# Archimedes’ Contributions in Mathematics Archimedes, also known as the “Father of Mathematics,” contributed a lot in the field of Mathematics. Not only Mathematics, he made many valuable inventions in the field of Physics as well. Apart from being a mathematician, he was also an astronomer and made many significant contributions in the field of astronomy. In the life span of 75 years, he introduced various laws and theories related to mathematics. Today, we all are able to measure area and volume of different shapes with the help of relevant laws, theorems, and formulae, but we don’t know from where all these laws and formulae came. Area and volume of different 3D and 2D shapes were given by Archimedes. We can not think of mathematics without him and his inventions. Many more inventions made by him are there that we still have to acknowledge. Let us briefly discuss his inventions one by one.

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## 1. Exact Value of Pi

We all must have studied and used the concept of Pi in mensuration. Pi is the ratio of circumference and diameter of a circle. We divide 22 by 7 to get the value of 3.14. But do you know who approximated this value? It was none other than Archimedes. To calculate this value, he used a 96-sided polygon, and he found the value of Pi approximately between 3.1408 and 3.1429 that is widely used by all of us. ## 2. Volumes and Areas

His true love for Mathematics made him to introduce formulae for volume and areas of different new shapes by using the previous shapes known to him by the exhaustion method. For example, to calculate the area of circle, he enclosed the circle into a polygon and then drawing on more polygon inside that circle. Then he tried to calculate the area of the polygons drawn. He tried the same procedure with other shapes as well like enclosing the circle into triangle, square, pentagon and other shapes known to him. He approximated the area of these shapes and reached to a final conclusion that gave him the area of circle. This method is known as Exhaustion method.

According to Archimedes, the area enclosed by a straight line and parabola is 4/3rd of the area of triangle enclosed in it that has the same base and height as the segment. He used the concept of infinite geometric series by taking common ratio as 1/4. He divided the area under parabola segment (area formed by the parabola and line) into a infinite no. of triangles whose areas formed a geometric series. He then calculated the sum of this geometric series and proved it to be the area of parabolic segment. ## 4. Infinitesimal

Archimedes was the first who invent integral calculus, i.e., around 2000 years before Newton and Leibniz. Infinitesimal is that quantity that tends to infinity, means it tends to something that does not usually exist or is unreal. With the evolution of this concept, some other related concepts came into existence like continuity, differentiability, limits and integration . ## 5. Sphere and Cylinder (Ratio of Volume and Surface Area)

Archimedes was the first who came up with the ratio of volume and surface area of sphere and cylinder. This is considered one of the most significant contributions of Archimedes to mathematics, and even Archimedes himself considered it to be his most valuable contribution to this field as he established a relationship between a cylinder and a sphere (enclosed in cylinder) having same height and diameter. He proved that volume and surface area of a sphere is 2/3rd of that of a cylinder. He was so influenced by this discovery that he requested to built his tomb in the shape of a cylinder having a sphere circumscribed in it . ## 6. Scientific Notation (Sand Reckoner)

He developed a method of counting numbers of zero in a given number. He proposed that number of grains of sand in the whole universe are 1 followed by 63 zeroes, which is very difficult to write manually. So, he introduced this method to represent big numbers easily. In scientific notation, above expression can be written as 1×{10}^{63}. In general, we write it as m×[ketax]{10}^{n}[/ketax], where 1≤m<10 and n is an integer. With the help of scientific notation, he was able to remove Greek alphabets from counting system.

## 7. Volume of Irregular objects

Archimedes is best known for his ‘EUREKA’ moment when he came up with the concept of ‘buoyancy’ which is considered one of the most valuable findings in the field of physics. With the help of buoyancy, he was able to calculate the volume of irregular objects. One day, he stepped into a bathtub full of water up to the edge. He noticed that as he stepped inside, some water was displaced from the bathtub. It was then that he realized that the volume of the body inside the water is equal to the volume of water displaced by it, which can be easily calculated.

## 8. Archimedes Death Ray

Archimedes was born in Syracuse, which was very prone to attack from the Romans. He tried to invent some instruments based on mathematical principles that would help people of Syracuse fight against the Romans. One of them was the Death Ray, which was made up of a number of parabolic mirrors. These mirrors were made up of highly polished bronze and copper metals. They are placed in such a way that the sun rays falling on the mirror converge to a particular point. Any enemy ship arriving at that point will get burn by the strong sun rays reflected by the mirror, and the ship will eventually sink into the sea. ## 9. Archimedes Catapult

As discussed above, he used his knowledge of geometry to build various instruments to save people of Syracuse from enemies, and one of them was catapult, which works on the principle of projectile. It is also called stone thrower. The arrangement of catapult include a beam, which is attached to a pouch in which the projectile is kept. Then, on the opposite side of beam, a heavy weight is attached, which when released, will project the stone kept in pouch to a far distance and can cause a lot of damage. Initially, in place of heavy weights, the beam was pulled manually.

## 10. Archimedean Spiral

It is a type of spiral antenna, which is designed in such a way that the distance between the turning of spiral is constant. It is used in various fields like in defense industry for covering a wide range of radio frequencies and in GPS also.