Calculus

by MIL E SMAT H IS

Contents

I GENERALITIES

1 CENTRALDISCOVERIES

2 DEATHBY MATHEMATICS

3 ELEVENBIG QUESTIONS

4 AREVALUATION OF TIME

5 ANGULARVELOCITY

6 UNIFIEDFIELDS IN DISGUISE

7 THEKINETIC ENERGY EQUATION

8 THEEQUATIONV=V0+AT iii

1

3

25

35

47

57

65

71

77

iv II CALCULUS

CONTENTS 85

9 AREDEFINITION OF THE DERIVATIVE87 9.1The Groundwork. . . . . . . . . . . . . . . . . . . . . . . . .91 9.2Historical Interlude. . . . . . . . . . . . . .. . . . . . . . . . 103 9.3The Rest of the Groundwork. . . . . . . . 111. . . . . . . . . . . 9.4The Algorithm114. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5The Derivation119. . . . . . . . . . . . . . .. . . . . . . . . . . 9.6Application to Physics123. . . . . . . . . . . . .. . . . . . . . . 9.7The Second Derivative – Acceleration. . . . . . . .126. . . . . .

10 THECALCULUS IS CORRUPT

11 DERIVATIVEFOR POWERS

12 DERIVATIVEFOR EXPONENTIALS

13 DERIVATIVEFOR ln AND 1/x

14 DERIVATIVEFOR log

15 VARIABLEACCELERATION

16 CALCULUSSIMPLIFIED

17 TRIGDERIVATIVES

133

139

151

179

195

199

211

233

CONTENTS

18 ANOTHERFATAL FLAW

19 NEWTON’SLEMMAE

v

239

245

20 THEEQUATIONa=v²/r257 20.1 Newton’sDerivation259. . . . . . . . . . . . . . . . . . . . . . . 20.2 TheCurrent Solution. . . . . . . . . . 264. . . . . . . . . . . . . 20.3 Feynman’sVariation268. . . . . . . . . . . . .. . . . . . . . . . 20.4 MySolution. . . . . . . . . . . . . . . . . . . . . . . . . . . .272 20.5 Implications. . . . . . . . . . . . . . . . . . . . . . . . . . . .281

21 CLARIFICATIONOFa=v²/r

287

It has been known for millenia that the Earth rests upon the back of a giant turtle. Only in recent centuries has this knowledge been added to. In 1794, in one of the high valleys of the Himalayas, one of the wise was asked, "Master, what does the turtle rest upon?" The Master answered: "It is turtles all the way down, my son." But now that scientists have ﬁnally succeeded in mapping the universe, a turtle controversy has arisen. It turns out that level 7,484,912 is occupied not by a turtle, but by a man dressed as a turtle.It is not known how this will affect our other equations.

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You probably aren’t used to having a book on science and math open with a joke. But a sense of humor is crucial to existing in a world where even our great-est accomplishments contain large elements of the absurd.Some contemporary thinkers are of the opinion that we are very near to a complete understanding of the universe.I am far from agreeing with them.We have made some wonderful discoveries and are due a small dose of pride, I suppose. But the things we don’t know so overwhelm the things we do that any talk of a full understanding is just bombast. Worse,it is hubris.It may even be a scientiﬁc sacrilege, with real curses attached to it.When we become too secure in our knowledge, we stop questioning. Failureto question is the ultimate scientiﬁc failure.Answers quit coming precisely when they aren’t sought, and they aren’t sought precisely when they are (erroneously) thought to be in hand. We are like the dog who discovers how to use the little ﬂap-door and now considers himself master of the house.

x

PREFACE

He lies in front of the ﬁre and congratulates himself for his cleverness. He would be better outside chasing rabbits.

