I had the pleasure of helping lead a field trip for 9th grade Geometry students at School Without Walls SHS that we call ‘Math on the Mall’ assisting with two colleagues from the SWW math faculty.

One of our goals is for the students to see how beautifully and geometrically this city was laid out by Pierre l’Enfant, Andrew Ellicott, and Benjamin Banneker about 230 years ago.

While there are lots of myths written and repeated about Banneker’s actual contribution, the fact is that he was the astronomer, who was responsible for determining due north, exactly, and the exact latitude and longitude of the southern tip of the original 10-mile-square piece of land. With no Internet or SatNav or even a telegraph or steam engine, but with a very nice refractor and highly accurate clock that he was entrusted with, but with no landmarks to measure from, he was able to do so, in 1790.

I was sad to see that exactly none of the students know which way was north – in a city where the numbered streets near the Mall and the rest of DC’s historic downtown were almost all laid out perfectly north-south, and the streets whose names begin with letters or words like ‘Newark’, and the streets along the Mall, are all laid out perfectly east-west. Very few of them had ever seen the Milky Way, though most had heard of Polaris or the North star.

Hopefully they will remember that in the future as they do more navigation on their own in this great city.

I challenged them to try to figure out why the angle of elevation of the North Star is the same as their latitude. Here is a diagram illustrating the problem:

This diagram is intended to help you understand why the North Star’s elevation above your horizon always gives you your latitude (if you live north of the Equator.

The big circle represents the Earth. The center of the earth is at E. The equator is AD.

YOU, the observer, are standing outside on a clear night. You see Polaris in the direction of ray BG. Line HCE is the Earth’s axis, and it also points at Polaris – which is so far away, and seems so tiny, but yet is also so large, that yes, parallel rays BG and CH do, for all practical purposes, point at the same point in the sky. Ray ED starts at the center of the Earth, passes through you at B, and goes on to the zenith (the part of the sky that is directly overhead). The horizon (BF) and the zenith (ray EB) are perpendicular. Also, line HCE (the earth’s axis) is perpendicular to its equator (segment AED).

Using some sort of angle measuring device, if you are out on the National Mall at night, you can very carefully measure the angle of elevation of the North Star above the local horizon, and you should ideally find that angle, FBG, is about 38.9 degrees, but we could also call it X degrees.

**Prove (i.e. explain) why your latitude (which is angle AEB) measures the same as angle FBG**.

**What are the givens?**

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Full disclosure: My daughter graduated from SWW two decades ago, and I taught there as well for a year and for 10 years at a school that is now associated with it: Francis (then JHS now a middle school).

The kids were nice back then, and they still are. I thought the teachers did a great job.

This is a DC public high school that you have to apply to.

Benjamin Banneker was an amazing person. There are a lot of **myth**s that have been attached to his work and accomplishments, which I am guessing might be because those people didn’t actually understand the math and astronomy that he did accomplish. The best book on him is by Silvio Bedini.

‘Math on the Mall’ was originated by Florence Fasanelli, Richard Thorington, and V. Frederick Rickey around 1990. I participated as a math teacher in a couple of those tours led by FF. Later, I wanted to take my students on a similar tour that would include a trip to see a number of the works of the geometer and artist Maurice C. Escher, and couldn’t find my copy of their work, so I made up my own, and added to it using the work of FF, RT, and VFR and suggestions from teachers and students. Later on, the Mathematical Association of America made something similar, which you can find here.

**My version** was on the website of the Carnegie Institution for Science for a number of years. See page 56 **on this link**. I need to find someone to cut out some of my excess verbiage and then trot it out to a publisher.