To
new evolution theory of Earth and Solar System
(Ice
Ago-Atlantis-Moon-Nibiru-:)
Emilio Spedicato
Department of Mathematics,
We propose a new
theory for the origin of the Moon. We assume that a large body P approached
Earth at the final time of last Ice Age. The body deformed our planet breaking
the oceanic crust. Large amounts of magma escaped heating the atmosphere,
melting much of the ice cover and killing many living beings. The Atlantis
civilization was terminated by this catastrophic event. Due to the geometry of
the close passage, P lost a satellite, which became our Moon. Moon's orbit was
probably initially closer to Earth, so that Moon looked bigger and brighter
than now. Earth had another satellite before getting the Moon, needed for
stability reasons and to explain varies. We suggest that Mars was the previous
satellite. It was removed by body P when Moon was captured. Mars for millennia
had an elliptic orbit. It got the present almost circular orbit only about 700
BC, as Velikovsky and De Grazia have claimed.
Finally we discuss the likely fate of P in terms of very recent
discoveries [1-32].
1. Astronomic theories on Moon's origin
Origin of Moon is
still an open problem in astronomy. Many theories have been proposed, all
having problems. Before lunar rocks were collected and analyzed, it was
generally believed that Earth and Moon formed coevally from the condensation of
gas and dust originated by an ancient supernova explosion. Such material slowly
coalesced, leading inter alia to the formation of so called rocky planets
between Sun and the gas planet Jupiter, while other giant gas planets formed
beyond; see the so called standard model of Whetherill [32]. After the
discovery of the Kuiper belt beyond the gas planets and the proposal of a still
farther Oort cloud, reaching half the way to the nearest star, it has appeared
that at least such remote structures, numbering trillions of objects, consist
partly of material captured by the Sun while it circles around the galaxy
centre of mass (in about two hundred million years). The Sun crosses the
galactic plane about every 30 million years, losing and acquiring material from
the so called molecular clouds common on the galactic plane.
The scenario of Whetherill
cannot be accepted as the standard one after the discovery in the last several
years of many (more than 200) planetary systems around other stars. Here some
large gas planets orbit very close to their star, at about 1 million km, while
in Whetherill scenario large gas planet should be about half a billion km away
from the star. Even considering that the observed systems are biased towards
those with large planets, only few small planets having been found, it is clear
that Whetherill model uses simplifications and hypotheses not true generally.
The mechanism explaining why large planets can revolve so close to a star is
due to Del Popolo [10, 11]. Moreover, additional computations by Meyer et al [14]
have provided a result almost unbelievable, namely that large gas planets do
not form in the previously estimated time of order 10 million years, but just
in a few centuries! And often the required coalescing of the gas and dust does
not occur! Moreover, quite strong arguments by Ackerman [1, 2],
including the analysis of the Shoemaker-Levy comet impact on Jupiter, suggest
that the so called gas planets have a solid inner part, consisting of hydrates
and clathrates of methane and other materials.
The problem of
satellite formation is similar but possibly richer of solutions. Indeed,
satellites could form coevally with the planet, but they could also arise from
impacts or capture of bodies originally on a different orbit or gravitationally
tied to another body. The formation of Moon coevally with Earth has now been
dropped since the isotopic composition of lunar rocks is different from that of
terrestrial rocks. A mechanism that would separate particles by their isotopic
ratio is not known. Hence one has looked at other possible origins, in
particular to the three body capture and to a large impact whose debris
coalesces around Earth. The last mechanism is preferred by most astronomers;
see for instance Boss [5] or Palme [18]. It assumes that a body of Mars-like
mass impacted Earth tangentially billions of years ago. The body essentially
vaporized together with part of Earth. Some of the vaporized material fell over
Earth, some escaped to space, and some condensed around Earth forming the Moon.
Notice that multiple Moon systems might arise under such conditions, see Canup
et al. [6], but after some time they would usually coalesce into just one. The
impact process has been simulated mathematically with different parameters and
suitable composition in order to explain the mineral composition of the Moon.
What can obviously be obtained by proper selections!
Another theory that
would explain a Moon differing from Earth in material composition is capture.
The simplest form of capture assumes that the Moon, originating somewhere else
in solar system (e.g. in Kuiper belt or in Oort cloud, objects of much analysis
by Clube and Napier [7]) or outside it, approaches Earth and, after entering
the so called Hill sphere, where Earth gravity dominates Sun's, is captured.
