zotero-db/storage/X2VBL6QK/.zotero-ft-cache

arXiv:2206.07491v1 [physics.pop-ph] 20 May 2022

Polaris: The Mathematics of Navigation and the Shape of the Earth
John P. Boyd Department of Climate & Space Sciences and Engineering University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109
jpboyd@umich.edu
June 16, 2022
Abstract For millenia, sailors have used the empirical rule that the elevation angle of Polaris, the North Star, as measured by sextant, quadrant or astrolabe, is approximately equal to latitude. Here, we show using elementary trigonometry that Empirical Law 1 can be converted from a heuristic to a theorem. A second ancient empirical law is that the distance in kilometers from the observer to the North Pole, the geodesic distance measured along the spherical surface of the planet, is the number of degrees of colatitude multiplied by 111.1 kilometers. Can Empirical Law 2 be similarly rendered rigorous? No; whereas as the shape of the planet is controlled by trigonometry, the size of our world is an accident of cosmological history. However, Empirical Law 2, can be rigorously veriﬁed by measurements. The association of 111 km of north-south distance to one degree of latitude trivially yields the circumference of the globe as 40,000 km. We also extend these ideas and the parallel ray approximation to three diﬀerent ways of modeling a Flat Earth. We show that photographs from orbit, taken by a very expensive satellite, are unnecessary to render the Flat Earth untenable; simple mathematics proves Earth a sphere just as well.
1 Polaris and the True North Star
The key measurement for navigation is the following.
Deﬁnition 1 (elevation angle) The elevation angle is the angular distance between the horizon and the target star. More precisely, is the angle between a ray from the center of the earth to the star and a second ray in the same longitude but in the equatorial plane. Alternatively, the elevation can be deﬁned as the complementary angle to the zenith angle ζ:
≡ π/2−ζ. A Synonyms for “elevation angle include“altitude angle” and simply “altitude”.
Deﬁnition 2 (zenith angle) The zenith angle ζ is the angular distance between directly overhead [observer’s zenith] and the target star. A second equivalent deﬁnition ζ ≡ π/2 − .
The “Elevation angle” is ninety degrees when the Pole Star is at zenith and zero degrees when the Pole Star is bisected by the horizon.
Both angles can be measured by a variety of “inclinometers” [angle-measuring devices] including the mariner’s astrolabe, quadrant, sextant and the Arab instrument known as the Kamal.
1

As explained below, angle measurements of the Pole Star are especially useful for navigation. However, one diﬃculty is that Polaris is 3/4 of a degree from the celestial North Pole today, and was more than three degrees away in Elizabethan times. However, mariners soon realized that by looking at stars near Polaris and using the set of empirical rules with the charming name of the “Regiment of the Stars”, the distance-from-the-North-Pole errors could be greatly reduced. Deﬁnition 3 (Regiment of the North Star) A set of rules that allow sightings of stars such as Kochab to correct for the diﬀerence between the position of Polaris and the true north celestial pole.
We use “Pole Star” denote a star that is not Polaris, but rather Polaris-after-theRegiment-of-the-North-Star has been applied. Our Pole Star is thus aligned with the earth’s rotation axis and is the true north celestial pole.
Practical star sightings incur additional errors because of the limitations of the sextant, the mariner’s astrolable and other inclinometers. Again, correction tables and instruments were developed many centuries ago [7, 8]. The ful twelve-minute video [11] is a careful discussion of all these corrections. For the videographer’s example, a sextant sighting of 29 degrees, 53.5 minutes is altered to a best estimate of 29 degrees, 21.3 minutes, a correction of about half a degree or about 30 nautical miles.
The importance of angles for navigation has driven the development of planar and spherical trigonometry [16, 17, 18, 19].
2

Table 1: List of Symbols

Ω is the angular frequncy of the Earth’s rotation. When two angles add up to π/2, the angles are said to

be “complementary”; the cosine of one member of the pair is equal to the sine of the other.

The

“declination [angle]” is the angular distance of a point north or south of the celestial equator.

d

length of a geodesic from the observer’s position to the North Pole.

D Dstar E disk E globe

distance from observer’s position to the Pole Star, Polaris distance from observer’s position to target star equatorial circumference equatorial circumference

L

thickness of the layer of atmospheric refraction

W W earth W star

scaling for the nondimensional distance-to-North Pole coordinate width (radius) of spherical earth width (radius) of star

η

angle between the central ray from star to earth and the least parallel ray

elevation [angle], angular distance between the horizon and the target star.

More precisely, is the angle between a ray from the center of the earth

to the star and a second ray of the same longitude but in the equatorial plane

= π/2 − ζ , synonym: “altitude angle”

app apparent elevation

ζ

zenith [angle], angular distance between directly overhead

[observer’s zenith] and the target star

ζ = π/2 −

ϕ

latitude, ϕ = π/2 − θ

θi

angle of incoming ray, relative to the zenith

θo

angle of outgoing ray

θ

nondimensional distance-to-North-Pole

θ

colatitude, θ = π/2 − ϕ, [sphere only]

ψ(θ0, θ1) angular width of arc between polar coordinates ψ(0, θ) = θ

Ω

(2π/86400) per second; angular frequency of earth’s rotation.

