Draait de zon om de aarde, of draait de aarde om de zon?

Mostafa Abdelkader and the Geocosmos

With the marginal exception of Euler’s and Leslie’s proposals, the hollow Earth

remained entirely outside of the scientific community’s consideration or even awareness

(except as a novelty; see Sexl 174-176) until 1982, when Mostafa Abdelkader proposed a

mathematically-based rationalization for the geocosmos, one of the mystical forms of the

hollow Earth idea that arose in the 19th century. To say that Abdelkader reintroduced the

idea to the modern literature of science is true. But to say that it had any noticeable effect

whatsoever on the world of mainstream science would be an overstatement. The reasons

lie in the ways that the practice of science as a conservative social construction, evolved

during the nearly three hundred years separating Halley from Abdelkader.

In 1692, nothing, really, was known of the nature of Earth’s interior, the boundary

between the nascent modern, materialistic world view and the entrenched superstition of

Christianity was vague, and the scientific community had not developed the system of

peer review that lies at the heart of modern scientific practice. Halley was able to publish

his theory in one of the premier scientific organs of the day, in part because of the valid

empirical data it contained (his list of compass variations held considerable value for

navigation) but also because of the general state of scientific knowledge at the time and

because his standing within the Royal Society meant that he could probably have

published pretty much anything he pleased.

By 1982, modern geoscience had evolved, matured, and developed a robust

description of Earth’s (non-hollow) interior based principally on evidence from seismic

waves. That understanding was developed and is maintained by the necessarily

conservative process of peer review, and in 1982 there were few venues where it is

possible to submit an idea as radical as the hollow Earth to serious review and

consideration by an audience of scientific peers. One of those was the journal

Speculations in Science and Technology.

Speculations in Science and Technology was one of a handful of serious-minded,

professional, scientific journals that have been established to examine topics and issues at

the fringe of modern science’s range of acceptable inquiry (a notable peer in this niche is

the Journal of Scientific Exploration). There are doubtlessly many in the scientific

community that would deny the journal all validity, and a great many more who don’t

even know it ever even existed. But Speculations was published from 1977 until 1998 by

respectable publishers (Elsevier and then Kluwer, both powerhouses in academic

publishing) and its contributors, reviewers, and editorial board members were generally

(though not always) practicing scholars, some of them quite distinguished, in legitimate

fields of science and philosophy. Nonetheless, the journal’s stated purpose was to provide

a forum for speculation on ideas that are outside the scientific mainstream (though not too

far: topics related to UFOs and Extra Sensory Perception, for example, were not

accepted).

So, while Halley’s theory entered mainstream scientific discourse at its core,

Abdelkader’s geocosmos did so at its fringe. Moreover, it arrived there from an origin in

religious mysticism. To appreciate Abdelkader’s proposal in its appropriate context, it is

useful to briefly consider the trajectory of hollow Earth ideas as they evolved among

pseudoscientists and mystics during the 19th and 20th centuries.

The conception of Earth as a hollow sphere in an otherwise Copernican universe

(as invoked by Kircher, Burnet, Halley, Euler and Leslie) is the most intuitive conception

of the hollow Earth. The geocosmos, in which Earth’s surface occupies the interior shell

of a hollow sphere containing the entire universe, requires considerably more

imagination. Its modern form originated in the mind of Cyrus Reed Teed, an Eclectical

physician and practicing “electro-alchemist” from Utica, New York (see Kafton-Minkel

and Gardner for accounts of Teed’s remarkable history). In 1869, Teed had a mystical

experience in which he received the revelation that he was the living incarnation of

Christ. He also came to understand that the Copernican conception of the universe was

backwards. According to Teed’s “Cellular Cosmogony,” Earth is a hollow sphere that

contains the entire universe. We live on the inside surface.

Teed changed his name to Koresh, established a religious cult (“Koreshenity”)

that grew to be national in scope, and eventually established a utopian commune Florida.

There, adopting the outer appearances of scientific inquiry, Teed and some of his

followers organized the Koreshan Geodetic Survey and conducted an experiment to

prove Earth’s concavity. Using a specially-constructed apparatus dubbed the

“rectilliniator,” the Survey spent five months in 1897 patiently moving the device along a

six kilometer-long stretch of beach. Not surprisingly, the results of the survey were

exactly as Teed predicted—Earth’s surface proved to be concave (Gardner, Fads… 24).

