First, watch the video segment regarding Archimedes’ approach to calculating the area of a parabolic sector. The calculus part is explained in one beautiful diagram but the premise regarding the successive triangles baffled me.

I didn’t know that the height of each successive layer of triangles would decrease by a factor of 4 โ I don’t remember ever learning something like this.

Thanks, Steven!

*they might therefore be located. The*

**exactly where***was mentioned: find the point with a tangent parallel to the previous chord using the ‘parallel sliding trick’.I realize*

**‘how it was located’***was the point of the Strogatz lecture; therefore, geometry was off point. The reader might be left wondering how to buy into the successive geometric factoring of the area of the triangles.*

**calculus**I calculated where such a tangent point would be found on the parabola. As I had guessed, it was at the midpoint of the ‘x interval’. And of course, Archimedes knew this. I think this adds an extra element of beauty and elegance to the problem and the solution.

I understand that Archimedes was solving this problem in the 3rd century BC โ long before alโKhwarizmi’s algebra (9th century), Descartes’ coordinate geometry (17th century), and Newton’s calculus (17th century).

*times the height but I prefer to state as follows:*

**base***times the height*

**width**And finally here is a derivation of easy interpolations, useful for heights or widths.