=a/2, one might wonder if perhaps the Egyptians did not know the general case after all.) Closely related topics: Ancient Egypt, The Pyramid, The Measurement of Area and Volume, and The Method of Exhaustion.23. Eves, Howard. On the Practicality of the Rule of False Position. Mathematics Teacher 51 (1958), 606--8.Eves shows how the method of false position can be simpler than our own methods by giving one example from the Ahmes Papyrus, three from the Greek Anthology of c. 500 AD, and two of his own. One of his examples is from surveying, and Eves says that it is the method a surveyor would probably use. In the other example of his own, he likens the rule of false position to the method of similitude in geometric constructions. Closely related topics: The Method of False Position, Ancient Egypt, Medieval Europe, Surveying, and Geometry.24. Altshiller-Court, Nathan. The Dawn of Demonstrative Geometry. Mathematics Teacher 57 (1964), 163--66.The author argues that it seems unlikely that the Greeks could have invented their notion of proof so rapidly and in isolation. Instead, he suggests that the notion of geometric proof was a secret that was jealously guarded from all but the "inner sanctum" of the Egyptian priesthood. (Of course, since his argument implies by its very nature that Egyptian proofs were unlikely to have been written down, this will be a hard argument to either prove or disprove.) Closely related topics: Geometry, Proof, Ancient Egypt, and Greece.25. Vitrac, Bernard. The Odyssey of Reason. UNESCO Courier (1989), 29--35.The development of Greek schools, the role of mathematics in Greek thought, "pure" and "applied" mathematics, the mathematical community that existed in the Hellenistic era. Includes a passage by Proclus on Geminus' classification of mathemata (the root mathema originally meant "that which is taught", so included all branches of knowledge). Closely related topics: History of Education, Greece, and Applied Mathematics (General).26. Meserve, Bruce E. The Evolution of Geometry. Mathematics Teacher 49 (1956), 372--82.Discusses the history of geometry starting with the Egyptians and Babylonians and continuing into modern times. The rise and decline of Greek geometry, the logical structure of Greek proofs. Contributions by the Islamic world on the parallel postulate. Contributions of Renaissance artists studying perspective. Analytic geometry. More on the parallel postulate. Non-Euclidean geometry. The development of projective geometry and algebraic geometry. The article concludes with a discussion of how computational technology might change the nature of mathematics. Closely related topics: Geometry, Analytic Geometry, Projective Geometry, Algebraic Geometry, Greece, The Islamic World, The Parallel Postulate, and Perspective.27. Jones, Phillip S. Irrationals or Incommensurables. I. Their discovery, and a "Logical Scandal". Mathematics Teacher 49 (1956), 123--27.The discovery of irrationals. Discusses an appealing theory, due to Kurt von Fritz, that the discovery of irrationals grew out of a study of the pentagram. Von Fritz is in support of the traditional theory that discovery or irrationals was due to Hippasus of Metapontum. Closely related topics: Irrationals, The Pentagram, and Hippasus of Metapontum.28. 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. Closely related topics: The Method of Exhaustion, Eudoxus, The Measurement of Area and Volume, and The Circle.29. Swift, J. D. Diophantus of Alexandria. American Mathematical Monthly 63 (1956), 163--70.Discusses the notation, the techniques, and also several problems in Diophantus' Arithmetic. The author finds that Diophantus' methods are similar to those of the Babylonians, and observes that "the work may be viewed as an episode in the decline of Greek mathematics or as the finest flowering of Babylonian algebra." One interesting problem seems to involve an approximation to a square root. Swift also discusses the transmission of Diophantus' work and the resurgence of interest in it in the 1500s and 1600s. There doesn't seem to have been much interest in it in the Hindu or Islamic world. Closely related topics: Diophantus, Indeterminate Equations, Diophantine Equations, and Sumerians and Babylonians.30. Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Closely related topics: Ptolemy (Claudius Ptolemaeus), Trigonometry, and Interpolation.Historical Exhibit 3.4 Swetz, Frank J. The Method of Archimedes. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. . 