WAVE IDENTITY    

CHAPTER 35           (Dec-2000)                   INDEX TO OTHER PAGES

  1. To understand the nature in which we abide and of which our bodies are made is completely incomprehensible.  This no doubt is something many may not agree upon, yet those with an understanding of nature will appreciate the sentiment. 

  2. In times passed I spoke of magnetic lines in the way of alignment, and I drew the atom in its nomenclature and as the wheels of nature.  And I spoke of matter and of motion and of coordinates, and of a great many things.  Much of which, as far reaching as they may seem are nonetheless elementary. 

  3. My habit in updating things has been to add new chapters, which comes to some duplication.  And this is in that same manner, to add something to the existing, rather than to rewrite the previously written.

  4. And so let us go once again into the “identity of a wave”. The things then we wish to question or debate upon is.

1.  Can or are wavelengths compressed or expanded upon themselves?

2.  Are wavelengths at all times continues or not?

3.  In what way does a wave identify itself on the spectra?

4.  Is there a dual nature in the nature of waves?

5.  How is a wave generated or initiated and/or arrested?

6.  Are all the waves of the EM spectrum alike or not?

  1. And to come to the major question,  “How any wave when altered is still able to show its chemical or nature of origin?” Which then may or should reveal unto us the how or what in the  “Identity of the wave”.

Are, or can waves be compressed upon themselves?

  1. For lack of experiment let us use reasoning to come to some solution.  Our question then can also be stated as. “What exactly happens to a wave when it enters a denser media?” and by figures 35-1 and 35-2, we will utilize the media of water at the index of 1.33, and keep our illustration roughly to scale except amplitude, the circular diameter of the wave.

  2.   Given a length of 7000a (figure 35-1 A-C) at a diameter of one-inch (one-inch equal to 2000a).  The retarded length would be 5263a C to L.  Assuming therefore that the length of the wave does not compress the actual length of the wave, as it was A-B-C, remains the same at C-K-L.  

  3. The diameter of the wave must therefore (according to our scale) increase from one inch to one and one/half inches.   This would place our new reading upon the spectra as 5263a (M) that formerly read 7000a. (D).

  4.  Next let us assume that the diameter does not increase, nor shall the wave be compressed upon itself, in which case the wave wraps itself more around the tube. This would be more than 1.5 times as noted by figure 35-1 C to E to H to G, and beyond. 

  

  1. The question then becomes what the wavelength as a crest to crest measure would be, and what the same would read upon the spectra?  

  2. Our supposedly 5263a measure has been reduced to a measure somewhere in the neighborhood of 3200a.  And yet this 3200a measure still reads 5263a upon our spectra since the angle is virtually the same. 

  3. How therefore can 3200a read 5263a?    The cause as we conjecture is in the change of diameter.  Yet this is very curious how for a mere 1.33 retardation the crest to crest becomes less than half of what it was before yet give us a much larger reading.  

  4. If on the other hand the wave did compress with no change in diameter it would pass C-F-J, as the actual (compressed) length in the measured or crest to crest length of 5263a (C-J).    

  5. But the same would then give us a reading of 7000a at the spectra since of course the angle remained the same as before - while in effect it is only 5263a long.  

  

  1. And so it is time to remind ourselves of the facts as we know them.    It is a fact that light in passing through water is slower than through air. And while we did not place a tape measure on the wave crests - mathematics clearly brought the length – by distance in time - down to 5263a.  

  2. Seeing thus how this wave-crest length were mathematically - how are we to certify ourselves of the real measure to be 5263a or 3200a?

  3. We can do the same thing by a different set of measures, to reduce the diameter by one half in figure 35-2.  In which case the retarded diameter did not increase by only fifty percent as before, but more than doubled from one half inch to one and one quarter inch.  And to place R-S-T into the one half inch diameter it becomes R-U-V-etc. reducing the crest to crest measure down to somewhere near 2100a.   

  4. This because the wave not being compressed must wrap itself more than twice around the circumference in order for the P-Q-R measure at 7000a to fit within the measure of 5263a.   In this case the angle R-U is somewhat less than R-S.

  5. As therefore different measures will place different angles it would enhance us to know the real diameter.    Figure 35-2,  depicts the 7000a wave at a diameter of 1000a.  But I believe this diameter to be less, small enough that if scaled upon our 8-1/2 x 11 sheet it would appear more like a straight line. 