Capture is theoretically impossible from the laws of the three body problem, as
I learned from celestial mechanics specialist Tom Van Flandern, now sadly
deceased, who attended Bergamo's two conferences organized by this author (the
1999 Conference on New Scenarios on the Evolution of Solar System and
the 2001 Conference on Fifty Years since of Velikovsky's Worlds in Collision).
Capture would be possible in a three body problem if other forces like a
braking atmosphere are acting, see Nakazawa et al [17]. The Japanese authors
obtained also the remarkable result that Moon's orbit would circularize in less
than a century! Another unexpected result of fast evolution of celestial
structures.
Capture is possible,
under fine tuning, as Van Flandern again told this author in an email of
5-11-07, not long before he died of cancer, in a 4-body scenario, like the one
that we propose below, apparently never considered before. It should be
possible also in a 5-body scenario that we will additionally discuss, since it
seems necessary to justify certain properties that a satellite gives to Earth.
A capture is also discussed by Ticleanu et al [31], who date the event
differently from us, but still at a quite recent time, about 40,000 years ago,
when the precession rate seems to have changed remarkably. Our proposal uses to
some extent human memory that may have survived some 10,000 years, but less
likely in our opinion 40,000.
Our scenario indeed
sets Moon's capture within human memory. We see the event in the framework of
the great catastrophe that from our approach ended rapidly the last Ice Age,
leading to the demise of an advanced civilization, the one referred by Plato as
the Atlantis civilization, see Spedicato [23, 24, 25], to
the destruction of many animal species and a large part of mankind. We recall
that De Grazia also proposed a relatively recent origin of the Moon. According
to him Moon was expelled from the part of Earth where the Pacific Ocean is now,
considered as a scar left by lunar departure. He additionally uses this event
to explain several geologic features of Earth. While the last word is not said
about his theory, it may have unresolvable difficulties, say:
-
which mechanism can produce the event, expelling
part of Earth with enough kinetic energy to settle into orbit;
-
how could life survive such an immense catastrophic
event;
-
why there is no significant difference in the
geology of the bottom of Pacific and other oceans, an observation made by
geologist Walter Alvarez of Berkeley University, one of the discoverers of the
K-T event.
In the following
we give our scenario for the capture of the Moon within a 4-body gravitational
system, and then we extend it to a 5-body scenario. Our idea was born from
religious and mythological traditions. Then we discuss a possible validation of
such a scenario in the framework of a nonlinear optimization problem.
2. Did Moon
originate in a 4-body capture event?
Velikovsky in his unpublished book, named In the beginning and
available on the website where his unpublished papers have been collected by
Jan Sammer, briefly considered the possibility that the Moon was recently
captured, in view of certain statements in the Bible and other sources. Here we
quote some of his references (V) and others found by (S) us in books dealing
with ancient traditions.
-
(S) The Chimu were a
people of the coastal civilization in Peru, predating the Incas civilization,
which took over only around the 14th century AD. For the Chimu the
Moon appeared at a certain time in the sky and had a father, the god
Pachacamac. This god was also god of the Incas, who were monotheist. The name
Pachacamac means He who creates, animates (camac) all, the universe (Pacha),
see Miccinelli et al [15].
-
(S) The Malekula are a
tribe of Melanesia, Vanuatu state, once cannibals. They claim that there was time when air was full of vapors
and it was almost impossible to see even at short distance. When the sky
cleared they saw that the sea level had arisen and many lands had disappeared.
Moreover, in the sky the Moon had appeared. See Bonatti [4].
-
(S) The Hindu claim
that the Moon appeared after the sea had boiled, see Daniélou [9].
-
(V) In the Bible, Job
addresses God as Lord of when there was the Moon and when there was not; see Psalm
72-5 and Job 25-5. Notice
that the book of Job has generally been considered as the oldest one in the
Bible, predating the five books written by Moses by a long time. In a future
paper we will argue that Job witnessed the Typhoon explosion over northern
Sinai, to be dated back to about 1940 BC. The Typhoon event possibly led to the
demise of the Ancient Kingdom of Egypt, the Assyrian empire founded by Sargon
the Great and the Indus-Sarasvati civilization.
-
(S) Censorinus in De die natali writes that Arcadians claim, but I do not believe it,
that before Moon existed the year had not 12 but 3 months.
