EARTH
TUNNELS
I
AM reasonably sure that most of my readers have never come across the name of
Maupertuis. If you want to remedy this in a hurry, just reach for an
encyclopediayou'll find him listed as "Maupertuis, Pierre Louis Moreau
de, (16982 1759), French mathematician and astronomer."
You'll find some remarks about his
life, too: that his king (Louis XV) sent him to Lappland to measure the length
of a degree of the meridian, and that he quarreled with Voltaireas who
didn't?and was an able mathematician. Also it is stated that he was the author
of a number of works such as Sur la figure de la terre, Lettre sur la comete
de 1742, and Astronomie nautique.
All of which may tend to make you
feel that Monsieur de Maupertuis may have been an important scientist in his
time but not worth much type nowadays, beyond an entry in the encyclopedia.
BUT
that is merely due to the fact that reference books often omit the really
interesting things about people. Perhaps Maupertuis was excessively vain (at
least all his contemporaries said that he was), and perhaps he was just a shade
more quarrelsome than the other learned men of his timebut he was also full of
amusing ideas and never hesitated to talk about them.
When, for example, his compatriot
and contemporary George Louis Leclerc, Comte de Buffon, speculated about Terra
australis incognita, the "Great Unknown Southland" which was then
supposed to exist somewhere in the South Seas, Maupertuis jumped in without
hesitation.
Buffon, in his speculations, had
drawn a parallel to South America. Coming from the Atlantic Ocean, he said, one
first encounters the South American lowlands, inhabited by primitive tribesbut
farther inland, where the mountains tower toward the sky, there was an area
where there had been a high culture with magnificent architecture. Similarly,
in the South Seas, we had found only the low islands, inhabited by primitive
peoples. So when we finally managed to penetrate to the hypothetical
Southlandwhich was assumed to have tall mountains toowe'd probably find
another strange and so far completely unknown culture.
Maupertuis agreed with Buffon so
far as the idea of an unknown southern continent was concerned. He also
subscribed to the assumption of high mountains in the unknown Southland for the
simple (and to us surprising) reason that it was not at that time known how an
iceberg is formed: it was thought that icebergs formed in rivers onlyhence
there had to be enormous rivers to produce the icebergs which had been seen.
And the large rivers, in turn, demanded high mountain ranges to supply them.
But there Buffon and Maupertuis
parted company. Maupertuis did not expect to find a high culture in the
Southlandor any culture at all! The culture of the Andes was high, he
admitted, but from there on things went downhill. The islands in the Pacific
had a lower and lower culture the farther one went, so obviously none was left
for Southland itselfits inhabitants probably still had tails! All this was
proclaimed, without any unnecessary hesitation, at Castle Sanssouci near Potsdam, at the dinner table of Frederic the Great.
Many years later, French and
German explorers might add deadpan to their reports that the natives were
tailless . . . to the puzzlement of readers who did not know just what this was
about.
MY
MAIN reason for telling about Maupertuis, though, is that he is the father of
an idea which has been repeatedly used in science fiction. He
"invented" the tube through the center of the earth.
Just what would happen if one had
a truly bottomless well?
Maupertuis was, as has been said,
an excellent mathematician. He was also one of the men of science who accepted
Sir Isaac Newton's teachings wholeheartedly. The problem of the bottomless well
became an exercise in mathematics.
Obviously, if you fell into such a
well, you'd fall faster and faster until you reached the center of the Earth.
But you would not stop there, of course, for by that time you would have
acquired a considerable speed, and would continue to "fall upward"
through the remaining half of the well. Gravitation would slow you down to zero
at the precise point of reaching the surface at the other end of the bottomless
welland your assistant at the other end could yank you sidewise onto solid
ground, before you had time to start falling again. The tripand the trip back,
if you had no assistantwould take 84 minutes and 22 seconds. The speed would
be highest at the Earth's center, where it would be (expressed in our
measurements) five miles per second.
Of course, the bottomless well
would have to be evacuated, so that the traveler would not be slowed down by
air resistance. And the whole thing would work even better only if one first
stopped the rotation of the Earth, for lacking that, one would have to dig the
well from pole to pole.
But in the pole-to-pole connection
itself there is some hidden trouble: the area of the South Pole is about one
mile above sea level, while the area of the North Pole is virtually at sea
level. So if you try it from north to south, you don't quite reach the other
end, which is one mile above your farthest point. And if you go from south to
north, you reach north polar sea level with enough residual velocity to rise
one mile into the thin air!
