 
## Reflections on the Michelson & Morley Experiment and the Ineluctable Self-Interview

By Jim Spinosa

ISBN 9781370895526

Published by Jim Spinosa at Smashwords

Copyright 2017 All rights reserved

Dedicated to Steven G. Spinosa "The Spinner Cares"

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Table of Contents

Introduction

1. The Michelson-Morley Experiment Probes the Luminiferous Ether

2. A Further Consideration of Certain Aspects of the Michelson-Morley Experiment

3. The Re-conceptualized Michelson-Morley Experiment Probes the Ether

Conclusion

The Ineluctable Self-Interview (Part 1)

Section A. Mold in the Refrigerator

Section B. Watching the Moonrise from the Mindanao Deep

Section C. Pink Sky and Green Stars

Section D. The Cave of the Day

The Ineluctable Self-Interview (Part 2)

Section A. The Stars Are a Sieve

Section B. Myrmecophagous Bridgette

Introduction

This reconsideration of the Michelson–Morley experiment demonstrates that the conceptual theories that underpin the experiment are flawed. Because the conceptual theories are flawed, the mathematics that accompanies the experiment is inappropriate. The conceptual theories have flaws that are serious enough to nullify both the results of the experiment and its conclusions. This reconsideration of the experiment will take into consideration the following important point: There were at least two definitions of the term ether when Albert Michelson and Edward Morley first performed their experiment. Many scientists considered ether a hypothetical, invisible substance that pervaded all of space and served as a medium for the transmission of light waves. That definition of the ether is used in this reconsideration of the experiment in chapters one and two of this e-book. Other scientists considered the ether to be a continuous expanse of empty space, extending in all directions, and chapter three of this e-book examines the experiment employing that definition. The conclusion of the e-book summarizes the experiment's shortcomings when each of the definitions are used.

1. The Michelson–Morley Experiment Probes the Luminiferous Ether

The Medium for the Propagation of Light Waves

First performed in 1887, the Michelson–Morley experiment used a complex apparatus to probe the nature of the luminiferous ether. The luminiferous ether is often called the aether or simply the ether. Ether wind is a misleading term because the prevailing opinion among scientists in the late 19th century was that the luminiferous ether was stationary under most conditions. These scientists thought the ether wind effect was caused by the motion of the earth through the stationary ether. It is similar to the experience produced by riding a bicycle on a windless day. As the bicycle rider pedals along, he feels as though a wind is blowing on his face. As the speed of the bicyclist increases, the wind increases in its apparent speed.

Another aspect of the Michelson–Morley experiment was that from its results many scientists believed that the absolute velocity of the earth would be determined. To accomplish the task of determining the absolute velocity of the earth, two precise time measurements needed to be made. They are the measurement of the duration of a light beam's round–trip journey along one arm of the apparatus and the duration of the light beam's round–trip journey along the other arm of the apparatus. These measurements had to be made when one arm of the apparatus was aligned with the motion of the earth and the other arm was perpendicular to the apparent ether wind that is to say perpendicular to the motion of the earth.

The centerpiece of the Michelson–Morley experiment was a device is called an interferometer; it is a device that divides a light ray into two beams and then brings them together again to cause interference. The recombination of these two beams of light produces interference fringes (bands of more intense color). When these fringes are counted, they give information about the light.

The main part of the apparatus consists of two identical arms fitted together to form a right angle. When one arm is aligned with the motion of the earth, the other arm will be perpendicular to it. Each arm is of equal length. At the arms' vertex a single ray of light is split into two beams by a half–silvered mirror. One beam travels along one arm, and the other beam travels along the other arm. A fully–silvered mirror is at the far end of each arm. Each half of the split light ray strikes its respective mirror and returns along its respective arm, and the two beams rejoin producing interference fringes.

If the earth was moving through ether that was stationary, it was mathematically determined that one light beam would take longer than the other to complete its round–trip journey. This conclusion was decisively drawn although the arms were of equal length. The round–trip journey of the light beam traveling in the arm aligned with the motion of the earth would take longer to complete than the round–trip journey of the light beam traveling in the arm that was perpendicular to the motion of the earth. If the ether surrounding the earth was partially dragged along by the motion of the earth, the overall outcome would still be the same. The only change would be in the size of the difference between the round–trip duration for one arm as compared to the other arm. Albert Michelson and Edward Morley believed that each of these possible outcomes could be detected and distinguished by their interferometer. Each outcome would produce a distinct set of slight changes in the interference fringes of the recombining light beams.

The entire apparatus was mounted on a granite slab. Floating in a basin filled with mercury, the granite slab could be rotated. It was hypothesized that the rotation of the apparatus would produce slight changes in the interference fringes. They would occur as the arm of the interferometer perpendicular to the motion of the earth was rotated until it exchanged position with the arm of the interferometer aligned with the motion of the earth.

Early in chapter 2 of Lillian R. Lieber's book, The Einstein Theory of Relativity, the formula t1 is introduced. It is the formula for the time it takes the light beam to make a round–trip journey in the arm of the apparatus aligned with the motion of the earth. Lillian R. Lieber writes, "Therefore the time required to travel from A to B would be a/(c–v), where a represents the distance AB, and the time required for the trip from B to A would be a/(c+ v). Consequently, the time for the round trip would be t1 = a/(c–v) +a/(c+ v) or t1 =2ac/ (c2–v2)." This short citation is from Lillian R. Lieber's book _The Einstein Theory of Relativity: A Trip to the Fourth Dimension_ (New York: Rinehart & Company, Inc., 1936, 1945) page 9. The common denominator for the denominators (c–v) and (c+ v) is obtained by multiplying (c–v) (c+ v) which gives us (c2–v2). The numerator 2ac comes from simplifying the numerator a(c+ v) +a(c–v), which gives us ac +ac or 2ac. The letter c is the velocity of light, and the letter v is the velocity of the apparent ether wind, which has the same magnitude as the earth's velocity if the ether is stationary.

The formula for the time, t2, it takes the light beam to make a round–trip journey in the arm of the apparatus that is perpendicular to the motion of the earth, is introduced by the author two pages later. Lillian R. Lieber writes, "So the time for the round trip from A to C and back to A, would be

t2 = 2a/ (c2–v2)½." This short citation is from page 11 of Lillian R. Lieber's book _The Einstein Theory of Relativity._

a = the length of the light path or perhaps it would be more accurate to say the length of some sort of hypothetical light path, which is the same in both arms of the apparatus.

c = the speed of light.

v = the velocity of the apparent ether wind.

When the formulas, t1 and t2, are mathematically analyzed, it is revealed that t1 is greater than t2. Therefore, it takes more time for the light beam in the arm of the apparatus aligned with the motion of the earth to make its round–trip journey than it does for the light beam in the arm of the apparatus that is perpendicular to the motion of the earth. Let's mathematically analyze the formulas t1 and t2. First, we will find a common denominator for the two fractions that appear in the formulas. The common denominator for the two fractions is (c2–v2). This gives us the following: t1 =2ac/ (c2–v2) and t2 = 2a (c2–v2)½/ (c2–v2). Since the denominators are now the same, we can ignore them and concern ourselves with the numerators only. For instance, when comparing the fractions, 3/5 and 4/5, we can ignore the denominators and compare the numerators only. Since 4 is larger than 3, we can determine 4/5 is larger than 3/5. The numerator of t1, is 2ac, and the numerator of t2 is 2a (c2–v2)½. The product 2a is the same in both numerators. We can further determine that if v is greater than zero, the value c is larger than the square root of (c2–v2), which is another way of expressing the term (c2–v2) raised to the ½ power or (c2–v2)½. Thus, t1 is greater than t2. To make this analysis two crucial requirements must be met. The first requirement is that a represents the same length in both equations. The second requirement, is that both equations contain the term (c2–v2), which can be raised to any particular power.

These requirements cause difficulties when analyzing the behavior of the light beam in the arm of the apparatus perpendicular to the motion of the earth that is to say perpendicular to the apparent ether wind. In this arm, the apparent ether wind should sweep the light beam downwind. That would make its light path longer than light path a. This is analogous to a swimmer who tries to swim across a river. To make his journey as short as possible, the swimmer will try to follow a line that runs perpendicularly to the shoreline and intersects his starting point. Nevertheless, inevitably, the current will sweep the swimmer downstream, which increases the distance of his journey. He actually ends up swimming along the hypotenuse of a right triangle, instead of one of its legs.

This difficulty is overcome in the formula t2 by using the term (c2–v2)½. This insures that the light path in formula t2 equals the light path in the formula t1. The term (c2–v2)½ gives us the speed of the light beam. We construct a right triangle in which the speed of light is the hypotenuse of the right triangle, and we denote it by the letter c. It represents the speed of a light beam that is not being influenced by the apparent ether wind. The letter v represents a leg of the right triangle and the speed of the earth. The speed of the earth and the speed of the apparent ether wind are the same; their directions are opposite though. The length of the other leg of the right triangle is obtained using the Pythagorean Theorem. It is represented by the term (c2–v2)½, and it gives us the speed of light under the influence of the apparent ether wind. The equation t2=2a/ (c2–v2)½ is essentially a restatement of the more general equation time = distance/speed. However, the use of the equation t2 =2a/ (c2–v2)½ introduces its own difficulties.

Before we continue any further, an excerpt from pages 11 and 12 of Lillian Lieber's book will show the confidence she has in the swimmer analogy. She states, "But what has all this [the swimmer analogy] to do with the Michelson–Morley experiment? In that experiment, a ray of light was sent from A to B: At B there was a mirror which reflected the light back to A, so that the ray of light makes the round trip from A to B and back, just as the swimmer did in the problem described above. Now, since the entire apparatus shares the motion of the earth, which is moving through space, supposedly through stationary ether, thus creating an ether wind in the opposite direction, this experiment seems entirely analogous to the problem of the swimmer." This short citation is from Lillian R. Lieber's book _The Einstein Theory of Relativity._

A further examination of the analogy between the light beam and a swimmer will reveal the difficulties with this explanation. The light path a is represented by a perpendicular line running across the river. The line begins at the swimmer's starting point. If the swimmer knows three facts, he can calculate the precise angle by which he must deviate from the perpendicular in an upstream direction to counteract the effects of the current. These three facts are the following: the velocity of the current, the speed at which he swims and the trigonometric equations known as the Law of Sines. The swimmer must also know that the current flows perpendicularly to a line measuring the width of the river. The swimmer swims upstream at an angle expressly chosen so that, when the current sweeps him downstream, his path forms a line that is perpendicular to the flow of the river and intersects his starting point.

The following is an example of the calculation a swimmer could make, and it should be noted that the velocity of the current is crucial to the calculation. Let us say there is a river that flows east to west. A swimmer on the river's south bank wants to swim to the north bank along a perpendicular line. The river is 20 miles wide, and we will imagine that the width of the river is side a of a right triangle whose base is on the opposite side of the river. The current flows at a constant rate of 3 mph, and the swimmer swims at a constant speed of 5 mph. The lengths of both side b, the base, and side c, the hypotenuse, of the triangle are unknown. It is known that the ratio b/c = 3 mph/5 mph. The ratio b/c equals this ratio: the speed of the current/the speed of the swimmer. This is true because of the principle of similar triangles. Solving the ratio for b gives: b = c (3/5) = (.6) c.

The Law of Sines is the following: a/Sine A = b/Sine B = c/Sine C. The angles A, B, and C represent the angles directly opposite the respective sides a, b, and c. We know angle C is 90o because the current flows perpendicularly to a line measuring the width of the river. The sine of 90o equals one. If we substitute (.6) c for b and 20 miles for a and one for sine C, we have: 20miles/sine A = (.6) c/sine B = c/1. Solving the equation for sine B we have sine B = .6. The solution is reached by cross multiplying the fractions (.6) c/sine B=c/1, which gives us (.6c) (1) =sine B (c). Next we divide each side of the equation by (c), and we generate .6=sine B. Therefore, angle B equals approximately 36.87o because the sine of 36.87o is approximately .6. A swimmer must swim at an upstream angle of approximately 36.87o from the perpendicular in order for the current to carry him back to the perpendicular.

A swimmer can swim at an angle that precisely deviates from the perpendicular. Swimming at this angle allows the current to carry him back to the perpendicular. The phrase, "carry him back to the perpendicular" is confusing the reader may think that the current is carrying the swimmer forward to the perpendicular. If we define forward as moving in the direction of the flow of the river then the current is carrying the swimmer forward to the perpendicular. A light beam can also precisely deviate from the perpendicular light path a. This deviation will allow the apparent ether wind to carry the light beam back to the perpendicular light path a. The term (c2–v2)½ is the proper term to use when the light beam is being carried back to the perpendicular by the apparent ether wind. Once the light beam is carried back to the perpendicular, it is mathematically changed from the hypotenuse of a right triangle into the leg of a right triangle. The Pythagorean Theorem determines that the length of the leg a of a right triangle is: a = (c2–b2)½.

The difficulty arises in determining the precise angle the light beam should deviate from the perpendicular. This angle must allow the apparent ether wind to carry it back to the perpendicular. To calculate that angle, the velocity of the apparent ether wind must already be known, but the velocity of the apparent ether wind is unknown. Unless the precise angle of deviation is already known, the formula t2 cannot be employed. It seems clear that we do not know the angle that the light beam (in the arm of the interferometer perpendicular to the motion of the earth) must deviate from the perpendicular in order for the apparent ether wind to blow it back to the perpendicular. Therefore, the equation t2=2a/ (c2–v2)½ is invalid.

However, there is a strategy that might make the equation t2=2a/ (c2–v2)½ valid. That strategy utilizes the rotation of the interferometer. This strategy will be discussed shortly, and its shortcomings will be pointed out.

Instead, a formula similar to the equation t2 must be used. We will call the new equation t2 prime, which we can denote as t2. The new equation is the following: t2=2b/ (c2+v2)½, and it must be used to describe the path of the light beam when it travels perpendicularly to the apparent ether wind. Its use does not require that the light beam travels along a known angle of deviation from the perpendicular.

b = the distance the light beam travels when swept downwind by the apparent ether wind–it is a longer distance than that of light path a.

c = the speed of light.

v = the velocity of the apparent ether wind.

A strategy can be employed that in order to use the formula t2 = 2a/ (c2–v2)½ it is not required that the precise angle of deviation must be known. The strategy is that since the interferometer rotates, it will pass through the precise angle of deviation necessary for formula t2 to be correctly employed. It is true that the interferometer will pass through the precise angle of deviation. It is analogous to a swimmer who swims across a river many times and with each trial he chooses a different angle as measured from the perpendicular. By trial and error he will discover the precise angle that allows the current to carry him back to the perpendicular.

However, when arm A of the rotating interferometer comes to the precise angle of deviation that allows the apparent ether wind to blow the beam of light (traveling along its arm) so that it is perpendicular to the apparent ether wind, the other arm, arm B, of the interferometer will not be perpendicular to that beam of light. The apparent ether wind does blow the beam of light in arm A so that it advances to its correct position. The _correct position_ for the light beam in arm _A_ is perpendicular to the apparent ether wind. The apparent ether wind does not blow the beam of light in arm B so that it advances to its correct position. The _correct position_ for the light beam in arm _B_ is aligned with the apparent ether wind. Therefore, the beam of light in arm B lags behind from its correct position.

Arm B of the interferometer is not perpendicular to the beam of light in arm A because the beam of light in arm A has been blown downwind by the apparent ether wind. It has been blown until it is perpendicular to the apparent ether wind while the beam of light in arm B of the interferometer has lagged behind, and so it will not be aligned with the motion of the earth. Therefore, the formula for the beam of light in arm B, t1 = 2ac/ (c2–v) cannot be correctly employed. Arm B of the interferometer is not perpendicular to the beam of light blown downwind by the apparent ether wind in arm A. Arm B of the interferometer is still perpendicular to arm A because the arms themselves are not affected by the apparent ether wind, only the light beams traveling along the arms are affected by the apparent ether wind.

The light beam traveling along arm A of the interferometer has been blown downwind by the apparent ether wind. Therefore, the light beam is no longer parallel to arm A. Since it is no longer parallel to arm A, it cannot be perpendicular to arm B because the arms are at right angles to one another. Since the light beam traveling along arm A is not perpendicular to arm B, it is not perpendicular to the light beam traveling along arm B.

It should be noted that the effect that the apparent ether wind has on the light beam traveling along arm B is very slight, but it actually increases the amount by which the beam of light in arm B lags behind from its correct position.

Though the rotating interferometer passes through the precise angle of deviation required for formula t2 to be correctly employed, it does so in a way that makes it incorrect to employ formula t1.

There is another problem with the argument. The mirror at the far end of arm A of the interferometer would need to be tilted at a precise angle so that once the light beam has struck the mirror, the returning light beam will be traveling at the proper angle for it to be blown back to the perpendicular by the apparent ether wind. The precise angle the mirror must be tilted to fulfill this requirement is unknown. Therefore, for all the reasons listed above, t2 is the incorrect equation to employ, and a formula similar to t2 is the correct equation to employ.

The equation t2' =2b/ (c2 +v2)½ as with the other equations already mentioned, is an expression of the general equation: time = distance/speed. The distance is 2b because the light beam makes a round–trip journey, and it is assumed that each leg of the journey has a length which equals b. (We will soon see this assumption is incorrect.) The speed of the light beam is (c2+v2)½. The speed is determined by the addition of vectors.

A vector is a mathematical expression denoting a combination of both magnitude and direction. In these equations, the vectors used express the speed and the direction of one of the three following quantities: a light beam, the apparent ether wind, or a light beam under the influence of the apparent ether wind. Assuming the earth does not drag or even partially drag the ether along with it, then the vector for the velocity of the apparent ether wind will have the same speed as the vector for the absolute velocity of the earth. However, the direction for the vector of the absolute velocity of the earth is the reverse of the direction of the apparent ether wind's vector. When the vectors, which represent the apparent ether wind and the velocity of a beam of light, are added they form a third vector. It results from the combination of the speeds and directions of these two vectors.

The following is an example of vector addition. Let us say there is a river that flows east to west at a rate of 3 mph. A swimmer is positioned on the river's south bank who is ready to swim to the north bank, at a speed of 4 mph. The river's current can be represented by an arrow 3 units long, pointing toward the west, and beginning at the swimmer's starting point. The swimmer can be represented by an arrow 4 units long, pointing toward the north. This arrow also begins at the swimmer's starting point. These two vectors form two sides of a parallelogram. The other two sides of the parallelogram can be constructed because they mirror the two sides already formed by these two vectors. Once we construct the parallelogram, we can draw the third vector. It represents the addition of the current's vector and the swimmer's vector. We begin the third vector at the swimmer's starting point. We end the third vector at the diagonally opposite corner of the parallelogram. The length of the third vector represents the speed of the swimmer under the influence of the river's current. The position of the third vector represents the direction of the swimmer under the influence of the river's current.

In this example, the third vector is the hypotenuse of a right triangle. One leg of the triangle is three units, and the other leg is four units. Using the Pythagorean Theorem we find the length of the hypotenuse/vector is (32+42)½ or 5 units. The length of the vector equals the speed of the swimmer under the influence of the river's current. It is 5 mph.

The same reasoning is used to learn the speed of the light beam under the influence of the apparent ether wind. The speed of the light beam is (c2 +v2)½ when the light beam begins its journey perpendicular to the apparent ether wind. Likewise, the speed of the light beam under the influence of the apparent ether wind is (c2–v2)½ when the light beam begins its journey at that one specific upwind angle from a line drawn perpendicularly to the apparent ether wind that we have designated as the known angle of deviation from the perpendicular.

For the reasons given before, we are compelled to use the term (c2+v2)½. When that term is used, deciding mathematically which of the two possible round–trip journeys will have the greatest duration is impossible. We cannot mathematically determine which will take longer: the journey of the light beam in the arm of the interferometer perpendicular to the motion of the apparent ether wind or the journey of the light beam in the arm of the interferometer aligned with the motion of the apparent ether wind.

It cannot be mathematically determined whether t1 = 2ac/ (c2 –v2)½ is greater than, less than, or equal to t2=2b/ (c2+v2)½. Since this is the case, the experimental result that the light beams in each arm of the interferometer take the same amount of time to complete their round–trip journeys is not in conflict with this new mathematical analysis. The original mathematical analysis of equations t1 and t2 conflicted with the empirical results of the Michelson–Morley experiment. Part of the original mathematical analysis was based on the legitimate assumption that the apparent ether wind had to have a velocity greater than zero. The most palatable way to resolve the conflict using the original equations was to assume that the velocity of the apparent ether wind was zero.

If the velocity of the apparent ether wind is zero, it must mean that the earth totally drags along the surrounding ether with it as it moves. What is true of the earth would be true for all matter. In this scenario, the ether surrounding the earth is the same as the air inside your car with the windows shut. It does not matter how fast your car is traveling; the air inside your car has a velocity of zero when measured from within your car. An anemometer inside your car will measure the air speed as zero. A weather vane inside your car will not detect a wind blowing from any direction.

The conclusion that the earth totally dragged along the surrounding ether was in direct conflict with the results of many other experiments. They all showed that the earth only partially dragged along the surrounding ether. However, the conclusion of the Michelson–Morley experiment that the earth must completely drag the ether along with it was drawn from an invalid mathematical analysis.

2. A Further Consideration of Certain Aspects of the Michelson–Morley Experiment

Since we have proven that the mathematical underpinnings of the Michelson–Morley experiment are faulty, we can now turn our attention to three other items. First, we must modify our notion of the interferometer's ability to measure the absolute velocity of the earth. Second, we must modify the term b in the equation t2. Third, we must examine the light path of the beam of light in the arm of the interferometer aligned with the motion of the earth.

Turning our attention to the first item, let's examine the following statement: Assuming the earth does not completely drag or even partially drag the ether along with it, then the vector for the velocity of the apparent ether wind as measured by the interferometer will have the same speed as the vector for the absolute velocity of the earth. However, the direction for the vector of the absolute velocity of the earth is the reverse of the direction of the apparent ether wind's vector.

This notion needs to be modified. A discussion of the properties of weather vanes and anemometers will explain certain aspects of the interferometer. A weather vane shows the direction of the wind, and an anemometer measures the speed of the wind. What happens if a powerful fan is positioned directly above or below a weather vane? When the fan is turned on, will the weather vane show the direction of this powerful, yet localized wind? No, a weather vane can only rotate around its vertical axis. It can only show the direction of winds that blow horizontally relative to the weather vane's vertical axis. The wind must blow horizontally relative to the weather vane's vertical axis or, at least, have a horizontal component. Only under these circumstances can the weather vane show either the wind's direction or the direction of its horizontal component. Like a weather vane the anemometer can only rotate around its vertical axis. It cannot measure the speed of a wind blowing from directly above it or from directly below it.

The interferometer, as well, can only rotate around its vertical axis. By comparing the effect of the apparent ether wind on a light beam's transit time in the arm of the interferometer aligned with the apparent ether wind to that of a light beam in the arm perpendicular to the apparent ether wind, the interferometer was supposed to produce a measurement of the apparent ether wind's velocity. If the apparent ether wind were to blow from directly above or directly below the interferometer, the speed and the direction of the apparent ether wind would not be registered by the interferometer. Or, if some kind of reading was produced, it would not be an accurate reading. The interferometer only measures the apparent ether wind's velocity if it is moving horizontally relative to the interferometer's vertical axis. Otherwise, the interferometer only measures the component of the apparent ether wind's velocity that is moving horizontally relative to the interferometer's vertical axis. It is entirely possible that the interferometer only detects a component of the earth's absolute velocity when it measures the velocity of the apparent ether wind.

Now, we can turn our attention to the second item, which is the term b in the equation t2. The formula t2'=2b/ (c2+v2)½ can be modified too more accurately represent the path of the light beam in the arm of the interferometer perpendicular to the apparent ether wind. When an arm of the interferometer is perpendicular to the apparent ether wind, the light beam in that arm begins its journey traveling perpendicularly to the apparent ether wind, and it is blown downwind by the apparent ether wind. At this time we are not concerned with the situation where the arm of the interferometer is angled upwind and the apparent ether wind blows the light beam to the perpendicular. When the light beam strikes the mirror at the end of the interferometer's arm, it is no longer perpendicular to the mirror. As with any light beam that strikes a mirror, when this light beam strikes the mirror, its angle of incidence will be equal to its angle of reflection. When the light beam begins its return journey, it is not traveling perpendicularly to the apparent ether wind; it is already traveling at a downwind angle before it is influenced by the apparent ether wind. Under the influence of the apparent ether wind, the light beam will be blown even farther downwind. The light beam's return journey does not measure length b because it travels a distance whose measurement is longer than length b. We can call this measurement length f. The time it takes the light beam to travel length f is given by the formula:

tf = f/ (c2 +v2 –2cv cos. F)½.

c = the velocity of light.

v = the velocity of the apparent ether wind.

Cos. F = the cosine of the angle directly opposite vector f. Vector f is the longest leg of an oblique triangle. The second longest leg is d, and it is equal in length to b. Leg d represents the path the returning light beam would have taken if it were not affected by the apparent ether wind during its return journey. Leg d represents the reflection of light path b according to the rule that the angle of incidence equals the angle of reflection. The shortest leg of this oblique triangle is e. It represents a measurement of the total distance that the light path f is blown in a downwind direction by the apparent ether wind. (The length of leg e does not include the distance that light path f is directed downwind by the optical principle that the angle of incidence is equal to the angle of reflection.)

tf = f/ (c2 +v2 –2cv cos. F)½.

In the equation for tf, listed above, the Law of Cosines is used to add the vectors c and v. The Pythagorean Theorem is not used because the vectors no longer form a right angle. The light beam is no longer perpendicular to the apparent ether wind when it begins its return journey.

A further difficulty is encountered. The light beam's return journey down the arm of the interferometer will not bring it back to the exact point from which it began to travel up the arm of the interferometer. It will return to a point downwind from its starting point. How does the light beam return from this downwind position? The light beam must travel back from its downwind position to recombine and interfere with the light beam returning from the other arm of the interferometer.

Even if a river had only a modest current, a swimmer would face the same type of situation swimming across the river and back. Once the swimmer returned to the side from which he started, he would need to swim upstream to reach his starting point. Swimming upstream would be slow going for the swimmer because he would have to contend with the full force of the current. The swimmer can consciously decide to swim upstream; however, that is not possible for a beam of light.

Once the light beam has somehow traveled the upwind distance necessary to reach its starting point, it must make a 90o turn so that it can recombine and interfere with the returning beam of light from the other arm.

A more accurate measure (but still incomplete) of the total time for the round–trip journey of a beam of light traveling perpendicular to the apparent ether wind is given by the formula:

t2 = tb \+ tf

Or

t2 = b/ (c2 +v2)½ \+ f/ (c2+v2 –2cv∙cosF)½

c = the velocity of light.

v = the velocity of the apparent ether wind.

Cos. F = the cosine of the angle directly opposite length f.

b and f = light paths of various lengths – f is longer than b and b is longer than a.

The results of the Michelson–Morley experiment require that t1 = t2. Solving this equation for v is difficult. Compared with the original equation for t2 the new equation, t2, is cumbersome and the equation still does not fully account for the light beam's return to its original position.

Most physicists in the 19th century believed that the ether was stationary or only partially dragged by the motion of the earth and that the motion of the earth through the ether produces an apparent ether wind. They should have concluded that under these conditions the Michelson–Morley experiment cannot produce results of any validity. The formula for the time it takes a beam of light to make a round–trip journey in an arm of the interferometer that is perpendicular to the apparent ether wind is incorrect. The modified version of the formula for the time it takes a beam of light to make a round–trip journey under the same conditions is cumbersome and incomplete. Solving the equation for the variable v is difficult. Since the conclusions of the Michelson–Morley experiment are invalid, they offer no empirical evidence that the luminiferous ether is totally dragged along by the earth. No empirical evidence exists to contradict the evidence, garnered from many ingenious experiments that the luminiferous ether is only partially dragged along by the earth. There is no longer any contradictory empirical evidence about the nature of the luminiferous ether. No need exists to introduce the Lorentz and Fitzgerald contraction of matter to explain away a contradiction that no longer exists.

The Lorentz and Fitzgerald contraction of matter explained the empirical result that t1 equals t2. It did so without making it necessary that the earth totally drags along the surrounding ether as it moves. If matter contracts along the axis aligned with the direction of motion, it would explain why t1 equals t2. Though according to the mathematical analysis they should not be equal. The mathematical analysis definitively claimed this: the time, t1, of a round–trip journey of a beam of light traveling first against the apparent ether wind and then returning with the apparent ether wind would be longer than the time, t2, of a round–trip journey of a beam of light traveling perpendicularly to the apparent ether wind.

The contradiction between the mathematical analysis and the empirical evidence could be explained by assuming a contraction of matter along the t1 axis. That is along the axis aligned with the direction of the apparent ether wind. The supposition of Lorentz and Fitzgerald is that the length of the arm of the interferometer aligned with the motion of the earth contracts just enough to make the round–trip times equal in each arm of the interferometer.

Once the original mathematical analysis is shown to be invalid, no need exits to introduce the Lorentz and Fitzgerald contraction of matter along the t1 axis to reconcile the empirical results with the mathematical analysis.

The third item is the path the light beam follows in the arm of the interferometer that is aligned with the motion of the earth. It has been pointed out that the light beam in the arm of the interferometer perpendicular to the motion of the earth does not travel distance a either on its outward journey or on its return journey. The distance a represents the length of the arm of the interferometer. It has also been pointed out how attempts were made to coerce the light beam to travel distance a on both its outward and return journeys. These attempts seemed doomed to failure for several reasons. For instance, referring to the beam of light on its outward journey in the arm of the interferometer perpendicular to the apparent ether wind, we don't know how to precisely adjust the half-silvered, beam splitting mirror to allow the light beam to be blown to the perpendicular by the apparent ether wind. This is because we don't know the velocity of the apparent ether wind. Tilting the half-silvered, beam splitting mirror would alter the light path of the outward bound beam of light. If the tilt of the beam splitting mirror altered the path of the outward bound beam of light so that it was perpendicular to the apparent ether wind, then would the tilt of the mirror at the far end of the interferometer need to be precisely adjusted to alter the light path of the returning light beam so that it too was blown to the perpendicular by the apparent ether wind? Tilting a mirror at the far end of one of the arms of the interferometer seems to be disallowed by the need for both mirrors at the far ends of the arms of the interferometer to have the same tilt, i.e., perpendicular to the vertical axis of the arm of the interferometer. If the beam splitting mirror were tilted so little (or in such a manner) that there was no need to tilt the mirror at the far end of the arm of the interferometer in order for the light beam on its return journey to be blown to the perpendicular by the apparent ether wind, then the light beam on its outward journey would not have the correct tilt to be blown to the perpendicular by the apparent ether wind. The Wikipedia article on the Michelson—Morley experiment seems to suggest that tilting the beam splitting mirror can cause the two beams of the split light ray that would otherwise be apart to come together and form interference patterns. The notion that tilting the beam splitting mirror can bring the light beams together is examined in the answer to the question about the Michelson—Morley experiment posed in the self-interview portion of this e-book.

We now must return to the third item, and state that the beam of light in the arm of the interferometer that is aligned with the motion of the earth does not travel distance a, the length of the arm of the interferometer on either its outward or its return journey. Yet, this fact is correctly compensated for in the equation for t1 so that it is as though the light beam did in fact travel distance a on both its outward and its return journeys. We may think of it as though on the light beams outward journey, the mirror at the far end of the interferometer is rushing away from the beam of light with the velocity v so that the effective speed of the light beam is (c-v). During the light beams return journey, the half-silvered, beam splitting mirror is rushing toward the beam of light with the velocity v so the effective velocity of the light beam is (c+ v). The reason for this is that unlike everything else in our everyday experience, the beam of light is not carried along by the motion of the earth. The apparent ether wind does not alter the direction of the light beam in this case. In this case the apparent ether wind is the similar to the velocity of the earth except their directions are opposite. It is the fact that the apparent ether wind alters the direction as well as the speed of the light beam that is traveling along the arm of the interferometer perpendicular to the motion of the earth that causes much of the dilemma.

3. The Re-conceptualized Michelson–Morley Experiment Probes the Ether

Frame of Reference at Absolute Rest

Another interpretation of the ether is possible. In this interpretation there is no apparent ether wind produced by the motion of the earth through the stationary ether. Since no apparent ether wind exits, no mechanism is present to blow a beam of light that is perpendicular to the motion of the earth downwind. In this scenario the ether is empty space and not a medium that is necessary for the propagation of light beams. In this empty space an observer at absolute rest can be present. The notion that the ether was empty space coexisted with the notion of the ether as a medium for the propagation of light. The notion of the ether as empty space was known to Albert Michelson and he was not averse to it.

Lee Smolin in his book The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next, which was published in 2006 by Houghton Mifflin of Boston and New York, comments on the attempts to find the absolute velocity of the earth. On page 37, he states, "Several attempts had been made to detect Earth's motion through the aether before 1905, when Einstein proposed special relativity, and they had failed. Proponents of the aether theory had just adjusted their predictions so as to make it harder and harder to detect Earth's motion." It seems odd that the "proponents of the aether theory" did not expose the flaws with the Michelson–Morley experiment. Perhaps, though, some kind of realization that the apparent ether wind version of the Michelson–Morley experiment was invalid did exist.

Lillian R. Lieber in her book The Einstein Theory of Relativity, published in 1945, does not suggest that the apparent ether wind interpretation of the experiment is invalid. At least, explicitly she makes no such suggestion, but her complete dependence on the analogy between the behavior of the light beams and the behavior of a swimmer is problematic especially since her explanation is strained at points. However, by the time Stanley Goldberg's book, Understanding Relativity: Origin and Impact of a Scientific Revolution, was published in 1984 the apparent ether wind interpretation of the experiment had been abandoned. It was replaced with a version that employed empty space and an observer at absolute rest. Lee Smolin claims that the proponents of the ether theory adjusted their predictions to make the earth's motion harder and harder to detect. However, the proponents of special relativity were not averse to changing the interpretation of the Michelson–Morley experiment in an attempt to keep it from being invalidated.

Appendix 5 of Stanley Goldberg's book is entitled, "Ether Drift Experiments: The Search for the Absolute Frame of Reference," and it contains a six–page description of the Michelson–Morley experiment. Several features in his description of the experiment deserve an examination.

An observer at absolute rest observes the split light ray following two paths. These two paths can be analyzed mathematically to generate the formulas t1 and t2. Stanley Goldberg's formulas for t1 and t2 are mathematically equivalent to the original formulas for t1 and t2 but other differences are present. For example, Stanley Goldberg's formulas are no longer derived through a process that includes the addition of two vectors that represent a light beam and the apparent ether wind respectively. A vector representing the apparent ether wind is not even present in Stanley Goldberg's scenario. Though, a vector representing the absolute velocity of the earth is present.

In Lillian Lieber's description of the experiment, the apparent ether wind could affect both the speed and direction of a light beam. In Stanley Goldberg's description, only the direction of a light beam can be affected, and this influence is apparently caused by the absolute motion of the earth.

Another distinction between Stanley Goldberg's formulas and Lillian Lieber's formulas is that the observer who observes the interference fringes is in a state of absolute rest. No mention is made of the fact that an observer at absolute rest would observe the experimental apparatus passing by at a speed of at least 18 miles/second. Also, the method by which an observer may attain the condition of absolute rest is not mentioned since it does not exist. Though it is most likely that it is not a requirement that the observer be at absolute rest to observe the interference fringes, rather the observer at absolute rest is used to make Stanley Goldberg's explanation clearer.

According to Stanley Goldberg, an observer at absolute rest observes that the light beam, traveling perpendicularly to the motion of the earth, behaves as though the directional component of the earth's velocity, v, has been imparted to it. The speed of the light beam is unchanged, and only the direction of the light beam is changed by an interaction with the earth's velocity. This change in direction allows the light beam to keep up with the lateral motion of the fully–silvered mirror at the end of the interferometer's arm. The light beam has a forward (upstream) direction imparted to it by the velocity of the earth or perhaps, by another unknown force. This forward motion from the earth combines with the light beam's motion, which is perpendicular to the motion of the earth, with the result being the light beam travels diagonally forward, but with no increase in speed. This violates a fundamental property of light observed by the Dutch astronomer W. De Sitter. The velocity of light is not affected by the velocity of the light source. Einstein writes on page 21 of his book, Relativity: The Special and the General Theory, which was republished in New York by Three Rivers Press in 1961. "By means of similar considerations based on the observations of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light." The velocity of the light beam includes both its speed and direction. It is not enough that the speed of the light beam does not depend on the velocity of the body emitting the light. The direction of the light beam cannot depend on the velocity of the body emitting the light, as well.

In Stanley Goldberg's conception, the beam of light perpendicular to the motion of the earth behaves like the well–known example of a ball tossed between two passengers on a moving train as observed by an observer at rest with respect to the earth. The train travels at a constant velocity. The passengers are in an enclosed car so the train's motion through the air cannot influence the tossed ball. Both stand on a straight line perpendicular to the motion of the train. This means that they are tossing the ball across the width of the train car. From the frame of reference of the two passengers the ball travels back and forth between them in a straight line at a constant speed. From the frame of reference of the observer at rest with respect to the earth the ball travels diagonally forward at a constant speed forming a pattern that resembles a saw's teeth. This is possible because the velocity of the train is imparted to the ball and the passengers who toss the ball. This behavior is impossible for a beam of light because the velocity of the light emitting source cannot be imparted to a beam of light. Stanley Goldberg tacitly acknowledges this when he states that the speed of the beam of light remains unchanged. He offers no explanation for the light beam's change in direction.

Actually, the light beam would appear to move at a diagonal angle in a direction opposite to the motion of the earth to an observer at absolute rest. This is similar to the way a light beam behaves because of the influence of stellar aberration. The astronomer who observes stellar aberration is not at absolute rest but merely at rest with respect to the motion of the earth. Unlike Stanley Goldberg's explanation, if we use stellar aberration as the model to decide what the observer at absolute rest observes, then his observations and those of an observer in motion are the same with respect to the light beam.

A thought experiment should make the situation easier to visualize. Let's assume two bicyclists are riding a tandem bicycle. They are traveling with a constant velocity. The bicycle represents the motion of the earth. The rider in the rear position has a large sack of balloons filled with helium. He releases them one after the other at a uniform rate. When he releases them, they travel straight up at a uniform rate; the moving bicycle does not impart any of its motion to the balloons in this thought experiment. The balloons represent a beam of light traveling perpendicularly to the motion of the earth, and the sack of balloons represents the source of the light beam that is traveling with the earth. At some point the bicyclists stop their bicycle and look around in every direction. Thus, they become observers at "absolute rest." When they look backwards, they observe that the balloons have receded from them along a rising diagonal line. With this example in mind, conceiving of how an observer at absolute rest would observe the light beam moving diagonally forward, instead of receding diagonally, is difficult. Calling the receding, diagonal light beam a "composite light beam" would be more accurate since it is not really a light beam.

If the light beam traveling perpendicularly to the earth's motion did not have the forward directional change imparted to it by the velocity of the earth, as Stanley Goldberg suggests, the light beam would strike the mirror, at the end of the interferometer's arm, downstream from dead center–its correct position. If the mirror was traveling with a very rapid velocity, the light beam would completely miss it. Let us say the light beam strikes the mirror, at the end of the interferometer's arm, downstream from its correct position by the length x. While the light beam was returning from the mirror to its point of origin, its point of origin would travel forward by the additional length y. Therefore, the light beam would miss its point of origin__ its correct position__ by a total length of x \+ y in the downstream direction. Thus, the split light beam does not recombine. We have previously encountered this type of problem with the apparent ether wind interpretation.

Even with Stanley Goldberg's assumption that a forward directional change can somehow be imparted to the light beam traveling perpendicularly to the motion of the earth, the light beams still do not recombine. A careful study of Figure 11 elucidates this point. As can be seen from Figure 11, the dotted diagonal line, which represents the light beam traveling perpendicularly to the motion of the earth, follows a distinctly diagonal path. However, it must be traveling perpendicularly to recombine with the other light beam– the light beam aligned with the motion of the earth. Note that the path of the light beam represented by the dotted diagonal line ends before it reaches the observer represented by the eye. If we were to continue the dotted diagonal line, we would observe that it passes to the right (upstream) of the observer represented by the eye. This would represent a clear indication that the split light beams do not recombine; therefore, it is left out of Figure 11.

Stanley Goldberg's interpretation of the experiment can be seen as an effort to overcome the shortcomings of Lillian Lieber's conception of the experiment enumerated in parts 2 and 3 of this e-book.

The long excerpt that follows is from Stanley Goldberg's description of the Michelson–Morley experiment, which includes his accompanying schematic diagram. The description of the paths that must be followed for the recombination of the light beams is important especially since the schematic diagram illustrates that these requirements are not met.

"The Michelson–Morley experiment depends on observing the behavior of fringes when light beams are combined and allowed to interfere. While all such instruments are known as 'interferometers,' that term more and more is reserved for the particular interferometers of the Michelson–Morley experiment. The apparatus is depicted, schematically, in Figure 11. Light is incident from the direction q onto the half–silvered mirror at 0. Part of the light is reflected from 0 to the fully silvered mirror at A and is reflected back to 0, while part of the light incident on 0 is transmitted through 0 to another fully–silvered mirror at C whence it, too, is reflected back to O. Part of the light arriving back from A is transmitted through 0 to the observer while part of the light arriving from C is reflected by 0 to the observer. It is thus possible for the observer (represented by the eye in Figure 11) to observe the fringe pattern resulting from the interference of the two beams of light. Figure 11 depicts the interferometer in three different positions as it moves through space. The dotted lines represent the paths of the two beams of light as seen by an observer at rest with respect to absolute space. We have arranged the apparatus so that the arm OC of the instrument is parallel to the x axis, the presumed direction of the earth's motion through space." This citation is from pages 357 and 358 of Stanley Goldberg's book _Understanding Relativity: Origin and Impact of a Scientific Revolution,_ which was published in Boston by Birkhauser in 1984. Figure 11 is from page 359 of his book.

Conclusion

Neither the apparent ether wind interpretation of the Michelson–Morley experiment nor the observer at absolute rest interpretation proves that the absolute velocity of the earth is impossible to detect. The flaws in the design of the Michelson–Morley experiment undercut the significance of the results, and the flaws in the mathematical interpretations undercut the significance of any conclusions drawn from any possible results. The fact that the experiment detects no change in the interference fringes can only be considered significant if we know the exact nature of the light path in each arm of the interferometer.

In the apparent ether wind interpretation of the experiment the formulas that describe the light paths are incorrect. The apparent ether wind interpretation assumes that the apparent ether wind blows the light beam in one arm of the interferometer in a downwind direction so that it is precisely perpendicular to the motion of the earth. The next assumption, made by the apparent ether wind interpretation, is that the beam of light in the other arm of the interferometer is now aligned with the motion of the earth. This assumption is incorrect. Instead, the proper assumption is that the light beam has not yet reached the position that would align it with the motion of the earth. The light beam lags behind the position that it must assume in order for the formula that describes its behavior to be valid.

Attempts to overcome this dilemma are unsuccessful. For instance, if we assume that a beam of light in one arm of the interferometer is aligned with the motion of the earth, the beam in the other arm will be blown downwind from the perpendicular by the apparent ether wind. A formula designed to describe the light path of such a light beam will be cumbersome and incomplete. Most important, it will not yield any decisive result when it is compared with the formula for a beam of light aligned with the motion of the earth.

In the observer at absolute rest interpretation of the experiment, the light beam perpendicular to the motion of the earth is compelled to behave in a way that violates the empirically observed behavior of light. The velocity of a light source is not imparted to a light beam emanating from it. Velocity includes both speed and direction.

If Stanley Goldberg's scenario for the behavior of a light beam was correct, a thin beam of light could strike a distant target provided the light source and the distant target were in alignment and moving with the same velocity. Let us say a light source and a distant target are both moving with the same velocity of 20 miles per second. They are separated by 186,000 miles as measured along a line that has the target and light source as its respective endpoints. The line measuring the distance is perpendicular to the axes of motion of both the light source and target. If the light source were to emit a microsecond burst of a thin beam of light along the perpendicular line that leads to the target, the light beam would strike the target. It would somehow hit the target though the target had moved 20 miles downstream from the endpoint of the perpendicular line along which the thin beam of light has traveled.

Both the apparent ether wind interpretation and the observer at absolute rest interpretation of the Michelson–Morley experiment are invalid. For an interpretation of the experiment to be valid, the paths of the two beams of light must be accurately described in a way that does not violate the laws of physics. Also, the manner in which the two beams of light recombine to form interference fringes must be described in a detailed manner.

The Ineluctable Self–Interview (Part 1)

Section A. Mold in the Refrigerator

1. Would you like to relate your idea of an extreme review of your e-book?

In Nuts & Bolts: Taking Apart Special Relativity the author claims to show that Einstein's special relativity theory is incorrect. Next, he'll be telling us the following prevarications: that miniature pieces of driftwood form inside decaying white birch trees, which have fallen to the forest floor, that twenty to forty slugs will crawl into an empty beer can and then are lethally poisoned by the last sip of beer that is left inside the beer can by the litterbugs who discard them along our roadsides, that moles can get fatally trapped inside glass beer bottles that have sunk into the ground, that wind can slow the flow of a mountain stream so that it flows with a pulsing rhythm, that groundhogs sometimes climb up saplings, which bend under their weight, that pine cones open and close long after they have fallen off the tree, that the negative terminal of a car battery is actually the positive terminal and vice versa, that a car's starter motor never actually breaks—it merely becomes short circuited by very small debris that enters through the vents (this only applies to starter motors that have vents), that very large lightning bolts shoot up from the earth and small segments of them start to fall back to earth just before they disappear, that the formations of black ice, which are embedded into the ice of lakes and ponds and make designs that look like an abstract spider, are caused by rainwater finding its way through the solid ice and not by the warmer water that lies under the ice, that when it turns bitter cold there's a rusty orange, mushy ice, which forms on the bottom of some streams, and if you scoop it out of the water, it turns bright white, that because of all the detergents found in suburban streams a new kind of brittle ice can form, which uses detergent bubbles as a template, that when a film of detergent lies on the surface of a roiling stream you can see the wildly dancing shadow of the line where the edge of the film meets the water though you can't see the line itself, that in the suburbs the trees are too close together because they're planted when they're small and never thinned out. Doesn't the author know there are examples of relativity in everyday life? Take a tape measure for instance: as they get old, the flange at the free end gets loose and slides backward so that an older tape measure will measure the same object as shorter than a new tape measure.

2. What associations have been helpful in advancing your literary career?

Without a doubt, the SFABC, the Science Fiction Association of Bergen County, which is helmed by Philip DeParto, has been truly advantageous to my writing endeavors. I have been able to overlook the fact that the group's acronym omits an o for the word of with ease because of the fellowship provided by the group.