180--181.Shows how Archimedes used his Method to discover the formula for the volume of a sphere. (Of course Archimedes also gave a rigorous proof using Eudoxus' Method of Exhaustion.) Closely related topics: Archimedes' Method, Archimedes, The Measurement of Area and Volume, and The Sphere.IV. European Mathematics during the "Dark Ages" The author quotes St. Augustine's as stating in 400 AD that "The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bounds of hell." However, mathematics was more developed and respected in the middle ages than this statement might suggest.31. Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Closely related topics: Vitruvius, Architecture, Symmetry, Proportion and the Golden Ratio, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.32. Schrader, Dorothy V. De arithmetica, Book I, of Boethius. Mathematics Teacher 61 (1968), 615--28.Paraphrases Book I of Boethius' De arithmetica, which is in turn based on the Arithmetica of Nichomachus. This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today. Boethius begins what we might think of as modular arithmetic (even and odd, and later evenly-even, evenly-odd, oddly-even), but the classification of numbers and parts of numbers soon acquires an unexpected complexity. The article gives an excellent introduction to the character of Medieval arithmetic/number theory. Closely related topics: Boethius (Ancius Manlius Torquatus Severinus Boetius), Arithmetic, Number Theory, and Nichomachus of Gerasa.33. Sanford, Vera. Counters: Computing if You Can Count to Five. Mathematics Teacher 43 (1950), 368--70.As the author points out, the words calculator and calculus come from the Latin calculus (a small stone). Small stones were used in early counting boards, which were something like loose abacuses. Similar counting boards were used into the 1700s. The author explains how to use one to add, subtract, and multiply using a Roman-numeral type system (so, for example, the counting board has rows for 1, 5, 10, 50, 100, 500, and 1000). Closely related topic: The Abacus.34. Miller, G. A. Gerbert's Letter to Adelbold. School Science and Mathematics 21 (1921), 649--53.Gerbert puts circles and squares inside an equilateral triangle, and attempts to explain why they give different answers for the area. We think of these answers as estimates, but Gerbert's letter contains no hint of a limiting process. Closely related topics: The Abacus, Gerbert, Pope Sylvester II, The Measurement of Area and Volume, and Limit.35. Schrader, Dorothy V. The Arithmetic of the Medieval Universities. Mathematics Teacher 60 (1967), 264--75.The history of the notion of the liberal arts, particularly in the middle ages. The role of arithmetic (computational and theoretical). The abacus of Gerbert. The computation of Easter. The influence of the Arabic texts. Different attitudes towards arithmetic at different times and in different places. An excellent introduction to the mathematics of the middle ages, though of course it omits much on topics such as geometry and astronomy. Closely related topics: The Liberal Arts, Arithmetic, Number Theory, Gerbert, Pope Sylvester II, Religion, Medieval Europe, and The Islamic World.36. Sleight, E. R. The Craft of Nombrynge. Mathematics Teacher 32 (1939), 243--48.As we are told, The Craft of Nombrynge is based on the Canto de Algorismo by Alexander de Villa Dei (1220). It explains how to add, subtract, double, and divide by two, but does not discuss general division or the extraction of roots. (The method of multiplication is essentially the galley method.) Topics are introduced from the Latin Canto, and the remaining text is given in English. Arithmetic (algorism) is attributed to a supposed King Algor of India. Closely related topics: Arithmetic, The Extraction of Roots, Alexander de Villa Dei, and England in the 1400s.37. Sleight, E. R. The Art of Nombryng. Mathematics Teacher 35 (1942), 112--16.The Art of Nombryng is from England in the 1400s, and is a translation of de Arte Numerandi, which was in turn written in the 1200s and is attributed to Sacrobosco. It explains how to do the basic operations of arithmetic, including mediation and duplication, and going as far as the extraction of square and cube roots. Closely related topics: Arithmetic, Sacrobosco (John of Holywood), and England in the 1400s.38. King, Charles. Leonardo Fibonacci. Fibonacci Quarterly 1 (1963), 15--19.A brief survey of the work of Fibonacci, Leonardo of Pisa. Closely related topic: Leonardo of Pisa (Fibonacci).39. McClendon, R. B. Leonardo of Pisa and His Liber quadratorum. American Mathematical Monthly 26 (1919), 1--8.The author discusses some of the most important work in Fibonacci's Liber quadratorum, and convincingly makes the case that Leonardo was the greatest genius in number theory between Diophantus and Fermat. Closely