1. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
Archimedes was one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics.
Archimedes is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed.
The details of Archimedes life are obscure; a biography of Archimedes mentioned by Eutocius was allegedly written by his friend Heraclides Lembus, but this work has been lost, and modern scholarship is doubtful that it was written by Heraclides to begin with.
Plutarch tells a slightly different account, relating that Archimedes boasted to Hiero that he was able to move any large weight, at which point Hiero challenged him to move a ship.
Athenaeus, likely garbling the details of Hero's account of the baroulkos, mentions that Archimedes used a "screw" in order to remove any potential water leaking through the hull of the Syracusia.
The greatest reputation Archimedes earned during antiquity was for the defense of his city from the Romans during the Siege of Syracuse.
Cicero mentions that Marcellus brought to Rome two planetariums Archimedes built, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated the other to the Temple of Virtue in Rome, and the other he allegedly kept as his only personal loot from Syracuse.
In Measurement of a Circle, Archimedes employed this method to show that the area of a circle is the same as a right triangle whose base and height are equal to its radius and circumference.
Archimedes then approximated the ratio between the radius and the circumference, the value of, by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step.
Archimedes first gives an outline of this proof in Quadrature of the Parabola alongside the geometric proof, but he gives a fuller explanation in The Method of Mechanical Theorems.
Archimedes made his work known through correspondence with mathematicians in Alexandria, which were originally written in Doric Greek, the dialect of ancient Syracuse.
The introductory letter states that Archimedes' father was an astronomer named Phidias.
The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy.
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner.
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.
The fluids described by Archimedes are not since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
Archimedes calculates the areas of the 14 pieces which can be assembled to form a square.
Reviel Netz of Stanford University argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square.
Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles.
Many written works by Archimedes have not survived or are only extant in heavily edited fragments: Pappus of Alexandria mentions On Sphere-Making, as well as a work on semiregular polyhedra, and another work on spirals, while Theon of Alexandria quotes a remark about refraction from the Catoptrica.
Archimedes confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work.
Sometimes called the father of mathematics and mathematical physics, historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.
The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented; Athenaeus recounts in his Deipnosophistae how Archimedes supervised the construction of the largest known ship in antiquity, the Syracusia, while Apuleius talks about his work in catoptrics.
Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.
The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri.
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.
Archimedes has appeared on postage stamps issued by East Germany, Greece, Italy, Nicaragua, San Marino, and Spain.