3. Who are your favorite authors?

Jorge Luis Borges is one of my favorite authors. I also like Michael Piazza, Charles Garofolo and Jonathan Wood. I still find it distressing that authors who write about the sciences and mathematics have seemingly failed to write skeptical books on the following topics: the Michelson-Morley experiment, Bell's inequality, stellar aberration, Gödel's incompleteness theorems and Einstein's theories of relativity.

4. What motivated you to become an indie author?

Sadly, the nature of my e-book, Nuts & Bolts: Taking Apart Special Relativity, is such that it will never be published by a traditional publishing house. I have found this out through the experience of having my e-book rejected by traditional publishers as well as by fringe publishers. Also, I have been told this by the resident expert at Elderberry Press. The notion that the most significant dilemma of our time is hidden knowledge seems to have gone out of fashion around the time of the Garden of Eden. Today, members of society generally believe something similar to what the wise Okies believed in The Grapes of Wrath. They believe that problems arise when deep emotional bonds are broken such as the bonds of family, the bonds of traditional culture or a spiritual bond to the land, i.e., nature.

5. How has Smashwords contributed to your success?

It is difficult to say. I don't know why people are downloading my free e-book. I've had only one radio interview, and that was on the show Night People on WFMU. Soon after that interview, Night People was demoted to a pod cast. I've had a few press releases published, but that was several years ago. My book promoting strategy was to write letters to anyone whom I thought might be interested in the e-book, provided I could obtain their mailing address. These were mostly people I read about in newspapers and magazines. I have a revised version of this e-book and another e-book, Bell's Inequality Untwisted, available for free at lulu.com. I have been using Pinterest to promote these e-books, and downloads of my books that I have garnered are much less than downloads I have garnered at Smashwords. Perhaps, downloads of my e-book are taking place because of word of mouth.

6. What is the greatest joy of writing for you?

I have the idea that some force is trying to keep certain questions from being asked. I see that force at work in the movies. Why are only certain types of movies made? Some say that it's the audience that demands only certain types of movies. But, then the question becomes after years and years why doesn't the audience develop a yearning for something different. I see that force at work in popular music. There was a time when Art Rock was popular. There were groups such as Yes, King Crimson and Henry Cow. These groups were difficult for many to understand, but afterwards why did popular music decline so rapidly? Strangely, I think it might have something to do with humor. If jokes such as the following: what's huge and purple and lives at the bottom of the sea? to which the answer is Moby Grape, typified the humor of an era as opposed to an era where Wayne and Garth make quips about whom "blew chunks" then the former era may have a chance for growth that the latter era lacks.

7. What do your fans mean to you?

I have two of them on right now. I've always believed in an attic fan that you could leave on for hours and hours on very hot days. Although, when the hot, summer sun beats down on the roof all day long, it's questionable how much cooling an attic fan can provide. Unfortunately, the fans inside air conditioners seem to develop mold and mildew. I once made a homemade filter for my air conditioner, and it got so fouled that it let only a small amount of air through. During a very severe heat wave, this clogged filter may have been the main cause of my air conditioner's failure. Strangely, the heat wave was so prolonged that for the first time in my life I became very ill from its effects. I was not able to sleep at night for many days and did not feel comfortable during the day. I developed a painful skin rash so my fans are very important to me.

8. What are you working on next?

I am working on a long essay in which I analyze Kurt Gödel's first incompleteness theorem, step by step and formula by formula. I reach the conclusion that Gödel's first incompleteness theorem is invalid. Douglas Hofstadter's award winning book, Gödel, Escher, Bach is a good example of a force that keeps certain questions from being asked. Because of its extraordinary length and complexity, apparently, no one has noticed that at the critical moment in his explanation of Gödel's first incompleteness theorem he employs a clever deception. He lets the variable a' represent two distinct equations. If we carefully inspect the use Douglas Hofstadter makes of these two distinct equations, we will see that the conclusions he draws, through his use of the equations, are invalid. And, thus, his entire proof of Gödel's first incompleteness theorem is invalid.

9. What inspires you to get out of bed each day?

My inspiration comes from the fact that there appears to be limits to the success that can be obtained through social engineering that is based on dogmatic principles, i.e., things can only get so bad.

10. Do you remember the first story you ever read, and the impact it had on you?

No, not actually. But, I do think boring books can serve an important function. For instance, I would nominate Jules Verne's Mysterious Island as a boring book. All the characters are responsible and intelligent so with only a minimum of difficulty they transform their desert island (a wild and uninhabited island as opposed to a dry, barren, sandy region often extremely hot) into a utopia. Now, our society has a great deal of responsibility and intelligence, yet no utopia has been produced. It may be that novels use an overabundance of irresponsibility to disguise the fact that the wisdom and intelligence displayed within the novel's pages is only the appearance of wisdom and intelligence. If we mentally correct all the irresponsible behavior that occurs in a novel, will those corrections make the wisdom and intelligence displayed within the pages of the novel seem invalid and self-serving?

11. How do you approach cover design?

My covers have a distinctly homemade appearance. I try to break the rules. If the rules say the book's title and the author's name must appear on the cover, I would produce the title and name as say a square block of letters containing six rows and six columns (provided, of course, that the sum of the number of letters in the book's title and author's name was 36) so that the traditional breaks between words would be confounded.

12. When you're not writing, how do you spend your time?

I often collect the trash that accumulates along our roadsides. Then I carry it home and throw it away unless I find a homemade bong, which I keep. A homemade bong is a plastic bottle that has had several holes melted into it probably with a cigarette lighter. One of the holes is for the homemade pipe stem and the other is for carburetion. When I collect five or so of these bongs, I take a picture of them and send it to the local newspapers. I claim to have discovered new and desultory evidence of the frightening effects of global warming and climate change.

13. Do you have any final thoughts about your fans?

Yes, I once had a fan that had a polarized plug, but the outlet I wanted to plug it into was made in the old style and would not accept the one widened prong of the polarized plug. The outlet was in a difficult location in which to do electrical work so I decided to file down the widened prong of the polarized plug. I never had any trouble with the fan. It leads one to question if there is any actual value to polarized plugs.

14. How can we gauge the intelligence level of our society?

I don't know. A temptation that must be avoided is to overemphasize the minor foibles that are present in society. That said, I've always been amazed by the fact that people continue to put out garbage bags for pickup by the garbage man that contain scraps of food in them. Don't they know the crows will peck holes in the bags or turkey vultures will tear the bags with their talons and make a mess searching for the food scraps? Perhaps, they think crows and turkey vultures are migratory and go south for the winter.

15. What would you have to have to consider yourself a success as a writer?

A squadron of tanks.

16. If you could change one thing in the world, what would it be?

I'd rename the kingfisher the jackass bird.

17. What's the most esoteric riddle you've come across in nature?

It's that the plant chrysanthemum and the plant mums are one and the same. It's hidden in plain sight. That's what makes it so difficult to figure out.

18. As an American, what is it hard for you to understand?

That Vince screwed Bret!!! Vincent Kennedy McMahon used trickery to force Bret Hitman Hart to lose the WWE title belt in Montreal, Canada. Bret Hart is one of the most famous Canadians, and the one thing he didn't want to do was lose the title belt in Canada.

19. How does your knowledge of science influence you as a writer?

Instead of saying an e-book's sales are miniscule, I can say they are in the horizontal limb of the exponential curve.

20. Can scientific problem solving be used to solve political problems?

I've always felt that the images of Jesus that believers find on pancakes, potato chips and grilled cheese sandwiches etc. serve as an important societal ameliorant. I think a group of experts should study the phenomena. They should find out which objects most often bear these images. Then they should search those objects for images of other religious icons such Buddha, Krishna and Mohammed. If for instance an image of Mohammed could be found on a grilled cheese sandwich, this could serve as an important bridge between Christianity and Islam.

21. What surprises you about the internet?

I find it amazing that the internet is not flooded with photos of the mushrooms Ravenel's Stinkhorn (Phallus ravenelii) and Netted Stinkhorn (Phallus duplicatus). As the first term of the binomial nomenclature of both mushrooms suggests these mushrooms bear a striking resemblance (a resemblance that so approaches subtle caricature as to make an argument for intelligent design) with regard to Homo sapiens to the nominative case of the binomial nomenclature.

22. What is your favorite figure of speech?

Zeugma is one of my favorite figures of speech because I'm not exactly sure what it means. According to the Webster's New World College Dictionary fourth edition © 2001 it is, "a figure of speech in which a single word, usually a verb or adjective, which is syntactically related to two or more words though having a different sense in relation to each."
Section B. Watching the Moonrise from the Mindanao Deep

23. What changes have you seen recently in your local community?

For several years, I noted that the storm drains alongside the roads that were lined with tall, leafy trees were completely clogged with leaves. During a heavy rain, the water, which was unable to enter the storm drains, would erode the sides of the road. I would try to keep the storm drains clear by scraping the leaves away with the side of my shoe. As I pushed the mat of leaves aside, the lower layers of which were always wet, I would always see a large number of earthworms. The only ones working to keep the storm drains unclogged were me and the earthworms, and both of us were in over our heads. Today, I saw a specially designed truck with a powerful vacuum hose that was unclogging the storm drains.

24. Is one of the keys to solving problems recognizing design flaws that society doesn't recognize as design flaws?

I wrote a letter-to-the-editor of the Bergen Record in which I suggested that Bergen County adopt the phrase Goose Toilet Bowl as its official nickname. The Bergen Record couldn't be bothered to publish my letter. It's this kind of blinkered provincialism that is hindering progress on a number of fronts. If it's true, and it may well be, that you can't solve certain problems until it is acceptable to make fun of them, we are in difficult straits.

25. As a conservative, who would you designate as one of the least perspicacious, conservative leaders?

Although he would probably use a greater sockdolager than perspicacious in his conversational speech, I find William F. Buckley to be among the least perspicacious conservative leaders. If we acknowledge that a political movement must appeal to the average voter in order to have power, we can see that William F. Buckley did almost nothing to endear himself to the average voter. In fact, his persona seemed to exude a disdain, if not contempt, for the average person. The following are two among many examples of his elitist character: his musical instrument of choice—the harpsichord—and his central philosophical influence—the instrumentalism of William James. This penchant for the unconventional and the obscure doesn't fully capture his elitist character. When William F. Buckley was the face of conservatism, it had waning popular appeal. In contrast, Ronald Reagan was a figure that connected to the masses. When he was the face of conservatism, its popular support waxed.

26. Can you think of a suitable replacement for the lackluster words emoticon and emoji?

All my engagements I will construe to thee,

All the charactery of my sad brow.

Julius Caesar Act II Scene 1 lines 308 & 309

27. How can you tell when a rapidly flowing stream is about to freeze over?

Ponds and the slow flowing sections of a stream will freeze over many weeks before the rapidly flowing sections of a stream freeze over. When the level of a stream drops a few inches overnight, this is a sign that much of the ground under the stream and alongside the stream has frozen. This greatly limits the flow of ground water into the stream. During the next cluster of days in which the temperature dips well below freezing, the rapidly flowing sections of the stream should freeze over, and that ice should link up with the ice of the already frozen, slow flowing sections of the stream.

28. What experiments could Facebook friends perform?

If an individual had a large number of followers, he could ask his followers to collect data. That data could be analyzed to shed light on particular questions. For instance, the followers could walk along the roads and streets near their homes for a week. They could note whether any lost gloves they came across were lost singly or in pairs. The common sense notion is that gloves are lost one at a time. I have been taking note of lost gloves for some time, and it is remarkable how many gloves are lost as pairs.

29. What magazine articles have you read recently?

I read an interesting article in Popular Science. There are several types of new cement that are vastly superior to traditional cement in strength. These new cements are used in some modern skyscrapers as a partial replacement for steel girders. Yet, in the article describing these new stronger cements, no mention was made of using these stronger cements to pave roads and thus reduce or eliminate the pothole problem. We're trained to accept progress in one field while at the same time not wondering about the lack of progress in another closely related field.

30. What have you learned about splitting logs into firewood?

Almost every depiction that I've seen of splitting logs into firewood has been idealized whether that depiction was in literature, TV or movies. Since it is often depicted as a manly activity, you often see a muscular man splitting log after log with only an ax. Actually an ax would easily get stuck in a log of moderate size because of the presence of knots that resist splitting along the grain. I find that splitting logs for firewood requires at least three log splitting wedges and a sledge hammer of moderate weight. Often two log splitting wedges will get stuck in the process of splitting a log of a large diameter. The artist Andrew Wyeth, who is noted for his realistic depictions of rural life, portrayed a log splitter, and his tools were a sledge hammer and one log splitting wedge; a log splitter using only one log splitting wedge is an unlikely occurrence. Log splitting is often depicted as a well-ordered and valued activity as it may have been in the past. Today, because it is anachronistic and noisy, it is often relegated to a messy corner of a piece of property.

31. What sight gags are you working on for your video?

I want to make some extravagant claim about wormholes such as their production locally for purposes of scientific study shouldn't be that difficult. Then I want to rake away about two inches of my compost heap hopefully revealing several large earthworms that will disappear into wormholes. The problem is that during the summer months the earthworms don't seem inclined to disappear into wormholes.

32. What are your thoughts regarding the roles of native plants and invasive plants in local ecosystems?

The idea of invasive plants may need to be rethought. What we may be witnessing is the ecological principle of homeostasis in operation. Here, in New Jersey, the local ecosystems may be solving the whitetail deer overpopulation problem. They are rebalancing themselves. As the deer population boomed they consumed many of the native plants that grew in the woods and open fields—the Cardinal flower being the most spectacular. What seems like an invasion by certain plants is actually the growth of plants that the deer will not eat. With less food available, the deer population will decline.

As a recent newspaper article stated, "native plants are preferred by wildlife to invasive plants." Replace the word wildlife with deer and we see that a successful campaign against invasive plants allows the deer population to boom, which is exactly what we don't want because the deer have consumed so many native plants. The article continued, "With more artificial habitats, parks and gardens, growing native plants rather than non-native varieties provides an easier transition for wildlife. . . ." Again replace the word wildlife with deer and we see that encouraging native plants means allowing the deer population to boom. Are the environmentalists deliberately trying to preserve our local eco-systems in a destabilized state to further their own agenda?

33. Is the Bible Code empirical evidence that Gödel's proofs are correct?

At first glance, the Bible Code's failure to use every letter of the Bible seems to distinguish it from the kind of mapping Kurt Gödel was concerned with. We can overcome this flaw by assigning all the unused letters and punctuation some kind of null symbol. Now, with this alteration, since all the letters and punctuation are used, the Bible Code appears more in line with Gödel's Proofs. Must the Bible Code be predictive to fulfill the demands of Gödel's Proofs? How meaningful is it to find unexpected and unconventional groups of letters that form meaningful phrases in the Bible? Another interesting point is that we could seemingly disallow the Bible Code with several simple rules two of which would be traditional words breaks must be observed and the formation of vertical words is not allowed. That there is no way to rule out his mappings has always been Gödel's profound claim, and it has always seemed quite spurious. The suspicion that for the Bible Code to be meaningful it must be predictive hints at the supernatural aura that surrounds Gödel's Proofs.

34. How do you regard the art movement known as abstract expressionism?

If you study it for some time, I think you will come to the conclusion that it is unintentionally hilarious. The pretentious verbiage that surrounds the paintings themselves is both amusing and amazing, not to mention annoying. For instance, an artist who drew human shadows in some of his later paintings is credited with reintroducing the human form into western art. As time goes by the paintings themselves are summed up best by a line the architect Doug Roberts (Paul Newman) says to his girlfriend Susan (Faye Dunaway) in the movie The Towering Inferno. Referring to the burned out skyscraper Doug Roberts intones, "Maybe they should leave it as it is—a kind of shrine to all the bullshit in the world." Abstract expressionism seems to have ushered in a worldview in which the opinion of the cognoscenti and/or clever advertising establishes the merit of an art object or for that matter any idea regardless of any intrinsic value it has. It is somewhat mysterious that the paintings made by today's artists can sell for astronomical prices. Of course, the explanation may be that the patrons simply like the artwork. But, if the patrons simply liked the artwork, it is odd that they show no apparent interest in the modern poetry that is imbued with the same kind of worldview. You would think that there would be gatherings where the poets (often referred to as language poets) would read their poems. These would be elegant gatherings where the poets would read their poems surrounded by the paintings of today's foremost artists. Their poems would employ their literary device of choice, which is a modern version of parataxis. According to the Webster's New World College Dictionary fourth edition © 2001 it is, "the placing of related clauses, phrases, etc. in a series without the use of connecting words (Ex. 'I came, I saw, I conquered')."

35. What do you think of the concept of the multi-verse?

I think there is only one universe and, of course the Leon verse, which is the second verse of "Beware of Darkness" that Leon Russell sings in The Concert for Bangladesh.

36. What is your analysis of the meth-amphetamine problem?

I find it interesting that meth-amphetamine is a terribly destructive drug both to the user's health and to those caught up in the violence of the drug gangs. Yet, meth-amphetamine is no more powerful than caffeine, and caffeine is a drug that is seldom characterized as having a deleterious effects either on the individual or on society. Is it wrong to think that there is some sort of mass irrationality involved in drug use? If drug users made an effort to buy their drugs from the least violent dealers, over time it would seem that the violent drug dealers would go out of business.

For a long time, I've believed that people who drink and drive or people who use drugs and drive do so because they want to drive while they're high. It is more difficult than we imagine to find interesting things to do whether or not we are high. In an attempt to make life exciting, I think people will drive when they are high. This notion has seldom been examined by those fighting the drug problem.

37. What is your favorite bumper sticker?

Years ago, I recall often seeing a bumper sticker consisting of two words Question Authority. My intuition is that the people who displayed the bumper sticker never questioned ideas that are associated with the liberal worldview such as global climate change. If my intuition is correct, this would be an example of irony. But, my unquestionably favorite bumper sticker that you see currently is the one in which the word "co-exist" is spelled with the familiar iconography of the various mainstream religions along with the peace sign and a symbol denoting harmony between male and female.

38. Can you recall any line of poetry?

I swallowed the pain of my childhood like a tree swallows barbed wire. What I recall is the criticism of the simile "like a tree swallows barbed wire." The criticism was that the simile was terrible because trees don't have mouths, and if they did have mouths, they wouldn't eat barbed wire; they would consume minerals dissolved in water. The flaw with that line of reasoning is that the poet is referring to another characteristic that trees have. As a tree grows it will very slowly engulf things like wire, rope or nails that are fastened to it. If a person never observes phenomena such as trees to which barbed wire has been nailed in order to construct a fence, there is no reason they would be aware of this phenomenon. Yet, the phenomena of trees engulfing various objects is present in cities as well as the countryside.

39. What can you say about the relativity of time?

Let's say I took the clear, plastic face off a clock and glued a wing onto the minute hand. Then when the minute hand was traveling from 12 to 6, I placed a fan below the 6 so that the flow of air impeded the progress of the minute hand. Next when the minute hand was traveling from 6 to 12, I moved the fan so that it was above the 12, and now the flow of air from the fan again impeded the progress of the minute hand. Would my simple experiment have slowed the passage of time, at least in my local vicinity? Or, would I merely have slowed a particular device that measures the passage of time?

There have been a number of experiments in which two atomic clocks have been set side by side and synchronized then one of the atomic clocks has been flown around the earth. When the two clocks are set side by side again the clock that has been flown around the earth has fallen behind the clock that was stationary with regard to the earth. The claim is that time itself slowed in the local vicinity of the moving clock and not that a particular device that measures time has slowed.

The triumph of relativity has been to take the common sense notion that measuring devices such as clocks will behave differently in different environments and incorporate it as a proof of the relativity of time. This is accomplished by redefining clocks as devices that define time instead of measure time. Defining clocks as devices that define time leads to certain problems such as does the deliberate slowing of a clock by using fans or other electromechanical devices mean that time itself is slowed. These problems are overcome by several strategies: (1) downplaying the notion that relativity requires that clocks define time instead of measuring it, (2) ignoring the problem of deciding which clocks and under which environmental conditions the flow of time is altered, (3) instead focus exclusively on the results of experiments involving atomic clocks in gravity fields of different strengths and on the results of experiments involving atomic clocks traveling a different speeds. Atomic clocks seemingly produce incontestable results with their ability to measure time to the trillionth of a second and perhaps, with greater accuracy than that.

40. What do you think of the notion that marijuana is not a gateway drug?

It seems completely false from my experience. If it is true, according to sociological data, what does that say about the curiosity I'd like to believe is inherent in human nature? Perhaps, people will say that humans are still curious by nature; it's just that marijuana smokers aren't curious about other drugs that will alter their perceptions. It seems to be another example of cognitive dissonance.

41. Is part of the difficulty with general relativity that its mathematical foundations are problematic?

Yes, there is a problem with the differentiation of tensors. If we differentiate a covariant tensor of rank two, sometimes we will get a tensor as the result, and sometimes we do not get a tensor as the result. In general the differentiation of a tensor does not give a tensor. This posed a problem because some mathematicians wanted a way to differentiate a tensor so that the result was always a tensor. They invented a method of differentiation they called covariant differentiation. This method of differentiation of a tensor always produces a tensor. But, is covariant differentiation a valid method of differentiation? Or is it merely an arbitrary method designed to get out of the dilemma posed by the fact that the traditional method of differentiation when applied to a tensor does not always result in a tensor? The traditional method of differentiation has substance and meaning. If it doesn't give the results that certain mathematicians desired when applied to tensors, can it be valid to arbitrarily design a new method of differentiation to get the results they desired?

42. Do you have any directions for your readers?

You are halfway through to finding your hat. Go past the road that goes toward Deer Lake. After turn onto Kingsland Road and go to the first pond you see on that road and the clue will be on a railing. . .

43. What hobbies do you have?

I want to collect all the catalytic converter covers, which I see lying along the roadsides, and turn them into some kind of sculpture.

Section C. Pink Sky and Green Stars

44. Is it true that in general relativity the mass of the sun is given as 1.47 kilometers?

I have not found that value used by Albert Einstein, but one of his foremost interpreters Arthur S. Eddington uses that value. Also, Lillian R. Lieber who wrote The Einstein Theory of Relativity, which is an attempt to make the very complicated mathematics of the relativity theories accessible to a wide audience, gives the mass of the sun as 1.47 km. Both Arthur Eddington and Lillian Lieber use that value to calculate the deflection of a ray of light as it passes close to the edge of the sun. Lillian Lieber uses that value to calculate the displacement of the Fraunhofer lines (spectral lines). The use of that value is implied in Arthur Eddington's calculation of the displacement of the Fraunhofer lines.

Also, in Lillian Lieber's interpretation of general relativity the term mass is introduced as a constant of integration, which seems odd for a quantity that seems to be a variable, but it may be mathematically correct to introduce the term mass in that manner.

45. What is a ghost tool?

Well, I've only come across one. Unless we count socket wrenches with a large hole in the center of the socket end of the wrench. This large opening allows long bolts to pass through the socket and the head of the socket wrench in order for the socket to reach the nut. I assume that such socket wrenches actually exist. A ghost tool is a tool that appears in a store's online catalog, but it seems the tool doesn't actually exist. Since I split logs for firewood using a sledge hammer and three wood splitting wedges, I was interested in finding some inexpensive hand-tool that would help me perform the task with less effort. The Home Depot online catalog carried a log splitting tool that consisted of a weight that traveled along a rod and some other paraphernalia I don't recall. The claim was made that this tool could slit logs into firewood. I don't see how a weight traveling along a rod could develop enough force to split a log. My local Home Depot had the price tag for the tool on the shelf that contained sledge hammers and other tools, but it seemed there was no tool that ever went along with the price tag. Perhaps, because people believe the internet can solve all our problems, the internet somehow generates ghost tools in order to sustain this illusion.

46. Have you seen anything recently that caught your interest?

As I was collecting roadside trash along Powerville Road, I came across several plastic bottles with curious holes burned in them. Perhaps, this is a new sign of the impending global climate change. There is a lot of decaying matter along certain stretches of that road. This could allow the formation of low level micro-clouds of the very potent greenhouse gas methane. These micro-clouds could act as a heat lens and focus infrared waves much as a magnifying glass focuses the wavelengths of visible light. The focused infrared waves would burn small holes in the plastic bottles.

47. Are there any aspects of quantum mechanics that you find troubling?

It is interesting to compare "one of the strangest features of quantum spin" as described by John Gribbin in his book Q is for Quantum with certain properties of complex variables as described by Murray Spiegel in his book Theory and Problems of Complex Variables. John Gribbin describes the very weird world of quantum physics where the quantum particle "is able to discern a difference between two identical copies of the universe." In his book Murray Spiegel describes many complex mathematical calculations that depend on the properties we ascribe to the square root of negative one, (-1), which is designated as the imaginary number i.

John Gribbin writes, "One of the strangest features of quantum spin is shown by the behavior of fermions. If an object like the earth turns in space through 360 degrees, it returns where it started. But if a fermion rotates through 360 degrees, it arrives at a quantum state which is measurably different from its starting state. In order to get back to where it started, it has to rotate another 360 degrees, making 720 degrees, a double rotation, in all. One way of picturing this is that the quantum particle "sees" the universe differently from how we see it. What we see if we turn through 360 degrees twice are two identical copies of the universe, but the quantum particle is able to discern a difference between the two copies of the universe."

I will paraphrase Murray Spiegel's calculations to avoid complex equations. Suppose we are given the function w, and w = the square root of z, in which z is a term that includes the imaginary number i. Suppose further that we allow z to make a complete circuit (counterclockwise) around the origin starting from point A. After a complete circuit back to A we have w=the negative of the square root of z. Thus, we have not achieved the same value of w with which we started. We have achieved the negative of the value we started with. However, by making a second complete circuit back to point A we obtain w= the square root of z, which is the same value of w with which we started.

I believe the strange features of quantum spin are merely the result of using certain equations that contain the imaginary number i as an exponent of e along with other complex mathematics such as polar coordinates.

48. What do you think of individuals who accept all the mainstream versions of all the major scientific theories?

How is it different from brainwashing? Well, except for the obvious facts that there is no hidden agenda and all the conclusions have been reached after long, painstaking, rational deliberation.

49. What is the hidden significance of the latest fashion trends?

For a long time, I tried to patch the holes in my jeans. The patches didn't work. Either they came loose in the wash or a new tear would form between two patches. It leads me to speculate that in the past patches only seemed to work because people ironed their clothes after every time they were washed. Thus, they resealed any loosened edges of the patch. Today, ironing seems like a boring and unnecessary task. Was it ever actually necessary? Were the wrinkles so pronounced and the creases so valuable? Creases don't seem important today. Was ironing something people were programmed to do? It's unfortunate that there's no time travel movie in which a modern character goes back to the 1950s and notices that all the clothes are ironed. It's odd that a revolutionary feminist movement began about the same time that women and society in general were questioning the value of ironing.

50. Do you have any thoughts on improving our various systems of education?

We should try to teach deer to swim. If we could teach a few deer, perhaps, the habit would spread throughout the entire species. The advantage would be that the deer ticks they carry would perish, if the deer stayed in the water for a significant length of time. The deer ticks in turn carry the Lyme disease causing bacteria, which would also perish. From an evolutionary perspective, deer may have avoided lakes and streams because they would be vulnerable to predators, when they were climbing the slippery banks or traversing muddy littorals. Since, for the most part, man is the only predator of deer today, if we failed to teach deer to swim, there doesn't seem to be a downside to genetically modifying deer so they would want to swim. Perhaps, groups of swimming deer would discourage the Canadian geese from using our lakes.

51. Where will you end up if you take a random drive consisting of alternate right and left turns?

There is no specific ending place, but you will probably encounter places that you have unwittingly avoided in your normal travels. You may encounter a localized low income neighborhood. It is not that you consciously avoided this neighborhood or these kinds of neighborhoods, but in your local travels you were more concerned going to stores, movies, restaurants and other places not located near low income neighborhoods.

52. Do flying saucers have brakes now?

Apparently they have a dual breaking system that is spring loaded. There are two plastic handles at opposite points on the circumference of the saucer. When the hinged handles are raised the bottom of each handle, which is a three inch plastic wedge, is lowered into the snow. When the handles are released a spring withdraws the wedge from the snow. And much like their science fiction counterparts these new saucers are made of a plastic that proved indestructible under repeated blows from my sledge hammer. I was breaking the saucer apart so it would easily fit into my garbage can. The repeated blows of my sledge hammer only served to make a strikingly decorative design appear on the plastic. A decorative design would probably have appeared on the flying saucers that were been subjected to intense investigation in science fiction novels if their authors had had a more literary bent.

53. Often the entire cumulus cloud is bright white while other times they are bright white except for their bottom layer, which is grey. Why is this?

I'm not sure, but I think the bottom grey layer occurs when there are many cumulus clouds in the sky. When there are many cumulus clouds in the sky, the sun will often be covered by a large cloud; the shadow of the cloud will cover a locally large swath of the earth. If you climb to the top of a moderately high hill and view the surrounding landscape, you may see the surrounding countryside dappled with cloud shadows. When the direct sunlight is cut off, clouds are illuminated by the blue light of the sky instead of the bright yellow of the sun. As the blue light of the sky passes through the upper layers of a cumulus cloud it diffuses. When the blue light of the sky reaches the base of the cloud, it has diffused to such an extent that the base of the cloud appears grey. One of the problems with this argument is the following: what do I mean when I say the direct sunlight is cut off? If the sun is overhead, can a cloud cut off direct sunlight to the surrounding clouds? The argument makes more sense when the sun is lower in the sky. If the sun is sinking in the west and is covered by a cloud, this could cut off direct sunlight to a great many clouds in the eastern portion of the sky.

54. What's the relation between the "Curse of the Bambino" and global warming?

It's interesting that thoughtful people, who believe in manmade global warming, are not in the least troubled by some well-known aberrations to which no unnatural causes are ascribed. Two that come to mind are: the failure of the Chicago Cubs to win a World Series championship since 1908 and the failure of the Boston Red Sox to do the same for 86 years from 1818 to 2004. It would seem that a strong argument can be made that there is no possible natural explanation for the failure of these two teams to win a World Championship in 104 years and 86 years, respectively. In fact, it may be possible to make a stronger argument that the World Series drought suffered by both these teams has no natural explanation than to argue that there is no natural explanation for global warming.

An operational description of the game of professional baseball, as a system with many interlocking levels of organization, lays the groundwork for forming the opinion that there is no natural explanation for such extended World Series championships droughts. One level of professional baseball's organization that we may overlook is the simple fact that professional baseball has rules that are stringently and fairly enforced. Recently, professional baseball introduced the use of the instant reply to review and change incorrect calls made by the umpires. In the 2014 season, the Miami Marlins introduced a new sculpture into their stadium. When in the beginning of the season, they won more games at home than on the road there was speculation that the sculpture had a hidden camera in it that allowed the Marlins to steal signals from the visiting teams. This is evidence of the kind of regulation that is possible of the professional baseball system. The concept of Money Ball is another indication of the way that any advantage that one team may possess can be negated by another team employing a more comprehensive strategy.

The fact that professional baseball can draft players from a myriad of colleges and high schools ensures that there is a super abundance of talented players for the teams to choose from. There are a great many reasons to suspect that professional baseball teams are evenly matched. Many people find the game of professional baseball somewhat boring. By listening to and analyzing the commentary of the baseball announcers one can build up a large stock of information that indicates long World Series droughts are unnatural. Appreciating the game through this perspective may make it more interesting.

55. Why do you say, "Nothing happens when the wind blows."?

It's a strange idea I have about insects and small animals. I note that during the warmer months humans enjoy and are most active on days when there is a vigorous breeze. A strong, steady breeze keeps away annoying insects, and it keeps you cool if you're laboring. Does it have the same invigorating effect on insects and small animals? A strong steady breeze may make it impossible for a mosquito to find nourishment. Such a breeze may be detrimental to insects, small birds and other small animals. They may be inclined to seek shelter from a strong breeze. If they are secreting themselves, all the other animals that prey on them would find no easy prey to feed on and so they might retire from activity as well. This trend could continue moving up the food chain.

56. Does the Riemann-Christoffel tensor distinguish between curved space and flat space?

Einstein's law of gravity takes the form of a tensor equation, Gst= 0, where s and t are indexes (subscripts) of G. It is derived from another tensor equation in which the Riemann-Christoffel curvature tensor Bastp is set equal to zero, where s, t and p are indexes (subscripts) of B and a is an index (superscript) of B. Through the tensor calculus operation known as contraction, the curvature tensor is transformed into Einstein's law of gravity. It would give Einstein's law of gravity added merit if the curvature tensor could tell us something profound about the geometry of space. Lillian Lieber in her book "The Einstein Theory of Relativity" claims that the curvature tensor does just that. She writes, "Given a Euclidean space the curvature tensor will be zero, whatever coordinate system is used, and conversely, given this tensor equal to zero, then we know that the space must be Euclidean." I have investigated this claim, and I cannot falsify it. The only question I can raise is about the validity of the tensor calculus operation known as contraction. This operation is used throughout tensor calculus, but it does appear to be based on a misuse of a technique used to find derivatives known as the chain rule.

57. Is Einstein's gravity equation, Gst= 0 where s and t are indexes (subscripts) misleading?

It may be possible to produce a version of the Einstein tensor Gst, which is also known as the Ricci tensor, that is equal to zero. We should note that s and t are indexes (subscripts). But, it does not seem necessarily correct or in other words that it must be true that we need to set the Christoffel symbols and the partial derivatives of the Christoffel symbols that make up the Einstein tensor equal to zero. Lillian Lieber writes in her book The Einstein Theory of Relativity, "Since Gst =0 does not necessarily imply that the B's [the components of the Riemann Christoffel curvature tensor] are zero, hence Gst =0 can be true even if the space is non Euclidean." "Gst =0 can be true even if the space is non Euclidean" is the important phrase. It can be true, but it is not necessarily true. It seems that the coefficients that Einstein chooses for his version of a non-Euclidean, four dimensional distance equation would make it difficult for Gst to equal zero because the coefficients consist of two variables x1 and x2, (where 1 and 2 are subscripts) and the variable x2 is used only once, which would make it difficult to cancel out the variable with its negative counterpart. Perhaps an analogy will make this argument clearer. The United States Constitution proscribes that a citizen must reach the age of 35 years before that citizen can obtain the office of the President of the United States. It would not be correct to assume from the previous statement that the age of any particular President was 35 years.

58. What previous answer has proven to be incorrect as these preposterous self-interview questions drag on?

The answer to the question about the Riemann curvature tensor appears to be in error; at the time I didn't understand how to correctly calculate the Riemann curvature tensor. Now, that I more fully understand how to work with the calculations that produce the Riemann curvature tensor, I believe that if we calculate the Riemann curvature for Einstein's version of ds as given in his theory of general relativity and then contract Riemann curvature tension to produce Einstein's equation for his law of gravity we will not obtain his law of gravity, which is Gst= 0 (where the subscript s could be replaced by the Greek letter sigma and the subscript t could be replaced with the Greek letter tau to make the equation more formal). Instead, I believe we would obtain Gst= some value other than zero. Einstein's value for ds as employed in his theory of general relativity can be found in The Einstein Theory of Relativity by Lillian R. Lieber. The methods for calculating the Riemann curvature tensor and contracting the Riemann curvature tensor can be found in Schaum's Outline of Tensor Calculus by David Kay. The calculations would seem to be quite lengthy in nature.

59. What are your thoughts on the various filters available for the intake duct of a forced, hot-air furnace?

Some time ago, I purchased the expensive, high tech, microbial trapping filters. I could never tell when they were so clogged that it was time to replace them. As an experiment, I didn't replace the filters for a long time. One cold, winter morning my forced, hot-air furnace came on, and its run seemed to extend for three or four times longer than normal. Then quite soon after it had shut off, it came on again. I knew something was amiss. It was highly likely that the filters were so clogged they were preventing an adequate amount of air to come through them. I replaced the clogged filters with the inexpensive, low tech filters. The furnace returned to its nominal status. Here I use the word nominal as it is used in Ron Howard's movie Apollo 13, which I suspect is an accurate reflection of NASA terminology. Thus, one of NASA's greatest accomplishments may be clouding the meaning of the word nominal. Perhaps, nominal is short for nominal variance from the ideal reading, but then it would seem more accurate to say normal variance from the ideal reading.

60. What is it about Taoism that you find interesting?

There seems to be some sort of enigmatic force and mysterious understanding that pervades Taoism.

61. How do you scare a snake?

In the morning, take a walk into the woods and sit under a tree. Wait until you hear the sound of a large, wood boring bumble bee drying its wings—drying its wings because they are wet from the rain that fell during the night. If you recall that no rain fell during the night, you are probably hearing the sound of a snake tasting the air with its tongue. Snakes seem to sense heat radiation with their tongues. If you see the snake, hide behind a large tree; this will disguise your heat signature. Next, you should move out from your hiding place for a short amount of time, and then move back. Do this a number of times. You can consider moving behind another large tree that is nearby and repeating your actions, or you could move from one large tree to another large tree circling around the snake. These actions should scare the snake. It may head back to its home, which could be a small hole in an old stone wall. The stone wall may have been made by farmers many generations ago. You could tap the snake on its tail as it disappears into its hole.

62. Do you have any rules of thumb regarding newspapers, magazines and books?

In the past, I regarded newspapers and magazines as unreliable sources accurate information. I found that books offered more accurate information with the caveat that many books were no more accurate than newspapers and magazines.

Section D. The Cave of the Day

63. The elemental and elementary mathematical concept of apples and oranges is both elegant and obvious. The concept tells us that only like quantities can be added together, subtracted from each other, multiplied together or divided into one another. You can be counted on to somehow muddy these waters. Right?

I don't know if I like the tone of that question. Lillian Lieber writes the following on page 315 of The Einstein Theory of Relativity: ". . . the 'dimensionality' of velocity is Length / Time; the 'dimensionality' of acceleration is Length / Time Squared. . ." If we say the acceleration of an airplane is 3 miles per minute per minute, we can rewrite it as 3 miles / minute / minute. We treat this quantity as a complex fraction that is similar to the complex fraction 100 / 5 / 4. This complex fraction we rewrite as (100 / 5) x (1 / 4) using the principle that to divide a complex fraction we invert the denominator and then multiply the numerator times the inverted denominator. We notice that we have rewritten the denominator of the complex fraction as 4 / 1 instead of merely 4. This is acceptable since we are dividing by one, which leaves the value 4 unchanged. We are also dividing like by like (a number by another number). When we perform a similar operation with the acceleration 3 miles / minute / minute, we are not dividing like by like. We are dividing a number, 1, by the unit of time measure, minutes.

We rewrite 3 miles / minute / minute as (3 miles / minute) x (1 / minute) and obtain 3 miles / minute squared. But, is it correct to rewrite the denominator of the "complex fraction," which is minutes, as minutes divided by one with one being a number and not a minute? Is it correct to denote the term 3 miles / minute / minute as a complex fraction?

64. You made an error in your last question and answer. Three apples can be multiplied by the number three and three oranges can be divided by the number six, but your notions about complex fractions were intriguing.

In the Mathematics Dictionary 5th ed. by Glenn James and Robert James part of the definition for the term fraction reads, "A simple fraction (or common fraction or vulgar fraction) is a fraction whose numerator and denominator are both integers, as contrasted to a complex fraction which has a fraction for the numerator or denominator or both." Note that the definition doesn't say a complex fraction has a simple fraction for the numerator or denominator or both, but merely a fraction. In the Mathematics Dictionary the broad definition for fraction is given as, "an indicated quotient of two quantities" so the prohibition of dividing apples into oranges could prohibit the division of minutes into miles, hence 3 miles / minute could not be considered a simple fraction.

65. Have we been lied to with impunity for decades by intellectuals or are some of us making mountains out of mole hills?

I don't know. Lillian Lieber on page 120 of The Einstein Theory of Relativity, makes the following statement, "any high school student knows that if x represents the length of an arc, and θ is the number of radians in it, then x / θ = 2π r / 2π . . ." Why does she say, "any high school student knows"? Upon reflection, it seems there is no way the statement could be true either now or in the 1940s when the book was written. Is Lieber exercising her ability to inveigle at a fundamental level or is it an example of something less sinister?

66. Can you give another example of disingenuousness by an intellectual?

Somewhere in the Norton Critical Edition of T. S. Eliot's The Waste Land, a critic writes that a certain line in the poem is a listing of the poet's personal synecdoches in order of increasing terror. I do not believe there can be personal synecdoches since according to Webster's New World College Dictionary a synecdoche is, "a figure of speech in which a part is used as a whole, an individual for a class, a material for a thing, or the reverse of any of these. (Ex: bread for food, the army for a soldier, or a copper for a penny)." For a synecdoche to be understood by a reader, there must be an underlying notion that is both widely known and accepted such as the notion that the phrase pig skin can be used for the word football.

67. How do you characterize the deleterious effects of the crumb-making ability of young children who are allowed to eat crackers in a car?

It is worse than the allied bombing of Germany during W.W.II. The only respite is calculating how long you could survive on the treasure trove of cracker crumbs secreted in every crevice of the car.

68. Can you make a mean spirited observation?

Appreciating nature —the plants and wildlife— is like being surrounded by people who refuse to use a toilet when defecating and being convinced overtime and through the compartmentalization of your own thoughts that it is beautiful.

69. You're on a roll, make another such observation.

Because no one questions whether Kepler's laws can actually be derived from Newtonian mechanics and because no one questions quantum mechanics, I'm willing to entertain the notion that physics is a kind of intellectual blather.

70. You like Fig Newtons, but you are skeptical of the Newtonian derivation of Kepler's three laws of planetary motion. This is a singular confirmation of the saying; a cobbler should stick to his last.

If you peruse section 5 of chapter 12 of Calculus with Analytic Geometry 4th ed. by C.H. Edwards, Jr. and David E. Penny entitled "Orbits of Planets and Satellites," you will find that their derivation of Kepler's three laws from Newtonian mechanics is suspect. I believe other derivations of the three laws from Newtonian mechanics are suspect, as well. We read in the sentence preceding Eq. (7), "We drop the factor 1/r. . . ." It cannot be correct to drop the factor 1/r especially since r = r (t), and, therefore, is an important function. This invalidates the derivation of Kepler's second law. Since Eq. (7), which is obtained by dropping the factor 1/r, is used in the derivation of Kepler's first and third laws, their derivations are invalid, as well.

There is also some confusing use of the term cosθ. It is used as both a constant and a variable in conjunction with unit vectors in the derivation of Kepler's second law. We are told that we must choose the coordinate axes so that at time t=0 the planet is on the polar axis and is at its closest approach to the sun. Under these conditions we are told that two unit vectors are equal: unit vector uθ = unit vector j. While these initial conditions are still presumably in force, we are told that the dot product of: unit vector uθ and unit vector j equals cosθ, and not one. It is true that under these initial conditions the dot product of the unit vectors equals cosθ, but under these initial conditions cosθ is not a variable as is required by the derivation, instead it is a constant. Angle θ is an angle of zero degrees. The cosine of angle θ under these initial conditions equals one. The dot product of the two unit vectors equals one under all conditions. The dot product does not equal cosθ under all conditions when we consider it as a variable as required by the derivation.

In the derivation of Kepler's third law, Eq. (16) is confusing. The left side of the equation comes from the equation for an ellipse that uses polar coordinates. In this instance the ellipse has only one focus and a directrix p. The focus is at the pole of the polar coordinate system. The term (major semi axis a) on the right side of the equation comes from the Cartesian coordinate equation of an ellipse in which there are two foci. The origin of the Cartesian coordinate system does not coincide with the pole of the polar coordinates. However, the equation is correct.

71. Why does Wikipedia state that absolute zero is the temperature at which molecular motion ceases?

Perhaps, because if Wikipedia stated instead that absolute zero was the temperature at which the vibrational motion of atoms ceases, it might allow some of their readers to develop the notion that atomic clocks slow as the temperature approaches absolute zero. Atomic clocks use the period of the uniform vibration of a particular atom as their fundamental unit of time. In Einstein's view, clocks do more than measure the passage of an absolute time (with varying degrees of accuracy), and instead, they define time—a time that his thought experiments determine to be relative. This leads to an interesting possibility, if atomic vibrational motion slows as the temperature approaches absolute zero. We could imagine a thought experiment that employs a series of many atomic clocks. Each atomic clock is isolated from the others in its own temperature-controlled chamber. As the temperature decreased in the long row of atomic clocks to near absolute zero, we could see the flow of time gradually slow and come to a near standstill as the temperature in an isolated chamber approached absolute zero. Would Einstein adopt the view that time was flowing at a different rate in each of the isolated chambers? Or, would he adopt the more common sense notion that the environment affects all measuring devices and clocks are merely a kind of measuring device? Since this is a thought experiment, perhaps we should add a second atomic clock to each isolated chamber. We can refer to the second clock as clock B. The series of B clocks could be modified in some way to precisely compensate for the lowered temperature. This would give us a long row of B clocks that all told the same time. You would then have a series of isolated chambers in which the flow of time was both gradually decreasing as measured by one set of clocks and flowing at a constant rate as measured by another set of clocks.

72. How do other text books present the derivation of Kepler's three laws of planetary motion?

Calculus and Analytic Geometry by George B. Thomas and Ross L. Finney provides the reader with a complex derivation of Kepler's three laws of planetary motion. In chapter 14 "Vector Functions and Motion" we find section .4 "Planetary Motion and Satellites," where they write, "We introduce polar coordinates in this plane and use polar coordinate equations to derive Kepler's three laws of planetary motion. The derivation, while lengthy in comparison to what we've seen so far, is one of the triumphs of vector calculus and the calculations in polar coordinates are typical of work done in orbital mechanics today." The derivation depends heavily on the cross multiplication of vectors. Cross multiplication of two vectors A and B gives us vector C whose length is the product of the lengths of A and B and the sine of the angle between them (the angular measure from the first vector to the second vector). Vector C is perpendicular to the plane of the given vectors and directed so that the three vectors in order A, B, C form a positively oriented trihedral. This leads to the interesting conclusion that when the vector representing the velocity of the earth is cross multiplied by itself the product is zero. This is so because the angle between the two vectors is zero degrees and the sine of zero degrees is zero. The mathematical fact that cross multiplication of two vectors in which the angle between them is zero gives us a product of zero is used several times in the derivation of Kepler's three laws. It is a mathematical fact, but is it reasonable? It would seem to indicate that if one man were pulling a heavy cart by means of a rope and another man joined him in his effort by tugging on the same rope the cross product of the vectors representing their efforts would be zero. Also, it would seem to indicate that if one large mass were attracting a distant and much smaller mass and then another large mass was introduced beside the original large mass (such that the two large masses and the much smaller distant mass formed a straight line), the cross multiplication of the vectors representing the attractive force of the two large masses would give us a product of zero. It seems cross multiplication of vectors is introduced into the derivation so that the product of vector forces can be set to zero although it is unlikely that this is a realistic outcome.

