The Origins of the Theory of The Hollow Earth

Edmond Halley (1656-1742), in Philosophical Transactions of Royal Society, raised the idea of a hollow Earth, with a solid sphere in the center. The publication was to explain the magnetic anomalies of the compass readings. Halley suggested that the atmosphere inside the earth was luminous (and possibly inhabited) and that the gas escaped was responsible for the Aurora Borealis.

 Edmond Halley with a diagram showing the multiple shells of his hollow Earth theory.

The Hollow Earth Was a Geometrical Necessity

For Halley, the Earth would be a shell of 500-mile thick, with three concentric spheres about the diameter of the planets Venus, Mars, and Mercury. Halley assumed that the spaces between the spheres in his model were representative of the atmospheres of them.

For Halley, the Earth would be a shell of 500-mile thick, with three concentric spheres about the diameter of the planets Venus, Mars, and Mercury. Halley assumed that the spaces between the spheres in his model were representative of the atmospheres of them.

Assuming as a unity the diameter of Earth, Mercury is 0.38 the diameter of Earth, Mars and Venus are 0.53 and 0.95. With these data, we can assume that Halley thought that in the very center of the Earth was the lowest of the spheres, representing Mercury. Then you should follow the sphere of Mars, which in turn should also be hollow to accommodate the Mercury sphere. Finally, the sphere of Venus, very close to the planet's surface, but also hollow to accommodate the field of Mars (which itself was inside that of Mercury). Thus, for Halley, the Earth is not completely empty but occupied by concentric hollow spheres, one inside the other areas [...] rotating at different speeds and oceans [...] filtered inwards" for the provisions that the Creator had taken". 

If we recall the geometric model of Kepler, a century earlier, in which he adapted the spheres of the first five orbits of planets within the five Platonic solids, we realize that what Halley did was similar to Kepler's model (1571-1630). Obviously, the Keplerian arrangement was had some solid foundations that are based on actual calculations of planetary orbits. What Kepler did is still valid because the orbits today have the same diameters of yesteryear, but what Halley did is now only a historical curiosity.

However, there is a common denominator between these two dedicated and famous astronomers: in both cases, they tried to find a justification for the existence of planets based on the harmony between the diameters of the planetary orbits. In both cases, the distances and diameters of the planets are not a product of long planetary evolution, but the planets are there because they were put there by the provisions that the Creator had made.

The arrangement of Kepler has a beauty that still one wonders why there is this amazing coincidence. In contrast, in Halley's model, the fact that the spheres corresponding to the planets are placed one inside each other, and the planet Mars is one of them, makes us question why the place Mars, although it is diameter smaller than Earth, its orbit is greater. This takes away the beauty and harmony of Kepler's model and makes us conclude that Halley's model is a model forced one.

The Hollow Earth Was a Geological Necessity

The idea that the Earth is hollow is a long story before Halley made it popular. To review some data, Archelaus ca. 500 (before the Christian era) was talking about how the Earth was "a swamp ... hollow in the center" and by 1616, another person named Balthasar van der Veen had also another theory that incorporated the idea of the hollow Earth. Griffin, in his article: What Curiosity in the Structure: The Hollow Earth in Science expands the list and includes proponents, advocates or hollow Earth mentioning Plato, Aristotle, Lucretius, Seneca, and Dante.

''Earthquakes and volcanoes, the holes in the ground, springs, and wells were sufficient to show that not everything is solid under our feet. The caverns and caves provide direct access to foreign inverted worlds beneath the surface of the Earth, while (in the absence of mechanistic understanding about how they form) fossils and other geological events reasonably suggest that they not only exist but hosts strange creatures", says Griffin.

The Hollow Earth Was a Theological Necessity

And then continues: "The need to reconcile the large number of geographical and geological information with the Mosaic accounts of creation and the Flood was the formulation of theories about the Earth a popular activity at the same time a moral necessity in the intellectual elite the seventeenth century, one that generated a scholarly debate and popular interest at the same time. Two books of that period - in particular, stand out: the Mundus Subterraneus by the Jesuit Athanasius Kircher, published in Amsterdam in 1664 and Sacred Theory of the Earth by cleric Thomas Burnet, in 1681".

It was while gathering data for his research of weather and topographic that Halley met with two difficulties which he described: "not easy to solve". The first was that "there is no magnet that I have heard with more than two opposite poles, while the Earth obviously has four, and maybe more. The second difficulty was that the poles were not fixed on the Earth".

