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A
Disproof of Newton’s Fundamental Lemmae
by
Miles Mathis
First
written November 2005
Newton
published his Principia
in 1687. Except for Einstein’s
Relativity corrections, the bulk of the text has remained
uncontested since then. It has been the backbone of trigonometry,
calculus, and classical physics and, for the most part, still is.
It is the fundamental text of kinematics, gravity, and many other
subjects. In this paper I will
show a simple and straightforward disproof of one of Newton’s
first and most fundamental lemmae, a lemma that remains to this
day the groundwork for calculus and trigonometry. My correction
is important—despite the age of the text I am critiquing—due
simply to the continuing importance of that text in modern
mathematics and science. My correction clarifies the foundation
of the calculus, a foundation that is, to this day, of great
interest to pure mathematicians. In the past halfcentury
prominent mathematicians like Abraham Robinson have continued to
work on the foundation of the calculus (see Nonstandard
Analysis). Even at this late a date in history, important
mathematical and analytical corrections must remain of interest,
and a finding such as is contained in this paper is crucial to
our understanding of the mathematics we have inherited. Nor has
this correction ever been addressed in the historical
modification of the calculus, by Cauchy or anyone else.
Redefining the calculus based on limit considerations does
nothing to affect the geometric or trigonometric analysis I will
offer. The first lemma in
question here is Lemma VI, from Book I, section I (“Of the
Motion of Bodies”). In that lemma, Newton’s provides the
diagram below, where AB is the chord, AD is the tangent and ACB
is the arc. He tells us that if we let B approach A, the angle
BAD must ultimately vanish. In modern language, he is telling us
that the angle goes to zero at the limit.
This
is false for this reason: If we let B approach A, we must monitor
the angle ABD, not the angle BAD. As B approaches A, the angle
ABD approaches becoming a right angle. When B actually reaches A,
the angle ABD will be a right angle. Therefore, the angle ABD can
never be acute. Only if we imagined that B passed A could we
imagine that the angle ABD would be acute. And even then the
angle would not really be acute, since we would be in a sort of
negative time interval. Newton is using A as his zeropoint, so
that we cannot truly cross that point without arriving in some
sort of negative interval, especially since we are talking about
the motion of real bodies.
[I have added this paragraph
after talks with many readers, who cannot visualize the
manipulation here. It is very simple: you must slide the entire
line RBD toward A, keeping it straight always. This was the
visualization of Newton, and I have not changed it here. I am not
changing his physical postulates, I am analyzing his geometry
with greater rigor than even he achieved. You
may take this link, where his second animation is the correct
one.]
If we are taking B to A and may not go past A, then
the angle ABD has a limit at 90^{o}.
When ABD is at 90^{o},
the angle BAD may not be zero. This will be crystal clear in a
moment when we look at the length of the tangent at the limit,
but for now it is enough to say that if angle BAD were zero, then
ADB would also have to be 90^{o},
which is impossible to propose. A triangle may not have two
angles of 90^{o}.
In Lemma VII, Newton’s uses the previous lemma to show
that at the limit the tangent, the arc and the chord are all
equal. I have just disproved this by showing that the angle ABD
is 90^{o}
at the limit. If ABD is 90^{o}
at the limit, then the tangent
must be greater than the chord. Please notice that if AB and AD
are equal, then ABD must be less than 90^{o}.
But I have shown that ABD
cannot be less than 90^{o}.
B would have to pass A, which would put us in a negative time
interval. If B cannot pass A (A being the limit) then the tangent
can never equal the chord, not when approaching the limit and not
when at the limit. This
verifies my previous assertion that the angle BAD cannot go to
zero. If the tangent is longer than the chord at the limit, then
this is just one more reason that the angle BAD must be greater
than zero, even at the limit. If AD is greater than AB, then DB
must be greater than zero. If DB is greater than zero, then the
angle BAD is greater than zero.
All this is caused by the fact that the angle ABD goes to 90^{o}
before
the angle BAD goes to zero.