73. Have people learned anything from smoking marijuana?

Is society different in any way that is attributable to many people smoking marijuana? I think one of the first notions you would learn from smoking marijuana is that taste and smell are subjective. When you first start smoking marijuana the taste, smell and sensation of drawing dense smoke into your lungs is unpleasant. But as you learn to appreciate the effects of marijuana the taste, smell and sensation become quite enjoyable. Strangely, I don't think the notion that taste and smell are subjective has entered society in any significant way. The reason for this is unclear, except perhaps for the difficulty people have in dealing with abstract concepts.

74. Is there a section of road in the outer suburbs where horns honk constantly?

Yes, I came across it putting up political signs (lawn signs) for Mayor Steve Lonegan. It is off Route 23 in the area where Route 23 boarders the Newark reservoir. There is a curved single lane tunnel on a back road. The tunnel goes under a railroad embankment. A car will come to a stop before the entrance to the tunnel and honk its horn to warn oncoming cars that it is entering the tunnel. Every car does this.

75. Do you have any further thoughts on marijuana?

Another notion that marijuana smokers could have learned is the value of frugality. It takes very little marijuana to get high. Often smokers will continue smoking their expensive marijuana well after the point of greatly diminished returns has been passed. This is another example of the difficulty in dealing with abstract concepts. Interestingly, I often read about teenagers and young adults who are stopped by the police for some driving infraction. The police smell the odor of burning marijuana, and they use this as a reason to increase their scrutiny of the occupants of the car, which often leads to an arrest. A possible solution to this problem would be the use of a strongly scented essential oil such as lavender to mask the odor of marijuana. Both the failure of marijuana smokers to refrain from driving while high and their failure to disguise the fact that they are getting high while driving are examples of their difficulties in dealing with abstract concepts. The failure of marijuana smokers to realize that the movie Reefer Madness presents valid insights into problems surrounding smoking marijuana is another example.

76. When using a handsaw to cut a wooden plank, is it possible to make the cut along the vertical axis precisely perpendicular to the horizontal plane formed by the surface of the board?

I don't know, but for years, making a precisely vertical cut has stymied me. I think it would be possible if you met the two following conditions: your two sawhorses were precisely level with each other and both sawhorses were placed on very flat section of the ground. If those requirements were met and you pushed and pulled the handsaw quite lightly, the vertical cut might be perpendicular to the horizontal surface of the board. The theory is that you are letting gravity draw the saw blade down along a line toward the center of the earth. That line is perpendicular to the surface of the board because of the care that went into the placement of the sawhorses.

77. How is a portion of Kurt Gödel's first incompleteness theorem like the ingredients list of Stove Top Stuffing Mix?

In subsection "2.4 Expressing metamathematical concepts" of his first incompleteness theorem there is a statement denoted as function 32. In function 32 Gödel skillfully uses comas and parentheses to make certain terms seem to mean notions that they do not actually mean. The ingredients list of Stove Top Stuffing Mix is organized using two kinds of parentheses and the attendant comas to make it seem that high fructose corn syrup is the eighth ingredient in terms of the amount present in the stuffing mix while actually it is the second.

78. Do you have any further thoughts on the Newtonian derivation of Kepler's three laws?

There seems to be a problem with the Newtonian derivation of Kepler's three laws as presented in various textbooks of calculus and analytic geometry such as Calculus with Analytic Geometry 4th ed. by Earl W. Swokowski. The problem arises with the derivative of the cross product of certain vector functions. The vector functions in question are the vector functions that describe the path of the earth or any other planet and the vector functions that describe the velocity of the earth or any other planet. The derivation of the vector cross product of these two vector functions is zero. For instance, in section "15.6 Kepler's Laws" of his textbook Swokowski writes, "d/dt (r x v) =0." The mathematical operation follows the rules for vector functions, but for all other types of functions the derivative of a constant is zero, not the derivative of the product of two functions with variables. The two functions are r (t) and v (t). The path of the earth is a variable since the distance from the earth to the sun varies. The earth is actually closer to the sun during winter (in the northern hemisphere) than during the summer. Kepler's second law which states that the vector from the sun to a moving planet sweeps out area at a constant rate coupled with the fact that the orbit of the earth is an ellipse indicates the velocity of the earth is a variable. How can the derivation of the product of two variables equal zero? In Cartesian coordinates you could have an equation y=x2(x-1)½. The derivation of the product of the two variables x2 and (x-1)½ is dy/dx = 2x(x-1)½ \+ x2(x-1)-½. It is not zero. This may explain why vector functions are merged into polar coordinates without formal transformation equations ever being presented. If formal transformation equations were given, the vector functions could be transformed into polar coordinates before the derivative was taken, and thus, perhaps, when the derivative was taken it wouldn't equal zero.

79. Do bread crumbs in a toaster behave as a recursive function?

Employing a minimum of sophistry, we could argue that with every slice of bread that is toasted more bread crumbs are deposited on the crumb-catching tray, which is located at the bottom of the toaster. That is at least until the toaster is cleaned. If we could make the argument that the increasing number of crumbs deposited on the crumb-catching tray of the toaster somehow influenced the number of bread crumbs produced by the toasting of each slice of toast, it seems we could describe the behavior of the bread crumbs by a recursive function. We could assume that the bread crumbs deposited on the tray act as a layer of insulation, and thus they trap heat around the slices of toast. We might also assume the layer of crumbs acting as insulation doesn't trap a significant amount of heat around the bi-metal device that regulates when the slice or slices of toast are done. We could assume that each slice of toast is now being exposed to a greater amount of heat while the duration of the exposure remains constant. That is to say, the duration of exposure to the heat of the toaster is determined by the position of the regulating lever of the bi-metal device and not by any influence the crumbs exert on the behavior of the bi-metal device since we are stipulating that the crumbs have no influence on the behavior of the bi-metal device. The insulating properties of the crumbs cause each slice of bread to be exposed to an increasing amount of heat. This could cause more bread crumbs to form, and thus we may have the elements necessary for the behavior of the bread crumbs to be defined by a recursive function. We must also stipulate that the operators of the toaster don't deviate from their normal operation of the toaster because they notice any increased browning of their slices of toast due to the insulating properties of the bread crumbs.

80. What is your view of the Wikipedia entry for the "Michelson-Morley experiment"?

Studying the Wikipedia article is instructive. At first glance, the description of the experiment is quite plausible. All the equations are mathematically sound.

There are two typos in the section entitled "1881 and 1887 experiments." In the 2nd paragraph of that section, .04 fringes is written instead of .40 fringes, and in the 3rd paragraph .02 fringes is written instead of .20 fringes.

If you ponder the section "Light path analysis and consequences," a curious thought may occur to you. It seems the beam of light perpendicular to the earth's motion is completely dragged by the aether, and the aether is in turn completely dragged by the earth's motion. In contrast, the beam of light parallel to the earth's motion travels through an apparent aether wind generated by the earth's motion through a stationary aether. An explanation of the complete dragging of the aether by the earth is called for since the Michelson-Morley experiment assumes a stationary aether. The explanation for the "inclined travel path" of the beam of light (i.e. the forward motion of the light beam as though it is being dragged by the aether which in turn is being dragged by the earth) perpendicular to the earth's motion is given in the same section of the article, "Light path analysis and consequences." The article states, "This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame." So, the article doesn't actually claim the aether is completely dragged by the earth with regard to the light beam perpendicular to the earth's motion. It is confusing. It would seem the light beam could never be contained within the interferometer rest frame. The light beam it seems would always be in the aether rest frame. The behavior of the aether rest frame could fall into one of three categories; it could be dragged by the earth, not dragged by the earth or partially dragged by the earth.

In the sub-section "Mirror reflection" of the section mentioned above, we find this statement, "For an apparatus in motion, the classical analysis requires that the beam splitting mirror be slightly offset from an exact 45 degrees if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed." Offsetting the mirror from 45 degrees would alter subtly the constant L. But not in the cases where the constant L represents the length of the arm of the interferometer perpendicular to the aether wind. It would alter L when L represents a hypothetical vector. This hypothetical vector L is the path the light beam would travel if it were at rest with respect to the aether in other words if there was no aether wind. This hypothetical vector L is the vector used in the calculations to determine the duration of the light path perpendicular to the earth's motion. This vector is added to the vector that represents the earth's motion to produce the "inclined travel path" of the beam of light perpendicular to the earth's motion. Offsetting the beam splitting mirror would seem to lengthen and alter the direction of the hypothetical path the light beam travels in the arm of the interferometer before we calculate the manner in which this hypothetical path is acted upon by either the earth's motion (apparent aether wind?) or the transformation to the rest state of the aether. The parallelogram of forces rule could still be used to add hypothetical vector L to the vector that represents the earth's motion to obtain the "inclined travel path." But, the Pythagorean Theorem could no longer be used to add the vectors together because the vectors would no longer be at right angles to each other since the beam splitting mirror has been offset from an exact 45 degrees. When the beam splitting mirror is at exactly 45 degrees, the hypothetical beam of light (vector L) is perpendicular to the velocity of the earth when the other portion of the split light beam is parallel to the motion of the earth.

Giving the subject further thought, would slightly offsetting the beam splitting mirror from an exact 45 degrees allow the two beams to emerge from the apparatus exactly superimposed? If we tilted the beam splitting mirror slightly, say to 44 degrees, then the beam of light reflected from the beam splitting mirror, the longitudinal beam, would be slightly tilted in the forward direction as it began its journey along the arm of the interferometer. The beam of light that passes through the beam splitting mirror, the transverse beam, would be unaltered in its direction as it began its journey along its arm of the interferometer. But, how would the transverse beam of light be effected by the slightly tilted mirror on its return journey along the arm of the interferometer? We must recall that this beam of light, the transverse beam, is reflected off of the back of the beam splitting mirror. The returning transverse beam, which we can also call the transmitted beam as opposed to reflected beam because it is transmitted through the beam splitting mirror at the beginning of its journey, is now reflected off of the back of the slightly tilted beam splitting mirror. The longitudinal beam or reflected beam strikes the slightly tilted mirror from the front side at the beginning of its journey, and it is tilted slightly forward. It follows that when the returning transverse beam strikes the back of the slightly tilted beam splitting mirror it will be tilted slightly backwards. Thus, whatever advantage was gained by having the longitudinal beam tilted slightly forward at the beginning of its journey will be negated by having the returning transverse beam tilted slightly backwards.

The Wikipedia article makes no mention that the beam splitting mirror is a double-sided mirror. But, being double-side would not would not change the fact that whatever advantage is gained by slightly tilting the beam splitting mirror and changing the direction of the longitudinal beam at the beginning of its journey is negated by the change in direction imparted to the returning transverse beam. If there were two beam splitting mirrors, then the one reflecting the longitudinal beam could be slightly tilted while the one reflecting the returning transverse beam could remain unchanged, but this would introduce a new set of problems. In the Wikipedia article, there is no indication that there are two beam splitting mirrors or that the beam splitting mirror is double-sided.

The sub-section "Mirror reflection" continues with this statement, "In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams." This is difficult to visualize. Let us imagine some object traveling at a very great velocity, many times greater than the velocity of the earth. Can we imagine that the contraction of the horizontal component of the beam splitting mirror will change the angle of the mirror by 20 degrees, by 30 degrees or perhaps more than 30 degrees?

Interestingly, assuming that both beams of light are dragged along by the aether that is in turn dragged along by the earth doesn't alter the final outcome of the experiment because the experimental results are around 40 times smaller than the theoretically expected results. The theoretically expected result is a fringe shift of .44 fringes and the experimental result is an average fringe shift of .01 fringes. My calculation of the theoretical result that would be obtained when both beams of light are dragged along by the aether, which is in turn completely dragged along by the earth, is a fringe shift of about .20 fringes, which is still 20 times larger than the experimental result.

There appears to be a problem with the theoretical calculation of the fringe shift. In theory the amount of fringe shift is a ratio. The numerator (top) of the ratio is the maximum total difference in the path lengths between the light beam that is perpendicular to the earth's motion and the light beam that is parallel to the earth's motion. The denominator (bottom) of the ratio is a particular wavelength of visible light namely 500 nanometers. Since the light source used in the experiment is white light whose wavelengths range from 400 nanometers to 700 nanometers, how can the choice of one particular wavelength of visible light be justified? Why not divide maximum total difference in path lengths by 600 nanometers or 700 nanometers? Why does dividing the maximum total difference in light path lengths by that particular wavelength give us the amount of fringe shift?

The maximum, total, path length difference between the two light beams can be calculated. The maximum length difference is only 220 nanometers. Light waves can interfere with each other in two ways. (1) Where the crest of one wave meets the crest of another wave or where the trough of one wave meets the trough of another, the two waves combine and form a bright spot of light. (2) Where a crest meets a trough, the two waves cancel, leaving a dark spot. With a maximum, total, path length difference of only 220 nanometers, it seems only light beams with a wavelength in the range of 440 nanometers could interfere with the cancelling type of interference (trough meets crest type of interference). The cancelling interference would occur because the light beams would be ½ a wavelength out of phase. To obtain the reinforcing type of interference (crest meets crest or trough meets trough interference) would we need wavelengths in the ultra-violet spectrum in the range of 220 nanometers? Then one suspects the reinforcing type of interference would occur because the light beams would one wavelength out of phase.

Another startling claim regards the interference fringes themselves. The interference fringes themselves change or shift under the influence of the changing of the total difference in the path length between the light beam that is perpendicular to the earth's motion and the light beam that is parallel to the motion of the earth. The total difference changes between a maximum and a minimum. While the interference patterns change, the article maintains that a part of the interference pattern does not change. A bright center fringe in the interference pattern doesn't change and is used as the zero point from which the changes in the surrounding fringes of the changing interference patterns can be measured. The article provides us with a quotation from Dayton Miller who writes, "White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all the readings." How is it that the center fringe of the changing interference patterns doesn't change while the surrounding fringes do change?

It is difficult to understand a portion of the lengthy caption for figure 5. The difficult portion reads, "c is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass (important since experiments were run with white light which has an extremely short coherence length requiring precise matching of optical path lengths for fringes to be visible; monochromatic sodium light was used only for initial alignment)"

This statement is confusing. At first glance, a compensating glass plate doesn't appear necessary. First, we must acknowledge that both the transmitted and reflected beam travel through 30 widths of glass as they make their along the light path dictated by the 4 mirrors located at each end of each arm of the interferometer. Therefore, each arm has a total of 8 mirrors. There are 4 mirrors at each end of each arm. On the outward journey, a light beam passes through 16 widths of glass as it is reflected from the mirrors located at the ends of each arm. On the return journey a light beam passes through 14 widths of glass as it is reflected from the mirrors located at the ends of each arm. How can we make the notion that a compensating glass is necessary make sense? For the notion that a compensating glass is needed to make sense, we must acknowledge that the beam of light from the source is not split into two beams (the transmitted and the reflected) until it reaches the backside of the beam splitter b. I assume beam splitter b is a half-silvered mirror. The transmitted beam isn't formed until the precise moment that it leaves the beam splitter b so at that point it has traveled through no widths of glass. The reflected beam by virtue of being reflected travels through one width of glass at the beginning of its journey. This is so because it is reflected from the backside of the half-silvered mirror and travels through one width of glass to escape the mirror. It is the un-split beam that enters the beam splitter b and travels to the very back of beam splitter b, a half-silvered mirror. The returning reflected beam passes through the beam splitter b as it heads toward the telescope. So it passes through one width of glass at the beginning of its journey plus 30 widths of glass as it makes its way up and down the arm of the interferometer plus one final width of glass as it passes through the half-silvered mirror (beam splitter b) as it heads to the telescope for a total of 32 widths of glass. The transmitted beam passes through no widths of glass at beam splitter b, and it passes through one width of glass at compensating plate c on its outward journey. It passes through 30 widths of glass as it makes its way up and down the arm of the interferometer. It passes through another width of glass at compensating plate c on its return journey. The total number of the widths of glass the transmitted beam passes through is 32.

The transmitted beam and the reflected beam both pass through 32 widths of glass. This forces us to conclude that the transmitted beam on its return journey doesn't enter into the half-silvered mirror (beam splitter b), but instead it is reflected from the backside of the half-silvered mirror (beam splitter b). The backsides of mirrors aren't usually associated with the reflection of light beams. How is a beam of light that impinges on the rear face of a mirror reflected at all? You may conclude that the dilemma can be overcome by assuming that the half-silvered mirror (beam splitter b) is a double sided, half-silvered mirror, but this does not appear to be the case.

81. Do you have any comments on Isaac Newton's proof of Kepler's second law?

Reading portions of Isaac Newton's original text from his Principia, which appear in Colin Pask's book Magnificent Principia: Exploring Isaac Newton's Masterpiece gives you an idea of how difficult Newton's language is for us today. But, putting that notion aside, in Chapter 8, Colin Pask gives an overview of Newton's proof of Kepler's second law. Colin Pask writes that Newton's proof uses only the simple geometry of triangles. Interestingly, the force of gravity, which Newton refers to as centripetal force, switches on and off repeatedly. Newton writes, "But when the body [planet] arrived at B, suppose that the centripetal force acts at once with a great impulse, and, turning aside the body [planet] from the right line Bc compels it afterwards to continue its motion along the right line BC. . . . By the like argument, if the centripetal force acts successively in C, D, E, etc., and makes the body, [planet] in each single particle of time, to describe the right lines CD, DE, EF, etc. . . . ." The word successively is different from the word continuously. It implies one gravitational impulse coming after the other in time. At point B gravity acts and changes the course of the planet, but while the planet is moving along the straight line BC gravity is not acting on the planet. If gravity were acting on the planet it would not continue on its straight line path. It would be diverted from that straight line path as it will be when it reaches point C. When the planet reaches point C gravity acts again and changes the course of the planet to line CD, but while the planet is traveling along the line CD gravity doesn't act on the planet. Newton writes that the length of the lines BC and CD etc. diminish in infinitum, but does that overcome the difficulty of gravity being switched on and off repeatedly?

82. Do you have any further comments on Isaac Newton's proof of Kepler's second law?

In Isaac Newton's proof of Kepler's second law, a planet sweeps out triangles of equal areas in equal amounts of time. The various lengths of the bases of these triangles of equal areas represent the varying velocities of a planet. From this, we obtain the notion that the greater the length of the base of a triangle then the greater the velocity of the planet at a certain point in its orbit. In a triangle in which the base of a triangle is formed by the line AB, the line AB represents the velocity of the planet at point A. The bases of the triangles, which is to say the velocity measuring lines, vary in length as the velocity of the planet varies. Interestingly, these velocity measuring lines or perhaps, we could refer to them as velocity vectors are not perpendicular to the position lines. The position lines are the lines connecting the force center (the sun) to the planet; they could also be referred to as the position vectors. Each of the velocity measuring lines cannot be perpendicular to the particular position line it is associated with because they form the sides of the triangles of equal area. A triangle cannot have two angles that are each 90 degrees. This point is shown in the original diagrams from Newton's Principia reproduced in C. Pask's Magnificent Principia and in C. Pask's own diagrams. The lines that measure the velocity of a planet, which are the bases of the triangles, vary in length as the velocity of the planet varies. Why isn't the velocity measuring line, the velocity vector, perpendicular to the position line, the position vector? Perhaps, the reason is because the area of a triangle is as follows: A= ½ bh. The area of the triangle equals one-half multiplied by the base multiplied by the height of the triangle. If the base represents the velocity, the velocity varies inversely with the height of the triangle, b= 2A/h. The velocity varies inversely with the heights of the triangles of equal areas; it doesn't vary inversely with the lengths of the sides of the triangles. The height of a triangle is perpendicular to the base of a triangle.

This poses a problem for the modern versions of Newton's proof of Kepler's second law. In the modern versions the acceleration vector is parallel to (superimposed on) the position vector. When the position vector is cross multiplied with the acceleration vector the product is zero. The acceleration vector is the derivative of the velocity vector and perpendicular to the velocity vector. If the velocity vector is not perpendicular to the position vector, the acceleration vector cannot be parallel to (superimposed on) the position vector. A modern version of Newton's proof of Kepler's second law is presented in Calculus and Analytic Geometry by George B. Thomas and Ross L. Finney. In diagram 14.23 the velocity vector is not perpendicular to the position vector. In the same diagram one of the two components of the velocity vector is perpendicular to the position vector. In diagram 14.25 "The force of gravity is directed along the line joining the centers of mass." That is to say the acceleration vector is parallel (superimposed on) the position vector.

Apparently, the difficulty presented by the velocity vector not being perpendicular to the position vector can be overcome in Newton's original proof by diminishing the areas of the equal area triangles "in infinitum," since as the bases of the triangles decrease in length the sides of the triangle becomes closer to the height of the triangle, which is to say closer to a line that is perpendicular to the velocity measuring line (the base of the triangle).

The Ineluctable Self–Interview (Part 2)

Section A. The Stars Are a Sieve

83. What are your thoughts on the Wikipedia article Romer's determination of the speed of light?

The Wikipedia article Romer's determination of the speed of light is confusing and instructive. The section 5.2 entitled "Cumulative effect" deserves special attention. The difference between the predicted (calculated) time of the emergence of Jupiter's moon Io from Jupiter's shadow and the observed time of the emergence of Jupiter's moon Io from Jupiter's shadow is given as 15 minutes for one particular observation. That particular observation occurred on April 29, 1672 at 10:30:06 a.m. It was the culmination of a series of observations. The April 29, 1672 observation was the 30th and the last in this series of observations. This series of observations lasted from March 7, 1672 to April 29, 1672. The observed delay in the appearance of Io is 15 minutes that is to say Io appears 15 minutes later than the calculations made at the beginning of this series of observations suggested it should appear. The observed delay in the appearance of Io is explained by the fact that the Earth has been moving further away from Io for the entire period of March 7 through April 29. The Wikipedia article doesn't give a figure denoting how far the Earth has moved away from Io during this time, but my calculations suggest the Earth has moved about 85 million miles away from Io during this time. But, this distance seems much too large; perhaps, we only want the vertical component of this distance. If we divide 85 million miles by the speed of light in miles per second, we obtain about 457 seconds or 7.6 minutes. Thus, the observed appearance of Io should have been delayed only 7.6 minutes. How can we explain a delay of 15 minutes? Perhaps, if the solar system itself along with the entire Milky Way galaxy were traveling at about slightly more than half the speed of light or approximately 95,000 miles per second, this would explain the delay of 15 minutes. This notion seems fanciful. Especially since the entire Milky Way galaxy would have been moving at this speed when the original calculations made in the several days after the observations began on March 7. The change in the position of the earth over the weeks of observations would somehow need to come into play. It should be noted the Wikipedia article states that the delay is 15 minutes; my calculations suggest the delay is slightly more than 16 minutes (16 minutes 5 seconds).

Let's examine how the apparent orbital period of Jupiter's moon Io is calculated. It is denoted as the apparent orbital period because the Earth is either moving toward Jupiter or away from Jupiter when such calculations are being made. And, that unaccounted for movement of the Earth introduces error into the calculations. The Earth was moving away from Jupiter during the period of March 7 to March 14 when the calculation of an apparent orbital period of 42 hours 28 minutes 31.25 seconds was made. If the Earth wasn't moving toward or away from Jupiter, an exact as opposed to an apparent orbital period could be determined. We would note the time Io emerged from Jupiter's shadow then we would note the time when Io next emerged from Jupiter's shadow. The duration of that time period would give us the orbital period of Io, provided, of course, that the observation of an intervening occurrence of Io's emergence from Jupiter's shadow had not been blocked by cloudy weather or some other factors. This is because the time it would take light from Io to reach the Earth would be the same for both the first noted observation of Io's emergence from Jupiter's shadow and the second noted observation of Io's emergence from Jupiter's shadow. For example, when Jupiter is at its closest approach to the Earth, it would take light from Io about 35 minutes to reach the Earth. So when we first noted Io's emergence from Jupiter's shadow it actually would have occurred about 35 minutes before our observation denoted it occurred. If the Earth was not moving, when we made our second observation of Io's emergence it would again take light from Io about 35 minutes to reach the Earth. So when we made our second observation of Io's emergence it would have again actually occurred about 35 minutes before our observations denoted that it occurred. The delay of about 35 minutes for each observation would cancel each other out, and we could determine the actual orbital period of Io. We are neglecting the motion of Jupiter in this explanation since its takes Jupiter slightly less than 12 years to orbit the sun and thus Jupiter has moved only a small fraction of its orbit during the weeks of observation.

But, the Earth is moving, and in the case we are examining the Earth is moving away from Jupiter so for each observation of Io's emergence it take light longer to reach the Earth than it did on the previous observation. To calculate the apparent orbital period of Io this fact is neglected for purposes of calculation. We begin (as Romer did) with our first observation of Io's emergence on March 7 at 7:58:25 a.m. and for calculation purposes we conclude (as Romer did) with the fourth emergence of Io on March 14 at 9:52:30 a.m. This gives us a total of 169 hours 54 minutes 5 seconds for four orbits of Io. We divide by four and obtain 42 hours 28 minutes 31.25 seconds as the apparent orbital period of Io.

We assume that each orbit of Io (from one emergence to the next emergence) will last 42 hours 28 minutes and 31.25 seconds. From March 7 at 7:58:25 am to April 29 at 10:30:06, Io makes thirty orbits of Jupiter. Therefore, it should emerge from Jupiter's shadow 30(42 hours +28 minutes + 31.25 seconds) or 1,274 hours 15 minutes 36 seconds after the date and time of March 7 at 7:58:25 a.m. But, when Io makes its 30th emergence from Jupiter's shadow in this sequence of observation, the date and time are April 29 at 10:30:06 a.m. This gives us a total time for 30 orbits (from one emergence to the next emergence) of 1,274 hours 31 minutes 41 seconds. So the observed 30th emergence of Io from Jupiter's shadow occurs 16 minutes 5 seconds after the predicted (calculated) time for the 30th emergence of Io from Jupiter's shadow.

The explanation for the delay is the cumulative distance the Earth has moved away from Jupiter in the 52 days from March 7 to April 29. The cumulative, extra distance the light from Io must travel before it reaches the Earth accounts for the delay. But, the cumulative, extra distance seems to be about 85 million miles (or perhaps much less) not the 180 million one would expect for a delay of 16 minutes. It is dubious that if the solar system and Milky Way galaxy were traveling between 95,000 miles per second and 100,000 miles per second that this might explain the delay which is variously calculated as 15 or 16 minutes. It seems the earth would need to change its position relative to the direction of the motion of the Milky Way galaxy in some extreme way. Another explanation could be that the emergence times of Io from Jupiter's shadow weren't noted correctly. During the 52 day period of observation, Jupiter traveled about 35 million miles, but since this is only about 1% of the distance Jupiter travels in its slightly less than 12 year orbit, the vertical component of this distance doesn't seem significant.

84. Can you find a flaw with the tensor calculus operation called contraction?

The tensor calculus operation called contraction is unique. There isn't anything analogous to it in everyday mathematics such as addition or multiplication. Tensors are a difficult category of items to understand. They are grouped in ranks according to Lillian Lieber's The Einstein Theory of Relativity while Glen and Robert James's Mathematics Dictionary prefers the term orders. A tensor of rank 0 is a scalar. A tensor of rank 1 is a vector. A tensor of rank 2 is formed from the combination of two vectors in a certain way according to L. Lieber. The Mathematics Dictionary defines a tensor as "an abstract object having a definitely specified system of components in every coordinate system under consideration and such that, under transformation of coordinates the components of the object undergo a transformation of a certain nature."

Tensors are made of components except for the tensor of rank 0 which is a scalar. A scalar is a number as distinguished from a vector or a tensor. Lieber gives temperature and age as examples of scalars. Tensors have indexes (indices). Indexes can be both superscripts and subscripts. L. Lieber favors using letters from the Greek alphabet to denote a tensor's superscripts and subscripts. The Mathematics Dictionary favors the use of letters from the English alphabet to denote a tensor's superscripts and subscripts. A tensor with both subscripts and superscripts is called a mixed tensor. A tensor with only superscripts is called a contravariant tensor. A tensor with only subscripts is called a covariant tensor.

The number of components that form a particular tensor is determined by two conditions. The first condition is the number of spatial dimensions under consideration. L. Lieber gives examples that are located in two, three and four dimensional space. As the number of spatial dimensions increases, the number of components of the tensor increases, as well. The more components a tensor has the more difficult the calculations with the tensor become. It is less demanding to perform calculations with tensors located in only two spatial dimensions. The second condition that determines the number of components that a tensor is formed from is the number of indexes (indices) that the tensor is denoted by that is to say the total number of subscripts and superscripts. L. Lieber writes, "In general in n-dimensional space, a tensor of rank two consists of n2 equations, each containing n2 terms in the right-hand member." The rank of a tensor is determined by calculating the total number of indexes. For instance, if a mixed tensor has two superscripts and one subscript the total number of indexes is three therefore its rank would be three. In a two dimensional space the tensor would consist of 23 equations or eight equations. And each of the eight equations would have 23 terms on the right-hand side of the equation. Each equation is a component of the tensor so the tensor would have eight components in a two dimensional space.

In a two dimensional space, we usually think of the components of a tensor of rank one (that is to say a vector) as the component along the x-axis and the component along the y-axis. In tensor calculus the x, y, and z axes are renamed. The x-axis is called the x1\- axis. The y-axis is called the x2-axis. The z-axis is called the x3-axis. The two components of a tensor of rank one (a vector) in a two dimensional space would be referred to as the x1 component and the x2 component.

There is another feature of tensor calculus known as the summation convention. The Mathematics Dictionary defines it as "the convention of letting the repetition of an index (subscript or superscript) denote a summation with respect to that index over its range. E.g., if (1, 2, 3, 4, 5, 6) is the range of the index i, then aixi stands for the sum of aixi over the range 1 through 6, which equals a1x1 \+ a2x2 \+ a3x3 \+ a4x4 \+ a5x5 \+ a6x6. The superscript i in xi is not the ith power of the number x, but merely an index which denotes that xi is the ith object of the six objects x1, x2,. . . x6." The summation convention seems straightforward, but with the repetition of more than one pair of indexes it can become confusing. Working out the details of the summation convention with two pairs of repeated indexes is challenging enough, but for the operation of contraction we are going to look at, there are four pairs of repeated indexes on the right-hand side of the equation and one pair of repeated indexes on the left-hand side of the equation. The operation of contraction itself in the example we are going to examine causes the tensor on the right-hand side of the equation to generate a repeated index that didn't exist at the beginning of the operation of contraction. Another difficulty with the operation of contraction is that on the right-hand side of the equation a repeated index exits between two of the partial derivatives. Very often a repeated index will exist between a partial derivative and the tensor that it multiplies. In the example we are going to examine, the operation of contraction begins with the mixed tensor on the left-hand side of the equation. It is a primed tensor with indexes that aren't repeated. Its indexes consist of two superscripts and one subscript. Lillian Lieber denotes her indexes using lower case Greek letters. The tensor itself is denoted by the uppercase letter A and it is further denoted as A prime because it has been formed in the primed coordinate system. We will denote the indexes using lowercase English letters. The mixed tensor has two superscripts a and b. The mixed tensor also has one subscript c. We can denote it as A prime with superscripts a, b and subscript c. The operation of contraction begins when the subscript c is replaced by an a thus forming a repeated index of superscript a and subscript a on the left-hand side of the equation. On the right-hand side of the equation replacing the subscript c with the subscript a produces an addition partial derivative with the index a thus we have a pair of partial derivatives with the repetition of the index a so we must sum on the partial derivatives that contain the index a. Plus, between the tensor and the three partial derivatives that are multiplying the tensor we have three pairs of repeated indexes.

I had written this much before I came to the conclusion that contraction is a mathematically sound operation as long as certain transformation equations are used.

85. Do you have any thoughts on David Wick's book The Infamous Boundary?

David Wick's book The Infamous Boundary, published in 1995, contains an interesting presentation of Bell's inequality. His succinct description is provided to the reader in the chapter entitled "Bell's Theorem." Remarkably, his description seems to be brilliant propaganda so brilliant, in fact, that I am not sure it is propaganda of the disinformation variety. He seems to be saying something very similar to the following: What is the probability of flipping three fair coins (each coin being flipped once) and obtaining a particular outcome. The particular outcome we want to know is the probability of obtaining is three heads; that is each of the three coins landing heads up. The probability of obtaining this outcome must be less than one. The following is where I believe a fairly stark error occurs in David Wick's account, which I am loosely paraphrasing. Since the probability of obtaining the outcome heads in the flipping of one fair coin is ½, all that we must do in order to obtain the probability of flipping three coins and obtaining three heads is add this probability for each of the three flips. Hence, ½ + ½ + ½ = 1 ½ which must be less than 1. But it isn't less than 1 much to the consternation of physicists. Thus, in a very similar manner David Wick arrives at his version of Bell's inequality. The actual probability of obtaining three heads in the flipping of three fair coins is 1/6. David Wick seems to violate one of the basic tenets of probability, which is the following: Determine the total number of possible outcomes and let this be the denominator of the fraction. Let the outcome(s), which you desire to know the probability of, be the numerator of the fraction.

86. Do you have any thoughts on F. David Peat's book Einstein's Moon: Bell's Theorem and the Curious Quest for Quantum Reality?

F. David Peat's book Einstein's Moon: Bell's Theorem and the Curious Quest for Quantum Reality, published in 1990, devotes several chapters to Bell's inequality, yet his explanation is no more convincing than David Wick's explanation in his book The Infamous Boundary. F. Peat introduces a formula for the hidden variable explanation of the correlation of paired particles that is too restrictive. His explanation is ambiguous at points. And, we should note, that it is only too restrictive in one out of four instances. But, that limited restrictiveness is enough to lower a number (the sum of six "probabilities") associated with the local-reality explanation of the correlation of paired particles. That limited restrictiveness lowers the number from a value that should be close to two to a value of zero. Local-reality is another term for the phrase: hidden variable explanation of the correlation of paired particles. The use of the term number refers to the value obtained from the summation of an inequality. It is an inequality that has six probability terms on the left-hand side of the less than or equal to inequality. Three of the terms are positive, and three of the terms are negative. Zero is on the right-hand side of the less than or equal to inequality. When the phrase probability term is used, it can be interpreted as a value that gives the correlation between the two detectors A and B. Interestingly, for the local reality explanation of the correlation of paired particles F. Peat employs a formula that is too restrictive while when he gives an example of a calculation of the correlation of paired particles using quantum theory, he has no difficulty letting all four angles equal 45 degrees. The four angles in the local-reality explanation all were required to be different and there was a further clever restriction, as well. There could only be four different angles used for the four different settings in question. Usually, there would be eight different angles employed for four different settings, that is to say two different angles for each setting (one angle for each detector). There would be an exception when both detectors were set to an angle of zero degrees.

87. Does the magnetic field of our sun bend a ray of light from a distant star that passes close to the edge of the sun?

The magnetic field of the sun is often described as disorganized so perhaps it would not affect a ray of light from a distant star. Interestingly, I recall reading somewhere that a beam from a sodium light source can be visibly bent by the introduction of a relatively weak magnetic field. One rarely, if ever, reads about magnetic fields bending rays of light. If magnetic fields do bend rays of light and this fact were widely known, it would have to be taken into consideration in the explanations given for the bending of a ray of light from a distant star that passes close to the edge of the sun during an eclipse and is thus photographed displaced from its normal position in the sky by the bending of its light rays under the influence of the gravitational field of the sun. The displacement from its normal position in the sky is very small. The effects of magnetic fields on starlight are never mentioned, as far as I know, in any of the descriptions of the famous eclipse experiments that are used to support Albert Einstein's theory of general relativity. These are experiments that use photographs taken of stars near the sun during an eclipse in order to detect a displacement of the stars from their normal positions in the sky due to the gravitational field of the sun. I wonder if the magnetic field of the earth could bend the light rays from a distant star.

88. Do you have any thoughts on the greenhouse gases?

The Britannica Book of the Year 1970 contains an interesting statement in the "Astronomy" section. In the subsection entitled "Infrared Astronomy" we read, "In order to study physical processes in cool stars, observations are required at infrared wavelengths, where most of the energy is emitted. Progress in this field was retarded by the opacity of the earth's atmosphere to much of this radiation and, even more seriously, by the lack of sensitive detectors for photons of this energy. The latter deficiency was rectified by the design of solid-state detectors cooled in some cases to liquid helium temperatures to reduce thermal noise. It is perhaps worth emphasizing that at room temperature the telescope, dome, and detection windows all radiate at maximum strength at a wavelength of ten microns. The problem is rather similar to attempting optical photometry with bright lights located in the telescope tube. However, by using sophisticated electronic techniques and working either from balloons or through 'windows' in the earth's atmosphere, considerable advances have been made, particularly during the last year." It is widely assumed that the greenhouse gases absorb light (photons) in the infrared wavelengths that is to say they absorb the radiation (heat) that the earth itself gives off. The earth gives off infrared radiation because it has been warmed by the sun which emits a great deal of radiation in the visible wavelengths of light. Presumably, the other gases in the atmosphere such as nitrogen, oxygen and argon which make up 99.93% of the gases in the atmosphere absorb infrared radiation to a much lesser extent than the greenhouse gases, if at all. John Houghton writes on page eleven of Global Warming: The Complete Briefing, "The gases nitrogen and oxygen which make up the bulk of the atmosphere neither absorb nor emit thermal radiation." On the previous page Houghton defines thermal radiation. He writes, "All objects emit this kind of [thermal] radiation; if they are hot enough we can see the radiation they emit. The sun at a temperature of about 6,000º C looks white; an electric fire at 800º C looks red. Cooler objects emit radiation which cannot be seen by our eyes and which lies at wavelengths beyond the red end of the spectrum—infrared radiation (sometimes called long-wave radiation to distinguish it from the short-wave radiation from the sun)." We should note that Houghton has said that all objects emit thermal radiation. Then on the next page he stated that nitrogen and oxygen neither absorb nor emit thermal radiation. Are we to conclude that molecules of oxygen and molecules of nitrogen are not objects? If we agree with Houghton that nitrogen and oxygen neither absorb nor emit thermal energy, then since the greenhouse gases make up less than .07% of the gases in the atmosphere, shouldn't the earth's atmosphere be transparent to infrared radiation? If we divided a large pane of glass into 10,000 equal sized squares and then blackened 7 of the squares, we would still consider the pane of glass transparent to visible light. Does the opacity of the earth's atmosphere to infrared radiation from cool stars suggest that the gases nitrogen, oxygen and argon absorb infrared radiation and perhaps to the same degree as the greenhouse gases?

89. Have you changed your mind about the tensor operation known as contraction?

After re-evaluating the tensor operation designated by either of the following two appellations contraction or the inner product, I have reached the conclusion that it is an invalid operation, but with the caveat that it does appear to be true for a mixed tensor of rank three, in a space consisting of two dimensions, when the coordinates are transformed by the rotation of the axes. If contraction is an invalid operation in most instances, it would undercut the mathematical foundations of Einstein's theory of general relativity since it is the contraction of the curvature tensor (which is also known as the Riemann-Christoffel tensor) that generates Einstein's law of gravity, i.e., the gravitational tensor. The curvature tensor is Bastr, and in Euclidean space, Bastr =0. If we rename/change the index called subscript r into an index called subscript a, then through contraction we obtain Gst, the gravitational tensor, and Gst =0 is Einstein's law of gravitation. The curvature tensor and the gravitational tensor are most often presented with lower case Greek letters as the indexes (subscripts and superscripts), but I am trying to use only the symbols commonly available on a keyboard. According to Lillian Lieber, one of the ways in which tensors are defined is by the transformation equations that are used to transfer the tensor from one coordinate system to another. The transformation equations presented by Lillian Lieber in her book The Einstein Theory of Relativity are rotational formulas for coordinates in a plane that is to say in a space of two dimensions. It can be confusing. We can begin with the primed coordinates (x/, y/) that belong to the old set of rectangular axes X/ and Y/. Next, we rotate a new set of rectangular axes, designated X and Y, through the angle theta with respect to the old set of rectangular axes, which we have designated X/ and Y/. The new unprimed coordinates are designated (x, y). The relationship between the new unprimed coordinates (x, y) and the old primed coordinates (x/, y/) is given by the following rotational formulas: x= x/cos. θ – y/sin θ and y= x/sin θ + y/cos. θ. (The reader may find it interesting and annoying that I have surrendered to the red squiggly line of my lap top's spell check, and I have often placed a period after the word cos in an attempt to placate it.) We can write the rotational formulas in another way in which the primed and unprimed x and y variables are replaced by primed and unprimed x1 and x2 variables, respectively. We refer to the variables as x subscript one and x subscript two; we append the terms primed and unprimed as the occasion calls for. The rotational formulas become the following: x1 = x1/cos. θ – x2/sin θ and x2= x1/sin θ + x2/cos. θ. It is this version of the rotational formulas that are referred to in the complex and condensed notation used to define tensors.

There are two important items to note: first, the use of both sin θ and cos. θ and second the strategic placement of the negative sign. These two items will turn out to be essential for the operation of contraction to be performed validly on a mixed tensor of rank three in a space of two dimensions. In the operation of contraction on a mixed tensor of rank three in a two dimensional space, we will repeatedly get the terms cos2 θ + sin2 θ. Since cos2 θ + sin2 θ equals one, this is one of the key factors that allow contraction to lower or reduce the rank of a mixed tensor by two. The other key factor that allows us to reduce the rank of the mixed tensor by two is the strategic placement of the negative sign. In the operation of contraction on a mixed tensor of rank three in a two dimensional space, we also repeatedly will get the terms cos. θ sin θ – cos. θ sin θ. Since cos. θ sin θ – cos. θ sin θ equals zero this will allow half of the equations generated through the summation (the summation convention) called for in the operation of contraction of a mixed tensor in a space of two dimensions to cancel themselves through being multiplied by zero. The other half of the equations will be reduced in rank by two during the required process of summation through the use of the identity cos2 θ + sin2 θ =1.

The three following facts about the rotational formulas for coordinates in a space of three dimensions should make a convincing argument that the operation of contraction is invalid for tensors in a space of three dimensions. 1. The rotational formulas do not employ both of the terms sin θ and cos. θ; they only employ the term cosine and it isn't the cosine of only one angle, theta. 2. The term cosine does not refer to a single angle, theta, but instead is matched with nine different angles A1, A2, A3, B1, B2, B3, C1, C2, and C3. 3. There are no negative terms that would allow many terms and hence equations to cancel one another.

According to the Mathematics Dictionary by Glenn James and Robert C. James the formulas for rotation of axes in a space of three dimensions are the following:

"If the direction angles, with respect to the old axes, of the new x-axis (the x/-axis) are A1, B1, C1; of the y/-axis are A2, B2, C2; and of the z/-axis, A3, B3, C3, then the formulas for rotation of axes in space are x= x/cosA1 \+ y/cosA2 \+ z/cosA3, y= x/cosB1 \+ y/cosB2 \+ z/cosB3, z= x/cosC1 \+ y/cosC2 \+ z/cosC3." To write the formulas in the condensed form used in tensor operations both the primed and unprimed x, y, and z coordinates would need to be replaced with primed and unprimed x1, x2, and x3 coordinates, respectively.

Lillian Lieber in her book The Einstein theory of Relativity denotes tensors using lower case Greek letters as subscripts and superscripts, i.e., indexes. I will use lower case English letters for the subscripts and superscripts of tensors. Instead of the familiar symbol for partial derivative, which seems to be similar to the lower case Greek letter delta, I will use the letter p followed by a period, p. Since I am using the symbols commonly available on a keyboard, the symbol I am using to indicate that a variable is primed is the forward slash raised to a superscript. The symbol that I am using to separate the numerator from the denominator of a partial derivative is also the forward slash (so it can be confusing), but it is neither raised to a superscript nor lowered to a subscript. Using this nomenclature, the formula for a mixed tensor of rank three is the following: A/abc = (p.xn/p.x/c) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as the following: The primed tensor A with superscripts a and b and subscript c equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript c multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with subscript b with respect to the variable x with the subscript m multiplied by the unprimed mixed tensor A with superscripts k and m and subscript n. We must further take into account the summation convention. It tells us that since the indexes k, m, and n occur twice on the right-hand side of the formula we must sum on these indexes. We should further note that since we are operating in a space of two dimensions the values of the indexes will range from one to two in other words a, b, c, k, m, and n take on either the value 1 or 2.

In the operation of contraction, the subscript c (or index c) is replaced with the index a on both the right-hand side and left-hand side of the formula. The formula that is generated is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as saying the following: The primed tensor A with superscripts a and b and subscript a equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript a multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with the subscript b with respect to the variable x with the subscript m multiplied by the unprimed tensor A with the superscripts k and m and the subscripts n. We must further take into account that since the indexes k, m, n and now a occur twice on the right-hand side of the formula we must sum on these indexes. Also, the index a occurs twice on the left-hand side of the equation so we must sum on the index a on the left-hand side of the formula, as well.

Lillian Lieber draws our attention to two of the partial derivatives that occur on the right-hand side of the formula. She analyses these two partial derivatives when the subscripts n and k have different values, and she also analyses these two partial derivatives when the subscripts n and k have the same value.

When the subscripts n and k happen to have different values Lillian Lieber concludes that (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk) = 0. Regarding the term (p.xn/p.xk), she tells us that because the x's are not functions of each other (but only of the x/'s), therefore there is no variation of xn with respect to a different x, namely xk. No variation of xn with xk means the partial derivative of equation xn with respect to xk equals zero. Thus, coefficients of the mixed tensor of rank three (Akmn) when n does not equal k will all be zero and this will make the terms drop out.

Lillian Lieber further tells us that when n=k, then (p.xn/p.x/a) (p.x/a/p.xk) = (p.xk/p.x/a) (p.x/a/p.xk) = (p.xk/p.xk) = 1. Thus formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) becomes A/aba = (p.x/b/p.xm) (Akmk) in which we must still sum on the right for k and m. We must recall that mixed tensor (Akmn) becomes mixed tensor (Akmk) because n=k. Thus, a mixed tensor of rank three is reduced through the operation of contraction to a tensor of rank one.

This is Lillian Lieber's argument, but it appears to be a misuse of the chain rule for partial differentiation.

