Thursday, March 21, 2024

Lisbon

Lisbon, the capital and largest city of Portugal, is one of the oldest cities in the world. It is on the far west coast of Europe and built around a natural harbor. It was an important place for the Celts, and then anyone who travelled by ship, first the Phoenicians and then the Romans.

The Moors, Moslems from North Africa, ruled Portugal as well as Spain. Lisbon was conquered by the Moors in 711. But Portugal was completely liberated by Catholics about 250 years before Spain. Lisbon was liberated, after a siege, in 1147. Neighboring Spain was not entirely liberated until 1492.

The Second Crusade was directed toward the Holy Land but also helped to liberate Portugal. The combat to liberate Portugal was not just in Portugal itself, Catholic forces also landed in the Moors' homeland of north Africa.

Lisbon became the capital of Portugal in 1225. What is so important about Lisbon is that it is the port from which the Age of Discovery began, the colonization of most of the world by Europeans. For good or bad, this completely changed the world.

The Age of Discovery was about sailing out into the Atlantic in search of new lands. It shifted the balance of the region away from the Mediterranean. The best-known figure from the Age of Discovery is Christopher Columbus. But it actually began with Portugal, with ships sailing from Lisbon. Columbus, with the idea of reaching the east by sailing west, approached Portugal and England with his idea, but was declined before convincing the king and queen of Spain.

The Age of Discovery began, in the late Fifteenth Century, with the Portuguese discovery of the Azores Islands and Madeira, all in the Atlantic. Vasco Da Gama then took the monumental step of sailing around Africa to reach India. The Portuguese were the first Europeans to establish contact with Japan. Another captain, named Cabral, crossed the Atlantic and landed on the coast of Brazil. It is uncertain whether Europeans established this first contact with South America by intentional cross-Atlantic exploration, or whether the ship was on the way around Africa and was knocked off course by a storm.

Portugal would come to rule a vast empire. The country was much smaller than the other colonial powers, Spain, France and, England, but it's empire was just as vast as theirs. We think of South America as speaking Spanish but did you know that more people in South America speak Portuguese? That is because Brazil speaks Portuguese and has nearly half the population of the continent and countries on the north coast of the continent, Guyana and Suriname, do not speak Spanish.

The first captain to sail completely around the world, Ferdinand Magellan, was Portuguese but was sailing for Spain. He stopped, and was killed, in the Philippines, and this is why the Philippines was ruled by Spain for over three hundred years. It was named for Philip II of Spain, who would also rule Portugal in a dynastic union but Portugal wanted to be a separate country and achieved that with the Portuguese Restoration War, ending in 1668.

The House of Aviz was the Portuguese branch of the House of Burgundy, until 1580 when there was a succession crisis due to the lack of a male heir. That was when Philip II of Spain gained control and the Philippine Dynasty, named for three successive kings named Philip, was the Spanish branch of the Habsburgs. This dynastic union of Spain and Portugal was known as the Iberian Union, as Spain and Portugal share the Iberian Peninsula.

This history would later be repeated in neighboring Spain, in the War of the Spanish Succession. The Spanish branch of the House of Habsburg would die out. The war for control would be between the Austrian branch of the Habsburgs and the French House of Bourbon. The Bourbons would win and the House of Bourbon still reigns over Spain today.

After Portugal gained separation from Spain in the Portuguese Restoration War, it was ruled by the House of Braganza, or Brigantine Dynasty, which lasted until the 1910 revolution in Portugal. In 1908 King Carlos I was assassinated and the Republican Party overthrew the monarchy two years later. What is known as the First Portuguese Republic then lasted from 1910 until a May 1926 military coup.

The First Republic had been unstable but, after it's overthrow in 1926, was followed by 48 years of authoritarian rule. First what was called the "National Dictatorship" and then the "New State", or the Second Republic. The leader associated with this time is Antonio Salazar. This was finally ended by the "Carnation Revolution" of 1974, bringing in the Portuguese Third Republic. The Carnation Revolution was nearly bloodless.

Notice that the modern histories of France, Spain and, Portugal all go by these numbered republics. A new republic basically means a new constitution.

The 16th Century Tower of Belem is the gateway to Lisbon from the sea. The first three images are from Google Earth and Street View.




There are multiple scenes following. To see the scenes, after the first one, you must first click the up arrow ^, before you can move on to the next scene by clicking the right or forward arrow, >,  After clicking the up arrow, you can then hide the previews of successive scenes, if you wish.

https://www.google.com/maps/@38.6916514,-9.2158698,2a,75y,215.59h,90t/data=!3m7!1e1!3m5!1sfDKIO5RXJnUAAAQvxgKVWw!2e0!6s%2F%2Fgeo0.ggpht.com%2Fcbk%3Fpanoid%3DfDKIO5RXJnUAAAQvxgKVWw%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D209.19179%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

The New Testament states that St. Paul had plans to one day visit Spain, although he apparently never got around to it. There was no nation of Spain at the time so this means the Iberian Peninsula. Could St. Paul have ever imagined that some day there would be a city called Lisbon. Ships sailing from it's harbor would establish a city on the other side of the world. That city would become the largest city in both the western hemisphere and the southern hemisphere and it would be named for St. Paul, Sao Paulo.

Alfama is the oldest part of Lisbon. This dates to the time of the Moors and mostly survived the great earthquake of 1755. The square is called Commerce Square. The arch entering the square is in commemoration of recovery from the earthquake. Lisbon's cathedral is nearby. The following five scenes are from Google Earth and Street View. The first two are of Lisbon Cathedral.









https://www.google.com/maps/@38.7140662,-9.1334887,2a,75y,296.85h,90t/data=!3m7!1e1!3m5!1sW4iM0DaGxadJQ05p5EITLg!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3DW4iM0DaGxadJQ05p5EITLg%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D302.9801%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

The following scenes begin on Avenida de Liberdade. The nearby park is Eduardo VII Park. The first two images are from Google Earth. The monument is of the Marquis of Pombal, the prime minister who led the recovery from the 1755 earthquake.

