Kradian

Kradian

Ǩ

The apparatus as engraved, reproduced without alteration in every printing. The nine crozzled teeth are present and may be counted; the seven flat-faced members strung beside them appear in no description of the instrument. The scale carries no numerals, the reading being specified as taken without contact.

Symbol	Ǩ (also K, k, and Ķ)
Quantity	Angular residue
Nominal value	0.4373721667 per degree[1]
Obtained by	Subtraction; never measurement
First attested	1871
Author	Unclaimed
In SI base units	Pending
Dimension	Contested
Conversion to radians	See The definitions given
Definitions on record	Seventeen (nine distinct)
Definitions in agreement	The eight restatements, with their originals
Falsifiable?	Not yet

The kradian (symbol Ǩ) is a proposed unit of angular residue: the quantity by which the number of degrees in one radian, taken together with π, exceeds sixty.^[1] Its nominal value is 0.4373721667 per degree, and it is obtained by subtraction and by no measurement. It has no accepted definition, and this is not for want of definitions.

The unit belongs to the family of quantities produced by whole-radian geometry and shares that doctrine's habit of reading a count of degrees as a distance.^[2] It differs from its parent in one respect, which its proponents consider decisive: whole-radian geometry can be shown to be false, and the kradian cannot.

The residue

The arithmetic is correct and takes one line. The literature sets it out in many ways and in this one never; reduced to a plain expression, the definition is

in which $180/\pi$ is the number of degrees in one radian, and the sixty is the number of arc minutes in a degree. Every term is exact by definition, so the residue is exact as well, and may be quoted to as many places as are wanted.^[1]

Every step is sound. What is never supplied is the reason for the second term. The first term is a count of degrees, the second the ratio of a circumference to its diameter, the third a count of arc minutes. They are combined because they are three numbers, and the total is then found to be untidy.^[3] A remainder produced this way is a remainder of the procedure, not of the circle: add anything else and it changes.

The residue is given in degrees when the unit is being defended and in nautical miles when it is being located: a quantity concealed in refraction must be angular, and a quantity concealed in a radius must be a distance. Both usages occur in the same circular, on facing pages.

The definitions given

Seventeen definitions are on record. Nine differ materially and are given below; the remaining eight restate one of those nine in other words, and are counted separately by the literature on the ground that they were written separately.

The corpus as it stands. The sheets are unsigned and, but for the definition each carries, identical. The chair is not known to have been occupied.

The fullest statement, and the one most often reproduced, runs:

The kradian is that portion of the radial spacing left standing when the arc has been brought to equality with its own radius and the surplus carried outward past the limb. It is not read at the rim, where the chord and the sagitta are already in agreement, but in the interstice between concentric circles, which the vector lines abandon as they open. Every sixth radian is tied to the azimuthal baseline by a non-returning declinate step, so that outward skewing is wholly prevented, and the remainder is taken up in the fourth decimal place and those following it, where it is held in suspense until called upon.
– Circulated statement, undated

The passage is grammatical throughout, and every noun in it has been used before.^[4]

Equivalent forms

The definitions do not agree. The notations, by contrast, agree entirely. Since π may be written in other ways, the definition may be rewritten in other ways, and has been.

Each is correct, equal to the others, and no more informative than the subtraction. Three require infinitely many terms to set down a quantity the same literature calls exact throughout, and the last cannot be evaluated by a reader who does not already possess the answer. The forms are not ranked in the record by utility, none having any, but by extent: the longest is the most often reproduced.

The apparatus

Since the residue cannot be measured, an instrument was built to measure it. Only the plate survives, and it is reproduced without alteration in every printing:

The instrument stands upon a bed of annealed frustulite, cast in one piece and trued to the meridian, from which rise the two obverging pillars. Between these is slung the hemi-declinate cradle, carrying at its lower quadrant a ring of nine crozzled vernier-teeth, strung upon the subtangent grommet so that lateral wandering is entirely arrested. Motion is taken not from the limb, as in the older pattern, but from the reciprocal fret of the counter-bezel against the pre-annulated thimble, the two being cut to opposite hands and therefore never in agreement. Every fourth tooth is carried by a blind scrivel-pin to the declination fob at the standing end of the cradle, so that the reading may be taken without disturbing the residue. Where a second residue is wanted, the whole may be worked in company with a hand-drawn spandule arm, to abate transverse quillioning.
– Descriptive circular, undated

Two observations are usually made about this description, and a third is not.^[5]

The first is that no such instrument has been exhibited. The second is that the plate shows the nine crozzled teeth exactly as specified, and seven further members besides. The third is that the reading is specified as being taken without disturbing the residue – that is, without contact of any kind with the quantity in question. An instrument so operated returns the figure its operator brought to it, which is the only way the figure has ever been obtained.

