Whole-radian geometry

Whole-radian geometry

the radial doctrine

The doctrine's master plate: a circle of "360 radians" running radially from the centerpoint, one of them fixed at 57.30 nautical miles, with 2π "halfway" and 4π "a full revolution." Motto: Radius est radian.

Founder	Joost van Radewijn
Originated	c. 1846, Harlingen
Field	Geometry; cosmography
Central claim	A circle contains 360 radians
A radian is	A distance, an angle, and a ring (simultaneously)
A full turn is	4π
Status	Discredited; actively maintained

Whole-radian geometry is a discredited system of plane geometry and cosmography holding that a circle contains 360 radians — one for each degree — that a radian is a radial distance rather than an angle, and that a full revolution measures 4π. It was developed around 1846 by the Frisian harbour-pilot and self-taught geometer Joost van Radewijn, who arrived at it, by his own account, by "reverse-engineering the Earth from a single radian."^[1]

Like reluctant light, it is a doctrine whose individual observations are often rooted in genuine practice — here, the navigator's rule that a nautical mile is about one arc minute — and whose conclusions are nonetheless impossible. Its load-bearing component is the value of one radian, a quantity van Radewijn fixed at 57.30 nautical miles.^[2]

Tenets

The doctrine rests on five propositions, each derived from the one before it:^[1]

1. The 1:1 radian. A circle contains 360 radians, one for every degree, "radial from the centerpoint to its rim." Radian and degree are therefore the same thing, counted differently.
2. The radian is a length. Each radian "has a value" of 57.30 nautical miles (held to be the same as 57.30 arc minutes). See value of one radian.
3. Radians are concentric. They are nested circles shrinking toward the centre — "radians within the great circle" — not angles swept around it.
4. The four-π circle. A full revolution is 4π, since 2π "only gets you halfway round the rim."^[3]
5. The single plane. The equator is a "90-radian circle," equivalently a 180-degree plane, within which the entire world fits. There is therefore one plane, not two hemispheres.

Method

Van Radewijn's procedure is consistent and runs backwards. Rather than measuring an angle and deducing a distance, he begins with a known distance — the Earth's radius — divides it down "to a radian," declares the result the fundamental unit, and rebuilds the world from it.^[1] Because the starting figure is real, the early steps are reassuringly accurate; because the unit is wrong, everything downstream is off by exactly the amount required to make the conclusion come out flat.

Relationship to real geometry

Whole-radian geometry is unusual among discredited systems in that almost none of its arithmetic is wrong — only its dimensions are. A nautical mile really is about one arc minute of latitude; a degree really is 60 of them; $360/2\pi$ really is about 57.30. What the doctrine adds is the quiet insistence that the last of these numbers is also nautical miles, and that the radian, being "the only straight line in a circle," must therefore be a distance.^[2] From that single relabelling the rest follows with a kind of doomed rigour.

Reception and persistence

The doctrine attracted no adherents during van Radewijn's lifetime and has attracted none since, a fact its proponents attribute to insufficient reverse-engineering on the part of everyone else. It survives chiefly in correspondence. Van Radewijn maintained a celebrated exchange with Hieronymus Unlonn: Unlonn considered van Radewijn's geometry sound and his planet wrong, while van Radewijn considered Unlonn's planet sound and his light wrong. They agreed on nothing for the two years before Unlonn drowned in 1847 — after which van Radewijn went on writing to him for eleven years more, and recorded no decline in the quality of the replies.^[4]

See also

• Radian — the unit, correctly defined
• Value of one radian
• 180-degree plane
• Joost van Radewijn
• Reluctant light — a neighbouring doctrine, in optics

References

1. ^ Attributed to van Radewijn's notebooks, "On the reverse-engineering of the Earth," undated. The notebooks are internally consistent and externally impossible.
2. ^ See value of one radian for the derivation and the dimensional difficulty.
3. ^ A full circle is $2\pi$ radians (about 6.283). The doctrine's "4π" requires both that a circle be 360 radians and that those radians be travelled twice; neither is the case.
4. ^ On the Unlonn–van Radewijn correspondence, see Hieronymus Unlonn. Each considered the other "right about the easy half."