While it is not clear whether or not Teed was consciously aware of it or not, his

geocosmos reflects the alchemical conception of the hermetic egg, the rotundum within

which, as Nelson (137) notes, microcosm and macrocosm— “cosmos, globe, and human

soul”—converge. Its genius lies in the fact that reconstitutes the geocentric universe (with

the comfortable reassurance that Earth, and thus humanity, occupies a privileged place in

a cosmos that is not only finite, but bounded at a humanly meaningful scale) in a way that

is still consistent with contemporary astronomy, provided one doesn’t look too closely.

Teed ensured that close examination would be unlikely by couching his theory within an

excruciatingly complicated cosmology and adopting the strategy of describing it in

impenetrable, scientific-sounding prose.

Teed died in 1908 (Koreshenity—including the commune of Estero, Florida—

persisted into the early 1950s), a decade or so before a German pilot named Peter Bender

came across several copies of the Koreshan’s Flaming Sword in a stack of American

magazines in a French prisoner-of-war camp during World War I. Bender was won over

by Teed’s geocosmos. After the war, he returned to Germany where he developed and

promoted the idea, which he dubbed the hohlweltlehre (“hollow Earth doctrine,”

sometimes also referred to as hohlwelttheorie). He abandoned the religious aspects of

Koreshenity and simplified Teed’s byzantine labyrinth of concepts and ideas to a simpler,

though still bizarre, mechanism to reconcile observed nature with the concave conception

of Earth.

Bender’s hohlweltlehre like other hollow-Earth theories before and since,

attracted its share of supporters, though none from within the ranks of mainstream

astronomers or Earth scientists. He was, however, able to muster enough political support

to manage two tests of his theory. The first of these was an attempt, in 1933, to build a

rocket and launch it straight up into the sky. If Bender’s hollow-earth idea was correct,

the rocket should have crashed into the opposite side of the planet. Instead, it failed to

launch and crashed a few hundred meters from its launch pad.

The second test came about through Bender’s connection (dating to his World

War I pilot days) with Hermann Göring and the interests of a group of German Naval

Research Institute officers who sought methods for locating enemy ships based on fringe

ideas such as pendulum swinging and the hohlweltlehre. These officers gained approval

to send an expedition to Rügen Island (in the Baltic Sea) to try and detect British ships

using powerful telescopic cameras pointed upwards across Earth’s concavity. Bender

claimed that the apparent convexity of Earth’s surface is due to the refraction of visible

light passing through the atmosphere. If Earth’s surface were concave, the officers

reasoned, photographs taken using infrared filters (infrared radiation is not refracted by

the atmosphere) should show parts of the North Atlantic and Baltic, and the positions of

British ships in those waters could be known. The failure of the Rügen Island experiment

proved embarrassing to the Nazi High Command, and Bender, his wife, and some of his

followers perished in death camps as a result.

Another German, Karl E. Neupert, published a pamphlet titled Mechanik des

Aethers, Gegen die Irrlehren des Kopernicus (“Mechanics of the Ether: Against the

Erroneous Teachings of Copernicus”) in 1901, and a book-length treatment titled simply

Geocosmos in 1942. Neupert collaborated with Bender until his unfortunate demise, and

after the war, he and another of Bender’s follower, Johannes Lang, continued to

publishing booklets and magazines on the subject promoting the idea. Neupert died in

1949, but Lang carried on, publishing a journal called Geocosmos into the 1960s. Neupert

and Lang, like Teed and his followers, distributed their writings widely, and at some

point, one of these copies caught the attention of Mostafa Abdelkader, who alone among

those who have encountered it was in a position to re-introduce the hollow Earth concept

back into the realm of mainstream science.

The key to the geocosmos model lies in reconciling the geometry of an internal

universe with observed phenomena such as the rising and setting of the sun and the

motions of other celestial bodies. Teed attempted this reconciliation by proposing an

absurdly complex clockwork model that invoked various gaseous layers within the

hollow of the planet and “refocalization” of the true Sun (which he said was light on one

side, dark on the other, and rotated like a beacon at the center of the universe) on the

upper layer of the atmosphere (Kafton-Minkel 94).

The simplest way to achieve such a reconciliation, however, is to abandon the

idea that light rays travel in straight lines, and have them travel in curves instead. The

simplest way to achieve this curvilinear behavior, in turn, is to simply perform a

mathematical mapping of the Copernican cosmos “outside,” into the geocosmos “inside.”

This is precisely what Abdelkader did, using a mathematical manipulation called

inversion to map the cosmos into the sphere of Earth.

