Contribution of Euclid

1. JANT, PALI, MAHENDRAGARH
COURSE :- PEDAGOGY OF MATHEMATICS
ASSIGNMENT ON :- Contribution of Mathematician (Euclid and Brahmgupta)
Submitted To:- Submitted By:-
Dr. Meenakshi Rawat Name:- Sasmita Barik
Assistant Professor Roll No. -221911
Department of Teacher’s Education B.Ed. Sem.- 2nd
Central University of Haryana Session:- 2022-24

2. INTRODUCTION
 Euclid was a great Greek mathematician.
 Although little is known about his early and
personal life, he went on to contribute
greatly in the field of mathematics and came
to known as the ‘Father of Geometry’, Euclid
is known to have taught mathematics in
Ancient Egypt during the reign of Ptolemy I.
 He wrote ‘Elements’ the most influential
mathematical works of all time, which
served as the main textbook for teaching
mathematics from its publication until the
late 19th or early 20th century.
 Elements aroused interest of the Western
World and mathematicians around the globe
for over 2000 years.

3. • Apart from being a tutor at the
Alexandria library, Euclid coined and
structured the different elements of
mathematics, such as Porisms,
geometric systems, infinite values,
factorizations, and the congruence of
shapes that went on to contour
Euclidian Geometry.
• His works were heavily Influenced by
Pythagoras, Aristotle, Eudoxus, and
Thales to name a few.
• Euclid used the ‘synthetic approach’ towards producing his theorems,
definitions and axioms

4. WORKS
 ‘The Elements’ (originally written in ancient Greek) became a standard work
of important mathematical teachings. It is divided into 13 books.
 Books one to six deal with plane geometry.
 Books seven to nine deal with number theory.
 Book eight is on geometrical progression.
 Book ten deals with irrational numbers.
 Books eleven to thirteen deal with three-dimensional geometry
In his Elements, he deduced the principles of ‘Euclidean geometry’ from a
small set of axioms.
Euclid also wrote works on perspective, conic sections, spherical geometry,
number theory and rigor.
In addition to his most famous work ‘Elements’, there are at least five works of
Euclid that have survived to this day.
EUCLID’S ‘THE ELEMENTS’

5. They seem to follow the same logical structure as followed in Elements.
They are ‘Data’, ‘On Divisions of Figures’, ‘Catoptrics’, ‘Phaenomena’
and Optics)
In addition to the abovementioned works, there are a few other works
that are attributed to Euclid but have been lost.
These works include ‘Conics’, ‘Pseudaria’, ‘Porisms’, ‘Surface Loci’ and
‘On the Heavy and the Light’.
1.His work on prime numbers
 Euclid’s lemma – which states a fundamental property of prime
numbers is that – If a prime divides the product of two numbers, it must
divide at least one of those numbers.
 The fundamental theorem of arithmetic or the unique-prime-factorization
theorem.
 Using Euclid’s lemma, this theorem states that every integer greater
than one is either itself a prime or the product of prime numbers and
that there is a definite order to primes.

6. 2. Euclid’s Lemma
Recall the definition of relatively prime integers:
Let a and b be any two integers, not both zero. Then a and b are said to
be relatively prime if gcd(a,b) =1.
Now consider the definition of prime integers:
Let p be an integer such that p>1. Then p is said to be prime if its only
positive divisors are 1 and p itself..
• The definition only applies to positive integers.
• One is not prime.
• Natural numbers that are not prime are said to be composite.
3. Euclidean algorithm – an efficient method for computing the greatest
common divisor (GCD) of two numbers, the largest number that divides
both of them without leaving a remainder.
• Euclid described a system of geometry concerned with shape, and
relative positions and properties of space.
• It was Euclid who put geometry into axiomatic forms (logically derived
theorems) His work is known as Euclidean geometry.

7. EUCLID’S AXIOMS
1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals then wholes are equal.
3. If equals are subtracted from equals then the reminders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than a part.
EUCLID’S POSTULATES
1. It is possible to draw a straight line from any point to another point.
2. It is possible to extend a finite straight line continuously in a straight line.
3. It is possible to create a circle with any centre and radius.
4. All right angles are equal to one another.
5. If a straight line crossing 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 the two right angles.

8. EUCLID’S CATOPTRICS
This book deals with the phenomena of reflected light and image
forming optical systems using mirrors. This arrangement is also
known as catopter.