-
(S) Apollonius Rodius claims in his Argonautica that the Arcadians are older
than Moon; a claim that considered
together with Censorinus's apparently indicates that their memory extended to
times before the Moon existed. We know little of the Arcadians traditions, so
we wonder if their memory extended to the first of the four Mayan catastrophes,
taking place about one thousand years before the end of Ice Age and our
proposed capture of the Moon. An event that probably had important consequences
in Europe that cannot be discussed here.
-
(V) Velikovsky in the
above quoted book cites several Greek and Roman writers for whom Moon appeared
within human memory, among them Democritus, Anaxagoras, Aristotle, Apollonius,
Plutarch, Ovidius, Hippolitus, and Lucian. He also observes that another claim
was that the Moon was more luminous in the past. I have been unable to check
the exact content of the statements by the above authors.
-
(S) An important
symbol in Islamic world, whose origin is not Arabic but Turkish (the Turkish
area has some of the oldest traditions in the world, only partly explored, as
the Epic of Manas of the Kirgisians
consisting of several million verses) is the well known large half Moon
having inside a small five pointed star. This symbol appears on the flag of
Turkey, and other Islamic states. A natural explanation will be given within
our scenario.
Human memory may go back several thousand years, possibly even over nine
thousand years as the memory of the great catastrophe that destroyed Atlantis
shows, if it is correct that such an event corresponds to the end of the last
Ice Age, as argued by Muck [16], Barbiero [3], Collins [8], Spedicato [23-26]
and others. In fact, even older memories may have survived. We can indeed
associate the most ancient catastrophe in the Mayan record, quoted above,
characterized by fire followed by a period of strong winds, with a great impact
over the Laurentide region that has been recently identified. It is dated to
around 10,900 BC, at the end of the Clovis period. That impact led to
widespread fires that left a layer of some 3 million square km of partially
burned material. It is called the black mat (I saw it in several
quarries near Beryl, in the Escalante desert, Utah, when I was guest of the remarkable
scholar Evan Hansen). The centuries following the event are believed to have
been a time affected by unusually strong storms, even in Europe, where whole
forests were often flattened. Such strong winds may also explain a strange
detail in the Atlantis story of Plato, namely that the city of Atlantis was
located in a place protected from winds, a statement never explained, as
far as I know. Even older memories, circa 19,000 and 27,000 BC, may be
associated with the first two Indian yugas, if their real chronology, like the
one given by other ancient traditions, is obtained by the decrypting technique
given by Spedicato (2012). The precise recollection by a Klamath tribe of a
strange eruption of volcano Hazama in California, now confirmed to occur about
5600 BC, i.e. almost 8000 years ago, has been given by Sivertsen [22] as an
example of memory preserved over several millennia.
In a monograph in Italian language, Spedicato [26], we have developed the
idea presented at the Milos conference of 2005 on Atlantis, that the most
likely cause of the end of the last Ice Age, the rapid melting of ices and the
increase of world temperatureš observed
by several authors, see e.g. Pedersen (Atlantis Milos conference communication)
andš for an Ice Age ending in three or
less years, e. g. Prof. Richard Alley of Penn State University, was the close
passage of a large body P near Earth. The tidal action of such passage slightly
deformed our planet. It broke the thin crust of the oceans bottom (
The appearance of the new satellite in the sky, the Moon, after these
events can be explained if the passing body had a satellite that was lost to
Earth, a fact possible in a 4-body context. Then there was a time with the
Moon and a time without, as Job and the Psalm declare. And the Moon
had a father, as the Chimu stated, to be identified with the large body P
passing near Earth. Incidentally the fate of this body is a very interesting
question. Here we just recall two possibilities. One is that it continued on
its previous probably very elliptic orbit, if not a hyperbolic one; we could
think of it as the mysterious Sumerian planet called Nibiru of usually given
period of 3600 years, but see later, for which however there is now no
evidence. The other possibility is that it should be identified with the
object, according to John Ackerman [1, 2], which impacted on Jupiter. The impact
formed an immense crater, the trace of which is the present Red Spot; from such
a crater immense amount of material was ejected, leading inter alia to the
formation, i.e. to the birth, of Venus-Athena. Ackerman here proposes a
mechanism for Venus-Athena recent birth not provided by Velikovsky. As observed
by Spedicato [28] the time of the event can be estimated quite precisely from a
Toltec statement that Venus was born circa 3800 years before the beginning of
their long year, estimated as 3114 BC (notice that precise dating of
Mayan-Toltec years is impossible since the numbers are given in days and
the number of days in a year cannot be considered constant for several
reasons). Thus we would have for Venus birth about 6900 BC (a date not too far
from 6500 BC, the fifth "day" of "creation" according to Talmudic scholars).
