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
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.
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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