Britain's King Edward VII visited Portugal in 1902. Did you know that the oldest military alliance that is still in force is that signed between England and Portugal in 1386? Of the four major colonial powers, England, France, Spain and, Portugal, England and Portugal are the only two that never fought each other. During imperial days, the two empires tacitly supported each other with the ships of one being able to find a friendly port in the other.



https://www.google.com/maps/@38.7239741,-9.1487943,3a,75y,232.26h,90t/data=!3m7!1e1!3m5!1s0afmWXgSWTeU-nGM3jSHhQ!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3D0afmWXgSWTeU-nGM3jSHhQ%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D225.22612%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

But Lisbon isn't all history, it is also a brilliant modern city.

https://www.google.com/maps/@38.7697109,-9.0953878,3a,75y,80h,100t/data=!3m7!1e1!3m5!1sbmJZ9_1HiMXNqN-LSoqprQ!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3DbmJZ9_1HiMXNqN-LSoqprQ%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D80%26pitch%3D-10%26thumbfov%3D100!7i13312!8i6656

This is an "everyday" area of Lisbon, some distance from the city center.

https://www.google.com/maps/@38.753349,-9.142843,3a,75y,71.02h,90t/data=!3m7!1e1!3m5!1sE22tpO6QcuYIT-OLbnN2sA!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3DE22tpO6QcuYIT-OLbnN2sA%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D66.196014%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

Across the Tagus River from Lisbon is the Sanctuary of Christ the King. This is very reminiscent of the Christ the Redeemer statue in Portugal's former colony of Brazil. The suspension bridge is the 25th of April Bridge, renamed after the Carnation Revolution of 1974 for the date the revolution began. It is Portugal's version of the Golden Gate Bridge. The first two images are from Google Earth.



https://www.google.com/maps/@38.6793386,-9.1712985,3a,75y,36.57h,90t/data=!3m8!1e1!3m6!1sAF1QipP3T0LU24iPqr-F79vLxVXyQLBwYZf1BfwtwsI!2e10!3e11!6shttps:%2F%2Flh5.googleusercontent.com%2Fp%2FAF1QipP3T0LU24iPqr-F79vLxVXyQLBwYZf1BfwtwsI%3Dw203-h100-k-no-pi-0-ya229.07344-ro0-fo100!7i8704!8i4352

South of Lisbon is Setubal, another ancient city that was important to the Moors. The Moors once ruled both Spain and Portugal. But Portugal was liberated at least 250 years before Spain was completely liberated. So while there are more Moorish structures in Spain there is also some in Portugal. The first two images are from Google Earth.




https://www.google.com/maps/@38.5247758,-8.8898076,3a,75y,155.85h,90t/data=!3m8!1e1!3m6!1sAF1QipO_D9Uq6j4NL747HpCxuUSZvcJMP1myzUWZGiiM!2e10!3e11!6shttps:%2F%2Flh5.googleusercontent.com%2Fp%2FAF1QipO_D9Uq6j4NL747HpCxuUSZvcJMP1myzUWZGiiM%3Dw203-h100-k-no-pi-0-ya160.6677-ro-0-fo100!7i5660!8i2830

Sintra is a very historic city along what is known as Portugal's Riviera. Unlike the French, Spanish and Italian Rivieras. the Portuguese Riviera does not face the Mediterranean. It faces the Atlantic Ocean, as Portugal has no Mediterranean coast. Wealthy people built extravagant homes along the Portuguese Riviera, I suppose it could be considered as Portugal's "Millionaires' Row".

https://www.google.com/maps/@38.8018452,-9.3815165,2a,75y,164.31h,90t/data=!3m7!1e1!3m5!1s21CgUMu7IEtswipVwsat4w!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3D21CgUMu7IEtswipVwsat4w%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D167.6538%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

These eight images, from Google Street View, are of Pena National Palace, in Sintra.









Posted by at No comments:

Portugal's Drug Policy

Like every other western country Portugal has a drug problem. What has gotten so much attention is it's way of handling illegal drugs. Almost everyone else considers a harsh policy of simply putting everyone in jail that has anything to do with drugs as definitely not the way to handle it.

The trend today, which Portugal has helped to pioneer, is to treat common drug users as people who need help. The dealers are the real criminals.

Decriminalizing drugs like marijuana does not necessarily mean endorsing drug use. It is simply choosing the lesser of two evils. The fact is that people are going to do drugs anyway. The best thing to do, for society as a whole, is to provide a safe place and offer clean needles. Life often consists of choosing the lesser of two evils, and this is an ideal example.

Harsh punishment does little to deter crime. Someone who is using or addicted to drugs probably isn't spending a lot of time thinking about the legal consequences. The reason for this new policy, in countries like Portugal, is that simply putting people in jail definitely isn't working.

Also remember that being "wrong" and being "illegal" are two different things. The purpose of the law is to facilitate the orderly functioning of society. Just because something is wrong doesn't necessarily mean that it should be illegal. It is completely unrealistic for everything that could be considered as wrong to also be illegal.

Let's review the investigation of "MARIJUANA", in the compound posting "Investigations", December 2018.M

MARIJUANA

All right, how much sense does this make? 

We know how destructive alcohol can be. It damages the health. Many deaths are directly attributable to the destructive health effects of alcohol. Countless people are killed and injured in accidents that are directly attributable to alcohol. 

Then there is the social effects of alcohol. It's overuse has destroyed endless marriages and families and careers.

Next, we have cigarettes. We do not even need to go into their destructive effects here. Cigarettes are not nicknamed "cancer sticks" for nothing. But cancer is only the beginning of their health effects. We used to be shown films in school of people dying in hospitals, pleading with young people not to start smoking.

