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PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

AMUSEMENTS IN THE HISTORYOF MATHEMATICSShirley B. Gray PhD a & Robert A. Mena ba Department of Mathematics and ComputerScience , California State University-Los Angeles ,5151 State University Drive, Los Angeles, CA,90032-8204, USA E-mail:b Department of Mathematics , California StateUniversity-Long Beach , 1250 Bellflower Blvd., LongBeach, CA, 90840, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Shirley B. Gray PhD & Robert A. Mena (1997) AMUSEMENTSIN THE HISTORY OF MATHEMATICS, PRIMUS: Problems, Resources, and Issues inMathematics Undergraduate Studies, 7:4, 317-328, DOI: 10.1080/10511979708965873

To link to this article: http://dx.doi.org/10.1080/10511979708965873

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Gray and Mena Amusements in the History of Mathematics

AMUSEMENTS IN THE HISTORYOF MATHEMATICS

Shirley B. Gray! and Robert A. Mena''

ADDRESS: (1) Department of Mathematics and Computer Science, Cali­fornia State University-Los Angeles, 5151 State University Drive, LosAngeles CA 90032-8204 USA. sgray@calstatela.edu and (2) Depart­ment of Mathematics, California State University-Long Beach, 1250Bellflower Blvd ., Long Beach CA 90840 USA. rmena@csulb.edu.

ABSTRACT: Dr. Gray teaches a one quarter course and Dr. Mena teachesa two semester sequence in the History of Mathematics. Their coursesemphasize diversity and multiculturalism. We have developed severalactivities that engage the students and promote discussion. The fol­lowing examples are included in this paper: (1) Crossword Puzzle;(2) Date the Event; (3) Quotes Matching; (4) Famous MathematicalExpressions; (5) Calendar Scramble; and (6) Map Puzzle.

KEYWORDS: Activities, names, dates, quotes, expressions, calendars,map.

The History of Mathematics extends over thousands of years and hasinvolved millions of scholars and students. No single course could ever dojustice to all of the people and events that have been part of this rich history.In courses at the California State University, both at Los Angeles and LongBeach, we have used a series of fun puzzles as reviews of our discussionsof the most important people, places, and discoveries in mathematics. Ourexperience has been that working these puzzles helps late 20th centuryAmerican students grasp the amazing range of time and geography oursubject encompasses. We hope that your students enjoy these puzzles asmuch as our students have.

Part I. CROSSWORD PUZZLEAlthough a History of Mathematics course is much more than a collection ofnames and facts, the latter still form an essential ingredient of any historical

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December 1997 Volume VII Number 4

ACROSS - (continued)40. The element associated withthe Platonic solid with eightsides.42. p;43. The artistic center of theGreek world.44. The author of the most im­portant book in the history ofmathematics.45. A synonym for proportion.48. The geometric mean of thesides of a is the side

of the square with the same area.50. o , ct, ct.51. What a good one of 91 Acrossmust eventually do.52. The name of the inscribedpolygon whose side is the radius.54. A, the conjunction.55. His formula produces thearea of a triangle from thelengths of the sides.56. The fourth astral body in theancient ordering.57. The arithmetic or the geo­metric or the harmonic.58. What A is not.61. one-__-one.63. Thus.64. What the side and the diag­onal of a square are not.67. The circumference of it was

measured by 2 Down.68. For example.69. 216 + 1 __ a prime.70. Twice, two-fold.73. H, to Don Quijote.

a .

19. Article.20. Modus Operandi.22. The first curvilinear shapesthat were squared.23. That man.24. Ego, superego and __.26. Epicycles were created asa commitment to this perfectshape in astronomy.27. Not subtract.28. Shade of color.31. City in the land between therivers.33. The inside angle of this poly­gon is 108°.35. The most fundamental con­stant in nature.36. The opposite of the power ofa number.39. Neither, .

course. In order to aid the student in the recollection of the names, weenclose a crossword used for the first period in early mathematics history.It particularly emphasizes the Greek times, although it dabbles in othercultures and periods.

ACROSS1. ... the two straight lines, ifproduced indefinitely, . . .3. The name of the Gou Gu The­orem in the West.7. Our knowledge of the math­ematics from this area stemmainly from a few papyri such asthe Ahmes papyrus.10. One of the curves studied byApollonius.12. The most important book inthe history of mathematics.16. ~.

