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Archimedes of Syracuse

Born: 287 BC in Syracuse, Sicily
Died: 212 BC in Syracuse, Sicily


tex2html_wrap_inline152 Archimedes, the greatest mathematician of antiquity, made his greatest contributions in geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz.

tex2html_wrap_inline152 He was the son of the astronomer Phidias and was close to King Hieron and his son Gelon, for whom he served for many years.

tex2html_wrap_inline152 He was an accomplished engineer but loved pure mathematics.

tex2html_wrap_inline152 Stories from Plutarch, Livy, and others describe machines invented by Archimedes for the defense of Syracuse. These include the catapult, the compound pulley and a burning-mirror.

tex2html_wrap_inline152 Among Archimedes most famous works is Measurement of the Circle, in which he determined the exact value of tex2html_wrap_inline162 to be between the values tex2html_wrap_inline164 and tex2html_wrap_inline166 . This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides. However, he required the proof of two fundamental relations about the perimeters and areas of inscribed and circumscribed regular polygons.

The computation. With respect to a circle of radius r, let


Further, let tex2html_wrap_inline170 denote the regular inscribed tex2html_wrap_inline172 polygons, similarly, tex2html_wrap_inline174 for the circumscribed polygons. The following formulae give the relations between the perimeters and areas of these tex2html_wrap_inline176 polygons.


Using n-gons up to 96 sides he derives


Archimedes -- Books Extant

On the Sphere and Cylinder

Measurement of the Circle

On Conoids and Spheroids

On Spirals

On Plane Equilibria, Two Books

The Sand-Reckoner

Quadrature of a Parabola

On Floating Bodies: two Books

Stomachion, fragments only

The Method

tex2html_wrap_inline152 Archimedes proved, among many other geometrical results, that the volume of a sphere is two-thirds the volume of a circumscribed cylinder.


This he considered his most significant accomplishments, requesting that a representation of a cylinder circumscribing a sphere be inscribed on his tomb.

tex2html_wrap_inline152 His fascination with geometry is beautifully described by Plutarch. Often times Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savors, with his figures he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.

tex2html_wrap_inline152 Archimedes literally invented the whole study of hydrostatics. In one particular result he was able to compute the maximum angle that a (paraboloid) ship could list before it capsized -- and he did it without calculus!

tex2html_wrap_inline152 The case of the fraudulent gold crown. King Hiero II commissioned the manufacture of a gold crown. Suspecting the goldsmith may have substituted silver for gold, he asked Archimedes to determine its authenticity. He was not allowed to disturb the crown in any way. What follows is a quote from Vitruvius (first cent BC):

The solution which occurred when he stepped into his bath and caused it to overflow was to put a weight of gold equal to the crown and know to be pure, into a bowl which was filled with water to the brim. Then the gold would be removed and the king's crown put in, in its place. An alloy of lighter silver would increase the bulk of the crown and cause the bowl to overflow.

There are some technical exceptions to this method. A better solution applies Archimedes' Law of Buoyancy and his Law of the Lever:

Suspend the wreath from one end of a scale and balance it with an equal mass of gold suspended from the other end. Then immerse the suspended wreath and gold into a container of water. If the scale remains in balance then the wreath and the gold have the same volume, and so the wreath has the same density as pure gold. But if the scale tilts in the direction of the gold, then the wreath has a greater volume than the gold.

tex2html_wrap_inline152 He explores very large numbers in the Sand Reckoner by determining the number of grains of sand required to fill the universe of Aristarchus. To do this he needs new numbers and notations for magnitude. He constructed numbers up to tex2html_wrap_inline190 .

tex2html_wrap_inline152 Archimedes discovered fundamental theorems concerning the center of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principal.

tex2html_wrap_inline152 In The Method Archimedes reveals how he discovered some of his theorems. The method is basically a ``geometric method of the lever." He balances lines as one might balance weights. This work was found relatively recently, having being rediscovered only in 1906.

tex2html_wrap_inline152 Archimedes' mechanical skill together with his theoretical knowledge enabled him to construct many ingenious machines.

tex2html_wrap_inline152 Archimedes spent some time in Egypt, where he invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world.

tex2html_wrap_inline152 Quadrature of a Parabola. Archimedes proved, with the Method of Exhaustion, that


external and gave two proofs.

tex2html_wrap_inline152 The Spiral. tex2html_wrap_inline208 Archimedes squared the circle using the spiral.

external He does this by proving that, in length, tex2html_wrap_inline206 . Note, PQ is tangent to the spiral at P and tex2html_wrap_inline208 is a right angle.

tex2html_wrap_inline208 He also determined the area of one revolution ( tex2html_wrap_inline210 ) of tex2html_wrap_inline212 to by



That is, the area enclosed by the spiral arc of one revolution is one third of the area of the circle with center at the origin and of radius at the terminus of the spiral arc.

tex2html_wrap_inline208 He also showed how to trisect the angles using the spiral. Simply construct circles of radii with constant successive differences. These circles will cut the spiral at equal angles. To trisect a particular angle, simply trisect a radial line segment corresponding to the values of the radii at the intersection of the spiral and lines that make the angle (with vertex at the origin). Construct the circles at the trisection points, with centers at the origin.


tex2html_wrap_inline152 Archimedes was killed during the capture of Syracuse by the Romans in the Second Punic War. Plutarch recounts this story of his killing: As fate would have it, Archimedes was intent on working out some problem by a diagram, and having fixed both his mind and eyes upon the subject of his speculation, he did not notice the entry of the Romans nor that the city was taken. In this transport of study a soldier unexpectedly came up to him and commanded that he accompany him. When he declined to do this before he had finished his problem, the enraged soldier drew his sword and ran him through.

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Next: About this document

Don Allen
Wed Feb 19 08:06:42 CST 1997