The angle ABD reaches the limit first, which keeps the angle BAD
from reaching it. BAD never reaches zero.
Of course this
means that B never reaches A. If B actually reached A, then we
would no longer have a triangle. The tangent and the chord are
equal only when they both equal zero, and they both equal zero
when the interval between A and B is zero. But the 90^{o}
angle at ABD prevents this
from happening. When that angle is at 90^{o},
the tangent must be greater than the chord. Therefore the chord
cannot be zero. If the chord is zero, then the tangent and the
chord are equal: therefore the chord is not zero. To put it into
a more prooflike form:
1) If the chord AB is zero, then
the tangent AD is also zero. 2) zero = zero 3) If AB = AD,
then the angle ABD must be less than 90^{o}. 4)
The angle ABD cannot be less than 90^{o}. QED:
AB does not equal AD; AB does not equal 0.
In fact, this
is precisely the reason that we can do calculations in Newton’s
“ultimate interval”, or at the limit. If all the variables
were either at zero or at equality, then we could not hope to
calculate anything. Newton, very soon after proving these lemma,
used a versine equation at the ultimate interval, and he could
not have done this if his variables had gone to zero or equality.
Likewise, the calculus, no matter how derived or used, could not
work at the limit if all the variables or functions were at zero
or equality at the limit.
Some will say that my claim
that B never reaches A is like the paradoxes of Zeno. Am I
claiming that Achilles never reaches the finish line? No, of
course not. The diagram above is not equivalent to a simple
diagram of motion. B is not moving toward A in the same way that
Achilles approaches a finish line, and this has nothing to do
with the curvature. It has to do with the implied time variable.
If we diagram Achilles approaching a finish line, the time
interval does not shrink as he nears the line. The time interval
is constant. Plot Achilles’ motion on an x/t graph and you will
see what I mean. All the little boxes on the taxis are the same
width. Or go out on the track field with Achilles and time him as
he approaches the finish line. Your clock continues to go forward
and tick at the same rate whether you see him 100 yards from the
line or 1 inch from line. But
given the diagram above and the postulate “let B go to A”, it
is understood that what we are doing is shrinking both the time
interval and the arc distance. We are analyzing a shrinking
interval, not calculating motion in space. “Let B go to A”
does not mean “analyze the motion of point B as it travels
along a curve to point A.” It means, “let the arc length
diminish.” As the arc length diminishes, the variable t is also
understood to diminish. Therefore, what I am saying when I say
that B cannot reach A is that Δt cannot equal zero. You cannot
logically analyze the interval all the way to zero, since you are
analyzing motion and motion is defined by a nonzero interval.
The circle and the curve are
both studies of motion. In this particular analysis, we are
studying subintervals of motion. That subinterval, whether it is
applied to space or time, cannot go to zero. Real space is
nonzero space, and real time is nonzero time. We cannot study
motion, velocity, force, action, or any other variable that is
defined by x and t except by studying nonzero intervals. The
ultimate interval is a nonzero interval, the infinitesimal is
not zero, and the limit is not at zero. The limit for any
calculable variable is always greater than zero. By calculable I
mean a true variable. For instance, the angle ABD is not a true
variable in the problem above. It is a given. We don’t
calculate it, since it is axiomatically 90^{o}.
It will be 90^{o}
in all similar problems, with
any circles we could be given seeking a velocity at the tangent.
The vector AD, however, will vary with different sized circles,
since the curvature of different circles is different. In this
way, only the angle ABD can be understood to go all the way to a
zerolike limit. The other variables do not. Since they yield
different solutions for different similar problems (bigger or
smaller circles) they cannot be assumed to be at a zerolike
limit. If they had gone all the way to some limit, they could not
vary. A function at a limit should be like a constant, since the
limit should prevent any further variance. Therefore, if a
variable or function continues to vary under a variety of similar
circumstances, you can be sure that it is not at its own limit or
at zero. It is only dependent on a variable that is.