This date is essentially the one found by the Tollmans [31], for Earth being
impacted sevenfold in different points, mainly on oceans, impacts easily
explainable by material ejected by Jupiter in Ackerman's model. And we can
propose why Venus was born from the foam of the sea, after the sea was
fertilized by the phallus of possibly Jupiter. Indeed one of the seven bodies
of the Tollmans might have been seen falling on ocean in a phallus form, say an
elongated form; recall that most asteroids have a potato-like form, not a
spherical form... The impact on the ocean led to magma come out followed by
vapors, foam. When vapors dissipated, a new body was seen in the sky,
Venus-Athena. Thus both for Moon and Venus we have a birth associated with
foams in the seas.
The capture of the Moon may be considered as a relatively minor event
compared with the gravitational effects of the passing body, which caused the
greatest catastrophe in Plato's list. However we are faced with the existence
of geological structures called varies
that are due to tidal effects of our satellite and exist since well before the
end of the last Ice Age. This problem can be solved if the capture of the Moon
meant also the loss of a previous satellite, responsible for the previous
varies. There are arguments that such a satellite evolved into Mars. Such a
loss would moreover lessen the effects of Moon's capture, especially if Moon
would set in a resonant orbit that favors stabilization of Earth axis, as is
indeed the case. Suggestions that the previous satellite was probably Mars are
found in the following items:
-
The quoted statement
in Censorinus can be interpreted as the existence, previous to the Moon, of
another satellite having about 3 cycles per year, hence providing 3 months per
year. By Kepler's third law we can estimate that such a satellite orbited
farther away than Moon, about 1 million km from us. Finding such passage of
Censorinus, which as far as I know escaped even the attention of Velikovsky,
was quite a surprise for me; I had believed impossible to get an estimate of
the distance of the previous satellite!
-
The Turkish symbol can
be interpreted as the new Moon looming larger in the sky and with well defined
phases. The star inside, having 5 points, suggests a reference to Mars, the 5th
body from the Sun, or even from Earth. Being smaller, it suggests that it
appeared in the sky smaller than the Moon. Thus the 5-pointed star used often
in human traditions may refer originally to Mars, associated with war and
violence notice that David and especially Solomon, man of peace and tolerance,
adopted a 6-pointed star.
Several features of Mars suggest a previous close relation with Earth, as
the similar rotation period and similar axis angle on the ecliptic. Moreover,
the evidence from Martian missions of water recently lost catastrophically by
Mars, some probably still surviving in the Martian soil, is quite compatible
with the removal that we prospect here. Notice that Mars would have been in the
habitable zone, hence life probably existed there. Possibly, not only a low
level life. Hence the many geometric features in Mars that suggest
artificiality, see several papers in Metares Bulletin edited by Tom Van
Flandern, or the recent book by Piccaluga [20], get one extra argument in favor
of their artificiality.
Now Mars has a diameter about twice that of Moon. Its distance can be
estimated from Censorinus and Kepler's third law to have been about 2.5 times
the present Moon distance. Thus, we estimate a reduction of its light by a
factor of 6 by distance, but an increase by a factor of 4 by surface size.
Hence, assuming a similar albedo, its luminosity had to be less than present
luminosity of the Moon; angular diameter was also smaller. But we have to
consider also statements in Greek and Latin authors that Moon was initially
more brilliant than now. This quite mysterious statement took suddenly a
meaning for me at the September 2009 Italian Archaeoastronomy Conference in
Florence. There from scholar Giuseppe Brunod I knew that about one hundred
petroglyphs found in Val Camonica (over 100.000 have been discovered) show by
signs associated to months that the year in the fourth millennium BC consisted
of 13 months. If this finding is correct, then it means:
-
Moon was closer, hence
looming larger and more luminous, from Kepler's third law;
-
its superiority over
the previous satellite was even more remarkable;
-
a strong reason
follows why number 13 was so important among many people, especially among
Toltec, Mayas and Aztec, appearing in their various calendars, architectonic
features and even rituals as the killing of the king by the Aztec if he had
been in power for 13 years. Also recall that the sacred Maya year, the tzolkin,
consisted of 260 days, i.e. 13 by 20. What the reason for 20, another
"sacred number", we leave to our forthcoming book Dictionary of ritual numbers.