Inversion is a geometric transformation that is useful for converting certain types

of otherwise intractable (or exceedingly complex) geometrical systems into forms that are

amenable to mathematical analysis. It is especially useful for transforming unbounded

regions into bounded ones; making the infinite finite, in other words. The geometry is

quite simple. To invert a plane with respect to a circle, for example, we simply map every

point outside the circle to a corresponding location within it. To invert the universe with

respect to a sphere, we simply map every point to some corresponding point within the

sphere, which is what Abdelkader proposes we do with respect to the sphere of Earth. But

this simplification both obscures the beauty and undermines the primary weakness of

Abdelkader's proposition. It is worth considering his proposition in some detail.

Abdelkader begins his paper with the proposition that Earth’s surface can be

considered a sphere (it is not, actually, but the slight equatorial bulge can be safely

ignored) of fixed radius with its center located within an absolute rectangular coordinate

system having x, y, and z axes. All points outside Earth’s surface can be denoted by X, Y,

Z and those inside the sphere by x, y, z. Abdelkader notes that in the Copernican system,

Earth rotates about its axis and revolves around the sun which, in turn, rotates around the

center of the Milky Way galaxy, and so on. By establishing the coordinate system in

relation to Earth’s center, however, Abdelkader has subtlety dispensed with the

Copernican universe and reestablished geocentrism: “We shall regard the earth as at

rest, so that all celestial objects are moving in the coordinate system (xX, yY, zZ)” (81).

Having prepared us, as a magician would, by framing the situation just so, Abdelkader

announces that he will perform the crux move of his trick: “In the following section, the

whole of space will be subjected to a purely mathematical mapping taking infinite space

outside the earth’s surface into its inside, and vice versa” (81). What follows are the

necessary mathematical manipulations.

The inversion operation is illustrated in Figure 2. Every point outside the sphere

of Earth maps to an analogous image point within it. “Thus,” Abdelkader explains (82),

“the earth’s surface is mapped into itself (with us living on the inside surface of a hollow

earth), all of outer space becomes embedded inside this hollow earth, with infinitely

distant points” mapping to the origin point of the sphere, and “objects such as stellar

galaxies and quasars distant several billions of light years, are shrunk to microscopic

size.”

After inversion, the moon, our closest celestial neighbor, maps to a sphere 955

meters across that circulates 6265 kilometers above Earth’s surface. The sun, on the other

hand, shrinks to about 2.5 meters across and recedes to a location just 253 meters from

the origin point (i.e. the center of the universe). Pluto shrinks to the size of a single

bacterium floating seven meters from the origin, while Alpha Centauri, the star closest to

our own Sun, becomes an infinitesimally small speck situated a mere millimeter from the

origin. Every other star and object in the cosmos, therefore, is contained in a sphere less

than two millimeters across that hovers 6371 kilometers above our heads.

Having inverted the Copernican cosmos to fit comfortably within Earth’s shell

(which becomes infinitely thick as a result of the inversion), Abdelkader goes on to

explore some of the implications of the transformation, first with regard to the shapes of

spheres and then the behavior of light. Because everything in the geocosmos shrinks with

distance from Earth’s surface, spherical bodies become slightly deformed in the direction

perpendicular to Earth’s surface (the Moon, for example, would be about one percent

smaller between the points nearest and furthest from Earth than it would be from pole to

pole).

The degree of deformation is relatively slight if we assume that the origin is, in

fact, a point. But Abdelkader notes that, while this assumption is perfectly acceptable in a

mathematical system, it is unrealistic in a physical one, so he substitutes a sphere of

arbitrary diameter for the origin point. If the radius of the origin sphere is very small

relative to the radius of Earth, the distortion is negligible. Larger radii for the origin

sphere, however, can result in a significant degree of distortion.

The changes in the behavior of light rays after inversion are perhaps the most

striking feature of Abdelkader’s model. In the Copernican cosmos, rays of light travel in

straight lines, as shown in 3A. Note that for an observer positioned where ray H intersects

Earth, E, (along the circle of illumination), the Sun would be visible on the horizon and

be seen as setting. For an observer positioned below ray J, it would be solar noon.

The inverse mapping preserves angular relationships, so that observers positioned

in the geocosmos would experience exactly the same phenomena as those in a

Copernican universe, as shown in Figure 3B. Ray H maps into e as ray h, and an observer

positioned at ray h’s intersection point would observe the sun on the horizon. Moreover,

because the Sun rotates around the origin, O, the observer would see it as setting, exactly

as does the observer in the Copernican cosmos (the Sun travels in a conical helix in the

geocosmos, which accounts for seasons). It is solar noon where ray j intersects Earth, and

halfway between solar noon and sunset below ray i. A person observing i would see the

sun as being somewhere between the horizon and the solar zenith at exactly the same

position in the sky as a person observing ray I in the Copernican universe.