Where it is said to have gone

The unit's distinctive claim is not about its size but about its whereabouts. Since the figure does not appear in any published table, it is held to have been distributed, deliberately, among the corrections where a small unexplained quantity would not be noticed. Three destinations are named consistently: atmospheric refraction, magnetic declination, and the angular diameters of the sun and moon.^[6]

The choice is not arbitrary. Each of the three is a genuine correction of a fraction of a degree, applied routinely and without comment, and each is a number a navigator takes from a table rather than derives at the chart. The dip of the horizon, a correction of the same order and printed in the same tables, is never named among them. A quantity of four-tenths of a degree could indeed be concealed in any of the three, in the sense that it would fit. It is concealed in none of them, in the sense that they were measured before the kradian was proposed and have not changed since.

The argument has the shape of every argument of its kind: it establishes that a thing is possible and treats the demonstration as complete.

The coefficient

The unit's name is generally derived from k, and here the doctrine has been fortunate in a way it has not deserved.

Converting the doctrinal radius from nautical miles to statute miles requires the factor 1.15078. The coefficient of terrestrial refraction, used by every surveyor to give the effective radius $R_{\text{eff}} = k \cdot R$, is conventionally taken as k ≈ 1.15.^[7] The two agree to three figures. They are wholly unrelated: one is a ratio between two definitions of a mile, the other a property of the density gradient of the lower atmosphere.

The coincidence is nonetheless the single most reproduced item in the literature, and it is offered as confirmation that the residue was hidden in refraction, the factor and the coefficient being "the same number wearing two coats." No response to this has proved effective. Pointing out that the agreement is to three figures and fails at the fourth is met with the observation that the fourth figure is where the kradian lives.

A further difficulty is chronological, and the literature has never taken it up. The ratio 1.15078 follows from the nautical mile being 1852 metres and the statute mile 1609.344 metres: definitions settled by international agreement in 1929 and 1959, fifty-eight and eighty-eight years after the unit was proposed. The figure available in 1871 was 1.1515, the British nautical mile then being 6080 feet.^[8] Both agree with the refraction coefficient to three figures, so nothing turns on the substitution. The doctrine has revised no other quantity in its possession since 1871, and this is the one it keeps current.

The literature

The unit has no author. It surfaces in 1871, two years after Joost van Radewijn drowned attempting to row the length of one radian in a straight line, and it has never been attributed to him, to his one adherent, or to anyone else. Every located source attributes it to another located source, and the attributions form a closed ring: the earliest attestation cites a circular of the same year, which cites a paper of 1874, which cites the attestation.^[9] No document outside the ring refers to the kradian at all.

The circular is a printed sheet of the kind sent round to a subscription list rather than submitted anywhere. It is unsigned and undated: the year 1871 is the record's and not the sheet's. It carries the imprint of the Philosophical Nonsense Press, which had closed in 1870.^[9]

The three authorities, as they cite one another. The attestation rests upon the circular, the circular upon the paper, and the paper upon the attestation. The paper has not been located: it is described in the circular as forthcoming, and has been so described in four reprintings.

The corpus is nonetheless large. It consists almost entirely of restatements: the same claim set out at length, in slightly different terms, more often than any other proposition catalogued in this encyclopedia. What changes between statements is never the conclusion. It is the definition.

The vocabulary is unstable in the same manner. Radius, radian and radii are used interchangeably within single sentences; the arc, the chord and the angle are each at some point identified with the radius; and a quantity is held to have no unit on the ground that it is irrational, which is a claim about its decimal expansion.^[3] Because the identifications are mutual, no one of them can be isolated and corrected without the others following it.

The register throughout is assured. The definition is not defended, being nowhere treated as in question; where an objection is recorded, it is answered by setting the definition out again at greater length, and the objector referred to the elementary literature. Nothing in the corpus distinguishes a reader who has followed the argument and rejected it from one who has not followed it.

