After exploring mathematically the problem with flat earth in a previous post it is much more easily seen visually.
Here again is one version of the Flat Earth: the Azimuthal projection. The thing to note in this projection is that the distance around the disk increases as you travel away from the North Pole. By the time you get to the South Pole it is more than 6x the distance from the North Pole. This is the only map with this distance property.
The Globe Earth model viewed from the South looks like this. As we move away from the Equator the distance around the globe shrinks to zero at the South Pole. This is the only map with this distance property. Unlike Flat Earth models it can only be drawn in 3D.
Distances around the globe are a copy of the North but in reverse. This is hugely different from the Azimuthal where distances around the Earth continue to increase away from the equator rather than reduce again.
The result of this divergence becomes very obvious in places like Australia.
Now look at actual distances on the maps. On the Azimuthal the distance across Australia is roughly the distance across Asia (about a little finger on my screen). On the globe the distance across Asia is more than twice the distance across Australia. There is a huge difference in distances.
As described more accurately in the previous post Google Maps uses a very simple distance estimating tool based on a globe model. I recreate it accidentally in the post. IF the world really was Azimuthal then this tool would give distances 2/3 too small in Australia which would be really noticeable. I spoke to some people who had lived in Sydney and they confirmed that the distance to Canberra is about 2 hours drive and not the 3 hours that an Azimuthal flat earth predicts.
It is fairly simple to show that the world is not Azimuthal Flat Earth. However there are an infinite number of other flat earth maps. And while each flat earth can be chosen to approximate the globe in some region, they all suffer from the same problem of huge divergence somewhere else. Given a particular Flat Earth model it is fairly trivial then to determine the region of divergence and get some simple car journey or shipping data to confirm which model is most practical.
Of course the Earth is not an exact Sphere any more than it is an exact Disk. But, approximating it to a sphere gives remarkably accurate results. I still cannot believe that Google only uses a simple Cosine Rule to estimate global distances, and that appears to be good enough across the globe. At least I've not heard people coming back from travels complaining their phones were out. This fact alone suggests the Earth really is very close to a sphere.
There was a time when we didn't travel much where a flat earth model really was good enough. But every region of the world would have its own flat earth map. To have a single model that works for all places we need to shift into 3D and have something like a global model.
The usual view of the world is that it is just there. But then we discover things and we realise that there are new things to find. This realisation makes us aware that we didn't know everything before. So as discoveries mount up the question starts to arise "can we know everything?", or "are discoveries and endless thing?" which is a similar question to "how big is the Universe?"
Discovery exists in every field of knowledge. Land discovery and exploration is a good model that fits all. We can always be sure that what we have discovered is definite, like the land on which we stand: that much is certain. But as discoveries mount up our understanding of what we have changes. we find a sea around the land and realise we live on an island, and then as discussed recently, we discover a boundary around even the sea which appears to be roughly a sphere and we realise we live on a planet. All the while where we stand does not change but out knowledge and ideas change and the context of what we have changes. Discovery is as much about discovering new lands as changing the context of what we have. Many films exploit this like for example "The Matrix" where even the certain reality of our senses has its context changed to a computer simulation by discoveries revealed by following Neo. Context, or interpretation, of what we have is what the thoughts do, they are a very powerful sense that we have. In meditation we learn to ignore it for a while and return to just what we have: our tendency can be to spend to much time in "interpreting" the world and allowing "context" to colour what is real. This leads ultimately to mental illness where our thoughts become so dominant that we can no longer just live with what is there, can no longer see the land on which we stand, and instead have our minds obsessed with thoughts like is the world round or flat, will we fall off one day, is there something solid beneath my feet and a million thoughts many of which lead to insecurity, ungroundedness, anxiety and fear. All that said, the mind sense, when controlled, is a formidable way to experience the world one that literally gives a new dimension to what we sense.
All this related to the question "what is out there?" by framing where that question applies. It does not apply to the world we commonly experience--that is just there--it is a question about the mind itself and its ability to provide interpretation and context. It is a question not about the land on which we stand--which is just there right now--but on what the ultimate context is: land, island, globe, planet, solar system, galaxy, galactic cluster, universe, multi-verse, multi-multi-verse... atom in the toe of a giant... computer simulation... thought in Gods mind... Where ultimately and I? The answer lies not in discoveries but in the mind itself.
