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Home , Carl Friedrich Gauss, John Napier, Maria Gaetana Agnesi, Niels Henrik Abel

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MATHEMATICAL VignetteS The People Behind the Proofs

f you teach math, maybe you love for its own When my students solve a virtual equation instead of merely sake; I do. Or maybe you’ve been assigned to teach the sub - factoring it as requested, I remind them of Christ’s injunction: ject. Have you felt that no matter how you teach math, it “‘If any man will sue thee at the law, and take away thy coat, let comes across as dry as the proverbial hills of Gilboa? Do him have thy cloak also. And whosoever shall compel thee to you find yourself discouraged at hammering basic skills go a mile, go with him twain.’” 1 Then I say, “In our Christian Iinto reluctant minds class period after class period? Been there! walk, we are to do more than required by the law. But in math - Done that! ematics, you should never do more than you are asked to do!” The good news is that we math teachers can enliven our As teachers, we often hear statements like, “I don’t under - classes by sprinkling our math instruction with bits of humor stand it!” or “It’s just not true.” I remember as a child trying to or biography to break the monotony. A joke or pun here and come to terms with my mother’s admonition that “two wrongs there, no matter how big a groan it elicits, at least recaptures don’t make a right,” with the fact that a negative times a nega - students’ attention. If we can throw in a (short) story now and tive is a positive. This was an irreconcilable philosophical par - then, our students will love us for it. After all, mathematics is adox to me. much more connected with the humanities than with the sci - A working at the Los Alamos National Laboratories ences, in spite of popular opinion to the contrary. If we can consulted the great 20th-century , John von bring out the human side once in a while, some of our students Neumann, about a problem. Von Neumann stated the problem may warm ever so slightly to the subject! was a simple application of the mathematical method of char - Allow me the liberty of suggesting some ways of doing this. acteristics. The physicist immediately confessed that he didn’t In , I often ask my students to factor something like understand this method. x2 – 5 x – 6. Distressingly frequently, I get the answer –1, 6. What Von Neumann replied, “Young man, in mathematics you they have done is create the virtual equation x2 – 5 x – 6 = 0 and, don’t understand things, you just get used to them.” 2 after factoring the expression, solved it for x. They have done I confess that I have never reconciled philosophically that a more work than I requested. Furthermore, I have to deduct negative times a negative is a positive. But I have long since got - points for their failure to precisely follow instructions. ten used to the concept.

BY WIL CLARKE

14 The Journal of Adventist Education • April/May 2013 http://jae.adventist.org We can also find a few tales to tell about mathematicians. When I find such a story, I write it on a sticky note and place it in an appropriate place in my teacher’s edition. These stories Carl Friedrich stimulate students’ interest. When you run across a mathemati - cian’s name in the textbook, it’s easy to go to the Internet and search for him or her. Beware—since the Internet is not refer - eed, there is always a danger that erroneous material can be and has been inserted. 3 Here are some examples of how I use math - ematician stories.

A Young Mathematical — When I talk about adding a of , such as 1 + 2 + 3 + 4 + . . . , I tell my students about Carl Gauss who, after misbehaving in class, was told to add the numbers between 1 and 100. 4 His teacher, hoping for some peace and quiet for a while, turned back to his work. However, almost immediately, young Gauss inquired, “Is it 5,050?” (This was the correct an - Blaise swer.) I then show his reasoning. This same precocious Gauss at the tender age of 3 corrected his merchant father’s payroll! 5 Including this anecdote not only allows me to introduce the techniques for calculating series, but also to show that a mathematician could have had behavior problems!

Religion and Mathematicians Many mathematicians were deeply religious. This may come as a surprise to many students. Here are a few examples.