But yet alcohol and cigarettes are perfectly legal in the western countries. 

What about marijuana (cannabis)? It's health effects are generally considered as not as bad as cigarettes and it's intoxicating effect no worse than alcohol. It is also believed to be less addictive than alcohol.

So why, at least until a relatively recent relaxing of the rules in some countries, is alcohol and cigarettes acceptable while possession or use of marijuana gets one thrown in jail?

How can marijuana be treated so differently from alcohol and cigarettes when the general opinion is that it is no worse than either?

The factor that does make marijuana different from alcohol is the way it is produced. Tobacco is grown on large plantations and the commercial production of both alcohol and cigarettes are industrial processes. Marijuana, in stark contrast, can be grown on virtually any plot of ground, or even inside under special lamps.

Have you ever stopped to think that this might be why marijuana is treated so differently by the law? Since I have no personal interest in any of the three, maybe I can give an unbiased opinion.

Manufacturers of alcohol and cigarettes are industrialists who will obviously have far more political power than anyone who grows marijuana. In America, legislation against marijuana seems to date from 1906. This was during the Gilded Age when capitalists had a tremendous amount of power.

Could it be that manufacturers of alcohol and cigarettes used their political power to eliminate marijuana as a competitor by making sure that it was illegal?

But then what about the countless thousands of people who have spent time in prison for marijuana? Was it all really because of Robber Baron tactics from the Gilded Age? Wouldn't that make it just about the greatest travesty of justice that the world has ever seen?
Posted by at No comments:

The Compass Meridian

To map the world it was necessary to first establish a system of latitude and longitude. Measuring latitude from any given point on earth is fairly straightforward. All that is necessary is to measure the angular altitude of the north star above a flat horizon, or the equivalent point in the southern hemisphere.

Measuring longitude is more difficult. We need to begin with a reference point. Latitude has natural reference points because the earth is divided in half by the equator as it revolves around the polar axis. But with longitude there are no such natural reference points so we have to define one. The reference line that we have defined to measure longitude by is the Prime Meridian, which passes through London.

Measurement of longitude became practical when John Harrison invented an accurate clock that didn't rely on a pendulum. Clocks that were based on pendulums were considered as unreliable as sea because the pitching and rolling of the ship in rough water might affect the timing of the pendulum.

The new clock could be set to Greenwich Mean Time (GMT). Wherever a ship was it could measure the local solar time by means such as a sundial. The difference between the two times reveals the ship's longitude. Since the earth rotates every 24 hours it rotates 15 degrees every hour so that four minutes equals one degree of longitude. If the local time is ahead of Greenwich Mean Time it means the ship is east of the Prime Meridian, since the earth rotates eastward. If the local time is behind Greenwich Mean Time it means west of the Prime Meridian.

There has to be some line of longitude where one day ends and the next begins. By convention it is the International Date Line, at 180 degrees longitude on the opposite side of the world from the Prime Meridian. The International Date Line is fortunately in the middle of the Pacific Ocean. It diverts so that it does not cross any land because it would be very inconvenient to have one's home and work or school in different days.

What I want to point out is another way that longitude might have been measured and I think it could have been done much earlier than waiting for the invention of an accurate clock that didn't rely on a pendulum. It could have been accomplished by using a magnetic compass. The ordinary magnetic compass seems to have faded into history but I think it still has a lot of possibility left. I explained one such use as a measurement tool in the compound posting "Measurement", September 2021, section 7) A VERY USEFUL TOOL.

The earth is a magnet, which is the basis of the compass, but magnetic north is not exactly the same thing as geographic north. The north star is very close to the north celestial pole and determining direction by the stars is more accurate than by a compass. We could have used this difference to our advantage.

The bottom of the yellow line in the following image from Google Earth is magnetic north and the top of the line is geographic north. 

The red dot is on Greenland. The yellow dot is on Quebec. The green dot is on Nunavut. The blue dot is on Alaska. The purple dot is on Siberia, and the white dot is on Scandinavia.

The span of the line, representing the difference between magnetic and geographic north poles may not seem to make a big difference. But if we could build a large magnetic compass, so that the differences in readings could be easily discerned, this would have enabled a mapping of the world long before it was done by means of a clock, as described above. A reading of north would be taken by the stars, and then by the compass. The difference between the two would reveal the longitude.

Instead of the Prime Meridian the line shown above would be extended in both directions as the Compass Meridian. Only when the two measurements were exactly the same would we be on the Compass Meridian. The greater the difference between the two measurements the further we would be from the Compass Meridian, in terms of longitude. The difference would reach a peak at the meridian line 90 degrees from the Compass Meridian and then begin to decrease.

Whether the Compass Error, as we could call it, was to the east or west of the star measurement would tell us whether we were east or west of the Compass Meridian. If magnetic north was showing as east of the stellar measurement it would mean that we were west of the Compass Meridian, and vice versa.

It is true that the magnetic poles are moving towards the geographic poles but that is not taking place very quickly and it wouldn't have interfered in the mapping of the world. The mapping of the world would be essential. It would be of little use to know exactly where we were, without an accurate map. But compasses would have made this possible much earlier than with clocks.

Posted by at No comments:

A Reminder About Kings

Vladimir Putin won an overwhelming victory in an election that the west is skeptical of. But let's remember what we saw in the posting "The Theory Of Kings", April 2022.

We think that monarchy has mostly gone out of style in the modern era, but nothing could be further from the truth. Kings are with us as much as ever, we usually just don't call them kings anymore. Monarchy has been the way societies have been governed for thousands of years and drafting a constitution isn't going to change that, except maybe by name.

Vladimir Putin is as much a king as the tsars. He just doesn't wear a crown. Kings in today's world tend to stick together, or at least not criticize each other much. This is why Vladimir Putin gets along with Donald Trump and Viktor Orban. Democracy hasn't worked out as well as was hoped and one reason is that kings are gone in name only.