So you might spend some time in
such a tunnel...
Just about half a century ago, a
Russian writer by the name of A. A. Rodnych (who later acquired fame as a
historian of aviation) demonstrated, that Maupertuis, and after him
Camille Flammarion, had failed to extract all the humor from the idea.
There was still a trick left and Rodnych described it under the title: Subterranean
self-propelled railroad between St. Petersburg and Moscow; Fantastic Novel, for
the Present in only three and moreover incomplete chapters.
The point was very simple: there
are no straight tunnels. If there were, you would not need fuel.
Suppose a perfectly straight
tunnel were built from Kansas City to San Franciscostraight enough to look
through, even though you might need a telescope to make pout the light from the
other end. This tunnel would be straight, but not horizontal: its center
would be closer to the center of the Earth than its two ends. Hence, from
whichever end you enter, you go down. A railroad car on a track through the
tunnel, with its brakes released, would begin to roll. Just as in the
bottomless well, the speed would increase and increase until the center of the
tunnel was reached, at which point the direction would be "up" and
the speed would begin to decrease. And if there were no friction in the
bearings and on the rails, and no air resistance (and also if the wheels could
stand the speed near the center without being torn apart by centrifugal force),
the railroad car would reach the other end of the tunnel with zero speed and
without having used a drop of fuel. Provided, of course, that both tunnel
mouths are the same distance from the center of the Earth, i.e. same elevation
above sea level.
The timetable for such a railroad
would not pose any problems : each and every train would arrive at its
destination precisely 42 minutes and 11 seconds after its brakes had been
released at the other end . . . whether the tunnel was 1000 miles or a mere 100
miles long.
SINCE
we are talking about gravity, let's consider the case of the young man whose
chair simply moved across the polished hardwood floor until it was next to that
of the beautiful brunette and who explained it by "attraction" and
blamed it all on Sir Isaac Newton.
Since I happened to be around, I
was called upon to testify that Newton actually had said that each particle of
matter in the Universe attracts every other particle. True, he did say that . .
. but if I had been the young lady, I would have protested bitterly at the
implication that I weighed several times as much as Mt. Everest.
Seriously : some people do wonder
why the law of universal gravitation is not noticeable in daily life. Science
says that all bodies attract each other, but most of the time it certainly
doesn't look that way. On the other hand, when two ships collide in a fog there
are always some people who believe that Newton's law was responsible. To get
the matter straight, we obviously need some figures:
If we have two pieces of matter,
each weighing one gram, and they are one centimeter apart, what is the
attraction between them?
Answer: about 1/15,000,000
milligram.
One milligram, of course, is the
thousandth part of a gram, and there are, 28 grams in an ounce. If one of the
two pieces of matter weighed 5 grams and the other 8 grams and they were still
one centimeter aparttheir mutual attraction would be 5 x 8 or 40 times as
large as that of the two one gram pieces. But if they were three centimeters
apart, you'd have to divide the attraction over the standard distance by 9 to
get the proper figure.
A nice big orange weighs about 200
grams; two of them almost make a pound, since a pound is equivalent to 450
grams. If we place two such oranges ten centimeters or just about four inches
apart, their mutual attraction then is 200 x 200 = 40,000 divided by 10 times
10, which gives 400 as the result. This result has to be multiplied by
1/15,000,000, the final result being 4/150,000 or not quite 1/40,000 milligram.
This, quite evidently, is far too little to be noticeable or even measurable.
As for the young man and the
beautiful brunette, their mutual gravitational attraction must have been
about 0.03 milligrams if they were originally 100 centimeters (about forty
inches) apart. But the friction of the chair on even the best polished hardwood
floor must have been well over twenty poundsso I'm afraid we must assume that
some other kind of attraction was responsible for that particular phenomenon.
How about something that is really
heavy, though say a middle-sized ocean liner of 25,000 tons weight?
Since ocean liners as a rule avoid
close contact, we'll say that they float 100,000 centimeters (6/10th of a mile)
apart. Making the same calculation as before, which is multiplying their
weights in grams, divided by the square of their distance in centimeters and
multiplied by the constant for one gram at one centimeter, we find that the
liners would attract each other with a force of 4.2 grams. Even if they were
only a hundred yards apart, the attraction would amount to just about one pound
hardly enough to move a ship. So when two ships collide in a fog, it just
means that they happened to be on a collision course.