Let's examine the chain rule as it is used in ordinary differentiation that is to say the differentiation of a function with two variables x and y. The following is an example: y = (x2 +1)3. We should note that the chain rule seems to be used when the variable x is raised to a power in more than one instance and in a somewhat complicated manner. The variable x is raised to the second power and then the sum (x2 +1) is raised to the third power. To use the chain rule, we must form the composite function y = u3 where u = x2 +1. To find dy/dx, we say that dy/dx = (dy/du) (du/dx). We see that in our example dy/du = 3u2 and du/dx = 2x. We substitute x2 +1 = u into 3u2, and we obtain an answer to dy/dx as follows: dy/dx = 3(x2 +1)2 (2x) =6x(x2 +1)2. The striking notion to consider is that the u in the derivative dy/du is the same as the u in the derivative du/dx. This isn't the case in Lillian Lieber's argument. To be fair, we must point out that Lillian Lieber is dealing with the chain rule as it applies to partial derivatives. The reader may think: perhaps, with regard to partial derivatives, the requirement that the u in the derivative dy/du is the same as the u in the derivative du/dx or more precisely that the u in the derivative p.y/p.u is the same as the u in the derivative p.u/p.x could be relaxed. Well, it actually can't be relaxed. The reader may think: perhaps the chain rule could be expanded so that it could be employed whenever partial derivatives occur in a manner that superficially resembles the chain rule. Another notion must be called into question. It is Lillian Lieber's false notion or at least overly simplistic notion that (p.x1/p.x2) equals zero since x1 doesn't vary with x2, instead x1 varies with x/1 and x/2.

Let's look at an example of Lillian Lieber's argument. We will substitute numbers into the subscripts of the following formula: (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk). First, we will substitute values in which n doesn't equal k, and we will see if the result is zero. Let n=1 and k=2 and a =2, thus the formula becomes the following: (p.x1/p.x/2) (p.x/2/p.x2) = (p.x1/p.x2). The formula reads as follows: the partial derivative of the equation x with subscript 1 with respect to the primed variable x with subscript 2 multiplied by the partial derivative of the primed equation x with subscript 2 with respect to the variable x with subscript 2 equals the partial derivative of the equation x with subscript 1 with respect to the variable x with subscript 2.

The equations referred to are the rotation formulas for the primed and unprimed coordinates x and y in a space of two dimensions when the primed and unprimed coordinate x is replaced with the primed and unprimed coordinate x1, as required and the primed and unprimed coordinate y is replaced by the primed and unprimed coordinate x2, as required. The two rotation formulas we need to solve our example are the following: x1 = x1/cos. θ – x2/sin θ and x/2 = – x1sin θ + x2cos θ. Using the rotation formula x1, we can find the first called for partial derivative. It is the partial derivative that occurs first on the left-hand side of the equation (p.x1/p.x/2) (p.x/2/p.x2) = (p.x1/p.x2). Thus, (p.x1/p.x/2) = –sin θ. Using the rotational formula x/2, we can find the second called for partial derivative on the left-hand side of the above equation. Thus, (p.x/2/p.x2) = cos. θ. We have run into the dilemma: [–sin θ] [cos. θ] should equal (p.x1/p.x2), in other words [–sin θ] [cos. θ] should equal zero if the chain rule for partial differentiation isn't being misused used by Lillian Lieber. According to Lillian Lieber, (p.x1/p.x2) equals zero since, as she falsely maintains, x1 doesn't vary with x2, instead x1 varies with x/1 and x/2. It is false, but at a simplistic level of analysis it is true. She is correct, x1 doesn't directly vary with x2 instead x1 varies with x/1 and x/2. But, she ignores the fact that in a circular fashion x/1 and x/2 vary with x1 and x2. So we can say x1 varies with x/1 and x/2 and x/1 and x/2 vary with x1 and x2.

The reader may think: Restrict θ to either 90 degrees or 0 degrees since the cos90° =0 and the sin0° =0, then there is no dilemma. But, θ represents the amount the new coordinate system is rotated with respect to the old coordinate system, and therefore it cannot be restricted.

The confusing use of the chain rule occurs in part because the x/2 in (p.x1/p.x/2) refers to the variable x/2 found in the equation x1 = x/1cos θ – x/2sin θ while the x/2 in (p.x/2/p.x2) doesn't refer back to the variable x/2 in the equation x1 = x1/cos. θ – x2/sin θ, but instead it refers to a different equation x/2 = – x1 sin θ + x2cos θ. As we saw in our example of the chain rule in ordinary differentiation, the chain rule works when the u in (dy/du) and the u in (du/dx) are essentially the same and can be equated as in our example u = x2 +1. We could write the partial derivatives another way in which they didn't resemble the chain rule. We could write the following: (p.f(x/1, x/2)/p.x/2) (p.f(x1, x2)/p.x2).

Now, let's examine the manner in which, in many instances, but not all, (p.xn/p.x/a) (p.x/a/p.xk) generates zero in a space of two dimensions. We should recall that it is part of the larger formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) in which we must sum on the indexes a, k, m, and n. We will concentrate on summing on the index a in a two dimensional space. That means, we hold the other indexes constant and let the index a first assume the value 1 and then the value 2 then we add the two terms together. We have already determined the value of (p.xn/p.x/a) (p.x/a/p.xk) when a=2, k=2, n=1. Now, we can arbitrarily assign values to the other indexes: let b=1, and m=1. Since we have already determined the value of (p.x1/p.x/2) (p.x/2/p.x2) as [–sin θ] [cos. θ] we can plug that into our formula along with the indexes we've chosen arbitrarily. We must recall we are summing on index a on both the left-hand and right-hand side of the equation. We should note that we can choose certain indexes arbitrarily with the caveat that we will cycle through all the possible combinations of the indexes all the while paying proper attention to the summation convention to arrive at the final answer. But, for now, we are examining only certain representative parts of the complete solution of the formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn).

With the index choices we have made we can now generate one of the partial representations of the formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). The partial representation is the following: A/111 \+ A/212 = (p.x1/p.x/2) (p.x/2/p.x2) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211). It seems confusing, but what is being done is that we are summing on the index a while holding all the other indexes at our arbitrarily chosen values. Also, I have begun the summation with the index a set at 2, and I then proceeded to set the index a at 1. This is only because I have already calculated the value we obtain when the index a equals 2. Thus we can rewrite the formula as A/111 \+ A/212 = [–sin θ] [cos. θ] (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211).

Now, we must determine the value of (p.x1/p.x/1) (p.x/1/p.x2). We need the rotation formulas for x1 and x/1. They are the following: x1= x/1cos. θ – x/2sin θ and x/1= x1cos. θ + x2sin θ. We see that (p.x1/p.x/1) = cos. θ and (p.x/1/p.x2)=sin θ. Now, we can plug this value into the formula, and we obtain the following: A/111 \+ A/212 = [–sin θ] [cos. θ] (p.x/1/p.x1) (A211) + [cos. θ] [sin θ] (p.x/1/p.x1) (A211). This can be rewritten as the following: A/111 \+ A/212 = (p.x/1/p.x1) (A211) [–sin θ cos. θ +cos. θ sin θ] or A/111 \+ A/212 = (p.x/1/p.x1) (A211) [0] or A/111 \+ A/212 =0. This is an example of how summing on the index a when index k doesn't equal index n allows many of terms and hence equations that determine the rank of the tensor to cancel each other. Let's point out that [–sin θ cos. θ +cos. θ sin θ] = [cos. θ sin θ – cos. θ sin θ] =0. Let's also point out that it is the number of partial derivatives that determine the rank of the tensor. Our formula has three partial derivatives, (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm), thus it is a tensor of rank three. That is to say it was a tensor of rank three before we began the operation of contraction. We have discovered that when the index k doesn't equal n the partial derivatives equal zero. If this were the only possible outcome, the tensor would reduce to zero, which is a scalar which is also known as a tensor of rank zero.

Next, we will show how summing on index a when index k equals index n produces the value one where just previously we obtained zero. We will let index k equal one, and also we will let index n equal one. We will maintain the same values as we have previously for the values of the other indexes and we will sum on index a. The formula when subscript c has been replaced with subscript a is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). Next, we will replace the letter indexes with the number indexes that we indicated we would use, and the formula we obtain is the following: A/111 \+ A/212 = (p.x1/p.x/2) (p.x/2/p.x1) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x1) (p.x/1/p.x1) (A211). As before, we will solve for (p.x1/p.x/2) (p.x/2/p.x1) and then we will solve for (p.x1/p.x/1) (p.x/1/p.x1).

To solve (p.x1/p.x/2) (p.x/2/p.x1), we need the rotational equation x1 and x/2. They are the following: x1= x/1cos θ – x/2sin θ and x/2 = – x1sin θ + x2cos θ. Thus, (p.x1/p.x/2) = –sin θ and (p.x/2/p.x1) = –sin θ so (p.x1/p.x/2) (p.x/2/p.x1)= [–sin θ] [–sin θ] = sin2 θ.

To solve (p.x1/p.x/1) (p.x/1/p.x1), we need the rotational formulas for x1 and x/1. They are the following: x1=x/1cos. θ –x/2sin θ and x/1= x1cos θ + x2sin θ. Thus, (p.x1/p.x/1) =cos. θ and (p.x/1/p.x1) =cos. θ. Therefore, (p.x1/p.x/1) (p.x/1/p.x1)= cos.2 θ. We can plug these values back into our formula and we obtain the following: A/111 \+ A/212 = sin2 θ (p.x/1/p.x1) (A211) + cos2 θ (p.x/1/p.x1) (A211). This can be re-written as the following: A/111 \+ A/212 = (p.x/1/p.x1) (A211) [sin2 θ + cos.2 θ]. This can be re-written as the following: A/111 \+ A/212 = (p.x/1/p.x1) (A211) [1].

Thus, we have encountered examples of the mechanism by which a mixed tensor of rank three in a space of two dimensions is reduced to a tensor of rank one through the operation of contraction. Every detail hasn't been explained, for instance, the manner by which A/111 \+ A/212 =C/1 hasn't been explained. But, what has been made clear is that Lillian Lieber's explanation of contraction involves an invalid use of the chain rule. By explaining the mechanism of contraction at the more generalized level of the partial derivatives, Lillian Lieber makes it appear that contraction would be a valid operation for all mixed tensors of at least rank three and above and in a space of two, three or more dimensions. As has been demonstrated, contraction is only be valid in a space of two dimensions with a mixed tensor of rank three where the transformation of the coordinates involves the rotational formulas. It has also been demonstrated that it is unlikely contraction would be a valid operation for spatial dimensions of three or more. This should undercut the mathematical foundations of Einstein's theory of general relativity.

In closing, we should examine three items: the definition of the chain rule for partial differentiation, the curvature tensor, and contravariant and covariant vectors.
We should examine the formula given in the Mathematics Dictionary as the chain rule for partial derivatives. It is the following: "p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp)." One of the examples given in Wikipedia under the heading chain rule should make this formula understandable. Wikipedia tells us that given u(x, y) = x2 \+ 2y where x(r, t) = r(sin(t)) and y(r, t) = sin2(t) then p.u/p.r = (p.u/p.x) (p.x/p.r) + (p.u/p.y) (p.y/p.r) = (2x) (sin(t)) + (2) (0) =2r (sin2(t)) and p.u/p.t = (p.u/p.x) (p.x/p.t) + (p.u/p.y) (p.y/p.t) = (2x) (rcos. (t)) + (2) (2sin(t) cos.(t)) = (2rsin(t)) (rcos.(t)) + 4sin(t)cos.(t) = 2(r2 +2) sin(t)cos.(t) = (r2 +2) sin(2t).

In the example above, we could attempt to use Lillian Lieber's argument and claim that both p.u/p.r and p.u/p.t equal zero since function u(x, y) varies with the variables x and y and not the variables r and t. But, we can see Lillian Lieber's argument isn't valid.

Let's apply the formula for partial differentiation to a formula we have examined. Given u(x1, x2) = x1cos θ + x2sin θ or in its more familiar form x/1= x1cos θ + x2sin θ where x1= x/1cos θ – x/2sin θ and x2= x/1sin θ + x/2cos θ. Now, using the formula for partial differentiation p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp) we obtain the following: p.x/1/p.x/2= (p.x/1/p.x1) (p.x1/p.x/2) + (p.x/1/p.x2) (p.x2/p.x/2) = [cos. θ] [– sin θ] + [sin θ] [cos. θ] =0 and p.x/1/p.x/ 1= (p.x/1/p.x1) (p.x1/p.x/1) + (p.x/1/p.x2) (p.x2/p.x/1) = [cos. θ] [cos. θ] + [sin θ] [sin θ] = cos2 θ + sin2 θ = 1. We can see that Lillian Lieber's argument that p.x/1/p.x/2 =0 because function x/1 varies with the variables x1 and x2 and not the variable x/2, isn't correct though it may superficially appear to be correct. We can see that Lillian Lieber's argument that p.x/1/p.x/1 =1 for which surprisingly she doesn't give an explanation also isn't correct except under the circumstances where p.x/1/p.x/1 = cos2 θ + sin2 θ. Using Lillian Lieber's argument we could insist that p.x/1/p.x/1 =0 since function x/1 varies with the variables x1 and x2 and not the variable x/1.

It turns out the curvature tensor itself is worthy of examination. Since it is inner multiplication, i.e., the inner product form of contraction that allows the formation of the curvature tensor itself, the curvature tensor should be considered invalid. We recall that the curvature tensor is Bastr. The final step in the complex derivation of the curvature tensor is the following: Astr – Asrt = (Bastr) (Aa), which upon multiplication of the right-hand side of the equation gives us Caastr. The multiplication that gives us Caastr can be thought of as a kind of inner multiplication. Using the operation of contraction we obtain Cstr, and thus the rule for the addition and subtraction of tensors is maintained. The rule in this case is that if a tensor with three subscripts is subtracted from another tensor with three subscripts the answer must be another tensor with three subscripts. Thus, Astr – Asrt = Cstr.

It is interesting to note that Lillian Lieber maintains that the coefficient of a contravariant vector is the reciprocal of the coefficient of a covariant vector. That is to say p.x/m/p.xs is the reciprocal of p.xs/p.x/m. They do indeed seem to have the characteristics of a reciprocal unless we recall the numerator of a partial derivative refers to a function while the denominator refers to a variable. Actually, when using the rotational formulas in a space of two dimensions the same formulas are produced for both contravariant vectors and covariant vectors.

The formula for a contravariant vector is the following: A/m= (p.x/m/p.xs) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following: A/1= (p.x/1/p.x1) A1 \+ (p.x/1/p.x2) A2 from which we obtain A/1= cos. θ A1 \+ sin θ A2 and A/2= (p.x/2/p.x1) A1 \+ (p.x/2/p.x2) A2 from which we obtain A/2= –sin θ A1 \+ cos. θ A2.

The formula for a covariant vector is the following: A/m= (p.xs/p.x/m) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following: A/1= (p.x1/p.x/1) A1 \+ (p.x2/p.x/1) A2 from which we obtain A/1= cos. θ A1 \+ sin θ A2 and A/2= (p.x1/p.x/2) A1 \+ (p.x2/p.x/2) A2 from which we obtain A/2= –sin θ A1 \+ cos. θ A2.

Thus, we can see that when using rotational formulas in a space of two dimensions a contravariant vector is essentially the same as a covariant vector. Lillian Lieber may want us to believe the relationship between the vectors is reciprocal in order to support her arguments concerning tensor calculus operation known as contraction.

90. What are your thoughts on the famous solar eclipse experiments of May 29, 1919 that measured the bending of the light rays from distant stars by the gravitational field of the sun?

The solar eclipse experiment was one of the three types of experiments that were suggested to provide rigorous tests for Einstein's theory of general relativity. The results from the solar eclipse experiment suggested that general relativity was correct. The reporting of these results by many of the world's leading newspapers helped catapult Albert Einstein into worldwide fame. The details of the experiment are difficult to understand.

It would be helpful to first to approximately calculate how large the disk of the sun is in the daytime sky. This approximate calculation is in degrees. We use the properties of triangles to make this calculation. We draw a line from the Earth to the center of the Sun. We say this distance is equal to 93,000,000 miles plus the radius of the Sun, which is 432,500 miles. We obtain 93,432,500 miles for the length of the longer leg of the right triangle we are constructing. The shorter leg of the right triangle is a line segment from the center of the sun to the edge of the solar disk that is perpendicular to the longer leg of the right triangle. Is length is 432,500 miles. Using the Pythagorean Theorem (a2 +b2 = c2) we can calculate the length of the hypotenuse of this triangle, which runs from the Earth to what we might view as the North Pole of the Sun. We have (93,432,500)2 \+ (432,500)2 = c2 = 8,729,819,112,250,000 and by taking the square root we obtain c = 93,433,501 miles. We now know all the lengths of the sides of the right triangle, and we know the degree measure of the right angle, which is 90 degrees. The acute angle that is the smallest angle of this right triangle gives us the degree measure of the radius of the solar disk as seen in the daytime sky, or in other words the degree measure of half the solar disk as seen in the daytime sky. Using the law of Sines we can determine this acute angle. We should note that the angles A, B, and C are opposite of their respective sides so that angle C is opposite side c and angle B is opposite side b. Side c is the hypotenuse of the right triangle so angle C is the right angle of the right triangle. Side b is the radius of the Sun, and it is opposite the acute angle we are trying to calculate. The law of Sines is a/sin A = b/sin B = c/sin C. We will use the following portion of the law of Sines c/sin C = b/sin B, which gives us 93,433,501/sin 90° = 432,500/sin B or since the sin 90° =1 we obtain 93,433,501 = 432,500/sin B. Therefore sin B = 432,500/93,433,501 or sin B = .004629. The angle with the sine of .004629 is the acute angle of .265°. This degree measure gives us the degree measure of half the solar disk as seen in the daytime sky so the entire solar disk would measure .53° in the daytime sky. This seems very small. I would have estimated the solar disk would be around 10°.

We can use a similar procedure to calculate the size of the moon in the sky. In this case, the lengths of the sides in miles of the right triangle are a = 239,937 and b = 1,080 and c = 239,939. So by the law of Sines, we have 239,939/sin 90° = 1,080/sin B from which we obtain sin B = 1,080/239,939 = .0045. The acute angle whose sine is .0045 is an angle of .258°. That is the degree measure of half of the lunar disk as it appears in the sky so the entire lunar disk would measure .516° in the sky. The degree measure for the sun is .53° while the degree measure for the moon is .516°. So in this approximation, the apparent size of the Moon is .014° smaller than the apparent size of the Sun.

Can we convince ourselves that these approximations are accurate? Let's take a paper disk with a radius of 1/8 inch. Let's connect a very thin, rigid, steel rod, which is 30 inches long, perpendicularly to the center of this disk. The 30 inch long rod and a radius of the paper disk form the legs of a right triangle so by the Pythagorean Theorem the hypotenuse would be (30)2 \+ (1/8)2 = c2 = 900.0156 and therefore c = 30.0003. Using the portion of the law of Sines, which we find useful, c/sin C =b/sin B we obtain 30.0003/sin 90° = (1/8)/sin B. Therefore sin B = (1/8)/30.0003 or sin B = .0042. The angle with the sine .0042 is .241° so the degree measure of the entire paper disk at a distance of 30 inches would be .482° which is slightly smaller than the degree measure for either the Sun or the Moon. This is interesting because if you go outside on the night of a full moon you can make the following observation: close one eye and extend one arm then hold a slip of paper about ¼ of an inch wide in the hand of the arm you have extended. The slip of paper ¼ of an inch wide should cover the moon. This indicates that our approximations are roughly correct.

What use can we make of our approximation that the degree measure of the sun in the daytime sky is about .53°? It seems a typical measure of the degree of radial displacement of a ray of starlight by the gravitational field of the sun for the solar eclipse experiment of May 29, 1919 was .87 arc seconds. In viewing reproductions of the solar eclipse photographs of May 29, 1919, it appears that the sun is about 2 inches in diameter. If we say that 2 inches is approximately .53°, then 1° is about 3.8 inches and 1 minute of arc is approximately .63 inches. One second of arc would be about .001 inches or about one thousandth of an inch. In the solar eclipse photographs from May 29, 1919, the readings for radial displacement of the stars are typically less than one second of arc. It seems that readings with a precision of about .1 arc seconds would be necessary. That would indicate precise measurements must be made on the order of .0001 inches or about one ten thousandth of an inch.

Increasing the size of the Sun in the photograph would decrease the precision of the necessary measurements, but the decrease in the called for precision is not that great. For instance, increasing the diameter of the sun to 8 inches leaves us with one tenth of an arc second equal to .00042 inches or about four ten thousandths of an inch. Increasing the size of the Sun in the photograph to 16 inches leaves us with one tenth of an arc second equal to .00084 inches or about eight ten thousandths of an inch.

Einstein gives us an indication that very small measurements are called for in the solar eclipse experiments. On page 145 of his book Relativity: The Special and the General Theory he writes the following: "For a ray of light which passes the sun at a distance of delta sun-radii from its center, the angle of deflection (a) should amount to a =1.7 seconds of arc/delta [sun-radii]. It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical ('curvature') of the space caused by the sun." So if a typical prediction of the deflection of a ray of starlight in the eclipse experiments is .87 arc seconds, a typical Newtonian prediction for the deflection of a ray of starlight would be .435 arc seconds. It seems you would need measurements on the order of a tenth of and arc second to distinguish between the Newtonian predictions and the predictions made by Einstein.

Another difficulty with the eclipse experiments seems to arise when we try to determine the precise method by which the eclipse photograph and the standard photograph are compared. Einstein seems to indicate an informative but perhaps imprecise method of comparison on page 146 of his book Relativity: The Special and General Theory. He writes, "In practice, the question is tested in the following way. The stars in the neighborhood of the sun are photographed during a solar eclipse. In addition, a second photograph of the same stars is taken when the sun is situated at another position in the sky, i.e. a few months earlier or later. As compared with the standard photograph, the positions of the stars on the eclipse-photograph ought to appear displaced radially outwards (away from the center of the sun) by an amount corresponding to the angle a." Before we continue, we should clear up one point. On page 145 of his book Relativity: The Special and General Theory, Einstein writes, "As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter." When a ray of starlight is bent toward the sun by the sun's gravitational field, it appears on photographs to have been displaced radially outwards, away from the center of the sun.

The imprecision in Einstein's comparison seems to arise because no method is given that allows us to determine the term delta, as it appears in the equation for the angle a. Delta is the distance in sun-radii that a ray of starlight would pass from the position in the sky we denote as the (eclipse-photograph-moment) center of the sun. Since the sun is no longer in that portion of the sky, it is not there to displace the ray of starlight radially outward, away from the center of the sun. It is from delta that we calculate angle a. Do we use the second photograph of the same stars and then somehow place the (eclipse-photograph-moment) center of the sun onto the standard photograph to determine delta?

It seems that we cannot use the eclipse photograph itself to determine delta because a ray of starlight that just grazed the edge of the sun would have a delta value of one solar radius and it would be displaced radially outward from the sun 1.7 arc seconds. If we used the eclipse photograph of the star to determine delta we would obtain a value for delta that would be approximately 1.0018 solar radii instead of the correct value of one solar radius.

How do we determine delta using the second photograph that was taken when the sun was not present? If the stars in the eclipse photograph formed a perfect circle and the sun was at the center of this perfect circle, it would be possible to calculate delta from the second photograph. The stars in the second photograph would also form a perfect circle. It would be possible to determine the center of this circle and thus we could determine delta or how many solar radii each star was from the sun. The stars in the eclipse photographs of May 29, 1919 don't form a perfect circle with the sun in the center instead they form a quite irregular pattern. So, in the second photograph, they will form a very irregular pattern, as well. Where do we situate the sun in this irregular pattern of stars in order to determine delta?

On further reflection, the stars in the eclipse photograph would not need to form a perfect circle around the sun in order to determine the proper center of the sun for the standard photograph. We would only need the two following conditions: 1. two stars that were the same distance from the sun and situated due north and due south 2. Two stars that were the same distance from the sun and situated due east and due west. There do not appear to be any pairs of stars that meet this condition in the eclipse photograph. It seems any two pairs of two equidistant stars that formed an orthogonal pattern could be used to determine the proper center for the sun in the standard photograph, but there do not seem to be any stars that meet this condition.

If we knew the precise coordinates of the sun at the precise moment when the eclipse photographs were taken, would this give us the information we needed? We would have the right ascension and declination of the sun, but how would this help us place the sun in the second photograph? The second photograph could have been taken when the stars in the eclipse photograph were at some distance from the precise coordinates of the sun when the eclipse photographs were taken. If we insist that the stars in the second photograph must be very near to the precise coordinates of the sun when the eclipse photographs were taken, we are still left with the question of precisely where should this irregular pattern of stars be in relation to the precise coordinates of the sun as it appeared in the eclipse photographs.

The stars in question with regard to the May 29, 1919 eclipse experiments are a portion of the constellation Taurus, the bull, known as the Hyades. They form a V-shaped cluster that marks the bull's face. It is interesting that Taurus is the second sign of the zodiac, yet in May the sun is in Taurus. Since the zodiacal calendar begins on March 21 with Aries, the ram, and the second sign of the zodiac, Taurus, runs from April 20 to May 20, you might suspect that by May 29 the sun would be in Gemini, but the signs have slipped westward about one full division, because the position of the earth's axis has changed in two millennium.

It seems the problem of placing the center of the Sun in its proper place in the standard photograph could be solved this way. The eclipse photographs were taken in the daytime in the spring. The astronomers were looking beyond the sun at what would be seen normally as the fall sky. Perhaps, one could find a fall day the length of whose night (sunset to sunrise) was equal to the length of the daylight (sunrise to sunset) on the spring day of May 29, 1919, the day of the eclipse. One could measure the time interval between sunrise and the taking of the eclipse photograph, and next allow the same time interval to pass after sunset of the fall day whose night is the length of the daylight length of the spring day May 29, 1919. At the appropriate moment take the standard photograph. This method assumes that the constellation Taurus would be rising in the eastern sky at sunset just as it was rising in the eastern sky at sunrise on May 29, 1919. This suggests another method to solve the problem.

Perhaps, another way to solve the problem would be to determine the day when the constellation Taurus rises in the eastern sky at the same time that the sun sets in the west. As Taurus makes its way across the sky wait the same interval of time as between sunrise on May 29, 1919 and the taking of the eclipse photograph, then at the appropriate moment take the standard photograph.

91. Have you observed any interesting feature of cement sidewalks recently?

I was walking along some side streets in Elmwood Park that are close to Route 46. Walking west on Martha Avenue, I crossed over Fleisher Brook. There I caught a glimpse of a Baltimore oriole where the brook ran under the street, and then turning north onto Miles Street, I crossed over the Garden State Parkway. In the vicinity of Miles Street there is an elementary school. I noticed that the sidewalks around the school were new, but despite that, some of the ends of the slabs of cement had heaved up and were an inch or so higher than their neighbor. Strangely, the edges of the raised slabs of cement had been ground by some power tool so that they were once again flush with their neighboring slabs. I suspect this was done because the sidewalks were near the elementary school where many young and trip-prone children walked and ran to and from school. The new slabs of cement that made up the sidewalk were white and not profusely dotted with the little, black, angular stones that you see on older sidewalks. I had assumed that new sidewalks were made of cement that did not have little stones added to it, but a close examination of the ground down portions of the new sidewalk revealed that it also was profusely dotted with little, black stones. It seemed remarkable that all of the little, black stones sank below the surface of the cement and were only revealed when the surface of the sidewalk had been ground away.

92. Can you comment on Arthur Eddington's explanation the deflection of a light beam by the gravitational field of the sun according to Einstein's theory of general relativity?

If we study Eddington's book The Mathematical Theory of Relativity, we find that in section 41, "The Deflection of Light," he gives a reader a concise, mathematical explanation of the deflection of a ray of light from a distant star by the gravitational field of the sun. His opening statement is, "For motion with the speed of light ds = 0, so that by eq. (39.62) ,  and the orbit given by eq. (39.61)  reduces to  eq. (41.1). The track of a light-pulse is also given by a geodesic with ds = 0 according to eq. (15.8) ds = 0. Accordingly the orbit given by eq. (41.1) gives the path of a ray of light."

The notion that ds = 0 for motion with the speed of light can be challenged by questioning whether Einstein adequately proved his equation  as presented in his special theory of relativity. But, if we assume ds = 0, then dividing  by 0 gives us  and dividing m, the mass of the sun, by infinity squared reduces the term  to 0. We should note that Eddington uses a clever calculation to determine that the mass of the sun is 1.47 kilometers.

Eddington opens the next paragraph with this sentence: "We integrate by successive approximation." Why must we integrate by successive approximation? Is it because the differential equation denoted as eq. (41.1)  cannot be solved? We should note that  , and r is a function of  . We could write a function such as  and since we know that u is a function of  , as well, we could also write  in which case r would be represented as .

Eq. (41.1) states the second derivative of the function u with regard to the variable  added to the function u equals three times the constant m, which is the mass of the sun, multiplied by the square of the function u. The problems with solving this differential equation are interrelated. The first problem is: how can we add the two terms on the left-hand side of the equation together so that they produce u2 on the right-hand side of the equation. This is especially difficult since finding the first derivative of an equation usually decreases the power the equation is raised to by one and finding the second derivative of an equation decreases the power the first derivative is raised to by one for a total reduction in the power the equation is raised to of two. For example, let  the first derivative is  , which reduces the power ϕ is raised to by one, the second derivative is  , which reduces the power ϕ is raised to by one, yet again for a total reduction of two powers. So continuing with our example, how can we manipulate  so that they equal  ? First, it would seem we need to get the second derivative and the function itself raised to the same power, and second, we need that power to be double that of the power of the function itself.

It is possible by some clever manipulation to add the two terms on the left-hand side of the equation together so that they produce u2 on the right-hand side of the equation, but here is where the interrelatedness of the problems occurs: producing the value u2 seems to make it impossible to produce the constants 3 and m. The value u2 does not need the constants 3 and m because they have already been provided by the right-hand side of the equation itself, and, in fact, any attempt to provide them will increase the value of the constants on the right-hand side of the equation to an improper value such as 6m2. But, in order for the addition of  to produce the constants 3 and m so that their sum can equal the right-hand side of the equation, they must somehow be provided for in the function u. If they are somehow provided for in the function u, they will appear in u2 and thus give the already provided for constants, 3 and m, an improper value. Also, the introduction of the necessary constants in the function u to produce the constants 3 and m upon the addition of the terms  seems to make it impossible to produce the value u2 because the introduction of the constants transforms the addition of the terms  into a form other than the one single desired term.

Let's see how we can produce the value u2 from  . We cleverly let  equal two equal quantities. We let  . When we differentiate, we will use the term  , but when we add terms and square the function u, we will use the term  . We should note that when we differentiate the constant 1, we obtain zero. Perhaps, it is not a completely rigorous way of trying to solve the equation. The first derivative is  or  . The second derivative is  or  . Thus  equals  + . We can see all the terms have common factor of  so we rewrite the equation as  and since  we obtain  . We should note that with , in order to correctly solve the differential equation we would want to obtain  . Since m=1.47 kilometers, we have made a close approximation with  . A close inspection of this approximate solution will hint at the difficulties in introducing any constants such as m, the mass of the sun, and R, the radius of the sun, into the solution

Eddington next writes, "Neglecting 3mu2 the solution of the approximate equation  is the straight line  (eq. 41.2)." Neglecting the term 3mu2 removes the two interrelated difficulties that make solving eq. (41.1) so difficult. Certainly, the equation  doesn't approximate the difficulties of finding a solution to eq. (41.1). The notion of solving differential equations by successive approximation seems highly questionable in itself. Where do we end the successive approximation? Do we end it with the complete second approximation as Eddington does? It seems the complete third approximation and the complete fourth approximation etc. would produce such complex and tangled equations that they would thwart any attempt to turn them into manageable rectangular coordinates as Eddington does with the second complete approximation.

There is at least one other solutions to the equation  . It is  . We should note the constant R, the radius of the sun, is correctly introduced as a constant of integration. If  , then the first derivative is  and the second derivative is  and thus,  . Using  as the solution to the differential equation  produces the essentially the same results as Eddington obtains from letting  .

If we introduce imaginary numbers into the solution set, we can solve the differential equation  with the equation  , where  . If we try to solve the equation  by letting  , we run into the same kind problems we ran into before when we try to make the second derivative of equation u plus equation u equal 3mu2.

Next Eddington writes, "Substituting this  in the small term 3mu2, we have  ." Eddington's argument that 3mu2 is a small term is presented in section 39. "Planetary Orbits." His argument seems to depend on the misuse of the concept of ratio and the surreptitious introduction of the velocity of light squared into a quotient he claims is a ratio in order to make it into a proper ratio. Eddington writes, "In eq. (39.61)  the ratio of  to  is  , or by eq. (39.62)  [we obtain]  ." We should recall that  . The difficulty is that  is not a ratio, it is a quotient. It is the result of dividing the dividend  by the divisor  . The dividend  has the units of measure  . The divisor  has the units of measure  . The units of measure should cancel in order for  to be a ratio. They do not cancel each other; the units of measure for  are distance2angular velocity2.

Eddington continues, "For ordinary speeds this [ ] is an extremely small quantity—practically three times the square of the transverse velocity in terms of the velocity of light. For example, this ratio for the earth is .00000003."

Since angular velocity squared multiplied by distance cubed equals transverse velocity squared multiplied by distance, we can write the equation  Thus  equals transverse velocity squared or  . The transverse velocity of the earth is 30 km. /sec. The transverse velocity of the earth squared equals 900 km.2/sec.2. So for the earth,  equals 900 km.2/sec.2. Therefore,  equals 2,700 km.2/sec.2. This is not a ratio, and it is not an extremely small quantity. Eddington turns this velocity squared into a ratio and an extremely small quantity by letting 2,700 km.2/sec.2 represent the dividend while the divisor is the velocity of light squared (300,000 km. /sec.)2. This gives us the ratio  which gives us the ratio .00000003. But, this ratio is only obtained by the surreptitious introduction of the velocity of light squared. Perhaps, Eddington goes to such great lengths to convince us that 3mu2 is a small term to assuage any doubts we may have about integration by successive approximation.

Eddington uses his method of solving differential equations, which he terms integration by successive approximation, to produce the complete second approximation which is the following:  equation (41.3). He multiplies each side of this equation by rR to obtain  . Next he transforms this equation into rectangular coordinates by letting  and  and  . He obtains the following equation (41.4)  . This is a very difficult equation to understand. It may be a hyperbola in non-standard form. Eddington uses a clever method to determine that the asymptotes of the curve are  .

The Mathematics Dictionary fifth edition by Glenn James and Robert C. James helps us spot an irregularity in the equation or at least in the modern interpretation of the radical sign. It states, "A square root of a number is a number which, when multiplied by itself, produces the given number. A positive (real) number has two real square roots. . . . The positive square root of a positive number  is denoted by  ." In order to produce the quantity  in the equation for the asymptotes  , Eddington would need to obtain both the negative and positive square root of the quantity  , but according to the Mathematics Dictionary the quantity  only calls for the positive square root of the positive number under the radical.

If we allow the quantity  to denote both the positive and negative square root of the positive number under the radical sign, we run into a problem. The equation  is no longer a function. We should note that the number under the radical sign is always a positive number because it is the sum of two numbers each of which is raised to the second power. The Mathematics Dictionary provides us with the definition of a function. A function is "an association of exactly one object from one set (the range) with each object from another set (the domain)." In examples we are familiar with the x-axis is usually the domain and the y-axis is usually the range. Let's take, for example, the function  . The x-axis is the domain of the function. The number 2 on the x-axis is associated with only one number on the y-axis, the range, and that number is +4. The number  on the x-axis is associated with only one number on the y-axis, the range, and that number is +4. It doesn't matter that the numbers +2 and  are both associated with the same number +4 on the y-axis, the range; it only matters that they are each associated with only one number on the y-axis, the range, and that one number happens to be +4. If the number +2 on the x-axis, the domain, were associated with two different numbers on the y-axis, the range, such as +4 and +9, the equation  would not be a function.

We can see that if in our example, we stipulate that the y-axis is the domain and the x-axis is the range, then our example is no longer a function. This is because one number +4 from the y-axis, which we have stipulated is the domain, is associated with two different numbers from the x-axis, which we have stipulated is the range, +2 and  .

Eddington's equation  is not in the standard form as is our example  so first we will let the y-axis represent the domain since the variable y is on the right-hand side of the equation just as x was on the right-hand side of the equation in our example  . Therefore, the x-axis will represent the range. This is the reverse of the way the domain and range are usually represented, but functions usually have the variable y on the left-hand side of the equation. A number from the domain, the y-axis, should only be associated with one number from the range, the x-axis. Let's examine the number 0 from the domain. If we let  , we find that it is associated with two numbers from range, the x-axis. They are  . So  is not a function when the y-axis represents the domain and the x-axis represents the range.

Now we let the y-axis represent the range and the x-axis represent the domain. This is the more familiar situation. We let 0 represent the variable x in the domain, the x-axis. When the number we choose from the domain is 0 that is to say  we find it is associated with two values in the range  So,  is not a function when the x-axis represents the domain and the y-axis represents the range.

So, it seems  is not a function as long as the quantity  represents both  , which it must to obtain the asymptote equation Eddington desires  Also, when we let  in both the equation  and the equation for its asymptote  , the equation and its asymptotes intersect.

Next, Eddington states without explanation, "The small angle between the asymptotes is (in circular measure)  ."

Lillian Lieber provides an explanation of how to obtain  from  in her book The Einstein Theory of Relativity. She uses the formula for finding the angle between two lines which is  . The desired angle is α, and the slopes of the two lines are m1 and m2. Using this formula she obtains  . From the tanα Lieber claims it is easy to find that  . There is an interesting feature in these calculations. Since tanα is equal to  , we can write  . This is so because in calculating the tanα from a complex fraction the denominator of the simple fraction that is the numerator of the complex fraction and the denominator of the simple fraction that is the denominator of the complex fraction can be constructed so that they cancel one another. This feature allows sinα to equal  which can be rewritten as  as long as we designate that  . When we calculate α from this equation α is equal to approximately 179 degrees. It seems this problem stems from the fact that the equation  generates a negative value for the angle α. Using the equation  the value for angle α seems to equal approximately  . This value of angle α presents a difficulty because angle α is one of the three angles of a triangle. Triangles are not usually composed of angles with a negative measure. This difficulty can be overcome by making sure that the slope m1 of line l1 is positive and recalling that 1+m1 (− m2) is a positive quantity.

Next, Lillian Lieber reminds us that since α is a very small angle, its value in radian measure is equal to sinα. So  and since m=1.47 kilometers and R= 697,000 kilometers the term  can be neglected and we have  . Thus, Lillian Lieber has demonstrated how it is possible to arrive at Eddington's assertion that  .

In conclusion, it seems the outstanding difficulties of Eddington's explanation of the deflection of light from a distant star by the gravitational field of the sun are that the differential equation  is not solved, except by Eddington's questionable method of successive approximation and the asymptotes of the equation  are calculated in a confusing manner. Plus, it is not clear if  is a function.

93. What are your early memories?

I remember catching a crow. I began by shaking the slender tree on which it was perched until it took flight. I followed the crow and shook every subsequent tree it perched on. I shook the trees until it was forced to take flight again and again. Finally, it became so tired that when it launched itself from a tree branch, instead of taking flight, it merely fluttered to the ground where I scooped it up. It was so long ago that the trees were different from the way they are now. They were all slender like saplings. They were like the stands of swamp maples you can see today. Swamp maples often grow in large groups in what was formerly a waterlogged section of a pasture that now has been allowed to return to a swamp.

I remember riding in the family car. It was around dusk; the season of the year escapes me. We were traveling on Route 46 near the intersection with Fox Hill Road. I looked out the window and saw another car. It was pulling a little red wagon behind it. The black wagon handle was tied to the bumper of the car with a brown rope. There were two little children in the wagon.

I remember walking home from a friend's house in the late fall. My brother and I were walking along Fox Hill Road as it was becoming dark. We were passing Leonard's Swamp, which was eliminated when Interstate Route 80 was built. I noted that the knees of my jeans were stiff. They had frozen. When I was playing in the sandbox at my friend's house, the knees of my jeans got wet, but the heat from my body kept the water from freezing. Now, that I was walking home, my jeans weren't pressed against my knees. The small amount of air space between my jeans and my knees allowed the knees of my jeans to freeze. I looked up and noticed the moon was in the sky; I hadn't noticed it was there before.

I remember falling through the ice on Leonard's Swamp. The water was shallow enough for me to walk to the shore. I didn't want to tell my parents that I had fallen through the ice so I sat outside on the milk box hoping my clothes would dry.

I remember germinating some acorns in a dented, nearly discarded saucepan. My father threw the dark water and the tangles mass of roots away. Perhaps, it reminded him of the way my mother often grew plants from cuttings of other plants, but it seems, to my father's irritation, she never kept her indoor plants well organized or neat in any way. Also, there was not much that could be done with germinated acorns. Today, I often think to myself that if I could only develop a taste for acorns, pinecones and English plantain, I could live off of the waste I remove from my lawn.

I remember trying to ride my tricycle down some cement steps that lead to our driveway. We lived in Rainbow Lakes then. I injured myself. I was rushed to the hospital where I believe I received three stiches to close the wound on my forehead. The scar is no longer visible. It is probably obscured by old acne scars.

I remember my mother filled many books with green stamps. Later we went to the store where you could purchase products, probably mostly household items, with the green stamps. I don't remember the items she purchased.

I remember showing my mother that if she repeatedly pressed down with her foot on a small, steel cylinder located on the driver's side on the floor of the car near the other foot pedals, she could pump a stream of water onto the windshield and then operate the windshield wipers so that the windshield would be cleaned. I observed over the years that she apparently never used the windshield washing mechanism, and has not till this day although today the windshield washing mechanism, specifically the pump, is now powered by an electric motor.

I remember we had a small, green, portable, gas fueled stove that we took with us on family trips. The idea was to use the stove at the various rest stops we stopped at to heat coffee and other items. The portable stove had two small burners and a red gas tank you had to pressurize with a small, hand-operated, air pump. The pump was small. The part you moved in and out of the tank to build up pressure was smaller than a pen.

I remember that I used to roll Arnold white bread into little balls, and then I would eat the little balls. Perhaps, this demonstrates the creativity, naiveté and only minor concern for hygiene of a child. I found salt from the sweat of my hands improved the taste of the bread.

I remember in the summer after school let out I wanted to walk around with bare feet. My hope was that I would develop calluses so that I could walk on my gravel driveway without discomfort. I didn't develop the necessary calluses.

I remember my father would work in his vegetable garden wearing an old pair of dress shoes. He may not have ever worn sneakers as an adult.

I remember I had a fishing pole that consisted of a rod and reel. I practiced casting with the fishing pole in the back yard. I had attached a heavy lure to the line, perhaps a lure they called a jitter bug. As I was reeling in the lure, it travelled along the lawn in jerks. My dog pounced on the lure, and the hook got stuck in his paw. We tried to get the hook out of his paw, but he would not hold still to let us work on his injury. He was in such pain that any attempt to remove the lure from his paw only increased his pain and made it impossible to hold him still. We took him to the veterinarian and the lure was removed.

I remember swinging on our backyard swing. It was homemade with a wooden seat and chains that were attached to the limb of a tree. I was swinging on the swing one day when all of a sudden the seat split. You might think that because I was holding onto the chains I would lower myself to the ground and bring myself to a stop by scrapping my feet on the ground. Instead, I flew into the air and landed on the ground; I was unhurt.

I remember one of the first science fiction books I read was Utopia Minus X. I was unfamiliar with the word utopia. For some reason, I pronounced it as up-tog-(sounding like log)-ah. I remember the plot featured elements of Einstein's twin paradox. I vaguely remember that an astronaut traveled away from the earth at near the speed of light and returned to the earth at near the speed of light. The astronaut had aged only a year or two, but on the earth 200 years had elapsed, and the earth had become a dystopia of sorts though perhaps the word dystopia was unheard of when Uptogah Minus X was written.

I remember constructing a box trap. It was very heavy. It had one wooden floor onto which the boards that formed the framework of the box were nailed. The boards that formed the framework for the top, back and sides of the box had rectangles of wire mesh nailed to them. Then there was another wooden floor, which was balanced on a fulcrum, above the main floor. This floor had a long, rusted, metal rod attached to it. When the animal moved to the rear of the trap, the second floor, which was balanced on the fulcrum, would lower at the rear end under the weight of the animal; this would draw the metal rod back slightly. This action would spring the trap since the metal rod held the heavy front door of the trap open. The slight backward movement of the rod allowed the heavy door to fall down trapping the animal in the box. I dragged the trap out into the woods, but not too far because it was heavy. When I came back to inspect the trap the next day, I saw that I had caught a skunk. I tried to think of a way to release the skunk without getting close enough to get sprayed by the skunk. I couldn't manage to open the door with a long stick or by any other means. Finally, I walked over to the trap and pulled the door open. The skunk slowly walked out of the trap without releasing any spray.

I remember getting stung by a bee. I was playing by the see-saw. I believe the bee's nest was in one of the pipes that made up fulcrum of the see-saw.

I remember that there was a hail storm on the 4th of July, 1976, the Bicentennial.

One winter morning after a snowstorm I was taking a walk by the shores of South Fayson Lake in Kinnelon. It was years before the bicentennial. Very near the shore, I saw a cat frozen to the ice. It was lifeless. When I climbed up the steep slope from the lake to Edgemere Terrace, a car slowed down, and the driver asked me a question. The driver and his passenger were an elderly couple. He asked me if I had seen a dog running loose. I told him I hadn't seen any dogs. Soon after he had driven away, I realized it was a dog I had seen frozen to the ice, not a cat.

I remember the cellar of our house in Fayson Lakes. At the top of the cinderblock walls there were four or five small, rectangular windows evenly spaced on three of the four walls. The fourth wall adjoined the garage so there were no windows along that wall. The windows were below the level of the ground. This problem was solved by digging away a semicircle of earth around the window that was about two feet deep. The semicircle had a radius of about 18 inches. The earth was held back by a sturdy semicircle of metal. Perhaps, the metal was aluminum, or perhaps, it was some rust proof alloy. As I recall, the metal semicircle had ridges to give it added strength. I often saw toads at the bottom of the window well. At the time, I was fond of constructing "forts." One of my forts was merely a hole in the ground covered with a few boards. It was located in a far corner of our backyard. Since I was happy inside my "forts," I believe I assumed the toads were in their version of "forts," and they were happy as well. I assumed they were happy catching insects or some other creatures for food and could come and go as they pleased. Looking back on the situation, it seems clear to me that the toads were trapped in the window wells because the sides of the window wells were too high for them to jump over. If I dig a hole in my yard several feet deep and with steep sides, I believe I will entrap a toad.

I remember a certain kind of Ace comb. It was short about four or four and a half inches long, but it was wide about three inches wide. It had short teeth about half an inch long that were very closely spaced. The teeth ran down both sides of the comb. Perhaps, it was designed to fit into the back pocket of a boy's jeans. I believe it was designed to help maintain a particular hair style. The style was to comb your bangs straight down over your forehead, then at the eyebrow line the hair was supposed to somehow make a small, sharp curl to the right. I believe the short and closely spaced teeth of the comb were supposed to help produce this tight curl.

