History of Early Geometry Millau Bridge Sir Norman Foster Millenium Park Frank Lloyd Wright Fallingwaters Frank Lloyd Wright Point, Lines, Planes, Angles History of Early Geometry

Geometry The word "geometry " comes from two Greek words geo and metron meaning "earth measuring." Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 years ago in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few. The most fascinating and accurate geometry text was written by Euclid, and was called Elements. Euclid's text has been used for over 2000 years!

History Thales of Miletus (624-547 BC) Pythagoras of Samos (569-475 BC) Hippocrates of Chios (470-410 BC) Plato (427-347 BC) Euclid of Alexandria (325-265 BC) Archimedes of Syracuse (287-212 BC)

Where These People Lived To understand the history, you need to know the locations of the places being discussed. Geography and geometry have a lot in common. Let’s look at that section of the world.

Mediterranean

Thales Thales of Miletus (624-547 BC) was one of the Seven pre-Socratic Sages, and brought the science of geometry from Egypt to Greece. He is credited with the experimental discovery of  five facts of elementary geometry (including that an angle in a semicircle is a right angle), but some historians dispute this and give the credit to Pythagoras.

Miletas Black Sea Situated at the mouth of the Meander in the south of the province of Ionia in Western Anatolia, the ancient city of Miletus was the oldest and the most powerful of the twelve Ionian cities in Asia Minor. It also founded over ten colonies on the shores of the Marmara and the Black Sea, while its commercial activities extended as far as Egypt. Its schools made a very great contribution to the intellectual and scholarly development of the Mediterranean world and one cannot talk of Miletus without mention of the great contributions to geometry and science made by Thales, one of the greatest scholars produced by the city.

Thales of Miletus "The skillful man is superior to the strong man." "Don't come to a conclusion before listening to both sides.“ "A small spark is enough to burn down a whole forest."

Pythagoras Pythagoras of Samos (569-475 BC) is regarded as the first pure mathematician to logically deduce geometric facts from basic principles. He is credited with proving many theorems such as the angles of a triangle summing to 180 deg, and the infamous "Pythagorean Theorem" for a right-angled triangle (which had been known experimentally in Egypt for over 1000 years). The Pythagorean school is considered as the (first documented) source of logic and deductive thought, and may be regarded as the birthplace of reason itself. As philosophers, they speculated about the structure and nature of the universe: matter, music, numbers, and geometry.

Note that Samos is much closer to Turkey than Greece.

Samos Greek Island

Fishing and resort island.

Hippocrates Hippocrates of Chios (470-410 BC) wrote the first "Elements of Geometry" which Euclid may have used as a model for his own Books I and II more than a hundred years later. In this first "Elements", Hippocrates included geometric solutions to quadratic equations and early methods of integration. He studied the classic problem of squaring the circle showing how to square a "line". He worked on duplicating the cube which he showed equivalent to constructing two mean proportionals between a number and its double. Hippocrates was also the first to show that the ratio of the areas of two circles was equal to the ratio of the squares of their radii.

Chios Note that Chios is closer to Turkey than Greece.

Chios

Plato Plato (427-347 BC) founded "The Academy" in 387 BC which flourished until 529 AD. He developed a theory of Forms, in his book "Phaedo", which considers mathematical objects as perfect forms (such as a line having length but no breadth). He emphasized the idea of 'proof' and insisted on accurate definitions and clear hypotheses, paving the way to Euclid, but he made no major mathematical discoveries himself. The state of mathematical knowledge in Plato's time is reconstructed in the scholarly book:  The Mathematics of Plato's Academy .

Athens The Parthanon

Euclid Euclid of Alexandria (325-265 BC) is best known for his 13 Book treatise "The Elements" (~300 BC), collecting the theorems of Pythagoras, Hippocrates, Theaetetus, Eudoxus and other predecessors into a logically connected whole. A good modern translation of this historic work is The Thirteen Books of Euclid's Elements by Thomas Heath.

Euclid Quote “There is no royal road to geometry.” When Pharaoh Ptolemy the ruler of Egypt said he wanted to learn geometry, Euclid explained that the Pharaoh would have to study long hours and memorize the contents of a fat book. Ptolemy complained that it would be unseemly and demanded a short cut. Euclid replied, “There is no royal road to geometry.” There wasn’t a shortcut to the learning process then and there still isn’t one. Homework needs to be done EVERY day.

Euclid Quote A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, he inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him three pence, since he must make a gain out of what he learns."  “The laws of nature are but the mathematical thoughts of God.”

The father of mechanics. Archimedes Archimedes of Syracuse (287-212 BC) is regarded as the greatest of Greek mathematicians, and was also an inventor of many mechanical devices (including the screw, the pulley, and the lever). He perfected integration using Eudoxus' method of exhaustion, and found the areas and volumes of many objects. A famous result of his is that the volume of a sphere is two-thirds the volume of its circumscribed cylinder, a picture of which was inscribed on his tomb. He gave accurate approximations to pi and square roots. BC. The father of mechanics.

Archimedes In his treatise "On Plane Equilibriums", he set out the fundamental principles of mechanics, using the methods of geometry, and proved many fundamental theorems concerning the center of gravity of plane figures. He discovered the 13 semi-regular (or "Archimedian") polyhedra whose faces are all regular polygons. Translations of his surviving manuscripts are now available as The Works of Archimedes. A good biography of his life and discoveries is also available in the book Archimedes: What Did He Do Beside Cry Eureka?.  He was killed in Carthage by a Roman soldier 212 BC.

Syracuse: Greece?

Greek Colony of Syracuse

What did one acorn say to another acorn? Gee I’m a tree. Groan?

Summary 1. Note that Geometry was not invented by the Greeks. Euclid, the father of Geometry, did not discover it all. Much work was done prior to him by Egyptians. 2. The three top ancient geometers were Pythagoras, Euclid, and Archimedes 3. Geometry means measure the earth.

C’est fini.