The figures show why the mutual
gravitational attraction of masses does not show in daily life. It becomes
important only if one of them is of planetary sizefor example when the man on
a slippery floor is attracted by the Earth.
MORE PRIME NUMBERS
NO
other item in my column has brought in such a large volume of mail as my piece
on prime numbers in the June 1953 GALAXY. Since more than a score of letters
and postcards22 by actual count, at the moment of writingqueried the
expression on p. 70. Iłll begin the discussion with that.
Of course, as eleven
correspondents stated or at least suspected, Fermat's expression suffered from
a typographical error. The exponent of the "2" is not 2n but 2n,
so that the expression reads correctly
2 (2^n)
The five readers who amiably
called me a bungler and ignoramus will please air their grievance with the
typesetter; my copy was correct, and I have a carbon copy of the article to
prove it.
Another twelve letters dealt with
a fact beyond my control, but one which I also regret. About a month after my
article was written, the NBSINA (National Bureau of Standards Institute for
Numerical Analysis) on the premises of UCLA (University of California, Los
Angeles) announced that SWAC (Standards Western Automatic Computer) had
established higher Mersenne primes than the famous 2127 1. The list
of higher Mersenne primes reads as follows:
2521-1
2607-1
21279-1
22203-1
22281-1
One Canadian reader thought that
he had found a proof of Goldbach's theorem. His reasoning was as follows:
disregarding the 2, which is the only even prime, a prime number must of
necessity be an even number plus one: P = E+1. Hence, if you add two primes p1
and p2 you really add E1 + E2 + 2 which
obviously must be an even number since you add three even numbers together.
Now this is proof, if one were
needed, that the sum of two primes must be an even number but this is not what
Goldbach said. Goldbach stated that every even number is the sum of two primes,
which sounds like the same statement, but actually is not.
To explain the difference, let us
assume that Goldbach's theorem is wrong. In that case, there should be at least
one even number which is not the sum of two primes but merely the sum of two
odd numbers, either of which, or both, are not primes. The proof to be found,
therefore, is that there is no even number which cannot be expressed as
a sum of two odd numbers both of which must be primes.
Among the correspondence there
were several letters asking me for a list of primes up to certain limits or
asking where such a list can be gotten. I don't know whether the list can still
be bought, but it should be in any reasonably large public library. Its title
is List of Prime Numbers from 1 to 10,006,721 by D. N. Lehmer; it
is a publication of the Carnegie Institution of Washington, Publication Nr.
165, released in 1914. The recent work on the big Mersenne primes can be found
in Mathematical Tables and Aids to Computation, Vol. 7. p. 72 (1953).
WILLY
LEY
ANY QUESTIONS?
Will you ,please
tell me how to determine the acceleration of a rocket if its weight and thrust
are known? In space, would the weight of a rocket affect its
acceleration in any way?
Richard Weed
201 Harper Avenue
Morrisville, Penna.
The formula for determining the
acceleration of a rocket is about as simple as a formula can be. It is P/W,
where P stands for the thrust and W for the weight. In practical application,
however, there are some minor complications which have to be taken into
account, one of which is the direction of the movement.
Let's assume that the rocket
has just taken off and is moving vertically upward. We'll say that its weight
is 100 lbs. and that its rocket motor develops a thrust of 300 lbs. 300 divided
by 100 is, of course, 3so the rocket's "absolute" acceleration would
be 3g.
If this were the whole story,
it would mean that the rocket is climbing at 3g (or accelerating 96 feet per
second), so that at the end of each second of flight its velocity would be 96
feet per second faster than at the beginning of that second. But (obviously) if
the motor were not working, the rocket would fall backwhich is to say that the
Earth's gravity swallows up one g of the "absolute" acceleration. Hence
the "effective" acceleration is 2g, and the formula has to be amended
to read P/Wlg. The interesting point here is that, if the rocket is manned,
the pilot feels the absolute acceleration while the speed increases
according to the effective acceleration.
The "true" effective
acceleration is influenced by air resistance, which will vary with the speed of
the rocket at a given moment and its altitude (or more precisely the density of
the air at that altitude). As important as that is the fact that the weight of
the rocket at the end of that second will be less than the weight at the
beginning of the second.