So we have 2 possibilities. Either the discoveries accumulate to a final point where there is nothing left to discover and our minds are the same as reality. Or, there is no final point and discovery is an endless thing.
I hope the preceding analysis forced a big enough wedge between what is real and beneath our feet, and what we think about that and the context in which it lies. They are different things. There is no reason to believe that every discovery of new land changes what we have: all men stand on the same small square of land no matter how much they know, or how much the own. This is reality. What we know and what we own are simply contexts provided by our brain. It should be quite obvious that discoveries need not ever end.
The SRH in this blog points much more precisely to why there is no end to discovery. Supposing we did discover everything, wouldn't the fact that we have discovered everything be a new thing to discover. We cannot put such a limit around discovery; such a limit would simply provide a boundary over which to explore. Anything boundary that the mind can fashion to try and end discovery simply becomes a new target of exploration.
Suppose physicists created a Grand Unified Theory to explain the basic physics of the universe. This is a boundary. The next exploration and question is why is it like that? Answers might be that we are just one of infinite multiverses and it is random, so we are just the one that ends up being like this. But then we have a whole branch of physics to do with the rules governing the creation of multiverses that is beyond anything we have ever dreamed of within this universe. Like Cantor discovered in mathematics, even something as simple as numbers has unbounded infinite complexity: you simply cannot put a boundary around it, because as soon as you create a boundary around numbers you can start counting those boundaries. There are an infinite number of infinities... etc until you head explodes.
It is not certain, but everything points to there being no final absolute context with which to explain what we have. Yes I live on an island, on a globe, in a solar system... but is that within a single universe of a multiverse or in a computer simulation of something stranger... there is no final answer. And, at the end of the day it is all just thoughts... albeit fascinating and illuminating thoughts.
So what of God. Many take God to be the unlimited expanse beyond the known. This is absolute correct as long as God remains unbounded within the mind. As soon as we give God a name, give Him qualities and dare start to Own him, or think we know Him then He is no longer God and just represents our own Ego. To fight in the name of God like so many have tried to do in the Past is a direct contradiction of our very knowledge of God. He is infinite and beyond description. How can we be anything but subservient to such a conception. we can certainly never presuppose to know God, or think we have special favour with Him. For all these people who would rise up in the name of God, sadly their fate is as much Satan as those who would knowingly honour Satan... at least Evil people have the one virtue that they are honest, unlike the virtuous who hold themselves up towards God in pure vanity.
There being no final answer is the best answer there can be. It reminds us that our thoughts are just there inside our heads, a very limited sense like our eyes and our smell. It is Mankind's ego and arrogance that makes us think our thoughts are more powerful than they are. Thoughts are limited and the discover of that is essential to the humility we need to live well and happily in this world.
Capitalism is the system of economics where people get a reward *just* for owning things.
If I own a field then I can work on that field and I own the produce from that field. This is the original sense of reward from ownership. However if I don't do any work then there is nothing to harvest and no reward.
Feudalism or Aristocracy pervert this logic. A Land Owner can let out his land to farmers who pay a rent. In Mediaeval times in UK this was 50% of the produce. Thus the Land Owner was rewarded for owning the land but without doing any work. This is Capitalism.
In UK the Land Owners were the Normans who conquered the country in 1066 and subjected the local population essentially to slavery. They built castles, violently repressed rebellions and stamped their authority on the land. Capitalism depends upon conquest to generate a ruling class of owners who live off the non-capitalists by rental cost.
We might argue why don't the peasants just go and farm on land that isn't owned. Very rapidly once rewards are set for ownership their is a battle for control of land and everything becomes owned. This is the major incentive for war. After war with no where else to farm, peasants are forced to rent land from the capitalists. In this sense while they are not in chains and are free to go where they like, in reality they have no choice but to rent land and work for capitalists as there is no other land, and so economically they are indistinguishable from slaves. Perhaps the only difference is that under slavery the slave owner is required to look after their slaves, and there is an incentive to do this well as slaves are expensive to replace. Under capitalism land owners do not look after their workforce which must fend for itself.
