Benjamin Banneker's Inscribed Equilateral Triangle

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Benjamin Banneker, the son of a former slave, was born in 1731 on a farm near Baltimore. From books loaned by a neighbor, Banneker taught himself surveying, astronomy, and mathematics. He later published several almanacs containing his astronomical observations. These almanacs widely distributed in Pennsylvania, Delaware, Maryland and Virginia. In 1791, Banneker received an appointment to assist in the survey of the boundaries of the Federal Territory - a ten-mile square now known as the District of Columbia. Banneker was also a social activist; he wrote a long letter to Secretary of State Thomas Jefferson likening the slavery of Negroes in the US to the enslavement of the American Colonies by the British. He attached his first Almanac as evidence that an African-American could be a distinguished scientist. Banneker died in 1806 on his Maryland farm. Banneker was honored on a 1980 US postage stamp.

In his journals, Banneker wrote and collected mathematical puzzles written in verse. These journals served as his notebooks for astronomical observations, his diary, and his math notebook. Unfortunately only one of his journals survived a fire on the day of his funeral. The mathematics in this journal consisted of six puzzles and two pages of mathematical writing. Banneker's six puzzles from the journal were published in an excellent biography of him written by Silvio Bedini. To my knowledge, much of Banneker's mathematics in his own handwriting has never been reproduced. I located a microfilmed copy of his journal at the Maryland Historical Society in Baltimore. The quality of the reproduction was poor, but I was helped by Mr. Omar Rumi of Kuala Lumpur, Malaysia, and my son, Quinn, a student at MIT. With the combination of Banneker's excellent penmanship, quality scanning and most of all painstakingly accurate photographic retouching on the part of Mr. Rumi, I am able to reproduce Banneker's actual handwriting of his mathematics.

Benjamin Banneker's Inscribed Equilateral Triangle - Introduction

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Benjamin Banneker, the son of a former slave, was born in 1731 on a farm near Baltimore. From books loaned by a neighbor, Banneker taught himself surveying, astronomy, and mathematics. He later published several almanacs containing his astronomical observations. These almanacs widely distributed in Pennsylvania, Delaware, Maryland and Virginia. In 1791, Banneker received an appointment to assist in the survey of the boundaries of the Federal Territory - a ten-mile square now known as the District of Columbia. Banneker was also a social activist; he wrote a long letter to Secretary of State Thomas Jefferson likening the slavery of Negroes in the US to the enslavement of the American Colonies by the British. He attached his first Almanac as evidence that an African-American could be a distinguished scientist. Banneker died in 1806 on his Maryland farm. Banneker was honored on a 1980 US postage stamp.

In his journals, Banneker wrote and collected mathematical puzzles written in verse. These journals served as his notebooks for astronomical observations, his diary, and his math notebook. Unfortunately only one of his journals survived a fire on the day of his funeral. The mathematics in this journal consisted of six puzzles and two pages of mathematical writing. Banneker's six puzzles from the journal were published in an excellent biography of him written by Silvio Bedini. To my knowledge, much of Banneker's mathematics in his own handwriting has never been reproduced. I located a microfilmed copy of his journal at the Maryland Historical Society in Baltimore. The quality of the reproduction was poor, but I was helped by Mr. Omar Rumi of Kuala Lumpur, Malaysia, and my son, Quinn, a student at MIT. With the combination of Banneker's excellent penmanship, quality scanning and most of all painstakingly accurate photographic retouching on the part of Mr. Rumi, I am able to reproduce Banneker's actual handwriting of his mathematics.

Benjamin Banneker's Inscribed Equilateral Triangle - Benjamin Banneker's Triangle

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Benjamin Banneker's Inscribed Equilateral Triangle - Transcription of Banneker's Problem

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Required the Lengths of the Sides of an Equilateral Triangle inscribed in a Circle whose Diameter is 200 perches, with a general Theorem for all such Questions

Solution of the above problem

10.00 ….. 3.142 …… 200
200
1000)628400
Lenght of the periphery 628.400
1/3 of the length of the periphery 209.466
1/3 of 1/3 of the periphery 69.822
2
2/3 of 1/3 of the periphery 139.644
Length of the Sides required 349.110

Note: In Banneker's journal there is a big X crossing out the last five numbers in his calculations and adjacent to each of the 175's in his figure is 349.110

This example deals with the problem of finding the lengths of the sides of an inscribed equilateral triangle in a circle of diameter 200 perches. A perch, a synonym of a rod, is a unit of measurement equal to 16½ feet. There are 320 perches in a mile. A surveyor could find the number of acres in a rectangular piece of land by multiplying the length in perches by the width in perches and dividing the product by 160.