The year got 12 months most likely after the second great catastrophe
quoted by Plato. Albeit almost no one analyzing Timaeus and Critias discusses
the second catastrophe, this is almost certainly the Biblical Flood,
corresponding to the Sumerian-Akkadian Flood of Ziusudra-Utnapishtim. This
event, for reasons not to be presented here, took place almost certainly in
3161 BC (again I thank Toltec for allowing me to solve the enigma: 3161 or
3171?, depending on whether the Hebrew calendar started from Noah's birth or
from his anointment as a priest of Melchisedec:). The cause of the Flood was
most probably a close passage of Mars. This planet, after its removal as Earth
main satellite, in the following millennia repeatedly approached Earth with
usually moderate catastrophic effects; see Patten and Windsor [19]. The 3161 BC
close passage was the most catastrophic, for reasons foreseen by Noah and
others. We believe that during the 3161 BC passage Mars had the final loss of
its waters, part of which reached Earth, being the biblical fountains of the
high. We also tend to accept Ackerman genial and bold proposal that sees in
this passage also the loss of Mars core, which became Hermes-Mercury. Mars
interacted with the Moon and Earth, with consequences described in the
well-known but enigmatic statement of Plutarch: Hermes stole one seventieth
of the light of the Moon passing it to the year whose number of days increased
by 5:.
The year passed from 360 days to about 365, the extra five days being
about one seventieth of 360 and being generally considered in ancient
civilizations as special days, devoted to festivals (by Mayas:) or with
prohibition for traveling, marrying, making contracts (by Egyptians):
For analysis of the final interactions of Mars and Earth, see the recent
monograph of De Grazia [12], written using material partly obtained while exploring
the Bolsena lake area. For a mathematical study that the final rounding up of
the orbits of Mars and Venus does not violate conservation laws, see Dixon
[13].
We conclude our scenario for the Moon observing that it was a bonus for
mankind after the terrible events that ended Ice Age. Mankind recovered only
after some 4000 years, at the time of the "creation" in Eden, in about 5500 BC,
an event soon followed by the explosion in human activity that defines the
Neolithic age.
Moon was seen as a gift sent by gods to provide light at night, at a
level not available when the satellite was Mars. Its name is clearly related to
light in the sky, see e.g.:
-
in Chinese, hue
lean, "lamp in the sky";
-
in Latin, and many
post Latin languages, luna, by metathesis and acceptable vowel
changes lu→ul→el, "light", na→an, "sky", light of the
sky.
There are, to my knowledge, no instances of the Moon having a bad
qualification in ancient records.
3. Mathematical modeling
of Moon's capture
Our scenario can be validated to some extent by mathematical simulation.
We can set it as a multiparametric problem, to be solved via nonlinear
optimization. Of course, we should make simplifications, and reduce the
dimensionality of the problem, being satisfied to have a first order solution
to the problem. We define here the problem features, letting to the future the
actual calculations that require many man-months of work.
As simplifying hypotheses, for the problem not considering Mars, we may
assume that:
-
the orbits of the
considered bodies lie on the ecliptic plane;
-
the Earth orbit is
circular, as well as the orbit of the satellite S of the object P (whose
passage near Earth ends the Ice Age and gives us the Moon);
-
all bodies mass are
concentrated in one point, thereby disregarding tidal deformations and their
specific rotation;
-
the only considered
gravity is by P, S, Earth and Sun;
-
only gravitational
forces are considered.
The following parameters cannot be provided by the above study. They must
be inserted at a number of discrete values, hereby solving the problem for
several cases:
-
the mass μ of the
body P, but see next section for a probable mass of about 10 Earth masses
-
the distance ∆
of closest passage of P to Earth
-
the radius ∂ of
the orbit of S, which should be larger than ∆ plus the diameter of Earth,
for a capture taking place of the far side of Earth with respect to P
-
the pointš ω wherefrom we start the approach of P
to Earth; it should be set at a value much greater than the radius of Earth
orbit
-
the velocity vector of
P in ω.