When I was young, I remember intermittently hearing a very high pitched sound. It seemed to rise in pitch until it went beyond the range of human hearing. I could never figure out the source of the sound. One day I heard the sound for a short time. A short time later, I heard the rumble of a truck, and I saw a home heating oil delivery truck make its way down the street. It occurred to me that the sound was a warning. It warned the oil delivery men when the in-ground oil tank was about to be filled with oil. The in-ground oil tanks had a vent and as the tank was filled with oil air rushed out of the vent. Perhaps, the tank and vent systems were designed so that when the tank was full air rushed out of the vent so rapidly that a high pitched whistle was produced. I never seem to hear that high pitched whistle anymore. Perhaps, it is because the in-ground oil tanks have been replaced with above ground tanks that have a visual indicator that warns when the tank is full, or perhaps, I have lost the ability to hear high pitched sounds.

I remember a neighbor a few years older than myself told me that he thought the Beatles album Abbey Road was recorded through oatmeal. It would be several years before I listened to the album myself. Of course, now, I believe he must have been referring to the song "I Want You (She's So Heavy)", which employed a pink noise generator for some of the effects.

I remember looking through the medicine cabinet for some acne treatment. I came across a liquid that seemed like a mixture of Pepto-Bismol and Calamine lotion diluted with alcohol. For some reason, I used it. It didn't work, but then again nothing ever actually worked.

Once, in elementary school during a spelling bee, I spelled the word circumference with an s. I was quite embarrassed. I never actually learned the art of spelling. It's more difficult than sharpening nail clippers and more practical than fixing light bulbs or repairing sponges.

I remember carrying my school textbooks home. I would have homework in five or six subjects so I would carry five or six textbooks home. They would be precariously positioned on my loose leaf notebook in order of size from largest to smallest. On top of the pile of textbooks, I would place a paperback dictionary; I believe it was a Scholastic Books dictionary. The cover of the dictionary was light blue.

I remember my mother used to pack me a lunch when I was in elementary school. She often prepared a ham and cheese sandwich for me. Also she often gave me a Hostess fruit pie. As I recall, the crust was coated with a sugar glaze. The fruit within the crust seemed very sugary, as well. It's odd that the world hasn't seemed to get markedly better merely from the reduction in the amount of sugar we consume.

I remember that there was a licorice candy lozenge advertised on TV called Good & Plenty. I believe the slogan was Good & Plenty is plenty good. On the commercial an animated train engineer named Choo-Choo Charley promoted the candy. It seems that shaking the candy lozenges while they were inside their box was a reasonable imitation of the sound of a steam powered train engine starting up and chugging along the railroad tracks.

I remember some of my friends and me were smashing small rocks into pieces by hitting them with a pipe. We would hit them with the open end of the pipe. Perhaps, we were influenced by having seen a pile driver in action driving telephone pole like logs into the ground to stabilize it for some sort of construction project. We either saw this on TV or in real life. Someone would hold the upper portion of the pipe then forcefully bring the lower end of the pipe into contract with the small stone. For some reason, I decided to hold a small rock while someone struck it with the pipe. Unfortunately, my left index finger was hit by the pipe. The fingernail seemed to take the force of the blow. The injury was not severe but it took a long time to heal. The fingernail turned black and fell off, but eventually I grew a new fingernail. The new fingernail was slightly different from the one it had replaced.

I remember having a gray, wool sweater—a pullover. As I recall, I put it in the washing machine and washed it in warm water. Next, I put it in the dryer; I don't remember the setting. When I removed it from the dryer, it had shrunk.

I remember that in school a teacher would often explain a particular topic then the teacher would pose the following question: Does everyone understand the topic I have just explained? The students that didn't fully understand the teacher's explanation were expected to raise their hands and divulge which portion of the teacher's explanation was unclear to them. As I recall, no student ever raised their hand and admitted they didn't understand certain aspects of the explanation. I have forgotten almost all the difficult topics that my various teachers explained. But, several of the topics have reappeared as problems I have encountered in the workplace, and no one in the workforce could solve these problems and often they weren't recognized as problems.

I remember I had a calendar on which every week I wrote down the tests I was going to take that week. There seemed to be four or five tests per week, but, perhaps, I am misremembering. There was a spelling test every Friday, I believe. I remember spending hours trying to memorize the spelling of the words, but I never actually became competent at the art of spelling. To this day, the spelling of words like necessary and successful give me difficulty.

When I was young, I was intrigued by the fact that potato sacks burned with a green flame. The sacks were made of thick paper and had a small window covered with netting that allowed you to see the potatoes that you were buying. Perhaps the potatoes were treated with some copper based preservative, and that is why they burned with a green flame. Copper sulfate is a blue substance that turns white when heated according to the dictionary, and it is a germicide. I have noted that the limbs of the Chinese chestnut tree produce a blue flame when burned. The blue flame may be produced by some oil in the wood.

I remember learning in school that Abraham Lincoln believed that when you could see the smoke from your neighbor's chimney the community was getting too crowded and it was time to move to an area with less people. When I began supplementing my home's heat with a wood stove, I noted that houses could be relatively close together and you would not see the smoke from your neighbor's chimney. There were several factors to consider. If your neighbor was burning well-seasoned wood in an efficient manner, there would be little smoke to begin with and what little smoke there was would quickly disperse into the atmosphere. Also, if there were tall trees and rolling hills in the area, these would serve to mask the smoke from a neighbor's chimney. It seems that a long, distinct plume of smoke rising from the chimney of a log cabin is something you see more often in paintings than in real life.

I don't remember too much about bullies. I recall once I was on my hands and knees; perhaps, I was playing with green, plastic army men. David Cutler was standing over me, and he clamped my head between his lower legs. He jumped up and down presumably applying some sort of "ear burn" to my ears. I recall that I would sometimes I would disparage myself by saying something such as "yes, I am stupid" if that is what I thought a bully was about to say. Somehow, I must have thought that beating them to the punch would solve the problem.

I remember when I was ill with a cold or the flu my mother would often give me some teaspoons of a medicine that she called blue moon presumably because of its blue color. As I recall, it was prescription medicine. The taste of NyQuil reminds me of the taste of blue moon.

I remember my older brother read Erich Fromm's The Sane Society as I recall it was in the late 1960s. He underlined all the portions of the book he thought were important. Perhaps, high lighting markers hadn't been invented yet, or at least they weren't widely available. He had underlined so much of the book's text that I thought he would have saved ink if he had underlined only the unimportant portions of the text. It has only recently been acknowledged that Erich Fromm was an agent of the U.S.S.R., or perhaps, it is still a contentious question as to whether he was or wasn't an agent of the U.S.S.R.

I remember my grandfather kept knives or at least knife blades stored in a jar of clear oil of some kind. These special knives or knife blades were so sharp that somehow merely exposing them to the air would impinge upon their sharpness.

I remember riding my bicycle at sunset. The sun had disappeared behind a ridge. The houses along the top of the ridge were still illuminated by the rays of the setting sun. There was one house that had a turret; I thought that a witch must live there.

I remember in elementary school there was a student who went skiing in Europe over the Christmas vacation. Perhaps, he went to Switzerland; wherever he went he came back with a dark tan. He seemed cool.

I remember my older brother had special silverware when we were young. Maybe it was merely a fork with a few swirling lines engraved in the handle. But, I resented that small amount of special treatment he received.

When my younger brother Steve was quite young, he was impressed with the moon landings. I believe he saw a photograph of a solar wind experiment set up on the moon by the astronauts. In the photograph, the experiment appeared to be not much more than a plastic bag attached to a pole that was stuck into the lunar soil. One winter day when the backyard was covered with snow so that perhaps it somewhat resembled a lunar landscape my brother set up his version of the solar wind experiment. He later told me that our father had expressed misgivings about his project. When my brother Steve was young, he would amuse himself by drawing a basketball court on a piece of paper. Then he would cut out the figures of some basketball players from pictures he had in some of his sport's books. He would then deploy the cut out figures on the basketball court he had drawn on a piece of paper. I can't remember exactly but I think myself and other family members looked upon such activities with disdain. I deeply regret my behavior and the behavior of some members of my family.

I remember when a law was passed in NJ that made it mandatory for both drivers and passengers to wear seat belts. At first, my mother would only drape the seatbelt across her waist; she would not clasp it shut. I can't remember if our seatbelts had shoulder straps at the time. Before the law was passed I must not have used a seatbelt, but now I have no memory of not using a seatbelt.

I remember the push to adopt the metric system that occurred in the mid-1970s. At the time, I was against conversion to the metric system. I thought that people could use our more cumbersome system with little difficulty. I thought they would have the time and the inclination to master the somewhat arcane details of the various conversions. Was I some kind of unconscious sadist? Now, everyone's life is so hectic there's no time to do anything at all let alone study something like avoirdupois weight. Incidentally though I would like to know why the Corning mining ton is 352 pounds heavier than the short ton and 112 pounds heavier than the long ton.

I remember when mini-vacs (small vacuum cleaners) first came out. They were powered by rechargeable batteries. I had two mini-vacs. I used them until the rechargeable batteries would no longer hold a charge. Then I purchased a number of rechargeable flashlight batteries. I strapped a number of them to the outside of each of the mini-vacs. Then on each min-vac I wired the batteries I had strapped onto it to the motor of the min-vac. It was cumbersome but the mini-vacs worked though I must have thrown them out years ago, perhaps, after the rechargeable flashlight batteries lost their ability to hold a charge.

I remember the drain of our kitchen sink used to get clogged. I would work on unclogging it with drain unclogging liquids, a plunger, a drain snake and other devices. I had heard that pouring Clorox bleach down a drain could unclog it so I tried that. After a good deal of effort, I gave up and called a plumber. He diagnosed the problem. He said the kitchen sink did not have a vent pipe. He said that by looking at the three windows and the door in the wall adjacent to the kitchen sink he could tell there was no room in the wall to run a vent pipe. So he reworked the kitchen drain so that it was connected to pipes that were connected to a vent pipe. When I studied the situation, I discovered the kitchen sink had always been connected to pipes that were connected to a vent pipe. If you think about it, three windows and a door in a wall are not an insurmountable barrier to running a vent pipe within a wall. Obviously, you would run the vent pipe in the section of the wall that was not occupied by the windows and door. With further study, I came to the conclusion that the problem had nothing to do with vent pipes. Instead, I believe, the portion of the kitchen drain pipe that was attached to the u-shaped trap was old; it was old and it was peeling off thin layers of metal. The thin layers of metal that the pipe shed would clog the drain. Then my efforts to unclog the drain would collapse the thin layers of metal into a plug of softened metal and kitchen waste. My efforts also scoured the pipe so there was no evidence it was peeling. At some point, I removed the pipe from the drain without first trying to unclog the drain. When I examined the pipe I saw that it was peeling. I replaced the peeling pipe with a section of plastic pipe. This seemed to solve the problem. The plumber could not diagnose the problem because my efforts to unclog the drain previous to his visit hid the evidence of an exfoliating pipe.

I remember not too long ago burning some old 78 rpm records in my wood stove. I don't know what the records were made of perhaps acetate. They gave off a thick, black smoke with a pungent smell. A neighbor came to the door to express her concern that something might be amiss with my wood stove. I convinced my brother to assuage the neighbor's concerns. He related to me that she was concerned because her son had asthma.

Some time ago, there was a long drought that lasted from the end of summer into the fall. The stream near my house, Beaver Brook, was reduced to a series of puddles of various sizes with water trickling between them. When the leaves fell, they covered the surface of the stream. I was walking by the stream when the moon was out during the day. From the right position, I could see the reflection of the moon under an upturned leaf floating on the chain of puddles the stream had become. I thought it was odd that you could see the reflection of the moon on a stream covered with leaves. On a smaller, nameless stream that is a tributary of Beaver Brook, I once saw a double reflection of the moon. The kind of reflection you see when you see the reflection of the moon through a double windowpane of glass. At that same point where I used to cross that nameless stream walking my dog, I used to encounter what I thought were small birds which I had startled into flight. One evening I turned around to look for the birds I had startled into flight, and I saw that they were bats.

Recently, I was taking apart an old mattress and box spring so I could put them in black, plastic garbage bags and put them out for the garbage men to take away. First, I worked on tearing apart the mattress, which was old and horribly dusty with a dust composed of very small, dirty beige particles. Breathing the dust irritated my eyes, throat and lungs. It took many bags, but I finally got the old mattress out for the garbage men to pick up. I waited many days, before I began work on the box spring. First, I cut and tore off all the fabric. Then I pulled off all of the padding. This left me with a wooden frame to which the springs and the spring's metal framework were attached. I used a bolt cutter to cut apart the springs and the metal framework that surrounded the springs. This activity somehow attracted a large spider. The spider seemed attracted by the vibrations the box spring was making as I cut it apart. Could the spider have suspected that it had come across a huge web? As I cut the metal framework of the box spring, did the spider mistake these vibrations for the struggles of an insect trapped by what it believed was a huge web? The cut up springs and metal framework were so heavy and bulky that I had to construct two homemade boxes. A homemade box is one large box made from two smaller boxes. I taped torn open black garbage bags to the homemade boxes and labeled them bulky to forewarn the garbage men.
Several years ago, I was weeding the wild plant garden that runs along the sides of my narrow pond. I glanced across the pond, and I noted that a patch of shadow appeared darker than normal. I returned to weeding the garden, but I thought: the shadow looks strange because a bear is hiding in the shadows watching me and the bear doesn't suspect that I know he is there. I had the idea that if I ran at the bear yelling loudly, he would be surprised and run away. I got up and walked slowly to the other side of the pond. Next, I charged down the path leading to the bear yelling loudly. The bear took off like a shot. I had always heard that bears could run very fast when they wanted to. It was out of sight, but I had learned from experience that when animals flee they often run up and over the steepest embankment so they can disappear from sight. I ran up and over the steepest small hill. As I reached the top of the small hill, I saw the bear 20 yards ahead of me. It was standing still, perhaps, it was as out of breath as I was. I looked at the bear, and it turned its head and looked at me then it walked away.

94. Is there a clever analogy to explain the fact that the tensor operation known as contraction is invalid?

The mathematical operation known as the contraction of a tensor is invalid. Contraction of a tensor is also known as the inner multiplication tensors. This argument can be demonstrated by a somewhat dense mathematical presentation. It is difficult to give a non-mathematical presentation of this argument and to explain why other physicists and mathematicians have not noticed this. Contraction of a tensor is invalid because it makes improper use of the chain rule of differentiation as applied to partial derivatives.

We can make an analogy to two of the tools often used in do-it-yourself automobile repair. The two tools used in do-it-yourself automobile repair can represent the two terms used in chain rule differentiation. The tools are the lug nut wrench and a large slot screwdriver. It seems that whenever I have the lug nut wrench and a large slot screwdriver at hand I am going to remove one of the wheels from my car. I need the large slot screwdriver because my lug nut wrench is an X-shaped version with a different sized lug nut socket at each end and thus no chisel end with which to pry off the wheel cover. I will use the large slot screw driver to remove the wheel cover. I will use the lug nut wrench to remove the lug nuts that hold the wheel on. The wheel removed from the car can be thought of as analogous to the result of employing the chain rule of differentiation. It is difficult, if not impossible, to think of an automotive repair job other than the removal of a wheel that would occur when these two tools are at hand. This may explain why is has seemingly not been noticed that the contraction of a tensor is invalid.

The mathematicians who invented tensor calculus by analogy came up with a use for the lug nut wrench and large slot screwdriver other than removing the wheel. But, instead of clearly stating that they had found another use for the lug nut wrench and large slot screwdriver, they proceeded as though they were removing a wheel from the car. Removing the wheel from the car would be analogous to simplifying a complex tensor, which is the result of applying the chain rule of differentiation. Thus the mathematicians who invented tensor calculus made it appear they had discovered a way to simplify a tensor when in fact they hadn't. If we wanted to make our automotive analogy more complex, we could say that instead of having the large slot screwdriver the mathematicians had one of those new types of screwdrivers that can be described as a kind of star screwdriver with many small points. But, again no one seemed to notice that instead of a large slot screwdriver they had a different type of screwdriver. This may be because whenever both a lug nut wrench and a screwdriver were at hand it was assumed that the screwdriver was a slot screwdriver that would be employed to remove the wheel cover.

The different type of screwdriver represents the second term in the chain differentiation. It seems to be directly related to the first term as in the standard chain differentiation, but in fact its relation to the first term is in name only. It is a screwdriver, but not a large slot screwdriver that is needed in order to pry off the wheel cover. It is a screwdriver, but not the proper kind of screwdriver.

95. Comment on the complex equations found in chapter 2, section 28 "Equations of a geodesic" of Arthur Eddington's The Mathematical Theory of Relativity.

In chapter 2, section 28 "Equations of a geodesic" Arthur Eddington's derives the equation(s) employed for finding the geodesic in any kind of space either flat (Euclidian) or curved (non-Euclidian). His argument is complex and seems oscillate between convincing and exasperating. He blurs the distinction between variables we are familiar with such as s, x1 and x2 and variables that seem to behave both as the familiar variables and as differential variables such as ds, dx1 and dx2. According to the Mathematics Dictionary by Glenn James and Robert C. James, differentials such as ds, dx1 and dx2 are variables. But, they seem to always accompany familiar variables; they aren't quantities that stand alone. They aren't normally raised to the second power. What is the derivative of ds2? There is also the problem of unclear functions. We know, for instance, that  is a function of the variables  , but are the  variables functions of the variable s? Are the  and  variables functions of the s variables, as well? If so, doesn't that make the coefficients represented by  indistinguishable at some level from the variables/differentials  and  they modify?

He begins section 28 as follows.

"We shall now determine the equation of a geodesic or path between two points for which  is stationary. This absolute track is of fundamental importance in dynamics, but at the moment we are concerned with it only as an aid in the development of the tensor calculus.

"Keeping the beginning and end of the path fixed, we give every intermediate point an arbitrary infinitesimal displacement  so as to deform the path. Since ,  and  . . . . (28.1). The stationary condition is  . . . .(28.2)."

Lillian Lieber's version of the stationary condition, i.e., of equation (28.2), which appears in her book, The Einstein Theory of Relativity, is  . She writes, "We know from ordinary calculus that if a short arc on any of these paths is represented by ds, then  represents the total length of that entire path." The  seems to indicate that we take total differential of the entire path length, which is represented by  . Next, we set the total differential equal to zero. The solution of the equation is the geodesic. Lillian Lieber's method of integrating a differential equation and then finding the total differential of that result seems similar to Eddington's method of taking the total differential of a differential equation and then integrating that result. There is the concern though that the equation  is not actually a differential equation, but it merely appears as such.

The equation  employs the summation convention. The summation convention tells us that whenever the same index (subscript or superscript) appears twice in the same term, then we must sum on that index. Since both  and ѵ appear twice in the term  we must sum on both those indexes. The equation  is a generalized version of the Pythagorean Theorem. It can represent all the variations of the Pythagorean Theorem that are possible in Euclidian space and all the possible versions of the Pythagorean Theorem that are possible in non-Euclidian space. In the standard Cartesian coordinate system the Pythagorean Theorem tells us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. We could write the equation  to represent this. Though we seldom, if ever, think of this fact, the coefficient of  is one and the coefficient of  is also one. The quantities represented by  include the coefficients in all the possible versions of the Pythagorean Theorem.

In the Pythagorean Theorem, which we are familiar with, the  terms would equal one. In Einstein's version of the Pythagorean Theorem, which is employed for both curved space and flat space,  is composed of the variables x and y. They are written as x1 and x2. We should note that when we use the equation  to generate the familiar Pythagorean Theorem the term  will represent zero as well as one. Although there are no coefficients of zero in the familiar Pythagorean Theorem, when we employ the equation  to generate it, we obtain the terms xy and yx. To rid the equation of these terms their coefficients must be zero. Of course, they would appear as x1x2 and x2x1 respectively. Just as x2 would appear as x1x1, and y2 would appear as x2x2.

The various values for the term  must be known beforehand and assigned to the term  . The term  is a tensor of rank two. For instance, to generate the familiar Pythagorean Theorem we would write  and  while  and  . In Einstein's version of the equation  , which he employs for curved space, many of the various occurrences of the term  are set equal to zero while four are composed of the variables x1 and/or x2 in a specific formulation.

The equation  appears to be the total differential of the equation  . Here we are assuming that  is not a differential equation, but the kind of equation we are familiar with but in non-standard form. The equation  is in a non-standard form for two reasons. The first reason is the appearance of  on the right side of the equation. In a more standard form, the equation would be written as  . The second reason is the variables themselves, ds,  and  . In a more standard form, they would appear as s,  and  . The variables cannot assume the form  and  because the coefficients that Einstein uses for his curved-space version of the Pythagorean Theorem are in the form of  and  . The coefficients of the variables must be distinguishable from the variables.

The equation for the total differential of a function of several variables is  . From this equation, we can see that   , or more precisely it generates  as well as generating a portion of  . We can also see that  and  . We can also see that  , or more precisely it represents a portion of  . We can also see that  and  . This allows us to conclude that  and  .

We can also see that  since  is formed from variables such as x1 and x2 in Einstein's version of the Pythagorean Theorem.

It is more difficult to accept that  and  since the equations do not mean the differential of the differential of  equals the differential of the differential of  and the differential of the differential of  equals the differential of the differential of  , respectively. Instead,  seems to indicate that the differential of the variable  is equal to the differential of the undefined quantity  , which could represent the kind of variable we are familiar with or it could represent an infinitesimal similar to  . In either case, its precise relationship to the preceding equations remains unclear. The same reasoning applies to  . It seems to indicate that the differential of the variable  is equal to the differential of the undefined quantity  which could represent the kind of variable we are familiar with or it could represent an infinitesimal similar to  . In either case, its precise relationship to the preceding equations remains unclear.

Eddington continues, "which becomes by (28:1)  ," what Eddington has done is perform three operations on the equation  . Then he has placed an integral sign before the new version's right-hand side. Then Eddington set that equation equal to zero, which is called for by equation (28.1)  . The three operations he performed on the equation are the following. First, he divided each side of the equation by 2. Next, he divided each side of the equation by ds. Then, he multiplied each side of the equation by ds/ds. On the left-hand side of the equation multiplying by ds/ds takes the form of multiplying by 1. After those three operations he integrated the right-hand side of the equation and set it equal to 0 as called for by equation (28.1).

Dividing each side of the equation by ds gives the equation the more standard appearance of finding the total differential of the equation  especially when that equation is put in the standard form of  . One of the first steps you would take in finding the total differential of  is to write  . Therefore  and  . Dividing the right-hand side of the equation  by ds accomplishes the same task since  and  . We should note both sides of the equation are divided by ds.

Eddington continues, "or, [by] changing [the] dummy suffixes in the last two terms,  . Applying the usual method of partial integration, and rejecting the integrated part since  vanishes at both limits,  ." Applying partial integration is where Eddington makes his error. He applies partial integration to the term  . The formula for integration by parts that he is apparently using is the following:  . Thus we see that  and thus  and  and  . In the Mathematics Dictionary by Glenn James and Robert C. James, we are told that u is a function of x therefore it follows that in the quantity  and in the quantity  there exists some function of the variable s, but this is not the case. An argument can be made that dxμ and dxѵ are functions of ds, but ds is not s. And, apparently, dxμ and dxѵ are no longer the variables we are dealing with. We are now dealing with the variables  and  . And, the term ds is treated as a variable that is not related to s in the equation with which we began  . In the Mathematics Dictionary, we are told that v is a function of x therefore it follows that the quantity  should be a function of the variable s, but it is not. In the Mathematics Dictionary, we are told that dx is the differential of the variable x therefore it follows that ds should be the differential of the variable s, but the variable s does not appear in the set of equations we are working with.

It is confusing. A term such as  seems to be appropriately read as the division of the variable dxμ by the variable ds. Perhaps, Eddington wants us to read it as the differentiation of the dependent variable  with respect to the independent variable s. What are we to make of a term such as  ? It seems  is the differential of the variable(s)  . The variables  form the coefficients of the variables that Einstein uses for his curved-space version of the Pythagorean Theorem. But, the term  also seems to be a stand in for the variables  and  . So should  be read as the division of the differential of the variable  by the variable ds? Or, should  be read as the division of the differential of the differential of the variable  by the variable ds? Or, should we read  as the differentiation of the variable  with respect to the variable s. Or, should we read  as the differentiation of the differential  with respect to the variable s.

Using these questionable and confusing methods Eddington does produce the equation for determining the geodesic, which is  . Interestingly, Eddington uses this equation for determining the geodesic in his next section "29. Covariant derivative of a vector." Perhaps, Eddington senses that his derivation of the equation for the geodesic is questionable. Therefore, he realizes that its use in deriving the covariant derivative of a vector would make that derivation questionable also. So, Eddington provides the reader with section "31. Alternative discussion of the covariant derivative." In section 31, Eddington provides the reader with a method of deriving the covariant derivative that does not use the equation for the geodesic.

96. What are your thoughts on the light pollution in the night sky?

It seems unfortunate but accurate that the term light pollution is used to describe the phenomenon. It seems to me that decades ago on very clear nights I could see the band of stars that forms the milky-way galaxy. I don't know if that band of stars is visible to the naked eye anymore from my front lawn. Since I live about 30 miles west of New York City, on certain cloudy evenings the lower reaches of the eastern sky seems to glow with pale, whitish red. Why does the sky glow with this particular pastel color? Perhaps, the light from all the different light sources merges together. Then the molecules in the atmosphere diffuse the higher frequency light waves, and the clouds, perhaps, absorb the higher frequency light waves. Distant mountains have a bluish tint because the air molecules diffuse the high frequency blue light waves. So, perhaps, the blue component of light from NYC is diffused and into the background while the longer wavelength components of the light such as the red light components of the light travel far enough to color the clouds a light red against a pale blue background. After I wrote the statements above, I was driving on Route 80 on a very foggy day, and I noticed the light from the highway light poles some of which are quite tall seems to be the same color as the light pollution I observed.

97. What are your thoughts on traffic jams and other forms of traffic congestion?

Perhaps, one of the obstacles to solving the problem is that subconsciously many commuters may not want to spend more time at home. The only thing more frustrating than sitting in a traffic jam is dealing with the intractable issues of their domestic lives.

Perhaps, years of driving in a city including a small city like Garfield, NJ would cause a person to go slightly insane. Over time the lack of parking and the congestion would get to you.

Driving east on Interstate Route 80 from Boonton to Garfield serves as a kind of prelude to this incremental insanity. The only interesting feature to see while making the drive is the moiré pattern made by the chain link fences that alight atop most of the bridges that cross the highway.

98. Since vinyl records have made a comeback, what other casualty of the digital revolution would you like to see resurrected?

I would like to see the return of the card catalogues, which were a hallmark of libraries until fairly recently. If only I could come up with some fatuous reason to claim the superiority of the card catalogue to the computer systems that are utilized now, I would be in the same league as the individual who came up with the claim that vinyl records have a warmer sound than CDs. That claim always seemed somewhat spurious to me. As a fan of rock 'n' roll, which it seems to me is almost uniformly recorded at a loud volume with a few exceptions such as Henry Cow, I found it difficult if not impossible to listen to classical music on vinyl records because the quiet passages were replete with surface noise.

I am not sure of the fate that has befallen plastic belts that seemed to come into fashion several years ago. I believe that since they proved to be much less durable than leather belts, and market forces lead to their elimination.

Thinking about all of this has lead me to ponder whatever became of my family's homemade shoeshine box. I remember polishing my shoes when I was young, but then much like many other people I began to wear sneakers almost exclusively, and it seems sneakers don't require polishing. I recall that my farther never wore sneakers; he would work in his vegetable garden wearing an inexpensive pair of dress shoes.

99. What are your thoughts on the pentagram?

I am surprised that the digits of all the angles contained within the pentagram and the various combinations of angles within the pentagram add up to nine. The angles that makes up the interior pentagon of the pentagram are each 108°, and one plus zero plus eight equals nine. There are five such angles for a total of 540°, and five plus four plus zero equals nine. The angles formed by the bases of the triangles that form the points of the star of the pentagram are each 72°, and seven plus two equals nine. There are two base angles for every triangular point of the star. The sum of the two base angles is 144°, and one plus four plus four equals nine. There are a total of 10 base angles for a total of 720°, and seven plus two plus zero equals nine. The angle that makes up the tip of the points of the star is 36°, and three plus six equals nine. There are 5 tips of the pentagram, and five times 36° equals 180°, and one plus eight plus zero equals nine. The tip of the pentagram plus one of the base angles equals 36° plus 72°, which equals 108°, and one plus eight equals nine. Of course, each triangle that makes up a point of the star has 180°, and one plus eight plus zero equals nine. There are five such triangles for a total of 900°, and nine plus zero plus zero equals nine. The sum of the angles for the points of the star and the interior pentagon is 900°+540°, which equals 1,440°, and one plus four plus four equals nine. The sum of the angles for two pentagrams is 2,880°, and two plus eight plus eight plus zero equals 18, but one plus eight equals nine. The sum of the angles for three pentagrams is 4,320°, and four plus three plus two plus zero equals nine. The sum of the angles for seven pentagrams is 10,080°, and one plus zero plus zero plus eight plus zero equals nine. The sum of the angles for thirteen pentagrams is 18,720°, and one plus eight plus seven plus two plus zero equals 18, but one plus eight equals nine. The sum of the angles for twenty-five pentagrams is 36,000°, and three plus six plus zero plus zero plus zero equals nine. The sum of the angles for forty-one pentagrams is 59,040°, and five plus nine plus zero plus four plus zero equals 18, but one plus eight equals nine. The sum of the angles for seventy-three pentagrams is 105,120°, and one plus zero plus zero plus five plus one plus two plus zero equals nine.

100. What are your predictions for the future?

Wild animals such as squirrels, rabbits and groundhogs will be trained to pick up roadside litter, and the unforeseen consequences will be alarming. Shopping carts will be equipped with backup beepers much like the annoying beepers that construction equipment and some pick-up trucks are equipped with today. Much like today, some household items will not get dusted because the occupants of the house are too busy and dusting is not that interesting an activity. Two crows or occasionally one single crow will mob a hawk, but no one will view it as intolerable bullying. The sleeves of men's shirts will still be too long, and no one will do anything about it such as make the cuffs of shirts adjustable in the same way that the insoles of shoes are adjustable that is by cutting off the excess insole with a pair of scissors.

101. Two questions: What is the justification for taking the covariant derivative of a tensor instead of the ordinary derivative of a tensor? Is the covariant derivative of a tensor valid?

The justification for taking the covariant derivative of a tensor instead of the ordinary derivative of a tensor is that the ordinary derivative of a tensor does not always produce a tensor as the result of taking the derivative. This justification is difficult to understand and most likely incorrect. The argument hinges on the nature of a particular term, which happens to be a second partial derivative, found in the equation for the first partial derivative of the transformation equation for an ordinary tensor of rank one. More specifically, the question is whether the second term on the right-hand side of the equation for the first partial derivative of an ordinary tensor, which happens to be a second partial derivative of a particular and appropriate transformation equation—drawn from the large pool of transformation equations, is always equal to zero. The answer seems to be that we can manipulate certain transformation equations so that the second partial derivative is not equal to zero, but I believe that in doing so we invalidate the function of these transformation equation's first partial derivatives. The role of these first partial derivatives is to represent certain coefficients. If we invalidate that role, we invalidate the correct operation of the transformation equation of a tensor. But, before we examine this question any further, let's look at the partial derivatives of some equations.

Let's find the first partial derivative of the following equation with respect to independent variable x. The equation is z = x + y.  . To find the first partial derivative of the equation z = x + y with respect to the independent variable x, we treat the independent variable y as a constant. The derivative of a constant when it stands alone is zero. Next, we find the derivative of the independent variable x, which is 1, so  . Let's find the first partial derivative of the following equation with respect to the independent variable x. The equation is z = 49x + 49y.  . Again, we treat the independent variable y as a constant so the entire term 49y is treated as a constant. The derivative of a constant or constants when they stand alone is zero. Since the derivative of x is 1, the derivative of the term 49x is 49. Let's find the first partial derivative of the following equation with respect to the independent variable x. The equation is z = 49rx + 49ry where r, x and y are all independent variables. We treat the term 49ry as a constant. The derivative of a constant when it stands alone is zero. We treat the r in the term 49rx as a constant so  .

Let's find the second partial derivative of the equation z = x + y with respect to the independent variable x. To find the second partial derivative of the equation with respect to the independent variable x, we take the derivative with respect to the independent variable x of the first partial derivative with respect to the independent variable x. The first partial derivative with respect to the variable x is  . The derivative of a constant when it stands alone is zero so  . Let's find the second partial derivative with respect to the independent variable x of the equation z = 49rx + 49ry. The first partial derivative with respect to the independent variable x is  . To find the second partial derivative with respect to the independent variable x, we find the derivative with respect to x of 49r. We treat the variable r as a constant so the derivative of 49r is zero. Thus,  .

The interesting point is that although one of the coefficients of the variable x is itself a variable the second partial derivative is still zero because we treat the variable as a constant when we are performing partial differentiation. Lillian Lieber gives her version of the justification for covariant differentiation in her book The Einstein Theory of Relativity beginning on page 183. She writes, "Applying this principle [the differentiation of a product rule] to the differentiation of  (equation 39) with respect to  we get:  (equation 40). Or, since  , by (equation 26), hence (equation 40) becomes  (equation 41). From (equation 41) we see that if the second term on the right, were not present, then (equation 41) would represent a mixed tensor of rank two. And, in certain special cases, this second term does vanish, so that in such cases, the differentiation of a tensor leads to another tensor whose rank is one more than the rank of the given tensor. Such a special case is the one in which the coefficients  in (equation 39) are constants, as in (equation 13) on page 147, since the coefficients in (equation 13) are the same as those in (equation 11) or (equation 10), and are therefore functions of  ,  being the angle through which the axes were rotated (page 141), and therefore a constant. In other words, when the transformation of coordinates is of the simple type described on page 141, then ordinary differentiation of a tensor leads to a tensor.

"But, in general, these coefficients are not constants, and so, in general differentiation of a tensor does not give a tensor as is evident from (equation 41).

"But there is a process called covariant differentiation which always leads to a tensor, and which we shall presently describe."

Equation 41  isn't the equation of a tensor because all the terms on the right-hand side of the equation are not multiplied by  . If  equals zero then  equals zero, and the second term vanishes. Thus, every term on the right-hand side of the equation is multiplied by  and (equation 41) is the equation of a tensor. It is a mixed tensor of rank two as Lillian Lieber writes. Is there a way to write (equation 41) so that it is a tensor regardless of the nature of the second derivative? Would it look something like the following:  ? With the term  we would want to cancel the first partial derivative  located outside the brackets by multiplying the term  by that first partial d0erivative  then we would to produce the second partial derivative by finding  . Can we bring  inside the parentheses of the term  ? In the above process, some utter hypocrisy has been engaged in by claiming that  equals one; that is to say the two fractions cancel each other. We have examined at great length the objections to what has been characterized in this interview as a misuse of the chain rule. That examination can be found in the answer to the question dealing with the tensor operation called contraction. This hypocrisy could be justified by claiming that the rules that are being employed in the above process are the rules that the tensor mathematicians claim to be following. Let's try writing the equation another way  . In this version of the equation we try to cancel  by placing a duplicate of  in the denominator of the appropriate fraction. Perhaps, we should have placed  in the denominator of the appropriate fraction giving us  . In a formal sense, all the equations that have three or more indexes that are the same in any one term are in violation of the summation laws and therefore invalid. An example would be three indexes that are  in a single term. It doesn't matter if the indexes are subscripts or superscripts. But, it is difficult to estimate how injurious it would be to alter such offending indexes. Let's leave this speculation behind and return to the definition of the partial derivative.

The Mathematics Dictionary by Glenn James and Robert C. James gives a definition of the partial derivative. The following is a portion of that definition, "the partial derivative is the ordinary derivative of a function of two or more variables with respect to one of those variables, considering the others as constants. . . . The partial derivative of x2 \+ y with respect to x is 2x; with respect to y it is 1." So, if the coefficients are not constants they would be treated as constants for the operation of partial derivation. Let's examine (equation 41)  . When we look at the second term on the right-hand side of the equation, we see that in finding the first partial derivative of the equation  , all the variables except  are treated as constants. In finding the second partial derivative all the variables except  are treated as constants. We would not expect the equation with the dependent variable  (a primed variable) to have as one of its independent variables  (a primed variable). We would expect the independent variables to be  (an unprimed variable). The purpose of this equation and others of this type is to rename or transform the various known representatives of the unprimed variables of the unprimed coordinate system into the as yet unknown representatives of the primed variables of the primed coordinate system. We cannot have primed variables as independent variables because we can't calculate and don't know the value represented by a primed variable until we calculate its value using the known representatives of the unprimed variables from the unprimed coordinate system. We can't calculate the unknown value of a primed variable on the left-hand side of the equation if the right-hand side of the equation contains a primed variable of an unknown value along with unprimed variables of known values. The conclusion that  will not be one of the independent variables leads us to the conclusion that all of the variables will be treated as constants. Therefore, they will be differentiated as though they were a stand-alone constant. The derivative of a stand-alone constant is 0. Thus, the value of  should be 0 not only when the coefficients of the independent variables are constants, but also when the coefficients of the independent variables are themselves variables provided that one of the coefficient's variables is not  . And, we have just explained why  cannot be one of the coefficient's variables.

It is also somewhat odd that we would want to find the derivative of (equation 39) with respect to the variable  . It is similar to finding the derivative of the equation z = x + y with respect to the dependent variable z. If we were to find the derivative of (equation 39)  with respect to the variable  , we would obtain, when the second derivative vanishes,  , which would seem to be a contravariant tensor of rank one.

Let's examine a pair of equations that change unprimed coordinates into primed coordinates. The equations are  and  Let's write the equations another way  and  . When we take a partial derivative such as  we obtain  in other words the partial derivative  is equal to the coefficient of  in the equation  . Thus, if we took the other similar partial derivatives we would obtain the other coefficients. For example  , which is the coefficient of  in the equation  . This is important because the term  in (equation 39)  represents these coefficients or similar coefficients depending on the context of the equation. The context of the equation is the number of spatial dimensions in which the tensor operates and other factors. The important point is that the term  must represent the coefficients of the variables that are designated by  in the equations designated by the dependent variable   For example, the partial derivative  is equal to the coefficient of  in the equation designated by the dependent variable  .

Now let's alter the equations  and  . We will alter these equations so that the second partial derivative with respect to the independent variable  will not be zero. The altered equations are  and  . We could write the altered equations another way  and  . When we take the first partial derivative of the equation  with respect to the independent variable  , we obtain  . The second partial derivative of the equation  with respect to the independent variable  is the following:  . The second partial derivative does not equal zero; it equals  . But, we should note a very important point. The first partial derivative of the equation, which was altered so that the second partial derivative does not equal zero, is not the same as the coefficient of the variable  in the equation designated by the dependent variable  . This means the first partial derivative is no longer an accurate representation of the coefficient that is used to turn unprimed coordinates into primed coordinates. We can expand this and say the first partial derivatives are no longer accurate representations of the coefficients that are used to turn unprimed coordinates into primed coordinates. This means that (equation 39)  is not valid if the second derivative of the equation does not equal zero. A question arises. Was it the particular way in which the equations, for transforming unprimed coordinates into primed coordinates, were altered that made the first derivatives such that they no longer were equal to the coefficients of the unprimed variables? It seems that any alteration of the equations such that the second derivative with regard to the variable  would not equal zero would make the first derivatives invalid representations of the coefficients of the independent variables of the transformation equations.

In Schaum's Outlines: Tensor Calculus the author David C. Kay provides a concise explanation of the inadequacy of the ordinary differentiation of a tensor in chapter 6 "The Derivative of a Tensor." We should make this note about his terminology, instead of primed and unprimed coordinates and tensors he uses barred and unbarred coordinates and tensors. In section 6.1 "Inadequacy of Ordinary Differentiation" he writes, "Consider a contravariant tensor  defined on the curve  . Differentiating the transformation law  with respect to t gives  which shows that the ordinary derivative of T along the curve is a contravariant tensor when and only when the  are linear functions of the  ." But, as we have demonstrated when the  are not linear functions of the  the term  will be an inaccurate representation of the coefficients of the unprimed variables or in this case the un-barred variables. We should point out that the  do not need to be linear functions of other variables; they only need to be linear functions of the  variables. Covariant differentiation contains terms that are essentially the same as  , that is to say, terms that represent the coefficients of certain variables. The terms used in covariant differentiation that we are concerned with are  and  . So analogously to the ordinary differentiation of a contravariant tensor, covariant differentiation will only produce a tensor when the  are linear functions of the  and the  are linear functions of the  . Thus, covariant differentiation shares the same shortcoming that ordinary differentiation has. Let's make it clear, covariant differentiation does not share the shortcoming that if the  are not linear functions of the  then the second derivative will not vanish. This is because covariant differentiation has developed a substitute for the term containing the second derivative. The shortcoming it shares with ordinary differentiation of a tensor is that the terms  and  are not accurate representatives of the coefficients they are supposed represent unless the  variables are linear functions of the  and the  variables are linear functions of the  . If this shortcoming makes ordinary differentiation inadequate, it must make covariant differentiation inadequate as well. But, we must remember that the specific shortcoming mentioned is that the second derivative does not disappear in ordinary differentiation of a tensor unless the  are linear functions of the  . To be scrupulously accurate only Lillian Lieber in her book The Einstein Theory of Relativity actually mentions the non-disappearance of the second derivative as a specific shortcoming. David Kay in his book Tensor Calculus presents an equation with a second derivative as one of the terms and states [this equation], "shows that the ordinary derivative of T along the curve is a contravariant tensor when and only when the  are linear functions of the  ." But, the implication is strongly present since, three pages further in section 6.3, David Kay gives us the terms that are used to substitute for the second derivative in covariant differentiation. The equation for covariant differentiation is the following:  .

The other shortcoming is that  will not accurately represent the coefficients it is supposed to represent unless the  are linear functions of the  , and it is not mentioned. Let's state the unmentioned shortcoming as it appears in the equation for covariant differentiation. The unmentioned shortcoming is that  will not accurately represent the coefficients they are supposed to represent unless the  variables are linear functions of the  and the  variables are linear functions of the  . And, that shortcoming is not mentioned. The justification of covariant differentiation is like a magician's trick by focusing our attention on the nature of the second derivative under certain conditions we fail to notice the change in the accuracy of the first derivative under these same conditions.

It should be pointed out that in equations such as  and  the  and  are essentially components of a contravariant tensor that could be designated by  or  . And, both,  and  are essentially components of a contravariant tensor that could be designated by  or  . But, there is a problem with the statement above. Lillian Lieber gives the equations for the components of a contravariant tensor as follows:  and  . So, we would want to say that the  and  are essentially components of a contravariant tensor that could be designated by  or  . And, both,  and  are essentially components of a contravariant tensor that could be designated by  or  . But, if  is equivalent  how can we find the derivative of  with regard to  ? Would we have to construct something like  ? But, is there a function that relates  to  ? The Mathematics Dictionary states that dx is a variable that is independent of the variable x.

When we undercut the justification for covariant differentiation, we undercut the justification for the Riemann-Christoffel curvature tensor and for Einstein's law of gravitation. But, an argument can be made that both ordinary differentiation of a tensor and covariant differentiation of a tensor are both equally unjustified or for that matter both are equally justified. That argument can be substantiated by claiming that all that is being done in covariant differentiation is the substituting of a term for the second derivative and then rearranging the terms of an ordinary differentiation of a covariant tensor of rank one. That claim is accurate. The problem is that the manufacture of the substitute term for the second derivative is questionable.

So let us leave behind the question of whether covariant differentiation of a tensor is justified by the shortcomings in ordinary differentiation of a tensor and turn our attention to the question of whether the operation known as covariant differentiation of a tensor is valid. If there is a fault, it may be that it can be traced to the over broad and contradictory definition of a tensor. We can say that by definition  is a tensor of rank two since we believe that the equation  is accurate. The equation states that  or perhaps we should write it as  is a scalar that is to say an ordinary number. An ordinary number is by definition a tensor of rank zero. And, furthermore is equal to a covariant tensor of rank two multiplied by a contravariant tensor of rank one and that product is itself multiplied by a contravariant tensor of rank one. Note that we have two contravariant tensors yet they have subscripts for their indexes instead of superscripts so you would suspect they are both covariant tensors of rank one. We could rewrite the equation as  . This makes it easier to see that by inner multiplication  , where  is a covariant tensor of rank one. Next, again by inner multiplication  , where  is a scalar which is to say a tensor of rank zero. So, since we believe the equation  is accurate by definition we must believe  is a covariant tensor of rank two.

In Lillian Lieber's The Einstein Theory of Relativity she provides us with the theorem that allows  to be classed as a tensor. It is located in the appendix of her book, which she titles, "Would You Like to Know?" In section III of the appendix titled "How To Judge Whether A Set Of Quantities Is A Tensor Or Not," she writes, "See if it satisfies the following theorem: A quantity which on inner multiplication by any covariant vector (or any contravariant vector) always gives a tensor, is itself a tensor." The term  undergoes inner multiplication by two contravariant tensors of rank one. A contravariant tensor of rank one is also known as a contravariant vector. The result of the inner multiplication is always the scalar  , which is to say a tensor of rank zero, so we must assume  is a tensor.

David C. Kay gives us a similar theorem in his book Schaum's Outlines: Tensor Calculus. In chapter 4 section 4.2 "Tests For Tensor Character," he writes, "The following statements are useful criteria or 'tests' for tensor character; they may all be derived as special cases of the Quotient Theorem. . . .(3) If  is invariant for all contravariant vectors  and  , then  is a covariant tensor of order 2." So, by one of the special cases of the Quotient Theorem,  is a covariant tensor of rank two.

The Mathematics Dictionary 5th edition by Glenn James and Robert C. James provides us with this definition of a tensor. "An abstract object having a definitely specified system of components in every coordinate system under consideration and such that, under transformations of coordinates, the components of the object undergo a transformation of a certain nature. Explicitly, let  be one of a set of functions of the variables   , where each index can take on the values  and the number of superscripts is  , the number of subscripts  . Then these  quantities are the x-components of a tensor of order r+s, provided its components in any other system  are given by  where the summation convention is to be applied to the indices  and  ."

