Pythagorean Theorem — or Pythagoras’ theorem: in any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
Pythagoras of Samos, the creator of one of the most famous equations of all time, has a rather obscure history. He wrote very little about himself, delegating the task of documenting his life and views to his followers. What we do know is that in his world of the mid-sixth century BCE, he was considered to be a thinker and a mystic.
In today’s world, Pythagoras’ school would be considered more like a religious cult than a philosophical establishment. His teachings included many eccentric doctrines including, the veneration for, and abstinence from, the eating of beans. He advocated reincarnation and the transmigration of souls. Most see him as the founder of the modern belief in numerology, later popularised by Nostradamus.
Pythagoras argued that the ultimate nature of reality is number, which he developed out of his theory of music. He claimed that music had a special power over the soul. The proof was found in the intervals between musical tones, which could be expressed as ratios between the first four integers, number 1 – 4. His discovery of irrational numbers did play havoc with his beliefs on the origin of the universe; however, they have proven to be a major and lasting development in mathematical thinking.
After his death, his followers split into two camps; one embraced his religious and mystical teachings, while the other pursued his scientific and mathematical philosophy. Ideas and beliefs, whether or not they prove to be valid, must be considered, lest we overlook the very insight that will bring us to the next stage of development.
“It is the mark of an educated mind to be able to entertain a thought without accepting it.”
This defines entrepreneur and entrepreneurship – the entrepreneur always searches for change, responds to it, and exploits it as an opportunity.”
Peter F. Drucker, Innovation and Entrepreneurship: Practice and Principles
The more I read about Thales of Miletus, the more I believe that he was the ancient version of the Renaissance man. Philosophy is a thinking exercise that usually involves a considerable amount of time. Most of the population of any age or society are involved in making a living and putting food on the table. Thales of Miletus, possessing an entrepreneurial spirit, cleverly dealt with the issue of time and money. It seems he made a fortune investing in oil-presses before a heavy olive crop harvest. All of which suggests that to be a philosopher and scientist in Ancient Greece, 7th century BCE, business skills are a notable asset.
Thales significance as a philosopher centers on methodology. He was the first thinker who tried to find common, underlying principles to account for the natural world, rather than relying on the whims of anthropomorphic gods. He sought to give a naturalistic explanation of observable phenomena that still has relevance in modern scientific exploration. Thales believed that the mind of the world is god, that god is intermingled in all things, a viewpoint that would shortly emerge simultaneously in a number of world religions.
Thales lived in the past, yet his thought process made him universal. He would thrive in any age.
“I cannot teach anybody anything. I can only make them think”
“Man is the only creature who refuses to be what he is.”
The other day, someone asked me, “What is your philosophy of life?”
We talk about philosophy as if it were something that could be summarized in one or two sentences. And yet, it generally takes a lifetime to identify with the reality. It is more than a thought, a response or a single activity. It is our entire worldview, our personal system of principles for guidance in practical affairs.
The “philosophy question” has been in the back of my mind over the past couple of weeks, especially as I was researching the ancient scientists. It seems that their philosophy was the precursor to their scientific investigations.
Philosophical discussions rarely have neat and tidy outcomes because the business of philosophy is to challenge prevailing assumptions and concepts in order to generate new perspectives on complex problems. This week will focus on beginnings. That is the only place that will give us the genesis of this worthy conversation that has spanned the history of humanity.
“Wise men speak because they have something to say; fools because they have to say something.”
“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”
Antoine de Saint-Exupery
Ptolemy’s Almagest has been on the scientist’s best sellers list for centuries. But that wasn’t his only success. Ptolemy’s other best seller was called simply, Geography. And that is the only thing simple about it!
Geography was the first time anyone had ever presented the world with a detailed mathematical explanation for calculating lines of longitude and latitude. Ptolemy had a gift for amalgamating the cumulative knowledge of the ancient world into a systematic layout and design. He built on the work of Hipparchus and drew on the compilation of the known sailing directions, called periplus, collected from sailors far and wide. The Periplus of the Eritrean Sea, written in the first century by a Greek merchant living in Alexandria, provided invaluable data on the trading routes as far east as India.
Ptolemy’s greatest innovation was to record longitudes and latitudes in degrees for 8,000 locations on his world map, the first of its kind. He developed two ways of drawing grid lines on flat maps to signify the lines of longitude and latitude on the curved surface of the globe. In hindsight, there were many inaccuracies, such as the equator being too far north and Asia stretching too far to the east. Yet, it was the standard for over 1,300 years. Christopher Columbus was inspired by Ptolemy’s view of the world when he set sail for Asia and unexpectedly bumped into America. There are always surprises along the way.
“As for me, I am tormented with an everlasting itch for things remote. I love to sail forbidden seas, and land on barbarous coasts.”
Herman Melville, Moby-Dick
I must down to the seas again, to the lonely sea and the sky,
And all I ask is a tall ship and a star to steer her by,
And the wheel’s kick and the wind’s song and the white sail’s shaking,
And a grey mist on the sea’s face, and a grey dawn breaking.
John Masefield, Sea Fever
You may remember our dear friend, Hipparchus from the “Narratives of Science” who was a stargazer as well as the mathematical genius who invented trigonometry. While that would be enough for any lifetime, even for a man of his intellect, it seems that he had a third occupation, that of a renowned geographer, over and above being an astronomer and mathematician.
In the midst of cataloguing all of the known stars, Hipparchus perceived the potent relationship between earth and the heavens. He was the first person to plot places on the earth’s surface using the concepts of longitude and latitude in his geographical positioning. He concluded that a geographic map must be founded on astronomical measurements of latitude, longitude and triangulation. Some even believe that Hipparchus was the inventor of the astrolabe, an elaborate instrument that would enable navigators to predict the positions of the Sun, Moon, planets and stars.
Three centuries later, Claudius Ptolemy, would compete his work by creating the first world atlas. The world of navigation would never be the same again.
“Skill’d in the globe and sphere, he gravely stands, And, with his compass, measures seas and lands.”
“One learns more from listening than speaking. And both the wind and the people who continue to live close to nature sill have much to tell us which we cannot hear within university walls.”
Yesterday was the anniversary of the passing of Thor Heyerdahl (October 6, 1914 – April 18, 2002). Anyone who reads about his Kon-Tiki expedition, the 8,000 kilometre voyage across the Pacific Ocean in a self-built raft, cannot help but think back to the legendary Polynesian navigators that lived thousands of years before the Kon-Tiki set sail.
The Polynesians developed a system of navigation that allowed them to make extended voyages across thousands of miles of trackless ocean, notorious for its capricious unpredictability. They travelled to remote islands throughout the southern Pacific, gaining knowledge from their natural world.
Navigators were held in high esteem within Polynesian culture. Each island established a guild of navigators which allowed oral traditions to be passed from mentor to apprentice, often in the form of a song. A navigator memorized everything, including the motion of specific stars as they would rise and set on the horizon. Much like our Viking friends, they understood weather patterns and the season of travel, cloud formations and the flight path of birds. Yet, the Polynesians used navigational techniques that made them distinctive. They watched the waves, deciphering from their direction and type which course to take. And they observed the soft shimmer spread on the horizon that came from islands just below the skyline.
Modern navigators are still baffled by the accomplishments of the Polynesian seafarers. Their navigational feats remain unequalled.
“For every minute, the future is becoming the past.”