We can see in that posting why having a constitutional monarchy is a good idea. A country that doesn't have a king, even a tame constitutional monarch, is much more likely to end up being ruled by someone who acts like a king. It is completely unrealistic to think that countries are going to be ruled by kings for thousands of years and then they are going to go away overnight.

The recent issue of photographs of the British Royal Family has been portrayed in the news as a crisis. But it really means that they are very important. Everything that the Royal Family does is scrutinized up and down. When you stop to think about it this is a tremendous compliment. When everyone pays attention to someone, whether that attention is negative or positive, it means that they have something that is worth paying attention to.

Posted by at No comments:

Thursday, March 14, 2024

Technology Reminders

Just a few reminders about technology.

We have gotten really dependent on GPS but the satellites are quite vulnerable. For one thing there has not been a really major solar storm since the advent of the GPS system, like the one in 1859, and it remains to be seen what will happen to the satellites if there was. The satellites could also be shot down. There is a potential low-tech supplement to the GPS system. Wherever humans settle there will be telephone poles. Why couldn't we just put a number on every telephone pole so that anyone could determine their location just by reading the number on the nearest telephone pole?

If a phone can include a flashlight then why couldn't it include a projector? Wouldn't it be good to be able to project the screen of your phone onto a wall, preferably in a darkened room, to show to a group of people?

A language has around ten thousand words that are in common everyday use. Why not assign each word a four digit code from 0000 to 9999, and display the codes on signs? When you were in another country you would only have to enter the codes into an app to get the words translated. This would only be for basic communication and it wouldn't be necessary to convey fine shades of meaning.

Complex wording cannot be effectively translated word for word from one language into another. This is because grammar and syntax is different and parts of speech may not be arranged in the same order. It is necessary to translate by sentences rather than by words, but there are many millions of possible sentences. Why not put computers to work collecting, categorizing and, numbering sentences? That would not only make translation between languages simple but would greatly compress the storage and transmission of data, it would only be necessary to send the numbers rather than the words.

The standard alphabetical order is ABCDE... When typewriters came along a new order was developed so that the typist could most easily reach the letters that were most used. This is the QWERTY... alphabetical order. But now far more people use phones than keyboards. Unlike on a keyboard someone texting on a phone uses only one hand. Why not have a new order with the letters that are most used, which would include the vowels, to the right side, since most people are right handed and hold the phone in their left hand, or text with their thumb while holding the phone in their right hand? The QWERTY order is of no use while texting, it would be better just to have the ABCDE... order.

Remember how detrimental traffic lights are. How much time and fuel is wasted, and how much damage done to the environment, while waiting for traffic lights? How many times have you seen a line of cars at a red light and no cars in the other direction with the green light? There has got to be a better way than this. With all the talk about "smart" technology, traffic lights are about as dumb as it gets. This is an example of what I refer to being "technically forward but system backward". When a new technology emerges a system of organization is set up. Progress is made in improving the technology but what is really holding it back is that we are still using the same primitive system of organization.


Posted by at No comments:

New Trigonometric Functions

Considering all the attention that the moon is getting with the upcoming eclipse this would be a good time to review the possibility of adding new trigonometric functions. I wrote this years ago, in the early days of this blog, but have added more to it including diagrams.

As you may know, the branch of mathematics known as trigonometry deals with triangles. Specifically, right triangles. That is, a triangle containing one 90 degree angle. The internal angles of a triangle always add up to 180 degrees. It is very useful for measurement of the world and universe around us.

Consider a straight line that we will call X. Now let's draw a line perpendicular to X and call it Y. From the point where X and Y meet, the origin, we can draw another line we will refer to as R, for radius. The line, R, can be drawn at any angle out from the origin from 0 degrees to 90 degrees. If it is drawn at 0 degrees, R will be one and the same as the line, X. If R is drawn at 90 degrees, R will be one and the same as the line, Y. If R is drawn at 45 degrees, it will divide the original angle XY into two equal angles.


We can also think of it as a square or rectangle. One side of the rectangle is the X side. The perpendicular side is the Y side. The line, R, is the diagonal line that could be drawn between opposite angles of the rectangle. You may notice that unless R is drawn at either 0 or 90 degrees, it must always be longer than either X or Y. If lines X and Y are equal then R will be 1.414 times as long as either, which is the square root of two.


The length of R, or the radius, as opposed to X and Y, depend on the relative lengths of X and Y. If these two axes are equal, R will form a 45 degree angle to X or Y in the origin. If X and Y in the rectangle are not equal, R will form an angle other than 45 degrees to X or Y.

In trigonometry, we define X as the horizontal axis and Y as the vertical axis. The angle of R is measured from the X axis and intersects the two axes at the origin. We have what we refer to as "trigonometric functions". These are the sine, cosine and, tangent. Each angle has a value for each of these three functions.

The sine is defined as the ratio Y/R for any given angle. The sine starts at zero at 0 degrees and goes to one at 90 degrees. The cosine is defined as the ratio X/R and does the opposite. It starts at one at 0 degrees and decreases to zero at 90 degrees. The tangent is defined as the ratio Y/X for a given angle. It starts at zero at 0 degrees, reaches one at 45 degrees and goes to infinity at 90 degrees.

There are the lesser-used so-called "Inverse Functions". The cosecant is the inverse of the sine. So, it is defined as the ratio R/Y. The secant is the inverse of the cosine and is defined as R/X. The cotangent is X/Y. You may notice that three of the six possible functions have the prefix co- in front of them. These co- functions are the ones whose values decrease as the angle from 0 degrees to 90 degrees increases. The trigonometric functions are based on a 90 degree angle. This, of course, makes sense because in our universe, the dimensions of space form 90 degree angles.

