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INTRODUCTION TO EUCLID`S GEOMETRY BY Al- Muqtadir Hussain,

Rasidul Islam , and

Muffajul Hussain

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Submitted on -Friday, October 04, 2013

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GEOMETRY AROUND US

Our daily life is filled with geometry—the pure mathematics of points, lines, curves and surfaces. We can observe various shapes and angles in the objects that surround us. Observe, for example, this table and its rectangular surface; the boomerang and its angular shape; the bangle and its circular shape.

 Euclid, an ancient Greek mathematician, observed

the various types of objects around him and tried to define the most basic components of those objects. He proposed twenty-three definitions based on his studies of space and the objects visible in daily life. Let us go through this lesson to learn each of Euclid’s definitions.

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PREFACE

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid , which he described in his textbook on geometry : the Elements Euclid's method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions (theorems ) from these. Although many of Euclid's results had been stated by earlier mathematicians,

Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system . The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory ,explained in geometrical language .This entire project aims

at the explanation of the very complicated Euclid geometry , so that the students are able to see and understand it in a better way. The entire work is done by three students of class 9, Al- Muqtadir Hussain,

Rasidul Islam , and Muffajul Hussain

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CONTENTS

Definitions of Euclid Introduction to

axioms Axioms- Axiom i and ii Axiom iii Axiom iv and v Axiom vi and vii

Introduction to

Postulates

Postulates- Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Conclusion

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DEFINITIONS OF EUCLID

Euclid gave the definitions of a few very basic attributes of objects that are normally around us. These definitions are listed below.

1. A point is that which has no part.

 

2. A line is a breadth-less length.

 

3. The extremities of a line are called points.

 

4. A straight line is one that 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 one that lies evenly with the straight lines on itself.

 

 

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8. A plane angle is the inclination to each other of two lines in a plane, which meet each other and do not lie in a straight line.

  9. When the lines containing the angle are straight, the angle

is called rectilinear.

  10. When a straight line set up on another straight line makes

the adjacent angles equal to each other, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands.

  11. An obtuse angle is an angle greater than the right angle.

  12. An acute angle is an angle less than the right angle.

  13. A boundary points out the limit or extent of something.

  14. A figure is that which is contained by any boundary or

boundaries.

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INTRODUCTION TO AXIOMS Euclid’s Axioms  Euclid assumed certain properties to be universal truths that

did not need to be proved. He classified these properties as axioms and postulates. The properties that were not specific to geometry were referred to as common notions or axioms.

  He compiled all the known mathematical works of his time

into the Elements. Each book of the Elements contains a series of propositions or theorems, varying in number from about ten to hundred. These propositions or theorems are preceded by definitions. In Book I, twenty-three definitions are followed by five postulates. Five common notions or axioms are listed after the postulates.

  In this lesson, we will study some of Euclid’s axioms.

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AXIOMS:

Things that are equal to the same thing are also equal to one another (Transitive property of equality).

If equals are added to equals, then the wholes are equal.

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

Things that coincide with one another are equal to one another (Reflexive Property).

The whole is greater than the part.

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AXIOMS I AND II Let us start with the first axiom which states that things

that are equal to the same thing are also equal to one another.

Let us suppose the area of a rectangle is equal to the area of a triangle and the area of that triangle is equal to the area of a square. Then, according to the first axiom, the area of the rectangle is equal to the area of the square. Similarly, if a = b and b = c, then we can say that a = c.

  Now, the second axiom states that if equals are added to

equals, then the wholes are equal. Let us take a line segment AD in which AB = CD. Let us add BC to both sides of the above relation (‘equals

are added’). Then, according to the second axiom, we can say that AB + BC = CD + BC, i.e., AC = BD.

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AXIOM III The third axiom states that if equals are subtracted

from equals, then the remainders are equal. Let us consider the following rectangles ABCD and

PQRS. Suppose the areas of the rectangles are equal. Now,

let us remove a triangle XYZ (as shown in the figure) from each rectangle. Then, according to the third axiom, we can say that the area of the remaining portion of rectangle ABCD is equal to the area of the remaining portion of rectangle PQRS.

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AXIOM IV AND V

The fourth axiom states that things that coincide with one another are equal to one another.

This axiom is sometimes used in geometrical proofs. Let us consider a point Q lying between points P and R of a

line segment PR, as is shown in the figure. We can see that (PQ + QR) coincides with the line segment

PR. So, as per the fourth axiom, we can say that PQ + QR = PR.

Now, the fifth axiom states that the whole is greater than the part.

Let us again consider the line segment PR shown above. We can see that PQ is a part of PR. So, as per the fifth axiom, we can say that PR (i.e., the whole) is greater than PQ (i.e., the part). Mathematically, we write it as PR > PQ.

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AXIOM VI AND VII The sixth and seventh axioms are interrelated. The former states that things

that are double of the same things are equal to one another, while the latter states that things that are halves of the same things are equal to one another. 

Let us consider two identical circles with radii r1 and r2. Also, suppose their diameters are d1 and d2 respectively. 