If
AB and AD have real values at the limit, then we should be able
to calculate those values. If we can do this we will have put a
number on the “infinitesimal.” In fact, we do this all the
time. Every time we find a number for a derivative, we put a real
value on the infinitesimal. When we find an “instantaneous”
velocity at any point on the circle, we have given a value to the
infinitesimal. Remember that the tangent at any point on the
circle stands for the velocity at that point. According to the
diagram above, and all diagrams like it, the tangent stands for
the velocity. That line is understood to be a vector whose length
is the numerical value of the tangential velocity. It is commonly
drawn with some recognizable length to make the illustration
readable, but if it is an instantaneous velocity, the real length
of the vector must be very small. Very
small but not zero,
since we actually find a nonzero solution for the derivative.
The derivative expresses the tangent, so if the derivative is
nonzero, the tangent must also be nonzero.
Some have said that since we can find sizeable numbers for the
tangential velocity, that vector cannot be very small. If we find
that the velocity at that point is 5 m/s, for example, then
shouldn’t the velocity vector have a length of 5? No, since by
the way the diagram is drawn and defined, we are letting a length
stand for a velocity. We are letting x stand for v. The t
variable is not part of the diagram. It is implicit. It is
ignored. If we are letting B approach A, then we are letting t
get smaller. A velocity of 5 only means that the distance is 5
times larger than the time. If the time is tiny, the distance
must be also.
There
is another way to analyze Newton's problem, and it may be the
most interesting of all (for some). In the Principia,
Newton's actual language in describing this problem (Lemma VI) is
this: "if the points A and B approach one another. . ."
Two things bear closer attention here. One, A cannot approach B
without messing up the geometry. If we start moving the point A,
we destroy our right triangle. What he means is what I have said
above: Let B approach A. To be rigorous, we should let one point
remain stationary and let the other point move. If we let both
move, we create unnecessary problems. The other thing to notice
is the word "approach". Newton is postulating motion.
As confirmation of this, we need only look at his title for this
section: "Of Natural Philosophy". Natural philosophy is
not pure math, it is physics. Newton is describing a philosophy
or study of nature, which we now call physics. Nature is not
pure, it is physical. Therefore this lemma must be a part of what
we now call applied mathematics. If this is so, then time must be
involved. As I have asserted above, Newton is studying a
diminishing interval in order to analyze curved motion. He uses
this analysis immediately afterwards to apply to an orbit, for
instance. So both motion and time are involved in Newton's
analysis. For this reason alone, his angle BAD cannot vanish.
That would be taking the problem to a zero time interval, and
there is no such thing as a zero time interval in physics. You
cannot study motion and then postulate a zero time interval,
since motion is defined by a nonzero time interval. If you have
a zero time interval, you have no motion, by definition. Simply
by using the word "approach", Newton has ruled out a
zero time interval. His interval can get smaller and smaller, to
any extent he likes, but it cannot vanish. By definition,
"approach" and "vanish" are mutually
exclusive.
But it gets even more interesting. Using the
limit concept alone, this problem cannot be solved at all.
Meaning, if we let our angle at R equal θ, then BAD = θ/2
and ABD = π/2 + θ/2.
If
we let θ go to zero, then BAD and ABD approach the limit
in the same way. The limit concept does not support my analysis.
No, it supports Newton's analysis, since historically it grew out
of his analysis. The limit concept fails to explain why we find
nonzero solutions at the limit for both the chord and the
tangent, and it fails because its analysis is faulty just as I
have shown Newton's analysis is faulty. The limit analysis treats
the entire problem as an abstract or puremath problem, whereas
it is a physical problem. Motion and time are both involved here.
What that means is that we must have a necessary time separation
between A and B. Since we have motion, we cannot have a zero
interval. If we do not have a zero interval, then we must have a
time separation. Stated that way, we arrive at. . . yes,
Relativity. If this is a physical problem, then A and B cannot
exist the same time, operationally. An event at B cannot be fully
equal to that same event as seen from A. If we think of the
measurement of an angle as a physical event instead of an
abstract geometric quantity, then angles in a diagram like this
must be analyzed from a physical point of view.