Since by the quotient theorem  is a tensor, it must follow the definition given above. In Arthur Eddington's book The Mathematical Theory of Relativity it appears to follow the definition. In section 31 of his book titled "Alternative Discussion of the Covariant Derivative," he begins with this equation  . Specifically, the tensor  is taking the role of the tensor  , which has been the focus of our discussion. The tensor  is also taking the role of tensor  in the above definition while the tensor  is taking the role of the tensor  in the above definition. Starting with the equation  Eddington begins a complex process from which he obtains the quantity that is substituted for the second derivative that is obtained in ordinary differentiation of a covariant tensor of rank one. The difficulty is that the equation  is incompatible with the demands being placed on it. Since  is a tensor it is an invariant, but unlike an ordinary tensor its components are also invariants. Therefore the components of the tensor  will be exactly the same as the components of the tensor  and not the components of  that have been multiplied by  . Perhaps, we should say multiplied by  in accordance with the summation convention. We must make an important point here. The equation  is not invalid. It is only incompatible with the demands being placed on it, and in that sense, it is invalid when we make a certain assumption. The assumption is that the transformation equations denoted by  and  that is to say the transformation equations where  and  are the dependent variables are the same as the transformation equations for the ordinary tensor in the particular spatial dimension we are interested in. Why is this assumption valid? It is valid because we are going to essentially substitute a value equal to  as determined by the equation  for the same value  as determined by the equation for the differentiation of an ordinary tensor in the same space as the equation  . Thus the  from the equation  must be the same as the  from the transformation of an ordinary tensor. As we noted before,  is a tensor, and therefore it is an invariant, but unlike an ordinary tensor its components are also invariants. Therefore the components of the tensor  will be exactly the same as the components of the tensor  provided you are not transforming from flat space to curved space or to a space with more dimensions as from two dimensional space to three dimensional space. So under most conditions we could let  from the equation  equal one and the equation would be valid, but we cannot let  equal one when transforming an ordinary tensor.

Let's examine the familiar example of the Pythagorean Theorem as it operates in two dimensional Cartesian coordinates. We will use the equation  . Here the tensor  is equal to  . Now, let's rotate the Cartesian coordinates an arbitrary number of degrees counterclockwise; let's say  . We will designate the number of degrees we have rotated the coordinate system by the Greek letter  We now have both the original unprimed coordinate system and the new primed coordinate system. To find the values for the tensor  in the new primed coordinate system we use the equation  The equations designated by  and  in two dimensions for an ordinary tensor are the following:  and  . Let's solve the equation for only one component of the tensor  . We will solve the equation for the component  . We must use the summation convention because  and  each appear twice in a single term on the right-hand side of the equation. Thus we obtain  . Since  and  , we obtain  . Solving this equation we obtain  or  . Thus  . Thus, for a rotation of the axes in two dimensions, the equation  is valid when using the transformation equations for an ordinary tensor. This reminds us of the situation concerning the operation of contraction, which we have previously examined; it too was valid in two dimensions. As we saw with contraction, an operation that is valid in two dimensions is not necessarily valid in three or more dimensions. As we shall see the equation  is not valid in three dimensions.

Let's examine the familiar example of the Pythagorean Theorem as it operates in three dimensional Cartesian coordinates. We will use the equation  . Here the tensor  is equal to  The Mathematics Dictionary describes a rotation in three dimensions, "In space, a rotation moves the coordinate trihedral in such a way as to leave the origin fixed and the axes in the same relative position. The coordinates of a point are transformed from those referred to one system of rectangular Cartesian axes to coordinates referred to in another system of axes having the same origin but different directions and making certain given angles with the original axes. If the direction angles, with respect to the old axes, of the new x-axis (the x′-axis) are A1, B1,C1; of the y′-axis are A2, B2,C2 ; and of the z′-axis, A3, B3,C3, then the formulas for rotation of axes in space are  ,  and  "

Let's solve the equation  for one of the components of  in three dimensional space. Let's choose the component  . The equation for the component  is  , but since  we obtain  . Solving the equation we obtain  , which gives us  . The tensor component  must equal 1, but it does not equal 1. It equals  . If we choose certain angles for A1, B1 and C1, we can make  equal 1 such as A1= , B1=  and C1= . But, the tensor component  must equal 1 for any possible angles that we choose, and not only for certain angles. Therefore, the equation  is invalid for the rotation of Cartesian coordinates in three dimensional space when using the transformation equations for an ordinary tensor. The equation  is the equation that is used to begin the process from which we obtain the quantity that is used to substitute for the second derivative, which we obtain when we find the ordinary derivative of the transformation equation  . Substituting the quantity we obtain from a process beginning with the equation  allows us to obtain the equation that is designated as the covariant derivative. But, since the transformation equation  is invalid when using the transformation equations for an ordinary tensor (with the exception we have already noted), the equation for the covariant derivative is also invalid. And, as we've already stated, we must use the transformation equations for the appropriate ordinary tensor for the substitution to make sense. The equation represented by the  from one equation must be the same as the equation represented by  from the other equation. There would be a way around this dilemma if we could say that the second derivative of all tensors was zero, then we could equate second derivatives where the equation represented by  from one equation was different from the equation represented by  from the other equation, but for the covariant differentiation of a tensor to be justified, that cannot be the case. This way around the dilemma would obviate the justification for covariant differentiation of a tensor.

Since the Riemann-Christoffel curvature tensor is derived from a handful of operations performed on the covariant derivative, the Riemann-Christoffel curvature tensor is invalid, as well. Since Einstein's law of gravity is derived from the contraction of the curvature tensor, it is invalid, as well.

The reader may ask can a substitute term for the second partial derivative be obtained without the use of the equation  . The answer is no. We need an equation that will produce Christoffel symbols of the first kind. These can only be obtained by differentiation of the equation  with respect to  . Then using the chain rule on the term  . These steps provide us with our differentiated equation. Next, we rewrite this equation two times, and with each rewrite of the equation we remember to permute the subscripts cyclically. Thus, we obtain three equations. We add the second and third equation together and subtract the first equation from that sum. Then we divide both sides of the resulting equation by 2.

The resulting equation, which has been briefly described in the above paragraph, contains two Christoffel symbols of the first kind. One is on the left-hand side of the equation and the other is on the right-hand side of the equation. The right-hand side of the equation also contains a second partial derivative. Through a somewhat complicated process the two Christoffel symbols of the first kind are changed into Christoffel symbols of the second kind. Once that has been accomplished, we place the second partial derivative on the left-hand side of the equation. The right-hand side of the equation containing among other quantities the two Christoffel symbols of the second kind is the entity we substitute for the second derivative to produce the covariant derivative. So it seems the  and the  of the equation  are essential to producing the Christoffel symbols of both the first and second kind.

It is instructive to examine the relationship between the Riemann-Christoffel curvature tensor and the covariant derivative. We start with a tensor  and find its covariant derivative with respect to  . The result is the tensor  on the left-hand side of the equation and two terms on the right-hand side of the equation. Next, we find the covariant derivative of the tensor  with respect to  . The result is tensor  on the left-hand side of the equation three terms on the right-hand side of the equation. Through some clever substitutions the three terms on the right-hand side of the equation are changed into seven complicated terms on the right-hand side of the equation. Now, we find the derivatives of the same tensor  but in the reverse order. First with respect to  and then with respect to  . The result is tensor  on the right-hand side of the equation and seven complicated terms on the left-hand side of the equation. Next, we subtract tensor  from tensor  . The result is  . The four terms inside the brackets are the Riemann-Christoffel curvature tensor.

102. Are there any questions you no longer wonder about?

In the past, I would often wonder why ducks and geese had such a proclivity for human foods such as slices of bread and chunks of rolls when they were so different from their natural foods. Then one day, I saw some brown ducks catching crayfish from a shallow section of the Rockaway River. In that section of the river, the riverbed was covered with black rocks. A duck would dip its head underwater and come up with a crayfish in its beak which was struggling to get free. It seemed to be an enormous struggle for the ducks to swallow the crayfish. Yet, the ducks were inclined to swallow crayfish after crayfish. After witnessing that I never wondered about their penchant for bread and rolls.

Section B. Myrmecophagous Bridgette

103. Do you have any advice for Bridgette?

Someday you will most likely meet an older person who was raised on a farm. They will probably tell you something such as "When I was young, I got up at 4:00 a.m. and milked forty cows before I went to school." You have the option of replying "I'm surprised you didn't have your slaves do it for you." Your statement would imply that you calculate that his age is so great that his childhood occurred before the Civil War.

Someday if you want to impress your driver's education teacher tell him that you have analyzed a complex exit off of an interstate highway such as the exit off of Route 80 East onto Midland Ave. in Saddle Brook, NJ. It is a complex exit because it is also an exit for the Garden State Parkway. If you visualize the length of Route 80 that is west of the exit as a rope and the path of the car once it exits Route 80 East as the trajectory of one of the ends of the rope, the trajectory is not complex enough to produce a simple knot. The end of the rope loops around and over itself and then under itself. Thus it produces something like the first half of a half hitch. Perhaps, you could find an exit that is so complex that an actual knot is produced by the trajectory of the car.

Someday you may be in a high school or college class in which the students are encouraged to express their feelings. You may be questioned about how you felt in elementary school when you were grouped with those students that were considered to be of diminutive stature. You could reply that it wasn't all bad and that you rode your Irish wolfhound as if it were a horse until you were in the third grade. You could say that you made a saddle for it by gluing together two doggy sweaters and stuffing straw in between the two layers of material.

Someday you may want to color or streak your hair. You don't have to be like everyone else and buy your hair coloring items at a store. You can color your hair with a dye made from the berries of wild plants. The dark purple berries of the pokeweed would make an excellent hair dye. The pokeweeds grow in waste places and along roadsides in the eastern and southern USA. They grow in many places in fact there are some pokeweeds growing behind the Garfield Library and according to An Instant Guide to Wildflowers they can grow as tall as nine feet. The berries are juicy and grow in great profusion on long sprays.

As I've already mentioned to you, there is a large statue of a jaguar stored in the large parking lot adjacent to several buildings located just over the Ackerman Bridge in Clifton. The statue depicts a jaguar walking on a large fallen tree. The buildings and the parking lot are on the north side of Ackerman Ave. You could pretend to be a princess in exile—Princess Bridgette. You could write an elaborate letter or a simple letter in which you state that in your opinion the jaguar statue should be donated to the Garfield Library because you feel that if the jaguar statue were placed on the library grounds would encourage young people to read. After you write the letter, you could tape it to one of the metal poles that makes up the open crate in which the statue is stored. There is also a submarine docked in the Hackensack River near the end of Bridge Street. It seems to be a WWII era submarine. You could write a letter in your Princess Bridgette persona and suggest that the submarine be moved to Dahnert's Lake in Garfield so that more people will have an opportunity to see it. You could tape the letter onto the submarine.

You could write a short story in which your older brother Rodrigo is kidnapped by radical Islamists. The short story begins with you writing away for a free copy of the Koran. Instead of receiving a free copy of the Koran, you receive a long letter from an individual who thanks you for your interest in the Islamic religion and who wishes to talk to you in person to tell you about Islam. Both you and Rodrigo to whom you show the letter agree the letter is highly suspicious. You put the letter away and never think about it again until one night after you have had a furious argument with Rodrigo. In your anger, you write a return letter to the individual who sent you the letter thanking you for your interest in Islam. In the letter you pretend to be Rodrigo, and you say that you are very interested in understanding more about the Islamic faith. Also, you provide a location where this individual and Rodrigo can meet. You go so far as to put the letter in an envelope and address it. But, you fall asleep at your desk with the letter beside you. Your mother comes into your room, and seeing that you are sleeping, she decides to mail the letter for you. You awaken from your sleep at your desk and go to bed. The next day, you wake up and have no inclination to mail the letter. Your anger at Rodrigo has dissipated. You notice the letter is no longer on your desk, but you surmise that you must have inadvertently knocked the letter behind your desk when you were going to bed. A few days later, Rodrigo is missing. You suspect the worst when a thorough search of your room fails to turn up the letter. The short story ends with Rodrigo in Islamabad studying the Koran. He is depicted as thinking that he always believed the Islamic religion began around 700 A.D., but now his study of the Koran has revealed that it began thousands of years earlier at about the same time as the Jewish religion began. We are left to wonder what other information he hasn't been told.

You could write a short story about a nightmare you had in which someone writes a letter of recommendation for you that is slightly skewered. The letter of recommendation could have the following content. I worked alongside Bridgette Tiape as a peer helper at The Happy Reading Tree Saturday School for many years. She felt the name "The Happy Reading Tree Saturday School" reeked of bourgeois sentimentality and lobbied perennially for it to be changed to The Melon Patch Mafia. She felt that name would harken to the primordial appellation of the city of Garfield, which was the Cadmus Melon Patch. She was great with the kids. She never sent one of them to the hospital for more than a few hours. In her defense, she always gave them fair warning of the consequences of their misbehavior. Although she would refer to herself in the third person, which may have been confusing to some of her younger charges, it seems she always got her message across. She would say things such as: When Bridgette gets disrespected, Bridgette gets mad. When Bridgette gets mad, her knife comes out. When Bridgette's knife comes out, some little troublemaker is going to get cut. I think her pedagogical philosophy was summed up by her oft repeated statement: "You get the point or you get the point." Some of her detractors claimed it was nothing more than an incoherent and tautological catch phrase, but I always thought of it as an adjunct to the first propositional axiom in Gödel's first incompleteness proof. Bridgette had her detractors; some of the less dedicated individuals referred to her as a grimalkin, but that was merely a misunderstanding. She was always an old soul. And, those that referred to her as a hellgrammite were merely engaged in name calling and not very accurate name calling at that unless anything small can be associated with those of diminutive stature. As far as I know, the rumors are false that occasionally the after effects of her Friday night celebrations would cause her to whisper to her brother Rodrigo, "Is Miss Evans speaking Spanish?" Bridgette would be an asset to any college or university she attended.

104. Are there any towns or cities that you are surprised are not adjacent to each other?

I have always wondered why Schenectady wasn't next to Metonym.

105. What is your opinion of the documentary film Time Dilation: An Experiment with Mu Mesons?

The following is a review of the documentary film Time Dilation: An Experiment with Mu Mesons 1962 PSSC: David Frisch, James Smith, and M.I.T. Physics, Physical Sciences Study Committee Films (PSSC) playlist: <http://www.youtube.com/playlist?list=PL_hx>.

David H. Frisch and James H. Smith performed a time dilation experiment with the subatomic particles known as mu-mesons, which are also called muons. The muon decay experiment was performed on Mt. Washington, New Hampshire at an elevation of approximately 6,300 feet (the elevation given for Mt. Washington by the Rand McNally Road Atlas 2016 is 6,288 feet) and at sea level in Cambridge, Massachusetts probably somewhere on the M.I.T. campus. The date was 1962. The experiment was said to have confirmed the time dilation predicted by Einstein's theory of special relativity. Reviewing the experiment 55 years later, it appears to be flawed. In this experiment, the distribution of muons acts as a clock. This means the number of muons counted per hour at an elevation of approximately 6,300 feet compared to the number of muons counted per hour at sea level acts as a clock. Muons are subatomic particles that are present in secondary cosmic rays. Primary cosmic rays consist of protons that travel at near light speeds. Primary cosmic rays come from outer space and strike the earth's atmosphere. At about 10 miles above the earth nearly all of the primary cosmic rays, which is to say protons, have been changed into secondary cosmic rays through collision with molecules of the atmospheric gases. The secondary cosmic rays consist of many different kinds of subatomic particles. The muons in secondary cosmic rays travel at nearly the speed of light. For the distribution of muons to act as a clock implies that the bombardment of the earth's atmosphere by primary cosmic rays is a constant since if the bombardment of primary cosmic rays were a constant, the production of secondary cosmic rays and hence muons would also be a constant. The bombardment of the earth's atmosphere by primary cosmic rays is neither a constant as implied in the documentary, nor wildly irregular. Therefore, it is difficult to gauge the significance of the inconsistency of the cosmic ray bombardment of the earth.

The experimental apparatus that was assembled on Mt. Washington performed several major tasks. A shell of solid iron bars about 2 ½ feet thick performed two essentials functions. This iron shell slowed down all the muons that struck it, and it differentially slowed down all the muons that struck it. This allowed the iron shell to imbue muons that traveled between .9950 and .9954 the speed of light with a specific property. That property was that they were at rest relative to the velocity of the earth when they exited the iron shell. Very soon after they exited the iron shell, they were captured by the scintillator which was also at rest with respect to the velocity of the earth. The muons that traveled at less than .995 the speed of light were slowed down by the iron shell and came to rest with respect to the velocity of the earth inside the iron shell in which they decayed. The muons that traveled faster than .9954 the speed of light were slowed down by the iron shell, but they still exited the iron shell moving faster than the speed of the earth. Thus, they were not be captured by the scintillator because they were still moving too rapidly.

The scintillator captured muons that were traveling between .9950 and .9954 the speed of light but were now at rest relative to the speed of the earth. These muons captured within the scintillator gave off a signal the moment they were captured by the scintillator, and they also gave off a signal at the moment they decayed within the scintillator. There were additional devices attached to the scintillator. These devices counted each muon that decayed while captured in the scintillator, and they measured the time elapsed from when a particular muon entered the scintillator until that particular muon decayed.

Muons decay quite quickly. The mean lifetime of a muon is 2.2 microseconds or 2.2 millionths of a second. According to the Webster's New World College Dictionary muons are unstable, negatively charged leptons with a mass of 105.7 MeV. About 207 times that of an electron. A muon decays into an electron, a neutrino and an antineutrino. But, if you listen closely to scientists one of them says that they are detecting positive mu mesons. He seems to say that they decay into a positron, antineutrino and neutrino. The experimental apparatus on Mt. Washington didn't measure the length of the lifetimes of muons. It measured the amount of time the muon existed once it entered the scintillator. The muon existed before it entered the scintillator since it had to be slowed down to a rest velocity with respect to the earth by the iron shell before it could enter the scintillator. A number of muons only existed about ½ a microsecond in the scintillator before they decayed. Most of the muons existed in the range of 1 to 2 microseconds in the scintillator before they decayed. In the range of 3 to 4 microseconds, a lesser number of muons existed before they decayed. The number of muons that existed in the scintillator in the range of 5 to 6 microseconds was smaller still. The number of muons that existed in the scintillator in the range of 7 to 8 microseconds was yet again still smaller.

The experimental apparatus on Mt. Washington was at an elevation of approximately 6,300 feet. An average of 564 muons were captured and decayed in the scintillator every hour. Some of the muons existed in the scintillator for as short as ½ of a microsecond while others existed for 6 or 7 or more microseconds. Of the group of muons captured by the scintillator, the scientists claimed that only those that existed for 6.3 microseconds or longer would make it to the surface of the earth (sea level) before they decayed. The experimental results predicted that 27 muons per hour should make it to the surface of the earth before they decayed. These 27 muons are from the specific group of muons that were selected by the shell of solid iron bars and captured by the scintillator. If the muons present at sea level were to undergo the same selective process as the muons at an elevation approximately 6,300 feet on Mt. Washington underwent, we would expect that the scintillator at sea level would capture and hold until they decayed 27 muons per hour, which additional devices would count and record. Why does it take a muon at an elevation of approximately 6,300 feet 6.3 microseconds to reach the earth? The scientists assumed that the muons were moving at 1,000 feet per microsecond, which is approximately the speed of light. If we say the speed of light is 983,571,587 feet/second, this gives us approximately 984 feet per microsecond. Since the muons in the group the scientists were interested in were traveling at .995 the speed of light, their speed would be about 979 feet per microsecond. So it seems that assessing the speed of the muons at 1,000 feet per microsecond would be a reasonable approximation.

It does seem like a reasonable approximation of the speed of the muon until we are informed of the manner in which the experimental apparatus is set up at sea level. The experimental apparatus at sea level must select muons that have certain characteristics in common with the muons that the experimental apparatus at an elevation of approximately 6,300 feet selected. One of the characteristics is that they must come to rest with respect to the velocity of the earth when they exit the shell of solid iron bars so that they can be captured and held by the scintillator until they decay. Another characteristic they must have is that at an elevation of approximately 6,300 feet they are traveling at .995 the speed of light. We were assured that at an elevation of approximately 6,300 feet the muons that entered the shell of solid iron bars at a speed of .995 the speed of light would be captured by the scintillator. At sea level we were informed that one foot of solid iron bars needs to be removed from the shell in order for the scintillator to capture and hold until they decay the muons the scientists were interested in, that is muons that were traveling at .995 the speed of light at an elevation of approximately 6,300 feet. We are told that the earth's atmosphere slows down the muons as they travel the approximately 6,300 feet—that is from the elevation of Mt. Washington to sea level. The muons are slowed down so much by the atmosphere that one foot of solid iron bars must be removed from the shell.

There you have the contradiction. We cannot assume that the atmosphere slows down the muons so much that one foot of the shell of solid iron bars must be removed to account for it, and at the same time calculate the time it takes the muons in question to traverse a distance of approximately 6,300 feet using the speed of the muons as 1,000 feet/microsecond, approximately the speed of light. The muons that the scientists were concerned with do not traverse the approximately 6,300 foot distance at a speed of 1,000 feet/microsecond, approximately the speed of light. The muons under the influence of the atmosphere are apparently decelerating at an increasing rate of deceleration for the entire course of their journey. This would seem to eliminate them from consideration as objects that experience time dilation according to the principles of special relativity since special relativity is concerned with objects that travel at a constant speed.

Let's examine some of the calculations made by the scientists. The scientists claim that the distribution of the muons acts as a clock. According to their calculations, 27 muons per hour (of the kind the scientists were interested in) should reach sea level. They should be captured and held until they decay by the scintillator. They should be counted and recorded by the additional devices. This calculation is based on the assumption that muons that are captured and held by the scintillator on Mt. Washington at an elevation of approximately 6,300 feet for 6.3 microseconds or longer have the ability to reach sea level while those captured and held by the scintillator for a shorter duration do not. That is, of course, if they hadn't been captured by the experimental apparatus. This calculation is based on another assumption, which is that the muons transverse the approximately 6,300 feet traveling at a speed of 1,000 feet/microsecond. And, as we have seen this assumption is contradicted by other statements made by the scientists. The scientists set up their experimental apparatus at sea level. The experimental apparatus specifically the shell of solid iron bars consists of one less foot of solid iron bars to account for the slowing of the muons caused by the approximately 6,300 feet of the earth's atmosphere. The experimental apparatus records an average of 412 muons per hour instead of the 27 muons per hour that were predicted by the scientists' calculations. The scientists consult a chart they have made of the length of time the muons existed in the scintillator during their various one hour runs. They look for a time reading in which 412 muons existed in the scintillator. When they add up all the muons that existed for .7 microseconds or longer, they come up with a total of 412 muons. If they added up all the muons that existed for .5 microseconds or longer, they would come up with a total that was larger than 412 muons. The number 412 is the number of muons that are captured and held by the scintillator until they decay during the course of an hour run at sea level.

The scientists divide .7 microseconds by 6.3 microseconds and obtain the fraction 1/9, which in decimal form is .111. This is different from the value you obtain by solving for the square root of the following quantity: 1 minus the fraction produced by dividing the square of the velocity of the muon by the square of the velocity of light. In mathematical terminology  or  where c=983,571,588 feet/second, the speed of light and v= 978,653,730 feet/second, the speed of a muon traveling .995 the speed of light. The value you obtain is .0998.

Let's make some assumptions in order to calculate an approximate speed for the muons the scientists were interested in as they journey from an elevation of approximately 6,300 feet to sea level. Since the shell of solid iron bars is about 2 ½ feet thick, let's assume that for every foot of iron bars that the muons travel through their speed decreases by a factor of 4/10. When the muons the scientists were interested in entered the shell of solid iron bars, their speed would be .995 the speed of light. When they had traveled through 1 foot of solid iron bars, their speed would be .597 the speed of light. When they had traveled through 2 feet of solid iron bars, their speed would be .199 the speed of light. When they exited the shell of solid iron bars, their speed would be near zero. Since the orbital speed of the earth is about18 miles/second that speed is near zero compared to the speed of light—186,000 miles/second. So, let's assume that when the muons begin their journey through approximately 6,300 feet of atmosphere they are traveling at .995 the speed of light, and when they reach sea level, they are traveling at .597 the speed of light. That would make the average speed of the muons .796 the speed of light or 782 feet/microsecond. The muons would thus take 8.1 microseconds to reach sea level. If we assume that the number of muons per hour captured and held until they decay by the scintillator at sea level remains the same (412 per hour), the calculation of the time dilation would be .7 microseconds divided by 8.1 microseconds or .086 in decimal form, which is different from the value predicted by the special relativity equation of .099.

If we assume that there is no time dilation, can we extract the number 27 muons per hour from the 412 muons per hour recorded at sea level? The film of the experiment does not provide us with the details of the lengths of time the various muons captured by the scintillator at sea level existed before they decayed. Let's assume it is similar to the lengths of time the muons captured by the scintillator on Mt. Washington existed before they decayed. Since we are assuming that muons the scientists were interested in have already existed more than 8.1 microseconds before they enter the scintillator at sea level, all those muons that existed in the scintillator at sea level for 4 microseconds or longer could be ignored. We can ignore them because this would mean the muon had existed for longer than 12.1 microseconds, and the average lifespan of a muon is 2.2 microseconds. By ignoring them we mean that we assume they were created at some point between 6,300 feet and sea level. We assume they were created by the collision of secondary cosmic rays and molecules of the atmospheric gases. If we estimate that the radius of the scintillator is 9 inches as many as 153,000 secondary cosmic rays should strike it per hour (ignoring the influence of the shell of solid iron bars), which is based on the measurement that 600 secondary cosmic rays strike a square inch of earth per hour. We assume they had a speed imparted to them by this collision such that when they were slowed down by the atmosphere and the solid iron bars they exited the solid iron bars at rest with the velocity of the earth. Using this kind of reasoning, we could ignore muons that existed in the scintillator at sea level for 3 microseconds or even 2 microseconds or at least a portion of the muons that existed for these lengths of time. Using this method we could extract the number 27 muons per hour from the number 412 muons per hour.

It is difficult to judge the validity of this experiment. You can view the contradictory calculations by employing a thought experiment. What would we see if we could view a clock on a muon that was slowed down by the atmosphere and the clock on another subatomic particle that was not slowed down by the atmosphere and traveled at .995 the speed of light? From the experiment you might conclude that their clocks would tell the same time.

106. Do you have any further comments on the documentary film Time Dilation: An Experiment with Mu Mesons?

Before Einstein's special theory of relativity and general theory of relativity became accepted by physicists, one of the goals of physics was to ascertain the absolute velocity of the earth. Does the documentary Time Dilation: An Experiment with Mu Mesons give us enough information to attempt to calculate the absolute speed of the earth? The absolute velocity of the earth would be a vector with two components: the absolute speed of the earth and the direction in which the earth traveled. We are given the speed of the muons that will eventually enter the scintillator and be at rest with respect to the velocity of the earth, but we are given their speed before they enter the shell of solid iron bars that slows their speed. At rest with respect to the velocity of the earth means traveling at the same speed and direction as the earth. The speed of the muons just before they enter the shell of solid iron bars is in the range of .9950 the speed of light and .9954 the speed of light, which tells us that the muons are traveling between 185,070 miles per second and 185,144 miles per second. When these muons exit the shell of solid iron bars and enter the scintillator the physicists in the documentary assure us they are at rest with respect to the velocity of the earth otherwise they would not stay in the scintillator long enough to decay. If we knew the range of speeds at which the muons were traveling when they exited the shell of iron bars and were captured by the scintillator, we would know an approximation of the absolute speed of the earth since these muons are at rest with respect to the velocity of the earth. Unfortunately, the documentary does not tell us the speed of the muons when they exit the shell of solid iron bars.

But with the information present in the Wikipedia article Time dilation of moving particles <https://en.wikipedia.org/wiki/Time_dilation_of_moving_particles> we can make an attempt to calculate the absolute speed of the earth. The Wikipedia article tells us that the muons that enter the shell of solid iron bars at sea level are traveling in the range of .9881 the speed of light and .9897 the speed of light, which tells us that they are traveling between 183,786.6 miles per second and 184,084.2 miles per second. We should recall from the documentary that these muons have been slowed by traveling through 6,257 feet of the earth's atmosphere, and that is equal to the amount they would have been slowed by traveling through approximately one foot of solid iron bars. Now, we can calculate the fraction of their speed by which the muons will be slowed when they travel through one foot of solid iron bars because it is equal to the fraction they have been slowed traveling through 6,257 feet of the earth's atmosphere. We subtract the muons speed when they reach sea level (before they enter the shell of solid iron bars) from their speed when they are 6,257 feet above the earth. We have 185,070 m/s—183,786.6 m/s =1,283.4 m/s, and we have 185,144—184,084.2=1,059.8 m/s. Next, we divide our results respectively by the two speeds of the muons when they were 6,257 feet above the earth, which gives us (1,283.4 m/s)/ (185,070 m/s) = .0069 and (1,059.8 m/s)/(185,144 m/s)=.0057.

From the documentary, we learned that at sea level the muons traveled through 1½ feet of solid iron bars. From the Wikipedia article, we learned that at sea level the muons were traveling between 183,786.6 m/s and 184,084.2 m/s. From this we can attempt to calculate the absolute speed of the earth since it will be equal to the speed at which the muons exited the shell of solid iron bars. We make the assumption that the fraction of speed the muons slow passing through one foot of solid iron bars when multiplied by 1½ equals the fraction of speed the muons will slow passing through 1½ feet of solid iron bars. So the muon traveling at 183,786.6 m/s at sea level will slow by (183,786.6 m/s)(.0069)(1.5)=1,902.2 m/s. Thus, when it exits the shell of solid iron bars it will be traveling at 181,884.4 m/s. And, the muon traveling at 184,084.2 m/s at sea level will slow by (184,084.2)(.0057)(1.5)=1,574 m/s. Thus, when it exits the shell of solid iron bars it will be traveling at 182,510.2 m/s. This gives the speed of the earth in a range between 181,884.2 m/s and 182,510.2 m/s.

This seems to be too great a speed for the earth to be traveling for it is approximately .98 the speed of light. If the earth is traveling that fast, a time dilation of approximately 1/9 seems too large when the "clocks" on the muons are traveling at approximately .99 the speed of light. But, there do not appear to be any obvious errors in the calculations.

The physicists in the documentary make another interesting claim. They claim that an observer traveling along with the muon would be entitled to claim that he was at rest and that the earth was racing by at .99 the speed of light. That claim would seem to entitle the observer to judge that earth's clocks were running at 1/9 the speed of the clocks located on his muon. That is to say for every one second that passed on earth nine seconds would pass according to his muon based clocks. It is difficult to conceive of how clocks located on a muon could run simultaneously both faster than and slower than clocks located on the earth.

107. Can you speculate on the nature of the mathematics used in certain niches of physics?
Is the mathematics used in certain niche areas of physics a kind of non-rigorous, anything goes affair? Where the motto could be _rules are meant to be broken._ Is this the gimmick that has escaped notice for so long? Who or what kind of system grants a scientist so much fame that he becomes a part of the history of science or more rarely a part of history? Once this fame has been granted, is there any way to put the genie back in the bottle? Is there any motive for anyone to ever want to put the genie back in the bottle?

Imagine the euphoria of being a famous scientist—the elite of the elite. Below you in status are the journeymen scientists who have studied and studied for decades to obtain a PhD, and now they ceaselessly labor on in anonymity. Below them in status are the ham and eggers, the little people that spend their lives filling out forms and watching movies. It seems every famous scientist has his circle of promoters. This is a mixed lot that consists of other scientists, science writers, journalists and others.

John S. Bell's famous paper, _On the Einstein, Podolsky, Rosen Paradox,_ was published in 1964. Lyndon Johnson was president of the USA at that time. An intimate portrait of Lyndon Johnson was published in 2005 by his friend and confidant Horace Busby. Busby's book _The Thirty-First of March_ unintentionally portrays Johnson as a collection of gimmicks. Perhaps, this portrayal was unavoidable because to modern eyes Johnson appears as a factotum of the press. Is it possible that everyone was a factotum of the press in those days?

What gimmicks does Bell employ? In his paper _On the Einstein, Podolsky, Rosen Paradox_ Equation number 3 is denoted as the quantum mechanical expectation value. Later in the text, equation 3 is denoted as the quantum mechanical correlation. The reader may think to himself equation 3 can't be both the expectation value and the correlation. But, if you work out the complex mathematics you discover that in this rare instance the expectation value is equal to the correlation. This works as a kind of trick to make you suspend your judgment about the mathematics that is to follow. But, if you carefully work out the details of the mathematical argument that Bell presents you discover his famous inequality is invalid—a chimera.

108. Is there a relationship between Gödel's first incompleteness proof and simple jokes?

One of the most difficult ideas in mathematics to understand is Gödel's first incompleteness proof. It is a proof that claims to make a fundamental discovery about the limits of set theory with regard to its ability to provide a logical groundwork on which all higher mathematics can be based. When I studied Gödel's proof, it seemed to me that it didn't make any sense. But, recently, I had the strange notion that it may operate under the same principles that jokes operate under. Here are two simple jokes. Why did the cookie go to see the doctor? The funny answer is that he went because he was feeling crummy. Another joke is the following: I've been on a diet for a week, and all I lost is seven days. Like many jokes, the first joke is based on the fact that words often have two or more meanings. "Crummy" in the context of the joke means both miserable and easily crumbled. In the second joke the humor comes from the thwarting of your expectations. You would expect a person to say something like "I've been on a diet a week, and I've lost five pounds." The speaker instead of saying he lost weight states that he lost only time, which is to say he wasted his time on the diet. The implication is that the diet didn't work, he didn't lose any weight, and perhaps he even gained weight. The funniness of the joke is heightened by the fact that the teller says the word "seven," which is a number, and he doesn't follow it with a unit of weight such as pounds as you would expect instead he follows it with the word "days," a unit of time measurement.

It may sound strange, but Gödel's first incompleteness proof may be based on the concept that you cannot restrict words or concepts to one meaning. Just as some of the words in jokes have more than one meaning, it seems Gödel insists that some terms in his proof have more than one meaning while at the same time he insists that they do not. Here is how his thought process may work. He invents a concept called Gödel numbers. Gödel numbers are the familiar positive integers, also known as whole numbers or counting numbers. But, Gödel gives them a second meaning. As Gödel numbers they represent variables and mathematical formulas and other mathematical terms. He takes away their original meaning as counting numbers. Counting numbers are no longer represented in the usual manner, and I believe they are called numerals instead of counting numbers. That is they are no longer represented as 0, 1, 2, 3, 4 . . . They are represented this way 0, s0, ss0, sss0, ssss0 . . . The way they are spoken about in everyday speech is very cumbersome. They are spoken of as zero, the numeral subsequent to zero, the numeral subsequent to the numeral subsequent to zero, the numeral subsequent to the numeral subsequent to the numeral subsequent to zero, the numeral subsequent to the numeral subsequent to the numeral subsequent to the numeral subsequent to zero and so on and so on.

Now, here in a simplified form is what Gödel may do. He constructs a formula such as x = 3. Then he finds the Gödel number of this formula. The Gödel number of even such a simple formula is a very large number. But, wait a minute, I have made a mistake; the number 3 is no longer a whole number in Gödel's proof. It is a Gödel number; it represents a formula or some other mathematical term. If I want to write the formula x = 3 as it would be written using the nomenclature he employs in his proof, I must write x = sss0. That may be Gödel's point because of the force of habit we must think of whole numbers as 1, 2, 3, 4. . .. It makes little sense for us to try to think of them as s0, ss0, sss0 and so on. And, it makes even less sense for us to try to think of mathematical terms and formulas as Gödel numbers such as 19,934,566,728,941,628,347,912. But, let's return to our example. Next, he formulates an instruction that says insert the numeral value for the Gödel number of the formula x = sss0 into the variable x of the formula. You may ask is it correct to have a variable x and an equal sign in the formula? Shouldn't these terms be replaced by Gödel numbers? Shouldn't the entire equation be represented by a Gödel number? We'll leave that question aside. Let's arbitrarily decide that the Gödel number of the formula x = sss0 is 500,002. So we write the numeral version or whole number version of the Gödel number 500,002 as s (followed by the letter "s" repeated 500,000 times) s0. We obtain the equation s (followed by the letter "s" repeated 500,000 times) s0 = sss0. This equation is incorrect because 500,002 does not equal 3. This is troublesome because all the equations produced in Gödel's system that contain no variables are supposed to be correct. But, there does appear to be a way out of the dilemma. You could formulate a simple rule such as you cannot insert the numeral value of the Gödel number of an equation into the variable of the equation unless you are sure the substitution produces equality, which it seems would be a very unlikely occurrence. This kind of rule would appear to solve the dilemma. Here is where the principle behind many jokes comes into play. Although we insert the numeral value of a Gödel number into the variable of our equation because of force of habit we think of the numeral value as the number 500,002 and that is the Gödel number our equation. So it seems we have two equations, the first is 500,002 =3 (written in the 0, s0, ss0 . . . notation) and the second equation is x = sss0 = sss0. In Gödel's proof, equations with variables in them are unprovable. The juxtaposition of the equations seems to be saying, "The equation Gödel number 500,002 =3 is unprovable because the Gödel number represents an equation with the variable x and equations with variables are unprovable or the equation 500,002 = 3 is unprovable because it is incorrect." This is a true statement since in Gödel's proof only formula such as sss0 = sss0 or ss0 + s0 = sss0 are considered provable. Thus we have generated a statement that we know is true namely "500,002 = 3 is unprovable and thus incorrect" It is unprovable in Gödel's proof because only formula such as 3 = 3 or more precisely sss0 = sss0 are provable. It seems to make sense in the same way a joke makes sense.

For instance, we can relate Gödel's proof to the first joke that was mentioned. The phrase, "Why did the cookie go to the doctor?" seems to be analogous to the instruction in Gödel's proof to insert or substitute the numerical value of the Gödel number of the equation in question for the sole variable in the equation. They are analogous because we don't usually think of a cookie going to a doctor and we don't usually think of substituting the numerical value of a Gödel number into the variable of an equation. The phrase that is the punch line of the joke "Because he felt crummy" is analogous to the production of two equations in Gödel's proof that when combined together seem to say something that we know is true, yet is unprovable in the framework of Gödel's proof or Gödel's system such as "500,002 = 3 is unprovable and thus incorrect."

109. Are recent astronomical discoveries providing evidence that may falsify Einstein's theories of special and general relativity?

It is easy to understand how an individual who has never seriously studied Einstein's theories can imagine they are in no danger of being falsified. It is never mentioned in articles in the mainstream news media. It is more difficult to explain the reluctance of individuals who have read books such as Lillian Lieber's _The Einstein Theory of Relativity_ to admit that his theories are in grave danger of being falsified or have already been falsified and are being propped up by speculation. Lieber specifically points out that the cosmological principle is used to produce Einstein's equations. Yet, in a very short article in _The Week_ in the "Health & Science" section of Feb. 1, 2013 we read that the discovery of two enormous clusters of quasars confound the cosmological principle, which holds that the universe is essentially homogenous (uniform). Many of the quasars lie at the center of their own galaxy. It is the size and proximity of the two clusters that confounds the cosmological principle. One of the astronomers says that the discovery makes it hard to say the universe is uniform.

110. Does Einstein's theory of special relativity subtly violate causality?

Is it possible that Einstein's notable rebuke of quantum mechanics for its dismissal of causality was not what it seemed to be? A pair of famous quotations is attributed to Einstein in this regard: "God does not play dice with the universe." and "Do you believe the moon actually disappears when you are no longer looking at it?" Perhaps, Einstein's rebuke was not aimed at the negation of causality, but at a bombastic negation of causality. When I first read a popular explanation of quantum mechanics' negation of causality more than five decades ago, it seemed to have a hint of desperation and bombast to it. As I recall, it was something along the lines of the following: although the odds are around a quintillion to one nothing prevents the atoms in a kitchen table from vibrating in such a way that the entire table moves an inch to the left. Since then, the explanation has become more refined, but there still seems to be a hectoring quality to quantum mechanics' claims dismissing causality.

Einstein may have felt that the modern scientific outlook called for the negation of causality, but only when it was done with the profoundest of subtlety. Interestingly, an individual who accepts Einstein's theories will never conclude that his notions negate causality. Only an individual who is convinced that special and general relativity are invalid might conclude these theories violate causality. This conclusion might be reached especially if the individual is familiar with the critique of special relativity presented in the French philosopher Henri Bergson's book _Duration and Simultaneity_ that was published in1920. Bergson does not claim that special relativity violates causality, but if we elaborate on his criticism that Einstein's special relativity employs ghost observers we can detect a subtle violation of causality.

With regard to special relativity, Bergson makes the claim that the observer on the moving train is a ghost observer, not a real observer–a projection that exists in the mind of the observer on the embankment only. Bergson claims that according to the conventions of special relativity any observer has the right to claim that he is not in motion–in other words that he is at rest. The observer on the railway embankment has the right to claim that he is not in motion–in other words that he is at rest (by convention) regardless of the fact that he is standing on the earth, which does seem to be in motion. This claim allows him to report that since he is midway between the two lightning strikes and since he sees that light rays from the two lightning strikes impinge on his mirrors, fixed at right angles to each other, at the same time, the lightning strikes occurred simultaneously. Einstein claims that the observer on the moving train does not see the lightning strikes as simultaneous because he is in motion. The observer on the train is rushing toward the light rays emanating from one lightning strike, and he is rushing away from the light rays emanating from the other lightning strike. Bergson points out that by the conventions of special relativity any observer has the right to claim he is at rest so the observer on the moving train has the right to claim he is at rest. Since the observer on the moving train was at the midpoint when the lightning strikes occurred and since, according to his point of view, he has not moved from the midpoint because he is a rest, he sees the lightning strikes as simultaneous.

Einstein's claim that the observer on the moving train sees the lightning strikes as non-simultaneous because he is moving violates causality because according to the conventions adopted by special relativity all observers have the right to claim they are at rest. Interestingly, the observer on the moving train could rightly claim that since he is at rest, it is the observer on the embankment and the embankment itself that is moving and therefore the observer on the embankment must see the lightning strikes as non-simultaneous. Thus, the observer on the embankment would be required to see the lightning strikes as both simultaneous and non-simultaneous. And by the same reasoning, the observer on the moving train would be required to see the lightning strikes as both simultaneous and non-simultaneous.

The profound subtlety of Einstein's violation of causality is that the violation of causality agrees with our common sense view of the world. Common sense tells us that an observer on a moving train who was at the midpoint between two lightning strikes at the moment of their occurrence and who since that moment has been rushing toward the light rays emanating from one lightning strike and away from the light rays emanating from the other lightning strike would not see the lightning strikes as simultaneous.

Since special relativity's violation of causality agrees with our common sense view of the world, we do not see it as a violation of causality. The violations of causality depicted in quantum mechanics often violate our common sense view of the world.

Probably Einstein's most forthright violation of causality occurs when states that of his own free will, he stipulates that the time it takes a light ray to travel from point A to the midpoint is equal to the time it takes a light ray to travel from point B to the midpoint, when points A and B and the midpoint are all located on the moving earth. But, this does not seem like a violation of causality because our common sense experience tells us that the earth is not in motion.

111. In your studies of Einstein's theories, have you come across any author whose writing might contain small portions that are clever communist propaganda?

The following excerpts are from Lillian Lieber's _The Einstein Theory of Relativity_. The book was published in at least two versions in the 1930s and 1940s. I suspect that the excerpt is clever communist propaganda.

"When Einstein warned President Roosevelt that such experiments [splitting the atom] might lead to the acquisition of terrific new sources of power by the ENEMY of the human race, the President naturally saw the importance of having these experiments conducted where there was some hope that they would be used to END the war and to PREVENT future wars instead of by those who set out to take over the earth for themselves alone! Thus the ATOMIC BOMB was born in the U.S.A.

"And now that a practical method of releasing this energy has been developed, the MORAL is obvious: We MUST realize that it has become too dangerous to fool around with scientific GADGETS, WITHOUT UNDERSTANDING the MORALITY which is in Science, Art, Mathematics—SAM, for short. These are NOT mere idle words. We must ROOT OUT the FALSE AND DANGEROUS DOCTRINE that SAM is amoral and is indifferent to Good and Evil. We must SERIOUSLY EXAMINE SAM FROM THIS VIEWPOINT.

"Religion has offered us a Morality, but many 'wise guys' have refused to take it seriously, and have distorted its meaning! And now, we are getting ANOTHER CHANCE—SAM is now also warning us that we MUST UNDERSTAND the MORALITY which HE is now offering us. And he will not stand for our failure to accept it, by regarding him merely as a source of gadgets! Even, using the atomic energy for 'peaceful' pursuits, like heating the furnaces in our homes, IS NOT ENOUGH, and will NOT satisfy SAM. For he is desperately trying to prevent us from merely picking his pockets to get the gadgets in them, and is begging us to see the Good, the True, and the Beautiful which are in his mind and heart. And, moreover, he is giving new and clear meanings to these fine old ideas which even the skeptical 'wise guys' will find irresistible. So DO NOT BE AN ANTI-SAMITE or SAM will get you with his atomic bombs, his cyclotrons, and all his new whatnots. He is so anxious to HELP us if only we would listen BEFORE IT IS TOO LATE!"

If my suspicion is correct and the excerpt is communist propaganda, does it mean that in that era we were awash in clever communist propaganda? The tone of Lieber's writing in the excerpt is interesting she moves from the whimsical, "don't be an anti-samite" to the hectoring, "we must root out the false and dangerous doctrine." She characterizes the Nazis as "those who set out to take over the earth for themselves alone!" Does that imply that it may be acceptable to take over the earth as long as you did it in the name of all of humanity?

What is she actually saying? It seems to be that science, art and mathematics offer us a new religion. As she writes, "Religion has offered us a Morality. . . And now, we are getting ANOTHER CHANCE." The manner in which she blithely dismisses our morality derived from our Christian religion is amazing, but apparently the fact that, "many 'wise guys' have refused to take it seriously, and have distorted its meaning!" is enough for her to jettison Christian morality. And what will replace Christian morality? SAM will replace Christian morality. SAM is science, art and mathematics. What are we to make of her claim, close to seventy years later, that, "SAM . . . is begging us to see the Good, the True, and the Beautiful which are in his mind and heart"? Has science, art and mathematics presented the world with a new and better morality that because of its spectacular clarity and profundity everyone finds it irresistible to adopt? It doesn't seem SAM has produced a new morality. It doesn't seem that SAM could produce a new morality. And, it doesn't seem SAM has made any effort to produce a new morality with the exception of psychology and sociology. These two efforts have been largely failures and largely dismissed. Many individuals believe these efforts were largely Marxist inspired. From my limited investigations, I believe that sometime around the mid-1970s sociologists abandoned the notion that sociology could make the world a better place. The gave up on the notion that they had the insights that would allow for the ending of poverty, the ending of war, the ending of crime and the ending of other social ills. Lieber claims that it is a "FALSE AND DANGEROUS DOCTRINE that SAM is amoral and is indifferent to Good and Evil." Are we sure it is a false doctrine? How would we know it is a false doctrine? It seems scientists, artists and mathematicians are for the most part liberals who seem to parrot liberal platitudes with amazing uniformity. Recently, I have been reading portions of _Borges and His Fictions: A Guide to His Mind and Art_ by Gene H. Bell-Villada. It seems that underlying many of Bell-Villada's ideas is the notion that all writers should think alike. Literary writers should have read certain number of the great writers from the past. They should agree as to who are the greatest writers of the modern era and their interpretations of these modern masters should be similar if not exactly the same. Whenever Jorge Borges diverges from orthodox opinion in these and other matters Bell-Villada finds fault with his ideas. The flaw with SAM as a source of a new and better morality is that SAM in an aggregate form is as narrow minded as any sectarian moralist, but SAM refuses to concede that he is narrow minded.