Rays K and L do not intersect Earth in the Copernican universe and, assuming

they do not intersect anything else, will continue traveling to infinity. In the geocosmos,

however, k and l travel in arcs that lead back to the origin. The rays never actually reach

the origin, however, because the inversion operation affects not only the direction of light

rays, but their velocities as well. The speed of light is constant in the Copernican

universe, but variable in the geocosmos, ranging from ca. 3x109 cm/second at the surface

of e to zero at O.

The result of these conditions, Abdelkader notes, is that “all observations and

estimates of the size, direction and distance of any celestial object would lead to exactly

the same results” for an observer on the outside of Earth in a Copernican universe “and

his image observer inside, whether situated on or above” Earth’s surface (86).

Furthermore, as the case of the speed light illustrates, all physical laws that apply in the

Copernican universe can be inverted to apply in a geocosmos as well, provided we

invoke appropriate conditions to support them. The movement of Foucault pendulums

and the Coriolis effect, for example, are explained conventionally as effects arising from

Earth’s rotation about its axis. As Abdelkader notes, it is meaningless to attribute motion

to Earth in the geocosmos, but these phenomena can be explained in a geocosmos by the

rotation of the origin sphere (this, in turn, he attributes to an “all-pervading perpetual

cosmic force;” page 88). This isomorphism between the geocosmos and the Copernican

universe is a critical feature of Abdelkader’s hypothesis, because it creates a situation in

which it is impossible to empirically refute the geocosmos as a valid model of the

universe on the basis of observational tests.

The bulk of Abdelkader’s paper constitutes, as he puts it (87), “the purely mental

operation of geometrically mapping outer space…into the hollow earth…, a perfectly

legitimate process of thought” to which “nobody could raise the slightest objection.”

Though Abdelkader seems to have been unaware of it, Roman Sexl invoked the

hohlweltlehre in exactly the same vein in a paper on geo-chronometric conventionalism

published in 1970. Sexl used the hollow Earth to show that topology of space-time is

conventional, rather than intrinsic (he uses the example of “flatland”—c.f. Abbott—for

the same purpose regarding dimensionality). But Abdelkader has a larger goal in mind,

and he departs from the realm of idle mathematical curiosity in the last two pages of his

treatise. “Consider now” he entreats us “the hypothesis that our actual universe is the

finite and not the infinite ” (87; emphasis in

original).

Abdelkader supports his proposition by arguing that observational evidence

suggests that our universe is Copernican, provided we are willing to accept the untestable

assumption that “light is propagated in straight lines for billions of years, so that the

positions of celestial objects are in their observed directions…” (87). His point is not that

this is an unrealistic assumption, but rather that it is empirically untestable and therefore

the assumptions underlying the geocosmos are no more or less unreasonable than those

on which the Copernican model depends. So, Abdelkader reasons, given the choice

between two unfalsifiable models, both of which depend upon untestable assumptions

and yield identical observational data there is no reason to accept the Copernican view a

priori.

Abdelkader suggests that “there is no way of ascertaining the truth or falsity of the

hypothesis that our actual universe is except by digging a tunnel right

through the earth’s centre. … If our universe is , a tunnel 12,742 kilometres

long brings us to the earth’s surface again. If our universe is , nobody

knows what lies underground” (87). In fact, such a tunnel (if it were possible to dig one)

would not necessarily solve the dilemma. As the drill creating the tunnel receded from

the surface, it would become larger and larger, eventually becoming infinitely large and

infinitely far from the surface. At that point, it would likely emerge from the opposite

direction (some mathematicians and philosophers disagree on this point) and begin

shrinking as it approached the surface, emerging at a location antipodal to its starting

point.

There are, however, other grounds on which to reject the geocosmos, principally

its complexity and the privileged position in the universe that it ascribes to Earth. Martin

Gardner has discussed these objections in an essay entitled “Occam’s Razor and the

Nutshell Earth” (16). Occam’s razor dictates that, given a choice between two theories

with the same explanatory and predictive power, we adopt the simpler one. Complication

is to be tolerated only if it yields a commensurate gain in explanatory or predictive

power. Non-Euclidean geometry and Einsteinian relativity, for example, are more

complicated than their Euclidean and Newtonian counterparts but provide greater

explanatory and predictive power at astronomical scales. The same is true of quantum

theory at the subatomic level. Abdelkader’s geocosmos carries a high cost in

mathematical complexity (Figure 4) but, as noted above, there is no way to empirically

determine which model, geocosmos or the Copernican universe, provides the better

description of the cosmos.