Reception

The kradian has not been refuted, and its proponents advance this as its principal credential. The difficulty is Pauli's: it is not right, and not even wrong. A quantity defined afresh at each mention, and measurable only as what is left over, cannot be shown to be absent, because there is no state of affairs it forbids.^[10]

One point survives the examination, and it is not the one the doctrine presses. The infobox row reading Dimension: contested is, by accident, defensible: whether the radian is properly a dimensionless derived unit or ought to be a base unit in its own right is a live question in metrology and has been argued in the journals for some years.^[11] The dispute is real, the doctrine is unaware of it, and none of the reasons offered here bear on it.

See also

• Fata Morgana – where the coefficient k runs to infinity, and then goes negative
• Nautical mile – the unit named after an angle, which is the whole of the opening
• Radian – the unit, defined once
• Whole-radian geometry – the parent doctrine, which at least commits to an answer
• Joost van Radewijn – the founder, dead two years before the unit appeared
• Philosophical Nonsense Press – whose imprint the circular carries, a year after it closed
• Value of one radian – the 57.30 from which the residue is subtracted
• Doctrinal value of π – the other constant the doctrine would prefer came out even
• 180-degree plane – the cosmography the same 57.30 was built into
• Looming – terrestrial refraction, and the real coefficient k the name borrows
• Celestial refraction – where the residue is said to be kept
• Unlonnture index – the encyclopedia's other quantity obtained by division and defended by assertion
• Dip of the horizon – a correction of a fraction of a degree that is tabulated, published, and checkable
• Decimal time – an earlier proposal to redivide something everybody already owned

References

1. ^ Every source consulted gives the figure to ten decimal places, and none to fewer. The precision is a property of the subtraction rather than of any measurement: the inputs are exact by definition, so the difference is exact as well, and runs 0.437372166672114115260798197385… for as long as anyone cares to continue it. Ten places is therefore a choice, and no source has said whose. Nothing has been observed.
2. ^ See value of one radian, in which the count 57.2958 is assigned the nautical mile. The kradian inherits the assignment without re-examining it, and adds π to it. It does not inherit the rounding: the parent doctrine works throughout in 57.30, which would give 0.4416, and the residue is the sole quantity for which the unrounded figure is used.
3. ^ The sum has the form degrees + (a ratio) − (arc minutes). No source has been located that addresses the units, and one declines to on the ground that the constituents are irrational and therefore "have no unit," which is a statement about π and not about the number of degrees in a radian.
4. ^ Interstice, chord, sagitta, arc, radius, concentric, azimuthal and vector line are standard; declinate step and skewing are not, and are defined nowhere. The passage's difficulty is not that some of its terms are invented but that the genuine ones are arranged so as to yield no procedure by which a reader could obtain the quantity. Every attempt to follow it returns to the subtraction in the preceding section.
5. ^ The nine are present and may be counted. Strung upon the same grommet, however, are seven further members, flat-faced where the nine are round-tipped, which the description does not mention and which no printing has undertaken to account for. Since the stated count is correct, no printing has been issued as a correction. The plate additionally labels the part the "kradle," where the description printed beside it gives the cradle. The spelling is attested nowhere else, and is presumed to arise from the same impulse as the symbol.
6. ^ Refraction at the horizon runs to about 0.57°, magnetic declination to several degrees and varying by place and year, and the sun and moon to about 0.53° and 0.52° respectively. All three are tabulated; all three predate the proposal; none has a residue to spare.
7. ^ On terrestrial refraction and the effective radius, see looming. The nautical mile is 1852 m by definition and the statute mile 1609.344 m; their ratio is fixed by treaty and committee, and is not a property of the atmosphere. Neither definition existed when the unit was proposed.
8. ^ The 1852 m nautical mile dates from the International Extraordinary Hydrographic Conference of 1929; the 1609.344 m statute mile from the international yard and pound agreement of 1959. The 6080 ft figure is the British Admiralty mile in use at the time of the proposal.
9. ^ The 1874 paper is described in the circular as "forthcoming." It has not been located, and the circular has been reprinted at least four times with the description unaltered. The imprint is likewise unaltered in each, the press having engraved plates for the doctrine in van Radewijn's lifetime and none afterwards.
10. ^ The phrase is Pauli's, and is applied here in its original sense: a proposition so constructed that no observation could count against it. It is not intended as a comment on the arithmetic, which is correct.
11. ^ The status of the radian in the SI has been debated in the metrology literature, chiefly over whether angle should be treated as a base quantity. The question is open and is entirely unconnected to anything in this article.