related topics: Leonardo of Pisa (Fibonacci) and Number Theory.40. Smith, Thomas M. Some Uses of Graphing before Descartes. Mathematics Teacher 54 (1961), 565--67.Briefly discusses how graphing was used before the 1600s. The De Configurationibus qualitatum of Nicole Oresme is particularly important in this regard. Oresme even points out that if the two axes represent time and velocity, then the enclosed area represents distance. Closely related topics: Graphing, Nicole Oresme, Dynamics, Force, and Motion, and Calculus.Historical Exhibit 4.2 Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Religion.V. Non-Western Mathematics41. Kokomoor, F. W. The Status of Mathmatics in India and Arabia during the "Dark Ages" of Europe. Mathematics Teacher 29 (1936), 224--31.A survey of some of the work in mathematics during the middle ages. The focus is on the Islamic world. Closely related topics: The Islamic World, India, China, Medieval Europe, and The Middle Ages.42. Rashed, Roshdi. Where Geometry and Algebra Intersect. UNESCO Courier (Nov., 1989), 37--41.The interaction of Islamic algebra with algebra and geometry. Ways in which Islamic methods anticipated discoveries in Europe that were centuries later. Examples include the solution of cubics with intersecting curves (al-Khayyam, often attributed to Descartes) and the notion of maxima and minima of an algebraic expression (al-Tusi). Closely related topics: The Islamic World, Algebra, Number Theory, Geometry, Analytic Geometry, and Calculus.43. Shloming, Robert. Th\^abit ibn Qurra and the Pythagorean Theorem. Mathematics Teacher 63 (1970), 519--28.Discusses the life and work of Th\^abit ibn Qurra, focusing on his work on the Pythagorean Theorem. Th\^abit gave two proofs of this theorem (both independently rediscovered in the early 1900s), and also a generalization to triangles that are not necessarily right-angled (independently rediscovered about 1665 by John Wallis). The author also discusses the Ishaq-Th\^abit translation of Euclid's Elements, which was the basis for the translation by Gerard of Cremona. Closely related topics: Th\^abit ibn Qurra and Pythagorean Triangles and Triples.44. Arndt, A. B. Al-Khwarizmi. Mathematics Teacher 76 (1983), 668--70.An introduction to the work of al Khwarizmi. Focuses on his algebra, the Al-Kitab Al-jabr wa'l muqabalah and its influence on the West. Closely related topic: Abu Abdullah Muhammed ibn Musa al Khwarizmi.45. Pazwah, Hormoz; Mavrigian, Gus. The Contributions of Karaji---Successor to al-Khwarizmi. Mathematics Teacher 79 (1986), 538--41.An introduction to the work of al Karaji (often known as al Karkhi). Includes a little on arithmetic, algebra, geometry, and surveying. Closely related topic: Abu Abdullah Muhammed ibn Musa al Khwarizmi.46. Struik, D. J. Omar Khayyam, Mathematician. Mathematics Teacher 51 (1958), 280--84.An excellent introduction to the work of Omar Khayyam. He discusses Omar's solution of cubic equations (by intersections of cubics), his possible understanding of the binomial theorem (occurring in the the work of al Kashi, and later in the work of Michael Stifel), his work on the parallel postulate (including his reduction to the cases of an acute angle, an obtuse angle, and a right angle; his proof of the parallel postulate rests on other axioms, including the axiom that a straight line can be indefinitely prolonged and on the Axiom of Archimedes), and his implicit recognition that a ratio can be expressed by ratios of integers to any desired degree of accuracy. He closes with his view of the real flesh and blood Omar Khayyam. Closely related topic: Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam).47. Eves, Howard. Omar Khayyam's Solution of Cubic Equations. Mathematics Teacher 51 (1958), 285--86.Shows how Omar Khayyam solved the equation x3+b2x+a3=cx2 using the intersection of a circle and a rectangular hyperbola. Closely related topics: Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Cubics, and The Conic Sections.48. Zimmermann, Francis. Lilavati, Gracious Lady of Arithmetic. Unesco Courier 51 (Nov., 1989), 18--21.Discusses Bhaskara's Lilavati, but the main interest really seems to be on the character and context of Indian mathematics. Excellent brief introduction. Closely related topics: India and Bhaskara.49. Aiyar, S. Balakrishna. The Ganita-S\=ara-Sangraha of Mah\=av\=\i r\=ac\=arya. Mathematics Teacher 47 (1954), 528--33.An overview of Mahavira's Ganita-Sara-Sangraha. The author makes the interesting observation that in Jainism, Mahavira's religion, mathematics was very popular, and was "accorded the status of one of the four anuyog\=as, which were the auxiliary sciences, the study of