  6. If for example we take the speed of light in free space at 299,800 km/sec, and we assume, as we have, that the velocity of constant is an unwavering 300,000 km/sec.    Then utilizing a wavelength of 7000a, the mean length of that wave would be 7004.66a.   This would place the diameter, or amplitude at but a few angstroms.  

  7. If therefore one is able to provide us with an accurate measure of amplitude we can then compute to record the actual and true velocity of the constant. 

  8. For as we should know; the velocity of constant at 300,000 km/sec is more an assumption or calculated guess than reality, and therefore not absolute.  If for example we take the speed of light in space at a relative velocity of 299,800 km/sec.  

  9. And we consider a general diameter of the lighter atoms at 5.3a, with the wavelength at 7000a.  This added with the angular in the circumference thereof becomes 3.14 x 5.3 to bring the mean length of the wave to 7016.64a.   

  10. If then we divided the velocity of 299,800 by 7000, and multiply this by 7016.64 it would bring the velocity of constant to 300,513 km/sec.  Or if we consider the diameter of larger atoms along which light is still known to pass to assume 50a, it would bring the mean length to 7150a.  

  11. Then by assuming the relative velocity at 299,400 km/sec, the velocity of constant would come to 305,816 km/sec.  These figures of course are mere examples, and roughly at that.  

  12. What therefore is the real velocity of the constant?  Or one might ask me how or why I figure this constant to be a constant, or even to question me for its very existence?   And yes I do indeed conceive and understand it to be - and of a constant, but as to these details I must elect to be silent.  

  13. If however we could somehow measure the velocity at which the magnetic flux passes, we might obtain it.  Or taking the actual real amplitude and length of the electrical wave by the relative velocity thereof, we might also obtain it.   

  14. I should caution us however to consider all the variables so as not to bring ourselves in error.  Thus I deem it possible for us to discover the actual true velocity of constant.

  15. The next item is the spectra, our reading of the angle of the wave by which length is determined, and by which shifts to the red or blue may be seen.  How for example can we (Figure 35-3)  determine the individual lengths of the waves of the incoming light that strikes the prism all at once together?   

  16. For we must also determine it a fact that while these waves arrive all bundled together, they do not all travel at the same relative velocity even though their velocity of constant may be the same.  The red waves are always on the fast lane passing the blue.

  17. The answer as we know is to separate the individual lengths from one another by refraction.   It is then that we set our scale from high to low to span – as we presume – the difference in the visible wavelengths.  But are we so sure?   For it may also be seen that we merely spanned the degree of the dispersion of the light.   

  18. The index for glass then being at 1.60, what if we passed it through diamond with an index at 2.42, would not the angle of refraction be more acute?

  19. Indeed it would be, but what is true for the long waves is true for the short ones as well, wherefore the degree of the dispersion should remain the same.  The angle of refraction for each individual wave then being according to each their own angle at which they arrive at the prism, we may rely upon our scale as being correct.

  20. The prism may reduce the relative velocity of all of the waves as they pass through, but in leaving these will regain the same.   And while the refracted waves may be compressed or pass through the prism by a larger diameter, this also is corrected again in passing from the prism back into the original media.  

  21. Wherefore we can not find anything wrong in the way that we read wavelengths.   This leaves us with this absolute, that we do in fact read wavelength by the angular formation or moment of its wavelike appearance. 

  22. I then said “appearance” since in all reality these so called waves are not really waves at all – but simply lines of movement (substantial or otherwise) passing forward around a circumference, a tubular formation. Their sine formation no more than appearance set on paper.

  23. And thus coming back at the quest if the lengths of light are or can be compressed, what shall the answer be?  Realistically it is difficult to conceive how a wave can wrap itself from a single turn to more than a double turn (as in figure 35-2) around the circumference considering how the index is no more than 1.33.    

  24. It is far more likely for the wave to assume a larger diameter.  But then for the wave to compress upon itself also seems plausible seeing how the angle C-F Figure 35-1 placed as A-X next to A-B can very well come out at the 5263 measure. 

  25. But the same is comparable to the scale drawn.  For in viewing the same by figure 35-2 having cut the diameter by one half - the separation of the comparison line of R-Y placed next to P-Q is reduced.  Moreover the drawing very well seems to invalidate the expansion in diameter.   

