Inventions
Archimedes is one of the most instrumental academic figures of his time and all time. He is considered to be one of the three greatest mathematicians of all time including Isaac Newton and Carl Gauss. His close relation to the Royal family meant he was able to investigate his mathematical theories without interruption. As well the king asked him to use his genius to help him with issues that occurred such as when the Punic War tried to besiege Syracuse. Archimedes had a combination of mathematical theories and inventions.
Mathematics Discoveries
Archimedes was responsible for the first major discovery and advancement in calculus around 225 BC. He made one of the most significant of the Greek contributions. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. Archimedes composed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas
A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ...
The area of the segment of the parabola is therefore
A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A.
This is the first known example of the summation of an infinite series.
Archimedes used the method of exhaustion to find an approximation to the area of a circle. This, of course, is an early example of integration which led to approximate values of π.
Among other formulas created by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution.
Other inventions
Although Maths was Archimedes main focus he also invented many devices to benefit the people of Syracuse.
Archimedes screw:
This was a device used for raising water. It has a screw running the length of the of a pipe which rotates and raises the water. The pipe is placed on a angle into a water source and a handle at the top is turned to rotate the screw. This device is still used today in some third world countries.
The pulley and lever:
Although leverage had been used to lift heavy weight for years, Archimedes was the first to fully appreciate how much weight one person could move. A quote from Plutarch;
King heiro was astonished, and begged him to put his proposition into execution, and show him some great weight moved by as slight force. Archimedes therefore fixed upon a three-masted merchantman of the royal fleet which had been dragged ashore by the great labours of many men, and after putting on board many passengers and the customary freight, he seated himself at some distance from her, and without any great effort, but quietly setting in a motion a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water.
This quote shows the enormity of weight that Archimedes was able to move without a large effort.
Archimedes claw
Archimedes used the pulley and lever system to create a war machine that picked up boats like a crane and rocking the boat back forth, making the occupants fall out. The claw would then drop the boat onto the rocks making destroying the boat completely.
Death ray
Death ray is one of the most debated of Archimedes war machines. The idea is that Archimedes used a giant mirror to set alight the galleys of the incoming boats. To achieve this Archimedes needed a mirror set at exactly the right spot to reflect the light so that the focal point of the reflected light was directed at the boat. This means the boats would have to remain in a stationary position long enough for a fire to start. However there is a theory that Archimedes just used the reflection of the sun to disorientate the enemy while Syracuse archers fired flaming arrows at the boat and by the time the attackers realised, it was already too late.
Mathematics Discoveries
Archimedes was responsible for the first major discovery and advancement in calculus around 225 BC. He made one of the most significant of the Greek contributions. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. Archimedes composed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas
A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ...
The area of the segment of the parabola is therefore
A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A.
This is the first known example of the summation of an infinite series.
Archimedes used the method of exhaustion to find an approximation to the area of a circle. This, of course, is an early example of integration which led to approximate values of π.
Among other formulas created by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution.
Other inventions
Although Maths was Archimedes main focus he also invented many devices to benefit the people of Syracuse.
Archimedes screw:
This was a device used for raising water. It has a screw running the length of the of a pipe which rotates and raises the water. The pipe is placed on a angle into a water source and a handle at the top is turned to rotate the screw. This device is still used today in some third world countries.
The pulley and lever:
Although leverage had been used to lift heavy weight for years, Archimedes was the first to fully appreciate how much weight one person could move. A quote from Plutarch;
King heiro was astonished, and begged him to put his proposition into execution, and show him some great weight moved by as slight force. Archimedes therefore fixed upon a three-masted merchantman of the royal fleet which had been dragged ashore by the great labours of many men, and after putting on board many passengers and the customary freight, he seated himself at some distance from her, and without any great effort, but quietly setting in a motion a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water.
This quote shows the enormity of weight that Archimedes was able to move without a large effort.
Archimedes claw
Archimedes used the pulley and lever system to create a war machine that picked up boats like a crane and rocking the boat back forth, making the occupants fall out. The claw would then drop the boat onto the rocks making destroying the boat completely.
Death ray
Death ray is one of the most debated of Archimedes war machines. The idea is that Archimedes used a giant mirror to set alight the galleys of the incoming boats. To achieve this Archimedes needed a mirror set at exactly the right spot to reflect the light so that the focal point of the reflected light was directed at the boat. This means the boats would have to remain in a stationary position long enough for a fire to start. However there is a theory that Archimedes just used the reflection of the sun to disorientate the enemy while Syracuse archers fired flaming arrows at the boat and by the time the attackers realised, it was already too late.