M North-Pole-to-equator distance

ρ

air density

r

radial coordinate in spherical coordinates

ν

Eratoshtenes parameter, (W earth + W star)/Dstar

refraction correction to the elevation angle

3

2 The First Empirical Law: Latitude Is Elevation

For at least two millennia, navigators have known how to determine their latitude. Knowing the latitude of the desired destination, the navigator could sail north or south to that latitude and then sail east or west to reach that destination. To do this, it was necessary to have a method of measuring angles. In early days Arabs used one or two ﬁngers at arms length to measure the angle between the horizon and Polaris. Later they used a Kamal, which is a piece of cord with knots tied in it. This could be used to measure the angle between the horizon and Polaris. A knot could be tied in the cord as a measure of the homeport latitude before leaving, so the desired latitude was premeasured. Arabs tied knots on the cord at intervals of one issabah, Arabic for “ﬁnger”, whic denotes 1 degree and 36 minutes.
In the 10th century AD, Arabs introduced to Europe the astrolabe and the quadrant. The quadrant spans 90 degrees and is divided into whole degrees. A plumb bob establishes the vertical. The quadrant was popular with Portuguese explorers in the 15th century. In addition to Polaris, they used observations of the Sun for determining latitude, particularly in the southern hemisphere.

— P. Kenneth Seidelmann in Sec. 7 of [13]

Waters’ treatise on navigation in the sixteenth and seventeenth centuries [20] discusses the Polaris-latitude connection on pgs. 41 to 50.
Royal Air Force navigators used “latitude equals elevation” as a tool in navigating Lancaster heavy bombers in night attacks on Germany during World War II as described in [9], written by a retired war-time RAF navigator.
Thus, many centuries of navigation by land and sea provides overwhelming evidence for the following.
Empirical Law 1: Latitude is the Elevation Angle of Polaris In the Northern Hemisphere, the latitude of the observer is approximately equal to the elevation angle measured by the observer, or in other words

≈ϕ

(1)

. We show below that this heuristic rule is in fact a provable theorem. Practical navigators are obliged to make small corrections to the observed or “apparent”
elevation angle to account for errors intrinsic to the sextant and aslo atmospheric refraction [11], observer height above the sea (“dip angle”) and so on. The observation and corrections collectively generate the “true” elevation angle denoted by ; it is this corrected elevation that appears in Empirical Law 1.

3 The Leveled Observer
The surface of the planet is corrugated with small scale curvature in the form of innumerable hills and valleys. A planar earth has zero large scale curvature, but a spherical earth is curved on a planetary scale as well as on a small scale. To obtain consistent star measurements, we assume a “leveled observer” who employs a sextant with a spirit level (or equivalent surveying tricks) applied so that a vector from the surface to his zenith is parallel to the “eﬀective gravity” (gravity plus the centrifugal force due to the planet’s rotation).

4

The eﬀective gravity is not constant; a mountain will alter the gravitational ﬁeld in its immediate vicinity. However, the variations in Earth’s surface gravitational ﬁeld are less than one part in 15,000 of its mean value. The corrugations will then have no eﬀect on our measurements or results except perhaps in the fourth decimal place. Geodesists and oil prospectors need to be concerned about these variations, but we need not.

4 Empirical Law 2: The Distance to the North Pole Is the Product of Colatitude With 6360 Kilometers

By the late 16th century, Englishmen knew that the ratio of distances at sea to degrees were constant along any great circle such as the equator or any meridian, 1 Sssuming that Earth was a sphere. Robert Hues wrote in 1594 that the distance along a great circle was 60 miles per degree, that is, one nautical, mile per arcminute (pg. 374 of [20]).
Edmund Gunter wrote in 1623 that the distance along a great circle was 20 leagues per degree. (Pg. 374 of [20]), which is the same statement in a diﬀerent length unit.
Pierre-Louis Maupertuis (1698-1759) was a major ﬁgure in physics for stating the Principle of Least-Action. In 1736, he was appointed chief of the French Geodesic Mission to Lapland to measure the length of a degree of latitude. Simultneously, a second team was sent to Ecuador. The measurements triumphantly conﬁrmed Newton’s prediction that Earth is a (slightly!) oblate spheroid. In 1799, the meter was deﬁned to be one-quarter of the length of a meridian divided by 10,000,000.
Let “geodesic distance” denote the arclength of the curve that connects two points by the shortest path that lies in the surface of the planet. This deﬁnition of the meter implies that one degree of latitude shortens the geodesic distance to the North Pole by (10,000 kilometers/90) ∼ 111 kilometers, consistent with Hues, Gunter and other navigators. Collectively, these and other astronomers over many centuries arrived at the following.
Empirical Law 2: Nautical Miles to the Pole and Degrees of Latitude Each increase in latitude ϕ by one degree decreases the geodesic distance to the North Pole from the observer by 60 nautical miles or equivalently 111 kilometers. In symbols, this is equivalently

π

d(ϕ) = W − ϕ

(2)

2

where W is a length scale determined in the next section. It is not possible to prove this empirical law in the same way that Empirical Law 1 will be elevated to a theorem below. However, it is possible to prove that Emprical Law 2 can be measured as true.