  26. For it is plausible that R-Y would give us a 5263a reading, while R-S is much to acute and more in the way of 2200a.   And the same goes for C-K that is comparable to a 3200a measure, while C-F is clearly a 5263 measure.

  27. What therefore shall the outcome be?   I gather we have convinced ourselves that wavelengths not only can but are compressed upon themselves. And rather than leaving it to assume that therefore the diameter may not or will not increase.

  28. I prefer to believe that both can go together.  And that there are certain criteria wherein the one will increase or decrease for the other, the matter coming down to what I spoke of in chapter 32, figure 32-17.

  29. We should however not always be so sure of ourselves; we are after all from the outside looking in.    We started with seven questions and we have yet to be guaranteed that our answer to the first is absolute.  

  30. We could for example disbelieve me and look at light in the nature of transverse waves.   But I will no longer join in that debate since it will lead nowhere.   And as to the actual collateral of the wave, this we merely speculate upon. 

  31. Thus what shall we say in order to complete upon this first question of ours?  We could refer to figure 35-4a to let wavelength angle “A” comes to a change in density, a greater density.  Here at atom X the angular trajectory is forced into a shorter turn around the circumference that interprets into a shorter crest to crest wavelength measure as it is wont to do under these circumstances.  

  1. This then may be interpreted that by angle B the wave will now take on a larger diameter if atom Y is larger than atom X and all those along which it passed just previously. But it also shows or means something else.  

  2. The wave at angle B will have or take on a relative velocity of only 225,500 km/sec, while wave angle “A was and is traveling a the relative velocity of 299,300 km/sec.  Obviously angle “A” will outrun angle B, or compress itself upon it as we might say, for the train of angle “A” must follow angle B.

  3. Thus it seems logical for a wave to compress upon itself. And the same may be true in reverse by figure 35-4b where angle D must go faster whereby to expand the wave into a red shift thereof.  

  4.   The question then becomes just how much or to what point the compression and/or expansion may be allowed?  If both of them go hand in hand then God’s universe may still be expanding, while I thought I shot down that bird.

  5. The next query was: “If wavelengths are at all times continues or not?”

  6. No doubt it must come to some embarrassment that electrons do not bob up and down to accommodate a line of light to pass by, or any wave for that matter.  From previous discussions we made it clear that such a phenomena is out of the question.  

  7. This leaves us with the line of light to pass through and by the media as a straight line winding itself around the perimeter of the atoms and/or molecules.

  8. But shall these lines (or waves as they are more commonly called) be continues or may they be in parts, either sporadic or by trains of?    In my estimation, and not that I have direct physical evidence thereof, I will nevertheless say; it can be everyone and all.    

  9. And as we look at the wave spectrum with its many variants we must remember that as all things are twofold so light in its nature and in its movement is twofold. 

  10. And when we are all done with discovering light for its movement and for its nature – we shall come to this conclusion to know for a certainty that we do not as yet know the true nature of light, nor its motivation

  11. And the same will be true for such things as matter, the material substance, and for the nature and/or cause to its movement.  Accordingly we are a long ways from home, nor does it seem to me that we apprehend the path leading thereto.

  12. By figure 35-5  let wavelength AMB be a full wavelet, and that it can and may very well continue as such, especially in the secondary ones.   For as we may recall from “The way we see,” I am speaking of what I called the regenerated ones of which the air must be filled during sunlight hours.  These are the more harmless and the most abundant of the waves, which come into our eye.  

  13. Looking directly into the sun or any bright light will bring us the primary waves, easily noted from the others since all wavelengths reveal their source.  This is why the rays of the sun reflected from a mirror still reveal the sun, yet the rainbow even though they are from the sun do not reveal the sun, but rather the water droplets through which they passed. 

  14. Not that we are able to behold a single droplet at that distance, but the individual waves in their color reveal their last point of contact at which they were either turned back or regenerated.   It this case it is turned back rather than regenerated through the interior of the droplet.  And that makes the difference as to why from a mirror it is the sun, but not from the droplet.

  15. By figure 35-5 part A-M of the line of light is passing on this side of the tube, while M-B is on the far side and therefore in a broken line.   I then placed the ideal of the spectra at point M, but in all reality any one part of this whole wavelet would show up on the spectra at the same measure.  