which helped the aspirant to the attainment of soul-liberation." Closely related topics: Mahaviracarya and Arithmetic.50. Martzloff, Jean-Claude. Pi in the Sky. Unesco Courier (Nov., 1989), 22--28.Very brief. Includes a bit on the influence of divination, astronomy/astrology, Confucianism, and Taoism on the development of Chinese mathematics. The emphasis on the answer rather than the proof shows a Taoist influence, "on the grounds that the fallacious arguments of the sophists showed its limits". Also a bit on how mathematics and mathematicians fit into Chinese society. Closely related topics: China, Divination, Astronomy, Astrology, Confucianism, and Taoism.51. Swetz, Frank. The Evolution of Mathematics in Ancient China. Mathematics Teacher 52 (1979), 10--19.An overview of Chinese mathematics, including the discovery of the lo shu magic square (thought to have a plan of universal harmony), square roots, the Chinese remainder theorem, and polynomials of high degree (including a quintic in x2). Closely related topics: China, Algebra, and Magic Squares.52. Swetz, Frank. The Amazing Chiu Chang Suan Shu. Mathematics Teacher 65 (1972), 423--30.A chapter by chapter survey of the Chiu Chang Suan Shu. One interesting example (of many) is a pursuit problem, which anticipated Alcuin's hound-pursuing-a-hare problem by nearly a thousand years (Chapter VI). Swetz' article leads one to understand how the president of the Bureau of Foreign Affairs in China (Prince Kung) might have felt justified in his claim (in the 1860s) that "Western sciences borrowed their roots from ancient Chinese mathematics. Westerners still regard their mathematics as coming from the Orient. It is only because of the careful, inquiring minds of the Westerners that they are good at developing something new out of the old... China invented the method, Westerners adopted it..." Closely related topic: The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art).53. Swetz, Frank. The "Piling Up of Squares" in Ancient China. Mathematics Teacher 70 (1977), 72--79.Chapter IX of the Chiu Chang Suan Shu has a series of interesting problems on the Pythagorean Theorem, many requiring a little resourcefulness to solve, even today. Two methods are used in Chapter IX. This article discusses one of these, the Chi-Chü, or "piling up of squares". This is a dissection method; thus areas are disassembled and reassembled in a different way. The author gives several examples. The last two are among the most interesting. They find the largest square and circle that can be drawn in a right triangle; only the case where the square includes the right angle seems to be considered. The methods are ingenious, and would make appealing classroom demonstrations. The Chi-Chü method is also used in problems that at first seem to have little to do with areas. Problem 14 is an example:54. Jones, Phillip S. From Ancient China 'til Today!. Mathematics Teacher 49 (1956), 607--10.Two men starting from the same point begin walking in different directions. Their rates of travel are in the ratio 7:3. The slower walks towards the east. His faster companion walks to the south 10 pu and then turns towards the northeast and proceeds until both men meet. How many pu did each man walk?The author also discusses problem 6, the famous problem of a reed in a square pond:In the center of a square pond whose side measures 10 ch'ih grows a cattail whose top reaches 1 ch'ih above the water level. If we pull the reed toward the bank, its top becomes even with the waters surface. What is the depth of the pond and the length of the plant?As the author observes, this problem is very similar to a much later problem of Bh\=askara, where even the ratios involved are the same:In a certain lake, swarming with red geese, the tip of a bud of a lotus was seen a span (9 inches) above the surface of the water. Forced by the wind, it gradually advanced and was submerged at a distance of two cubits (approximately 40 inches). Compute quickly, mathematician, the depth of the pond.The question of Chinese influence on Indian mathematicians is still unsettled. One can't but wonder how the Chinese became so amazingly successful with the Chi-Chü method. The author mentions the possibility that familiarity with the tangram exercises may have contributed to their skill. Excellent article. Closely related topics: The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), Pythagorean Triangles and Triples, and The Tangrams.Discusses Chinese remainder problems and their connection with topics such as the Euclidean algorithm and continued fractions. The history is not examined in depth. Closely related topic: Chinese Remainder Problems.