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csPresented By – Akshay Kumar Kushawaha

Class – 10th, B Roll No. 03 Radiant Academy

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Euclid was a Greek mathematician, often referred to as the "Father of Geometry". Euclid was born in 365 B.C. He went to school at Plato's academy in Athens, Greece. He founded the university in Alexandria, Egypt. He taught there for the rest of his life. One of his students was Archimedes.Euclid was kind, fair, and patient. Once, when a boy asked what the point of learning math was, Euclid gave him a coin and said, "He must make gain out of what he learns." Another time, he was teaching a king. When the king asked if there was an easier way to learn geometry Euclid said, "There is no royal road to geometry." Then he sent the king to study.In his time he was thought of as being too thorough. Now, in our time, we think he wasn't thorough enough. Euclid died in 275 B.C.Euclid's most famous work was the Elements. This series of books was used as a center for teaching geometry for 2,000 years. It has been translated into Latin and Arabic.The Elements were divided into thirteen books, which subjects are as follows: Books 1-6= plane geometry, books 7-9= number theory, book 10= Eudoxus's theory of irrational numbers, and books 11-13= solid geometry. More than 1,000 editions of Elements have been published since 1482. Elements were popular until the 20th century.

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Euclid’s Biography

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Continued………….Born: c. 365 BCBirthplace: Alexandria, EgyptDied: c. 275 BCLocation of death: Alexandria, EgyptCause of death: unspecifiedOccupation: Mathematician, EducatorNationality: Ancient GreeceExecutive summary: Father of geometryUniversity: Plato's Academy, Athens, GreeceTeacher: Library of Alexandria, Alexandria, Egypt Asteroid Namesake 4354 Euclides

Lunar Crater Euclid (7.4S, 29.5W, 11km dia, 700m height)

Eponyms Euclidean geometry Slave-owners

Author of books: Elements (13 volumes) Data (plane geometry) On Divisions (geometry) Optics (applied mathematics) Phenomena (astronomy)

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EUCLID’S DEFINITONS Some of the definitions made

by Euclid in volume I of ‘The Elements’ that we take for granted today are as follows :-

A point is that which has no part.

A line is breadth less length. The ends of a line are points. A straight line is that which

has length only.

Euclid's construction of a regular dodecahedron.

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Continued………….A surface is that which has length and

breadth only.The edges of a surface are lines A plane surface is a surface which lies evenly

with the straight lines on itself o Axioms or postulates are the assumptions

which are obvious universal truths. They are not proved.

o Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning .

EUCLID’S AXIOMsSOME OF EUCLID’S AXIOMS WERE :- Things which are equal to the same

thing are equal to one another. i.e. if a=c and b=c then a=b. Here a, b and c are same kind of

things. If equals are added to equals, the

wholes are equal. i.e. if a=b and c=d, then a+c = b+d Also a=b then this implies that a+c =

b+c .

Continued….. If equals are subtracted, the

remainders are equal. Things which coincide with one

another are equal to one another. Things which are double of the same

things are equal to one another The whole is greater than the part.

That is if a > b then there exists c such that a =b + c. Here, b is a part of a and therefore, a is greater than b.

Things which are halves of the same things are equal to one another.

EUCLID’S FIVE POSTULATES

EUCLID’S POSTULATES WERE :-POSTULATE 1 :- • A straight line may be drawn from any

one point to any other pointAxiom :- • Given two distinct points, there is a

unique line that passes through themPOSTULATE 2 :- • A terminated line can be produced

infinitely

Continued….. POSTULATE 3 :- • A circle can be drawn with any centre

and any radius POSTULATE 4 :- • All right angles are equal to one another POSTULATE 5 :- • If a straight line falling on two straight

lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

THEOREMS WITH PROOF THEOREM :- Two distinct lines cannot have more than one point

in common PROOF :- Two lines ‘l’ and ‘m’ are given. We need to prove

that they have only one point in common Let us suppose that the two lines intersects in two

distinct points, say P and Q That is two line passes through two distinct points

P and Q But this assumptions clashes with the axiom that

only one line can pass through two distinct points Therefore the assumption that two lines intersect

in two distinct points is wrong Therefore we conclude that two distinct lines

cannot have more than one point in common

Euclid Division lemmaTHEOREM :-Given positive integers a and b, there

exist unique integers q and r satisfying a = bq + r, ≤ r < b.

Thank You