Euclid's Window

EUCLID'S WINDOW

LIST OF BOOKS TO READ THAT ARE REFERENCED:


Elements - Euclid
Arithmetica
Conic Sections
The Decline and Fall of the Roman Empire - By Voltaire and Gibbons
Topographia Christiana - Concept that the World is Flat
Discourse by Renee Descartes

THE GOLDEN AGE - THE STORY OF EUCLID

Thales - predicted the first solar eclipse and stopped a war. Learned from the Egyptians and Babylonians about math. Told Pythagoras to go there, which he listened and spent many years of his life there.

Found by Pythagoras - One of the first natural laws of science discovered was that sound was directly related to the length of a string on a harp. Half notes relative to length of strings on a harp. compared to Jesus often. Walked on water, calmed the storms, thought to have been raised from the dead. Spent a lot of time on the mount.

Socrates, Plato, Aristotle, Alexander the Great - Alexander encouraged interbreeding with nations not from Greece. This likely increased the gene pool and created smarter races (but perhaps questionable as this would have to occur over several generations to take full effect). Also he had envisioned to create a great educated metropolis and call it Alexandria. The city was eventually built and a great museum housed over 200,000 scrolls (no such thing as books at that time).

Euclid
Abraham Lincoln studied Euclid
Kant defended him
Spinoza praised him
Opened a school in Alexandria, scorned materialsim
Famous for works on Conics (Cone and a plane)
Lookup Conics again and Calculus
Newton had assumed that the orbit of Mercury stays constant forever. Some scientists later proved that it deviated a very small amount every 100 years. Einstein proved why. So Newtonian theory broke down. Similarly the logic proof asked to Bertrand Russell as it was stated in logic that if there is one fallacy then anyone can prove that 1 and 2 are equal. Once someone stated to him ok. Prove to me that you are the pope if 1 and 2 are equal. His response was after a few seconds.... the Pope and myself are 2. Therefore the pope and I are one.

Euclid wrote the book 'The Elements'
The idea of the parallel postulate. One line and one dot outside of the line. There only exists one parallel line that you can draw on that plane.
Ptolemy I, II and III built a huge museum in Alexandria that essentially became the center for all Greek thought going forward and acted as a library. Many scrolls were written here through forced labor (copies of the Old Testament).

Eratosthenes - 212 BC. Had estimated the circumference of the earth. By taking a location that casted no shadow (the summer solstice in Aswan) and compared that with the angle of the shadow. His student measured the distance between Aswan and Alexandria, the metropolis with the great museum. According to the laws that Euclid had found and the circumfernce of a circle he found the arc length to be 1/ 50 to that of the earth. And estimated that the circumference to be 25,000 miles, which was off by 4%.

Alexandria existed until 400 AD - The life of Hypatia coincided with the fall of Alexandria. She was likely the first successful female mathematician and was taught under her father Theon who was a student of the school in Alexandria. She surpassed him in knowledge it is claimed and became a knowledgeable philosopher regarding the works of Aristotle and Plato. A Roman became jealous of her power and started a rumor that she was which. She was eventually dragged and beaten from her chariot into a church; skinned alive and killed. Her body parts were dismembered and separated to various parts of Rome. Likely could have been the earliest feminist. Her works were destroyed along with the 200,000 scrolls that existed in that museum. Some works were passed down. Soclaphes was rumored to have written over 100 plays. 7 of which became published eventually. There was never a great mathemetician that came out of Rome and actually the work Topographia Christiana was put into publication by a Roman who had said the world was flat based on Christian beliefs not logic.

Voltaire and Edward Gibbon wrote a book called the Decline and Fall of the Roman Empire.

THE STORY OF DESCARTES

Next great mathematicians wouldn't come until several centuries later. The concept of maps and cartography advanced mankind forward.

Nicole d'Oresme - invented the graph. Proved the Merton Rule. Speed vs. Time. The area of the graph under the curve is distance travelled. We eventually find out that the slope of any line is acceleration. But that a constant speed (rectangle) graphed against a constant acceleration (triangle) will equate if the half way point the two are going the same speed. Had first notion of relativity. A person standing a ship moving quite fast moves his arm up and down vertically. Anyone standing on the ship will see it move up and down vertically. Anyone standing on the dock will see a diagonal movement. Often called Galilean Relativity. He thought that the earth may rotate and move around the sun (like Galileo figured out), but later detached himself from this notion as he became more involved with the church. Psalm 93:1 was the verse he thought contracted the rotation of the earth. Never wrote but was famous for agreeing with Socrates that all he knew was that he knew nothing. His progress didn't go very far until Descartes.