17. The zero numeral comes fromthis region.18. To the early Greeks, a prod­uct of three numbers represented

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e........CIlElill

...cl....CIl~....0

i:'0....~ill

...cl....

.9(I)....=ill~

0)El .-lill M(I)

='El<

CIl

=ill~-e~

~...C>

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December 1997 Volume VII Number 4

ACROSS - (continued)76. The element associated with25 Down.77. A major work by 35 Across.79. I ___ , I understand.81. 365.2425 days according toGregory.82. Every.

DOWN2. See 67 Across.3. The most influential as­tronomer of antiquity.4. The scientific center of theGreek world.5. The civilization that overtookthe Greek.6. 232 + 1 is prime.8. Ratio.9. The Egyptians most remark­able achievement was the compu­tation of its volume.11. It has a chord.13. Breadthless length.14. Antonym to always.15. objects have thesame shape.21. Not even.23. He performed 22 Across.25. The Platonic solid withfewest sides.29. Eins.30. The Platonic solid with thefaces having the most number ofsides.32. It and the straightedge con­stituted the Euclidean tools.34. The product of two numbersto a Greek.37. You to Julius.38. The author of Achilles andthe Tortoise.41. The most influential writeron number theory.

320

ACROSS - (continued)84. What this puzzle is about.89. The last of the Greek geome­ters.90. Half-chord.91. Procedure named after al­Khwarizmi.

DOWN - (continued)45. Sun God to Nefertiti.46. The Platonic solid with themost number of sides.47. The best scientific mind ofantiquity.49. Not the hypotenuse.53. A very modern word forlength.57. Hypatia was killed by one.59. Top of the deck.60. Modern examples of pros­thaphaeresis.61. T.

62. One of the building blocks ofall numbers.65. The residual of a base twentyancient system is reflected inFrench when this number iscalled four-twenties.66. The smallest base in whichone can divide easily by the firstsix numbers.71. That is.72. Piece of a circle.74. What the modern symbol E

means.75. He considered circles to bethe most perfect of curves.76. Comes after 88 Down.78. x-80. 11.82. Not a science.83. He gave us 'If F::l ~SS.

85. Abbreviation for ~.

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Gray and Mena Amusements in the History of Mathematics

DOWN - (continued)86. The seventh and last one af­ter 88 Down.

DOWN - (continued)87. Surjective or __ to.88. It follows DO ***

Part II. Date the Event

Construction, trade, warfare and technology have greatly influenced theHistory of Mathematics. Earlier, the technology of printing, and today, thepower of the computer and calculator, are altering our professional endeav­ors. A chronology is important for a grasp of developments over centuriesof time. Match the date to the event impacting mathematics.

2600 B.C . 1. Al-Khowarizml's treatise, Hisb al-jabr w'almuqbalah, the source of our word "algebra."

150 B.C. 2. Crelle'" Journal published. First journaldevoted exclusively to mathematical research.

105 A.D. 3. Great Pyramid at Gizeh, near Cairo, Egypt,constructed.

ca. 400-500 4. Nine Chapters on the Mathematical Artwritten in China. The oldest textbook onarithmetic.

ca. 820 5. Earliest successful attempts in India to finda general solution for the indeterminate equa­tion ax + by = c.

1202 6. Johannes Gutenberg's invention of printingwith movable, metallic type.

ca. 1450 7. Fibonacci introduces the Hindu-Arabicnumber system in Europe.

1826 8. First use of hemp, tree bark, fish nets andrags to produce paper in China.

1972 9. Altair 8800, first successful personalcomputer.

1975 10. HP-35; first hand-held scientific calculator.Question 9 was a Jeopardy question.

Part III. QUOTES MATCHING

Quotes adorn our libraries and public buildings. It is nice to recognizethe verbal hallmarks of our masters. The following are quotes by or aboutmathematicians. Match each expression to its author.

