Transcript Euclid - Pegasus @ UCF

Euclid
Very little is known about his life
Professor 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 algebra
Book III - Circles, chords, secants,
tangents and measurement of associated
angles
Euclid’s Elements
Book IV - Construction of regular polygons
Book V - Eudoxus’ theory of proportion
Book VI - Theory of proportion to plane
geometry
Euclid’s Elements
Books VII,VIII,IX - Elementary number
theory
Book X - Irrationals
Books 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
1
x  a 1  5
2
3. Construct the isosceles triangle ABC
with sides x,a and a.
4. Circumscribe a circle about the triangle
5. Complete the pentagon
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
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.