Tag Archives: math history

Lunchtime at the Fish Pond

This is a problem from the 629 AD work of Bhaskara I, a contemporary of Brahmagupta.

“A fish is resting at the northeast corner of a rectangular pool. A heron standing at the northwest corner spies the fish. When the fish sees the heron looking at him he quickly swims towards the south (in a southwesterly direction rather than due south). When he reaches the south side of the pool, he has the unwelcome surprise of meeting the heron who has calmly walked due south along the side and turned at the southwest corner of the pool and proceeded due east, to arrive simultaneously with the fish on the south side. Given that the pool measures 12 units by 6 units, and that the heron walks as quickly as the fish swims, find the distance the fish swam.”

Answer.

See Lunchtime at the Fish Pond for a solution.

Two Men Meet

This is another problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website Convergence.

“A square walled city measures 10 li on each side.  At the center of each side is a gate.  Two persons start walking from the center of the city.  One walks out the south gate, the other the east gate.  The person walking south proceeds an unknown number of pu then turns northeast and continues past the corner of the city until they meet the eastward traveler.  The ratio of the speeds for the southward and eastward travelers is 5:3.  How many pu did each walk before they met? [1 li = 300 pu]”

Answer.

See Two Men Meet for a solution.

Cheshire Cat Paradigm

I have been meaning to focus on this aspect of mathematics for some time.  It is a topic I elaborated in my “Angular Momentum” post. But I also think it has something to do with the difficulties that normal folks have with elementary math, in particular, numbers.  I thought I would dub it the Cheshire Cat Paradigm, involving the Cheshire Cat’s grin.

See the Cheshire Cat Paradigm.

(Updates 6/7/2025, 7/19/2025, 7/20/2025) Contra Concrete Algebra, James Tanton Videos, Imaginary Numbers Continue reading →

Horses to Qi

This is a challenging problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website.

“Now a good horse and an inferior horse set out from Chang’an to Qi.  Qi is 3000 li from Chang’an.  The good horse travels 193 li on the first day and daily increases by 13 li; the inferior horse travels 97 li on the first day and daily decreases by ½ li.  The good horse reaches Qi first and turns back to meet the inferior horse.  Tell: how many days until they meet and how far has each traveled?”

The solution involves common fractions, which the Chinese were already adept at using by 100 BC.

Answer.

See Horses to Qi for a solution.

Making Arrows

This is an interesting problem from 180 BC China.

“In one day, a person can make 30 arrows or fletch [put the feathers on] 20 arrows.  How many arrows can this person both make and fletch in a day?”

It turns out the solution to this problem led me into the history of numerator/denominator (aka common) fractions, a subject I had been finding difficult to track down.

Answer.

See Making Arrows for a solution.

Alcuin’s Corn Problem

Alcuin of York (735-804) had a series of similar problems involving the distribution of corn among servants.  Since the three propositions were the same format with only the numbers changing, I thought I would present them in a more concise form:

“Proposition

A certain head of household had a number of servants, consisting of men, women, and children, among whom he wished to distribute quantities, modia, of corn.  The men should receive three modia; the women, two; and the children, half a modium.

(a)  If the head of household has 20 servants and wished to distribute 20 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(b)  If the head of household has 30 servants and wished to distribute 30 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(c)  If the head of household has 100 servants and wished to distribute 100 modia of corn among them, let him say, he who can, How many men, women and children must there have been.”

I will give Alcuin’s solutions first, followed by my more expansive solutions that rely on our familiar symbolic algebra that was not available in Alcuin’s time.

See Alcuin’s Corn Problem

A Divine Language

I have just finished reading a most remarkable book by Alec Wilkinson, called A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.  I had read an essay of his in the New Yorker that turned out to be essentially excerpts from the book.  I was so impressed with his descriptions of mathematics and intrigued by the premise of a mature adult in his 60s revisiting the nightmare of his high school experience with mathematics that I was eager to see if the book was as good as the essay.  It was, and more.

The book is difficult to categorize—it is not primarily a history of mathematics, as suggested by Amazon. But it is fascinating on several levels.  There is the issue of a mature perspective revisiting a period of one’s youth; the challenges of teaching a novice mathematics, especially a novice who has a strong antagonism for the subject; and insights into why someone would want to learn a subject that can be of no “use” to them in life, especially their later years.

Wilkinson has a strong philosophical urge; he wanted to understand the role of mathematics in human knowledge and the perspective it brought to life.  He was constantly asking the big questions:  is mathematics discovered or invented, what is the balance between nature and nurture, why does mathematics seem to describe the world so well,  what is the link between memorization and understanding, how do you come to understand anything?

See A Divine Language

Symbolic Algebra Timeline

As I am sure is common with most mathematicians, I had become interested in the history of the development of mathematical symbols, first for numbers (numerals) and then for algebra (symbolic algebra).  Joseph Mazur’s book Enlightening Symbols provided an excellent history of this evolution.  His focus on the development and significance of symbolic algebra in the Renaissance was especially illuminating.  I also augmented Mazur’s information with details from Albrecht Heeffer’s work.

Such a subject cries out for a timeline to appreciate the order and timing of discoveries, which Mazur provided, concentrating on the Renaissance.  I decided to both simplify Mazur’s version and expand it to cover the evolution of numbers and their notation, as well as to set the whole enterprise in the context of historical periods.

See Symbolic Algebra Timelines

Rotating Plane Problem

Here is another challenging problem from the first issue of the 1874 The Analyst, which also appears in Benjamin Wardhaugh’s book.

“3. If a line make an angle of 40° with a fixed plane, and a plane embracing this line be perpendicular to the fixed plane, how many degrees from its first position must the plane embracing the line revolve in order that it may make an angle of 45° with the fixed plane?

—Communicated by Prof. A. Schuyler, Berea, Ohio.”

Part of the challenge is to construct a diagram of the problem.  I used techniques for a solution that were barely in use when this problem was posed in 1874.  The contrast between then and now is most revealing.

Answer.

See the Rotating Plane Problem for solutions.

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