return to homepage 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 proof-like 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 t-axis 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 non-zero interval. The circle and the curve are both studies of motion. In this particular analysis, we are studying sub-intervals of motion. That subinterval, whether it is applied to space or time, cannot go to zero. Real space is non-zero space, and real time is non-zero time. We cannot study motion, velocity, force, action, or any other variable that is defined by x and t except by studying non-zero intervals. The ultimate interval is a non-zero 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 zero-like 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 zero-like 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 non-zero solution for the derivative. The derivative expresses the tangent, so if the derivative is non-zero, the tangent must also be non-zero. 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.
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 zero-like limit
(such as the angle 90 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 point-event, 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.
If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |