MATHS PROJECT

To Mrs. Pooja mam

BY.Nihas Kamarudheen.k

CLASS-IX-C

EUCLID

KNOWN AS THE FATHER OF

GEOMETRY

ABOUT EUCLID

Euclid, also known as the Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptomeli1 (323–283 BC). His E is one of the most influential works in the history of mathematics, serving as the main textbook for teaching (especially geometry) from the time of its publication until the late 19th or early 20th century. ] In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor."Euclid" is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory".[

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 . The diagram accompanies Book II, Proposition 5.

ELEMENTSAlthough many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

Definitions of Euclid 1. A point is that which has no parts .1. A line is breadth less length .

2. A straight line is a line which lies evenly with the points on itself.

3. A surface is that which has length and breadth only .

4. The edges of a surface are lines .5. A plane surface is a surface which lies evenly with the straight lines on it self ..

Euclid’s Axioms

1. Things which are equal to the same thing are equal to ne another .

2. If equals are added to equals ,the wholes are equals

3. If equal are subtracted from equals ,the remainders are equals .

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

5. The whole is greater than the part .

6. things which are double of the same thing are equal to one another .

7. things which are half of the same thing are equal to one another .

Euclid’s postulate's

A Straight line may be drawn from any one point to any other point . 1.

Note: that this postulates tell us that at least one straight line passes through the two distinct points ,but it does not say that there cannot be one such line.

2. A terminated line can be produced indefinitely.

Note: that we call a line segment now a days what Euclid call a terminated line. So , according to the present day terms , the second postulate says that a line segment can be extended on either side to form a line .

A

B

3. A circle can be drawn with any centre and any radius.

4. All right angles are equal to one another .

5. If a straight line falling on two straight line 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 .

A

B

C

D EF

Line ,Ray ‘N’Line segmentLine :

A straight line extends indefinitely in both directions . It has no definite length .Ra

y: A ray is a part of line with on end & the other

part extends indefinitely .

Line segment : A line segment is a

part of line having two end points .

Prove that equilateral triangle can be constructed by any given line segment .

In the statement above, a line segment of any length is given say AB

Using Euclid’s postulates 3 you can draw a circle with point A as the centre and AB as the radius. Similarly, draw another circle with point B as the center and BA as the radius. The two circles meet at a point say C .Now draw the line segment AC and BC to form a triangle ABC .

AB =AC, since they are the radii of the same circle

AB=BC,(radii of the same circle)

From these two facts and Euclid’s axiom that things which are equal to the same things are equal to one another, you can conclude that AB=BC=AC

A B

i

A B

C

ii A

C

Biii

So, ABC is a equilateral triangle

Euclid’s AchievementsEuclid is best known for his treatise on mathematics,

The Elements. His book is one of the most influential works in the history of mathematics, serving as the main textbook for teaching math, especially geometry, from the time of its publication until the late 19th or early 20th century. The Elements is a mathematical and geometric treatise consisting of 13 books written in Alexandria at around 300 BC. It is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry, such as definitions, propositions, theorems, constructions, the five postulates (axioms), and mathematical proofs of the propositions. Axioms are statements that are

THE

END

(𝑥+𝑎 )𝑛=∑𝑘= 0

𝑛

(𝑛𝑘)𝑥𝑘𝑎𝑛−𝑘

(1+𝑥 )𝑛=1+𝑛𝑥1 !

+𝑛 (𝑛−1 ) 𝑥2

2 !+…