Euclid

Very little is known about his lifeProfessor at the university of Alexandria“There is no royal road to geometry”

Euclid’s Elements

No work , except the Bible has been more widely used

Over 1000 editions since first printed in 1482

No copy of Euclid’s Elements has been found that dates to the author’s time

First complete English translation, 1570

Euclid’s Elements

A highly successful compilation and systematic arrangement of works of other writers

The work is composed of 13 books with a total of 465 propositions

Contrary to widespread impressions, it is not devoted to geometry alone, but contains much number theory and elementary (geometric) algebra.

Euclid’s Elements

Book I - Definitions, Pythagorean Theorem

Book II - Geometric algebraBook III - Circles, chords, secants,

tangents and measurement of associated angles

Euclid’s Elements

Book IV - Construction of regular polygonsBook V - Eudoxus’ theory of proportionBook VI - Theory of proportion to plane

geometry

Euclid’s Elements

Books VII,VIII,IX - Elementary number theory

Book X - IrrationalsBooks XI,XII,XIII - Solid geometry

Proposition I-6

If in a triangle two angles are equal to one another, then the opposite sides are also equal.

Proposition II-1

If there are two straight lines, and one of them is cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the uncut straight line and each of the segments.

Proposition II-11

To divide a given straight line into two parts so that the rectangle contained by the whole and one of the parts is equal in area to the square on the other part.

Proposition III-16

The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed.

Proposition IV-10

How to construct an isosceles triangle with each base angle equal to two times the vertex angle.

Construction of the Regular Pentagon

1. Take an arbitrary line segment; let a be its length

2. Construct a line segment of length

3. Construct the isosceles triangle ABC with sides x,a and a.

4. Circumscribe a circle about the triangle

5. Complete the pentagon

x=12a −1+ 5( )

Proposition VII-31

Any composite number is measured by some prime number.

Proposition VII-32

Any number is either prime or measured by some prime.