---------------------
for PD more importasnt is if it looks better
sort of a youg woman model is bettr scientifically than an old person
anyway Mr mpc
my circlon model is not al all th elink you quoted
it i s described schematically at the** appendix**
of my abstract of my model book
it does not look like a young model woman
but anyway
see my model at the appendix:
http://sites.google.com/site/theyporatmodel/an-abstract
ps about my whole ** book***
PD (and some others )stole it
and is very eager to hide
and obfuscate that fact !!....
ATB
Y.Porat
--------------------------
As you will see, Bernouilli was not posing an abstract mathematical
puzzle, for the mere sake of befuddling others, the solution to this
problem led to important discoveries in mechanics, as well as
metaphysics.
Bernoulli's attack on this problem began with what he called "Fermat's
metaphysical principle", that light always seeks out the path of least
time. It was a discovery of the ancient Greeks, that when light was
reflected from a mirror, the path it took was the shortest length.
However, when light was refracted by traveling through different
media, such as water and air, the path of the light was not the
shortest length. Fermat, discovering that the velocity of light was
slower in denser media, demonstrated that the light changed its
direction at the boundary between the two media, so as to follow the
path of least time. This, of course, was consistent with the Greek
discovery. In reflection, since the light travels through only one
medium and therefore doesn't change velocity, the shortest path, is
also the path of least time. But, when there's a change in medium, the
light travels the shortest path in space-time, or the path of least-
time.
Bernoulli's approach was to follow the light, so to speak, to the path
of least time. If the path of a ray of light traveling through a
medium, whose density is continuously changing, according the same
principle as that of a body falling under gravity, the the least time
path of the light, will be the same as the least time path of the
body.
But, how to discover the path, when we only know the principle of
change, and have no positions to which to orient? At each moment along
the light's path, the light would be changing its speed and direction,
such that its overall travel took the least time. Thus, similar to the
motion of a planet, at each such moment, the light was ceasing to be
what it was, and becoming what it will be. At each moment, the
position of the light was a function of the principle of maintaining
the least-time path.
Fermat had shown, that as light moved from a rarer to a denser medium,
it slowed down, and its path became more vertical. For example, if
light were traveling through air to water, the angle its path made
with a vertical line, changed at the boundary between the air and
water. If the angle its path made with the vertical in the air
changed, the angle it made with the vertical under the water changed
accordingly. However, the two angles did not change proportionally.
Rather, they changed such that the sines of the angles were always in
the same proportion.
So, at each "moment of becoming" along the light's path, the light's
velocity and trajectory were changing, such the sine of the angle the
light's path made at that moment, was always proportional to the sines
of the angles at all such "moment's of becoming."
To find the brachistichrone, Bernoulli thought of the medium in the
following way:
"If we now consider a medium which is not uniformly dense but is as if
separated by an infinite number of sheets lying horizontally one
beneath another, whose interstices are filled with transparent
material of rarity increasing or decreasing according to a certain
law; then it is clear that a ray which may be considered as a tiny
sphere travels not in a straight but instead in a certain curved path.
This path is such that a particle traversing it with velocity
continuously increasing or diminishing in proportion to the rarity,
passes from point to point in the shortest time."
Under this idea, at each horizontal sheet, the speed and direction of
the light changes. The principle under which its speed and direction
changes at each horizontal sheet, Leibniz called, the differential.
The totality of all such differentials, (what Leibniz called the
integral), is the sought after brachistichrone curve.
From one "moment of becoming" to the next, the position of the light
changes, as it passes vertically from one density to the next. Each
such vertical change in position is accompanied by a horizontal change
in position, that corresponds to the sine of the angle of inclination
at each "moment of becoming". (Bernoulli's geometrical construction of
the above can be found on p. 652 of Smith.) Bernoulli adopted Leibniz'
notation for these ideas, calling the vertical change, dy, the
horizontal change, dx, and the resulting change in the path of the
light, dz. The proportion between the vertical and the horizontal,
dy:dx, and the resulting change in the path, dz, is a function of the
rate at which the density of the medium is changing.
Bernoulli shows, that if the density of the medium is changing
according to the rate at which a body falls under its own weight,
(specifically, that the velocity changes according to the square root
of the vertical distance) then the resulting curve is a cycloid.
"...you will be petrified with astonishment when I say that this
cycloid, the tautochrone of Huygens is our required
brachistochrone..." he declared.
But, Bernoulli noted that this was not a discovery of a particular
physical phenomenon, but a metaphysical discovery of a universal
principle:
"Before I conclude, I cannot refrain from again expressing the
amazement which I experienced over the unexpected identity of Huygen's
tautochrone and our brachistochrone. Furthermore, I think it is
noteworthy that this identity is found only under the hypothesis of
Galileo so that even from this we may conjecture that nature wanted it
to be thus. For, as nature is accustomed to proceed always in the
simplest fashion, so here she accomplishes two different services
through one and the same curve, while under every other hypothesis two
curves would be necessary the one for oscillations of equal duration
the other for quickest descent. If, for example, the velocity of a
falling body varied not as the square root but as the cube root of the
height falalen through, then the brachistochrone would be algebraic,
then tautochrone on the other hand transcendental; but if the velocity
varied as the height fallen through then the curves would be
algebraic, the one a circle, the other a straight line."
http://www.wlym.com/drupal/node/286
thus:
on the wayside, I am not "top-posting;"
I am self-publishing on an infinitessimal scale. and,
I certainly did know that it was not you,
who was whining about to top-post or not to -- and
I am rather set, for decades, in making fun
of this so-called nettiquette-cum-obsessive-repulsive-
strain-dysorder.
so, you've "solved" the problem of quantum gravity, but
you have yet to coin a proper name for de entrain!... really,
not a bad stab at it (whoosh,
goes the dysplaced and/or entrained aether-shockwave,
in front of the blade & behind it .-)
> You can't distinguish between my posts and the other poster who is
> asking you not to top post? Unless you post something relevant, I'm
> done replying to your posts. Take care.
thus:
incidentally, Oberon, husband of Titanya and
King of the Fairies, is twirling around Uranus!
> Not sure what is going on with the heliopause. Haven't had a chance to
> think about it much yet.
thus:
doctor Einstein's essay seems quite confuzed
about the electromegnetic properties of matter, but
that was a while before our standard textbookoid concepts
were put out from the Texas Schoolbook Suppository.
thus:
he is giving a lot of credit to Lorentz, who may
be more responsible, after all, for the time-space crack-up
than doctor Minkowski; can you say,
Most useless formalism of Century 20.1?
however, the real problem is your persistent use
-- with whomever else from the past & future --
of the the concept of vacuum, as Pascal first thought of it,
which is really, strictly relative or active (as in,
That giant sucking sound, you hear, when you're trying
to read this ****).
> http://www-groups.dcs.st-and.ac.uk/~history/Extras/Einstein_ether.html
--Brit's hate Shakespeare, Why?
http://wlym.com/campaigner/8011.pdf
--Madame Rice is a Riceist, How?
http://larouchepub.com/other/2009/3650rice_racist.html
--The Riemannian Space of the Nucleus, What?
http://www.21stcenturysciencetech.com/Articles_2009/Relativistic_Moon.pdf
--In perpetuity clause in healthcare bill, Where?