There are two additions that I wish to make to trigonometry. I have noticed that, as useful as trigonometry is, it could be even more useful. The trigonometry that we use now is merely the 90 degree set of functions. The nature of the space we inhabit brings two more sets of meaningful trigonometric functions into being that we are not using as of yet.

45 DEGREE TRIGONOMETRIC FUNCTION

The 90 degree set of functions will always remain the most useful simply because that is the angle at which the dimensions we inhabit intersect. But I have noticed two more useful trigonometric functions. If we multiply the sine by the cosine of an angle from 0 to 90 degrees, we find that the product starts at zero at 0 degrees, peaks at 0.5 at 45 degrees and, goes back to zero at 90 degrees. It would be more convenient to multiply the product by two to have it peak at one at 45 degrees.

What we thus obtain is a trigonometric function of a different set than the traditional functions. This new function is obtained by multiplying two of the original 90 degree functions but it is based on an angle of 45 degrees rather than 90 degrees. It may be true that the spatial dimensions of our universe are based upon an angle that we have defined as 90 degrees. But the nature of this space also causes a number of everyday situations to fit into a description based on a 45 degree function.

To give a few examples of the usefulness of the 45 degree trigonometric function, consider the following. You are at one corner of a rectangle, say an athletic field. You wish to go to the opposite corner of the rectangle. How much travel distance will you save, expressed as a ratio, by cutting diagonally across the field instead of going around the perimeter?

The answer depends on the ratio of the lengths of the two perpendicular sides of the rectangle. If the opposing sides are equal, the direction to the opposite corner will be 45 degrees and the efficiency of the savings will be at a maximum. We can express this efficiency as 1. If the rectangle has one side vastly longer than the perpendicular side so that it is a long thin strip, the efficiency will be much less than 1.

If one side could be infinitely long and the side perpendicular to it was infinitely short, the efficiency of a diagonal crossing would be 0. The efficiency of the diagonal cut, peaking at 1, can be expressed as a ratio, the short side divided by the long side. It is at a maximum when the two perpendicular sides are equal so that the angle of the diagonal is 45 degrees.

At that point, if we multiply the sine of the 45 degree angle by the cosine of the angle, we come up with 0.5 as a result. For convenience, again, we will obtain the Trigonometric Product by performing this multiplication and then multiplying it by a factor of two, so that it starts at 0 and peaks at 1 in the same way as we are accustomed to with the 90 degree functions. It peaks at .5 because 45 degrees is .5 of 90 degrees.

Let's look at another application of the 45 degree function, the Trigonometric Product. Suppose you have a stack of fence panels and you wish to enclose the maximum possible area with these panels. We could refer to the problem as area enclosed per given length of perimeter. The enclosed area would be at a maximum when the two perpendicular sides were equal and would be expressed as short side/long side.

An example of the Trigonometric Product for 45 degree functions is that of a cannon being fired into the air. It provides yet another example. It's range on level ground or sea would be at a maximum when the cannon was aimed equidistant between the horizontal and the vertical, in other words at 45 degrees. It's impact point on the ground could be brought closer than the maximum range by aiming it either higher or lower than 45 degrees. If the cannon were fired straight upward, at 90 degrees, it would theoretically fall right back down on its starting point and it's range would be zero.

If you have ever studied calculus, you may have noticed by now that this 45 degree function can replace quite a bit of calculus and is much simpler and easier. Calculus uses a graphed curve to find maxima and minima of the curve. We seek to find which point on a curve in calculus is the one in which the curve stops it's climb or descent and begins to move in the other direction.

You have seen how the peaking of the Trigonometric Product in the 45 degree function does the same thing. Just picture the zero to one and back to zero again of the function as a calculus graph. And to find such things as distance traveled, it is much easier to calculate the area of a right triangle of certain given angles than it is to find the area under a graph in calculus.

A simple way to illustrate this is multiplying two numbers that sum to a certain number, such as 10. The product will be greatest when the two numbers are equal, 5 x 5 = 25, in the same way that the Trigonometric Product is greatest when the sine and cosine are equal, at 45 degrees where the X and Y axes are equal.

180 DEGREE TRIGONOMETRIC FUNCTION

Now that we have the original 90 degree functions and the new 45 degree function, let's add the 180 degree function. Just as the distance across a rectangle involves the 45 degree function, the distance across a circle involves the 180 degree function.

Put another way, the familiar 90 degree functions involve pre-existing equal dimensions. The new 45 degree function involves pre-existing potentially unequal dimensions. The 180 degree functions involve the relationship between lines and pre-existing circles. In 90 degree functions, the radius, R, draws a circle. In the 180 degree function, the circle already exists as our "field" and we draw a chord across the circle.

We could call the original 90 degree functions, the "primary functions" and the new 45 and 180 degree functions, the "secondary functions". Imagine a circle, such as a circular park. Suppose you were at one point on the circle and wished to go to another point on the circle. How much efficiency would you gain by taking a shortcut directly across the circle to the destination point?

The answer would depend on how far ahead was your destination point and thus how much of the circle you were to cut out. The closer the destination point was to the present point, the less would be the efficiency of cutting straight across. The efficiency would be at a maximum if you were cutting directly across the circle to the point diametrically opposite you.

The proportional distance saving starting at A and going to B would be greater than going A to C in the following diagram. How much greater would be described by the 180 degree function. We can call this trigonometry because the proportional distance savings can be graphed as an isosceles triangle, starting at zero, reaching a peak midway and then going back to zero.


This could also be described in terms of a right triangle in that going from A to C and then to B, instead of directly from A to B, would be 1.414 times as far, with C being at a 90 degree angle from A or B. 1.414 is the square root of 2 and if the X and Y axes of a right triangle are equal in length then the length of the radius, or diagonal, will be 1.414 times either one. Remember the Pythagorean Theorem for a right triangle, A squared equals B squared plus C squared, with A being the diagonal or radius.