Often peasants accept the rule of land lords because in the wars that capitalism causes they need protection from attack by neighbouring groups. The best of a bad world is to pay your rent, avoid trouble from your Knight and let that Knight do the politics and fighting to stop your land being taken by someone else. This has been perverted in the modern age (beginning with Edward III in UK) where the Knights no longer protect the peasants for the rent they pay, but now the peasants fight to protect the Knights (80 million killed in WW1+2) who then take their rent! This absurdity is called National Patriotism.
Archaeological evidence suggests that war began with the rise of farming 7000 years ago, and escalated into the Bronze age. Once people stopped being nomadic and settled down, and once they started to generate wealth from the land and accumulate that wealth, it became a natural step for people to try and steal that land and so conflict began leading to huge Bronze Age kingdoms with vast lands, thriving trade, complex systems of ownership and powerful armies and kings. One of the earliest Western epics is Homer Iliad detailing one such Bronze age battle. In my own view what we have today is that same system which evolved in the great neolithic farming revolution and the kingdoms which followed. There is a lot of talk but essentially nothing has changed in 7000 years.
I hope I have made a case for Capitalism being the system by which people rent resources from owners, and those owners gain an income simply from being the owners of those resources. This is the main feature of the capitalist property system. While property is a right for all people in a property system, it only really benefits those who have more property than they need, so that they can start to rent out property and gain an income. This translates into finance where people with more money than they need can accumulate wealth in shares of companies (that is a share of the property of the company) and in so doing receive a dividend - essentially a rental reward from the part of the company that you own. Dividends are paid first to Capitalists before bonuses are paid to the workers, who actually make it possible, because Capitalism naturally is designed to primarily benefit capitalists.
Having Capitalists in an economic system is like having parasites. For every piece of work done there is rent to pay to someone. And even capitalists must pay rent to the parasites on their back. Up near the top of the food chain, and the most visible parasites, are the Banks. Almost everything is owned by banks, and we all pay a huge amount of rent to the banks. This is the ultimate achievement of capitalism. A class of people who get paid an income simply for lending out money. This was illegal in the Bible and one can see why. It is the road to serfdom. Hayek may argue that Socialism is the "Road To Serfdom", but the irony is that the work he uses, Serfdom, is none other than Capitalism itself. The Jewish people have often been caught up in this, as their laws only prohibit usury to other Jews not gentiles, while the gentles banned it universally. As a result the Jews could lend and made a vast profit in Mediaeval times bankrolling the European super powers. Serfs and Kingdoms naturally became rather unhappy at times from having these parasites around and it became convenient to remove Bankers at times. Sadly the Jews have too often been right in the firing line. But the problem was never Jews, but the system of Capitalism itself which twists and turns and finds scapegoats (ironically a Jewish custom), and feeds the masses propaganda, but ultimately never gives up on the structure of Rich and haves renting to the Poor and have-nots and creating an income in the parasitic process. Far from being the perpetrators, Jews have been at the forefront of criticising and raising questions about capitalism for over two centuries. But in all that time Capitalism has simply grown and grown.
I needn't add much on the excesses of the modern Capitalist Banking system that would make any Mediaeval Knight's eyes bleed. The entire world is in dept to International Banking. In the 80s the call was to Cancel 3rd World Debt, the problem in the new Millennium has only got worse and now the call is the Cancel 1st World Debt. There is almost no household in this world that is not paying rent to capitalists. Via the system of mortgages, Banks own 50% of all property in the UK and receive around 4% rent on that - this is an astronomical amount of income for a few private capitalists. It should be added that almost like a bucket shop, banks simply take money deposited and feed it straight out again in loans. They are just middle men - they don't actually do anything. And it is worse because the system of Fractional Reserve means that they only need to keep enough cash sitting around to handle the daily through flow of deposits and withdrawals... this means that they can write loans for almost any amount even far exceeding what they actually own themselves. This is the Capitalist Wet Dream gaining an income on things you don't even own!