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December 1997 Volume VII Number 4

__ Plato

__ Kepler

__ Pythagoras

__ St. Augustine

1. Inscription over the Academy door:"Let no man ignorant of geometry enterhere."2. "Know thyself."3. "All is number."4. "Eureka, Eurekal" "Give me a place tostand, and I will move the earth."5. "The good Christians should beware ofmathematicians and all those who makeempty prophecies. The danger already ex­ists that the mathematicians have madea covenant with the Devil to darken theSpirit and confine man in the bonds ofHell."6. "There is no royal road to geometry.""Give this man a coin since he must makea profit from what he learns."7. "Let no one who is not a mathematicianread my works."8. "Geometry has two great treasures: oneis the Theorem of Pythagoras; the other,the division of a line into extreme andmean ratio. The first we may compareto a measure of gold; the second we mayname a precious jewel."9. "Still it moves."10. "If I have seen farther than other, itis because I have stood on the shoulder ofgiants."11. "Cogito ergo sum."12. "Marks of the ABSURD GEOM­ETRY, RURAL LANGUAGE, SCOT­TISH CHURCH POLITICS AND BAR­BARISMS OF JOHN WALLIS." (title ofmathematics paper)

Part IV. MATCHING MATHEMATICALEXPRESSIONS

__ Galileo__ Newton to Hooke

__ Descartes__ Hobbs

__ Leonardo da Vinci

__ Archimedes__ Euclid__ Thales

Each of the following mathematical expressions is inextricably associatedwith a name, mainly of their creator. To know the expression is to know asignificant contribution. Match the expression to the name.

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Gray and Men a Amusements in the History of Mathematics

1. Maria Agnesi2. Archimedes3. Leonhard Euler4. Pierre de Fermat

5. Benoit Mandelbrot6. The Golden Ratio

....._~~ - 2

P =a9x n + yn = zn

e"; + 1 = 08a3

y = 4a'+:r'f(x) = z2 + c

Part V. CALENDAR SCRAMBLE

The calculation of the number of days in the year has, from ancient times,been an important motivator for arithmetical activity and notation. Thefollowing form a small sample of the variety of cultures and solutions thathave been involved with the problem. Choose one item from Column A, oneitem from Column B, one item from Column C, and one item from ColumnD, that best match the number on the left .

a)

b)

c)

d)

e)

g)

#

365

365.25 or365t

365 .244407or 365">;'...

365 .2420

365 .2424 .. . or365"13,

365 .2425 or365·'/_

365 .242193

A

The number of daysIn the ear In

today'smeasurements

Omar Khayyam'sCalendar

most ancient solarcalendars

the Julian Calendar

the GregorianCalendar

the ChineseCalendar

the Mayan Calendar

B cWhere

Mexico­Guatemala

Egypt

Rome

Rome­Alexandria

China

The World

Persia

D

When

Twentieth

Century

Eleventh

Century

Fir5t

Century

Be

Ancient

Ninth

Century

Si>eteenth

Century

EI<jhth

Century

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December 1997 Volume VII Number 4

Part VI. MAP PUZZLE

Famous mathematicians are often associated with a certain city or univer­sity. Select the number on the map that corresponds to each of the followingpeople.

Niels AbelArchimedesDescartesEuclidEulerGalileo

324

GaussKeplerNewtonPascal, Mersenne and CauchyPythagoras

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IIIU

~SQJ

...c:l....c'd~....0

~0

00....~ ~

QJ H...c:l 0....

Z.ElIII 0....= ZQJ

LQS < c:"lQJ~III =;::l 00 0S ~ ....

< ~

~ ::l

~ -000 00

Z "'0< ~

~c'd ~= 0QJ ~

~ U-e

~~

~

~ ~...o ~

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December 1997 Volume VII Number 4

Part II. Date the Event

3, 4, 8, 5, 1, 7, 6, 2, 10,9 in that order.

Part III. Quotes Matching

1, 4, 6, 2, 8, 7, 3, 5, 11, 12, 9, 10 in that order.

Part IV. Matching Famous Mathematical Expressions

6, 2, 4, 3, 1, 5 in that order.

Notes:

1. The curved named "Witch of Agnesi."

2. The curve in polar coordinates named "Spiral of Archimedes."

3. Combines in one expression the three most important constants andthe two most important numbers in mathematics.

4. Fermat's Last Theorem, also associated with Andrew Wiles' proof.

5. The mathematical expression used for generating fractals, i.e., theMandelbrot Set.

6. The Greek letter "~" is used to honor Phidias, the builder of theParthenon.

Part VI. Map Puzzle

7 (Oslo, Norway), 3 (Syracuse in Sicily, Italy), 10 (The French town has beenrenamed " Descartes" ,whose picture is on the 100 franc note), 2 (Alexandria,Egypt), 11 (Basel , Switzerland. Euler's picture is on the 10 Swiss francnote), 4 (Pisa/Florence, Tuscany), 6 (Gottingen, Germany. Gauss' picture ison the 10 mark note), 5 (Graz, Austria), 8 (Cambridge, England. Newton'spicture is on the 1 pound note), 9 (Paris, France), 1 (Greek island of Sam08)in that order.

