The Couple Talks about the Volume of the Truncated Cone

On June 7, my wife and I joined a bus trip to Exhibition of Paintings from State Pushkin Museum and Flower Festival Commemorative Park. We had lunch at a French restaurant, whose building was a nationally designated Important Cultural Property. At the lunch table, a couple younger than my wife and I took the seats in front of us. Let's call them Mr. and Ms. N. A glass containing water from the well of the restaurant and some ice cubes was prepared for each person from the beginning of the lunch course (see the photo above).

At some stage of the lunch course, Mr. N took the glass and asked Ms. N if she could guess how to calculate the volume of the solid of such a form. She said, "Add a small cone to make it a large cone. Calculate the volume of the large cone and subtract the volume of the small cone from it." Mr. N replied that it would be very cumbersome. Then, he said that the volume can be obtained as the mean of volumes of three cylinders having the same height as the solid. His voice was so low that I was unable to hear the radii of the three solids, but from the movement of his hands, I supposed that he referred to the radii of the top and bottom circles of the truncated cone and a certain mean of the two. He additionally stated that we could obtain the formula by integration of the circular area.

I had never heard of the formula for the volume of the truncated cone, and thought it wonderful that Mr. N learned it and remembered it for some reason. However, I also wondered why he who spoke of a more complex method of integration said that his wife's simpler method was cumbersome. After returning home, I calculated the volume by Ms. N's method and easily found that the third radius mentioned by Mr. N was the geometric mean of the radii of the top and bottom circles.

To see the formula and the derivation of it, visit here. The explanation is in Japanese, but readers might easily follow equations by looking at a diagram included. The third method mentioned there by the use of Pappus-Guldin theorem (also known as Pappus' centroid theorem; the second theorem is relevant here), however, might be a little difficult to understand, if you have never heard of that theorem.

Leo Tolstoy and Mathematics

Leo Tolstoy. By Scan by User: Gabor [Public domain], via Wikimedia Commons.

These days I am reading Leo Tolstoy's autobiographical novels Childhood, Boyhood and Youth by Japanese translations in Shinchō World Literature Volume 16 (1972). In the first half of Youth, there is an episode that the hero, "I", passes the entrance examination to the mathematics department of a university.

The examination is made in the following manner: Each applicant takes a piece from the problem cards held by a professor and answers to the problem chosen, in front of the professor. In the mathematics examination, there were two professors. The hero was called at the same time as another applicant, and they secretly exchanged the cards they chose. The card the hero initially took was of the problem about "combinatorics," which he thought difficult. However, the question he got by exchanging the cards was about "Newton's binomial theorem," which he had tried to solve just before the examination. Thus, the hero was able to answer it perfectly.

Reading this, you may wonder if Tolstoy studied in the mathematics department. In fact, Tolstoy began studying law and oriental languages at Kazan University and left university in the middle of his studies (Ref. 1). Therefore, the story of the entrance to the mathematics department is fictional.

I learned "combinatorics" and "the binomial theorem" in senior high school. However, the latter was not Newton's but the basic one. The basic "binomial theorem" describes a formula for the algebraic expansion of nth power of a binomial x + y, where n is a positive integer. Isaac Newton generalized the formula to allow real exponents, and the formula can be generalized further, to complex exponents (Ref. 2).

It was probably at university that I learned about generalized binomial theorem. Therefore, Newton's binomial theorem seems to be too difficult for the entrance examination of university. It is also strange that the hero thought Newton's binomial theorem easier than combinatorics. This is because we learn combinatorics as preparation for studying the basic binomial theorem. Thus, I guess that problems of mathematics were also invented by Tolstoy on the basis of his superficial knowledge of these terms of mathematics. What do you think?

References

1. "Leo Tolstoy," Wikipedia: The Free Encyclopedia (December 8, 2012 at 19:10).
2. "Binomial theorem," Wikipedia: The Free Encyclopedia (November 25, 2012 at 04:39).