Math History Tidbits: Agnesi, Euler, and China

I’ve fallen behind on my project of transcribing my Alexandria Jones stories. Finally, here are a few more tidbits from math history, along with links to relevant Internet sites and a few math puzzles for your students to try.

I hope you find them interesting.

Math in the Trinity

Since these historical tidbits were published in the December edition of my old math newsletter, I included a short Bible verse for Christmas:

Being in very nature God, He did not consider equality with God something to be grasped, but made Himself nothing…

— Philippians

Maria Agnesi

Alexandria Jones’s mother, Maria, is named for Maria Gaetana Agnesi (1718-1799). Maria was one of the first women to be accepted as a serious mathematician.

Maria Agnesi was both beautiful and a genius. As a child she mastered French, German, Greek, Hebrew, Latin, and Spanish — as well as her native Italian.

On at least one occasion, Maria worked mathematics while sleepwalking. When she got stumped, she left the problem on her desk and went to bed. The next morning, she found the correct solution neatly written on her paper.

After teaching calculus to her younger brothers, Agnesi wrote what became Europe’s most popular calculus textbook for the next 50 years.

It took much skill and sagacity to reduce, as the author has done, to almost uniform methods these discoveries scattered among the works of modern mathematicians and often presented by methods very different from each other. Order, clarity and precision reign in all parts of this work. … We regard it as the most complete and best made treatise.

— Académie des Sciences book review

Included in her textbook was a discussion of the curve which, because of another writer’s typographical mistake, has come to be called the Witch of Agnesi.

But there was something Maria loved more than math — she longed to become a nun, but her father forbid it. After her father died, she turned their house into a hospice and spent her fortune helping the poor and infirm. She died in poverty, a patient in that same institution.

The Great Leonhard Euler

Now I will have less distraction.

— Leonhard Euler (1707-1783)
[upon losing the use of his right eye]

Alex’s little brother is named for Leonhard Euler (pronounced OIL-er), who had humility and a sense of humor, as shown in the above quote. But most of all, he had a creative way with numbers. In fact, Leonhard Euler changed the face of math for all time. In addition to several bookshelves full of new work, he revised almost all the math that had been done before his time. Among other things, Euler gave us the symbol , e for natural logarithms, i for imaginary numbers, and for summation.

Euler first learned mathematics from his father, a protestant minister who had studied under Jacob Bernoulli. When Leonhard entered the University of Basel at age 14:

… I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. … True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand …

That’s good advice for any student. If you want to study the Euler way, here are several forums where you can ask questions.

One puzzle Euler worked on is easily accessible to middle school students: The Bridges of Königsberg.

Residents of the city enjoyed taking a stroll across the river and from island to island on Sundays after church. How could they walk over all of the bridges without crossing any single bridge twice?

Here is another fun puzzle for your students:

• What is the sum of the series ?

The answer has been given as 0, or 1, or 1/2.

Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived — namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken — has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.

— Leonhard Euler

Magic Squares in Ancient China

Explore the history of Chinese mathematics at the MacTutor history pages. For instance, the earliest known mention of a magic square comes from the Chinese book called I Ching, or the Book of Permutations (c. 1150 BC).

My newsletter didn’t have room for much Chinese history, just this background story about Alex’s Christmas gift for Leon…

In the old days, Chinese people called the Yellow River valley “Middle Kingdom” — the center of the world. According to legend, the Emperor Yu (c. 2200 BC) was traveling the Yellow River [or perhaps the Lo River, see this comment] one day when he spotted the divine tortoise Lo-shu on the bank. Lo-shu’s back was decorated with a magic square, which Emperor Yu adopted as a good-luck charm.

Can you solve these puzzles from Chinese math history?

Tidbits

For awhile, the symbol p was used to stand for the circumference of a circle. In 1706, an Englishman named William Jones used as we do today — to stand for the ratio of the circumference to the circle’s diameter. Few people picked up on that idea, however, until Leonhard Euler decided to use in 1737.

Almost 1500 years ago, the Chinese writer Tsu Ch’ung-chih estimated to be about equal to 355/113. How close was he?

Look for the answer at the Historical Overview of Pi.

To Be Continued…

Read all the posts from the November/December 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.

• Alexandria Jones and the Christmas Present Quandary