If Halley believed that the earth had more than two poles in his time was because the tabulated data of geomagnetic positions was very limited, incomplete and fragmented, "especially near the poles", says Griffin. So not surprisingly, Halley believed that the Earth had more than two poles. But what most contributed that Halley postulated a hollow Earth was Newton's error in the estimate of the relative densities of the Earth and Moon. "Newton's estimate of a dense Moon provided Halley the key to solving the dilemma with which he lived for eight years". For Halley, the Earth was not only hollow but also had an inner sphere that floated under us.

After pointing out the possible arguments to his theory, Halley displays other arguments. He invokes the rings of Saturn as natural analogy and evidence that nested bodies may share a common center and hold in place by gravity.

''In the days of Halley", Griffin continues, "the question of utility was a significant issue that could not be ignored". That is, he could ask what is the use of a sphere within our hollow Earth and was considered a reasonable question, to which Halley replied: "Why do we think that this prodigious mass of matter will only serve to maintain the surface? Why not best to assume that it has been mandated by the Supreme Wisdom that it can also serve as a surface for the use of living creatures ..."

All this was previously answered Bernard Bovier de Fontenelle, in his book The plurality of worlds where he argued that there must be life on every planet; it is impossible to imagine any other use. Halley, influenced by this thought must have thought that same as above should be like below: if there are spheres outside to sustain life, the ultimate function of an inner sphere should be the same. To the problem of light to these inner worlds, "Halley professes humble ignorance noting that 'there are many ways that we do not know to produce light ..."

To Sir John Leslie is credited with saying that there are two suns, which gave them the names Pluto and Proserpina. Cotton Mather was influenced by these ideas and included them in his book, Christian Philosopher, and Leslie's ideas were the inspiration for Jules Verne's Journey to the Center of the Earth.

Jules Verne's Journey to the Center of the Earth is perhaps one of the best known and disseminated of his prolific career. Journey to the Center of the Earth was published in 1863, but more than a century before, in 1741, Ludvig Holberg, a Danish writer to which Humboldt refers, published a Latin work titled Nicolai Klimii iter subterraneum (Journey to the underworld Nicolai Klimii). Holberg's work seems to be closer in time to the ideas of Halley and Leslie.

Jules Verne's Journey to the Center of the Earth

However, the idea that Leslie suggested that in the center of the Earth there must be two suns, is widespread, but in an intensive search on what was actually proposed by him, or from a reliable source "resulted in that apparently he did not. Alexander von Humboldt, in his book Cosmos: A Sketch of the Physical Description of the Universe, Vol 1, says that Leslie "has cleverly designed the core of the earth full of imponderable matter which has an enormous force of expansion. These arbitrary and adventurous conjectures have led in circles completely unscientific, more fantastic notions. Hollow Earth has been slowly becoming alive with plants and animals, and two small subterranean spinning planets Pluto and Proserpina, were imaginatively supposed to shed light on his tenuous". That is, Leslie apparently, never proposed the Suns attributed to him. 

The Hollow Earth Was an Astronomical Necessity

Halley concluded that the Earth must be hollow by purely astronomical reasoning: his reasoning was based on the comparison of relative densities of the Earth and Moon, but based on erroneous calculations of Newton who then corrected them, but Halley continued to maintain his idea of Hollow Earth. 

On the other hand, Leslie also came to the conclusion that the Earth must be hollow, but by deductions other than those of Halley. Leslie was based on "the theory of compression of bodies" in which it was assumed that water is incompressible, but Leslie believed that the water he could be.

The common denominator in both theories, Halley's and Leslie's, is that both were based on false premises, but, curiously reached the same conclusion; an outcome even more surprising because it is believed that Leslie did not know the theory of Halley.

According to Griffin, the Suns at the center of the Earth are a must for Leslie because if the earth was hollow, space could be filled simply with air, because air is compressible because it would have ended in disaster. Instead of that substance, "the vast underground cavity should be filled with some highly diffusive, with an amazing elasticity or internal repulsion between molecules ... This leaves only one possibility: ... the only fluid known to possess this characteristic is the Light itself".

This leads us ... the most important and impressive conclusion. The large central concavity is not that dark and depressing abyss that the fanciful poets had painted for us. On the contrary, this spacious internal vault should contain the purest essence of ethereal light in its most concentrated state, glowing with a strong and pervasive effulgence splendor. 

Sir John Leslie. 

The Largest Library of All Times

Among Borges' fiction stories there is another one also related to books and infinity titled: La Biblioteca de Babel (The Library of Babel), an incredibly complex and labyrinthic library that has all the books of the universe. This story has been the subject of deep study among many writers especially by the English mathematician William Goldbloom Bloch, professor of mathematics at Wheaton College, in his book: The Unimaginable Mathematics of Borges' Library of Babel published by Oxford University Press.

