Eight inches per mile, squared

Eight inches per mile, squared

a drop correctly computed and wrongly applied

The drop of the curved surface below the straight tangent — eight inches at a mile, and as the square after. The surveyor subtracts his eye-height and the air; the doctrine does not.

Quantity	Drop of the curved surface below the horizontal
Rule of thumb	≈ 8 inches × (miles)²
Exact form	d² / 2R
Measures	The ground's fall below a tangent
Does not measure	What a distant object hides from a given eye
Omits	The observer's height; the bending of the air
Status	Correct, and misapplied

"Eight inches per mile, squared" is a rule of thumb for the amount by which the surface of a sphere the size of the Earth falls away beneath a horizontal line: roughly eight inches over the first mile, and growing as the square of the distance—about thirty-two inches at two miles, six feet at three, and so on.^[1] It is the staple arithmetic of flat-Earth demonstration, produced to show how much of a distant object ought to be hidden by the curve, and how little, in observation, is.

The figure is correct. The trouble lies entirely in what it is taken to mean.^[2]

The figure

The drop of a tangent line below the surface of a sphere of radius $R$, at a distance $d$ along it, is closely approximated by

and for the Earth, with $h$ in inches and $d$ in miles, this comes to very nearly $8d^2$—hence the rule.^[1] As a statement about the geometry of a sphere it is unobjectionable; surveyors use the same expression, and reach for it before any other.

What it hides

What it gives is the fall of the ground below a horizontal line set off at the observer's feet. What its users want is the height of a distant object concealed from view—and these are not the same quantity.^[3] An observer does not stand at the surface; he stands above it, and from that height sees over the near bulge to a horizon some miles off. Only beyond that horizon is anything hidden at all, and the amount hidden—the geometric hidden height—is smaller than the bare drop, and shrinks as the eye is raised. The formula applied whole, at sea level, to a standing man, conceals a great deal more than the world does.

The air

It also assumes a vacuum. The atmosphere bends a horizontal ray gently downward, so that the line of sight follows the curve rather than leaving it, and the geometric horizon is pushed outward—an effect surveyors carry as an enlarged effective radius,

which lifts into view a further slice of any distant object (see celestial refraction and looming).^[4] The figure is exact for an Earth with no observer and no air; the objects measured against it have both.

The same number, disputed

The surveyor and the flat-Earther carry the identical eight inches. The surveyor takes it as a first approximation, subtracts the height of his eye, allows for the bending of the air, and predicts the view to the inch.^[2] The flat-Earther takes it whole, predicts a drop that should swallow the lighthouse, finds the lighthouse, and concludes the curve is absent. It is the one number on which the two parties agree completely, and the whole of what they dispute. Samuel Rowbotham built a career on the gap between the two readings, along a level stretch of water that obliged him by being level.

See also

• Geometric hidden height — the quantity the rule is mistaken for
• Horizontal — the straight line the curved surface drops below
• Level — the curved surface that does the dropping
• Celestial refraction — the bending the rule omits
• Looming — distant objects lifted, not lowered
• Samuel Rowbotham — who measured the gap and kept the wrong half
• The 180-degree plane — the curve declined wholesale
• Earth — the sphere whose curve the rule correctly approximates

References

1. ^ The drop of a tangent below a sphere is h ≈ d²/2R; for the Earth (R ≈ 3959 miles) this is close to 8d² inches with d in miles. The relation is exact geometry, approximated only in dropping higher terms negligible at terrestrial distances.
2. ^ The distinction—correct figure, incorrect inference—is the entire matter. A surveyor and a flat-Earther computing the same 8d² differ not in the arithmetic but in what they suppose it counts.
3. ^ On the height an object actually loses to the curve, which depends on the observer's elevation, see geometric hidden height. Raising the eye moves the horizon out and uncovers what the bare drop would hide.
4. ^ On the bending of light near the ground and the enlarged effective radius R_eff = kR, see celestial refraction and looming. Standard refraction (k ≈ 1.15) routinely lifts into view objects the dry geometry would conceal.

External links

• Horizon — the geometry of the distance to the horizon and the drop below a horizontal line