Origins of Geometry


Where did geometry originate?

Most people assume the Egyptians started it

a) Through their practical surveying
b) The 'leisure class' of priests pursued it for fun.

Serious deficiency in Egyptian geometry - lack of clear cut distinction between relationships that are exact and those that are approximate only.

They made however the first exact statements about the interrelationship among geometrical figures (no theorems or formal proofs given however).

Examples of Egyptian Geometry

Egyptian rule for area of a circle

'' (4.5) = 64

'' = 3.1605

Ahmes said the area of circular field diameter 9 = area of square with side 8.

i.e. Egyptian taking = 3.1/6. But no hint that Ahmes was aware that they were not exactly equal.

Egyptian rule for the circunference of a circle.

area of circle area of circumscribed sq
___________=__________________
circum.of circleperimeter of circum.sq

this is exactly correct (better!)

Conclusions on Egyptians

Knowledge displayed in Ahmes and Moscow papyri is mostly practical and calculations were the most important thing.

Rules of calculation concern specific concrete cases only.

The Babylonians

Until 1936 it was thought that they were not as good as the Egyptians - took area of circle = 3r.

In 1936 a group of mathematical tables were unearthed at Susa. One tablet compares the areas and the squares of thesides of the regular polygons - pretty accurate - indicate that = 3.1/8 which is as good as the Egyptians.

Babylonians still only interested in geometry from point of view of finding numerical approximation to use in mensuration. Also they didn't distinguish between exact and approximate.

Don't know if Egyptians used "Pythagoras Theorem" but the Babylonians used it widely.

eg. A reed stands against a wall. If the top slides down 3 units when the lower end slides away nine units, how long is the reed?

Babylonians new angle in semicircle = 90 degrees.

But like Egyptians they had no feeling for proof.

The Beginnings of Greek Mathematics

Thales

The Greeks took the geometrical knowledge of the Babylonians and Egyptians and developed it.

They added the logical structure to Geometry.

Seems likely that Thales was the first Greek to demonstrate or prove theorems.

He is said to have proved

(i) The angle in semicircle = 90 degrees (Thales Tn)
(ii) A circle is bisected by a diameter
(iii) The pairs of vertical angles formed by 2 intersecting lines are equal
(iv) The base angles of isosceles triangle are equal
(v) If 2 triangles are such that 2 angles and a side of one are equal respectively to 2 angles and a side of the other, then the triangles are congruent..

Pythagoreans

Pythagoras formed a cult and mathematics was closely related to the love of wisdom.

Pythagoras theorem was probably derived from the Babylonians - did Pythagoreans present a proof?

Did the Pythagoreans write the first 2 books of the Elemens of Geometry by Euclid? - could have!

Anaxagoras

While in prison for his theories of nature he tried to solve the famous problem "Find a square exactly equal in area to a circle constructed using compasses and straight edge alone" - very different from Egyptians and Babylonians.

Just after Anaxagoras died in 428BC there was a plague and to stop the plague the Athenians decided that the size of the cubical altar to Apollo had to be doubled - led to the famous problem - "construct with compasses and straight edge the edge of a 2nd cube having double the volume".

About the same time there was another tproblem - construction the trisection of a given angle.

3 problems known as "3 classical problems" of antiquity - since proved insoluable.

The discovery that line segments are incommensurable

2 quantities are incommensurable when they do not have a ratio such as a whole number has to a whole number.

Greek community stunned to find that line segments are incommensurable.

The Age of Plato and Aristotle (c 4 BC)

The pythagopreans had defined a point as "unity having position" but Plato said it was the beginning of a line. The definition of a line as 'breadthless length' seems to have originated from Plato.

Aristotle started to look at definitions and hypotheses.


The Golden Age of Greek Mathematics

Introduction

History of Mathematics Module

Links to other History of Mathematics sites


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These pages are maintained by M.I.Woodcock.