No, that chart and equation are NOT correct, it is an equation for a parabola, not a circle!

No, the chart and equation are correct. They are approximations.

If you notice a parabola is approximately the same as a circle for small displacements.

If you need an example, look here:

https://www.desmos.com/calculator/s7as1l6oafFor a more math based approach, consider this image:

There are a few different options for distance and height.

Lets take the simplest one as distance along the surface (d1) and height perpendicular to the surface (h1).

Well, for a given distance we get the angle as a=d1/R.

Cos(a)=R/(R+h1).

so sec(d1/R)=(R+h1)/R

R+h1=R*sec(d1/R)

h1=R*(sec(d1/R)-1)

You could computer this exactly, or alternatively just use an approximation.

sec(x)~=1+x^2/2.

Using that instead:

h1=R*(d1^2/(2*R^2)+1-1)

h1=1^2/(2*R)

Notice that this is a parabola.

If instead you want to use h3, that has:

cos(a)=(R-h3)/R

R-h3=R*cos(a)

h3=R*(1-cos(a))

Then using the small angle approximation for cos of cos(x)=1-x^2/2 we end up with:

h3=R*(1-(1-d1^2/R^2)/2)

h3=d1^2/(2*R)

Just like the first.

And then if you want to use d2 or d3, the small angle approximation is that x~=sin(x)~=tan(x), so they are effectively the same, and with d3 and d2 effectively the same, h3 and h2 are as well.

Based upon the diagram, Row Boat appears to be using d2 and h1.

This gives us yet another way to calculate it.

R^2+d2^2=(R+h1)^2=R^2+2*R*h1+h1^2.

Noting that h1 is tiny, this simplifies to:

d2^2=2*R*h1

h1=d^2/(2*R).

So no, that parabola is just fine.

As for why it is 8 inches per mile squared, we want to output a height in inches, using a distance in miles.

If we use a distance in miles and radius of Earth in miles, we will get a height in miles which needs to be converted to inches by 63360.

So the number we need is 63360/(2*R). The radius of Earth is 3,958.8 miles, which gives us 8.002424977 inches per mile squared. So the 8 actually underestimates it until you get to quite some distance.

As a comparison of it all:

d2 | a (milli radians) | a (degrees) | d1 | d3 | H Row Pg | h Row | h par | h1 | h2 | h3 |

1 | 0.25 | 0.01 | 1.00 | 1.00 | 8 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 |

2 | 0.51 | 0.03 | 2.00 | 2.00 | 32 | 32.00 | 32.01 | 32.01 | 32.01 | 32.01 |

| | | | | | | | | | |

3 | 0.76 | 0.04 | 3.00 | 3.00 | 6 | 6.00 | 6.00 | 6.00 | 6.00 | 6.00 |

4 | 1.01 | 0.06 | 4.00 | 4.00 | 10 | 10.67 | 10.67 | 10.67 | 10.67 | 10.67 |

5 | 1.26 | 0.07 | 5.00 | 5.00 | 16 | 16.67 | 16.67 | 16.67 | 16.67 | 16.67 |

6 | 1.52 | 0.09 | 6.00 | 6.00 | 24 | 24.00 | 24.01 | 24.01 | 24.01 | 24.01 |

7 | 1.77 | 0.10 | 7.00 | 7.00 | 32 | 32.67 | 32.68 | 32.68 | 32.68 | 32.68 |

8 | 2.02 | 0.12 | 8.00 | 8.00 | 42 | 42.67 | 42.68 | 42.68 | 42.68 | 42.68 |

9 | 2.27 | 0.13 | 9.00 | 9.00 | 54 | 54.00 | 54.02 | 54.02 | 54.02 | 54.02 |

10 | 2.53 | 0.14 | 10.00 | 10.00 | 66 | 66.67 | 66.69 | 66.69 | 66.69 | 66.69 |

20 | 5.05 | 0.29 | 20.00 | 20.00 | 266 | 266.67 | 266.75 | 266.75 | 266.75 | 266.74 |

30 | 7.58 | 0.43 | 30.00 | 30.00 | 600 | 600.00 | 600.18 | 600.17 | 600.19 | 600.16 |

40 | 10.10 | 0.58 | 40.00 | 40.00 | 1066 | 1066.67 | 1066.99 | 1066.96 | 1067.02 | 1066.91 |

50 | 12.63 | 0.72 | 50.00 | 50.00 | 1666 | 1666.67 | 1667.17 | 1667.11 | 1667.24 | 1666.97 |

60 | 15.15 | 0.87 | 60.00 | 59.99 | 2400 | 2400.00 | 2400.73 | 2400.59 | 2400.87 | 2400.31 |

70 | 17.68 | 1.01 | 69.99 | 69.99 | 3266 | 3266.67 | 3267.66 | 3267.40 | 3267.91 | 3266.89 |

80 | 20.21 | 1.16 | 79.99 | 79.98 | 4266 | 4266.67 | 4267.96 | 4267.52 | 4268.40 | 4266.65 |

90 | 22.73 | 1.30 | 89.98 | 89.98 | 5400 | 5400.00 | 5401.64 | 5400.94 | 5402.33 | 5399.54 |

100 | 25.25 | 1.45 | 99.98 | 99.97 | 6666 | 6666.67 | 6668.69 | 6667.62 | 6669.75 | 6665.50 |

120 | 30.30 | 1.74 | 119.96 | 119.94 | 9600 | 9600.00 | 9602.91 | 9600.71 | 9605.12 | 9596.30 |

Notice how it holds quite well? Even at the 120 miles, the way he is expressing it still has it underestimating.