So Capitalism has come a long way from the original argument that someone owns the fruits of his labours (which the Bible says he doesn't anyway), to owning his produce of his land, to owning some of the produce of people who work on your land, to owning some of the income on loans backed by money that you don't even own.
What does all this mean to the average person. The visible impact of all this money sinking down the tree of ownership, is the branches get even fatter and fatter the further down you go and the leaves become skinnier and skinnier. It leads to what is called Inequality. In a perfect system where everyone earned the fruits of their own labours directly we would have a relatively equal system. Governments may intervene to enforce law and order, and provide a universal safety net but even without this the system would be fairly equal. But as soon as people begin to buy things just to rent them out, the drains start to flow and money gets leaked down the tree. The result is the the poor getting poorer and the rich getting richer.
In this fairly detailed exposition I hope it is visible that there are very good reasons why capitalism has come to exist and take over the world (staring 7000 years ago). That wars are an integral part of capitalism and that inequality is its result. The problem that remains is to understand why people keep on doing it. It might be that individually we always thing we'll be the lucky ones and get that life style of luxury for ourselves so we borrow to achieve it, not realising that rents on people borrowing is exactly the income that makes capitalists fat. It is like the person who says buy my book to find out how to get rich, only to discover once you've parted with your money that he is making his money from book sales. It is a self generating system, a mirage that keeps tricking people over and over again to lose their money and stay poor. Sadly the only way out for people to wake up, but we're 7000 years in and that hasn't happened yet.
Suppose we lived in a world where every contact with other people came out absolutely fair and quits. It would mean that the result of every contact was the same as before the contact, and we would die with exactly the same value we were born with. All that would happen in our life is that we would exchange things of equal value and no one would win or lose. It would make exchange pointless.
Our desires are always changing, and two people never want the same thing. It means that given any commodity C each person will want it a different amount. In a system of barter anyone who wanted C more than the owner could get it if they had a commodity D which the owner of C wanted more than C. In a very busy market, with people shopping around for best deals, exchanges will approach fairness.
Exchange thus happens because there is no absolute value, and as a result each exchange, while in absolute a loss of a commodity to the seller and the gain to the buyer, results in sum greater than the parts because each person now has something they want more than before.
So one way to wealth is to surround our self with what we want, and that is easiest if we want what no one else wants, and hardest if we want what everyone else wants. We already argued there is no absolute value, so no fixed reason why anyone should want anything more than another. We run into the alternative problem tho that if no one wants something then no one is going to be bothered to make it, so there won't be any around. This is reflected in supply/demand prices. In a free market, if we can chose and we want to easiest life, it is best to want what is either coming into fashion early, or what has just gone out of fashion. But then we have the much more complex problem that fashion itself is desirable because it sets social status which is itself desirable. Anyway I digress.
If we were born with nothing however we wouldn't ever be able to trade. We can't make something out of nothing. We would need say a 1p piece to get started. Then buy a 1p sweet. Don't consume it but instead go somewhere where there are no sweets and sell its opportunity cost for 10p. Go back and get some more sweets and do again to get 100p. Then buy some water and go to a desert and sell it for £5 to someone thirsty and so we could go on. But we need to start somewhere. In this example we are also able bodied, must have some food and water to sustain us during the exercise, be free from persecution, must have air, a healthy enough environment, we need to have been born, we need time... there is a huge amount of wealth that we are born with that is often over looked.
So in being born we are already very wealthy. Now the point of writing this was just to question whether our wealth ever really changes. Do we die with the same wealth we were born with, and does anything really change?
Yes, our desires changes, and as a result we exchange different things. As a child we are reluctant to exchange time and prefer mucking around with friends. As we grow older we exchange more and more time in the market place to accumulate things, especially the commodities needed to have a family our self. Time of course is something everyone is born with, we can't buy that itself. But what we can do, if we are materially wealthy, is then exchanging our wealth for other people's time. So we don't walk to the shops anymore, we pay someone to do that for us and this frees up some time.
