MATHEMATICS project

MATHEMATICS PROJECT: 

MATHEMATICS PROJECT SUBMITTED BY :-Tanmoy Pramanik Class IX–A JNV Kokrajhar TOPIC :- EUCLID’S GEOMETRY

INTRODUCTION TO EUCLEID’S GEOMETRY: 

The word Geometry comes from the Greek word ‘geo’ meaning earth and ‘metrein’ meaning to measure. Geometry appears to have originated from the need for measuring land. This branch of mathematics was studied in various forms in every ancient civilizations like in Egypt Babylonia, China, India, Greece, the Incas etc. The Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions . The knowledge of geometry was also used by them to calculating volumes of granaries and for constructing canals and pyramids. They also knew the correct formula to find the volume of a truncated pyramid. INTRODUCTION TO EUCLEID’S GEOMETRY

Development of geometry in ancient India: 

Indus Valley Civilization :-In the Indian Subcontinent the excavations at Harappa and Mohenjo-Daro etc. show that the Indus Valley civilizations (3000BC) made extensive use of geometry .It was a highly organized society .The cities were highly developed and very well planned . For example ,the roads were parallel to each other and there was an underground drainage system .The houses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and practical arithmetic . The bricks used for constructions were kiln fired and the ratio length : breadth: thickness, was found to be 4:2:1. Development of geometry in ancient India

Slide 4: 

VEDIC PERIOD :- In ancient India, ,the sulbasutras (800BC to 500BC)were the manuals of geometrical constructions. The geometry of the Vedic periods originated with the construction of Altars (vedis) and fireplaces for performing Vedic rites. The location of the sacred fires had to be in accordance to clearly laid down instructions about their shapes and areas, if they were to be effective instruments. Square and circular altars were used for household rituals, while altars whose shapes were combination of rectangles, triangles and trapeziums were required for public worship. The Sriyantra consists of nine interwoven isosceles triangles. These triangles are arranged in such away they produce 43 subsidiary triangles. Though accurate geometric methods were used for the constructions of the altars, their principles behind them were not discussed.

The beginning of proofs : 

The beginning of proofs A Greek mathematician ,Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts)by its diameter. One of Thales’ most famous pupils was Pythagoras (572BC), whom you have heard about. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300BC. At that time Euclid , a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called “Elements”. He divided the ‘Elements’ into thirteen chapters , into thirteen chapter each called a book. These books have influenced the whole world’s understanding of geometry for generations to come.

Euclid's Definitions, Axioms, and Postulates : 

Euclid's Definitions, Axioms, and Postulates A solid has shape, size, position , and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another, and are said to have no thickness. The boundaries of the surfaces are curves or straight lines. These lines end in points. Euclid began his exposition by listing 23 definitions in Book1 of the ‘ Elements ‘.A few of them are listed below 1)A point is that which has no part. 2)A line is breadth less length . 3)The end of a line are points . 4)A straight line is a line which lies evenly with the points on itself. 5 )A surface is that which has length and breadth only. 6)The edges of a surface are lines. 7)A plane surface is a surface which lies evenly with the straight lines on itself.

Contd…: 

Contd… Postulates:- Euclid used the term ‘Postulates' for the assumptions that were specific to Geometry. Axioms:- Common notions called axioms, on the other hand , were assumptions used throughout mathematics and not specifically linked to geometry. Some of the Euclid’s Axioms are:- 1)Things which are equal to the same things are equal to one another . 2)If equals are added to equals ,the wholes are equal. 3)If equals 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 things are equal to one another. 7)Things which are halves of the same things are equal to one another.

Postulates: 

Postulates Postulate 1:-A straight line can be drawn from any one point to the other. Axiom5.1:-Given two distinct points, there is a unique line that passes through them.Fig-8a P Q Postulate 2:-A terminated line can be produced indefinitely -------------- --------------- A B

Postulates: 

Postulates Postulate3:- A circle can be drawn with any centre any radius. Postulate4:- All right angles are equal to one another. Postulate5:- 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 lass than two right angles.

Euclid’s Geometry: 

A system of axioms is called consistent, if it is possible to deduce from these axioms a statement that contradicts any axiom or previously proved statement. So when any system of axioms is given, it needs to be ensured that the system is consistent. After Euclid stated his postulates and axioms he used them to prove other results . Then using these results he proved some more results by applying deductive reasoning. The statements that were proved are called Propositions or Theorems. Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions and theorems proved earlier in the chain. Euclid’s Geometry

Equivalent Versions of Euclid’s fifth postulates.: 

Equivalent Versions of Euclid’s fifth postulates. Euclid’s fifth postulate is very significant in the history of mathematics. We know that, no intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180 0 .There are several equivalent versions of this postulates .One of them is ‘Playfair’s Axiom.’ m P l

Information: 

Information This PowerPoint Presentation is Prepared By:-Tanmoy Pramanik Class:-IX-A Roll No:-38 Guided By:-Sir. Ajay Kumar Singh (PGT Maths )