It's hard to know what to say about Lieber's ideas. She personifies science, art and mathematics into a sort of god-king, which is how she seems to view Franklin Delano Roosevelt. As she writes in the dedication of her book, "Franklin Delano Roosevelt who saved the world from those forces of evil which sought to destroy Art and Science and the very Dignity of Man."

On pages 87 and 88 she writes, "On the other hand, since actual measurements are local and not universal, and that only certain THEORETICAL RELATIONSHIPS are universal, the Einstein Theory shows also that practical measurement alone is also not sufficient for exploring the universe. In short, a judicious combination OF THEORY and PRACTICE, EACH GUIDING the other–a 'dialectical materialism'–is our most effective weapon." "Dialectical materialism" is a term that is often associated with Marxism.

On page 309, she writes, "And knowing that the laws are MAN-MADE, we know that they are subject to change and we are thus PREPARED FOR CHANGE." This is a clever turn of a phrase. The actual laws that govern the behavior of the physical universe would not seem to be man-made. If mankind has discovered any scientific laws that are the actual laws that govern the behavior of the physical universe they would not be subject to change. Perhaps, any of the imprecise versions of these laws that mankind has discovered could be said to be man-made and thus subject to change. The Marxists in general found the concept of God to be anathema to their beliefs so they would prefer that the laws that govern the behavior of the physical universe to be man-made. They also wanted to change almost everything about our society, and they wanted to keep on changing society. They wanted relentless and perpetual change.

112. Can Gödel's first incompleteness proof be interpreted and viewed as an item of popular culture?

The book cover for my free eBook _Fathoming Gödel_ consists of a photograph of two pairs of old shoes ensconced in a shallow box of red bricks, and the title page photograph is of a swamp with hundreds of dead trees. These images accurately reflect my opinion of Gödel's first incompleteness proof.

Is Gödel's mathematical logic like picking up sticks in a forest? The forest is littered with sticks, but you convince yourself that you can pick up enough sticks to create a park that you and other people can enjoy. Then you lower your expectations and hope to create a small clearing that you alone might enjoy. Time passes, tinged with madness, and all that remains is the forest and sticks.

Is Gödel's mathematical logic like The Beatles' album _Sargent Pepper's Lonely Hearts Club Band?_ It's not actually a concept album, but since so many people claim that it is one, and not only that, but the first concept album in rock and roll, then in a way it is a concept album notwithstanding that it is not a concept album.

If you read _I Am a Strange Loop_ by Douglas Hofstadter, he seems to admit that Gödel's first incompleteness proof is invalid, but then he seems to go on to say that it is true by analogy. And, perhaps, he makes the further claim that all truth comes from analogy. This neatly sidesteps the claim that I believe was made by Aristotle that analogy is the weakness form of argument.

Perhaps, the explanation of a lesser mystery will shed light on the greater mystery of Gödel and his proofs. For years, I wondered why the rock and roll group The Band had managed to accrue such overwhelming accolades from their peers as well as from the critics in the music press. The Band put out a few fine albums, but I think the actual reason they were so lauded is that they were too obscure to be appreciated by the average rock music fan. It is difficult to believe that being too obscure for the average fan would be of paramount importance to the elite, but it seems it is. So it is with Gödel. It is the impenetrable obscurity of his proofs that by far makes the greatest contribution to his mystique. Those who claim to understand his work and can convince others of this fact have an automatic elevation in their status. And, to rise in the pecking order is the most sublime achievement society can offer. I recently learned that snapping turtles spend the winter months buried in mud at the bottom of ponds. This act would seem to rival Gödel in the obscurity competition.

It seems no one ever bothers to ask of Gödel and his proofs and other creators of obscure theories the following question: Why wouldn't friendly individuals produce at least one and perhaps several straightforward explanations of their complex ideas? Sometimes, it seems that in the past famous people were little more than gimmicks. If we ever achieve the world depicted in the _Star Trek_ universe, the question will be why weren't there many books written that suspected the validity of Gödel's proofs.

113. Is there a relation between causality and a hostility expressed toward the repressiveness often associated with the Christian religion?

It is well known that the followers of quantum mechanics dismiss causality in the realm of extraordinarily small particles whose behavior they are intent on explaining. While it is often assumed that Albert Einstein is a champion of causality in opposition to the doctrines of quantum mechanics, this may not be entirely true. He may want to dismiss causality, but in a more subtle manner than is done in quantum physics.

Why would scientists want to dispose of causality? Causality seems to be the most useful tool that scientists can employ. Perhaps, we should look at modern politics for an answer. One of the most shocking aspects of politics today is that it seems former hippies and the progeny of hippies want to institute an all-encompassing nanny state in the U.S.A. They seem remorseless and tireless in their quest for ever more power to control every aspect of our lives. If we go back in time 2,000 years, the founder of the Christian religion, Jesus Christ, seems to have much in common with the hippies. In fact, he might be termed the doyen hippy. Is it possible that the religion he founded over time became as repressive and controlling as our modern nanny state? If that is the case, what would it have to do with causality?

The Christian Church derives its power to control our behavior through the operation of causality. If we sin, we are punished by God for our sins. If we do not sin, we are rewarded by God for not sinning. If our sins are great, our punishment is great, and if our sins are negligible, our punishment is small, as well. This punishment or reward will be handed out on Judgment Day by Jesus Christ. The dead will be brought back to life. All the living individuals and all the formerly dead individuals will be judged by Jesus Christ. The sinners will be punished and the good individuals will be rewarded. It is a cause and effect scenario. The cause is the nature of an individual's behavior and the effect of that behavior is either punishment or reward. Good behavior is rewarded, and bad behavior is punished.

If we see cause and effect, i.e., causality operating in the natural world, it would argue that causality would operate in the spiritual world. If we believe God created the natural world and that it seems to operate according to the principle of causality, then it is likely that if God created the spiritual world, as well, it operates according to the principle of causality. But, what if the natural world doesn't operate according to the principle of causality? Why would God create a natural world that doesn't operate according to the principle of causality and a spiritual world that does operate according to the principle of causality?

When philosophers such as David Hume call into question the principle of causality, are they actually trying to restrict the ability of the Christian Church to control our behavior? The argument could be that since the natural world doesn't operate according to the principle of causality, it is unlikely that the spiritual world operates according to the principle of causality. They were both seemingly created by God. Therefore, on Judgment Day, Jesus Christ would not hand out punishment and reward based on the nature of our behavior. This would call into question the correctness of the Christian Church's controlling every aspect of our behavior in an effort to have us reap the rewards of a heavenly existence on Judgment Day. For those who found the restrictiveness, of the Christian Church in controlling every aspect of their lives, unpleasant this notion would be appealing.

Did David Hume find the restrictiveness of the Christian Church distasteful? According to _The World Book Encyclopedia,_ "David Hume (1711-1776) was born in Edinburgh, Scotland and spent most of his life writing. Occasionally, he served on diplomatic missions in France and other countries. His major work, _A Treatise of Human Nature_ (1739, 1740), attracted little attention when it was published. But Hume's fame grew, especially in France, after he published more works on philosophy, religion, and history." _The World Book Encyclopedia_ continues, "Hume, an agnostic, argued that the existence of God could not be proved. He said that even granting God's existence, nothing could be absolutely known about His nature."

Cause and effect or causality would seem to be something that we all can agree on. It wouldn't seem to matter what other beliefs we held we could all agree on causality. It's something we use in our everyday lives. This summer I was working outside. After some time had passed I was thinking this job is taking much longer than I thought it would and it is a very bright and hot summer day. Soon after that I noted that the skin on the back of my legs began to feel tight and it had a reddish hue. I thought my legs are getting a mild sunburn. That to me is an example of cause and effect. It can also be called common sense. If someone doesn't operate within a common sense framework, we tend to think they are somewhat crazy. A quotation often attributed to Einstein is, "Doing the same thing over and over again and expecting a different result is the definition of insanity." It is interesting that we don't think of Charlie Brown in regard to his perpetually trying to kick the football that Lucy holds for him as being insane, but perhaps we should. Why would a philosopher such as David Hume introduce notions into his philosophy that would seem to have a hint of insanity to them? _The_ _World Book Encyclopedia_ states the following. "Hume was famous for his attack on the principle of _causality._ This principle states that nothing can happen or exist without a cause. Hume believed that although one _event_ (set of impressions) always preceded another, this did not prove that the first event caused the second. The constant conjunction of two events, he said, built up the expectation that the second event would take place after the first. But this was nothing more than a strong belief or habit of mind taught by experience. One could never prove there were causal connections among impressions."

It seems Hume wants to have it both ways. In a way he admits the operation of common sense, but he doesn't want to call it causality. It seems like philosophical hair splitting. If we begin to imagine the amount of coincidence in his world, we can see the ridiculousness of his notions. For instance, whenever we're outside and we begin to feel cold we could check the thermometer and note that the temperature has gone down. Would Hume be forced to conclude that feeling colder and noting that the temperature reading of the thermometer has lowered were coincidence? Would he say it is impossible to prove a cause and effect link between changes in the hotness and coldness of our local environment and the readings given by a thermometer? With regard to temperature, we know that the wind can make us feel colder although the temperature on the thermometer may not have lowered. There is a kind of subtlety to common sense that philosophers such as Hume may have found distasteful because it was in competition with their own brand of subtlety.

_The World book Encyclopedia_ states, "According to Hume, three principles connected associated ideas with each other: (1) _resemblance,_ (2) _contiguity,_ and (3) _cause and_ _effect_. . . . In cause and effect, if one unit constantly preceded another, thought of the first resulted in thought of the second." It is interesting that Hume seems to accept cause and effect, but claims the relationship cannot be proven. Setting the bar for proof at such a height is a problem that seemed to confuse modern philosophers such as Karl Popper. It may be a kind of useless perfectionism.

Or it could have been an attempt to undercut the power and control the Christian Church exerted over individual's everyday lives. The argument would be we can't prove that causality exists in the natural world. This undercuts the argument that causality exists in the spiritual world. This calls into question the notion that God will judge us on Judgment Day and reward those whose behavior was good and punish those whose behavior was bad.

Is there a link between some of the nonsense we hear today and the notion that questioning causality will undercut the power and control of the Christian Church and traditional morality? We might say there was a method to David Hume's madness. And that same method may be on display in such famous nonsensical statements as, "You must pass the bill so you can find out what's in the bill."

We could also speculate that there is some connection between both socialism and communism and the arguments of philosophers such as David Hume against causality. Communism is typically an atheistic ideology that is openly hostile to the concept of God and religion in any form. Socialism tends to find traditional morality distasteful because it is too conservative and not progressive. Both of these ideologies would tend to embrace a notion that undercut the power of the Christian Church. But by embracing ideas such as David Hume's arguments against causality they are rejecting common sense and introducing a kind of insanity into their thinking. If the proponents of socialism and communism were to embrace the idea of common sense perhaps they could make their ideologies perform in a more robust manner.

It is interesting that we use common sense or cause and effect relationships often in our everyday lives. It seems the use of common sense can bring us a modest amount of pleasure. Why would a philosopher or anyone else try to minimize such activities? For example, I was taking a walk recently, and I found myself strolling across a Walmart parking lot. I noticed that as in most parking lots the asphalt paving had a number of cracks running through it. What caught my attention was that the white directional arrows on the parking lot seemed to have an enormous profusion of cracks running through them. My first thought was it must have something to do with the white color of the arrows as opposed to the black color of the asphalt. I know that the color black absorbs heat while the color white tends to reflect heat. But, then I examined some directional arrows painted yellow, they didn't seem to have any more cracks running through them than the surrounding pavement. That led me to surmise that it may not be the color of the directional arrows that causes the cracks. The white directional arrows seemed to be painted onto the asphalt using a plaster-like substance that was about one-third of an inch thick. The yellow directional arrows seemed to be painted on the asphalt with a thin layer of yellow paint. This led me to conclude that it was the plaster-like material that was the source of the profuse cracking and not the color that the arrows were painted.

114. Can you give a brief description of Ole Rømer's Experiment in which he attempted to measure the speed of light by observing the eclipses of Jupiter's moon Io?

Here is a brief description of Ole Rømer's experiment. He knew that both the Earth and Jupiter orbited the Sun, as did the other planets. He knew that the Earth and Jupiter were closest together when they were in opposition. In opposition means the Sun, the Earth and Jupiter are all in a straight line and in that order. After the Earth and Jupiter are in opposition the Earth moves further and further away from Jupiter. Rømer knew that Jupiter had at least four moons. The closest of the four known moons to Jupiter was called Io. As Io orbited around Jupiter, it passed into and out of Jupiter's shadow. This is known as the eclipse of the moon Io by the shadow of the planet Jupiter. From Earth, Rømer could observe the emergence of Io from Jupiter's shadow. The time period from one emergence of Io from Jupiter's shadow to the next emergence would give an apparent time period for Io's orbit around Jupiter. It would be an apparent time period and not an actual time period because it would not take into account the movement of the Earth either away or toward Jupiter during the time period between one emergence and the next. It would be a fairly accurate measure of the time period of Io's orbit because as we know today the Earth moves at a speed of 19 miles per second in its orbit around the Sun and the speed of light is 186,000 miles per second.

Rømer reasoned that if he measured the duration of Io's apparent orbit from one emergence to the next at a time shortly after the opposition of Jupiter and the Earth, the durations of Io's apparent orbits would increase for several months as the Earth moved away from Jupiter and its moon Io. Jupiter was in opposition to the Earth on March 2, 1672. Rømer measured the apparent duration of Io's orbit on March 7, 1672 and again on March 14, 1672. During that time Io had made four orbits of Jupiter. The average apparent duration of one orbit was 42 hours 28 minutes 31.25 seconds. Rømer reasoned he could calculate when Io should emerge from Jupiter's shadow after 26 additional orbits based on the duration of Io's orbit as determined by these two initial measurements. He knew that the actual observed time that Io emerged from the shadow of Jupiter after 26 additional orbits would be after the time predicted by his calculations because the Earth would have been moving away from Jupiter for the entire time it took for Io to complete 26 additional orbits. The difference between the observed time of Io's emergence from Jupiter's shadow and the calculated time for Io's emergence from Jupiter's shadow would be the time it took a light beam to cover the additional distance the Earth had moved away from Jupiter in the period of time for 26 additional orbits of Io. The emergence of Io from Jupiter's shadow for the last of the 26 additional orbits took place on April 29, 1672 at 10:30:06. From this measurement an apparent orbital period of 42 hours 29 minutes 3 seconds was calculated. The difference seems minute—32 seconds—but it meant that the emergence on April 29 was occurring 15 minutes after it was predicted.

The difference between the observed time of Io's emergence from Jupiter's shadow and the calculated time for Io's emergence from Jupiter's shadow after 26 additional orbits was 15 minutes. Rømer needed to calculate the additional distance the Earth had moved away from Jupiter in the time it took to complete the 26 additional orbits of Io. To make this calculation Rømer would need to know the radius of the Earth's orbit and the radius of Jupiter's orbit. He did not know these values, but we know these values so we can make our own calculations.

115. Can you calculate the speed of light using Rømer's data as given in the Wikipedia article "Rømer's Determination of the Speed of Light" combined with modern astronomical data?

With observations dating from around 1668 to 1678 with particular emphasis on observations from March 7, 1672 until April 29, 1672, Danish astronomer, Ole Rømer attempted to measure the speed of light by analyzing the eclipse data from Jupiter's innermost Galilean moon Io. To measure the speed of light accurately using this method Rømer needed to know the radius of the Earth's orbit and the radius of Jupiter's orbit. He did not know accurate values for these two radii. Today, we accurately know these two values. When we calculate the speed of light using Rømer's data as it is presented in the Wikipedia article "Rømer's Determination of the Speed of Light" combined with modern data, it seems that the speed of light is about 46,000 miles per second. The accepted value for the speed of light is 186,000 miles per second. The discrepancy is so large that it seems there is no simple explanation. Although there seem to be at least two discrepancies in the Wikipedia presentation of Rømer's data, that do seem significant.

How can we use the eclipse data of Jupiter's moon Io to measure the speed of light? If Jupiter orbited the Earth as was assumed in the pre-Copernican era, there would not be a significant increase or decrease in the distance between the Earth and Jupiter, and hence there would not be a significant increase or decrease in the distance between the Earth and Jupiter's moon Io. An example of this concept is that both the Earth and Jupiter orbit the Sun and there is not a significant increase or decrease in their distance from the Sun. If Jupiter orbited the Earth, the observed length of time it took Io to make one orbit of Jupiter would always be the same; it would remain constant. Because the speed of light is not instantaneous, when we observed Io entering Jupiter's shadow, it would have already been in Jupiter's shadow for a number of minutes owing to the time it takes light to travel the distance from Jupiter's moon Io to the Earth, which at Jupiter's closest approach to the Earth is approximately 390,000,000 miles. Also, when we observed Io emerging from Jupiter's shadow, it would have emerged from Jupiter's shadow many minutes prior owing to the time it takes light to travel from Io to the Earth. But, if we observed the time from one emergence of Io from Jupiter's shadow to the next emergence, we would obtain an accurate time period for Io's orbit of Jupiter. Likewise if we observed the time from one entrance of Io into Jupiter's shadow to the next entrance, we would obtain an accurate time period for Io's orbit. This is because the two delays in observing two emergences of Io from Jupiter's shadow would be equal and thus cancel each other. Each time we observed Io emerging from Jupiter's shadow it would actually have emerged from Jupiter's shadow the same number of minutes before. Likewise, the two delays in observing two entrances of Io into Jupiter's shadow would be equal and thus cancel each other. Each time we observed Io entering into Jupiter's shadow it would actually have entered into Jupiter's shadow the same number of minutes before.

But Jupiter does not orbit the Earth so the distance between them is not approximately constant, and so the distance between the Earth and Io is not approximately constant. Rømer used the fact that the distance between the Earth and Io is not approximately constant and the fact that the orbital period of Io is constant to measure the speed of light. Here is what Rømer did. He knew that the Earth and Jupiter were closest together when the two planets were in opposition. The term "in opposition" means that the two planets are on the same side of the Sun and the two planets and the Sun form a straight line. Jupiter was in opposition on March 2, 1672. Around the time of the opposition, the emergence of Io from Jupiter's shadow cannot be observed. The explanation for this is that from the Earth, it is not possible to view both the entrance of Io into Jupiter's shadow and the exit of Io from Jupiter's shadow for the same eclipse of Io because one or the other will be hidden by Jupiter itself. At the point of opposition both the entrance and exit of Io from Jupiter's shadow are hidden by Jupiter itself. Rømer had to wait several days before he could observe the emergence of Io from Jupiter's shadow. Although he had to wait several days to make his observations, the Earth and Jupiter were still relatively close together. From observation made on March 7, 1672 at 07:58:25 and March 14, 1672 at 09:52:30 Rømer calculated that the orbital period of Io was 42 hours 28 minutes and 31.25 seconds. Io made four orbits of Jupiter in the period from March 7 at 07:58:25 until March 14 at 09:52:30. Here is the first discrepancy in Wikipedia's presentation of Rømer's data. Four orbits of Io at 42 hours 28 minutes and 31.25 seconds per orbit amounts to 169 hours 54 minutes and 5 seconds. The time period from March 7, 1672 at 07:58:25 until March 14, 1672 at 09:52:30 falls short of this amount by 12 hours even when we make the time period as long as possible by making March 7 a morning observation occurring at 07:58:25 a.m. If we change the March 14 at 09:52:30 (p.m.) observation into a morning observation occurring one day later on March 15 at 09:52:30 a.m. the 12 hour discrepancy vanishes. Using this data he could calculate when Io would emerge from Jupiter's shadow 30 orbits later using the date March 7, 1672 at 07:58:25 a.m. as the starting time for his calculations. This calculation would be based on the notion that the distance between Jupiter's moon Io and the Earth was approximately constant. But, the distance from the Earth to Io is not approximately constant. For the time period in question, the distance between the Earth and Io was always increasing. Much of the increase in the separation of the two planets was caused by the motion of the Earth, but some portion of it was caused by the motion of Jupiter. Any delay in the actual emergence of Io from Jupiter's shadow as compared to the calculated emergence of Io from Jupiter's shadow 30 orbits hence would be due to the increase in distance that light had to travel from Io to reach the Earth. Since by using modern data we can calculate the increase in the separation of the two planets, we need only to divide Rømer's figure for the time difference between the calculated emergence and the actual emergence into the value for the increased separation between the two planets, and we have calculated the speed of light.

The last exit of Io from Jupiter's shadow in this series of observations was on April 29, 1672 at 10:30:06 p.m. By this time, Io had completed 30 orbits around Jupiter since March 7 at 07:58:25 a.m. The exit from Jupiter's shadow on April 29 occurred 15 minutes later than the predicted exit of Io from Jupiter's shadow. Here is where the second discrepancy in the Wikipedia data occurs. If we multiply the March 7 through March 14 calculation of the orbital period of Io, which is 42 hours 28 minutes and 31.25 seconds according to Wikipedia, by 30 we obtain 1,274 hours 15 minutes and 37.5 seconds. If we multiply the orbital period of Io obtained through using the entire series of observations, which is 42 hours 29 minutes and 3 seconds according to Wikipedia, by 30, we obtain 1,274 hours 31 minutes and 30 seconds. The difference between the two orbital periods is 15 minutes and 52.5 seconds, which is very close to the value given by Wikipedia. The problem is that the time period from March 7 at 07:58:25 a.m. until April 29 at 10:30:06 p.m. is only 1,261 hours 54 minutes and 5 seconds. The period of time is too short by more than 12 hours. There could be a number of reasons for the two discrepancies. For instance, I could have incorrectly determined the length of the two time periods. Another explanation might be that there are two typos in the Wikipedia article. Also, there could be some other explanation for the two discrepancies.

Let's accept the Wikipedia value that the difference between the calculated time for the exit of Io from Jupiter's shadow and the observed time of the exit of Io from Jupiter's shadow after 30 orbits is 15 minutes. The reason for the time difference is that the Earth and Io are moving apart. The distance light has to travel to reach the Earth from Io increases slightly with each of the 30 orbits, and hence the apparent length of each of the 30 orbits increases slightly. The cumulative increase is 15 minutes.

Let's calculate the cumulative increase in the separation between Io and the Earth. Let's use 93,000,000 miles for the radius of the Earth's orbit. Let's use 483,500,000 miles for the radius of Jupiter's orbit, and let's assume Io is approximately the same distance from the Sun as Jupiter. At opposition on March 2 the distance between the Earth and Io will be 483,500,000 – 93,000,000 = 390,500,000 miles. Let's calculate the distance between Io and the Earth on March 11, which is about midway between March 7 and March 14. The three bodies, the Earth, the Sun and Io form an obtuse triangle. We know the length of two sides. One side is the radius of the Earth's orbit. The other is the radius of Jupiter's orbit. The third side we can calculate using the law of cosines: c2 = a2 \+ b2 – 2abcosC or c2 = (93,000,000)2 \+ (483,500,000)2 – 2(93,000,000) (483,500,000) cos. C. We calculate C by transforming the time periods into degrees of orbit. We arbitrarily choose the position of Earth and Jupiter at the point of opposition as the line designated 0 degrees. On March 11 the Earth has moved for 9 days away from the March 2 line of 0 degrees. From that we obtain 9/365.25 = x/360 and x = 8.87 degrees. Jupiter has also moved from the line of 0 degrees by 9 days, but 9 days is a much smaller fraction of Jupiter's orbit so we obtain 9/4,333 = x/360 and x = 0.75 degrees. So on March 11 angle C equals 8.87 degrees – 0.75 degrees = 8.12 degrees. The cosine of 8.12 degrees is 0.9899. So the law of cosines gives us c2 = 8.649 x 1015 \+ 2.337 x 1017 – 8.902 x 1016 = 15.333 x 1016. So c equals 3.916 x 108 or 391,600,000 miles. This indicates that on March 11 the Earth and Io are separated by 391,600,000 miles.

By how many additional miles are they separated by on April 29? The only value that changes is the measure of angle C. Let's calculate how many degrees the Earth has moved from the 0 degree line by April 29. We obtain 58/365.25 = x/360 and x = 57.16 degrees, and the Sun has moved 58/4,333 = x/360 from which we obtain x = 4.82 degrees. Therefore C equals 57.16 degrees – 4.82 degrees = 52.34 degrees and the cosine of that is 0.611. So c2 = 8.649 x 1015 \+ 2.337 x 1017 – 5.49 x 1016 = 18.74 x 1016. So c equals 4.329 x 108 or 432,900,000 miles.

We subtract the first c from the second c, and we obtain 432,900,000 miles – 391,600,000 miles = 41,300,000 miles. The Earth and Io are separated by an additional 41,300,000 miles on April 29 when compared to their separation on March 11. Since it takes light beams an additional 15 minutes or 900 seconds to travel that distance, the speed of light is 41,300,000 miles/900 seconds, which equals a speed of 45,889 miles per second.

_116. Is there a difficulty in making sense of some of the statements in the Wikipedia article titled_ "The Michelson-Morley Experiment" _?_

"For an apparatus in motion, the classical analysis requires that the beam-splitting mirror be slightly offset from an exact 45° if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed. In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams."

The above quotation is from the article in Wikipedia titled "The Michelson-Morley Experiment." It is difficult to understand. At first, you might think that since there is only one reality, if the beam-splitting mirror needs to be slightly offset from an exact 45° in order for the longitudinal and transverse beams to emerge from the apparatus exactly superimposed in the classical analysis, the beam-splitting mirror would need to be offset in the relativistic analysis, as well. Or, if the beam-splitting mirror becomes more perpendicular because of the Lorentz-contraction by precisely the amount necessary to compensate for the angle discrepancy of the two beams, you would not need to slightly offset the beam-splitting mirror from an exact 45°. Certainly, it can't be the case that before the Lorentz-contraction was hypothesized the beam-splitting mirror had to be offset, but after it was hypothesized it no longer had to be offset. The phrases "classical analysis" and "relativistic analysis" can be confusing. It seems we are concerned with the set-up of the experimental device, the interferometer, and either it is necessary to offset the beam-splitting mirror or it is not necessary to offset the beam-splitting mirror to make the longitudinal and transverse beams emerge from the apparatus exactly superimposed, one on the other.

At some point, you begin to think you are reading decades-old propaganda that has been imperfectly refurbished, but that can't be the case. Let's think of some understandable examples to try to solve this dilemma. In the 1860s, around the time of the Michelson-Morley experiment, the ether wind was thought to be generated by the passage of the earth through the ether. It is similar to when you hold your arm out of the car window when you are traveling around 60 mph. The air pushes your arm backwards. The flow of a river is often described as being analogous to the ether wind. Let's say you want to row a boat across a river that is flowing from east to west. When you stand on the shore, the river flows from right to left. Let's say you want to row across the river following a perpendicular line. To accomplish that task you start rowing across the river by fixing your sight on an object that is in a straight line across the river and on the opposite shore. If you continue rowing toward that object, you will be swept downstream by the current of the river. You will eventually row across the river, but you will end up downstream of your destination, which is to the left or west of your destination. You can overcome this problem by rowing across the river at an upstream angle that compensates for the current of the river. You set your sights on an object to the right of your intended destination, which is to the east of your intended destination. As you row across the river toward this object the current sweeps you downstream, which is to the and west toward your intended destination.

This is what I believe is meant by the phrase, "the beam-splitting mirror needs to be slightly offset from an exact 45°, if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed." There are some strange aspects to this phrase. Let's return to our analogy of rowing across a river. If you can calculate the precise upstream angle or by trial and error determine the precise upstream angle in which you need to direct your boat in order to cross the river in a straight, perpendicular line to reach your intended destination, it would seem that you could also calculate the speed of the current. So, this seems to indicate that if you know precisely how much the beam-splitting mirror needs to be offset from exactly 45°, you could calculate the speed of the ether wind. And, the speed of the ether wind is the speed of the earth because the ether wind is caused by the earth moving through unmoving ether. But, the claim that is made by proponents of the experiment is that the Michelson-Morley experiment cannot determine the speed of the earth. Or, more precisely, the claim is made that the Michelson-Morley experiment cannot determine the absolute velocity of the earth. But, it seems the experiment could determine both the absolute speed and direction of the earth's motion.

Another interesting aspect of the phrase, "the beam-splitting mirror needs to be slightly offset from an exact 45°, if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed" is asking ourselves in which direction does the beam-splitting mirror need to be offset. Does the angle need to be increased to say 46°, or does it need to be decreased to say 44°? Here is a simplified description of the experimental apparatus. The beam of light from the light source travels left to right, which can be thought of as west to east. The direction of the earth's motion is also left to right, which can be thought of as west to east, as well. Therefore, the ether wind is blowing in the opposite direction, right to left, which can be thought of as east to west. The source light beam strikes the beam-splitting mirror, which is inclined at 45°. What does inclined at 45° mean? Let's imagine the beam-splitting mirror on a table. The small angle that it makes with the table is 45° so, of course, the large angle it makes with the table is 135°. In the apparatus set up of the Michelson-Morley experiment (as it is usually depicted in diagrams) going from left to right, first we encounter the large 135° angle, and then we encounter the smaller 45°. This means the light beam from the source first encounters the 135° angle. If we were traveling along with the source light beam, we would see a "hill" in front of us sloping upward at 45°; we would not see a "tunnel entrance" with its roof sloping downward at 45°. The source light beam hits the mirror. I am going to call the small angle it makes with the mirror the angle of incidence. The angle of incidence is 45°, and the large angle is 135°. We can think of the angle of incidence as the bottom angle and above it is the large angle of 135°. Now, a law of optics comes into play, which states that the angle of reflection is equal to the angle of incidence. The source beam travels left to right and hits the mirror. The reflected beam travels straight upward and makes a 45° angle with the mirror. Here is how it can be pictured. The source light beam hits the mirror. The bottom angle it makes with the mirror is 45°. The reflected light beam leaves the mirror, and the top angle it makes with the mirror is 45°. The angle in between the bottom angle and the top angle is 90°. So the total of the degree measure of the angles is 45° + 90° + 45° = 180° as it must be because the mirror is a "straight line."

The reflected light beam is traveling perpendicular to the beam of light from the light source. Since the beam of light from the light source was traveling in the same direction as the earth, the reflected light beam is traveling perpendicular to the motion of the earth. The ether wind travels in the direction exactly opposite to that of the earth so the reflected beam of light is traveling perpendicular to the ether wind. The ether wind is blowing right to left, which can be thought of as east to west. The ether wind will blow the reflected beam of light downstream that is to the left (to the west). How can we adjust the mirror so the reflected beam of light veers off at an upstream angle so that the ether wind will blow it to the perpendicular instead of downstream? Let's change the inclination of the mirror from 45° to 30°. Now, if we were traveling with the source beam of light, we would see a less steep hill as we approached the mirror. The angle of incidence would be 30°, and therefore, the angle of reflection would be 30°. Remember the angle of incidence can be thought of as the bottom angle, and the angle of reflection can be thought of as the top angle so therefore the middle angle would be 120°. The reflected light beam is no longer traveling at a perpendicular angle to the source light beam, the motion of the earth and the motion of the ether wind. We have made the reflected light beam veer upstream by 30°, since 90° + 30° = 120°, now the ether wind can blow it to the perpendicular.

This answers the question in which direction does the beam-splitting mirror need to be offset. Does the angle need to be increased to say 46° or does it need to be decreased to say 44°? The answer is the angle needs to be decreased to some value below 45°. Now, we can examine this phrase, "In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams." But, it seems we have just learned that the beam-splitting mirror must become less perpendicular, not more perpendicular to compensate for the angle of discrepancy between the two beams. It can be difficult to visualize how length contraction makes the beam-splitting mirror more perpendicular. The slope of an angle can be thought of as rise over run. In the case of a 45° angle the rise and the run are equal. Let's say the rise is eight and the run is eight. The rise over the run is 8/8 or one. This is also the tangent of the angle. If we find the arc tangent of one using a calculator (where it is often indicated by the phrase, "tangent raised to the negative 1 power"), we will see that it is 45°. The Lorentz-contraction is a contraction of length. A contraction of length mean a contraction of the run since run is the distance along the x-axis and rise is the distance along the y-axis. Let's say that with the Lorentz-contraction the rise over run is eight over six or 8/6, which equals 1.33333. If we find the arc tangent of 1.33333, we obtain 53.13°. So it seems length contraction does make the beam-splitting mirror more perpendicular. The beam-splitting mirror has changed from a 45° angle to a 53.13° angle.

Now, in our analysis of the experiment increasing the incline of the beam-splitting mirror above 45° will increase the degree to which the reflected light beam is shifted downstream. We should not use the phrase "blown downstream" because the relativistic analysis discounts the existence of the ether. How can we explain the claim that "Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams." The explanation is that apparently to overcome the problem posed by the Lorentz-contraction of the beam splitter it was somehow decided that the reflected beam of light would move upstream instead of downstream. Thus, when the Lorentz-contracted beam splitter shifted the beam downstream it could be shifting the beam to the perpendicular since the beam was upstream of the perpendicular. But, how could the reflected beam move upstream? The ether wind was supposed to make the reflected beam move downstream, and even if you discount the existence of the ether wind, what would make the reflected light beam shift upstream? Light waves do not move with the object that emits them. Since the light source was attached to the earth, the reflected light beam would not travel upstream along with the earth. In fact, since the reflected light beam not moving upstream as the earth moves upstream, it would appear as though the reflected light beam would be shifted downstream even without the influence of the ether wind. The reflected light beam may appear to be shifted downstream in a different manner than the way it was blown downstream by the ether wind. There seems to be little or no justification for having the reflected beam of light shift upstream.

There seems to be a great advantage in having the reflected light beam mysteriously shift upstream so that it can be brought back to the perpendicular by the Lorentz-contraction. That advantage is that it makes it impossible to determine the speed of the earth by determining the speed of the ether wind. With the classical analysis there was always the danger that once it was learned that the beam-splitting mirror was offset from exactly 45°, someone would suggest that the amount of offset was indicative of the speed of the ether wind. Just as the upstream angle to which you direct your rowboat is indicative of the speed of the current of the river. The faster the current of the river the larger the degree measure of the upstream angle must be in order to offset the influence of the current.

There seems to be a problem with notion of offsetting or changing in any way the beam-splitting mirror in either the classical analysis or the relativistic analysis. It seems as though the beam-splitting mirror must be a double sided mirror. This aspect of the beam-splitting mirror seems never to be mentioned. But the beam-splitting mirror reflects the source beam of light which is traveling from left to right, and it also reflects the returning transverse beam of light which is traveling in the opposite direction right to left. The beam-splitting mirror at 45°reflects the source beam of light vertically upward, and it reflects the returning transverse beam of light vertically downward to the instrument that observes the interference patterns of the light waves. Upward and downward refer to directions associated with the usual diagrams of the experiment. Now, if the beam-splitting mirror is adjusted so that the reflected beam veers off from the perpendicular at an upstream angle (where it is blown back to the perpendicular by the ether wind), then the returning transverse beam will veer off from the perpendicular at a downstream angle (where it will be blown further downstream by the ether wind) this would seemingly make it difficult for the returning reflected beam of light to become superimposed on the returning transverse beam of light.

It is complicated, but we must remember that the beam-splitting mirror is a half-silvered mirror that allows light beams to pass through it as well as reflecting light beams. Let's assume that outgoing reflected beam of light veers off at an upstream angle and is blown back to the perpendicular by the ether wind. The returning reflected beam of light will have been reflected by a mirror at the end of the arm of the interferometer that is at 90° to the light beam. As the beam makes it return journey it will be blown downstream by the ether wind. It will pass through the half-silver beam-splitting mirror and continue downward at a nearly perpendicular angle toward the instrument that observes the interference patterns. The returning transverse beam of light traveling horizontally will be reflected by the offset beam splitting mirror. It will veer off from the perpendicular at a downstream angle where it will be blown even further downstream by the ether wind. So, it seems the returning reflected light beam will be blown downstream by the ether wind while the returning transverse light beam will have two sources directing it downstream the ether wind and the offset beam splitting mirror.

There are further difficulties that arise once you offset the beam-splitting mirror because it is difficult to surmise the exact course followed by the reflected light beam. But, it does seem clear it would be difficult for the beams to become superimposed. It should be noted that the calculations that form the mathematical basis of the experiment are based on the beam-splitting mirror being inclined at exactly45°. Also, it is difficult to surmise exactly the course that the reflected light beam is following when the beam-splitting mirror is at precisely 45°.

_117. Is it possible to make a brief yet understandable comment on the complex equations found in chapter 2, section 28 "Equations of a geodesic" of Arthur Eddington's_ The Mathematical Theory of Relativity _?_

In chapter 2, section 28 "Equations of a geodesic" Arthur Eddington's derives the equation(s) employed for finding the geodesic in any kind of space either flat (Euclidian) or curved (non-Euclidian). His argument is complex and seems oscillate between convincing and exasperating. He blurs the distinction between variables we are familiar with such as _s, x_ 1 and _x_ 2 and variables that seem to behave both as the familiar variables and as differential variables such as _ds, dx_ 1 and _dx_ 2 _._ According to the _Mathematics Dictionary_ by Glenn James and Robert C. James, differentials such as _ds, dx_ 1 and _dx_ 2 are variables. But, they seem to always accompany familiar variables; they aren't stand-alone quantities. They aren't normally raised to the second power. What is the derivative of _ds_ 2 _?_ There is also the problem of unclear functions. We know, for instance, that _g_ μѵ is a function of the variables _x_ σ _,_ but are the _x_ σ variables functions of the variable _s?_ Are the _x_ μ and _x_ ѵ variables functions of the _s_ variables, as well? If so, doesn't that make the coefficients represented by _g_ μѵ indistinguishable at some level from the variables/differentials _dx_ μ and _dx_ ѵ they modify?

He begins section 28 as follows.

"We shall now determine the equation of a geodesic or path between two points for which  is stationary. This absolute track is of fundamental importance in dynamics, but at the moment we are concerned with it only as an aid in the development of the tensor calculus.

"Keeping the beginning and end of the path fixed, we give every intermediate point an arbitrary infinitesimal displacement  so as to deform the path. Since ,  and  . . . . (28.1). The stationary condition is  . . . . (28.2)."

Lillian Lieber's version of the stationary condition, i.e., of equation (28.2), which appears in her book, _The Einstein Theory of Relativity,_ is the following:  . She writes, "We know from ordinary calculus that if a short arc on any of these paths is represented by _ds,_ then  represents the total length of that entire path." The  seems to indicate that we take total differential of the entire path length, which is represented by  . Next, we set the total differential equal to zero. The solution of the equation is the geodesic. Lillian Lieber's method of integrating a differential equation and then finding the total differential of that result seems similar to Eddington's method of taking the total differential of a differential equation and then integrating that result. There is the concern though that the equation  is not actually a differential equation, but it merely appears as such.

The equation  employs the summation convention. The summation convention tells us that whenever the same index (subscript or superscript) appears twice in the same term, then we must sum on that index. Since both  and ѵ appear twice in the term  we must sum on both those indexes. The equation  is a generalized version of the Pythagorean Theorem. It can represent all the variations of the Pythagorean Theorem that are possible in Euclidian space and all the possible versions of the Pythagorean Theorem that are possible in non-Euclidian space. In the standard Cartesian coordinate system the Pythagorean Theorem tells us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. We could write the equation  to represent this. Though we seldom, if ever, think of this fact, the coefficient of  is one and the coefficient of  is also one. The quantities represented by  include the coefficients in all the possible versions of the Pythagorean Theorem.

In the Pythagorean Theorem, which we are familiar with, the  terms would equal one. In Einstein's version of the Pythagorean Theorem, which is employed for _both_ curved space and flat space,  is composed of the variables _x_ and _y._ They are written as _x_ 1 and _x_ 2 _._ When Einstein's version of the Pythagorean Theorem is employed for flat space the variables _x_ 1 and x2 that compose  cleverly turn into the familiar constant one through the use of the exponential function in which the exponent (either _x_ 1 or _x_ 2) is set to zero thus causing the entire function to equal one. We should note that when we use the equation  to generate the familiar Pythagorean Theorem (not Einstein's version of the Pythagorean Theorem used for both curved and flat space) the term  will represent zero as well as one. Although there are no coefficients of zero in the familiar Pythagorean Theorem, when we employ the equation  to generate it, we obtain the terms _xy_ and _yx._ To rid the equation of these terms their coefficients must be zero. Of course, they would appear as _x_ 1 _x_ 2 and _x_ 2 _x_ 1 respectively. Just as _x_ 2 would appear as _x_ 1 _x_ 1 _,_ and _y_ 2 would appear as _x_ 2 _x_ 2.

The various values for the term  must be known beforehand and assigned to the term  . The term  is a tensor of rank two. For instance, to generate the familiar Pythagorean Theorem we would write  and  while  and  . In Einstein's version of the equation  , which he employs for curved space, many of the various occurrences of the term  are set equal to zero while four are composed of the variables _x_ 1 and/or _x_ 2 in a specific formulation.

The equation  appears to be the total differential of the equation  . Here we are assuming that  is not a differential equation, but the kind of equation we are familiar with but in non-standard form. The equation  is in a non-standard form for two reasons. The first reason is the appearance of  on the right side of the equation. In a more standard form, the equation would be written as  . The second reason is the variables themselves, _ds, dx_ μ and _dx_ ѵ _._ In a more standard form, they would appear as _s, x_ μ and _x_ ѵ _._ The variables cannot assume the form _x_ μ and _x_ ѵ because the coefficients that Einstein uses for his curved-space version of the Pythagorean Theorem are in the form of _x_ μ and _x_ ѵ _._ The coefficients of the variables must be distinguishable from the variables.

The equation for the total differential of a function of several variables is  . From this equation, we can see that   , or more precisely it generates  as well as generating a portion of  . We can also see that  and  . We can also see that  , or more precisely it represents a portion of  . We can also see that  and  . This allows us to conclude that  and  .

We can also see that  since  is formed from variables such as _x_ 1 and _x_ 2 in Einstein's version of the Pythagorean Theorem _._

It is more difficult to accept that  and  since the equations do not mean the differential of the differential of _x_ ѵ equals the differential of the differential of _x_ ѵ and the differential of the differential of _x_ μ equals the differential of the differential of _x_ μ _,_ respectively. Instead,  seems to indicate that the differential of the variable  is equal to the differential of the undefined quantity  , which could represent the kind of variable we are familiar with or it could represent an infinitesimal similar to  . In either case, its precise relationship to the preceding equations remains unclear. The same reasoning applies to  . It seems to indicate that the differential of the variable  is equal to the differential of the undefined quantity  which could represent the kind of variable we are familiar with or it could represent an infinitesimal similar to  . In either case, its precise relationship to the preceding equations remains unclear.

Eddington continues, "which becomes by (28:1)  ," what Eddington has done is perform three operations on the equation  . Then he has placed an integral sign before the new version's right-hand side. Then Eddington set that equation equal to zero, which is called for by equation (28.1)  . The three operations he performed on the equation are the following. First, he divided each side of the equation by 2. Next, he divided each side of the equation by _ds._ Then, he multiplied each side of the equation by _ds/ds._ On the left-hand side of the equation multiplying by _ds/ds_ takes the form of multiplying by 1. After those three operations he integrated the right-hand side of the equation and set it equal to 0 as called for by equation (28.1).

Dividing each side of the equation by _ds_ gives the equation the more standard appearance of finding the total differential of the equation  especially when that equation is put in the standard form of . One of the first steps you would take in finding the total differential of  is to write  . Therefore  and  . Dividing the right-hand side of the equation  by _ds_ accomplishes the same task since  and  . We should note both sides of the equation are divided by _ds._

Eddington continues, "or, [by] changing [the] dummy suffixes in the last two terms,  . Applying the usual method of partial integration, and rejecting the integrated part since  vanishes at both limits,  ." Applying partial integration is where Eddington makes his error. He applies partial integration to the term  . The formula for integration by parts that he is apparently using is the following:  . Thus we see that  and thus  and  and  . In the _Mathematics Dictionary_ by Glenn James and Robert C. James, we are told that _u_ is a function of _x_ therefore it follows that in the quantity  and in the quantity  there exists some function of the variable _s,_ but this is not the case. An argument can be made that _dx_ μ and _dx_ ѵ are functions of _ds,_ but _ds_ is not _s._ And, apparently, _dx_ μ and _dx_ ѵ are no longer the variables we are dealing with. We are now dealing with the variables _x_ μ and _x_ ѵ _._ And, the term _ds_ is treated as a variable that is not related to _s_ in the equation with which we began  _._ In the _Mathematics Dictionary,_ we are told that _v_ is a function of _x_ therefore it follows that the quantity  should be a function of the variable _s,_ but it is not. In the _Mathematics Dictionary,_ we are told that _dx_ is the differential of the variable _x_ therefore it follows that _ds_ should be the differential of the variable _s,_ but the variable _s_ does not appear in the set of equations we are working with.

It is confusing. A term such as  seems to be appropriately read as the division of the variable _dx_ μ by the variable _ds._ Perhaps, Eddington wants us to read it as the differentiation of the dependent variable _x_ μ with respect to the independent variable _s._ What are we to make of a term such as  ? It seems  is the differential of the variable(s) _x_ σ. The variables _x_ σ form the coefficients of the variables that Einstein uses for his curved-space version of the Pythagorean Theorem. But, the term  also seems to be a stand in for the variables _dx_ μ and _dx_ ѵ _._ So should  be read as the division of the differential of the variable  by the variable _ds?_ Or, should  be read as the division of the differential of the differential of the variable  by the variable _ds?_ Or, should we read  as the differentiation of the variable  with respect to the variable _s._ Or, should we read  as the differentiation of the differential  with respect to the variable _s._

Using these questionable and confusing methods Eddington does produce the equation for determining the geodesic, which is

. Interestingly, Eddington uses this equation for determining the geodesic in his next section "29. Covariant derivative of a vector." Perhaps, Eddington sensed that his derivation of the equation for the geodesic was questionable. Therefore, he realizes that its use in deriving the covariant derivative of a vector would make that derivation questionable also. So, Eddington provides the reader with section "31. Alternative discussion of the covariant derivative." In section 31, Eddington provides the reader with a method of deriving the covariant derivative that does not use the equation for the geodesic.

118. Can you describe a scientific concept you didn't understand when you first studied it and perhaps, still don't understand today?