Traditionally people would say that wealth was created by hard work. But the ability and time to do this is something we were already given, a wealth we already had. We are exchanging those no longer needed qualities for things we want.
Now enter money. If we trade our personal wealth for accumulating money we can be sure that in the future we can exchange that money for things we want. If you couldn't spend money it would be worthless, so we don't want money itself but only what we can exchange for money.
The question is whether we can get more in the future than what we spent getting the money. An example would be accumulating money while able bodied to exchange when we are ill. We should over do this however: when ill what we really want is able bodied, but what did we do when able bodied if we only saved it for when we were ill. I guess there are other things we want other than able bodies, so when ill we can enjoy these at the expense of our "able bodiedness" that we spent on working.
So far it appears that indeed via trading we do generate more satisfaction. But unclear when we die whether there has actually been any change of value compared with what we were born with.
As usual I'm in a hurry to hammer out a seedling idea. To be developed in future...
Abstract
I will show here how distances between line of longitude increase much faster in the Un Azimuthal Flat Earth model than they do in the Spherical model. The distance data provided by Google is determined to derived by the same models presented here so cannot be used to distinguish which model is correct. However with Flat Earth providing results rapidly diverging from spherical this method provides a simple way to test which geometry the planet has. At the end of the post I expand the model to find distances in all direction. I know people living in Sydney who say the distance to Canberra takes around 2 hours to drive. 2 hours at 70mph is 224km. This is almost exactly the result expected by the round Earth model proposed and half that predicted by the Flat Earth model. There are a huge number of Flat Earth models, indeed any manifold geometry is a candidate, nevertheless none of them have the property that distances in all directions relate to reality. There is no known flat map where the distance from Sydney to Canberra looks the same as the distance from London to Sheffield. This is the problem to solve for Flat Earth theory. I then examine the philosophy of Flat Earth and provide a spiritual interpretation of its desire for boundlessness with brief guidance how to enter the infinite. A bit of theory
To be clear about words first. A line of Longitude is a N/S line, or a spoke in the wheel radiating out from the North Pole. A line of Latitude is an E/W line, or a rim of the wheel or clock, going around the North Pole. A line of Longitude joins points of the same Longitude, and a line of Latitude joins points of the same Latitude. Perhaps confusingly then the Latitude of a point is its position along a spoke of Longitude, and the Longitude is the position around the clock of Latitude. There is actually a deep relationship between the ideas of clock faces and latitude.
The key thing to note is that the lines of latitude expand evenly from the North Pole. Compare this with looking down on a Sphere where they bunch up as you approach the equator.
Using the first model (there are others) we divide the diameter of the Flat Earth Disk up evenly. If the disk has a diameter D, then the diameter of a circle of latitude (φ) with φ ranging evenly, North Pole to Disk Edge, π/2 >= φ >= -π/2 is:
dl = D (π/2 - φ )/π [1.0]
This gives the distance around a line of latitude:
π dl = D (π/2 - φ) [1.1]
The distance of a point from the North Pole (rl) in both models in terms of latitude is half the diameter in 1.0:
rl = dl/2 = D/2 (π/2 - φ)/π = r (π/2 - φ) [1.2]
Where D/2π = r [1.3]
In the spherical model r is interpreted at the radius of the sphere. In the disk r is a mysterious constant.
Compare this with the distance around a line of latitude (φ) on a sphere with radius r:
2 π r Cos(φ) [2.0]
The significance of this is when comparing the distance between places of same latitude (E-W of each other) at different longitudes (distance from the North Pole).
Cardiff and London are E-W of each other and selected here on Mapfrappe. They have latitude (φ) = 51.5 degrees (0.90 radians)
If we move to latitude (φ) 5.6 degrees (0.10 radian), the longitude of London runs through Accra in Ghana, and the longitude of Cardiff runs through Ayame.