In The Library of Babel Borges writes:

…each book contains four hundred ten pages, each page forty lines, each line approximately eighty black letters. There are also letters on the front cover of each book; these letters neither indicate nor prefigure what the pages inside will say.

Image credit: Rice+Lipka Architects.

Based in these numbers, Bloch computed that “each book consists of 410×40×80 = 1,312,000 ortho-graphic symbols; that is; we may consider a book as consisting of 1,312,000 slots to be filled with ortho-graphic symbols.”

Borges continues the description of the orthographic symbols used for all the books of this enormous library: “… all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet.” Hence, each book can at most use 25 different characters. Later, he adds: “In the Library, there are no two identical books.” 

So, ultimately, there are, says Bloch: about

10^1,834,097

books in the library.

William Bloch made this interesting computation for us when he asked himself: Could our universe possibly contain the Library? “… if the universe consisted of nothing but sand, it would hold at most 10⁹⁰ grains of sand”.

Borges' short story The Library of Babel is a good example of how extremely big numbers can fall beyond our comprehension, but nevertheless, they are not close to infinity. Well, no number, no matter how big it is, is ever near to reach infinity.

And then adds, “… suppose that each book is shrunk to the size of a proton, that is, shrunk to about 10^-15 meters across … then a cubic meter of the universe could hold

10¹⁵ × 10¹⁵ × 10¹⁵ = 10⁴⁵ 

books.”

How does this compare to the size of the universe? This is Bloch's answer to this question:

"It would take

10^1,834,013 

universes the size of ours to hold just the books of the Library.”

In other words,

The Library of Babel will extend far beyond millions and millions, and ... millions of parallel universes.

Image credit: Gerd Altmann via Pixabay.

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The Curious Infinite of Jorge Luis Borges

Previous to the Argentine writer Jorge Luis Borges, (1899-1986) many writers, philosophers, and mathematicians have for a long time played and juggle with the language trying to analyze and decipher the infinite.

Jorge Luis Borges, the Argentinian writer, author of the famous short story The Book of Sand.

In The Book of Sand, a translation from the Spanish original El Libro de Arena, Borges shows us an unexpected view of his understanding of the infinitude.

In this short story that bears the same as the title of the collection of short stories where it first appeared, Borges is visited by a Bible salesman that after a brief introduction offers him a special book: "I don't only sell Bibles. I can show you a holy book I came across on the outskirts of Bikaner. It may interest you."

What kind of book can a Bible salesman offer Borges—the writer—that could capture his attention? He was a professor of English literature and librarian with a huge collection of books of his property, so he was accustomed to seeing every kind of book. Thus, what was so interesting in a book that on its spine the title was the simple unexciting and dull phrase: Holy Writ - Bombay?

Borges opened the book at random just to find that the pages were worn, written in an unintelligible tightly printed text with poor typography. But this was not a surprise for him, the true surprise was in the numbering of the pages.

I noticed the one left-hand page bore the number (let us say) 40514 

and the facing right-hand page 999.

Was this a misprint? Was it a typographical error? There was only one way of finding the truth: look for the numbering in other pages. And that he did, but

"I turned the leaf; it was numbered with eight digits". 

What should we expect after the number 999? The number 1,000, of course, but an eight-digit number is greater than 9,999,999; so the "error" persisted. In fact, no book on the earth is so big, so the first conclusion that comes to our mind is that from the very beginning of the book there was no intention in numbering it correctly.

He told me his book was called the Book of Sand, 

because neither the book nor the sand has any beginning or end.

We can reproduce this unordered numbering with any book from a bookcase: choose a book, erase or overwrite all the page numbers and substitute each one with a random one, no matter how big or small.

But there was an additional attribute of The Book of Sand that we cannot do with any book from any bookstand, bookshelf, or library: you see each page only once. This happened to Borges with one of the illustrations of the book:

It was at this point that the stranger said: 

"Look at the illustration closely. You will never see it again."

Borges' metaphor of choosing the sand to convey the idea of non-repeating events is perfect. Go to the beach —any sandbox is useful for our example— take a handful of sand, drop all of the grains and keep only one in your hands; the sentence applies perfectly: look at it closely because you will never see it again.

The book was so mysterious and weird that it even lacked the first page, not because there was no number 1 in the first page, but because in some inexplicably, or magical, puzzling and perplexing way there appeared more and more pages between the book cover and the "first page".

The stranger asked me to find the first page. 

I laid my left hand on the cover and, trying to put my thumb on the flyleaf, 

I opened the book. It was useless. 