The 180 degree function involves the sine multiplied by the cosine that is the basis of the 45 degree function. If we cross a circle by it's diameter as illustrated by the red line in the following diagram and consider the starting point as zero degrees and the destination as 180 degrees, at any point on the diameter the distance to the outside of the circle, by a line perpendicular to the diameter as shown by the blue line in the following diagram, the distance to the outside of the circle is given as a proportion of the diameter by multiplying the sine and cosine of the angle, which is between the zero and 180 degrees.


So, we could say that the expression of efficiency begins at 0 if our destination point is the point immediately ahead of our present position and goes to 1 if our destination point is on the diametrically opposite side of the circle. In other words, 180 degrees ahead. Thus, we have a new trigonometric function that starts at zero for 0 degrees and goes to 1 for 180 degrees.

Such a straight line from a point on a circle, across the circle to another point on the circle, is known as a chord. In a chord, the angle of the outside circle occupied by the chord is equal to the sum of the instantaneous angles formed in the two places where the chord intersects the circle. Obviously, the instantaneous angles formed by chords cannot be over 90 degrees in a circle because 90 multiplied by two equals 180, which is the number of degrees cut off the circle by the longest possible chord, which is a diameter.

By "instantaneous", I mean the immediate angle at only the contact point of the circle. Suppose you had an infinitely large circle. The intersection of a diameter line or a chord would form a clearly measurable angle with the circle. The difference between a chord and a full diameter of a circle is that the instantaneous angles formed in a diameter line are perpendicular, 90 degrees, and in a chord are less than 90 degrees.

This is the basis of the 180 degree function. The instantaneous angle of the radius, R, and the circle thus formed is always 90 degrees in the 90 degree functions. In our new 180 degree function, it varies from 0 to 90 degrees. The instantaneous angle is equal to the angle between the diameter and the chord, which can range from 0 to 90 degrees. Or half the angle of the outside circle that is within the chord, which can range from 0 to 180 degrees.

The length of the radius, R, always stays the same in the 90 degree functions. However, the length of the chord can vary from 0 to 1 in the 180 degree functions. A length of 1 would, of course, be equivalent to a diameter of the circle. As the efficiency becomes greater, the length of the chord becomes longer until at an efficiency of 1, the chord has it's greatest diameter and has become a diameter of the circle. In terms of distance, the efficiency of cutting diametrically across a circle would be 2/pi or 1.57. Now for the actual formula for the 180 degree trigonometric function. It is the sine of the angle, from 0 to 180 degrees.

As we draw a chord of varying lengths across part of a circle from our starting point, we notice that the increase in the length of the chord is much greater from 0 to 90 degrees than it is from 90 to 180 degrees. If a diameter (180 degrees) is considered as having a value of 1, a chord will have a length of .707 when it covers 90 degrees of the circle. It will increase in length only .293 more as it goes from 90 to 180 degrees.

This is simply because as the length of the chord increases in the second quadrant (90-180 degrees), it is decreasing in the first quadrant from .707 when the chord is 90 degrees to 0.5 when it is 180 degrees. At 180 degrees, of course, the chord of 1 will consist of 0.5 length in each of the two quadrants.

This new function is useful whenever there are interior lines forming a chord or a diameter in a circle. For example, two planets in their orbits. Or a specific point on the earth's surface to a particular point on the moon's surface at a particular instant in time. Of course, the orbits of the planets tend to be ellipses instead of perfect circles.

But this function can be easily modified for two ellipses by defining the aphelion and perihelion (furthest and closest points of approach) in terms of relative distance and expressing the orbits as comprising 360 degrees. The difference in horizontal plane of two orbits can easily be expressed in terms of 90 degree trigonometric functions.

To summarize, the 45 degree function describes a radius from an origin consisting of a perpendicular x and y axes. The function equals the sine multiplied by the cosine of the angle of the radius from the horizontal x axis and then multiplied by 2. This function starts at zero at 0 degrees, peaks at 1 at 45 degrees and goes back to zero at 90 degrees. The 180 degree function describes a straight line between two points on the inside of a circle. It starts at zero for an infinitesimal chord and goes to 1 for a diameter, or 180 degree chord.

The efficiency as well as the length of a given chord from zero to the maximum (diametrical) value are both expressed by the same number from 0 to 1. The function is given by the sine of the angle. We could call these two new functions, the "Compound Functions". The sine, cosine and, tangent are the Primary Functions. The cosecant, secant and, cotangent are the Inverse Functions.

The new 180 degree function could possibly be called the "Planetary Function" because it is ideal for describing the directional relationship between two planets, one rotating inside the orbit of the other. Actually the best way to illustrate this function is the view of the moon from the earth. I believe this is also a better way to give an example of trigonometry than anything concerning the traditional 90 degree functions.

The area of the moon that is lighted, as seen from earth, is a function of the angular distance in the sky between the earth and the sun. This can only be described by my 180 degree function and not by any of the 90 degree functions. The proportion of the moon that appears illuminated to us starts at zero at new moon, goes to complete at full moon and then back to zero at new moon. The proportion of the moon that appears as lit from earth is defined as the ratio of Line A to Line B.


New moon is, of course, when the moon is in the same place in the sky as the sun and full moon is when the moon is 180 degrees opposite the sun. In the above diagram, N is where the moon is at new moon and F at full moon. Take the proportion of the moon that appears lit by the sun, multiply by 180 degrees and that will give the angular distance in the sky between moon and sun.

I believe at this point that there are no more sets of useful trigonometric functions to be found beyond the original 90 degree functions and the new 45 and 180 degree secondary functions that we see here. A meaningful 270 degree (3/4 of a circle) set of functions could not exist because it could not have an equivalent rise and drop and would be a repetition of existing functions. 360 degree functions would be linear and equivalent to 0 degree functions. We could actually define 0 degree trigonometric functions as those functions in only one dimension.

