Notes From “Euclid’s Window” by Leonard Mlodinow (Part 1)

“The Story of Geometry from Parallel Lines to Hyperspace”

Page 6 “The Egyptians had always had death, but with wealth and settlement, they now also had taxes. Taxes were perhaps the first imperative for the development of geometry, for although in theory the Pharaoh owned all land and possessions, in reality temples and even private individuals owned real estate.” Ahh “death and taxes”! Geometry was developed to calculate land area and water levels for taxes. Crazy.

“rope stretcher” used to perform surveying of land. “Hypotenuse”, Greek for stretched against.

Here’s another interesting thing! The Babylonians did not write equations. It was all word problems. Kind of makes it difficult to make any changes. “The oldest know use of the plus sign for addition occurs in a German manuscript written in 1481.” To think that this type of math is such a recent thing in history.

Page 17 “To the Greeks, not the Egyptians, goes the credit for the idea that brings romance and metaphor to mathematics: that space can be a mathematical abstraction, and, just as important, that the abstraction can apply to many different circumstances.” Leave it to the Greeks to abstract on something! Science and philosophy had not parted ways just yet.

Page 18 “The Pythagorean theorem, too, must have seemed magical.” “It’s a discovery that would surely rate a headline on the front page of the New York Times: Surprising Regularity Discovered in the Right Triangle,” and in smaller print, “Applications Still Years Away.”

Page 21 “There is also a distinction between intelligent talk and blather, a distinction that Pythagoras did not always make.” Part of why I like this book is the commentary! Pythagoras had some pretty wild ideas from his fascination with numbers. It just goes to show you that someone can be a genius in some aspects but a complete nut in others.

Page 26 “Today people are murdered for many reasons – love, politics, money, religion – but not because somebody squealed about the square root of 2. To the Pythagoreans, though, mathematics was a religion, so when Hippasus broke the oath of silence, he was assassinated.” Holy cow they took math seriously. I’ve heard before that Pythagoras had quite the cult going and was a bit of a nut but this takes the cake. And why? Why keep math such a secret?

Page 29 Euclid’s Elements was lost and found several times, as well as re-written. And Euclid never claimed to have “discovered” them, only wrote them down.

Page 38 Euclid’s postulate of parallel lines was a universally accepted “fact” for centuries, but eventually someone questioned it.

Page 41 Eratosthenes used geometry to calculate the circumference of the earth in 212 BC! That’s pretty amazing. He couldn’t even see the earth or use a calculator.

Page 43 Claudius and Ptolemy (not the king) and Hipparchus made a model of the earth centric universe that lasted until Copernicus (1500). So for 700 years the earth centric model served the scientists just fine. It explained mostly everything, eclipses were predicted and the model looked just right. The science was settled. And then this crazy Copernicus comes and says it’s all wrong. Is there any wonder that people were angry at him? Doesn’t this sound like today? I’ve always heard that it was the crazy non-scientific Christians that were most angry at Copernicus because his theory would disprove the Bible. But the “science” they were following pre-dated the new testament, not the old but was anyone but the Jews listening to that when the science was decided? It seems to me that the scientists had a problem with changing the model.

Page 45 “the principal translator of Euclid into Latin was a Roman senator from an old established family, Anicius Manlius Severinus Boethius, a kind of ‘Reader’s Digest’ editor of Roman times. Boethius abridged Euclid’s works, creating the kind of treatment suitable for students preparing for a multiple choice test.” Much like today, the Roman’s were not interested in creating/promoting individuals that take care of themselves. They wanted subjects that could work and think basically but rely on the rulers to take care of the bigger stuff. It would make sense that they wouldn’t really be using mathematics to liberally broaden minds, but to get them to know enough to imagine they are free.

Here’s another note I made in the margin while reading about the growth of the Roman empire. Christian leadership grew out of the Roman Empire, control by fear of God instead of government, they combined the ideas to serve man. This is not Christianity. It is not what Jesus preached. It is mankind using religion as a weapon against mankind, to control.

Page 54 We’ve moved on to Descartes now. People act as though we have evolved past religion but we’ve only replaced it with modern science. Now to question the “science” is to commit heresy and you are punished in similar ways, though not as violently (usually). Science changes its theory and law just as quickly as religion. If you don’t think that is true, just question something like global warming, GMO’s, and evolution.

Page 64 The 13th and 14th centuries. “The weather gods, too, were in a foul mood. Europe at the time was at the start of a wet and cold period so distinctly miserable that today it is called the little ice age In the Alps, glaciers advanced for the first time since the 8th century. In Scandinavia, ice floes blocked the North Atlantic shipping lanes. Crops failed. Agricultural productivity plummeted. Famine was widespread.” It’s just too bad they didn’t know to fix climate change with some new laws and taxes.

Page 73 Ahh the graph. I can’t believe this had to be invented. How crazy is that?! “The power of graphs in helping the non-mathematician analyze patterns of data stems from this same connection of data to geometry. The human mind easily recognizes certain simple shapes – lines and circles, for instance. When looking at a collection of points, our mind tries to fit them into one of these family patterns. As a result, we notice geometric patterns when data is graphed that we might easily overlook when staring at a table of numbers.” Awesome, right? Shown a table of numbers, most people would not see anything important. Some would, but even more will see the point if put into a line or circle.

The whole time I’m reading this book I’m thinking, “If only I had been assigned this book in high school instead of more numbers to look at!” This is the kind of thing that make homeschooling better than regular school. We have the chance to learn what our minds are drawn to, especially since we have the internet full of new resources.

Page 78 Reading about Oresme and his “I indeed know nothing except that I know nothing.” The more we learn and discover, the more we find to learn and discover. I doubt it will ever end. But I hear and read about scientists who decide that the science is settled and there is no reason to look any further. That doesn’t sound like a scientist to me.

Decartes early life reminds me that everyone has a purpose in this world and we should respect life and try to preserve it as much as we can. His mother died after giving birth to him and he was a sickly child but his father nursed him at home and kept him in bed, still learning and exploring through books and tutors.

I’m wondering if there is really any point to studying Euclid anymore. I tried it on my own a few years back and found it tedious and never really got past the first few points. Since the discovery of curved space, Euclid’s theorems no longer stand anyway, right?

Page 98 Ptolemy (2nd century) “assumed the alternate form of the postulate, then derived the original from it. What should we think of Ptolemy? Did he live in an intelligence-free zone? Should we picture him racing to his friends, exclaiming, ‘Eureka! I found a new form of proof: the circular argument.’ Mathematicians would not make this mistake twice. They would make it over and over and over.”


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