Transcript Euclid - MCS193

Euclid
Very little is known about his life
Professor at the university of Alexandria
“There is no royal road to geometry”
Axioms
The axioms, or assumptions, are divided
into three types:
Definitions
Proposition
Common notions
All are assumed true.
Definitions
 The definitions simply clarify what is meant by technical
terms. E.g.,
1. A point is that which has no part.
2. A line is breadthless length.
3. When a straight line set up on a straight line makes the
adjacent angles equal to one another, each of the equal angles
is right, and the straight line standing on the other is called a
perpendicular to that on which it stands. …
4. A circle is a plane figure contained by one line such that all
the straight lines falling upon it from one point among those lying
within the figure are equal to one another.
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.
The Controversial
Proposition V
d
e
a
c
f
b
g
 5. That, if a straight line falling on two straight lines
make the interior angles on the same side less than two
right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles
less than the two right angles.
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


The Common Notions
 Finally, Euclid adds 5 “common notions” for
completeness. These are really essentially logical
principles rather than specifically mathematical ideas:
1. Things which are equal to the same thing are also equal to
one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are
equal.
4. Things which coincide with one another are equal to one
another.
5. The whole is greater than the part.
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
References
 http://www.es.flinders.edu.au/~mattom/science+society/lectures/illu
strations/lecture8/thales.html
 http://schools.techno.ru/sch758/geometr/Euclid.htm
 http://www.smokelong.com/images/euclid.jpg
 W. P. Belinghoff, F. Q. Gouvêa. (2002). Math Through The Ages.
Oxton House Publishers, LLC: Farmington, ME.
 http://www.whatreallyhappened.com/DECLARATION/us_declaratio
nE.jpg
 http://www.cooperativeindividualism.org/lincoln-abraham.jpg
 http://www.siu.edu/~pulfrich/Pulfrich_Pages/lit_pulf/nick_thm/Parab
ola.gif