5 A Trivial Derivation of the Radius of the Earth

The distance d along the geodesic distance (great circle arc) from the observer to the North Pole has no obvious length scale. Nevertheless, it is still convenient to write d as the product of a nondimensional distance and an arbitrary length scale W . The latitude ϕ is known for millions of cities and geographical features and can be easily determined by a sextant measurement of the elevation (Pole Star) via ϕ = (Pole Star). It is therefore convenient to use latitude as the nondimensional measure of distance from the observer to the North Pole. The formula for d becomes

gineqnarray d(ϕ) = W (π/2 − ϕ) .

(3)

1[ A meridian is a Great Circle of constant longitude; it passes through both poles.]

5

Empirical Law 2 requires that the pole-to-equator geodesic distance is the length associated with one degree of latitude (111.1 km) multiplied by the number of degrees of latitude (90) between the equator and pole, yielding d(π/2) = 10, 000km. Eq. (3) becomes

π

10, 000km = W

(4)

2

which demands that

W = 6360 km

(5)

= W earth

(6)

Note that W = W earth was obained without any presuppositions of planetary shape or curvature; the radius of the spherical Earth has emerged spontaneously.

6 Parallel Ray/Distant Source Approximation

“The historical assumption of the sun rays parallelism is far from being a spontaneous assumption for the students.”

— by Nicolas D´ecamp and C´ecile de Hosson on pg. 919 [6]

Whether light from a star falls upon the Earth or the Moon as a beam of parallel rays or as a beam of divergent rays has a profound impact upon the consequences. A perfectly parallel beam is impossible, but a beam of almost-parallel rays will nonetheless generate a much diﬀerent situation than a beam of strongly divergent rays. It is therefore essential to quantify the meaning of “almost-parallell”.
Let Dstar denote the distance from the observer’s position to the target star. Let W star and W earth be the widths (radii) of the light-emission source (star) and absorber (Earth). A ray from the center of the star to the center of the surface of the earth (“central ray”) is shown as the dashed line in Fig. 1.
The ray which is the “least parallel” (of those hitting the planet) in the diagram is the solid ray with the arrowhead. Its vertical range, as represented by the right side of the triangle in the lower half of Fig. 1, is the radius of the star plus the radius of the planet. Let η denote the angle in radians between the central ray and the least parallel ray. This is an upper bound on the angles between rays from star to planet.

Deﬁnition 4 (Nearly Parallel Rays) All the rays from the star to the Earth will be “nearly parallel” if and only if the bound-on-angles η 1.

It will prove useful to deﬁne the “Eratosthenes parameter” as

W star + W earth

ν≡

Dstar

[Eratosthenes parameter]

(7)

Theorem 1 (Spread of Angles and the Eratosthenes Parameter ν) In terms of the Eratosthenes parameter ν,

W star + W earth

η = arctan

Dstar

(8)

= arctan(ν).

(9)

6

Wearth+Wstar

Dstar
η
Hypotenuse
Figure 1: Top: Schematic of an emission source [star or the Sun] (yellow disk), Earth (blue disk) and light rays. The dashed line with an arrowhead is the path of a ray from the center of the star to the center of the Earth; we shall call this the “central ray”. The degree of parallelness of other rays is measured by the angle between the ray and the central ray. The least parallel ray that hits the Earth is one which, while traveling a distance Dstar ( horizontal in the diagram) from the star to the Earth is moving in the perpendicular direction (vertical in the ﬁgure) from one side of the star to the opposite side of the planet. This implies that this least-parallel ray is the hypotenuse of the triangle in the lower half of the diagram. The horizontal top side of the triangle is the path of the central ray with length Dstar. The angle η between these two rays is an upper bound on the angle between the central ray and all other rays that pass from star to planet.

Proof: The Law of Sines applied to the triangle in Fig. 1 yields [18]

W star + W earth

η = arctan

Dstar

,

(10)

which is (8). The second line follows from recognizing that the argument of the arctangent is just the Eratosthenes parameter ν as deﬁned by (7).

Theorem 2 (Angles for Nearly Parallel Rays) If the distance to the star Dstar is large compared to the larger of the radius of the radiating star and the radius of the target planet, then

1.

ν1

(11)

2.

η≈ν

(12)

3.

η1

(13)

7

This implies that all rays emitted by the star and falling on the planet will be “nearly parallel” as deﬁned above.
4. In the limit ν → 0 or equivalently, Dstar → ∞, the rays are parallel.