In this book I propose solutions for several of the greatest errors currently exist-ing in physics and mathematics. I do not propose to solve all the greatest errors, of course, or even to know what they are.I only present the ones that have be-come known to me in my years of research.Many may ﬁnd my list surprising or even shocking, since I do not seem to choose problems that are commonly acknowledged to exist.Rather I choose problems that are believed to have been solved. This,I realize, can have the appearance of caprice or insolence, but I have simply gone where my nose leads me.I suspect that the whole history of science has moved in much the same way, so I will not apologize for seeing problems where I see them.

Lest I be dismissed as a crank before my ﬁrst equation hits the page (and this sort of dismissal has become pandemic in the ﬁeld), I rush to add that I am not a so-called classicist, bent on refuting Relativity and Quantum Mechanics 1 simply because they disturb my sense of balance or my love of Newton.I attack Newton as well, long and – I like to think – shockingly. Beyond that, I am convinced of time dilation and length contraction and the necessity of transforms. I simply do not believe that Einstein provided the correct transforms.Likewise, I believe in the accuracy and usefulness of many of the equations of QED. But QED is still in large part a heuristic math posing as a theory.Even Feynman admitted this before he died, to the chagrin of most in the ﬁeld. QED is not “the ﬁnal solution” until it is ﬂeshed out with a coherent theory.I believe, contra current wisdom, that QEDwillbe provided with a coherent theory, one that makes sense even in the macro-world.

I am not a classicist, nor am I in any of the other dissenting groups that are op-posed to the standard interpretation of Einstein.That is to say, I am not proposing supra-luminal theories or any other theories that go beyond the math and theory

1 I am also not any sort of conspiracy theorist.I do not believe that Einstein plagiarized anyone, not even his own wife.I have no special regard for German philosophy or special disregard for Jewish scientists.I am not here to bury Einstein or to praise him.I am here to mathematically evaluate his equations.I ﬁnd it a shame that the ﬁeld has already been so muddied by politics and other petty misunderstandings that an objective critique has become a near-impossibility.

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of Einstein.I am not proposing any new particles, forces, ﬁelds, or maths.All the major chapters and ﬁndings in this book deal with straightforward mathematical analysis of famous historical papers and theories. For the most part, this analysis is high-school level algebra applied to these papers.In critiquing the calculus, some rather subtle number theory is used, but no higher math at all. This means that this book is unlike anything you have read or heard of before.It is not al-lied to the status quo, but it is also not allied to any of the dissenting groups.It is completely outside the 20th century argument, since it cannot be said to be ultimately pro-Einstein or contra-Einstein, pro-Newton or contra-Newton.It is pro-Einstein in that his theory (and Lorentz’s and Poincare’s, etc.) is shown to be correct in many important ways.However, it is contra-Einstein in that my alge-braic corrections falsify some fundamental assumptions and equations.How you would classify my correction is therefore more a matter of your own allegiances than mine, since I have none.

This book differs from all the other critiques I have seen of current theory in that my arguments are not mainly philosophical or even theoretical.They are mathematical. Irerun the original equations in the original papers and show where the speciﬁc mathematical errors are.In this I believe I may be the ﬁrst. Especially as regards Relativity, there has been a massive amount of criticism and absolutely no mathematical proof to back it up. A few mathematical variants have been put forward, some with a certain amount of validity; but no one has shown where Einstein’s math is wrong in itself.Herbert Dingle, perhaps the most famous critic of Einstein in the 20th century, said in the 60’s that he was astute enough not to search for mathematical errors in the theory.Whether his astuteness was based upon the recognition of his own mathematical limitations or upon some other factor is less clear. I suppose current wisdom is that because they are assumed to have been combed by everyone from Bohr to Feynman, the equations must now be unassailable. But nothing in this world is unassailable, as Einstein’s refutation of Newton was supposed to have proved.Newton survived two hundred years of geniuses before Einstein appeared.If Einstein had been cowed by genius, I would now have nothing to critique.But Einstein did not see problem-solving as an attack upon genius or upon the status-quo, or as the solution to his career aspirations; he saw it simply as problem solving, let the cards fall where they may.

My critique of Relativity was begun to solve a problem – that of the Pioneer