# Euclid’s Contribution in Mathematics Euclid or Euclid of Alexandria was a great Greek mathematician. He is also known as the Father of Geometry. There is no authentic information about his death and birth dates, but it is assumed that he was born and brought up in the era of Ptolemy. He gave many theorems in mathematics, more precisely in geometry. He is mainly known for his work in the field of geometry. He used to study the work of the mathematicians who preceded him. His main motive was to synthesis their work, i.e., rearranging their work to find something new. In his findings, he mostly used a deductive approach. He used the facts that were known to him and derived various theorems out of them, and he compiled his findings in his most famous book titled ‘The Elements.’ Many of the facts mentioned in this book were already known, but initially, they were just statements. Euclid imparted proofs to those axioms. His book ‘The Elements’ serves as a textbook in many schools to date.

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## 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.

### Meaning of Axioms

Mathematics is often known to be the study of logic. Every problem defined here is supposed to have a solution. Mathematical problems are mostly solved with the help of theorems, formulae, and axioms. Although theorems and formulae require proper mathematical proof, there are some statements in mathematics that are accepted and self-evidently true; these are known as axioms. Euclid gave 10 axioms and subdivided them into 5 axioms and 5 postulates. The difference between axiom and postulate is that postulates are meant for a specific field like geometry, whereas axioms are applicable in every field of science.

## 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.

## Euclid’s  ‘The Elements’

#### Euclid’s ‘Elements’ contains 465 propositions, 93 problems and 372 theorems in thirteen volumes. Let us mention some widely used results.

1. Euclid gave the proof of a fundamental theorem of arithmetic, i.e., ‘every positive integer greater than 1 can be written as a prime number or is itself a prime number’. For example, 35= 5×7, etc.

2. He was the first one to state that ‘There are infinitely many prime numbers, which is also known as Euclid’s theorem. He further elaborated this statement by saying that ‘for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (suppose P=36 and 36= 2×3×6, where {p}_{1}=2, {p}_{2}=3,{p}_{3}=5 (6-1), which shows that there is a prime no. 5 which was not there in the list of factors initially. It is a process, which can be repeated infinitely.

3. First four perfect numbers were identified by Euclid only. Perfect numbers are the numbers that are sum of all of their divisors, excluding the number itself. For example,

6= 1+ 2+3

28 = 1+2+4+7+18

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

8128 = 1+2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

He also observed some very interesting properties of perfect numbers.

a) They are triangular numbers, so they are the sum of all consecutive numbers up to their largest prime factor. For example, 6=1+2+3, 28=1+2+3+4+5+6+7, 496=1+2+3+…..+30+31, 8128=1+2+3+…..+126+127

b) Their largest prime factor is a power of 2 less one, and the number is a multiple of this number and power of 2 of the previous factor.

For example, 6 = 21(22 – 1); 28 = 22(23 – 1); 496 = 24(25 – 1); 8,128 = 26(27 – 1)

4. He gave the proof of the Pythagoras theorem. Although there were hundreds of proof already in existence his proof was exact and lucid. He used only those propositions and definitions that he had already proved to be true. He used geometry instead of algebra to derive this proof. 5. Maybe the Golden ratio (also called the golden mean or golden section) was first known to Pythagoras, its perfect explanation was given by Euclid. Two quantities a and b are said to be in the Golden ratio if (a+b)/a=a/b=φ, where the Golden ratio is denoted by the Greek letter φ. It comes out to be an irrational number whose value is 1.618…..Its application in real life is that it is used to determine the proportions of natural and man-made objects. The golden ratio also exists in nature in the form of a spiral arrangement of leaves and other parts of plants.

6. He also formulated Euclidean Algorithm, which states that

The greatest common divisor (GCD) of two integers A and B is the largest integer that divides both A and B.”

For example, Let A= 36 and B=48, then GCD (36, 48) is calculated as:

If A≠0, B≠0, then by Long division method we can write 48=1×36+12, then we will calculate GCD (36,12) as GCD (36,48)=GCD (36,12)

Here A=36 and B=12, then by long division method, we can write 36=12×3+0. So, GCD (36,48)=3. We will stop this process when either A or B will become 0.

## Phrases by Euclid

What is asserted without proof, can be denied without proof.”

There is no real way to Geometry.”

## Findings beyond ‘The Elements’

### Data

It is comprised of 96 geometric propositions and all of them follow the same format.

### On Division (of Figures)

It was extracted in 1915 from Arabic and Latin versions. It comprises problems of dividing figures into various ratios by drawing a number of straight lines.

### Euclidean Catoptrics

This book deals with the phenomena of reflected light and image forming optical systems using mirrors. This arrangement is also known as catopter. ### Phaenomena

This book deals with mathematical astronomy. It means how stars and the moon move. He used a mathematical approach to study when stars rise above the horizon and when they set below it.

### Optics

In the view of Euclid, the approximate height of an object is measured in terms of the angle formed by the straight lines passing from the top and bottom of the object reaching the observer’s eye. He also proposed that nearer objects appear to be bigger and seem to move faster. Adding further he discovered how to measure the height of distant objects from their shadow. ## Lost Works

### Conics

This is one of the most studied topics of Euclidean geometry. Conic is defined as a curve that is obtained by cutting (known as the cutting plane) the surface of double cones. According to Euclid, there are three types of conics, i.e., ellipse, hyperbola and parabola. However, the fourth type of conics was introduced by Apollonius, i.e., circle. ### Pseudaria (“fallacies”)

This book contains the text that will help beginners learn about the fallacies of geometrical reasoning.

### Surface Loci

This book deals with the study of loci on the surfaces or loci which are themselves surfaces. It is assumed that his findings might relate to quadratic surfaces.

### Porisms

Euclid was the first to talk about Porisms. These are neither theorems nor problems. They lie between theorems and problems. These are three different classes as specified by Euclid. However, this definition of Porisms was rejected by Pappus.