Some will
think I am overcomplicating this problem, or inventing esoteric
solutions, but consider this fact: Newton's gravitational studies
and proportionalities came out of this same book, the Principia,
indeed this same section. Is it not strange that Einstein's
Relativity corrections have been applied to gravity but not to
the orbit? The diagram above is a preliminary study of the orbit,
and underlies a=v^{2}/r, and yet it has never benefitted
from a Relativity analysis until now. We think that gravity
causes the orbit, and yet we do a Relativity analysis of gravity
but not of the orbit. Very strange.
The way that
Relativity solves this problem once and for all is that it gives
us a way of separating θ/2 at B and θ/2 at A.
According to the limit analysis, both angles should diminish in
the same way. But because they are spatially separated, they
cannot act the same. According to Relativity, we must pick a
point and measure everything from there. We must study the
problem from A or B, but we cannot study the problem from both
places simultaneously. Since we have given the motion to point B,
we must let that be our point of measurement. In other words, in
this problem, we exist at B. The event is at B. Let that
event be π/2 + θ/2 going to the limit. θ goes to
zero, so ABD goes to 90^{o}. Of course BAD is also going
to zero, but there is a time lag. As seen or measured from B,
information from A must be late, and vice versa. Therefore, as
measured from B, the limit at B must be reached before the limit
at A. Or, since I have shown that limits are never really reached
anyway, especially when those limits are at zero, it would be
more rigorous to say that θ/2 is smaller at B, as
measured from B, than θ/2 at A. Given time separation,
equal angles are not quite equal.
Of course, many people
will not like this analysis. Some will find it fascinating and
others will find it to be gibberish. Honestly I prefer the
simpler explanation myself: we cannot propose a zero time
interval, therefore the angles cannot vanish, therefore the lines
cannot be equal. No matter how small we go, in order to talk of
motion we must have a real time interval. As long as we have a
real time interval, we have a triangle. As long as we have a
triangle, we have a tangent that is longer than the chord. We
"approach" the limit, we do not "reach" the
limit. That said, I believe the Relativity analysis is also
correct. Either analysis gets the right answer, using ideas that
are physically correct and physically real. To be consistent, if
we apply time separations to the gravitational field, we must
also apply them to the orbit. Gravity cannot physically cause the
orbit, Relavity applying to gravity but not to the orbit. Since
Newton's whole section in question here is physical, we must
either apply Relativity to all of it, or to none of it. Einstein
updated Newton's analysis of gravity, and I have just done the
same for the orbit.
Conclusion
My finding in
this paper affects many things, both in pure mathematics and
applied mathematics. I have proven, in a very direct fashion,
that when applying the calculus to a curve, the variables or
functions do not go to zero or to equality at the limit. This
must have consequences both for General Relativity, which is
tensor calculus applied to very small areas of curved space, and
quantum electrodynamics, which applies the calculus in many ways,
including quantum orbits and quantum coupling. QED has met with
problems precisely when it tries to take the variables down to
zero, requiring renormalization. My analysis implies that the
variables do not physically go to zero, so that the assumption of
infinite regression is no more than a conceptual error. The
mathematical limit for calculable variables—whether in quantum
physics or classical physics—is never zero. Only one in a set
of variables goes to zero or to a zerolike limit (such as the
angle 90^{o}). The other variables are nonzero at the
limit. For QED, this means that when the Planck limit is reached,
length and time limits are also reached. Neither time nor length
variables may go to zero when used in momentum or energy
equations of QED. In fact, beyond the logic I have used here, it
is a contradiction to assume that values for energy would not
have an infinite and continuous regression toward zero, but that
values for length and time would.
This is not to say that length and time must be quantized; it is
only to say that in situations where energy is found empirically
to be quantized, the other variables should also be expected to
hit a limit above zero. Quantized equations must yield quantized
variables. Space and time may well be continuous, but our
findings–our measurements or calculations—cannot be.
