Archimedes : Mathematical Biography

The son of an astronomer named Pheidias, Archimedes was born around 287 B.C. on the Island of Sicily, then part of the Greek empire but now part of Italy. Not only was he renowned as one of the greatest mathematicians of all time, but he was also an accomplished physicist, engineer, astronomer, and philosopher.

Archimedes spent most of his life in his birthplace of Syracuse, Sicily but also studied at the University at Alexandria (on the North coast of Egypt) under the successors of Euclid. Although very few original works survive, many other sources attribute a great number of discoveries to Archimedes. He was most proud of his method of finding the volume of a sphere, being ²/₃ the volume of the smallest cylinder that can contain it. In fact he was so proud that he had a sphere and cylinder inscribed on his tombstone. He also created the field of hydrostatics, founded the fundamentals of mechanics, discovered the law of levers, and calculated the centers of gravity for flat surfaces and solids.

Legend famously portrays Archimedes discovering buoyancy whilst taking a bath and being so excited that he ran through the streets shouting “Eureka!” (“I have found it!”) An example of Archimedes demonstrating how revolutionary his work was, taking the step from the concept of “equal to” to the concept of “arbitrarily close to” or “as close as desired”, which is very close to what we now see as differential calculus.

During the 2^nd Punic War, Archimedes made himself very popular by inventing war machines to defend the city from the Roman siege (214-212 B.C.) His machines included catapults, ropes, pulleys, and hooks reportedly used to lift and smash Roman boats against the rocks. He was also rumored to have suggested the use of large mirrors to focus the rays of the sun onto the Roman ships, thereby setting them on fire. However, modern tests have shown this to be unrealistic and it is probably more accurate that the story was somewhat exaggerated from the sunlight being used to dazzle the Romans.

Legend says that Archimedes was killed by a Roman soldier while he was drawing in the sand. Apparently, the latter’s shadow fell over his work and Archimedes angered the soldier saying, “don’t disturb my circles”.

Given:

AB and BC make up a broken chord with BC>AB.

M is the midpoint of arc ABC

MF is perpendicular to BC

Archimedes proved that F is the midpoint of the broken chord ABC.

Proof:

BC>AB                                                                                   Given

M is midpoint of arc ABC                                                          Given

MF is perpendicular to BC                                                         Given

AM=CM                                                      M is midpoint of arc ABC

angle BAM = angle BCM       Both inscribed angles intersect chord BM

Δ ABM is congruent toΔ ECM                                   Side-Angle-Side

BM = EM                                    Corresponding sides of congruent Δs

ΔMBE is isosceles

ΔBFM is congruent to ΔEFM                                       Hypotenuse-leg

BF = FE                                      Corresponding parts of congruent Δs

AB = EC                                     Corresponding parts of congruent Δs

AB+BF = FE+EC

One of Archimedes’ most well-known inventions, this device is still used today in some parts of the world to elevate water from a source like a river or lake and up to irrigation systems.

A Screw fits inside a hollow pipe or tube with the lower end immersed in the water. When the screw is turned, the water slides up the tube until it falls out of the top. It is not even necessary for the screw to rotate. The screw and tube can be fixed together and if the entire tube were to rotate, the effect would be the same although, presumably this would use more energy thus being less efficient.

Given a parabola and a line from a point on the parabola parallel to its axis, and a line that is tangent to the parabola, Archimedes proved that the resultant triangle has three times the area than that formed by the parabola and resultant secant line.

• Make point D the midpoint of line AC.
• Assume D to be the fulcrum of a lever JB.
• JD=DB
• From one of his earlier findings, point I is the center of the triangle ABC (DI:DB=1:3)
• Therefore, if the weight of the triangle is centered at point I and the weight of the section of parabola is centered at point J, then the law of levers says that the lever is in equilibrium. (mass 1 x distance from fulcrum1 = mass 2 x distance from fulcrum 2)
• The torque at point I will be equal to the torque produced by an infinitely small slice of the triangle, parallel to the parabola’s axis, and centered where it intersects the lever (In this example, the section EH)
• Therefore, if the weight of the triangle cross section is centered at G, then the corresponding parabolic segment’s cross-section centered at point J, then the lever will be in equilibrium.
• This is the same as, EF:GD = EH:JD = EH:DB and EF:EH = AE:AB

The definition of the Archimedean spiral is the locus of points representing a particle moving away from a fixed point with constant speed along a line that rotates with constant angular velocity. Using polar coordinates it can be represented by the equation r = a+bΘ where the variable a affects the rotation of the spiral while the variable b affects the constant gap between successive rotations.

The above was made using the equation r = ¹/₂ + ¹/₂Θ

If Q<0 the spiral rotates in the opposite direction.

The thinking behind this method seems very logical: If a polygon is inscribed within a circle, then its perimeter must be less than the circle’s circumference. Similarly, if another polygon is circumscribed outside a circle, its perimeter must be greater than the circles circumference. Measuring both polygons’ perimeters and dividing by the circle’s diameter gives a bound for the value of . By using polygons with increasing numbers of sides, that bound can be made more accurate. Indeed, using an infinite number of sides we can theoretically get an infinitely accurate bound for .

Starting with a hexagon, Archimedes kept adding sides until he had polygons of 96 sides which gave him a range of 3¹⁰/₇₁ < π < 3¹/₇ or 3.14084 <  < 3.142858

Firstly, the law of the lever says,

                                          M1xD1 = M2xD2

The plane PS is perpendicular to GF and cuts the sphere, cone, and cylinder to form circles with radii PR, PQ, and PS respectively. If circles of PR and PQ are placed on the lever GEF at point G with E as fulcrum, then the system is in equilibrium with the biggest circle remaining centered at point P.

From this, Archimedes found, V_sphere+V_cone = ¹/₂V_cylinder

Euclid had already shown that the volume of a cone is ¹/₃ that of the cylinder so the volume of the sphere is ¹/₂ - ¹/₃ = ¹/₆ the volume of the cylinder.

This is the discovery that it is rumored sent Archimedes running naked through the streets yelling “Eureka!” and is commonly referred to as buoyancy.

• A solid lighter than a fluid will be immersed enough  that the weight of the solid will equal the weight of the fluid displaced
• Similarly, a fluid heavier than a fluid will sink and its apparent weight in the fluid will be less by the same amount as the weight of the fluid displaced.

A History of Mathematics by Carl B. Boyer

Calculus by Anton, Bivens, and Davis

A History of π by Petr Beckmann

University of St. Andrews