On the maps these lines of Longitude, the spokes, are straight and radiate out from the North Pole as seen in the flat earth map. The Flat Earth map is actually best to visualise this, but the same logic applies to the sphere model too. It doesn't matter how long the spokes are, the distance along the wheel rim between two spokes is always the same proportion of the whole wheel (the line of latitude). This means that if we know the distance between spokes along the rim of one wheel size we can work out the distance between the same spokes on a wheel of any size by scaling or stretching the wheel. Latitude (φ) is the measure of wheel size in these models. So if we know the distance between two points on the rim (same latitude) then using φ we can scale the wheel to any latitude and find the distance between points on those spokes (longitude). What is interesting is that the stretch is different depending upon what model you use for the shape of Earth. So by looking at which stretch works best we can determines the best model for the Earth shape.
2 π R Cos(φ2) / [2 π R Cos(φ1)] = Cos(φ2) / Cos(φ1) [4.0]
So on Flat Earth this stretch from London/Cardiff to Accra/Ayame is (from 3.0):
(π/2 - 0.1) / (π/2 - 0.9) = 2.2
So we'd expect the distance between Accra and Ayame to be 2.2x the distance between London and Cardiff on the flat disk.
However on the sphere the stretch from the other equation is (from 4.0):
Cos(0.1) / Cos(0.9) = 1.6
On a sphere we'd expect the distance between Accra and Ayame to be 1.6 times the distance between London and Cardiff.
So which is it? The general model at the end uncovers the fact that Google Maps simply uses the same spherical equations presented here to estimate distances. But if Google Maps was giving false distance data we would expect it to have been noted by now. It is interesting that to my knowledge people aren't finding their car milometers to be hugely different from their google map journeys.
I also note that Flat Earth is giving a distance between Accra to Ayame to be 37.5% larger than Spherical Earth. This is a huge distance. Even if someone in the area drives the journey with only 25% accuracy it would give us enough precision to test which model is more accurate.
In the general model below it is shown that the divergence grows so that Sydney to Adelaide is twice as far in the Disk model as Globe model with the Globe model fitting the data perfectly.
Summary
The idea of comparing distances between lines of longitude down the latitudes can be used anywhere on Earth and if you have colleagues in places far away North or South of you then you can test this directly. Ideally a long straight road between lines of longitude that you can drive to actually measure the distance would be perfect. You can also uncouple lines of longitude and compare distances at different latitudes since a degree of distance at any latitude is the same around the line of latitude. This way we can test this even with people not North or South of you.
Obviously for places close you need to measure very accurately. In UK lines of longitude passing through North Scotland and Devon provide only 3.5% difference between Flat Earth and Sphere models so to test in UK measurement would need to be very accurate.
Conclusion
It would appear that this method reveals a big difference between flat earth and spherical models which can be used to test which is more accurate. I note that the existing online Google Maps resource uses the same distance model presented at the end of the article. To my knowledge this hasn't been challenged over its accuracy as is used every day my millions. However no solid data has been found at this stage to determine once and for all which model is more suitable for modelling the Earth's geometry.
After Thought
So what is the reason for Flat Earth model anyway? Originally it was the most obvious description of the world. When we walk a distance we feel it is flat. Curvature only becomes relevant when we cover great distances and for most people this didn't happen until the 20th Century. I have met someone in Cumbria whose mother was the first woman ever in her family to leave the valley in which the family lived. Curvature was not an issue for this family.
Today Flat Earth has a new meaning. It represents the idea of unlimited borders. The problem with the globe model is that space is limited and horizons all point home. In Flat Earth looking out means we look to the infinite. A huge irony is that Flat Earth tends to reject the infinite cosmos model and thinks that looking up is into a finite dome. But ultimately the sensibility is to reject what we have been told and to experience the wonder of the unknown and infinite. In a world where we are overwhelmed by information, authorities, experts and the known such a simple system of rejection once again gives us control and meaning in the world.
It is however a dangerous move to gain freedom so crudely. Others can simply just reject the authority of the flat earth and so it goes on endlessly.
True freedom is quite different. What people don't realise is that they are already entirely free and that our mind is always infinite and unbounded already. There is simply nothing to do, nothing to believe, no faction to side with and no one to join. We are already entirely free.