  16. Since again in all reality the line of light that unto us appears wavy is in fact a perfectly straight line.  This is enhanced as we take the tube and lay it out to a flat sheet wherein the A-B wave passes A2 to B2.

  17. Why therefore does the line have to be continues when any burst or part of a wave such as C-D will pass exactly the same way, by the same angular and linear momentum?   For whether it be a full wavelet, or a quarter wavelet (as figure 35-5 C-D is to illustrate), as long as the angular moment thereof is the same as the angular moment of A-M, it will register on our spectra as one and the same.

  18. The lamps in our homes do not provide us with continues wavelengths.  These are turned on and off sixty times each second, wherefore in effect we are receiving sixty wave trains per second.  

  19. This may be depicted by figure 35-5-Z  showing only two wave trains, and how each of these trains are not continues but sectional like unto a train, boxcars coupled together yet separated from each other.   Or by any other fashion like unto figure 35-5-Y to receive multiple wavelengths also by a train thereof.  

  20. When we hear sound or speak on our cell phones most of the connecting wave or line is empty, or devoid of waves, train or otherwise.   For at the speed of three hundred thousand kilometers per second none of us could speak fast enough to fill a line.

  21. Then you might have heard it said how in order to get better reception we’ll create a carrier wave and then superimpose the tune or message upon it. And yes this can be done and is done.  But let us take a look at the line of light line Q-R in figure 35-5, to superimpose a wavelet upon a wave.   

  22.  This all looks fine, we’ll take a super fast chopper like E, and at some magnitude put a dent in the line.   Will then this line Q-F-G-K-R continue like that?  The answer is, No it will not, and the reason is quite simple.

  23. Every point or part of this wave travels at one single constant of velocity.  But the wave travels this velocity by an angular momentum as well as a linear momentum.  Each different wavelength therefore with a different angular momentum will take a longer or shorter time to travel any given distance.   This time-frame velocity is what we have come to call the “relative velocity” (Vr).  

  24. If therefore line Q-R has a Vr of 299.000 km/ps, and by E we force it unto G, which means a greater angular moment around the tube – then Q over F into G will instantly take on a slower relative velocity. Accordingly, the line that was formerly Q-R will be split apart at point G, with K-R to continue separately.   

  25. This means that KR will outrun Q-G.   For with Q-G taking on a Vr of 250,000 km/ps it will come to lag behind KR by 49,000 km each and every second of time.

  26. It is of course vain to superimpose by way of a dent in this kind of a wavelength.   For it took a whole atom or molecule at a fractional instant of time to even create this wavelength as is or was.   And not likely shall we be able to put dents in it.   

  27. More than likely with the impulse of E we would miss the line and simple create another wavelet into that particular angular moment. And thus we would have a newly created wavelength to ride or pass independent of line Q-R.

  28. This I conveyed to show in what way the lines of light travel and how we can and cannot impose upon it.  For if we take another wavelength such as S by figure 35-5 which is longer (less angular moment) than Q-R, it will come to catch up with it, and either join with it or cancel one or the other or both out.   

  29. This however is to be considered twofold for primary as well as secondary waves.   Normally when wave-crests arrive in time they re-enforce each other, while out of time they cancel each other. This is true in the nature of transverse waves, but in other waves such as those of light also.

         Summarizing

  1.  We have been speaking of wave transmission on the order of light, and in recap perhaps we should summarize the various wave formation in their various mode of transmission to gain an overall view of them.  

  2. In the first place let us not under-estimate the potential of a wave as a mere coordinate.   For while a wave may be a mere coordinate, yet a pulse of this coordinate can bring a great amount of mass to bear.  

  3. Take for example a tsunami, where the shockwave, a mere pulse, while it travels through the ocean does not cause much water to move, yet its potential is such that at reaching a shore it can built up the water to a hundred feet high.   The secret here is in the unique and close relation that coordinates form with the media in which it is a coordinate “in” as well as “of”.  

  4.  Or the coordinates of light to warm the entire earth, and if multiplied to burn the same. Or the power of the tide - mere lines of movement to lift the waters.  Or the whole earth, or the galaxy for that matter as matter in motion by coordinates.

  5.   The spectrum of waves should not only be the EM spectrum but all wave propagation.   By figure 35-6, in the EM section there is no essential movement nor displacement of the media, other than the movement that is always there namely the wheels of nature and the velocity of constant.  