Thus when you measure a straight line or express a linear distance without using trigonometry, you are actually using what we might call the 0 degree function. Although the very definition of the word "trigonometry" means triangle and this is not possible in one dimension. We could, however, refer to measurement of a one-dimensional line as making use of the 0 degree function and say that trigonometry only applies to two-dimensional space.
Posted by at No comments:

Thursday, March 7, 2024

Eastern Spain, Andorra And, Monte Carlo

Eastern Spain is where the Catalan language, as well as Spanish, is widely spoken. Spain began on it's eastern coast, as it faced toward the Mediterranean and was settled by Phoenicians and then Romans. But later, when the Age of Discovery began, the emphasis moved to Spain's Atlantic coast as ships went to and from the western hemisphere.

Alicante is an ancient city on Spain's Mediterranean coast, that originated with the Phoenicians. The castle above the city, Santa Barbara, is from when Spain was ruled by the Moors. The first two scenes of the Castle of Santa Barbara are from Google Earth.



There are multiple scenes following. To see the scenes, after the first one, you must first click the up arrow ^, before you can move on to the next scene by clicking the right or forward arrow, >,  After clicking the up arrow, you can then hide the previews of successive scenes, if you wish.

https://www.google.com/maps/@38.3459147,-0.4902567,3a,75y,121.32h,90t/data=!3m7!1e1!3m5!1ssiubh4nG3QShGpfwmBZ64w!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3Dsiubh4nG3QShGpfwmBZ64w%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D127.80274%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

Valencia began as a Roman settlement. It was a walled city during the era of rule of Spain by the Moors.

There have been so many legends spun around objects and people that are in the Bible, although the legends themselves are not in the Bible.

Start with the Magi, who travelled from the east to bring gifts to the baby Jesus. It is not stated how many there were, but since they brought three gifts it was presumed that there were three. They have been the subjects of endless legends.

Then there was the Queen of Sheba, who made a long journey from her homeland to visit King Solomon. At least we know where she was from, as we saw in the visit on this blog, "Yemen". In our visit to "Ethiopia", we saw that the Solomonic Dynasty of the country is based on the founding legend that the first emperor of Ethiopia, Menelik I, was the son of Solomon and the Queen of Sheba.

Then there is the Shroud of Turin, believed to be the garment worn by Jesus at His crucifixion.

More legends have been spun around the Ark of the Covenant. It was the portable wooden housing for the tablets of the law that were obtained by Moses on Mount Sinai. It was put in Solomon's Temple when that was completed. In all probability, the Ark of the Covenant was destroyed or irretrievably lost when the Babylonians, led by Nebuchadnezzar, destroyed the Temple, and led the Jews into exile, in 586 B.C., as the Ark is not mentioned after that. Nevertheless, an entire industry has grown up around legends that it somehow survived and is hidden in various locales, a church in Ethiopia for one.

But the biblical legend that involves Valencia is the Holy Grail. At the Last Supper, which Jesus shared with the Apostles prior to His crucifixion, Jesus used a cup. That cup is referred to today as the Holy Grail.

https://en.wikipedia.org/wiki/Last_Supper#/media/File:Última_Cena_-_Da_Vinci_5.jpg

The Holy Grail is believed to have survived. The trouble is that there are so many "Holy Grails" that we cannot be sure which one, if any of them, is the actual one that Jesus used. Anyway, one of the most popular cups that is claimed to be the real Holy Grail is in the Cathedral of Valencia, and the following scenes of the city begin there. The first two scenes of Valencia Cathedral, with the display of the supposed Holy Grail, are from Google Earth and Street View.



https://www.google.com/maps/@39.475706,-0.3747804,2a,75y,180h,90t/data=!3m7!1e1!3m5!1sKCerDOfI3byoxTPTNnEp8g!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3DKCerDOfI3byoxTPTNnEp8g%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D188.11734%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

Teruel is a medieval city in the mountains of eastern Spain.

https://www.google.com/maps/@40.3438185,-1.1068423,3a,75y,96.77h,90t/data=!3m7!1e1!3m5!1s21sTTeYO19-0_eisjeW8jw!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3D21sTTeYO19-0_eisjeW8jw%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D98.84786%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

The city of Zaragoza originated before Roman times, but then was an important Roman city. Some of the Roman walls are still there and it was also an important city to the Moors. These three scenes are of the Aljaferia Palace, from Google Earth and Street View, that was a combination of palace and castle.




These three scenes, from Google Earth, are of Our Lady of Pillars Basilica in Zaragoza.




These two scenes, from Google Earth, are of the Cathedral of the Savior in Zaragoza.



The following scenes begin at the Aljaferia Palace, in Zaragoza, built by the Moors.

https://www.google.com/maps/@41.6558421,-0.8968851,3a,75y,225.12h,90t/data=!3m7!1e1!3m5!1s1eBfZ16vZjrOVaAaWy6eIw!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3D1eBfZ16vZjrOVaAaWy6eIw%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D216.88333%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

This is more of the old part of Zaragoza.

https://www.google.com/maps/@41.6512729,-0.8796382,3a,75y,113.32h,90t/data=!3m7!1e1!3m5!1svPSQAxnCY7rt5pAPg1i_ew!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3DvPSQAxnCY7rt5pAPg1i_ew%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D119.98623%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

There is a famous medieval castle in the city of Sitges.

https://www.google.com/maps/@41.2371264,1.8058756,3a,75y,88.59h,90t/data=!3m7!1e1!3m5!1sRl567A-796OYcZSBa3_ptw!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3DRl567A-796OYcZSBa3_ptw%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D88.124626%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

Barcelona is Spain's second-largest city, after Madrid. It is the heart of Catalonia, where there was a separatist movement for an independent country. The area is prosperous and well-industrialized but suffered from having supported the losing Republican side in Spain's civil war of the 1930s.