"The Discovery of Stellar Aberration" by Albert B. Stewart is an interesting article that appeared in _Scientific American_ in the early 1960s. The first few times I read the article, I didn't completely and accurately grasp the concept he was explaining. Even now, I may not have a satisfactory understanding of stellar aberration. It is difficult to pinpoint the sources of my confusion. The second paragraph of his article begins with the following sentence: "The displacement, or aberration, of starlight can be detected by direct observation because the earth is not always moving in the same direction." The fifth sentence in the same paragraph reads, "The actual displacement of starlight because of aberration cannot be directly observed, but the changes in this displacement can." I don't want to quibble over his use of a comma to show that two parts of a sentence are being contrasted (". . . observed, but . . . ") when the contrast between the two parts might not be considered that stark. But I do want to point out that the first sentence of the paragraph and the fifth sentence of the paragraph seem to be contradictory. Perhaps, it is statements such as those above that led to my confusion.

Stellar aberration was discovered by James Bradley and Samuel Molyneux. They observed the star Gamma Draconis beginning December 3, 1725 and continued their observations the following year and throughout much of 1727, accumulating a total of 80 position measurements over two years. They discovered that Gamma Draconis wobbled approximately 40 arc seconds a year. The wobble had a 365-day cycle so it was suspected that the wobble was linked to the Earth's orbital motion around the Sun.

The timing of the wobble was contrary to what was expected for a parallax shift: The star's maximum southward deviation occurred in March, not in December; its maximum northward deviation occurred in September, not in June.

Beginning in August 1727 and continuing over the course of a year Bradley observed many other stars. The wobble was not unique to Gamma Draconis; it occurred in every star he observed.

Here is the explanation of stellar aberration. Whenever an observer views a star, the orientation of his telescope is determined by the combined effect of the velocity of light and the orbital velocity of the Earth. Were the earth stationary, as Ptolemy or Tycho Brahe would have it, a telescope would show the star in its true position; the tube would be aligned parallel to the incoming light rays, which travel unimpeded down the tube's length. However, since the Earth moves, the telescope is constantly swept along with the planet while starlight streams down the tube. Therefore, to center a star in the eyepiece, the telescope must be tipped slightly in the direction that the Earth moves; otherwise the star's light will be swept up by the telescope's inner wall before the light reaches the eyepiece. Thus, the observer sees the star, not in its true position, but skewed a maximum of approximately 20 arc seconds in the direction of the Earth's motion.

The telescope Bradley and Molyneux used was 24 feet in length. The wobble or southward creep of Gamma Draconis began in December and didn't halt until March at which time the star was approximately a full 20 arc-seconds south of its December position. Relative to its starting orientation in December the telescope was now tipped at an angle of approximately 20 arc seconds, less than 6/1,000 of a degree; the eyepiece now sat 3/100 of an inch sideways from where it had initially been.

The speed of light is approximately 1 foot per nanosecond. The speed of the Earth in its orbit around the Sun is approximately 18.5 miles per second. The telescope Bradley and Molyneux used was 24 feet in length, therefore, a light beam would take 24 nanoseconds to travel the length of the tube. Since the Earth is moving at a speed of approximately 18.5 miles per second in its orbit around the Sun, the Earth moves about 3/100 of an inch in 24 nanoseconds. The telescope is swept along with the Earth. The light beams inside the telescope are not swept along with the Earth because they are not affected by the motion of the Earth or any other object. The lens at the top of the telescope focused the light beams on the eyepiece, but by the time the focused light beams reached the eyepiece, it had moved about 3/100 of an inch.

There is a discrepancy. The Earth and therefore the eyepiece of the telescope travel approximately 3/100 of an inch in 24 nanoseconds, yet according to the results of Bradley and Molyneux the eyepiece of the telescope only has to be adjusted 3/100 of an inch in about three months in order to keep the star centered in the eyepiece.

Here is the formula that describes stellar aberration: sinδ = v/c sinθ. Another way of writing the formula would be sin(delta) _=_ v/c sin(theta)

δ (delta) = the angle of aberration

v = the speed with which the earth orbits the sun

c = the speed of light

θ (theta) = the angle the telescope forms with the ground which in the instance of Gamma Draconis will always be very close to 90 (degrees).

v/c  1/10,000

sin(20 arc-seconds) = sin(.0056 (degrees)) = .000098

θ (theta) = 90 (degrees)

sin(90 (degrees)) = 1

If we substitute these values into the formula, the result is: .000098  1/10,000 (1). The formula will only be valid for aberration angles of 20 arc-seconds or very nearly 20 arc-seconds. In the instance of Gamma Draconis the aberration angle is near 20 arc-seconds only in the months of March and September for the rest of the year it is a substantially lower value.

The error I made in the above analysis is I did not realize Bradley and Molyneux were only measuring the north-south component of stellar aberration. If we combined (added together) the north-south component of stellar aberration and the east-west component of stellar aberration it would always equal approximately 20 arc seconds. To add together the components, it seems you would need to employ the Pythagorean Theorem. When the negative values were squared, they would lose their negative value. It is the same principle that is at work when we draw a circle with a one inch radius centered at the origin of the x and y axes that is to say using Cartesian coordinates. The value of the radius is always positive one. For example, at 210 degrees the y component is -0.5 and the x component is -0.866025403. When each value is squared and then added together the value is one. Measuring the north-south component of stellar aberration is somewhat analogous to measuring the values of the x-axis for our circle of radius one centered at the origin. Strangely, the x-axis is usually pictured as a horizontal line that seems to run in a west to east direction.

For our circle of radius one the maximum negative value of the x-axis component would occur at 180 (degrees). The value of the x-axis would be -1, and the value of the y-axis would be zero. The maximum positive value of the x-axis would occur at 0 (degrees). The value of the x-axis would be +1, and the value of the y-axis would be zero. Often when we picture the way the Earth revolves around the Sun, we picture the Sun centered at the origin of a Cartesian coordinate system and the Earth moves in a circle around this center point. We picture the Earth at 90 (degrees) or 12 o'clock in December. In March, we picture the Earth at 0 (degrees) or 3 o'clock. In June, we picture the Earth at 270 (degrees) or 6 o'clock. In September, we picture the Earth at 180 (degrees) or 9 o'clock.

With stellar aberration, the maximum northward value, which is typically denoted as +20 arc seconds, occurs in September. In our familiar superimposing of the orbital motion of the Earth onto the x and y axes of the Cartesian coordinate system, September gives us the maximum negative value of the x-axis. With stellar aberration, the maximum southward value, which is typically denoted as -20 arc-seconds, occurs in March. In our familiar superimposing of the orbital motion of the Earth onto the x and y axes of the Cartesian coordinate system, March gives us the maximum positive value of the x-axis.

Since stellar aberration is caused by the velocity of the Earth, perhaps it is better to think in terms of the velocity of the Earth as opposed to in terms of the position of the Earth. We can think of the velocity of the Earth in terms of components of the x and y axes. Further we can image that Bradley and Molyneux were measuring the vertical or y-axis component of the velocity of the Earth. In March, the Earth would have a maximum component of vertical velocity or velocity measured along the y-axis. It would have little or no component of velocity along the horizontal or x-axis. The Earth can be imagined as heading southward along the negative y-axis with a maximum component of y-axis velocity. In September, the Earth would have another maximum component of vertical velocity or velocity measured along the y-axis. Again, it would have little or no component of velocity along the horizontal or x-axis. The Earth can be imagined as heading northward along the positive y-axis with a maximum component of y-axis velocity. In this scenario, I believe, the x and y axes are centered on the Earth.

In his article "The Discovery of Stellar Aberration," Albert B. Stewart provides us with a brief excerpt of a letter written by James Bradley, which was read to the Royal Society on January 9 and 16, 1729. James Bradley writes, as reproduced in Albert B. Stewart's article, "'At last I conjectured that all the phaenomena hitherto mentioned proceeded from the progressive motion of the light and the earth's annual motion in its orbit. For I perceived that, if light was propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction than that of the line passing through the eye and object; and that when the eye is moving in different directions, the apparent place of the object would be different.'"

The sentence, "For I perceived that, if light was propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction than that of the line passing through the eye and object; and that when the eye is moving in different directions, the apparent place of the object would be different," is confusing. We could rewrite it as follows: "For I perceived that, if light was propagated in time, the apparent place of a fixed object (a star) would not be the same when the eye is at rest, as when it is moving in any other direction except if the eye were moving along with the earth and looking at a fixed object (a star) that is located along a line of sight that is exactly the same as the direction of the earth's motion; and that when the eye is moving in different directions, the apparent place of the object would be different," in an effort to make it less confusing.

In the next paragraph of his article Stewart writes, "This hypothesis perfectly accounted for the movements Bradley had noted. The star Gamma Draconis appears farthest south in March, when observed at sunrise, because at that time of year the earth's motion about the sun gives the telescope its maximum velocity in a southerly direction. The star appears farthest north in September, when observed at twilight, because at that time of year the earth's orbital motion gives the telescope its maximum velocity in a northerly direction."

I agree that the telescope would have its maximum velocity in a southerly direction in March, but I disagree that the time of day would be sunrise. I believe the time of day would be the middle of the night. If we picture the Earth orbiting around the Sun in March, its orbital velocity is at a maximum in a southerly direction. Half of the Earth is illuminated by the Sun while the other half is in darkness. If we recall that the Earth is a sphere, sunrise occurs on the sphere occurs when an observer is moving from darkened half of the sphere to the lighted half of the sphere because of the action of the daily rotation of the Earth on its axis. At this time of day, the rotational velocity of the Earth would be along the horizontal axis, not along the vertical axis. In the middle of the night, the rotational velocity of the Earth would be along the vertical axis in a southerly direction. The addition of the two velocities along the vertical axis in a southerly direction would produce the maximum velocity in a southerly direction.

If we picture the Earth orbiting around the Sun in September, its orbital velocity is at a maximum in a northerly direction. Half of the Earth is illuminated by the Sun while the other half is in darkness. If we recall that the Earth is a sphere, twilight occurs on the sphere occurs when an observer is moving from lighted half of the sphere to the darkened half of the sphere because of the action of the daily rotation of the Earth on its axis. At this time of day, the rotational velocity of the Earth would be along the horizontal axis, not along the vertical axis. In the middle of the day, the rotational velocity of the Earth would be along the vertical axis in a northerly direction. The addition of the two velocities along the vertical axis in a northerly direction would produce the maximum velocity in a northerly direction.

It is also interesting to note that the entire solar system including the Sun and the Earth orbits around the center of the Milky Way Galaxy at a speed of approximately 130 miles per second. It seems odd that this velocity has no influence on stellar aberration. Since the orbit of the Earth is circular, at some point the direction of its orbital velocity around the Sun should coincide with the direction of the solar system's orbital velocity around the center of the Milky Way. And six months later the Earth's orbital velocity around the Sun should be opposed in direction to the solar system's orbital velocity around the center of the Milky Way.

119. Do you have any comments on any of the equations that helped catapult Einstein into worldwide fame?

The equation _r_ 2 _[d(phi)/ds] = h_ is from _The Mathematical Theory of Relativity_ by Arthur Eddington. I have replaced the lower case Greek letter _phi_ that appeared in Eddington's original equation with the name of the letter _phi_ spelled out using letters from the English alphabet since it should be easier for eBook platforms to reproduce this notation without error. I should note that the first variable is _r_ squared in case the superscript is not accurately reproduced. Eddington's book was first published by the Cambridge University Press. The version I have is published by BiblioLife, LLC. The equation is one of the many erroneous equations that I believe infest Eddington's book. As you might expect, the equation as it is first introduced is not in error. It is not until the equation is re-introduced a few pages later that the error occurs. It is re-introduced at a critical juncture. It is used to facilitate the calculation of the bending of a ray of light from a distant star by the gravitational field of the sun. This is one of the three very important and testable predictions that was made by Einstein's theory of general relativity when it was first introduced to the scientific community. The subsequent measurement of the bending of light from a distant star by the gravitational field of the sun made during the eclipse of May 1919 by Eddington and others catapulted Einstein into worldwide fame. The error occurs when _ds_ is made to equal zero. This introduces division by zero which is not allowed. It is an invalid procedure. Eddington uses this illicit division by zero to make the claim that when _ds_ equals zero the quantity _h_ equals infinity. Then the _h_ that equals infinity is introduced into the term _m/h_ 2. Thus the term _m/h_ 2 is reduced to zero since it is mass, _m,_ over infinity squared. To make the claim that Einstein's general relativity does not provide a valid derivation of the equation used to calculate the bending of a ray of light from a distant star by the gravitational field of the sun is not as farfetched as it sounds at first.

120. If the scientific community ever falsifies Einstein's theories, do you have any notion of the kinds of colorful phrases and simple but convincing arguments that will be deployed to gain the public's acceptance of this momentous change?

I think colorful phrases such as "the hundred years of sleep" and "the five score and fourteen years of a masquerade" will be employed. A phrase like "another century of snake oil and physics may never recover" has a nice ring to it. "Is there something about physics that attracts such diabolically clever charlatans?" is a phrase that has a pleading quality to it that the public has been conditioned to find appealing. The denunciations of his theories could take a more acerbic turn with statements such as the following: Einstein's theories were always more psychosis than science, and it is surprising that anyone paid attention to his ideas at all from the very beginning. Or the always utilitarian sartorial comparisons could be leveled against his theories such as Einstein's theories are similar to women who wear shoes that are uncomfortable and hurt their feet because it is fashionable, and yet neither they nor anyone else ever says anything about it. Perhaps, statements that are almost certainly apocryphal will be attributed to Einstein to highlight the determination with which he and his minions silenced opposing views. He could be credited with saying, "I'll make a raft from the bones of my critics and float it on their blood." Of course, more sober analysis would need to be deployed, as well. That analysis might begin along these lines: After prolonged deliberations, all the parties involved agreed that for special relativity to be consistent and reasonable it must allow that distant clocks must be capable of being synchronized by means other than light waves.

121. Do you find it instructive to note the treatment Einstein's theories receive in popular culture?

The hit TV show from the '60s _Rowan and Martin's Laugh-In_ is seldom mentioned these days, and a segment of the show called "News of the Future" is all but forgotten. Also, all but forgotten is Walter Cronkite's serious, contemporaneous version of "News of the Future" called something like _The Year 2000._ In those days, the chiliasm seemed so far into the future. I remember that in one episode of the show Walter Cronkite expressed a concern about Einstein's relativity theories. He felt that if by the year 2000 Einstein's theories were not taught in advanced physics classes in high school and in physics classes in college, then there would be something amiss with the difficulty level of the theories, if not the substance of the theories. Walter Cronkite, at least, feigned enthusiasm for the notion that fundamental truths should be accessible to a wide audience. Looking back, it seems propaganda had a much firmer grip on our minds when it was wedded to an unambiguously glorious vision of the future. It seems that then (just as it is today) most of those who professed a concern for the welfare of the average individual were unshakably in the camp of the elite ruling class.

For those of us who don't find Einstein's theories palatable, it seems light beams play the role of a trickster. On a moving body, such as the earth, sometimes an observer or a mirror will be racing away from an oncoming light beam and other times under the same conditions the observer or mirror will not be racing away from an oncoming light beam. Another way of making the same statement would be the following paraphrase of Henri Bergson's idea. If an observer standing on the railroad embankment, which is to say standing on the moving earth, has the right to claim he is at rest, then it seems we must allow the observer on the moving train the right to claim that he is also at rest and that he is neither racing toward one light beam nor racing away from the other light beam. Thus, both the observers see the lightning strikes as simultaneous contrary to Einstein's famous claim that his thought experiment demonstrates the relativity of simultaneity.

I am struggling to read Tim Maudlin's remarkably dense book _Philosophy of Physics: Space and Time._ As a professor of philosophy at New York University, he finds special and general relativity to be valid theories, at least his versions of the theories. On page 135, in a chapter titled "General Relativity" he writes, "The Strong Equivalence Principle is slightly fiddly." I note the use of the apparently made up word "fiddly." In chapter four "Special Relativity" on page 67, he writes, "Einstein himself presented the [special relativity] theory as the consequence of two principles: (1) the equivalence of all inertial frames and (2) the constancy of the speed of light. From these two principles, after some fashion, we derive the _Lorentz transformations,_ which are a set of equations relating one set of coordinates to another. But already we can see that this approach to understanding the theory has run seriously off the rails." He seems to be saying Einstein's approach to understanding relativity is flawed. I note the use of the phrase "after some fashion" to describe the derivation of the Lorentz transformations from the two principles. Could this be an oblique reference to (or even an acknowledgment of) the claim that I and others have made that the mathematics Einstein uses to derive the Lorentz transformations in his special relativity paper _On the Electrodynamics of Moving Particles_ is invalid?

To excite people's imaginations perhaps we need statements that are rasher but whose point is unambiguous such as: You could disprove Einstein's theories with three walkie-talkies and a firecracker. Or, if you are thinking of the eclipses of Jupiter's moon Io by the shadow of the planet Jupiter you might rashly proclaim that Einstein's theories could be disproved with a modest telescope and a stopwatch. I am reminded of a book I read about Islamic culture that was published in the '60s. Although I suspect the editors and contributors to the book were familiar with the questionable behavior of the Prophet Mohammad such as approving of the mass decapitations of the males of a subjugated tribe and his marriage to a very young girl, the only critique they offered of Islam was that it was a businessman's religion. Perhaps, in that same vein, we should refer to Einstein's theories as a rock star's science.

122. How did growing up in Boonton Township influence your writing style?

There are those that opine that improvements in the world's sanitation systems have provided more benefits to mankind than all the discoveries of modern medicine. So, it may be that living in a house with a septic system of its own as opposed to a system in which large sewers called interceptors carry sewage to a large wastewater treatment plant had some subtle effect on my thinking. One of the interesting aspects of living in Boonton Township is you can tell where the sewer system ends because after a certain point there are no more manhole covers on Powerville Road. It seems that when I am driving at night, especially on a rainy night, it is nearly impossible to keep from running over manhole cover after manhole cover. It can be quite irritating especially since I've only seen men working through an open manhole cover once, and that was in Garfield. Do we actually need all the entrances to the interceptors that are covered with manhole covers? In more salubrious times, when the junction box of my septic system was in need of repair, I didn't resort to the _Yellow Pages_. I didn't spend time scanning the various ads under septic system repair and finally call the business that appeared to promise the lowest cost along with a reasonably competent repair. I repaired it myself. I was concerned about the stench and what the neighbors might think, but I went ahead anyway. I had read that if a home owner used a garden hose to flush out the pipes that either led into the leaching field or back into the septic tank itself there was a danger. If the pump supplying the garden hose with water was to shut off due to mechanical failure or a power outage, there was a chance that the liquid from the septic system could be drawn back into the pipes that supplied drinking water to the house. Thus, your drinking water could become contaminated. The solution proffered by the article I had read was a one way valve that you could attach to your garden hose. With the utilization of this precaution, I was ready to flush out the junction box along with the one pipe that led into it and the three pipes that led out of it. Once this was accomplished, it was fairly simple to repair the junction box by enclosing it in a partial shell of plywood and cement and making sure the lid fit on tightly. Many years later when I was having professional repairs made to the junction box, the repair person told me that I could pay about $1,000 to have some debris removed from the area or he could bury them where they were. Without hesitation, I said that he should bury them here. There's something strange about life. Should we actually expect people to question Einstein's theories? Perhaps, we shouldn't question his theories until society is more adept at self-correction. For instance, with all the engineering competence in our society, it seems there is still no design for a chainsaw that doesn't leak out the oil that is used to lubricate the bar (blade) of the chainsaw when the chainsaw is not in use. It all seems like a dull, sad and lonely Edward Gorey play.

123. Why is writing books so difficult?

One reason is that there are so many grammatical rules that it is impossible to remember them all. It seems some rules have exceptions based on historical precedents such as the fact that the name _Jesus_ in the phrase _Jesus' parables_ is made into the possessive form by the addition of an apostrophe while the name _Boris_ is made into the possessive form by the addition of an apostrophe and an _s_ as in _Boris's parables._ It also seems many rules have addendum that aren't actually exceptions but instead nuanced changes that are a response to slight changes in the grammatical situation that is under consideration. The following two shorts stories may in a way demonstrate how much attention must be paid to grammar even when dealing with subject matter that is entertaining.

Amaya and the Mole People Save Garfield

Amaya was having a bad day, and it had started yesterday when she was at the Happy Reading Tree Saturday School. Everyone was having snacks. It was around quarter after twelve, and almost everyone was having a slice of pizza and a Capri Sun juice box. On that particular day, Amaya wasn't in the mood for a slice of pizza; she wanted a bag of chips or maybe more than one bag of chips. She asked one of the peer helpers, "Can I have chips instead of pizza?"

He replied, "Sure, follow me," and they went to the front of the room where the snacks were located. When she saw the selection of snacks, she couldn't decide between a snack bag of either onion rings or spicy barbeque chips. So, the peer helper suggested, "Take 'em both; just don't let Ms. Evans see you." Well, of course, that plan didn't work. The gimlet-eyed founder of the Happy Reading Tree School Ms. Irene Evans saw Amaya with the two bags of snacks, and in a flash she went into full diatribe mode. She fulminated, "You know these snacks are donated, and we're always in danger of running short of them. What if everyone took two bags of snacks, then there might not be enough snacks to go around, and then someone would have to go without a snack. Then that person wouldn't be happy, and all because Amaya couldn't choose just one snack bag of chips. Then the mother of the unhappy student might complain to the head of the library that the brochure for the Happy Reading Tree Saturday School says that every student attendee gets a free book and a free snack, but her son didn't get a free snack while all the other students did. And, if the head librarian got enough complaints like that, he might say to Ms. Evans that there have been too many complaints made by unhappy mothers about the reading program so he is going to have to cancel the program."

Amaya had heard this diatribe before. Only this time she made a connection. Perhaps, it was because she had just watched a DVD called _A_ _Charlie Brown Thanksgiving_ with her parents. She recalled that in the cartoon, parents and teachers never spoke with words, but instead their words were replaced by a trombone going wah, wah, wah . . . wah, wah, wah. And, Ms. Evans was like one of the teachers in the Charlie Brown cartoons. She was saying words, but somehow the words were like a trombone going wah, wah, wah. Perhaps, if she had been reading the Charles Dickens' classic _Oliver Twist_ or watching a movie version, she would have seen the similarity between her situation and the famous scene in which an orphan asks one of the staff members of the orphanage, "Please, Sir, may I have some more gruel."

That's the way Amaya's life went. No matter what she did, it seemed there was always someone giving her a hard time. And, it was every day, day after day, until finally it got her down. But, this morning something happened that changed everything and at the same time changed nothing. A large spaceship landed in the Walmart parking lot early Sunday morning. It must have crushed a few cars and some light poles when it landed; it was that big. Amaya could see it out of the window of her second-floor bedroom. Her house on Passaic Avenue was just west of the Walmart parking lot. She thought it wasn't likely the spaceship could have crushed Ms. Evans's car because she had learned that after many years of living in Garfield Ms. Evans had moved away and now lived 45 minutes west of Garfield in Boonton Township.

Garfield is a small city shaped somewhat like a trapezoid. It is surrounded on three sides by rivers–the Passaic and the Saddle rivers. The Walmart parking lot was near the mouth of the Saddle River where it flowed into the Passaic River. Over the years, near its mouth, a great deal of litter had accumulated along the banks of the Saddle River, and consequently not many people strolled along its banks in this area. Amaya sometimes strolled along the litter strewn banks of the river to get away from it all. She occasionally saw a nesting Canada goose or a huge buck deer. It seemed they didn't mind all the litter. If Amaya had thought more about it, perhaps, she would have realized the animals came here to get away from people just as she did. Beneath the Walmart parking lot was a huge drainage tunnel. It was large enough for her to walk into it, and she always meant to explore it, but she always forgot to bring a flashlight when she went walking along the riverbank.

When the space aliens first arrived, they had put a force field around the entire city of Garfield, and for a few days no one could get in or out of the city. The Walmart Shopping Center was located in a huge depression. It was surrounded on one side by the Saddle River while on the other side there was a huge retaining wall made out of very large, cement blocks.

The space aliens could fly; they had wings, but the sound their wings made was loud, very loud like the sound of a large truck that you pass on the highway with your car's windows open. Their faces were like human faces, their hair was like women's hair, and their teeth were like lions' teeth. They had scales that looked like metal. On their heads, they had a yellow band that glowed and perhaps controlled a force field of some kind. Every morning they swarmed out of their spaceship and over the huge retaining wall of white, cement blocks. They entered the city of Garfield like an enormous swarm of huge yellow jackets that was as dense as black smoke. Some people said that they could bite like a yellow jacket and that the bite would hurt and swell for five months. But, actually, they attempted to repair things and make things better. But, at first, they did quite get the hang of things.

For example, it seemed confusing to them that sometimes tree limbs would grow around wires. It could be a strand of barb wire, telephone wires or electric wires that the tree limbs would grow around. Then human workers would come along and cut the tree limb so that just a small chunk of dead wood was left suspended from the wire. This small chunk of dead wood would make the wire sag the slightest bit. It seems the aliens couldn't tolerate the slightly sagging wire. And, they would take measures to correct the situation. There was a hurricane fence that ran along the north side of Hudson Street as part of the fence that fenced off an environmental, clean-up site from the public. It was topped with three strands of barb wire, and a tree limb had grown around the center strand. Some workers had come along and cut the limb so that a large chunk of wood still clung to the center strand of barb wire and made it sag. The aliens tried to fix this. First, they used a gear clamp to raise the chunk of wood slightly higher so the barb wire wouldn't sag, but the gear clamp that they used was too large and didn't hold the chunk of wood tightly enough to the fence so the chunk of wood slipped back down. Later, they corrected this problem by using a smaller gear clamp.

The aliens tried to solve little problems and big problems. They strictly enforced the speed limits and other traffic laws. That annoyed some people, but others thought that it was a good thing. Besides, what could anyone do? People were happy that they let a select number of people through the force field so they could leave Garfield to go to work and for other reasons. Their way of enforcing the traffic laws was more educational than punitive. They would reason with an individual that had violated some traffic law, or if there wasn't an opportunity for discussion, they would hand out a lengthy pamphlet explaining the justification for a particular traffic law. For instance, it seemed every morning at the corner of Lanza Avenue and Prospect Street there were always drivers that would park in the yellow-lined no parking zone. This violation made it difficult for drivers turning left onto Lanza Avenue from Prospect Street to see the oncoming traffic. No matter how often the aliens explained the danger of this traffic violation to drivers, it seemed they never learned. This led some suspicious individuals to conclude that the aliens' educational tactics were a prelude to vaporization. They surmised the aliens would get tired of trying to teach drivers the proper way to drive their cars, and instead they would just vaporize the offending drivers and their vehicles.

Amaya's life didn't change that much with the presence of the space aliens. People were always telling her what to do and not to do the things she wanted to do. One Sunday, she felt worse than usual. She hadn't been allowed to do anything that she wanted to do at the Happy Reading Tree. She decided to take a walk down by the Saddle River to calm her nerves. This time she remembered to bring a flashlight so in case she wanted to explore the drainage tunnel she could. After walking along Hobart Place, she cut into the Walmart parking lot and headed for the drainage tunnel. It was huge and dark, but since she had a flashlight, she decided to investigate it. She went deeper and deeper into the tunnel. Then she came to a smaller tunnel that branched off from the main tunnel.

She decided to take it although the sides of the tunnel seemed to be made of dirt, not cement. She hadn't wandered far down the tunnel when she met a mole man. She felt more scared than she ever had before in her life. The mole man held up his right hand, and said, "Fear not, I am a living creature, I am alive, and not a demon from hell." His head and his hair were white as white wool, white as snow; his eyes were red like flames. He carried a long crowbar in his hands, which some people call a pry bar. Although the mole man was scary, he reminded her of a Big Bird, hand puppet she had been given at the Happy Reading Tree. At first, she didn't like the puppet, but she changed her mind when she noticed the puppet's hair was like her hair.

After a while, Amaya had the courage to ask the mole man a question. She queried, "How did you learn to speak English?"

He replied, "The others and I learned to read by studying all the litter that blew across the Walmart parking lot and came to rest on the banks of the Saddle River."

Amaya said, "There are others like you? What do you call yourselves?"

"You would not understand the name we go by so you might as well think of us as mole men, and think of me as the leader of the mole men."

Amaya thought and then retorted, "You're the leader, but you still must have a name."

"Call me Oscar Mega," he said.

Oscar Mega gave Amaya a short tour of the tunnels where the mole men lived. They all seemed to be wearing white robes, which Amaya thought was strange. Also, they didn't seem to be doing anything except standing around having quiet conversations with each other. They seemed somber and sad. For a second, she wondered if Ms. Evans had somehow scolded them for some silly infraction of the rules, but then she thought these weren't the kinds of people Ms. Evans would scold. Amaya knew about feeling sad.

She thought for a while, and she finally remembered a joke that had to do with tunnels. It wasn't a very funny joke, but she thought who knows folks who live underground might find it funny especially since there was always a trickle of water seeping into the tunnels where they lived. She knew that kind of thing would get on her nerves after a while.

She went up to a group of mole men, and said, "Do you want to hear a joke about tunnels?" Without waiting for an answer, she launched into the joke.

"Bridgette used to be a peer helper at the Happy Reading Tree School. Somehow she managed to marry a very wealthy Frenchman, and they moved to the Cote D'Azur in the south of France. Sometimes she would hang out with her friend Bianca watching the yachts sail by on the Mediterranean. One day Bianca said, 'Do you ever get nostalgic for your old hometown, Golffield?' To which Bridgette responded, 'It's Garfield, not Golffield, and whenever I feel homesick for Garfield, I just visit Paris.' A startled Bianca responded, 'Paris?' Bridgette replied, 'Well, the sewers of Paris.'"

Surprisingly, the mole men found the joke funny, but not because of the punch line that compared Garfield to the sewers of Paris. They were amused by the notion that someone would make the mistake of calling Garfield, Golffield. It seemed the mole men were literal minded individuals, and it amused them that someone would be so whimsical as to incorrectly identify an entire city.

Amaya had an inspiration. She had an idea. She asked the mole men, "Why don't you dig a big hole under the alien space ship? Then connect the big hole to a tunnel that leads to the Saddle River. So, that when the alien space ship falls down into the hole, it will be washed into the Saddle River. Then the river can float the space ship out into the ocean and away from here."

Oscar Mega looked at Amaya and asked a simple question, "Why should we do that?"

Amaya was stumped; she blurted out the first thing that came to her mind, "If Amaya's not happy, then no one is happy."

Oscar Mega thought for a long moment and asked, "And, you're not happy?"

"No, I'm not," she stated emphatically.

"Then we'll do it."

Perhaps, because the mole men had only recently learned English or perhaps for some other reason, it seems the mole men found figurative language opaque. Amaya went home shortly after that, and she didn't tell anyone of her adventure. She knew enough about people to know that no one would believe her. She visited the mole men sporadically.

Every time she visited the mole men, she was more and more amazed by the scope of their excavations. She wondered what they did with all the dirt they dug out of the ground. It seemed they had dug a tunnel under the Saddle River that followed its course northward for many miles. The hole they dug under the alien spaceship was enormous–many times the size of the large craft. During one visit she was surprised to learn that at the bottom of the hole under the alien spaceship there was a huge drain that was covered by more than twenty sewer grates. She asked Oscar Mega, "Won't all those sewer grates block the spaceship from being washed into the Saddle River?"

Oscar Mega explained, "We're not washing the spaceship into the Saddle River. That plan wouldn't work. What would stop the aliens from coming right back and landing in the Price Rite parking lot or the Century Fields? We're going to use the principle of cavitation. We're going to fill this giant hole with a rushing torrent of water mixed with sand and small sharp stones. We'll swirl the spaceship around in this giant hole until the grit in the water has worn the hull of the spaceship so thin that they will have to return to their home planet for repairs right away." And, that is just what they did.

It was on a Thursday night after three weeks of very rainy days. Amaya heard a loud rumble coming from the north and the ground began to shake. The sound got closer and closer. She went to her window. She could make out the alien spaceship in the dim light. Then suddenly, it sank out of sight. For six days, it was swirled around in that giant hole. The riverbed of the Saddle River was dry for miles in the northern direction because the water had been diverted underground into one of the mole men's tunnels. On the seventh day, the ground collapsed that had covered the entire hole except where the spaceship had sank days before.

This allowed the spaceship–with its damaged hull just barely holding together–to take off heading for its home just as Oscar Mega had said it would. Amaya and the mole men had saved Garfield.

Bridgette and the Borrowed Foxes

This is a story that I heard. I think most of it is true.

Money was always on Bridgette's mind. She was always trying to think of ways to make money. In a way, she couldn't help it. When the PNC bank, at the corner of Outwater Lane and River Road, was being renovated, the workers rerouted the air-conditioning system through two, large, flexible, aluminum hoses similar to those used to connect a clothes dryer to an outside vent, only much larger. The hoses were pleated like an accordion. She couldn't help thinking they were large enough for her to squirm through. If she could get to the roof of the bank and find the hoses, she could slit one open and squirm through it to make her way into the bank. Her plan didn't go any further than that because she was sure they always locked the vault at night.

A more practical idea she had was to borrow a snow-blower and make money by clearing the neighbors' driveways during the winter months. One day, she noticed that one of her neighbors had two snow-blowers in his driveway. Since money was potentially involved, she worked up the nerve to ask the neighbor why he had two snow-blowers. He replied, "I got a new snow-blower so that my girlfriend, Irene, could help me out by using the old snow-blower. I learned later there was no way in hell that my girlfriend was going to use a snow-blower. So, I keep the old snow-blower as a backup." Bridgette saw this as a perfect opportunity to bring her plan to fruition. The neighbor was agreeable to her idea. He would lend her the snow-blower, but first he felt obligated to make sure she knew how it worked. He explained how it worked to her. It didn't seem that complicated. She noticed there was one piece of equipment that seemed to have a plug on it whose function he didn't explain. She asked, "I think I understand how to operate it, but what the hell is this plug inside this plastic box for?"

He replied, "It's the plug for the electric starter. It lets you start the snow-blower by plugging it into an electrical outlet. I never use it. It's like other new things like e-mail. Other people think they're great, but I have no use for them." He continued, "It's so cold during the winter that I have to wear two hats to keep warm. If it gets so cold, I have to use the electric starter to start the snow blower, I would probably have to wear three or four hats to keep warm. That's just too many damn hats." His last piece of advice was strange. He talked rapidly saying, "Sometimes when you're using the snow-blower and the wind is howling and snowmobiles are making a racket and the snowplows are on the road the squirrels started acting crazy. A scared squirrel might run right into the blades of the snow-blower. When that happens, do you know what time it is?" Without waiting for her reply he said, "Lunchtime." It sounded strange to her, but you probably would take a break after something like that happened.

Toward the end of November there was a huge snowfall–more than 12 inches. Bridgette got the snow blower started and set off to clear the neighbors' driveways that had hired her services. The snow was deep and heavy. The snow blower was struggling to clear the driveways. At times, she had to push the snow blower into the snow or else the wheels would just spin while the machine stayed in place. She thought about trying to put snow chains on the wheels, but she wasn't quite sure how to do it. Besides she was working at a steady pace, and she didn't want to take a break. The wind was howling and blowing snow in her face. It was bitter cold. She hadn't thought snow-blowing driveways would be so hard. She should have charged more money.

It was so bad that one of her gloves froze to the frame of the snow blower. Fortunately, she had a spare pair of gloves. She tried to remove the frozen glove, but she couldn't. She didn't know if it was because she was exhausted or because it was so cold the glove was frozen solidly to the frame. She didn't want people to think she was too incompetent to remove a frozen glove, but she couldn't think of a solution to the problem. Then she remembered something she had seen on the reality show _Ice Road Truckers_. She tied a length of rope around the glove. She dropped the other end of the rope in front of the snow blower then she held in the clutch and the auger drive. She pushed the snow blower to get it moving because the snow was deep. The auger blades caught the rope and fed it to the impeller, which grabbed the rope with enough force to tear the glove loose.

That little victory gave her the perspective to look at her situation from a calmer viewpoint. It was getting late. It was also getting colder and windier. She would never finish the jobs she had been hired to do without help, if only she had some strong animal to help her pull the snow blower through the deep snow. The only thing she could think of was something her friend George had told her. He claimed to have found a fox that he could communicate with. It sounded unbelievable, but a desperate situation calls for desperate action.

She made her way to her friend George's house. She told George her problem. She asked him, "Do you think the fox you found, the one you say is intelligent and you say you can talk to, could help pull my snow blower through the snow drifts?"

George, looking at the snow replied, "No way, Poindexter but the intelligent fox I found hangs out in the park behind my house with a pack of seven other foxes. So, may be, if you tied them all together, they could pull the snow blower."

So that's what they did. George convinced the intelligent fox to help Bridgette, and the intelligent fox convinced the other seven foxes to help, as well. Using ten lengths of long rope, they managed to tie all eight foxes to the snow blower. There was still one minor problem. Bridgette had to clear one lane of snow on every driveway so that there was pavement cleared of deep snow for the team of foxes, since they couldn't get good traction in the deep snow.

Another problem was the foxes tended to pull the snow blower to one side, but she could counter that pull by throwing her weight to the other side. With the help of the intelligent fox and his friends, she completed snow-blowing all the driveways she'd been hired to do. She was heading for home when the wind began howling louder than the sound of explosions in a movie theater.

Suddenly, she heard a loud crack. A telephone pole snapped and fell across the street dragging its black wires with it and snapping several of them. The wires were whipped around by the wind sending arcs of electricity across the street. She looked at the intelligent fox. She shouted over the howling wind to him, "That tears it. I'm going through it. I'm gonna put the snow blower in fifth gear. You and the other foxes pull me as fast as you can. I'll release the rope just before we come to the wires and you cut south and head for the park."

The intelligent fox gave her a quizzical look or maybe it was just that gusting wind was so strong that it stretched the skin of his muzzle into a distorted smile or maybe he found the acrid smell from the burning of the insulation of the electric wires many times more unpleasant than Bridgette did because of the sensitivity of his olfactory system. Whatever the case, Bridgette continued, "I only know one damn thing: you're either an Argonaut or an Argo-nothing." For a second, the fox bared all its teeth. With that she slipped the shift lever into fifth gear and released the clutch. The snow blower lurched forward and the foxes raced forward like an arrow. She saw a sparking wire twisting right toward her. She released the foxes and doubled herself over the handle bars of the snow blower so her feet were off the ground. Her head rammed into the hot engine block and she could smell something burning. She didn't know whether it was just her hat or her hat and her hair. For an instant, she wished she had worn two hats or maybe even three. She shifted her weight just slightly with the hope that the scraper bar would be lifted off the road and only the rubber tires would be making contact with the ground. She saw a flash of bright white light.

The next thing she remembered was the blower bobbing up and down like someone's head might do when they fall asleep sitting at the kitchen table. She pushed herself back over the handle bars of the blower. She slowed the snow blower to third gear, and soon she was in front of her own driveway. She looked back and the power lines were still twisting in the gusting wind. She put the snow blower in the garage. She went into her house and took off her jacket and boots. She collapsed on her bed exhausted without taking off her hat.

124. Have you ever written a long, boring letter in which you attempted to dissect the grammar and punctuation of some document of only ephemeral importance?

Greetings Evelin,

I saw your photograph in the _Garfield Middle School Yearbook 2018-2019_. You looked very fetching although the photograph was small. Your hair was pulled back behind your ears; you looked very confident. There seemed to be a very small, reddish orange smudge under each of your eyes. Your sister, Vanessa, also looked quite sophisticated in her photo wearing her white cap and gown. I noted that only one boy in the entire sixth grade wore a suit and a tie. You should congratulate that boy for having the courage to dress differently from all the other kids. You could tell him that next year, if you have the courage, you are going to wear a clip-on tie and other accessories that are out of the ordinary.

I read the one-page "Introduction" by Principal Richard "Riggs" Rigoglioso. His grammar and punctuation reminded me of your writing style. It also reminded me that I didn't begin to learn the nuances of grammar and punctuation until many years after I had graduated from college. Everything is so rushed in middle school, high school, college and the working world that almost no one has time to learn the details of grammar and punctuation. Also, the rules of grammar and punctuation become so involved and boring that you actually can't remember all the details.

I thought you might find it interesting to read my critique of your principal's "Introduction" to the _Garfield Middle School Yearbook 2018-2019._ To begin with, he is using the block paragraph style. In the block paragraph style, the first line of the paragraph should not be indented. Your principal follows this rule, but another rule is that there should be a blank line between paragraphs; your principal doesn't follow this rule. Perhaps, he is trying to fit his "Introduction" onto a single page to save money or to leave more space for the photographs of the students.

His first sentence is quite long, but I believe it contains several errors. He writes, "As we reflect on the 2018-2019 school year, please join the Garfield Middle School staff and I in acknowledging and celebrating the many wonderful experiences and opportunities that will not only shape our students' futures, but also be permanently etched in their minds whenever they fondly look back on their time at Garfield Middle School." The sentence seems to be in the imperative mood. There are three moods: the indicative, the subjunctive and the imperative. Most people don't know about moods in grammar. Most people write, without thinking about it, in the indicative mood, which is the mood that states or questions facts. For instance, I didn't realize the _Lord's Pray_ begins in the indicative mood until recently: "Our father who art in heaven, hallowed be thy name. . . ." The use of the phrase "hallowed be thy name" instead of "hallowed is thy name" indicates the imperative mood. At least, I think it does. The imperative mood is used to express a command, wish or entreaty. So, you can see why it would be appropriate for a principal to use the imperative mood, which he uses many times in his "Introduction." The grammar book titled, _English_ _Grammar Simplified,_ states, ". . . the imperative is most often used without noun or pronoun, but a pronoun of the second person (singular or plural) is usually understood and may be readily supplied." When your principal writes, "please join the Garfield Middle School staff and I . . . ." the second person plural is understood. In other words, "You, the students and parents, please join the Garfield Middle School staff and I . . . ." This means the correct phrase is "please join the Garfield Middle School staff and me." It wouldn't sound correct to say, "Please join I for dinner." You would say, "Please join me for dinner." The subject of the sentence is the unnamed "you" whom you are asking to dinner; therefore, the "I" written by your principal is the object of the verb "join," not the subject, and it should be in the objective case, "me."

I note your principal likes to use pairs of words such as "acknowledging and celebrating," "experiences and opportunities" and "shape and etched." It seems to me the phrase, "that will not only shape our students' futures, but also be permanently etched in their minds," contains two verbs "will shape" and "be etched." I think it would be better to write, "that will not only shape our students' futures (no comma) but also _will_ be permanently etched in their minds." I made both of the verbs agree by putting them both in the future tense and I deleted the comma because these two parts of the sentence aren't being forcefully contrasted. The word "whenever" is a conjunctive adverb. It is used to link two clauses. I'm not sure but a conjunctive adverb may link an independent clause and a subordinate clause. It seems the two clauses, "(1) . . . please join the Garfield Middle School staff and I in acknowledging and celebrating the many wonderful experiences and opportunities that will not only shape our students' futures, but also be permanently etched in their minds (2) whenever they fondly look back on their time at Garfield Middle School," should be joined with a semicolon and there should be a comma after "whenever." It should read, ". . . please join the Garfield Middle School staff and _me_ in acknowledging and celebrating the many wonderful experiences and opportunities that will not only shape our students' futures but _will_ also be permanently etched in their minds _;_ whenever, they fondly look back on their time at Garfield Middle School."

In the next sentence, "It is a good time to reflect on what we have achieved, while also looking forward to an even more successful school year in 2019-2020!" I would omit the comma before the prepositional phrase, "while also . . . ."

Here is another long sentence with a very minor error, "It has been amazing seeing all of you progress, both academically and socially, while also growing into the mature and great students." The word "the" before mature should be omitted. The phrase, "both academically and socially," is a parenthetical phrase, and it is correctly enclosed within commas. The sentence that begins with the word "Remember" is in the imperative mood, and so is the sentence that begins with the words "Be proud."

Here is another long sentence with two minor errors, "This past year has had so many incredible moments that I can't possibly begin to list them all, but I want to especially thank our wonderful staff, and families for supporting all our exciting activities, for they have been an integral part of it all." The comma between "wonderful staff, and families" should be omitted. You wouldn't write, "I like classic rock, and electronic music." The comma between "activities, (comma) for they" should be omitted. The clause, "they have been an integral part of it all," could be made into a separate sentence. In the next sentence, the words in the parentheses "(well mostly)" should be punctuated (well, mostly) because you use a comma before and/or after an interjection. The word "well" is an interjection. There should be another comma between "(well, mostly)" and "which." The rule is "that" is the first word of a phrase or clause that is essential for the sentence to make sense or to mean what you want it to mean. When the information is not essential, use "which" and do use commas. The hyphen used in the phrase, "never-ending curiosity" is correct. The rule, which I never knew about until recently, is use hyphens in compound adjectives if they come before the noun they describe. This is one of the many rules I can never remember.

In the next paragraph there are seven sentences in the imperative mood. They are the sentences that begin with the following words: "Don't," "Always," "Be kind," "Enjoy," Get to," "Try to," "Finally, trust." He ends this paragraph with another long sentence, but it is not a complete sentence. He writes, "So in quoting Milton Berle 'If opportunity doesn't knock, build a door.' and the great E. E. Cummings; 'It takes courage to grow up and become who you really are.'" The sentence seems to need a subject and a verb. Perhaps, he got carried away with the seven sentences in the imperative mood. The sentence could be written as follows: "In closing, I would like to offer you two pertinent insights. It was Milton Berle who said, 'If opportunity doesn't knock, build a door.' And it was the great E. E. Cummings who wrote, 'It takes courage to grow up and become who you really are.'" It is interesting to note that the poet E. E. Cummings legally changed his name to e. e. cummings.

The last sentence is another long sentence. He writes, "So, on behalf of Vice Principals Mr. Gray and Mrs. Mazzola, the Garfield Middle School Staff and myself we wish all of our students an enjoyable . . . ." At first, I thought the use of the word "myself" was incorrect, but after checking with the grammar book _Painless Grammar,_ I now believe it is correct. Now, I think there should be a comma after the introductory phrase, "So, on behalf of Vice Principals Mr. Gray and Mrs. Mazzola, the Garfield Middle School Staff and myself, . . . ." Also, I believe instead of writing, "we wish" he should have written, "I wish." A single individual would say _I speak on their behalf._ He wouldn't say _we speak on their behalf._

As you can see from what I have written, it is difficult to correctly punctuate long sentences. Can we extrapolate anything else from this critique? You might think your principal would have the free time to have a teacher from the English department polish his writing, but it seems everyone has so much work to do that no one has any free time. Also, if you went through my critique with an exacting attention to every detail of punctuation and grammar, you would find many mistakes, as well. One reason for this is that I am not sure how to punctuate a sentence fragment when it is in quotation marks. Another reason might be that the rules of punctuation might lead to contradictions.

Sincerely, Jim Spinosa

