The Death of Alexander the Grat led to internal strife but by 306BC control of the Egyptian portion of the empire was in the hands of Ptolemy 1.

He established a school at Alexandria and Euclid became a teacher there.

5 works of Euclid have survived 'Elements', 'Data', 'Division of Figures', Phaenomena' and 'Optics'.

Euclid was a good teacher - no new discovery is attributed to him, he just wrote 'Elements' as a textbook.

University students were being presented with a textbook ('Elements') which gave them the fundamentals of elementary mathematics (geometry and algebra).

'Elements' is divided into 13 books of which the first half dozen were at elementary plane geometry next 3 on numbers, book X on incommensurables and the last 3 on solid geometry.

Definitions do not really define because they use words which are no better known than the word being defined.

eg. The Euclidean definition of a plane angle as "the inclination to one another of 2 lines in a plane which meet one another and do not lie in a straight line" is not very good because inclination has not being previously defined and is not better known than the word "angle".

Following the definitions Euclid listed 5 postulates and 5 common notions (axioms).

Postulates - something to be demanded

Axioms - known and accepted as obvious

**Postulates** - Let the following be postulated

- To draw a straight line from any point to any point.

- To produce a finite straight line continuously in a straight line.

- To describe a circle with any centre and radius.

- That all right angles are equal.

- That if a straight line falling on two straight lines makes the interior angles on the same side less than 2 right angles, the 2 straight lines if produced indefinitely meet on the side on which the angles are less than 2 right angles.

- Things which are equal to the same thing are also equal to one another.

- If equals be added to equals the wholes are equal.

- If equals be subtracted from equals the remainders are equal.

- Things which coincide with one another are equal to one another.

- The whole is greater than the part.

Postulates 1 and 2 guarantee neither the uniqueness of the straight line through 2 non-coincident points nor even its infinitude - Euclid often assumes this.

But most tightly reasoned logical arguments so far developed.

Included in book 1 are the theorems on congruence of triangles, on simple constructions by straight edge and compasses, on inequalities concerning angles and sides of a triangle on properties of parallel lines and parallelograms.

Book closes with a proof of Pythagoras Theorem and its converse.

eg. proposition 1 "if there be 2 straight lines and one of them be cut into any number of segments whatever, the rectangle contained by the 2 straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

This theorem which asserts

AD(AP + PR + RB) = AD.AP + AD.PR + AD.RB is a geometrical statement of

a(b + c + d) = ab + ac + ad

(AD and AB are the 2 straight lines - implied is the 90

In Euclid's day magnitudes were pictured as line segments satisfying the axioms and theorems of geometry. In other words they had geometrical algebra.

Propositions 12 and 13 are geometric formulations of the Cosine Rule.

Last proposition is the familiar statement that if from a point outside a circle a tangent and a secant are drawn, the square on the tangent is equal to the rectangle on the whole secant and the external segment.

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