  6. In the collateral section with water waves there is the essential movement of the media but not the displacement thereof, although the same can become a displacement, as we know from the tides and the tsunami.  

  7. In the case of sound waves there is no essential movement of the media, even though it can be brought to movement and/or displacement.   This is of course with the exception of the cores of the atoms to vibrate back and forth transmitting the impulses along the “grid” of the media.    

  8. As then sound waves are longitudinal, the transverse movement being in line with the direction in which the sound travels, water waves are transverse the wave movement being at right angle to the direction of the wave.    

  9. In this case we can say that the wave activates or forces the molecules to their tune.   But we may also be in error if we do not replace that sentence with something more accurate.    For it is not the wave that directs the molecules to their tune, but rather the impulse did – with the wave being the product of the impulse.

  10. And so there is a distinct difference here.  For again in the propagation thereof it shall neither be the wave to direct the molecules, nor the molecules to design the wave, but rather in the combination of the two.   

  11. For as the impulse, the original shockwave caused the molecules to move about by a certain shock, a go and stop movement, that very movement became the wave.  The wave or the line thereof is therefore not unique, nor is the movement of the molecules, but the two in unity as we might say.

  12.   The spectrum therefore in all of its wave formations is the variant “in” and “upon” and “of” the coordinates of the media.  The magnetic wave for example with its single half/wave and essentially straight-line formation may be called the “standing wave”.  

  13. Electricity or the currents of electrical force on the other hand is and should be called an angular or rotational magnetic force.  It, the electricity, is nothing more than magnetic lines twisted over and over into many multiples of that original half/wave resembling the figure of eight.

  14. The forward movement of electricity is none other than its twisting design propagated by the 3M, even as the shockwave in water is rolling design propagated by the 3M. 

  15. And so we have looked at the top and the bottom of the wave spectrum.  The top end, the most fundamental end of the spectrum, is the one that is always affixed to its source. And with it we include the electrical wave since it being of the first - cannot subsist without its source, it being held in the integral coordinate of a media (capacitor, battery), or mechanically held such as with a generator.  

  16. This leaves us with the intermediate, which is the top end of the EM section.  And this as well as the bottom end of the spectrum classify waves that once initiated travel without being tied to their source. That segment, from the very long to the short also “depend on” and “are part of” the first most fundamental wave namely the magnetic line of movement.

  17.   And in reference to these we might liken the magnetic lines unto the string upon a bow to shoot an arrow. Or like unto the strings upon a guitar in which case the wave becomes one that is transverse.  In the case of the guitar the atoms along with the cores on the grid of the media are moved causing sound to be heard, which is not so on the more fundamental scale.   

  18. While in the EM segment there is no essential movement of the media, nor of the cores – {(no transverse ideal)} or else we would be deafened by the sound of the myriads of radio transmission passing through the air. It is as I said a variant in and upon and of the coordinates of the media.

  19. And thus to come back upon our quest in the continuity of waves, the magnetic wave is always continues and cannot be cut upon, yet it can be modulated the examples of which are the short, long, and electric waves.  The first ones in the EM segment that become independent of the magnetic wave are the shorter wavelengths, such as those of light and there about.  

  20. As for the long and short waves, and even those of the electric can and shall be continues if and when so generated on a continual bases.  Or the same (excepting the electrical) may be as short as a single wavelet traveling down the line.  

  21. If per example in the area of the long waves we generated one kilometer wavelengths, and knowing the velocity of the line that we are modulating is Vc, we would then have to generate wavelets at the rate of 300.000 per second in order to have them continues.

  22. But now we made a little fib, or we are confusing the scenario of the long wave with those of light waves.  For although the velocity of the line that we are modulating may be Vc (the constant of magnetic waves) this does not necessarily mean that the modulation is taken away at that speed.  

  23. Nor shall we say at such relative velocity as we account with light?  The answer would be yes and no, yes if we are dealing with the strings of magnetic but no on the order of mechanical innovations. 

  24. Assuming a 128-mm transverse wavelength with an amplitude of 32-mm having a mean length of 140 mm. If therefore the train of this modulation is taken away by the carrier whose velocity is that of the constant, the relative velocity would be at apr. 274,285 km/sec.  Or if the amplitude increased to 64 mm the wavelength remaining the same, the mean length would come to some 200 mm, with the Vr at 192,000 km/sec. 