Such freedom is bewildering to those brought up on a diet of knowledge and fixed beliefs. Flat Earth is ironically exactly the type of prison that those adopting flat earth think they are escaping. NASA is the governmental authority fooling the people, we reject such thought control and seek freedom in the dogma of the flat earth. Of course really this becomes just another dogma. Why pin your existence on any silly thought or model? This is not football and no one cares which team wins.
To annihilate the overload of information, to escape the talking heads and authorities which try to enforce themselves upon our world, to enter the infinite and unbounded existence of True Mind we must first learn to abide in the infinite space of the Present Moment, undistracted and unburdened by worries of Truth or Authority or Freedom. We need only focus on what nourishes our soul and what is sees as Good, with no worries of politics, persuasion or correctness at all.
See this random video. Let go of thoughts of what the Sun is, what shape the sky, why the waves ripple, what is creating those sounds these are all only thoughts. Instead observe only what is in the senses: what does the bright orb look like, there is a shimmering line before it, each ripple is different from any ripple ever produced in this universe, look at each one and see its uniqueness, watch them move toward the shore and how each one arrives differently, there are no ripples in the sky: examine what is in the eyes. There are sounds! What are they like? Soft sounds, long sounds, pulsing sounds or smooth sounds, simply examine what is in the ears. When more practiced examine and see what is in the feelings and even what is in the thoughts (which obviously right now is thoughts about what is being written). In the video thoughts may be "near thoughts" like "what is making that sound?" and we think "bird": watch that thought happen. We didn't have that thought a minute ago, now we do, sometime soon we won't have it again. It has happened, it is just a thought - it isn't a bird! That sound is not actually a bird, it is a sound, birds don't exist in our ears, we think "bird" separately in our thoughts (this is very advanced contemplation, will take time). Or there are "far thoughts" like I don't have time to do this or I have money worries, watch those thought happen and come back to the senses. There may be "far feelings" like anxieties or worries: observe these exactly the same, they come and go. Worries have a habit of attracting thoughts, stirring up a thousand reasons and thoughts why we need take action and get rid of the worries. Just for now, free yourself and ignore them, with the promise to attend to them tomorrow or later, if necessary be reckless with all worries, be Jesus dying on the Cross, that's as bad as it gets, how bad really are our feelings or concerns? Eventually simply let thoughts and feelings drift away from centre stage and allow time happening in our senses to be there instead. We are only limited and bounded by these thoughts and feelings; they are only our tool for encountering the world, like flat/round earth they are mental models but are not our master. Let them all go now. Enter instead gradually and imperceptibly into the sense world that surrounds and fills you. Then you have taken the first step to gaining mastery of your mind and seeing infinite freedom.
The General Model
The maths above is made easy by considering only distances along lines of latitude. You can however develop the model for distances in any direction.
Given two co-ordinates: (φ1, λ1) and (φ2, λ2) the distance between them on the Disk can be calculated using 1.2 to convert the coordinates to radii with the absolute difference in longitude (Δλ) and the Cosine Rule thus:
[5.0]
On the sphere given two co-ordinates: (φ1, λ1) and (φ2, λ2) the angle between them along a Great Circle is given by:
[6.0]
the distance (d) between the points on a sphere with radius r is then given by:
[6.1]
(Wiki notes that this is formula is not accurate for points close together and better, but more complex, functions are available.)
Using Mathematica equation 5.0 reduces to equation 3.0 for two pairs of points with same radius each and sharing Δλ. Curiously 7.1 only approximates to 4.0 under the same conditions. (Why is this?)
We know that there is a big divergence between the models approaching the equator from above. The distance from Ayame, Cote d'Ivoire (5.604497° N, 3.156939° W) to Accra (5.6037° N, 0.1870° W) using these equations is:
Using 5.0 for disk model: 486.39km
Using 6.1 for sphere model: 328.67km
Google Maps gives a distance of 328.67 km. So we have established that Google Maps is using the same equation presented here.
What we have established is that the disk model gives a result 48% more than the sphere model. This is easy to test. It appears we just need data from countries far from the North Pole to distinguish which model is best.
The divergence grows going into the Southern Hemisphere.