Every time I tried, a number of pages came between the cover and my thumb. 

It was as if they kept growing from the book.

Trying to find the last page of the book was equally frustrating for Borges as baffling can turn to be for us to find the first and the last grain of sand on a beach.

Now find the last page.

Again I failed. In a voice that was not mine I 

barely managed to stammer: "This can't be".

A book with no first and last page is nothing more nor less than an infinite book. That was The Book of Sand: an infinite book. Somehow, the book was infinite in pages, but not infinite in weight, nor in volume. The book was not infinitely big, it was an ordinary book, but with the particularity that its pages were constantly appearing and disappearing, new pages substituting existing ones.

The strange salesman that visited Borges was aware of his astonishment with the bizarre book he was holding in his hands: the same thing happened to him—that's the reason why he called it a "devilish book". Thus, ceremoniously he told Borges:

If space is infinite, we may be at any point in space. 

If time is infinite, we may be at any point in time.

Borges' concept of infinitude is different

In mathematics, we usually associate the infinite with the sequence of the natural numbers: 1, 2, 3 ... We say that the natural numbers are infinite because they don't ever end. To every number no matter how big it is, we can always add 1 to find a bigger number. Thus, there is no way of reaching a limit, of reaching an end, there is no last natural number. The natural numbers are the best example of the most elementary idea of infinitude. But the sequence of natural numbers is far from Borges' idea of what is the infinite: we already saw that the Book of Sand had no first page; it had no page 1.

But equally important as not having a first page is the fact that the book's page numbering followed no ordered sequence; any number can follow any other number at any moment. The page numbers were random at its purest stage.

If as he says, when he first opened the book he saw the page number 40514 followed by the page number 999, at some other time the same number 40514 may be followed by, let's say, the number 23089.

Can we say then that the book's numbering is just a scrambled number sequence? No, we cannot compare the book's numbering with a scrambled number series because in this case or ordering we always have a first number: the first number we choose for the scrambled sequence. But Borges' book had no first page, so he is not writing about unordered sequences of natural numbers: his metaphor is something beyond that.

Borges' Book of Sand confronts us with a different concept of infinitude: an infinite too far beyond our mental conception, an infinite that avoids any ruling, an infinite that escapes any possible ordering or any possible prediction. To Borges, the infinite is the kingdom where the chaos reign; the infinite is the source of every possible finiteness.

Borges chooses a simple short story to convey his idea of the infinitude because for him the infinite is not only unreachable, but any part of it is also inconceivable. The simple random numbering of a small book like the Holy Writ, that can be held in our hands is enough to take us to the vertigo of the infinitude.

Borges and Cantor: two minds where the infiniteness meet

Georg Cantor, the founder of the Set Theory in mathematics, discovered many other types of infinities. In mathematics there are many manipulations that can be done with the infinite; the infinitude of the natural numbers is just the simplest of them. One of the many breakthroughs in this field came when the mathematician George Cantor (1845-1918) introduced a more complex notion of infinitude with what he called the transfinite numbers which are an infinite class of infinities. For Cantor, there is a ladder of infinitudes, where the infinitude of the natural numbers is just the simplest of all the infinities. For him, this ladder of ever-growing infinity has no end. To this obscure but interesting field of mathematics belongs the abstract field called transfinite arithmetic.

Borges had a literary mindset with deep interests in mathematics; that's the reason why he exposes so excellently—and in a very short story—how the infinite is beyond our comprehension. On the other hand, Cantor was the pure mathematician that worked on the abstract concepts of the theory and the cardinality of sets. From there he discovered that the infinites are infinites in themselves.

The first of the "infinitude of infinites" discovered by Cantor is the one called the Aleph. From there he also introduced the so-called the Continuum. The connection I try to establish between Borges and Cantor is that in The Book of Sand we can begin to understand what is the Continuum without recurring to deep mathematics.

Imagine that you open that "devilish book" and by some unknown power you can write the sequence of the page numbering as it appears page by page. Now close the book and reopen it again and repeat the process again and again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but it is possible to prove, that your "list" of all possible orderings of the natural numbers is not countable, not even infinitely countable.

Thus opening an closing The Book of Sand is an act of delving into the Mathematical continuum.

The Sorites Paradox

The Sorites Paradox also called a Sorites problem, is an argument that arises when a situation is presented in vague terms, or poorly defined circumstances. The name “sorites” derives from the Greek word “soros” meaning “pile” or “heap”.

Curiously, this paradox arose when trying to determine when an amount of grains of sand constitute a “heap of sand” or a “pile of sand” or when there are not enough grains to call it a heap. This paradox is attributed to the Greek philosopher Eubulides of Miletus.