  25. If we take a wavelength at 256 mm with amplitude remaining at 64 mm the mean length would be 288 mm with Vr at 266,666 km/sec.  If the length were 512 mm, amplitude remaining at 64 mm, the mean length would be 528 mm, with the relative velocity increasing to 290,909 km/sec.

  26.   Thus we see how amplitude verses wavelength directly plays upon the speed at which the wave will travel for distance in time.  All this is absolute for the shorter waves in which the octave of visible light is seated, but shall this be so for the long and short waves such as radio waves etc. as well?   

  27. For here there is somewhat of a difference between the wavelengths of light and those comparable to radio broadcasting.    Which is not only in the fact that one is assumed transverse with the other by a circular mode, but that light-waves appear as an entity on their own while the longer lengths are in all essence modulations on what for all practical purposes are standing waves.

  28. The essence of the standing wave in the velocity thereof at the constant is the same as that for light. The speed therefore that exists upon and by which for example the short wave in figure 35-6 moves outward is at the full 300,000 km/sec.  At what relative velocity therefore shall the sine formation move outward?  

  29. For this sine formation is what I view as a mechanical implementation, in contrast to light-waves, which I do not view as such.   If then as with light - different wavelengths, or amplitudes in this case, travel at different relative velocities, how are they to be on a continues formation?

  30. Utilizing speculation on my part, considering how the waves are mechanical modulations on a standing wave-grid – I estimate that separate formations may travel at distinct velocities.   And that a combination in amplitudes in a train thereof may travel at a single relative velocity that is in the combination thereof.  

  31. Mechanical implementations of sine formations like those on a rope or a string that is traversed to create the same have different velocities depending on the rigidity and nature of the string or standing wave as it may be called.   

  32. The zero velocity grid that binds or associates all media to each other allows sound to travel at different velocities.  And even the media, as with water waves will transmit the sine formation at speeds comparable to that of our jet airliners.

  33. Zero velocity grid then means the imaginary lines that extend between the cores of all media, the whole grid as such at which no velocity is pronounced.   This in contrast to the Vc grid, also referred to as magnetic lines of force, or the magnetic grid, or flux, or simply Vc.   

  34. It then is well to realize how Vc is an entire grid of movement in all directions that proceeds never through but always around the perimeter of atoms and/or molecules.    Which in effect is more accurate than to confine ourselves to merely the flux formation of the earth’s magnetic field.

  35. The drawback in being mechanical for the longer waves, and for the range that in figure 35-6 we termed short waves - is their distance and straight-line trajectory.    For what we may call straight in the distance from the earth’s surface to the ionosphere is but a hands throw for the nature of the magnetic media.    

  36. For as these in fact are no more than a modulation of the magnetic fabric so their distance, trajectory, reflection, refraction, and otherwise distortion is directly relevant thereto.  

  37. Like unto light that may be distorted by a thin layer of warm air above a roadbed – so wavelengths on the longer end may be distorted by the curvature and/or change in the magnetic fabric.  The shorter the length therefore the more fundamental it becomes and the more apt it shall be to retain the same.

          Conclusion

  1. All this now may have answered a number of the questions at which we started out, but it still leaves us with light in the nature thereof, and the true identity of these waves.   Of the first six questions the answer to number 2 is both.  And to answer number 4, we saw a twofold nature in the formation of waves. 

  2. And the answer to question number 6 is both a yes and no, alike in some respects and different in other respects.  And for number 5, we may have all sorts of equipment to produce sine wave modulations taking note of crystals and all, but who is to describe in detail the how thereof? 

  3. To state that with a certain voltage at a certain rate I am shaking a line is of little use in the fundamental scenario thereof.  And to speak of electrons bouncing lose, or gaining or losing energy, or jumping levels, is like the priests falling down before an image praying to an idol that is less competent than they themselves are.   

  4. Or to speak of mono-polarities in all the elaborate cunning thereof is like fancying disfigured visitors from outer space with their flying saucers and all – something that exist only in the imagination of them that have yet to acquire education.

  5.   We are affected by reality and we should consider it in the same way.  And while speculation may be of some interest we should not set standards by it.   

  6. We came to question the identity of the lines of light to find  the identity thereof, and we have yet to find it.  Let us therefore by another chapter take another route to look into “coordinates.”

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