Proof: Recall that the deﬁnition of the Eratosthenes parameter is (7 ) ν ≡ (W star + W earth)/Dstar. When Dstar is large compared to the radii of the star and planet, it follows that ν 1, which is the ﬁrst proposition.
The Taylor expansion of the arctangent function, (4.4.42) of [1], transforms (8), η = arctan (ν), into

η=

∞
(−1)j

1

ν 1+2j

1 + 2j

j=0

(14)

Retaining just the term of smallest degree gives, with a relative error of O(ν2), η ≈ ν, which is the second proposition of the theorem. This in turn requires that small ν implies small η, which is the third proposition. The fourth part of the theorem follows from taking the limit in the approximation η = ν + O(ν3) and then recognizing that the limit of “nearly parallel waves”, diﬀering in angle from the central ray by no more than η, is parallel waves.
The nearly parallel approximation may be alternatively labeled the “Distant Source” approximation because it becomes more and more accurate as Dstar increases.
For all stars, ν is tiny because the distance to the star is huge compared to the diameter of the star.
For Polaris itself, Dstar = 4 × 1015 km and W star = 3 × 106 km yield

ν ≈ 0.75 × 10−9

(15)

For the Sun and earth combination, Dstar = 1.496 × 108 km, and W star = 6.96 × 105 km,

ν ≈ 0.0046 ≈ 1/215

(16)

The smallness of ν vindicates Eratosthenes, who assumed parallel (or nearly parallel) rays from the sun in his measurement of the radius of the earth.[6, 21, 10]

Theorem 3 (Flat Earth Parallel Rays) On a Flat Earth when the rays from the star are parallel, the elevation of Polaris (and every other star) is independent of the observer’s latitude.

Proof: Obvious from the left half of Fig. 2. The Flat Earth-parallel ray prediction that every observer sees Polaris at 90 degrees of elevation is in gross contradiction to observations. Thus, the Flat Earth is possible only if the parallel ray approximation fails, which requires that the distance to the star is of comparable magnitude to the larger of the radii of the earth and the star. In contrast, if the parallel ray approximation is accurate, observers on a spherical earth measure diﬀerent elevation angles depending on latitude. Let us suppose that all observations are made on the same meridian as that of the target star. The stars appear to rotate about the Pole Star; a time-lapse photograph gives a so-called “star trails” plot in which each star (except the Pole Star) traces a circle. Each point on the star trail gives a diﬀerent angle; the desired elevation angle of a star is the angle of the point on its star trail which is closest to the observer’s zenith, i. e., the highest point on the star trail.

8

Flat Earth

B C
A
Sphere Planet

Figure 2: Parallel rays falling on a Flat Earth (left) and a spherical planet (right). All the observers on the Flat Earth measure exactly the same elevation angle of a given star, regardless of observer location. In contrast, observers at diﬀerent points on the sphere measure diﬀerent elevation angles. The elevation angle for observer C on the right is around 90 degrees where the elevation angle for observer A is near 0; for him the rays are low, just above the horizon.
Fig. 3 shows why diﬀerent observers on a meridian of a spherical planet obtain diﬀerent angles for the beam of approximately parallel rays from a target star. The horizontal direction for each observer can be represented by a vector that is orthogonal to the local eﬀective gravity vector. The crucial point is that as we move from one observer to another, this vector must rotate, and the rotation angle must be added to the elevation angle of the previous observer to obtain the elevation angle at the location of the current observer as shown in the ﬁgure. Thus, observer B measures an elevation angle 90 − ψ where ψ is about 45 degrees while observer C has an elevation angle of C = 90 degrees. The diﬀerence in elevation angle arises because B and C are separated by about one-eighth of the circumference of the planet or equivalently by an angle of 360/8 degrees. In general, the measured elevation angles of two observers diﬀer by an angle Ψ which is 360 degrees multplied by the geodesic distance between the two observers divided by the circumference of the earth.

9

B C
ψ

Figure 3: Parallel rays falling on a spherical planet. Two rays from the center of the planet through the observers deﬁne an angle ψ. Observers B and C will measure diﬀerent elevation angles for the star. Because the rays from the star (and indeed any star) are parallel, the diﬀerence C − B is due entirely to the fact that a line segment [red] parallel to the horizontal direction of Observer B is the same as for Observer C except for a rotation through the angle ψ.

7 Elevation Is Latitude

7.1 Conversion of an Empirical Law to a Theorem
Theorem 4 (Elevation Angle Is Latitude) Let ϕ denote latitude and denote the “elevation angle”, also called the “altitude angle” or “altitude”. If the Pole Star is observed from a spherical planet and the distance to the Pole Star is large compared to the radius of the Pole Star (allowing the parallel rays approximation), then after is corrected for the small errors of the sextant, the slight displacement of Polaris from the northern Celestial Pole, and atmospheric refraction, etc.

ϕ=

(17)

Proof: To handle general positions of stars on the Celestial Sphere, rotate the latitudinal coordinate so that C is at a latitude of 90 degrees N. in the new coordinate ϕ˘. (Longitude is not modiﬁed.) Observer B is at a ϕ˘ that diﬀers from C’s by ψ degrees. Therefore,

B = 90 − ψ [degrees]

(18)

10

But the latitude of observer B is also 90 − ψ. It follows that latitude and elevation angle are always equal, which is the proposition of the theorem.
This theorem is identical with Empirical Law 1, which is thus converted from an empirical relationship — highly probable but vulnerable to disproof by as yet undisovered counterexamples — to a mathematical theorem.