What did Banneker do to solve this problem? First of all he calculated the circumference, which he calls periphery, by multiplying an approximation of \(\pi\) (3.142) by 200. He did this by multiplying 3.142 by 200 to get 628.400 and then dividing that by 1000 to get 628.400 (How often have do we, as teachers, pull our hair out when we see our students multiplying or dividing by powers of 10?) He then finds 1/3 the length (misspelled in his journal) of the circumference and later 2/9 of the circumference. He adds those numbers together to get 349.110, but clearly he knew that the side of the equilateral triangle had to be less than the diameter. He crossed out his work and then labeled the sides of triangle 175 which is approximately half of 349.110.

Using the properties of a 30º-60º-90º triangle and a circle of radius 100, one can see that the length of the side of the equilateral triangle is 173.205 which shows that Banneker's solution is within 1% of the actual one. The side of the equilateral triangle is the square root of 3 times the radius of the circumscribed circle.

How did Banneker figure this out without using the properties of a 30º-60º-90º triangle or trigonometry? Banneker calculated 5/9 of the circumference to get 349.110 and by taking half of that he essentially computed 5/18 of the circumference. Since (5/18)(2\(\pi\))100 is approximately equal to 174.533, Banneker's method is quite good. It would have been even better if he had taken exactly half of 349.110 to get 174.555. Banneker's method essentially uses (5/9)\(\pi\) to approximate the square root of 3. Perhaps this approximation was a rule of thumb that surveyors used in the 18th century.

Benjamin Banneker's Inscribed Equilateral Triangle - How to Use with Students

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Distribute this example of Banneker's handwriting to your students and ask them to figure out what he did and why his method "works." Banneker used his journals to record astronomical observations, as his diary, and as his math notebook. Is the work in your students' math notebooks as clear as Banneker's? Included below are two of Banneker's puzzles in his own writing for your students to read and solve.

Banneker's puzzles can be solved by middle and high school students. Banneker didn't use symbolic algebra to solve these problems and these problems illustrate that there are some problems which are most easily solved without algebra! The mathematics department at my high school sponsored a contest centered on these puzzles. Each grade was assigned a problem to solve and there was one problem open to all students. The students had a number of weeks to solve the problems and Savings Bonds were awarded to randomly chosen correct entries. The contest was quite popular and brought added attention to Banneker's mathematics.

Editor's note: For more problems from Benjamin Banneker’s notebooks, see the author’s article, “The Mathematical Puzzles of Benjamin Banneker,” at the College Board's AP Central website:
http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/34224.html
For more information about Banneker, including images of his Almanac and of a letter to him from Thomas Jefferson, see the website, Mathematicians of the African Diaspora:
http://www.math.buffalo.edu/mad/special/banneker-benjamin.html

Benjamin Banneker's Inscribed Equilateral Triangle - References

Author(s): 
John F. Mahoney (Benjamin Banneker Academic High School)

Adams, Daniel. Scholars Arithmetic or Federal Accountant, 1802, Leominster, MA

Bedini, Silvio A. The Life of Benjamin Banneker: The First African American Man of Science, 2nd ed. Baltimore, MD. Maryland Historical Society, 1999

Benjamin Banneker Association: www.bannekermath.org

Eglash, Ron, "The African Heritage of Benjamin Banneker", in Social Studies of Science, Volume 27, Issue 2 (April 1997)

Fasanelli, Florence D, "Benjamin Banneker's Life and Mathematics: Web of Truth? Legends as Facts; Man vs. Legend," a talk given on January 8, 2004, at the MAA/AMS meeting in Phoenix, AZ.

Lumpkin, Beatrice, "From Egypt to Benjamin Banneker: African Origins of False Position Solutions", in Vita Mathematica: Historical Research and Integration with Teaching, MAA Notes, Vol 40 (1996).

Lumpkin, Beatrice, "Mathematical Puzzles and Exercises from Banneker's Manuscript Journal" an unpublished manuscript. Dr. Lumpkin taught at Malcolm X College in Chicago.

Mahoney, John F., "Benjamin Banneker's Mathematical Puzzles" in NCTM's Mathematics Teacher, Vol 96, #2, February, 2003.

Mahoney, John F., "Benjamin Banneker and Single Position" in NCTM's Mathematics Teaching in the Middle School, Vol 10, #7, March, 2004

Mahoney, John F., "Benjamin Banneker and the Law of Sines" in NCTM's Mathematics Teacher, Vol 98, #6, February, 2005.

Tyson, Martha E. Banneker, The Afric-American Astronomer, Philadelphia Friends Book Association, 1884. (Copy at the Maryland Historical Society)