The distance between Sydney and Canberra is given by Google as 228.564km.
Obviously the Sphere model presented here agrees. But the Disk model gives a distance of: 468.023km.
Google Directions give the driving distance as 259km. It is easy to test whether cars can drive between Canberra and Sydney in under 400km and distinguish which model is better.
This is readapted from Hegel's brilliant one line dismissal of Kant's Critique of Pure Reason. You cannot use a telescope to find out if the telescope you are using is working (meaning if reason or the telescope are faulty then your investigation and results will be faulty... except I note at "fixed points of reason" where reason makes no difference and the fractals that even a faulty reason under self investigation will make) Anyway...
Once upon a time, when a boy was too young to remember, he was given a wonderful telescope which he strapped to his head and spent his days marvelling at the spectacular universe above his head. As he grew up he became quite proud of his miraculous ability to see deep into space and report on things that no other could see. Time came when he wanted to know how he had this special ability, different from other people, and he came to investigate it. He searching far and wide in the universe but could see nothing that indicated why he should be so special. Eventually one day he asked someone why he had this special ability and was told about the telescope still strapped to his head. Taking it off he suddenly realised that the ability was not his own but depended upon this device now in his hands. He put it in a box under his bed and walked away seeing, for the first time in his life, the world without the telescope in the way. In time he became an astronomer and returning to the box under the bed he took out the telescope reattached it and continued to study the stars. Nothing had changed except now he knew that the special ability was not his own but was simply there because he wore a telescope.
So time passed and the boy, now a man, had an accident with the telescope which damaged his eyes and he went blind. He asked a friend to put the telescope safe in a box and his eyes were removed and preserved in case of medical developments. Now he was living in the world without a telescope in the way but also for the first time without eyes in the way. Remembering taking off the telescope he saw that the miraculous power of the sight that he had enjoyed was not his own but depended on the organs that had once been set in his head. In time a wonderful technology was developed which restored his sight and as before he was able to continue his life as an astronomer, but now in the knowledge that this ability of sight was not his own but came with his eyes.
As the man became an old man he learned the sad news that his own son had been killed in an accident at a major telescope site--his son too had become an astronomer. He was once again thrown back into the life without children that he had lived before. Unlike with the telescope and his eyes he could remember this time and had always cherished the joy of having a son. But, the event brought him to question what will happen when he loses his own life to death. He would never look his son in the eye again for those eyes were gone and when he died his eyes would go again too. But he lost far more of his son that that: he could never talk to him again, hear his thoughts, go to the football with him, share his life's successes and failures: everything was gone. And likewise he realised when he died all this things would be gone too.
But a curious insight stirred in him. When he lost the telescope he realised it was not his, and he could go on afterwards free from it. And when he lost his eyes he realised they too were not his own, and he could go on afterwards free from them. And when he lost his son, he was reminded what he already knew, that the son was not his own really, and now they could go on afterwards free from each other. Now conceiving his own death he realised it was not his own either, and he realised at death he would go on free from even his own life and existence.
But like with the telescope, and his eyes, once he had realised they were not his own he took them up again and continued as before, but free and safe in the knowledge that he was only wearing all these things just for a time.
And so in a lazy moment he might think: so what does happen after death? What is "going on after death" like? I went on when I lost my eyes, how might I go on when I have lost my life?*** And then he remembers these thoughts are not his own, he only wears them for a while and he just lets them go. At that moment he knows what going on after death is like.
And so the man became quite famous for his ability to see things that other could not see, peering deep into the nature of the world and of life and death. But now the man was not proud of his ability for he saw most clearly that it was not his own and he would not possess it forever.
*** In more detailed discussions he would examine the experience of first removing that telescope. Some things changed and some did not. He could no longer see tiny things up close, but his consciousness of what he saw was just as clearly without the telescope. Likewise when he lost his eyes he could no longer see things at all, but his consciousness was still just as clear as before. When we sleep our consciousness is lost, but we are not dead. There is a place (many in fact) before death where we live without consciousness. When I die... the man realises understanding life without life is like his younger self not seeing the telescope.