7.2 Sextant Corrections
As noted in the introduction, the accuracy of the sextant measurement can be usefully improved by applying a number of corrections. The details can be found in any handbook of celestial navigation [5, 4] or videos such as [12, 11]; it would take us too far aﬁeld to describe them here. Instead, the correction for atmospheric refraction [next subsection] will have to illustrate the idea of corrections for all species of corrections.
The important point is that most of these errors can be greatly reduced by using the tables of the Astronomical Almanac, a joint production of the U.S. Nautical Almanac Oﬃce, U. S. Naval Observatory, Her Majesty’s Nautical Almanac Oﬃce and the U. K. Hydrographic Oﬃce. The printed version contains precise positions over time of astronomical objects both natural (stars, planets, etc.) and man-made (artiﬁcial satellites) for a given year. This ephemeris serves as a world-wide standard for such information. The online version [http://asa.hmnao.com/] extends the printed version by providing data in machine-readable form. A good source for the theory behind the Almanac is [15]; the history and pre-history of navigational almanacs is described in [14]. The videos [12, 11] walk the reader through examples.

7.3 Atmospheric Refraction
When the material properties of the medium vary, this induces variations in the speed of light, smoothly curving or discontinuously bending the trajectories of light rays as discussed further in Subsection 11.1. In particular, density variations in the atmosphere alter the speed of light. Unfotunately, the density of the atmosphere varies with the seasons, local weather phenomena and the exponential decrease of density required by hydrostatic equilibrium. Without density measurements, the best one can do is to apply a simple analytic approximation that represents a time-average of refraction.
Let app denote the apparent elevation angle. Let denote the true elevation (or synonymously,“ true altitude”). Let denote the correction to app. The simplest analytical formula for refraction is, in radians,

= −0.000293 cot( app)

(19)

which is given on the ﬁrst page of [2]. It is not recommended when app is smaller than 15 degrees.
Bennett [2] compares many analytic approximations for the refraction angle ( app). We apply his method H, a more complicated formula than (19), which is widely used, notably in the U. S. Naval Observatory’s Vector Astrometry software.

7.31

= 0.000291 cot app + app + 4.4

(20)

Note that the argument of the cotangent function must be computed in degrees and then converted to radians.
Once the refraction has been calculated, the best approximation to the elevation is

≈ app +

(21)

11

where app is the “apparent” elevation and is the “true” elevation, which is the sextant measurement modiﬁed by all non-refractive corrections. This is also the best approximation to latitude, ϕ ≈ .
Fig. 4 and Table 2 show that the refractive correction is less than an arcminute for all app greater than 45 degrees, which implies a navigational error, if the refractive correction is omitted, of only one nautical mile. The correction grows as the elevation angle decreases, which is why navigators are advised to measure Polaris and other celestial objects only when the target is at least 15 degrees above the horizon. However, is a good approximation to latitude even when Polaris is within a degree or two of the horizon.

latitude (degrees)

Latitude versus apparent elevation
90

75

60

45

30

refraction

15

uncorrected

error

15 30 45 60 75 90
apparent elevation (degrees)

Latitude versus apparent elevation

10

1

Error (degrees)

error (degrees)

8

6

4

refraction

uncorrected
2

2

4

6

8 10

apparent elevation (degrees)

0.5
0 2 4 6 8 10
apparent elevation (degrees)

Figure 4: Corrected and uncorrected elevation angles and their diﬀerence on the full range of the elevation angle, app ∈ [0 degrees, 90 degrees] (top). Not that the uncorrected and refraction-corrected curves are graphically indistinguishable and the dotted error curve is invisible underneath the horizontal axis. Bottom left: the same but on the reduced range app ∈ [0 degrees, 10 degrees]. The lower right graph plots − app, also on the shortened range [0, 10 degrees].

latitude (degrees)

12

Table 2: Refraction corrections using the formula “H” of Bennett [2]
Note that refraction in nautical miles (rightmost column) is numerically identical to the refraction in arc minutes.

app [degrees] 0.5 1 2 5 10 15 30 45 60 75

[degrees] 0.021 0.59 1.70 4.84 9.91 14.9 29.97 44.98 59.99 75.0

[radians] - 0.008 -0.01 -0.0071 - 0.0029 - 0.0016 -0.0011 -0.00050 -0.00029 -0.00017 -0.000077

[degrees] -0.48 -0.41 -0.30 -0.165 -0.090 -0.061 -0.029 -0.017 -0.0096 -0.0044

[nautical miles] -28.8 - 24.3 -18.2 -9.89 - 5.39 -3.63 -1.72 -1.00 -0.57 -0.27

13

8 Why Flat Earth Needs a Nearby-Polaris and Nonparallel Light From the North Star
On a Flat Earth, the horizontal levels of all Flat Earth observers are parallel. If the light from a star like Polaris is also approximately parallel, then, as seen in Fig. 2 all Flat Earth observers will see Polaris at the same point in the celestial sphere. This is in gross contradistinction to what is actually observed, ≈ ϕ, Theorem 4.
Therefore, Flat Earth requires that Polaris be so close to Earth — a few thousand kilometers or less— so that its rays are not approximately parallel. Diﬀerent observers will see diﬀerent elevation angles.2 The close-Polaris hypothesis allows the computation of Dstar, the distance to Polaris, by triangularization as described in the next section.
If, on the contrary, Polaris is very distant, then the following corollary to Theorem 2 shows that calculation of Dstar is impossible.
Corollary 1 (Triangularization Fails in the Parallel Limit) In the limit of parallel rays, it is impossible to obtain an estimate of Dstar from triangulation because the relevant triangles are inﬁnitely long and narow, degenerating into rays. This conclusion is independent of the shape of the planet.
Proof: Take the limit, for ﬁxed W star and W earth, of Dstar → ∞ in Fig. 1.

