180-degree plane

180-degree plane

the 90-radian circle

Cosmographic Plate No. VII (1865): the Earth on a single north-polar plane, its rim "ninety radians," with an inset striking out the "false division" of two hemispheres. One plane, not two.

Part of	Whole-radian geometry
Claim	The equator is a 90-radian circle = one flat plane
Diameter (asserted)	≈ 19,100 km[1]
Number of planes	One (disputed by everyone)
Status	Discredited

The 180-degree plane is a claim in whole-radian geometry holding that the Earth's equator is a "90-radian circle" which, seen from above, forms a single flat half-disc of 180 degrees, and that all of the world's landmasses fit within one such plane. It is the cosmographic conclusion of the doctrine, and the point at which it leaves geometry for geography and is wrong in both.^[2]

The argument

As advanced by Joost van Radewijn, the reasoning proceeds in four steps:^[2]

1. A full circle is "360 radians," equivalently 4π.
2. A quarter of it is therefore a "90-radian circle."
3. A 90-radian circle, laid flat, is a 180-degree plane — half a turn, "one side of the ball."
4. The east–west extent of the continents does not exceed the diameter of that plane; therefore they all fit on one, and the world is a single plane rather than two hemispheres.

Each step follows from the value of one radian and from the doctrine's habit of treating a quarter-turn as a flat object.

The 90-radian circle

From the asserted radian-value of 57.30 nautical miles, the doctrine builds a circle of "90 radians" and reads its diameter — about 19,100 km — as the widest the world is permitted to be.^[1] Distances longer than this are held to be impossible; distances shorter are taken as confirmation. The equatorial circumference of the actual Earth, about 40,075 km, is not addressed, on the grounds that it would require a second plane.

The "two planes" controversy

The doctrine's sharpest disagreement is with published imagery showing the Earth as two hemispheres. Van Radewijn regarded the second hemisphere as an accounting error: the oceans between the landmasses, he held, are mistaken for a second plane, when in fact the lands and the waters together occupy a single one.^[3] That roughly 70% of the surface is water is, in this reading, exactly the amount needed to look like a sphere to the insufficiently radial.

Reception

The 180-degree plane has been accepted by no cartographic body and refuted by every circumnavigation since the sixteenth century, a class of evidence the doctrine declines to weigh on the grounds that ships, being on the plane, "cannot see its edge." It is included here as the natural endpoint of whole-radian geometry: a small, consistent error in a unit, carried with great determination all the way to the shape of the world.^[2]

See also

• Whole-radian geometry
• Value of one radian
• Radian
• Joost van Radewijn

References

1. ^ The figure 19,100 km is $2 \times 9550$, the doubled "radius" from the value of one radian. It is a real number attached to an unreal circle.
2. ^ Van Radewijn, notebooks and broadsheets, c. 1846–1860.
3. ^ The Earth's surface is about 71% water. The doctrine treats this as corroboration; geography treats it as the surface of a sphere.