All this applies to empty space
too, except that (a) there will be no air resistance, (b) the value for the
thrust is about 15 per cent higher than the sea level value of the same rocket
motor and (c) at a sufficient distance from the Earth the value for g may be
noticeably less than the sea level value. For a height of 250 miles, for
example, g has dropped from about 32 ft/sec2 to 28.5 ft/sec2.
Does a rocket which takes off
toward the west have to attain a higher velocity relative to a point on the
ground to reach orbital velocity than a rocket headed east?
William J. Hunt
2325 NE 32nd Avenue
Portland 12, Oregon
Let's take this one step by step.
In von Braun's orbit1075 miles
above mean sea levelthe rocket will have to have a velocity of 4.4 miles per
second. If it has that velocity in the orbit, it doesn't matter whether it goes
around the equator heading east or heading west, or along a meridian from pole
to pole, or at any odd angle in between.
Repeat: If it has that velocity
in the orbit . . . but first it has to acquire this velocity.
For simplicity's sake, let's
suppose that take-off is at the equator. The equatorial diameter of the Earth
is almost 8000 miles, hence the length of the equator is that figure multiplied
by "pi" or, in round figures, 24,000 miles. Since the Earth turns
once in 24 hours, a point at the equator moves in an easterly direction at the
rate of 1000 miles per hour or about 0.28 miles per second.
So if your rocket heads east it
has, relative to the center of the Earth, a speed of 1000 miles per hour before
it even starts. If you wanted to head west, you would not only lose that 1000
mph, but you would have to "kill" it firstso that you lose 2000
miles per hour, or around 0.56 miles per second. Half-a-mile a second is a
considerable speed, even for a rocket, so take-off in an easterly direction is
usually assumed.
What does the term
"Doppelganger" mean?
Cpl. D. A. Freeman, USMC
Fleet Post Office
San Francisco, Calif.
This is originally a German
word, composed of two words each of which is hard to translatewhich is, of
course, the reason why it was adopted rather than translated.
"Doppel" can mean
"twice," and also "double" (the amount) or
"duplicate."
"Ganger" is best
translated as "walker."
The whole means : "a
walking duplicate."
The superstition attached to
the word is that some people have such a "duplicate" walking around,
and when they meet it face to face they know that they are doomed, with usually
only three days of grace left to them. But in everyday German conversation, the
term "Doppelganger" can be and often is used without mystical
connotations. A man saying to a friend, "I met your Doppelganger
today," merely means that he met somebody or saw somebody who looked just
like his friend.
Why is the nautical mile longer
than the ordinary mile? Is there any relationship between the nautical mile and
the metric kilometer? And is there such a thing as a "metric mile"?
James A. Monahan
(no street given)
Chicago, Illinois.
If anybody knows the origin of
the English mile, I wish he'd write me, for as far as I know the mile1760
yards or 5280 feet longis a unit which just happened.
The nautical mile, however, has
a reason; its length is 1/60th of one degree at the equator.
As for the kilometer, it also
has a reason: it is 1/10,000th of the distance of a point on the equator from
either poleOr rather it is supposed to be, for more recent measurements have
shown a small deviation from that figure.
There is no "metric
mile"; but for a while geographers used a unit which they called the
"geographic mile," which corresponded to 1/15th of a degree at the
equator.
The various "national"
miles are as uncertain in origin as the English mile, though some of them
happen to come fairly close to the old geographic mile. To compare them, the
kilometer has been used as the unit in the following table:
1
English mile 1.609 km.
1
nautical mile 1.852 km.
1
geographic mile 7.420 km.
1
German mile 7.500 km.
1
Danish mile 7.582 km.
1
Swedish mile 10.688 km.
1
Norwegian mile 11.295 km.
The Russian verst measured
1.066 kilometers (or 0.6629 miles), but like all the other old miles given, it
is now obsolete. The only units in use internationally now are the kilometer,
the English mile, and the nautical mile; but the French nautical mile is three
times as long as the nautical mile of everybody else. Don't ask me why.
HOLD IT!
We
mean your collection of GALAXY, naturally, which will really dress up your
library when they're kept in our handsome gold-stamped binders, instead of just
being allowed to accumulate. Arranged according to date, easy to hold,
protected from rough handling, your back issues of GALAXY will give you
continued rereading pleasure . . . and increase constantly in value. Each
binder holds six issues and costs only $1.50 postpaid from Galaxy Publishing
Corp., 421 Hudson Street, New York 14, N. Y.
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