Blaise Pascal When we teach the or combinatorics, we Augustin-Louis Cauchy encounter the Pascal Triangle. (See Figure 1 on page 16.) Each entry in the table is obtained by adding the two num - bers immediately above and to the right and left of the position. As I teach about these connections, I often give my students a brief account of Pascal’s spiritual struggle with mathematics. (1623-1662) grew up in a God-fearing home. Very early, he showed an interest in mathematics. His disapprov - ing parents removed all math books from their home, which only whetted his appetite. By age 14, Pascal had invented a digital in - strument to help his father calculate the taxes he had to collect. By the age of 16, he had presented his first paper on mathematics, including elements of projective . 6 However, he had an uneasy relationship with mathematics. His early training led him to believe that doing mathematics was wrong. And so, several Maria Gaetana Agnesi times in his life, he confessed this interest in math as a sin and abandoned it—only to be attracted to it again. Finally, during an electrical storm, the horses pulling Pascal’s carriage bolted and left the carriage hanging precariously over the edge of a bridge. Pascal was rescued, unharmed, which he

http://jae.adventist.org The Journal of Adventist Education • April/May 2013 15 regarded as a direct communication from God. This time, he tended much of ’s work. The witch is a gave up his dalliance with mathematics forever and started she discussed in an unpublished commentary on Guillaume writing religious works. His work Pensées , which describes his L’Hospital’s work. 12 thoughts about God and His dealings with humanity, has be - Far from being a witch, Agnesi was deeply religious. At the come a classic that is well known throughout the world. 7 age of 20, she desired to enter a convent. Her father forbad it. Later, she was appointed to the chair of mathematics at the Augustin-Louis Cauchy University of . There is no record of her ever lecturing Another religious mathematician was Augustin-Louis there, however. This is undoubtedly due to the fact that women Cauchy. did not lecture at universities in those days. An example of her For more than 100 years after invented , brilliance: As a child, Agnesi mastered at least seven languages. 13 mathematicians tried unsuccessfully to provide a rigorous After publishing her mathematical works and after her father’s foundation for it. When young Augustin-Louis Cauchy first death, she turned her intellect to studying . In her later started teaching, he expressed an interest to apply the rigor of years, Agnesi devoted her attention to ministering to the poor, to the calculus. He wrote his own calculus homeless, and sick. She worked very closely with the Christian textbook, Cours d’Analyse, to do ex - ministry of the Blue Nuns. There are actly that. 8 conflicting reports as to whether she We trace the branch of mathemat - Figure 1. 1 actually entered their order later in ics called analysis to Cauchy’s Pascal Triangle 1 1 life. 1 2 1 ground-breaking work. Mathemati - 1 3 3 1 cians regard Cauchy with great es - 1 4 6 4 1 Sir teem today, although this was not al - 1 5 10 10 5 1 When teaching , we ways so. 1 6 15 20 15 6 1 eventually get to the product-to-sum A devout Catholic, Cauchy used 1 7 21 35 35 21 7 1 identity: this devotion as a standard by which 1 8 28 56 70 56 28 8 1 cos x cos y = ½ [cos( x + y ) + he criticized his colleagues regarding cos( x – y )] their alleged laxness in religious mat - This formula led John Napier to an ters. Other mathematicians hated Figure 2. “Aha” experience. He asked himself, him for this, and some even called “Why can’t I use this formula to create him mad. 9 It appears that this critical a table of numbers that will allow me religious trait kept Cauchy from get - to turn multiplication into addition?” ting at least one prestigious appoint - After over 20 years of hard work, he ment he had sought. 10 created a table of . 14 How - I use this story when teaching cal - ever, this phenomenal effort possibly culus or analysis to encourage stu - contributed to his losing his mind to - dents to stand for the right even if it ward the end of his life. is unpopular. The table of logarithms he created allowed anyone to turn multiplication into addition and division into subtraction. Maria Gaetana Agnesi When we think about addition versus multiplication of four- Maria Gaetana Agnesi is another mathematician whose digit numbers without a , we suddenly realize what a story I introduce to my students. Every calculus student has en - marvelous invention Napier had made, which enabled even the countered the bell-shaped curve called the Witch of Agnesi, sea captain unskilled in mathematics to calculate his position which looks something like Figure 2. in the vast ocean much more accurately. If the captain of the The infamous name for this curve comes from John Col - Mayflower had had access to this table, he probably would have son’s mistranslation of the Italian word versiera , a term used landed the pilgrims in Virginia—their goal—instead of Mas - for a rope on a sailing vessel. The name witch has stuck, despite sachusetts, and the history of America would have been very its inaccuracy. 11 different! Maria Agnesi became very interested in mathematics as a In fact, the use of logarithms enabled the British navy to child. She read Newton eagerly and eventually published a two- land their ships and army accurately anywhere on earth. volume text Analytical Institutions (Instituzioni Analitiche ad If your students enjoy trivia, you can share the fact that uso Della Gioventu Italiana ) in which she presented and ex - Napier spelled his name in several different ways. However,