9 Triangularization On the Flat Earth for a Local Polaris

Let D denote the distance from the North Pole to the Pole Star. Equivalently, D is the length of a geodesic from the star to the North Pole. Because the star is at the zenith of an observer standing on the North Pole (a “leveled observer” as deﬁned earlier), the line segment is perpendicular to the surface of Flat Earth.
Let d denote the distance from the observer’s position to the North Pole. Let denote the elevation angle.
Fig. 5 shows a triangle whose vertices are (i) the observer’s position on the surface of the planet (ii) the North Pole and (iii) the Pole Star. It is a good ﬁrst approximation to pretend Polaris is the Pole Star because the error is only three-quarters of a degree.
The “Law of Triangles” [18] then shows that the ratio of the two shortest sides is equal to the tangent of the angle between these two sides, or in other words

D = d tan( ) ↔ d = D cot( )

(22)

We shall apply the left equation in Flat Earth Case I and the right equation [d(D, )] in Flat Earth Case II.
2This means jettisoning several independent measurements of the distance to Polaris by parallax and other methods that show the actual distance to Polaris is over 400 light-years. However, a central tenet of all Flat Earth sects and denominations is that all space agencies and professional astronomers are part of a vast conspiracy and hoax, and their publications and ﬁndings must be discarded. Flat Earth must be approached as fruitful for what-if scenarios, but is otherwise the exponential of the exponential of exponential of delusion.

14

Zenith angle ζ
D

Elevation angle ε

d

Figure 5: Triangulating the Pole Star on on a Flat Earth.

10 Flat Earth Case I : d [distance to pole] is a linear function of elevation angle

arctan(D/d) ⇔

D = arctan
d

Pole-Star-on-the-Flat-Earth

(23)

where D is the distance to Polaris and d is the distance from the observation point to the North Pole. This can be rewritten to give the unknown D in terms of the known quantities d and (d):

D = d tan( ) ⇔ D = d tan ( (d)) Pole-Star-on-the-Flat-Earth

(24)

Substituting

π

d(ϕ) = W − ϕ

(25)

2

where W = 6360 km and replacing elevation angle by latitude ϕ yields

π

D(ϕ) = W (π/2 − ϕ ) tan (ϕ )

(26)

2

This function is plotted in Fig. 6. Table 3 lists numerical values of the distance to Polaris

for diﬀerent latitudes of the observer.

For a Flat Earth evangelist, the very bad news is that Eq. 26 predicts a diﬀerent location

for Polaris for every latitude. An observer at 75 degrees N. latitude sees Polaris more than

6220 km above the North Pole whereas an observer in the tropics at 15 degrees N. sees Polaris

only 2233 km above the Flat Earth. One is reminded of the cartoon character Multiman,

who could almost instantly create an army of clones. Even his magic powers fail here,

however: placing clones of Polaris at both 2233 km and 6220 km above the planet would

mean the observer now sees two Polarises. Since the number of observers is not restricted

to two but may be arbitrarily large, we are immediately confronted with an inﬁnite number

of yellow giant clones of Polaris.

To amplify the absurdity, max(D) = W = 6360 km. This implies that Polaris is always

near and local, but can a star so close (with a distance of 10,000 km at most)

15

1

0.8

0.6

0.4

0.2

0

0

15

30

45

60

75

90

latitude (degrees)

Figure 6: Flat Earth Case I: The plot shows the distance D from the North Pole to the Pole Star, scaled by W = 6360 km, so that the nondimensional quantity plotted is D/W = (π/2 − ϕ) tan(ϕ). The ﬁgure assumes that d = W (π/2 − ϕ).

1. be a point source 2. be a star of the ﬁrst magnitude over the entire daylight sky 3. while invisible at night?

16

Table 3: Distance D from the North Pole to the North Star, Polaris. The star cannot be in multiple places at the same time, so this Flat Earth model fails.

Distances D have been independently checked and conﬁrmed in [3].

Analytical Form Fourier Latitude (degrees) , ϕ (radians)

d

1 degree

(1/180)π

9889

15 degrees

π/12

8333 km

45 degrees

π/4

5000 km

75 degrees

(5/12)π

1667 km

89 degrees

(89/180)π

111.11 km

D [Distance to Polaris] 173 km 2233 km 5000 km 6220 km 6360 km

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11 Flat Earth Case II: Polaris Location D is Independent of Latitude

Polaris cannot simultaneously be at every possible distance D above the North Pole from 0 to 6360 km. Can this unreality be ﬁxed?
The linear relationship between d, the distance from the observer to the North Pole, and π
latitude ϕ, which is d(ϕ) = W − ϕ as derived in the previous section, is supported 2
by at least half a millenia of experience at sea by those whose lives literally depended on it! Nevertheless, in the spirit that desperate illnesses often require drastic remedies, we now explore a nonlinear relationship d(ϕ).
The right equation in Eq. (22) gives a relationship between d, D and . If D is to be a constant so that Polaris has a single, unique distance from the planet, independent of the observer’s latitude, then the constraint from (22) becomes, with D as a constant,

d(ϕ) = D cot(ϕ).

