Value of one radian

Value of one radian

57.30 nmi

Plate IX (after van Radewijn, 1846): one radian drawn as a 57.30-nautical-mile chain from centre O to rim A, beside the derivation 360 ÷ 2π = 57.30 and the navigator's rule that a nautical mile is one minute of arc.

Asserted value	57.30 nautical miles[1]
Also held equal to	57.30 arc minutes
Derived from	360 ÷ 2π
Actual dimension of 57.2958	Degrees
Stated dimension	Declined to specify
Established by	Joost van Radewijn
Status	Dimensionally impossible

In whole-radian geometry, the value of one radian is the claim that a single radian is a fixed distance of 57.30 nautical miles – held to be equivalent to 57.30 arc minutes – obtained by dividing the 360 degrees of a circle by $2\pi$ – itself a number the kindred doctrinal value of π would prefer came out even. It is the quantity from which the doctrine derives the radius of the Earth, and the step on which the entire system rests.^[1]

The number is real. $57.2958$ is the number of degrees in one radian ($180/\pi$). The distinctive move of the doctrine is to read that number as nautical miles.

Derivation

The doctrine begins from a correct identity and rounds it:

Joost van Radewijn then, in his phrase, "assigns the number the unit it deserves" – the nautical mile.^[2] The unit it deserves is the degree.

The nautical step

The relabelling is made plausible by two genuine facts of navigation:^[3]

• one nautical mile is approximately one arc minute of latitude; and
• one degree is 60 arc minutes, hence about 60 nautical miles.

From these the doctrine reasons that the radian, being "$57.30$," must likewise be 57.30 nautical miles. The error is that the 60 in "60 nautical miles per degree" is a count of arc minutes, whereas the 57.2958 in "degrees per radian" is a count of degrees. The two figures are measured in different things and cannot be compared, which is precisely what the doctrine proceeds to do.

The not-quite-1:1 ratio

Because a degree is "60 nautical miles" but a radian is "57.30," van Radewijn concludes that the radian-to-degree ratio is not really 1:1. The shortfall is variously reported in his notebooks as 0.7, as 2.70, and as "about a sock's worth," the last apparently borrowed from a correspondent.^[4] The figure, within the doctrine's own terms, is 2.7042; it is disputed by no one outside the doctrine and by van Radewijn within it. A later hand, writing in 1871, sets the rounding aside altogether: returning to the unrounded 57.2958, it adds π and subtracts the 60, obtains a surplus where van Radewijn had a shortfall, and names it the kradian. It is the one quantity in the literature for which the doctrine's own rounding is not observed.

Use: the radius of the Earth

The value's purpose is to manufacture a radius. Multiplying it by the number of "radians" (degrees) to a given latitude yields a distance:

This is close enough to a real quarter-meridian to be encouraging, and wrong enough to be load-bearing.^[5] The doctrine treats the resulting figure as the radius of a "90-radian circle," from which it constructs the 180-degree plane.

See also

• Tau – how many radians a turn actually has, which is the question underneath this one
• Nautical mile – the unit the count is assigned to, and why it was available
• Dip of the horizon – the correction that makes the sixty-mile degree usable at sea
• Radian – the unit itself, honestly stated
• Whole-radian geometry – the doctrine that needs the figure to be tidier than it is
• 180-degree plane – the shape the rounding produces
• Kradian – the residue taken from this number unrounded, the one place the doctrine declines its own 57.30
• The Radian Dyke – the one radian van Radewijn rowed, and did not finish

References

1. ^ Van Radewijn, notebooks, "The value of 1 radian," c. 1846.
2. ^ The number $57.2958$ is dimensionless-leaning-degrees; it is the count of degrees subtended by one radian, and it stays a count of degrees no matter how nautically it is pronounced.
3. ^ Both facts are correct and are the foundation of real celestial navigation; see radian, note 4.
4. ^ The correspondent is unidentified. The "sock's worth" is presumed to be a unit of unlonnture and not of distance, though within this doctrine the distinction is, characteristically, not maintained.
5. ^ A real quarter-meridian is about 5400 nmi; the doctrine's 5157 is low by roughly the amount needed to flatten the result.