16 The Journal of Adventist Education • April/May 2013 http://jae.adventist.org never once did he spell it “Napier” as we do today. In his formal writing, which was always in , he always spelled it “Nepe - rus.” Sir John Napier was born in 1550 at the beginning of the Sir John Napier Scottish . After becoming an evangelistic Protes - tant, Napier wrote A Plaine Discovery of the Whole Revelation of Saint Iohn, which he regarded as his greatest work. 15 He thus can be placed as a direct intellectual ancestor of the Millerite movement and Adventism. 16

Sir When doing math, we often use the calculator’s square-root button, but have you ever wondered how the calculator finds the square root so quickly? The algorithm it uses comes from one invented by Sir Isaac Newton to solve difficult equations numerically. Newton is, of course, better known for his discoveries relat - ing to calculus and gravity. 17 During the winter of 1665/1666, Sir Isaac Newton England was struck by a vicious strain of plague. The closed, and Newton, soon to be one of its most famous students, went home to the family farm. The year 1666 has come to be known by the Latin term annus mirabilis or the “miracle year.” 18 During that brief period, Newton developed calculus, the theory of gravitation, the laws of motion, and his theory of light. We have a delightful story from Newton’s youth. After the death of his stepfather, who hated and banished him, Newton went home. His mother asked him to cut a hole in the barn door so the cat could get in and out. After cutting the hole, Newton noticed that the cat had a bunch of kittens, so out of concern for them, he cut a smaller hole next to the larger one Augustus De Morgan so they could get in and out, too! An avid student of the prophecies of Daniel and Revelation, Newton wrote The Prophecies of Daniel and the Apocalypse, which was published posthumously. 19 So he, too, is a direct an - cestor of Adventism. LeRoy Edwin Froom quotes frequently from both Napier and Newton in his definitive four-volume work The Prophetic of Our Fathers .20

Augustus De Morgan Augustus De Morgan was the first mathematician to give a rigorous formulation of the principle of mathematical induc - tion. This is an extremely powerful tool that is often used in proving theorems in all branches of mathematics. His name is also attached to a set of laws in logic and set theory. De Morgan seemed to appear out of obscurity. Although he was regarded as a brilliant student of mathematics at Cam - bridge University, in order to receive his M.A. there, he had to sign a document that he believed certain theological doc - trines. 21 He did not accept the wording of one or more of these

http://jae.adventist.org The Journal of Adventist Education • April/May 2013 17 doctrines, so he refused to sign. For this , he never re - Wil Clarke is Professor of Mathematics at ceived his M.A. He published several mathematical books but La Sierra University in Riverside, Califor - was barred from the upper levels of mathematical research by nia. He earned a B.A. in mathematics from his decision. He stands as an example of someone who lived by Andrews University in Berrien Springs, his conscience. Michigan, and a Ph.D. in mathematics at He stated in his will: “I commend my future with hope and the University of Iowa. He has also taught confidence to Almighty God; to God the Father of our Lord mathematics at Ikizu Secondary School in Jesus Christ, whom I believe in my heart to be the Son of God Tanzania, Helderberg College in South but whom I have not confessed with my lips because in my time Africa, and Atlantic Union College in Massachusetts. such confession has always been the way up in the world.” 22

