Hailed as the most popular mathematician and inventor in ancient Greece, Archimedes is responsible for several important works in geometry, arithmetic and mechanics. Probably one of the most well-known stories attributed to him is his exclaiming of “Eureka!” after having made his most popular discovery (now known as Archimedes’ principle). However, that story has been deemed apocryphal as is the story of him using lots of mirrors to burn Roman ships that were laying siege on Syracuse. Despite several stories circling about of his life that can’t be proven, it has to be said that most of them to reflect his many interests, particularly in catoptrics (a branch of optics dealing with reflection of light from mirrors), mechanics and pure mathematics.

Archimedes was born in Syracuse, a Greek city-state on the eastern coast of Sicily. He spent part of his life in Egypt but moved back to Syracuse where he remained until his death. His works were published in the form of correspondence, particularly with Conon of Samos and Eratosthenes of Cyrene – both Alexandrian scholars. When the Romans laid siege on Syracuse in 213 BCE, Archimedes played a vital role in the defense of the city by building war machines which delayed Syracuse’s capture. However, the city-state eventually fell to Roman general Marcus Claudius Marcellus and Archimedes was killed during the sack of the city.

There were several important discoveries and works that Archimedes achieved during his life, and these accomplishments include.

### Archimedes’ Principle

There is a popular anecdote regarding how Archimedes invented a way to determine the volume of an irregularly shaped object. Vitruvius, best known for his multi-volume work De architectura, tells of Archimedes being asked to determine whether silver was added by a goldsmith in making a pure gold votive crown (for a temple) for King Hiero II. The story goes: while getting into the bath, Archimedes noticed the water level rise as he got in then realized he could use that to determine the volume of the crown.

Since water is incompressible, submerging the crown would mean losing an amount of water equal to its volume. Dividing the mass of the crown by the volume of water displaced gives the density of the object. If the result is less than that of gold, then surely the material used in making the crown were less dense. After making this discovery, he allegedly screamed “Eureka!” and having forgotten to dress, ran naked on the streets.

True enough, the crown was indeed mixed in with silver. However, that story doesn’t appear in any known works of Archimedes. Plus, the exact method described in the story has been called into question, particularly on aspects of accuracy.

Instead, Archimedes comes up with a solution which is now known as Archimedes’ principle. It’s a principle in hydrostatics which he describes in his treatise On Floating Bodies. It states that a body that is immersed in fluid will experience a buoyant force that is equal to the weight of the body it displaces. The crown problem can still be solved using this principle: the crown would be balanced with a gold reference sample with the apparatus immersed in water. The scale would tip accordingly between the two samples to indicate a difference in density.

### Archimedes’ Screw

A number of Archimedes’ engineering work was a result of answering the needs of Syracuse, his home city. Athenaeus of Naucratis, a Greek writer, described how Archimedes was commissioned by King Hiero II to design a ship that could be used for luxury travel and as a naval warship (Syracusia).

The Syracusia, according to Athenaeus, could carry 600 people and included garden decorations, a gymnasium and a temple dedicated to Aphrodite. Given a ship this size would leak a lot of water through the hull, Archimedes’ screw was supposedly developed in order to remove the accumulated water.

The machine developed by Archimedes featured a screw-shaped blade that revolved (this was contained in a cylinder). The machine was turned by hand and could also be used for the transfer of water from a low-lying body of water into irrigation canals.

The SS Archimedes, which was the first seagoing steamship that used a screw propeller, was launched in 1839 and was obviously named in honor of Archimedes and his work on the screw.

### Treatises

Archimedes wrote in Doric Greek which was the dialect of Syracuse in ancient times. Compared to Euclid, his written work hasn’t survived that well and seven of his treatises are known to exist through references made to them by other authors.

These are his surviving works:

**On the Sphere and Cylinder **

This treatise identifies the relationship between a sphere and a circumscribed cylinder of the same height and diameter.

**On the Measurement of the Circle **

Consists of three propositions written to Dositheus of Pelusium, a student of Conon of Samos. The second proposition gives the approximate value of pi.

**On Conoids and Spheroids **

Features 32 propositions again addressed to Dositheus. Here, Archimedes presents calculations for the areas and volumes of sections of conoids, paraboloids and spheres.

**On Spirals **

This work contains 28 propositions still addressed to Dositheus and defines what is now called the Archimedean spiral. This is the locus of points that correspond to the locations of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity over time.

**On the Equilibrium of Planes **

Also called Centers of Gravity of Planes. This is comprised of two volumes and contains Archimedes’ explanation of the Law of the Lever. (Archimedes is attributed with inventing the lever, which he didn’t, but he did provide this explanation.)

**The Quadrature of the Parabola **

Comprised of 24 propositions addressed to Dositheus where he proves that the area of any segment of a parabola is 4/3 of the area of the triangle that has the same base and height as that segment.

**The Sand-Reckoner **

A treatise addressed to Gelon, the son of Hiero. Although written for the layman, it presents profound mathematics. Basically, Archimedes tries to count the number of grains of sand that will fit inside the universe. It’s Archimedes’ only surviving work where he presents his views on astronomy.

**The Method Concerning Mechanical Theorem **

This work uses infinitesimals and explains how breaking a figure into an infinite number of infinitely small parts can determine its area and volume.

**On Floating Bodies **

A treatise where Archimedes describes the law of equilibrium of fluids. It features his principle of buoyancy: “Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.”