We can formulate our problem as a multistage heuristic parametric
nonlinear unconstrained optimization problem in two variables. Let υ1,
υ2 be the velocity components of P in ω, they being
our variables in the optimization. We want to select them so that P passes as
close as possible to the selected distance ∆ of a certain point on Earth
orbit. We can do this by minimizing overš
υ1,š υ2 šthe square of the norm of the difference
between ∆ and the position ρ(υ1,š υ2, τ) at time
τ of P computed by solving the Newtonian equations of motion. Accurate
solution of such equations is important and to this purpose the new very
accurate Trigiante-Brugnano algorithms may be used.
At this moment we have just solved a 2-body problem, the orbit of P being
expected to be elliptic. Once values of υ1, υ2
are obtained by the minimization process we repeat the computations adding S in
the computations with a ∂ greater than ∆ plus Earth diameter. We
try different initial positions of S in its orbit till, at the time of the
closest distance to Earth orbit, S will lie inside Earth orbit. At this moment
we have solved a 3-body problem. It would be useful to follow the trajectories
by a graphic representation of the system.
Then we add Earth to the scenario, trying for it different positions
along to initially circular orbit (it will of course undergo a perturbation),
till it gets in a position at a distance of about ∆ from P, with S
expected to be inside the orbit, hence closer to Earth than to P. Here we have
solved a 4-body problem.
Finally the distances of S from P and Earth will be monitored by further
application of the dynamical equations, to verify if S remains in the vicinity
of Earth.
As formulated above, the problem is multiparametric in only two
dimensions as an optimization problem. It might be possible to include further
parameters in the optimization procedure. It is also possible to consider other
formulations.
Once the above model provides numerical evidence that capture is possible
under some parameter range, one could study the behavior of Mars assumed as
being initially our satellite, in order to derive the parameters of its orbit
after it is lost by Earth under the pull of P. Notice that Earth might have
ended up with two satellites!
4. On the mass of P and
its final destiny
The mythological information provides no clue about the mass of P, which
could have been smaller or greater than the mass of Earth. However, a recent
unexpected discovery suggests that P might have a mass about 10 times that of
Earth. The discovery comes from a probe that recently passed near Jupiter and
Saturn, measuring the mass of their cores, see Scranton [21]. It turned out, to
the great surprise, that Jupiter mass is smaller, about 70% of Saturn. This can
be explained assuming that Jupiter core lost mass after an impact with a large
object. Studies by David Shiba and Shulin Li of Beijing University indicated a
likely mass of the impacting object as ten times Earth mass. The impact must
have spewed material in the space, some being trapped around Jupiter, some
traveling long distances in part of the solar space, some possibly coalescing
as planet Venus as Velikovsky and Ackerman have proposed. It is also likely
that Jupiter moved to the present orbit from another one, possibly a closer
one, a question to be studied, and that could result in a new explanation of
the asteroid belt. The proposed impact is possibly seen in the Greek myth of Goddess Metis [the terrible] being pregnant
of Athena and entering the body of Jupiter, from whose head Athena was
expelled. The time of the event, if it as really happened should be about
6900 BC, from a statement in a Toltec code related by Ixtichihuatl, see
Spedicato [28], and from geological evidence collected by Alexander and Edith
Tollman [31].š The impacting object could
have been the one that lost a satellite to Earth. If it was Nibiru of Sumerian
mythology, with a claimed orbital period of 3600 years, the real period might
actually have been only 20 years, implying that it most likely crossed Jupiter
orbit. The number 20 follows from using the decryption technique discussed in
Spedicato [27], i.e. a division by 180, which applies to many Asian
chronologies.
Acknowledgements. The idea that Earth could have acquired Moon
recently came from Velikovsky's book In the beginning and by the quoted
statements of the Chimu, Malekula, Hindu and Censorinus. The idea that Mars was
a previous satellite belongs to Marco Fagone, author of a website on Mars.
Stimulating discussions and important documents have been provided by Alfred De
Grazia and his wife and collaborator Amy de Grazia. The book by Laird Scranton,
read on a train to Rome to deliver a communication about the origin of Moon,
provided the unexpected support to our general model in addition to the
possible mass of P, a problem considered before as unsolvable. Thanks are also
due to Laird Scranton for the invaluable information found in his book, and to
Leroy Ellenberger for pointing my attention to that book.
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