Niels Henrik Abel NOTES AND REFERENCES In algebra classes, students learn to solve equa - 1. Matthew 5:40, 41, KJV. 2. Gary Zukav, The Dancing Wu Li Masters: An Overview of the New tions. About 200 years after Christ, Diophantus developed the (Toronto, Canada: Bantam Books, 1980), p. 208. procedure for solving quadratic equations. He was the first 3. See http://en.wikipedia.org/wiki/Wikipedia, where concerns are ex - mathematician to use variables to represent numbers. In the pressed about the “quality of writing and the amount of vandalism” that can middle 1500s, Giralomo Cardan (or Cardano) published the be found there. methods for solving cubic equations in his book Ars Magna . 4. Various accounts use different numerical sequences, cf., E. T. Bell, Men Shortly thereafter, mathematicians discovered how to solve of Mathematics (New York: Simon & Schuster, 1937), p. 221; Carl B. Boyer, re - vised by Uta C. Merzbach, A (New York: John Wiley & quadratic (fourth-degree) equations. This started a grand Sons, 1991), 2nd edition, p. 497; and G. Waldo Dunnington, Carl Friedrich search for the general solution of a fifth-degree equation. Gauss: Titan of (New York: Exposition, 1955), p. 12. The young Norwegian mathematician Niels Abel concluded 5. Biographical Dictionary of Mathematicians (New York: Charles Scribner, this search once and for all by proving that it is impossible to 1991), vol. 2, p. 861. solve fifth or higher degree equations by the method of radicals. 6. Howard Eves, Great Moments in Mathematics After 1650 (Washington, D.C.: Mathematical Association of America, 1983), p. 4. Abel and another young mathematician, Evariste Galois, at - 7. Bell, , op. cit., p . 83; or http://www. christianity.com/ tacked the problem independently in the novel fashion of c hurch/church-history/timeline/1601-1700/blaise-pascals-conversion- showing that it couldn’t be done! 23 The set of abstract algebraic 11630126.html. commutative groups are called Abelian groups in his honor. He 8. Bruno Belhoste, Augustin-Louis Cauchy: A Biography (New York: was so brilliant that he was able to graduate from the University Springer-Verlag, 1991), p. 66. of after attending for only one year. 24 Abel lived honorably 9. Bell, Men of Mathematics , op. cit., p. 275. 10. Ibid., p. 290. though in great poverty, dying of tuberculosis when he was 11. Louise S. Grinstein and Paul J. Campbell, Women of Mathematics (New only 26. York: Greenwood Publ. , 1987), p. 4. Interestingly, the Abel family had a unique heraldic crest. It 12. Lynn M. Osen, Women in Mathematics (Cambridge, Mass.: MIT Press, consisted of a snake in the Tree of Knowledge of Good and Evil. 1974), p. 42. Adam reaches out with his right hand to receive a fruit from 13. Biographical Dictionary of Mathematicians, op. cit., vol. 1, p. 22. 14. Howard Eves, Great Moments in Mathematics Before 1650 (Washington, Eve’s left hand. Meanwhile, Eve is plucking another fruit with D.C.: Mathematical Association of America, 1983), pp. 183ff. her right hand. Niels Abel used this crest, explaining that his 15. Biographical Dictionary of Mathematicians, op. cit., vol. 3, p. 1774. surname came from the son of the Edenic pair. 25 16. LeRoy Edwin Froom, The Prophetic Faith of Our Fathers (Washington, These anecdotes show that using stories and biographical D.C.: Review and Herald Publ. Assn., 1942), vol. 2, pp. 455-462. sketches can lend interest to your instruction and help motivate 17. James Gleick, Isaac Newton (New York: Pantheon, 2003), p. 34. students. Recently, at a funeral, I met a former student who 18. Biographical Dictionary of Mathematics , op. cit., vol. 3, p. 1792. 19. Frank E. Manuel, The Religion of Isaac Newton (London: Oxford, reminisced about how much she enjoyed the calculus class she 1974), p. 99. took from me “because of all the stories you told.” 20. Froom, The Prophetic Faith of Our Fathers , op. cit., vol. 2, pp. 658-669. I encourage you to start creating a list of anecdotes about 21. Biographical Dictionary of Mathematicians , op. cit., vol. 1, p. 591. people who have influenced whatever subjects you teach. A 22. James R. Newman, The World of Mathematics (New York: Simon & website that will help you get started is http://en.wikipedia. Schuster, 1956), vol., 4, p. 2368. 23. Peter Pesic, Abel’s Proof (Cambridge, Mass.: MIT Press, 2003), chap. 6. org/wiki/List_of_Christian_thinkers_in_science. 24. Biographical Dictionary of Mathematicians, op. cit., vol. 1, p. 2. As you add to your list of anecdotes, please think of com - 25. Go to http://www.flickr.com/photos/henrik_abel/2094989701/. For piling them for an article in THE JOURNAL OF ADVENTIST EDU - other Coats of Arms of scientists and mathematicians, try http://www. CATION . I’m sure others will enjoy reading about them, too. i numericana.com/arms/index.htm.

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