Euclid zVery little is known about his life zProfessor at the university of Alexandria z“There is no royal road to geometry”

Axioms zThe axioms, or assumptions, are divided into three types: yDefinitions yProposition yCommon notions zAll are assumed true.

Definitions zThe definitions simply clarify what is meant by technical terms. E.g., y1. A point is that which has no part. y2. A line is breadthless length. y3. 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. … y4. 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 z5. 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 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 zFinally, Euclid adds 5 “common notions” for completeness. These are really essentially logical principles rather than specifically mathematical ideas: y1. Things which are equal to the same thing are also equal to one another. y2. If equals be added to equals, the wholes are equal. y3. If equals be subtracted from equals, the remainders are equal. y4. Things which coincide with one another are equal to one another. y5. The whole is greater than the part.

Euclid’s Elements zNo work, except the Bible has been more widely used zOver 1000 editions since first printed in 1482 zNo copy of Euclid’s Elements has been found that dates to the author’s time zFirst complete English translation, 1570

Euclid’s Elements zA highly successful compilation and systematic arrangement of works of other writers zThe work is composed of 13 books with a total of 465 propositions zContrary to widespread impressions, it is not devoted to geometry alone, but contains much number theory and elementary (geometric) algebra.

Euclid’s Elements zBook I - Definitions, Pythagorean Theorem zBook II - Geometric algebra zBook III - Circles, chords, secants, tangents and measurement of associated angles

Euclid’s Elements zBook IV - Construction of regular polygons zBook V - Eudoxus’ theory of proportion zBook VI - Theory of proportion to plane geometry

Euclid’s Elements zBooks VII,VIII,IX - Elementary number theory zBook X - Irrationals zBooks XI,XII,XIII - Solid geometry

References zhttp:// strations/lecture8/thales.html zhttp://schools.techno.ru/sch758/geometr/Euclid.htm zhttp:// zW. P. Belinghoff, F. Q. Gouvêa. (2002). Math Through The Ages. Oxton House Publishers, LLC: Farmington, ME. zhttp:// nE.jpg zhttp:// zhttp:// ola.gif