Euclidean Geometry


c.625 - 545 B.C. Thales of Miletus

Thales was of Phoenician descent who lived in an Ionian city (a Greek colony). The popular phrase "Know Thyself" is credited to him. Aesop tells a story about Thales. It seems one of his mules, loaded with salt for trade, realized accidentally that if he (the mule) rolled over in the stream, his load became very light (because the salt dissolved). The mule did this act on several occasions, prompting Thales to come up with a plan of discouragement. Thales loaded the mule bags with sponges. Now the water did not dissolve these, instead the poor mule's load became heavier.

Thales is hailed as a great mathematician and astronomer. He was the first to introduce logical proof based on deductive reasoning instead of experiment and intuition.


580 - 500 B.C. Pythagoras of Samos

Pythagoras is credited with a lot of discoveries in mathematics which he himself probably did not find. He began his own school, which was not uncommon in this time. But his school was different in that its aims were political, philosophical, and religious. He began the school with about 300 young aristocrats. The "community" was as a secret society or fraternity. The school regulated diet, ways of life, and method of education. The student studied number theory, music, geometry, and astronomy. These four subjects were known as the "quadrivium" in the middle ages. To these were added the trivium logic, grammar, and rhetoric (subjects associated with the use of language). These seven liberal arts became the "proper and necessary" course of study for the educated.

The disciples (students) of Pythagoras were either listeners (which meant you listened to lectures from behind a curtain for three years without speaking) or mathematici (which meant you were initiated into the inner circle and were confided the main doctrines of the school). Supposedly, at least 28 women were part of the inner circle (women were forbidden in this day to attend public meetings). Pythagoras was alleged to have married one of his students, Theano, when he was 60 years old. But some claim she was his daughter, not wife. In any case she was a very bright student.

All members of the society were sworn to secrecy. No students could claim discoveries. Pythagoras never wrote anything down, so all learning was passed on orally as his own.

He was first to teach the earth is a sphere and center of the universe.

An incredibly interesting society, you must read more. Go to these links:
the society
the music in the numbers
astronomy rocks
harmony



Problems of Antiquity

c. 460 B.C. Squaring the Circle

Problem description: Given any circle, find a square with the same area. This problem was attempted from 460 B.C. (Hippocrates) to 1882 (Lindeman).

This problem was due to the fact that they had no number for pi. In fact, this is what led to the approximations of pi.

The first attempts at the value of pi (of course, they didn't call the number pi, and probably didn't really understand that it was a single number) were rude approximations. The first sign of these were in the Ahmes Papyrus (dealing with area of circle = (d - 1/9d)^2). The number 3 is an even older approximation (used in Early Chinese works, early Hindu work, medieval manuscripts, the Talmud, and the Bible - I Kings 7:23, II Chronicles 4:2 ).

Most used the method of exhaustion to find pi. This method said to inscribe a polygon in a circle, find its area. Now double the sides successively until the approximation exhausted the area between the polygon and the circle.

Other very interesting links to pi:
enhanced pi page
pi page
music of pi
pi in the sky
uselessness of pi

 

c. 425 B.C. Trisection of an Angle

Problem description: Trisect an arbitrary angle using only compass and strightedge. This problem was contemplated from 425 B.C. (Hippias) to 1792 (Gauss) when it was finally proven impossible. Apollonius made great progress in the problem. Click here to see his construction.

 

c. 370 B.C. Duplicating the Volume of a Cube

Problem description: Given a cube, find the length of a new cube that has twice the volume. This problem was attempted from 370 B.C. (Eudoxus) to 1770 (Newton) to 1869 (Montucci).



460 - 380 B.C. -- Hippocrates of Chios

He was the father of Greek medicine. Began his life as a merchant and ended as a teacher. He was robbed of his money and went to Athens to prosecute the offenders. He lost the respect of his peers when he was robbed because some felt he must not have any "common sense" if he allowed that to happen. He had to stay there for many years and attended the lectures of philosophers. He may have been influenced by Pythagoras.

Proclus reported that Hippocrates developed elements of geometry a full century before Euclid.

Hippocrates made great strides in the quadrature of the circle problem.



330 - 275 B.C. -- Euclid

Euclid wrote the Elements which is essentially our textbook geometry of today. This is considered the first time demonstrative geometry came to be (Thales had completed some elementary proofs, but these were lost). Euclid saw the need for uniformity. He did not necessarily arrive at all the theorems himself, but he did prove them, and put them into a workable order. Through the translation of Boethius (510) the Elements were known in the Dark Ages.

More about the Big Guy



287 - 212 B.C. Archimedes of Syracuse

Archimedes attempted to find pi in 225 B.C. by method of exhaustion. He claimed the area of a circle is equal the the area of a right triangle, one leg being the circumference of the circle and the other being the radius of the circle. Also, he said the ratio of the area of the circle to the squatre on the diameter is 11:14. He inscribed and circumscribed regular polygons up to 96 sides and showed that the area of the circle lies between the results. He claimed pi was between 3 1/7 and 3 10/71.



262 B.C. - 190 B. C. -- Apollonius of Perga

He is known as the "Great Geometer" because of his extrordinary Conic Sections. This is a collection of 8 books, the first 4 books deal with the general elementary theory of conics, while the later books are devoted to more specialized investigations.


276 - 197 B.C. -- Eratosthenes

Eratosthenes was very intellegent. He was a distinguished mathematician, astronomer, geographer, historian, philosopher, poet and athlete. He had the nickname of Beta, which has many stories of origin. Some say it was because he was looked upon as a second Plato. Others say it was because he failed to top his contemporaries in any one branch (always second best). In old age,he reportedly became almost blind and committed suicide by voluntary starvation.

Eratosthenes computed the Earth's circumference in 230 B.C. He used arc of the great circle extending from Syene to Alexandria. He approximated the circumference to be about 25,000 miles. How close is that?

 


c. 250 -- Liu Hui

Liu Hui uses a polygon of 384 sides to derive 3.141024 < pi < 3.142904 in his commentary on the Chinese Nine Chapters.

 

 


476 - 550 -- Aryabhata

He approximated pi to be 3.1416 and claimed this to be the relation of the circumference to the diameter.


598 - 670 -- Brahmagupta

He approximated pi to be the square root of 10, at least he called this the "neat value."


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