(27)

as illustrated in Fig. 7. However, as ϕ → 0 (the equator), d diverges to inﬁnity. Since d(0) = ∞, the Northern Hemisphere, latitude ϕ ∈ [0, π]) is mapped to the entire
inﬁnite plane. Where, then, to put the Southern Hemisphere? One could perhaps put the Southern Hemisphere on the other side of the inﬁnite plane. Then a journey across the equator — easy in our reality — would become a journey of inﬁnite length. The PanAmerican Highway from Alaska to Chile would be stretched to a never-ending strip of pavement.
We are left with insoluble diﬃculties:
1. The Northern Hemisphere is an inﬁnite disk.
2. There is no place to put the Southern Hemisphere.
3. limϕ→0 d(ϕ) = ∞.
4. The nonlinear relationship between d and ϕ is refuted by many centuries of navigation on land, sea and air.

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3
2.5
2
d/W
1.5

distance d from observer to North Pole
D=1000 D=5000 D=10000

1

0.5

0

0

15

30

45

60

75

90

latitude (degrees)

Figure 7: Flat Earth Case II. The distance d from the North Pole to the observer at latitude ϕ, scaled by W , as illustrated for three diﬀerent values of the distance D (in kilometers) to Polaris.

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11.1 Flat Earth Case III: Flat Earth with Refracted Light Rays

Another possibility is that atmospheric refraction of the light from Polaris can remove the absurdities of the two previous Flat Earth models. An important constraint is that refraction is zero in a vacuum. Therefore, refraction will be conﬁned to a layer of thickness L where L is at most a few kilometers. Equivalently, refractive light-bending occurs only in the lower and middle atmosphere. Since the distances D from the North Pole to Polaris have been a few hundred to a few thousand kilometers in the earlier sections, we shall make the plausible assumption that

L << D

(28)

We shall suppose for simplicity that the index of refraction (i) is equal to one, its vacuum value, everywhere above the refractive layer and (ii) varies discontinuously at z = L:

n=

1, nbot(ϕ),

z>L z≤L

(29)

where z is height above the surface of the planet. Snell’s Law is usually stated in terms of the angles θi and θo for the ray incoming from
the Pole Star and the outgoing, refracted wave, respectively, as shown in Fig. 8. Denoting the indices of refraction in the top and bottom layers by ntop and nbot, respectively, Snell’s law is

sin(θo) = ntop

(30)

sin(θi) nbot

However, we previously employed angles that are the complements of θi and θo, δ = π/2 − θi and = π/2 − θo. Furthermore ntop = 1. Making these substitutions and invoking the identity that the sine of an angle is equal to the cosine of its complementary angle, Snell’s Law can be rewritten as

cos (δ(ϕ))

nbot(ϕ) = cos( (ϕ)

(31)

To obtain expressions for δ(ϕ) and (ϕ), we execute the following steps:

1. Empirical Law 1 is

=ϕ

(32)

2. Empirical Law 2 gives the Polaris-to-North-Pole distance as

π

d=W − ϕ

(33)

2

3. Trigonometric identities applied to the triangle with vertices at the North Star, the North Pole and the observer’s position gives

Wπ

δ = arctan

− ϕ)

(34)

D2

Then

Wπ

cos arctan

−ϕ

D2

nbot(ϕ) =

cos(ϕ)

(35)

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θi
δ ϕ
θo

Figure 8: Snell’s Law of Refraction for Flat Earth Case III for the Pole Star. The horizontal dotted line is the boundary [interface] between the two layers of air; the dashed line is a vertical guideline perpendicular to the interface. θ1 and θo are the angles of the incoming and outgoing light rays. φ is colatitude

The index of refraction observed in the atmosphere is very close to one. The National Institutes of Science and Technology oﬀers an Engineering Metrology Toolbox at emtoolbox.nist.gov. This includes a calculator to compute the index of refraction for air as a function of temperature, humidity and pressure. One ﬁnds that even for extreme pressure, density and temperature, the index of refraction is less than 1.004. Furthermore, the atmosphere slows down the incoming light; the index of refraction can never be less than 1. Thus, the physically realizable values of the index of refraction are

1 ≤ nbot ≤ 1.004

(36)

Fig. 10 shows the required index of refraction in the lower, refractive layer for the Flat Earth to have a single, unique distance to Polaris. For almost all observer latitudes, the needed index of refraction nbot is unphysical, violating (36).
A little reﬂection will show that the simplifying assumptions made earlier have no eﬀect on the conclusion: Refraction fails to salvage the Flat Earth.