Most people in Barcelona understand the Catalan language, although Spanish is spoken as the national language. Much of the infrastructure of the city was remodeled for the 1992 Summer Olympics, and many factory buildings were demolished.

This is a view, from Google Earth, of Barcelona Cathedral.


Barcelona has it's own version of the Arc De Triomphe. Image from Google Street View.


The famous tree-lined pedestrian street in central Barcelona is La Rambla. These views of the city begin there. The first two views of La Rambla are from Google Earth. The monument in the traffic circle is of Christopher Columbus.



https://www.google.com/maps/@41.3808637,2.1736347,3a,75y,303.42h,90t/data=!3m8!1e1!3m6!1sAF1QipMxkMjAg-ReLCjA6u5J4-0id-xexsTffX9L-gcv!2e10!3e11!6shttps:%2F%2Flh5.googleusercontent.com%2Fp%2FAF1QipMxkMjAg-ReLCjA6u5J4-0id-xexsTffX9L-gcv%3Dw203-h100-k-no-pi-3.8606253-ya40.68314-ro3.1813312-fo100!7i5792!8i2896

The best-known building in Barcelona is the fantastic cathedral, La Sagrada Familia, although it is not yet complete. Work on it has been going on for well over a century. The following scenes begin inside there. The first two scenes are from Google Earth and Street View.



https://www.google.com/maps/@41.4036578,2.1743887,2a,75y,76.27h,90t/data=!3m7!1e1!3m5!1s9NtOtEElfYfgoG8fAtCGFA!2e0!6s%2F%2Fgeo2.ggpht.com%2Fcbk%3Fpanoid%3D9NtOtEElfYfgoG8fAtCGFA%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D71.959366%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

In the Mediterranean, east of mainland Spain, are the Balearic Islands. The three largest of the islands are Mallorca (or Majorca), Menorca and, Ibiza. The islands are very popular with sun-seekers from northern Europe. The largest city on the islands is Palma, on Mallorca. These three views, from Google Earth and Street View, are of Palma Cathedral. There is a royal palace next to the cathedral.




On a hill above the city of Palma is Bellver Castle. But this is from after the time of the Moors. Image from Google Earth.


This is a look at Palma. Be sure not to confuse Palma, on the Balearic Islands, with Las Palmas, on the Canary Islands.

https://www.google.com/maps/@39.5715229,2.6519563,3a,75y,294.9h,90t/data=!3m7!1e1!3m5!1srp369mQg6ntcAXtj3KdHlQ!2e0!6s%2F%2Fgeo1.ggpht.com%2Fcbk%3Fpanoid%3Drp369mQg6ntcAXtj3KdHlQ%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D301.06146%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

ANDORRA

In the Pyrenees Mountains, between France and Spain, is a small nation that is essentially a city-state and is known as Andorra. There is a legend that it was founded by Charlemagne. The population is less than 80,000 people.

Like Spain, to the south, Andorra has been under Roman and Visigoth influence. Also like Spain it is a parliamentary democracy, unlike France which uses the presidential system. It has been a nation since the 13th Century and is a very highly-rated place to live. It is also considered as a tax haven. Although Andorra is an independent country, the French President and a Spanish bishop are it's co-princes.

Here is a look at the major urban area of Andorra.

https://www.google.com/maps/@42.5070981,1.5303157,3a,75y,106.29h,90t/data=!3m7!1e1!3m5!1sWyDiZq4VxRLJ-E--eWmATg!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3DWyDiZq4VxRLJ-E--eWmATg%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D111.62398%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656

MONTE CARLO ( MONACO )

Monaco is also essentially an independent city-state. It is not contiguous with Spain, being on the Mediterranean coast of France. Monte Carlo is the central part of Monaco. The population of Monte Carlo is less than 40,000 people and it is possible to walk across it.

Monaco has a popular royal family. It has been led by the House of Grimaldi, originally a noble family from Genoa, since 1297. The royal family still lives in the same original palace, unlike most royal families which periodically build new palaces. On high ground above the city, these three scenes of the famous Palace of Monaco are from Google Earth.




Long before Meghan Markle married into the British Royal family, another American actress married into the royal family of Monaco. Her name was Grace Kelly and she married the Prince of Monaco, Rainier III, in 1956. The present Prince of Monaco, Albert II, is their son.

Unfortunately, Princess Grace had something in common with another British royal. Like Princess Diana she was killed in a car accident, in 1982.

With it's economy suffering, Monaco tried a novel idea to bring in money. In 1863, it opened a casino. While it was not an immediate success, Monte Carlo is now world-famous for it's casino, being a playground for the wealthy, and several other casinos have followed. This is the prototype for cities everywhere that have tried to boost their economies with a casino. This image, from Google Earth, is of Monte Carlo's fabled casino.


Due to the success of the casino, there are no income taxes in Monaco. This has drawn wealthy people from all over the world and about a third of it's population is at least a millionaire. It has also become a global banking center. Charles De Gaulle once blockaded Monaco over it's allowing French citizens to avoid taxes.

The original casino is Casino de Monte Carlo. The opera house was designed by the same architect as the Palais Garnier, in Paris.

https://www.google.com/maps/@43.7403379,7.4276649,3a,75y,121.28h,90t/data=!3m7!1e1!3m5!1ssaPt-DL-WxEIwY4EWJw9Mg!2e0!6s%2F%2Fgeo3.ggpht.com%2Fcbk%3Fpanoid%3DsaPt-DL-WxEIwY4EWJw9Mg%26output%3Dthumbnail%26cb_client%3Dmaps_sv.tactile.gps%26thumb%3D2%26w%3D203%26h%3D100%26yaw%3D130.05719%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656
Posted by at No comments:
Subscribe to: Posts (Atom)