Historical Tidbits: Alexandria Jones

[Read the story of the pharaoh’s treasure: Part 1, Part 2, and Part 3.]

Here are a few more tidbits from math history, along with links to relevant Internet sites or books, and three more math puzzles for you to try. I hope you find them interesting.

Next time, a new adventure (sort of)…

About the Names

Everyone in Alexandria Jones’s family takes his or her name from someone (or in Alex’s case, something) important in the history of mathematics.

Alexandria Jones is named after the Egyptian city of Alexandria, founded in 332 BC by Alexander the Great. As his army marched across the land, Alexander founded many cities, and he named most of them after himself. (But he did name at least one city in honor of his horse, Bucephalus.)

The Egyptian Alexandria, near the mouth of the Nile River, became a great center of learning. Among the scholars of Alexandria:

• Ptolemy, who came up with a mathematical system of astronomy.

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.

— Claudius Ptolemy

• Eratosthenes, who invented a “sieve” to find prime numbers and calculated (accurately!) the earth’s circumference.

[Eratosthenes] was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others.

— Thomas Little Heath

• Euclid, who wrote the best-selling math textbook ever.

There is no royal road to geometry.

— Euclid of Alexandria

Ramanujan

Alexandria’s dog, Rammy, is named for[See Alexandria’s Dog is Now a Teacher.] Srinavasa Ayengar Ramanujan, a self-taught mathematical genius from India, who lived in the early 1900’s. Ramanujan loved playing around with numbers, looking for patterns. Because of his limited schooling, he often “discovered” math truths, only to find out they were first stated by Pythagoras and other famous mathematicians.

An equation means nothing to me unless it expresses a thought of God.

— Srinivasa Ramanujan

Fibonacci

Leonardo Fibonacci (from whom Alexandria’s father takes his name) traveled all around the Mediterranean world. When he returned to his hometown of Pisa, Italy, in AD 1200, he brought back “new” ideas about math. He rediscovered concepts that had been lost (to the Europeans) since the fall of Rome. Even more important, Fibonacci brought to Europe the Arabic numbering system — including the notion of zero.

[W]hen I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else…

— Leonardo Pisano Fibonacci

Puzzles from History

Diophantus

Diophantus, a mathematician who lived in Alexandria around AD 250, invented the idea of a variable — a symbol that stands for an unknown number. He is famous for his study of Diophantine equations, where the variables have to be replaced by whole numbers.

Can you solve the epitaph of Diophantus?

This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh; and in the fifth year after his marriage, He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.

Pappus

Pappus of Alexandria lived sometime around AD 300. He wrote a summary of all the mathematics that was known in his day. He called it the Collection. Many works of the ancient Greek mathematicians survive only because Pappus wrote them down in his book. He also wrote about his own studies in geometry and other subjects.

Try this geometric construction of Pappus’s Invariant for yourself, from a fun book called Math Charmers:

• Draw a straight line, and mark on it any three points. Label them A, B, C (in roughly left-to-right or top-to-bottom order).

• Draw another straight line anywhere on your paper, and mark on it any three points. Label them D, E, F (in corresponding order). Suggestion: The first time you try this, do not let your lines cross each other.

• Draw straight lines to connect A-E, A-F, B-D, B-F, C-D, C-E. Do NOT connect A-D or B-E or C-F.

• Mark the points where the lines connecting corresponding points cross each other. That is, mark where A-E meets B-D, and mark where A-F meets C-D, and mark where B-F meets C-E. Label these new points G, H, I. Warning: If your original lines crossed each other, and you chose points on both sides of the intersection, you may need a LARGE sheet of paper to extend these lines until they meet.

• No matter how you draw the original lines or where you make the points on them, if you follow these instructions, the three points G, H, and I will always be collinear. That is, you can always connect them to make a new straight line.

Fibonacci

Fibonacci wrote a book called Liber Abaci, which included many word problems. The following problem may be his most famous. It leads to a set of numbers called the Fibonacci series.

Can you discover the Fibonacci series?

Assume that baby rabbits take 2 months to mature, and that as soon as they are mature the rabbits start producing a new pair (one male and one female) of babies every month, and that no rabbits ever die. If you have one pair (male and female) of baby rabbits on January 1st, how many rabbits will you have on Christmas day?

[Edited to add: Answers to the puzzles about Diophantus’s age and Fibonacci’s rabbits are posted here.]

To Be Continued…

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

• Alexandria Jones and the Secret of the Pharaoh’s Treasure