A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes Thoughtful
by Domenico Fetti (1620)
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.
The Death of Archimedes (1815) by Thomas Degeorge
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The Archimedes' screw can raise water efficiently.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola.
Frontpage of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases.
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest.
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost.
The Fields Medal carries a portrait of Archimedes.
Artistic interpretation of Archimedes' mirror used to burn Roman ships.
Painting by Giulio Parigi, c. 1599.
Bronze statue of Archimedes in Berlin

Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.

- Mathematical analysis

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; 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
A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

3 related topics

Alpha

Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

During the Hellenistic period, this method was further developed by Archimedes, who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus.

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

The study of series is a major part of calculus and its generalization, mathematical analysis.

Greek mathematician Archimedes produced the first known summation of an infinite series with amethod that is still used in the area of calculus today.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

Geometry is, with arithmetic, one of the oldest branches of mathematics.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Archimedes (c.

Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.