Meaning, we can imagine shrinking ourselves down and using tiny
measuring rods to mark off subareas of quanta. But we cannot
calculate subareas of quanta when one of our main
variables—Energy—hits a limit above these subareas, and when
all our data hits this same limit. The only way we could access
these subareas with the variables we have is if we found a
smaller quantum.
As I said, there has also been confusion
on this point in the tensor calculus. In section 8 of Einstein’s
paper on General Relativity, he gives volume to a set of
coordinates that pick out a point or an event. He calls the
volume of this point the “natural” volume, although he does
not tell us what is “natural” about a point having volume.
General Relativity starts [section 4] by postulating a point and
time in space given by the coordinates dX_{1}, dX_{2},
dX_{3}, dX_{4}. This set of coordinates picks out
an event, but it is still understood to be a point at an instant.
This is clear since directly afterwards another set of functions
is given of the form dx_{1}, dx_{2}, dx_{3},
dx_{4}. These, we are told, are the “definite
differentials” between “two infinitely proximate
pointevents.” The volume of these differentials is given in
equation 18 as dτ = ∫dx_{1}dx_{2}dx_{3}dx_{4}
But we are also given the “ natural” volume dτ_{0},
which is the "volume dX_{1}, dX_{2}, dX_{3},
dX_{4}". This natural volume gives us the equation
18a: dτ_{0} = √gdτ
Then Einstein says, “If √g were to vanish at a point of the
fourdimensional continuum, it would mean that at this point an
infinitely small ‘natural’ volume would correspond to a
finite volume in the coordinates. Let us assume this is never
the case. Then g cannot change sign. . . . It always has a finite
value.” According to my
disproof above, all of this must be a misuse of the calculus, a
misuse that is in no way made useful by importing tensors into
the problem. In no kind of calculus can a set of functions that
pick out an pointevent be given a volume—natural, unnatural,
or otherwise. If dX_{1}, dX_{2}, dX_{3},
dX_{4} is a pointevent in space, then it can have no
volume, and equation 18a and everything that surrounds it is a
ghost. In the final analysis
this is simply due to the definition of “event”. An event
must be defined by some motion. If there is no motion, there is
no event. All motion requires an interval. Even a nonevent like
a quantum sitting perfectly still implies motion in the
fourvector field, since time will be passing. The nonevent will
have a time interval. Every possible event and nonevent, in
motion and at rest, requires an interval. Being at rest requires
a time interval and motion requires both time and distance
intervals. Therefore the event is completely determined by
intervals. Not coordinates, intervals. The
point and instant are not events. They are only event boundaries,
boundaries that are impossible to draw with absolute precision.
The instant and point are the beginning and end of an interval,
but they are abstractions and estimates, not physical entities or
precise spatial coordinates.
Some will answer that I have just made an apology for Einstein,
saving him from my own critique. After all, he gives a
theoretical interval to the point. The function dX is in the form
of a differential itself, which would give it a possible
extension. He may call it a point, but he dresses it as a
differential. True, but he does not allow it to act like a
differential, as I just showed. He disallows it from
corresponding to (part of) a finite volume, since this would ruin
his math. He does not allow √g to vanish, which keeps the
“natural” volume from invading curved space.
Newer versions of this same Riemann space have not solved this
confusion, which is one of the main reasons why General
Relativity still resists being incorporated into QED.
Contemporary physics still believes in the pointevent, the point
as a physical entity (see the singularity) and the reality of the
instant. All of these false notions go back to a misunderstanding
of the calculus. Cauchy’s "more rigorous" foundation
of the calculus, using the limit, the function, and the
derivative, should have cleared up this confusion, but it only
buried it. The problem was assumed solved since it was put more
thoroughly out of sight. But it was not solved. The calculus is
routinely misused in fundamental ways to this day, even (I might
say especially) in the highest fields and by the biggest
names.
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