ARCHIMEDES

Archimedes

Archimedes was a Greek mathematician mostly known as the greatest mathematician of his age. Although there are not many existing detail about his life, according to the reports, Archimedes lived in Syracuse city in Sicily between about 287 BC-212 BC. It is also reported that Archimedes may have studied for a while at Alexandria, Egypt as a student of Euclid.

During the Second Punic war, Archimedes invented ingenious war machines against the Romans. The most well known war machine that he invented was Archimedes Screw which is still being used in developing countries. However, Romans won the war and Archimedes was killed by a Roman soldier although his life was spared by the Roman general Marcellus.

Archimedes had many inventions and contributions to the existing inventions in concepts such as; Law of The Lever, The Hydrostatic Principle, The Sand-Reckoner, Measurement of the circle, Angle Trisection, Area of a Parabolic Segment, Volume of a Paraboloidal Segment, Segment of a Sphere, Semiregular Solids and Trigonometry

Integral Calculus

Archimedes is regarded as the father of Integral Calculus with his method of exhaustion approximately 2000 years before Newton and Leibniz developed the idea of Integral Calculus. Archimedes proposed many invaluable ideas in finding the area of a paraboloic segment and volume of a paraboloidal segment which are the concepts of modern integration.

Method of Exhaustion

Method of Exhaustion refers to finding the area of a shape by inscribing it inside a sequence of polygons whose areas converges to the area of the containing shape. Archimedes used this method to find the area of a paraboloic segment which refers to the region bounded by a parabola and line. The most popular theorem of Archimedes by using method of exhaustion is The Quadrature of the Parabola. The theorem is:

parabola

"Consider a certain inscribed triangle in a paraboloic segment. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord. A line from the third vertex drawn parallel to the y axis divides the chord into two equal segments. The theorem claims that the area of the parabolic segment is 4/3 of the inscribed trianlge."

Proof:

proof

The main idea of the proof is the dissection of the parabolic segment into infinitely many triangles, as shown in the figure to the right. Each of these triangles is inscribed in its own parabolic segment in the same way that the blue triangle is inscribed in the large segment.

Archimedes proves that the area of each green triangle is one eighth of the area of the blue triangle. From a modern point of view, this is because the green triangle has half the width and a fourth of the height.

ea1

Each of the yellow triangles has one eighth the area of a green triangle, each of the red triangles has one eighth the area of a yellow triangle, and so on. Using the method of exhaustion, the area of the parabolic segment is given by

Area=T+2( T 8 )+4( T 8 2 )+8( T 8 3 )+... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadk hacaWGLbGaamyyaiabg2da9iaadsfacqGHRaWkcaaIYaWaaeWaaeaa daWcaaqaaiaadsfaaeaacaaI4aaaaaGaayjkaiaawMcaaiabgUcaRi aaisdadaqadaqaamaalaaabaGaamivaaqaaiaaiIdadaahaaWcbeqa aiaaikdaaaaaaaGccaGLOaGaayzkaaGaey4kaSIaaGioamaabmaaba WaaSaaaeaacaWGubaabaGaaGioamaaCaaaleqabaGaaG4maaaaaaaa kiaawIcacaGLPaaacqGHRaWkcaGGUaGaaiOlaiaac6caaaa@4EC0@

Here T represents the area of the large blue triangle, the second term represents the total area of the two green triangles, the third term represents the total area of the four yellow triangles, and so forth. This simplifies to give

Area=( 1+ 1 4 + 1 16 + 1 64 +... )T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadk hacaWGLbGaamyyaiabg2da9maabmaabaGaaGymaiabgUcaRmaalaaa baGaaGymaaqaaiaaisdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG ymaiaaiAdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOnaiaaisda aaGaey4kaSIaaiOlaiaac6cacaGGUaaacaGLOaGaayzkaaGaamivaa aa@495D@

To complete the proof, Archimedes needed to show that

1+ 1 4 + 1 16 + 1 64 +...= 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU caRmaalaaabaGaaGymaaqaaiaaisdaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaaGymaiaaiAdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG OnaiaaisdaaaGaey4kaSIaaiOlaiaac6cacaGGUaGaeyypa0ZaaSaa aeaacaaI0aaabaGaaG4maaaaaaa@44F9@

proof

Archimedes proves this equation using a geometric method demonstrated in the picture to the right. In this picture, each successive shaded square has one fourth the area of the previous square, with the total shaded area being the sum. The sum of the shaded areas can be represented as

1 4 + 1 16 + 1 64 +... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGinaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaGa aGOnaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI2aGaaGinaaaacq GHRaWkcaGGUaGaaiOlaiaac6caaaa@40CB@

As you see, there are three congruent shapes; one purple and two yellow and the total areas of these shapes converges to the area of the unit square. As a result, the purple area covers one thirds of the unit square.

Archimedes Documentary on the Lost Secrets of Archimedes