EUCLID’SGEOMETRY

Snehal BhargavaIX - D

GEOMETRYGeometry is a branch of mathematics which deals

with questions related to shape, size, relative position of figures, and the properties of space.

A mathematician who works in the field of geometry is called a geometer.

Geometry is the study of angles and triangles, length, perimeter, area and volume.

Few of the known geometers are Euclid, Archimedes, René Descartes,

Euler and Gauss.Euclid

HISTORY OF GEOMETRY

Early geometry was a collection of principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various

crafts.

In the 7th century BC, a Greek mathematician used to solve problems such as calculating the height of pyramids and the

distance of ships from the shore.

Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful andinfluential textbook of all time, introduced the

axiomatic method and is the earliest example of the format still used in mathematics today, that of

definition, axiom, theorem, and proof.

The oldest surviving Latin translation of the Elements 

EUCLIDEuclid   was a Greek mathematician, who is known as

the "Father of Geometry".

His Elements is one of the most influential works in the history of mathematics, In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from

a small set of axioms.

Euclid’s life is not known. Nothing is known about his birth or death. Even description of Euclid's physical appearance made

during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's

imagination. It is believed that Euclid may have studied at Plato's Academy in Athens.

Few other works of Euclid include works on perspective, conic sections, spherical geometry, number theory and rigor.

EUCLID’S GEOMETRYIt includes sets of axioms, and many theorems

deduced from them.

Euclid was the first to show how these theorems could fit into a comprehensive deductive

and logical system.

It has 13 books, of which, books I–IV and VI discuss plane geometry; books V and VII–X deal with

number theory and books XI–XIII concern solid geometry.

Fragment of Euclid's Elements

BASIS OF EUCLID’S GEOMETRY

• The Elements is based on theorems proved by other mathematics supplemented by some original work.

• Euclid put together many of Eudoxus' theorems, many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors.

• Most of books I and II were based on Pythagoras,  book III on Hippocrates of Chios, and book V on Eudoxus , while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.

• Euclid often replaced misleading proofs with his own.• The use of definitions, postulates, and axioms dated back to

Plato.• The Elements may have been based on an earlier textbook by

Hippocrates of Chios, who also may have originated the use of letters to refer to figures

CONTENTS OF THE BOOK

• Books I–IV and VI discuss plane geometry.– Many results about plane figures are proved.– Pons Asinorum i.e. If a triangle has two equal angles, then

the sides subtended by the angles are equal is proved.– The Pythagorean theorem is proved.

• Books V and VII–X deal with number theory.– It deals with numbers treated geometrically through their

representation as line segments with various lengths.– Prime Numbers and Rational and Irrational numbers are

introduced.– The infinitude of prime numbers is proved.

• Books XI–XIII concern solid geometry.– A typical result is the 1:3 ratio between the volume of a

cone and a cylinder with the same height and base.

EUCLID’S DEFINITIONS• The Elements begins with a list of definitions.• It has been suggested that the definitions were added to the

Elements sometime after Euclid wrote them. Another possibility is that they are actually from a different work, perhaps older.

• Though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.

• Euclid deduced a total of 131 definitions. There were 23 in Book I, 2 in Book II, 11 in Book III, 7 in Book IV, 18 in Book V, 4 in Book VI, 22 in Book VII, 16 in Book X and 28 in Book XI.

EUCLID’S DEFINITIONSSome of the definitions are:1. A point is that which has no part. (D1, B1)2. A line is breathless length. (D2, B1)3. Equal circles are those whose diameters are equal, or whose radii are

equal. (D1, B3)4. Circles are said to touch one another which meet one another but do not

cut one another. (D3, B3)5. A straight line is said to be fitted into a circle when its ends are on the

circumference of the circle. (D7, B4)6. A ratio is a sort of relation in respect of size between two magnitudes of

the same kind. (D3, B5)7. Magnitudes which have the same ratio be called proportional. (D6, B5)8. The height of any figure is the perpendicular drawn from the vertex to the

base. (D4, B6)9. An even number is that which is divisible into two equal parts. (D6, B7)10.A solid is that which has length, breadth, and depth.(D1, B11)11.Parallel planes are those which do not meet. (D8, B11)

EUCLID’S AXIOMS & POSTULATES

• Euclid assumed certain properties, which were not to be proved. These assumptions are actually ‘obvious universal truths (Axioms)’. He divided them into two types:

1. POSTULATES – He used the term ‘postulate’ for the assumptions that were specific to geometry.

2. COMMON NOTIONS– Common notions, on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry.

EUCLID’S AXIOMS• Euclidean geometry is an axiomatic system, in which

all theorems ("true statements") are derived from a small number of axioms.

• These are ‘self-evident truths’ which we take to be true without proof.

• Axioms have been chosen based on our intuition and what appears to be self-evident. Therefore, we expect them to be true.

EUCLID’S COMMON NOTIONS/AXIOMSSome of the common notions are:

• Things which equal the same thing also equal one another.– If a=b and b=c, then a=c

• If equals are added to equals, then the wholes are equal.– If a=b, then a+c = b+c

• If equals are subtracted from equals, then the remainders are equal.– If a=b, then a-c=b-c

• The whole is greater than the part.– 1 > ½

• Things which are double of the same things are equal to one another.– If a=2b and c=2b, then a=b

• Things which are halves of the same things are equal to one another.– If a= ½ b and c= ½ b, then a=b

EUCLID’S POSTULATES• Each postulate is an axiom—which means a statement

which is accepted without proof— specific to the subject matter. Most of them are constructions. 

• ‘Postulate’ is actually a verb. When we say “let us postulate”, we mean, “let us make some statement based on the observed phenomenon in the Universe”. Its truth/validity is checked afterwards. If it is true, then it is accepted as a ‘Postulate’.

EUCLID’S POSTULATESEuclid gave 5 postulates for plane geometry: 1. A straight line may be drawn from any one point to any other

point. The first postulate says that given any two points such

as A and B, there is a line AB which has them as endpoints.2. A terminated line can be produced indefinitely.

The second postulate says that a line segment ( terminated line) can be extended on either side to form a line.

3. A circle can be drawn with any centre and any radius.4. All right angles are equal to one another.

This postulate says that an angle at the foot of one perpendicular, equals an angle at the foot of any other perpendicular.

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.

EUCLID’S PROPOSITIONS

• 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 using his axioms, postulates, definitions and theorems proved earlier in the chain.

• There were 48 propositions in Book I, 14 in Book II, 37 in Book III, 16 in Book IV, 25 in Book V, 33 in Book VI, 39 in Book VII, 27 in Book VIII, 36 in Book IX, 115 in Book X, 39 in Book XI, 18 in Book XII and 18 in Book XIII.

EUCLID’S PROPOSITIONSFew Euclid’s Propositions are:

1. In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. (P1.5)

2. If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. (P3.18)

3. If magnitudes are proportional taken separately, then they are also proportional taken jointly. (P5.18)

4. If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides. (P6.5)

5. If two numbers are relatively prime to any number, then their product is also relatively prime to the same. (P7.24)

6. If two similar plane numbers multiplied by one another make some number, then the product is square. (P9.1)

7. Any cone is a third part of the cylinder with the same base and equal height. (P12. 10)

Px. y – x = Book No. ; y = Proposition No.

THANK YOU

“I tell you that I accept God simply. But you must note this: If

God exists and if He really did create the world, then, as we all know, He created it according to

the geometry of Euclid.” - Ivan, in The Brothers Karamazov, by

Fyodor Dostoyevsky (1821-1881)