The paradox goes like this: if a heap of sand is reduced by taking single grains of sand one after another, at what exact point does the pile of sand ceases to be considered “a heap”? Removing a single grain of sand at a time does not turn a heap into a non-heap, so the paradox is to find at what point repeating the process of taking away grains one by one turns the heap into a non-heap.

We can also study this paradox by going in an opposite direction; let's start with nothing of sand. Add a single grain of sand: this is obviously not a heap of sand. Adding one more grain is not enough to build a heap of sand; so, what about three grains of sand? How many grains do we need to make a heap?

Obviously, at some amount of grains, we will exclaim: “Now I have a heap of sand!” But if you go backward and extract just one grain of sand out of the heap you just created, can you still make the same exclamation? This is the paradox.

How Do You Imagine the Multiverse?

Before my formal college background in mathematics, my vision of the infinite was a simple never-ending series of natural numbers without nothing in the way that could stop it.

Later, after getting acquainted with the theorems of Georg Cantor and his revolutionary vision of the transfinite numbers as a stair of many other super-infinities, my mind was furthermore expanded.

As time went on, a little short story titled The Book of Sand came into my hands. I never suspected that there was another twister waiting to squeeze my mind again. The simplicity of this short story by Jorge Luis Borges viewing the infinite as an incomprehensible chaos eternally randomizing and reconfiguring itself changed my mind forever. Borges’ infinity is the opposite of Cantor’s infinity; however, both are shapes of infinity.

Fortunately, the realm of infinity is not a private property of mathematics, nor of literature, nor of any field of knowledge. So, the science of physics—in its modality of cosmology—also raised its voice and said: here I am, I also have something to say about infinity! There are infinite parallel universes too! This universe is part of a multiverse!

But how do you imagine the multiverse? Do you imagine it as in Cantor-infinite style, or do you imagine it as in the Borges’-infinite-style?

The Fourth Dimension According to Charles Hinton

Surely at some time of your life, you have heard about traveling to the fourth dimension. The are many books—famous books—and many movies—famous movies—around this topic. This means that the topic is of interest to everybody, and not only for a selected group of mathematicians and physicists.

When people are confronted with the subject of higher dimensions, or when asked if they believe in it or not, their answer will vary depending on if their background is more religious than scientific or vice versa.

The fact is that for many people "the fourth dimension" is a place—similar and different to our surrounding three-dimensional world around us. Visualizing it as "place" enable us to enter or exit from it like when going to some kind of theme park. Going to such a place will enlighten and will empower us in such a way that from that experience on to forever we'll be talking about the experience like sacred events that should not be shared with anybody except those "chosen" to experience a trip like ours in the future.

For the privileged beings of this hyperworld, we are seen somewhat as toys that can be manipulated at their will.

In the literature arena of the books, Charles Hinton (1853 - 1907), a British mathematician, occupies the place of one of the initiators among the public the fourth dimension subject. He studied the topic in a reasonable systematic approach, recurring to simple examples and explaining his ideas in nonmathematical terms.

Selected Papers of Charles Hinton
about the fourth dimension

The Selected Papers of Charles Hinton about the fourth dimension is a good compilation of many of his works in a simple nonmathematical language.

Hermann Schubert Probes the Fourth Dimension

Mathematical Essays and Recreations

Hermann Schubert was a German teacher and textbook author. Mathematical Essays and Recreations is a small collection of articles ranging from the foundations of the number system, the foundations of algebra up to an extensive essay about the fourth dimension.

In the first article: Notion and definition of number, Shubert gives a brief account of how the concept of number evolved within the human mind.

In another of his essays: Monism in arithmetic, the author explores and writes about the elementary rules of algebra and about the importance that mathematical systems can operate with defined rules for any kind of number without making exceptions for some of them. For example, we can define the commutative rule of addition for the natural numbers, as in 2 + 3 = 3 + 2, and the same number should apply if instead, we use natural an imaginary numbers combined, like 2 + 3i = 3i + 2. Following his exposition, we can easily see why the class of complex numbers is such a robust field in mathematics, and how it can be derived from the natural numbers following easy and consistent steps. On the other hand, we can also see why the quaternions—when compared to the complex numbers—lack such popularity and why this class of number is so limited in applications and acceptance.

The third essay in this Datum edition is about the fourth dimension. But contrary to other science authors that rarely touch the fourth dimension from the metaphysical point of view, Schubert is not afraid to confront both doctrines. When dealing with the fourth dimension from the mathematical point of view, Schubert goes carefully from the definition of the point to the definition of dimension to a many-dimensioned space.