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Gray and Mena Amusements in the History of Mathematics

Part V. Calendar Scramble

A B C D

# Number of days in Where Whenthe year in

a) 365 most ancient solar Egypt Ancientcalendars

b) 365.25 or Julian Calendar Rome- 1st Century BC365 1 Alexandria

c) 365.244407 Chinese Calendar China 8th Centuryor 365~

d) 365.2420 Mayan Calendar Mexico- 9th CenturyGuatemala

e) 365.2424 . or Omar Khayyam's Persia 11th Century365!. Calendar

f) 365.2425 or Gregorian Calen- Rome 16th Century365!k dar

g) 365.242193 today's measure- The World 20th Centuryments

RELATED WORLD WIDE WEB SITES

Web Sites. There are now a large number of web sites featuring history ofmathematics. As a beginning , we suggest the following:http://vww-groups.dcs.st-and uk/ history/http://alephO.clarku.edu:80/ djoyce/nathhist/mathhist/htmlhttp://archives.math.utk.edu/index.htmlhttp://vww.scottlan.edu/lriddle/women/women.htmhttp://sunsite.unc.edu/expo/vatican.exhibit/exhibit/

Hain-Hall. htmlhttp://lcweb.loc.gov//rr/rarebook/guide/ra057001.jpghttp://forum.8warthmore.edu/math.history

REFERENCES

1. Boyer, C. B. 1991. A History of Mathematics, 2nd ed.. Revised byU. C. Merzbach. Princeton NJ: Princeton University Press.

2. Burton, D. M. 1995. Burton's History of Mathematics: An Introduc­tion, 3rd ed. Dubuque IA: Wm. C. Brown.

3. Davis, P. J. and R. Hersh. 1981. The Mathematical Experience.Boston MA: Houghton MifHin.

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December 1997 Volume VII Number 4

4. Eves, H. 1992. An Introduction to the History of Mathematics, 6thed. New York NY: Saunders College Publishing.

5. Calinger, R. ed. 1995. Classics of Mathematics. Englewood CliffsNJ: Prentice-Hall.

6. Calinger, R. ed. 1996. Vita Mathematica: Historical Research andIntegration with teaching. Washington DC: Mathematical Association ofAmerica.

7. Dauben, Joseph W. 1989. Georg Cantor: The Battle for TransfiniteSet Theory, A videotape. AMS-MAA Joint Lecture Series, Atlanta, Georgia­January 1988, American Mathematical Society.

8. Fauvel, J. and J. Gray, eds. 1987. The History of Mathematics: AReader . London: Macmillan/The Open University.

9. Gillispie, C. C. ed. 1970. Dictionary of Scientific Biography. NewYork: Scribners. 18 v.

10. Hurd, S. P. 1991. Egyptian Fractions: Ahmes to Fibonacci to Today.Mathematics Teacher. 84: 561-8.

11. Katz, V. J. 1993. A History of Mathematics: An Introduction. NewYork: Harper Collins.

12. National Council of Teachers of Mathematics. 1991. Student MathNotes. March.

13. Swetz, F. J. 1989. Using Problems from the History of Mathematicsin Classroom Instruction. Mathematics Teacher . 82: 370-77.

14. Swetz, F. J. 1995. Some Not So Random Thoughts about the Historyof Mathematics: Its Teaching, Learning and Textbooks. PRIMUS. 5(2):97-107.

15. Struik, D. J. ed. 1969. A Source Book in Mathematics, 1200-1800.Cambridge MA: Harvard UDiversity Press. Reprinted by Princeton Univer­sity Press, 1986.

BIOGRAPHICAL SKETCHES

Shirley B. Gray received her PhD from the University of Southern Californiain 1990. She has published fractal and HyperCard programs. Currently sheis producing and directing videotapes on using a graphing calculator. Shehas taught a History of Mathematics course many times. In the Summer of1996, she attended the Institute in the History of Mathematics and Its Usein Teaching, sponsored by a grant to the MAA by the NSF.

Robert Mena has been teaching at CSULB since 1988 after 15 years atthe University of Wyoming. His favorite courses are the History of Mathe­matics, Statistics and Combinatorics.

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