So what does the geocosmos provide in return for the computational burden it

imposes? For Abdelkader, the answer is a sense of psychological comfort. At the end of

his paper, the detached language of mathematics and minimalist rhetorical presentation

give way to prose that conveys a barely-contained sense of angst that is rare in the

published discourse of modern science. The first paragraph of his conclusion bears

quoting in its entirety:

For one who dogmatically insists on believing the unprovable hypothesis

that light propagates in straight lines over distances of billions of lightyears,

the universe must be the universally accepted Copernican system. If

one is open-minded enough to get rid of one’s attatcment to this dogma,

then the only alternative universe is Geocosmos. The former, with its

incredibly gigantic stellar galaxies and other celestial objects distant

billions of light-years, and its stupendous energy sources, scattered

aimlessly throughout space, reduces the earth and the solar system to

nothing in comparison; whereas in the latter, the earth’s surface is the

finite boundary of the whole universe contained within it. Since both

universes are equally possible, there is no valid reason for astronomers,

astrophysicists, and other scientists to confine their attention exclusively to

the study of , totally dropping the competitive

out of their consideration. Probably the majority of these

scientists have never even heard of ; it is never mentioned in

the proliferating books on astronomy, either the technical or the popular

ones, as far as the author is aware. (88 emphasis in original)

For Abdelkader (like his Koreshan and hohlweltlehre forebears), the geocosmos

banishes the incomprehensible void of outer space to a speck contained within Earth’s

interior, simultaneously rendering the cosmos humanly comprehensible and restoring

Earth’s pre-Copernican place of privilege in the cosmos. If, as most mathematicians

believe, the idea of an inverted universe cannot be empirically refuted, is there really

anything wrong with this? Does it matter?

From a practical standpoint, accepting the geocosmos would have little or no

effect on most of us. We experience the universe as Euclidean space with Earth’s surface

or (occasionally) the Sun as our reference framework, and we can pass our entire lives

without ever having to take an Archemedian perspective that views the framework itself.

The same cannot be said for the “astronomers, astrophysicists, and other

scientists” Abdelkader lambastes for failing to give the geocosmos its due. The

geocosmos model simply does not solve any scientific problems they face, and pre-

Copernican nostalgia and apeirophobia are apparantly not widespread enough within the

space science community to justify the burden it would impose. Even if it were, the

geocosmos would not necessarily provide a cure. Abdelkader’s inversion banishes the

topology of the Copernican universe, but does nothing (except axiomatically) to

undermine the Copernican principle.

The Copernican revolution taught us that we should not assume that we occupy a

privileged place in the cosmos. Inversion does not suspend this principle except by fiat,

and as one of Gardner’s correspondents points out (On the Wild Side 21), even if the

geocosmos is a valid model, there is no reason to expect the universe to be inverted with

respect to our little planet. There are, for example, an estimated 1010 galaxies in the

known universe. Assuming that each of these contains 1011 stars, as does our own galaxy,

and that each of these stars is orbited by a mere ten spherical bodies (planets, their

moons, comets, asteroids, and small bits of rock or ice—any spheroidal body will do),

there must be 1022 objects in the universe (let us be clear here—this is a one followed by

twenty two zeros) to choose from. The probability that any one of them, including Earth,

is the preferred body is only 1/1022, which is vanishingly close to zero. Moreover, there is

no reason why the inversion must be done in relation to a physical body at all. It is

equally plausible to simply perform the inversion around an arbitrarily chosen spherical

region of space, in which case the choice of regions and spheres is limitless. Regardless

of which sphere we choose, if it is anything other than Earth, our planet becomes even

smaller and less significant than ever.

The only way to retain Earth as the preferred body is to simply assume

geocentrism, as Abdelkader has done. But if we are willing to indulge in this sort of

axiomatic reasoning, why not take the logic a step further, to egocentrism? If banishing

the extrasolar universe to a two-millimeter sphere provides relief from a feeling of

cosmic insignificance, then surely inverting the universe with respect to one’s own eye

(remember—any spheroid will do) must be more satisfying still.