12 Summary
For at least half a millenia, navigators have used the empirical rule that the elevation of the Pole Star (Polaris) is equal to latitude. We have shown that proposition can be elevated to a proved theorem.
A second ancient empirical law is that the distance in kilometers from the observer to the North Pole, the geodesic distance measured along the spherical surface of the planet, is

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the number of degrees of colatitude multiplied by 111.1 kilometers. This cannot be proved a priori, but we show that this relationship can be established by measurments.
However, once Empirical Law 2, that is, that distance to the pole is colatitude multiplied by 111 km, is conﬁrmed by measurement, we prove that this rigorously and uniquely determines the circumference of the Earth to be 40,000 km.
We also extend these ideas and the parallel ray approximation to three diﬀerent ways of modeling a Flat Earth. All fail. As noted by many previous videographers, too numerous to list, triangularization gives diﬀerent distances from Earth to Polaris for each latitude. Absurd!
Photographs from space, taken on a very expensive satellite or astronauts on the moonm are unnecessary. Simple mathematics and ancient empirical laws prove Earth a sphere just as well.
References
[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
[2] G. G. Bennett, The calculation of astronomical refraction in marine navigation, J. Navigation, 35 (1982), pp. 255–259.
[3] Blue Marble Science, Mitchell from Australia and Trigonometry Just Don’t Get Along. video, September 27 2021. Channel=Blue Marble Science , duration=15:50.
[4] D. Burch, Celestial Navigation: A Complete Home Study Course, Second Edition, Starpath Publications, 2d ed., 2015. 296 pp.
[5] T. Cunliffe, Celestial Navigation: Learn How to Master One of the Oldest Mariner’s Arts Paperback – Illustrated, Fernhurst Books, 3rd ed., 2010. 80 pp.
[6] N. De´camp and C. de Hosson, Implementing Eratosthenes’ Discovery in the Classroom: Educational Diﬃculties Needing Attention, Science and Education, 21 (2012), pp. 911–920.
[7] H. C. Freiesleben, Early pole star tables, J. Navigation, 8 (1955), pp. 285–288.
[8] J. B. Hewson, A History of the Practice of Navigation, Brown, Son and Ferguson, Glasgow, 1951.
[9] P. Hoare, Navigating Lancasters in WW II, J. Navigation, 60 (2007), pp. 201–215.
[10] T. V. Papathomas, Celestial illusions and ancient astronomers: Aristarchus and Eratosthenes, in Proceedings of SPIE 5666, Human Vision and Electronic Imaging X, B. E. Rogowitz, T. N. Pappas, and S. J. Daily, eds., New York, 2005, SPIE, SPIE, pp. 4–8.
[11] Practical Navigator, Basic latitude by Polaris. video, Sept 27 2017. Channel=Practical Navigator , duration=11:59.
[12] Rigging Doctor , Can You Really FIND Your COORDINATES With a SEXTANT? — Sailing Wisdom. video, August 6 2021. channel=Rigging Doctor, duration= 12:28.
[13] P. K. Seidelmann, Before the nautical almanacs, in The History of Celestial Navigation: Rise of the Royal Observatory and Nautical Almanacs (Historical & Cultural Astronomy), P. K. Seidelmann and C. Y. Hohenkerk, eds., Springer, Cham, Switzerland, 2020, pp. 1–20.
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[14] P. K. Seidelmann and C. Y. Hohenkerk, eds., The History of Celestial Navigation: Rise of the Royal Observatory and Nautical Almanacs (Historical & Cultural Astronomy, Springer, Cham, Switzerland, 2020.

[15] S. E. Urban and P. K. Seidelmann, eds., Explanatory Supplement to the Astronomical Almanac, University Science Books, New York, 3rd ed., 2012.

[16] G. Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton, New Jersey, 2009. 352 pp.

[17]

, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton

University Press, Princeton, New Jersey, 2013. 208 pp.

[18]

, A Very Short Introduction to Trigonometry, vol. 626 of Very Short Introductions,

Oxford University Press, Oxford, 2020. 192 pp.

[19]

, The Doctrine of Triangles: A History of Modern Trigonometry, Princeton Uni-

versity Press, Princeton, New Jersey, 2021. 392 pp.

[20] D. W. Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, New Haven, Connecticut, 1958. 696 pp.

[21] Yazdan and his class of tweens, Measure the circumference of the earth like Eratosthenes 240 BC. video, Apr 3 2019. channel=Mr Yazdan, duration=3:59.

23

Polaris

D-L

δ elevation angle
above refraction layer
L d-dL

observer
ε dL

Figure 9: Flat Earth Case III. The downward-propagating light ray from Polaris (broken line segment with chevrons) has an elevation angle δ in the near-vacuum region z ∈ [L, D]. At z = L [dashed line], the density jumps discontinously with the density larger in the thin layer z ∈ [0, L]. The light bends at the density jump so that the measured elevation angle is altered by refraction from δ (for z > L) to give the smaller angle seen by an observer on the ground. The observer is a distance d from the North Pole. Note that although it is diﬃcult to show these triangles to scale [and these triagles are therefore omitted from the graph], D L and d dL. These inequalities make it possible to accurately apply triangle formulas with the approximation L = 0, i. e., the refractive layer were absent.

24

3

2.5

index of refraction

2

1.5

D=10,000 km

1

D=5000
0.5

D=1000

0

0

15

30

45

60

75

90

latitude (degrees)

Figure 10: Flat Earth Case III. nbot(ϕ) for three diﬀerent geodesic distances D of Polaris from the North Pole. Values with the index of refraction less than one, which are not physically realizable, are marked with disks.

25