**Archimedes of Syracuse** (c.287 BC – c.212 BC) was the greatest Greek mathematician, physicist, engineer, inventor, and astronomer.
Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in
physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing
innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes
designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of
exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate
approximation of

*?*. He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious
system for expressing very large numbers.

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero
describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the
sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest
of his mathematical achievements.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted
him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus, while commentaries on the works of Archimedes
written by Eutocius in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes'
written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance, while the
discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained
mathematical results.

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of
birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner,
Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes
was related to King Hiero II, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been
lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. During his youth,
Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Cyrene were contemporaries. He referred to Conon of Samos as his
friend, while two of his works (The Method of Mechanical Theorems and the Cattle Problem) have introductions addressed to Eratosthenes.

Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse
after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the
city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on
the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death
of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes
was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was
reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed
The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was
supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "Noli turbare circulos meos", but there is no
reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.

According to Plutarch, Archimedes was simply we shall gain mathematics. It forgot about food, did not care at all of itself. The works of
Archimedes concerned almost to all areas of mathematics of that time: it posesses wonderful researches on geometry, arithmetics, algebra. So,
it has found all semiregular polyhedrons which now carry its name, significantly has developed the doctrine about conic sections, has given a
geometrical way of decision of the cubic equations of a type

_{},
which roots he found by means of intersection of a parabola and a hyperbole. Archimedes has lead also full research of these equations, that
he has found, at what conditions they will have the valid positive various roots and at what roots will coincide.

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same
height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its
bases. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. He had heard stories about the tomb
of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse,
in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the
verses that had been added as an inscription. A tomb discovered in a hotel courtyard in Syracuse in the early 1960s was claimed to be that of
Archimedes, but its location today is unknown.

The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege
of Syracuse given by Polybius in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as
a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built
in order to defend the city.

The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular
shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used, and
Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. Archimedes had to solve the problem
without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath,
he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of
the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume.
By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower
than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery
that he had forgotten to dress, crying "Eureka!" (meaning in Greek "I have found it!"). The test was conducted successfully, proving that
silver had indeed been mixed in.

The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has
been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have
instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise "On
Floating Bodies". This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it
displaces. Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing
the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples
would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since,
besides being very accurate, it is based on demonstrations found by Archimedes himself".

A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of
Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel,
carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity. According to
Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite
among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was
purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a
cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The
Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw (shown in the
figure below) described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging
Gardens of Babylon. The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named
in honor of Archimedes and his work on the screw.

The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker",
the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking
ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the
feasibility of the claw, and in 2005 a television documentary entitled Super weapons of the Ancient World built a version of the claw and
concluded that it was a workable device.

The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire.
Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon. The device, sometimes called the "Archimedes heat ray",
was used to focus sunlight onto approaching ships, causing them to catch fire.

This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. Rene Descartes rejected it as false,
while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. It has been
suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a
ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace. A test of the Archimedes heat ray
was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this
occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a
plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into
flames within a few seconds. The plywood ship had a coating of tar paint, which may have aided combustion. A coating of tar would have been
commonplace on ships in the classical era. In October 2005 a group of students from the Massachusetts Institute of Technology carried out an
experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100 feet (30 m). Flames broke
out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was
concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show Myth
Busters, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In
order to catch fire, wood needs to reach its auto ignition temperature, which is around 300 °C (570 °F). When Myth Busters broadcast the
result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of
time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the
east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. Myth Busters also pointed
out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire
at short distances. In December 2010, Myth Busters again looked at the heat ray story in a special edition featuring Barack Obama, entitled
President's Challenge. Several experiments were carried out, including a large scale test with 500 school children aiming mirrors at a mock-up
of a Roman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reach the 210 °C (410 °F) required catching fire,
and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, dazzling, or
distracting the crew of the ship.

While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier
descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas.
According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth".
Plutarch describes how Archimedes designed tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that
would otherwise have been too heavy to move. Archimedes has also been credited with improving the power and accuracy of the catapult, and
with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into
a container after each mile traveled.

Cicero (106–43 BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129 BC.
After the capture of Syracuse c.212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms used as aids in
astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and
Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other
to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius
Philus, who described it thus: "When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze
contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that
position which was its shadow on the Earth, when the Sun was in line".

This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the
construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism,
another device from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have
required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available
in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient
Greeks.

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch
wrote: "He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life".
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction (in Latin
"reductio ad absurdum"), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the
answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of

*?*. He did this by drawing a
larger polygon outside a circle and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more
accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value
of

*?* lay between

(approximately 3.1429) and

(approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to

*?* multiplied by the square of the radius of the circle (

*?r*^{2}). In On the Sphere and Cylinder, Archimedes postulates
that any magnitude when added to it enough times will exceed any given magnitude. This is the Archimedean property of real numbers.

In "Measurement of a Circle", Archimedes gives the value of the square root of 3 as lying between 265?153 (approximately 1.7320261) and
1351?780 (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this
result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused John Wallis to remark
that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his
method of inquiry while he wished to extort from them assent to his results".

In "The Quadrature of the Parabola", Archimedes proved that the area enclosed by a parabola and a straight line is 4?3 times the area of a
corresponding inscribed triangle as shown in the figure below.

He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:

If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the
two smaller secant lines, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + • • • which sums to 1?3.

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse. The written work of Archimedes has not survived as well
as that of Euclid, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of
Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the
now-lost Catoptrica. During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. The
writings of Archimedes were collected by the Byzantine architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes
written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thabit
ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. 1114–1187 AD). During the Renaissance, the Editio Princeps (First Edition) was
published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. Around the year 1586 Galileo Galilei invented
a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.