This is truly an experiment that you can perform at home. You need not perform a

single calculation—simply declare that the cosmos is contained within your eye, and it is

done. Revel in knowing that you have given new truth (not to mention ownership) to

Walt Whitman’s claim “I am vast, I contain multitudes,” and no empirical test can refute

the proposition. Thrill to the fact that your brain is now the largest object in the universe,

and the question of what came before you and what will follow now have universal

importance. Experiment to your heart’s content, though it might be wise to keep the

knowledge secret, hidden away in your own little hollow world.

——————————————————————————–

Works Cited

Abbot, Edwin A. 1992. Flatland: A Romance of Many Dimensions. New York: Dover.

Abdelkader, Mostafa. “A Geocosmos: Mapping Outer Space Into a Hollow Earth.”

Speculations in Science and Technology 6 (1983): 81-89.

Burnet, T. The Sacred Theory of the Earth. (1690/91) London: Centaur Press, 1965.

Crowe, Michael J. The Extraterrestrial Life Debate, 1750-1900. Cambridge: Cambridge

University Press, 1986 (1999 Dover reprint).

DeCamp, L.S. and W. Ley. Lands Beyond. New York: Rhinehart and Co, 1952.

Drake, Ellen. Restless Genius: Robert Hooke and His Earthly Thoughts. New York:

Oxford University Press, 1996.

Gardner, Martin. Fads and Fallacies In the Name of Science. New York: Dover, 1957.

—– On the Wild Side. New York: Prometheus Books, 1992.

Godwin, J. Arktos: The Polar Myth in Science, Symbolism, and Nazi Survival. Kempton,

IL: Adventures Unlimited Press, 1996.

Halley, Edmund. “A Theory of the Variation of the Magnetic Compass.” Philosophical

Transactions of the Royal Society xiii (1683): 208-228.

—–. “An account of the cause of the change of the variation of the magnetical needle

with an hypothesis of the structure of the internal parts of the Earth.”

Philosophical Transactions of the Royal Society xvi (1692): 563-587.

Hooke, Robert. “Lectures and Discourses of Earthquakes and Subterraneous Eruptions,”

(1668-1700). Transcribed, annotated, and with an introduction by Ellen Tan

Drake in Restless Genius: Robert Hooke and His Earthly Thoughts. New York:

Oxford University Press, 1996.

Kafton-Minkel, Walter. Subterranean Worlds: 100,000 Years of Dragons, Dwarfs, the

Dead, Lost Races and UFOs from Inside the Earth. Port Townsend, Washington:

Loompanics Unlimited, 1989.

Kollerstrom, N. The Hollow World of Edmond Halley. Journal of the History of

Astronomy 23 (1992):185-192.

Leslie, Sir John. Elements of Natural Philosophy: Including Mechanics and Hydrostatics.

Edinburgh: Oliver and Boyd, 1829.

Nelson, Victoria. “Symmes Hole, Or the South Polar Romance.” Raritan 17 (Fall 1997):

136-166.

Peck, John W. “Symmes’ Theory.” Ohio Archaeological and Historical Publications 18

(1909), 28-42.

Sexl, Roman U. “Universal Conventionalism and Space-Time.” General Relativity and

Gravitation 1 (1970): 159-180.

Stanton, William. The Great United States Exploring Expedition of 1838-1842. Berkeley:

University of California Press (1975).

Symmes, John Cleves. Circular No. 1. Reprinted in Peck (30) and Kafton-Minkel (61).

Zircle, C. “The Theory of Concentric Spheres: Halley, Mather and Symmes.” Isis 37

(1947), 155-159.

Figure Captions.

(…)

Figure 2. Abdelkader’s inversion. Any point P outside Earth’s sphere is mapped to point

p inside the sphere according to the simple relation xX = a2 where x is the distance

between the surface E and p, X is the distance from E to X, and a is Earth’s radius (for

simplicity’s sake, Earth is considered to be a perfect sphere, though in reality it is slightly

flattened at the poles). We can obtain the distance x for any point P in the cosmos by x =

a2/X.

Figure 3. The behavior of light rays in a Copernican universe (3A) and Abdelkader’s

geocosmos (3B). Both diagrams are diagramatic only, and not to scale.

Figure 4. A ray of light passing through two points (X1, Y1, Z1) and (X2, Y2, Z2) follows a

straight line defined by the two equations in 4A. After inversion, its path is transformed

into a circle (or, if it intersects Earth’s surface, an arc thereof) passing through the origin

and defined by the equations in 4B. Based on Abdelkader’s equations 11-13.

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