As the circles are identical, their radii are equal.  ∴ r1 = r2  Now, as per the sixth axiom, we can say that 2r1 = 2r2  ∴ d1 = d2  Hence, we can say that if two circles have equal radii, then their diameters

are also equal. Now, instead of taking the radii as equal, let us say that the diameters of the

two circles are equal. Then, as per the seventh axiom, we can say that the radii of the two circles are also equal.

 

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INTRODUCTION TO POSTULATES

Certain things are considered universal truths that need not be proved. Consider, for example, the following: the sun rises from the east; Sunday comes after Saturday; March has 31 days. These things are universally true; hence, they do not need to be proven. Similarly, certain geometrical properties are regarded as universal truths. Euclid identified and presented such properties in the Elements. The properties specific to geometry were classified by him as postulates. In Book I, twenty-three definitions are followed by five postulates. Let us learn these postulates in this lesson.

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POSTULATES

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

"To produce [extend] a finite straight line continuously in a straight line."

"To describe a circle with any centre and distance [radius]."

"That all right angles are equal to one another." The parallel postulates : "That, if a straight line

falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

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1ST POSTULATE Postulate 1: A straight line may be drawn from any point to any other point.

Proof: Finally proved only yesterday, we must refer to the third and second postulate in order to fully prove this one. In order to prevent accusations of lack of rigor, I will use the still incomplete third postulate only in those cases where it may be applied.

Take any two collinear points A and B, where collinear means it is possible to draw a straight line between them. It is possible therefore to draw a straight line between them.

Now any points on the line AB must also be collinear, for otherwise a straight line could not have been drawn. Hence, it is also possible to draw a line from A to any point upon the line.

Now, let us rotate the line, such that the collinear point A is the centre of the circle so produced.

Now, it is possible to draw a straight line from A to any point in the circle. This is because the radius of the circle is a straight line, and upon rotation, it covers all the points in the circle, implying that a straight line can be drawn from all the points covered by the radius to the center of the circle, which is A.

Therefore all points in the circle are collinear to A i.e. they produce a straight line to A.

It is easy to show that all points in the plane are collinear: merely extend the radius infinitely, so the resultant circle encompasses the entire region.

Repeating the above for any point in the circle, we see that it is possible to draw a straight line from that point to any other point in its circle, and so on.

From the information above, we can deduce that all points are collinear to each other, or

It is possible to draw a straight line from any one point to any other point.

Hence proved.

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2ND POSTULATE Postulate 2: A finite straight line may be extended

indefinitely.

Proof: There are an infinite number of points in a region.

This implies that there are an infinite number of collinear points, as any operation with infinity that does not involve another infinity results in infinity. By collinear, I mean points between which a straight line may be drawn. ( I clarify this in order to prevent accusations of using a circular argument with the first postulate)This implies that a line may be extended infinitely.

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3RD POSTULATE Postulate 3: A circle may be drawn with any center and any radius.

Note: By the term "collinear", I mean that it is possible to draw straight line from it another specific point.

Proof: This is a little trickier to prove, so I divided the problem down into two parts. I will first prove that a circle may have any radius.

Taking point A as centre, we may look at the radius as a line. By Postulate 2, we know that line may be extended indefinitely.

Therefore, the radius may be extended indefinitely.

This implies that a circle may have any radius.

The second part is to prove that a circle may have any center.

Taking any collinear point, we see that it is possible to draw a straight line between this and any other straight line.

By rotating the line by 360 degrees, we obtain a circle.

This implies that any collinear point may be the center of a circle, as the straight line that can be drawn may be considered a radius, and rotating the radius produces a circle.

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4TH POSTULATE Postulate 4: All right angles are equal to each other.

Proof: Let us assume that this is not true, and all right angles are not equal to each other.

This instantly leads to a contradiction, as it implies that a triangle may have more than one right angle.

Therefore, by reduction ad absurdum, we see that all right angles must be equal to each other.

Another proof is by looking at the definition of a right angle. A right angle is any angle equal to 90 degrees.

We know that 90 = 90 = 90 ...

We see therefore that all right angles are equal to 90 degrees and as 90 degrees is equal to 90 degrees,

this implies that all right angles are equal to each other.

Hence proved.

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5TH POSTULATE Postulate 5: If a straight line falling on two straight lines makes the interior

angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

Proof: This means to say that the two lines meet on the side whose sum is less than 180 degrees.

From the diagram, we see that if we extend the lines indefinitely, we eventually get a triangle.It has been proved that the sum of the angles of a triangle sum to 180 degrees. This implies that the two angles formed by the third line which goes through the other two lines) cannot be equal to 180 degrees, as it would then violate the angle sum property of that triangle.The triangle is only possible if the two angles are not equal to 180 degrees or more. This implies that the lines may only meet on those sides where the angles together sum up to less than 180 degrees.

Therefore, this implies that no line which produces an obtuse or acute angle may be parallel to l. Therefore, the only other angle possible is a right angle, which taken in conjunction with the other interior angle forms 180 degrees, implying that any line which produces 90 degrees with the perpendicular is always parallel to line l.

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CONCLUSION

We am thankful to Sir Rabin for contributing their valuable suggestions in improving this PowerPoint presentation. We are confident that this presentation would help to provide different pleasures.

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