Edward Tufte's 'Visual Display of Quantitative Information'

Rene Descartes - Created Cartesian coodinate system. Equation of circle x2+y2=r2 (r2 being a constant), thus he used algebra. He was agitated by the Greek's inefficient use of geometry or explanations. He thought there should be a better system. He met Isaac Beekman by accident by solving a public math problem quite easily. It wasn't necessarily the coordinate system that was interesting (as the idea was already made in maps during Ptolemy's time). It was his use of the co-ordinate system. Came up with ax + by + c = 0 (essentially y= mx + b). He was scared to publish anything because of the Catholic Church. First to explain the physics of the rainbow and the refraction of light. Galileo lived in the same era and published 'The Dialogue of Two Chief Systems'. The book was burned and he was forced to renounce it and he imprisoned indefinitely. This work delayed Descartes book Discourse until 1637. He was physically attacked for believing that reason and observation could determine truth. He believed that people could control nature and that all cures would eventually be found and that people would eventually find eternal life. Throughout his life he was famous for his lack of emotion. He did not marry but had an affair with a lady named Helen, whom they had a daughter together. She died of illness at 10 years of age and devastated Descartes. He had later been invited by the Queen of Sweden (23 years of age and very well educated); she promised him to that an academy would be dedicated in his name and that he must be a personal tutor to her when he arrived; 1649 was one of the coldest winters in Sweden and he caught pneumonia and died on February 11th, 1650. "It seems to me that men's thoughts freeze here in winter just like the water..."


THE STORY OF GAUSS

He had been the one that defied 2000 years of Greek geometry. Or rather Euclid's 5th rule; the parallel postulate. He had discovered curved spaces which would essentially result in Einstein's theory of relativity and his assumptions of curved space. Born 50 years after Newton died, Gauss considered one of the greatest mathemticians to have lived. From Germany born 1777. At the age of 2 or 3 could do mathematics and no one had taught him. (Father adding his payrolls and son found a mistake, father didn't know he was watching). Went to a very strict school where children were whipped; Gauss had easily excelled way past his classmates. He was asked to add all the numbers from 1 to 100, which he easily did in minutes and did not write anything down; the answer being 5050. His teacher was astonished but Gauss had used the logic of multiplying 50x101.... a short cut to a sum of a series; which was a password used in Pythagoras' days. His teacher found the most advanced textbook at the time and Gauss thought the math was too easy at the age of 9. His teacher felt he could teach him no more so passed him on to his grad student. As he became a teenager of age 12, he started to wonder if space was actually curved. and started criticizing the 2000 year old mathematical law of Euclid. His father was a farmer and mother uneducated but mother was very proud of him and when he was able to go to University, his father wanted him to be a farmer but his mother stood firm on Gauss going to further his education. Gauss' supporter was a duke that paid for his school and afterwards; the duke died because of Napoleonic wars. Ironically, Napoleon spared the city which Gauss was from because Napoleon did not want to destroy the city where the greatest mathematician lived. He fell in love with Johanna who brought him tremendous joy.

His quote when he fell in love: "For 3 days, that angel, almost too celestial for this earth, has been my fiancee. I am superabundantly happy... Her cardinal trait is a quiet devout soul without one drop of bitterness or sourness. Oh, she is much better than I .... I had never hoped for this bliss; I am not handsome, not gallant, I have nothing to offer except a candid heart full of devoted love; I despaired of ever finding love."

She died later after their 3rd child was born. Shortly after 2 of his children died. He remarried but never found the same joy ever again as his first love. His quote near death:
"Lonesome, I sneak about the happy people who surround me here. If for a few moments they make me forget my sorrow, it comes back with double force..... Even the bright sky makes me sadder...."

Huge leaps forward in Geometry; although would be very secretive of his work. Created a space called hyperbolic geometry. Gauss feared Kant and his followers the most, who died in 1804.