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3100 BCE
Mesopotamia
The earliest known unambiguous examples of written records—dating from Egypt and Mesopotamia about 3100 BCE—demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Greeks gathered and extended this practical knowledge and from the combination of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth. -
3000 BCE
The first record of Geometry
Geometry was used by the Ancient Greeks and Egyptians for calculating astronomic formulas and creating some of the famous buildings in their civilizations such as the Egyptians with their pyramids. -
2900 BCE
Ancient Egypt
Around 2,900 B.C. Ancient Egyptians began using their knowledge to construct pyramids with four triangular faces and a square base. -
Period: 1700 BCE to 1500 BCE
Babylon
In Babylonian clay tablets (c. 1700–1500 BCE) modern historians have discovered problems whose solutions indicate that the Pythagorean theorem and some special triads were known more than a thousand years before Euclid. A right triangle made at random, however, is very unlikely to have all its sides measurable by the same unit—that is, every side a whole-number multiple of some common unit of measurement. -
600 BCE
The Early Greeks Create Modern Geometry
It was the early Greeks (600 BC - 400 AD) that developed the principles of modern geometry beginning the Thales of Miletus (624-547 BC) Thales is credited with bringing the science of geometry from Egypt to Greece. Thales studied similar triangles and wrote the proof that corresponding side of similar triangles are in proportion -
500 BCE
Written Numerals
Greeks started using written numerals -
Period: 460 BCE to 370 BCE
Bridge of Asses
The ancient Greek geometers soon followed Thales over the Bridge of Asses. In the 5th century BCE the philosopher-mathematician Democritus (c. 460–c. 370 BCE) declared that his geometry excelled all the knowledge of the Egyptian rope pullers because he could prove what he claimed. By the time of Plato, geometers customarily proved their propositions. -
Period: 410 BCE to 485 BCE
Last Euclidian Scribe
The last great Platonist and Euclidean commentator of antiquity, Proclus (c. 410–485 CE), attributed to the inexhaustible Thales the discovery of the far-from-obvious proposition that even apparently obvious propositions need proof. Proclus referred especially to the theorem, known in the Middle Ages as the Bridge of Asses, that in an isosceles triangle the angles opposite the equal sides are equal. -
400 BCE
Herodotus
The origin of geometry lies in the concerns of everyday life. The traditional account, preserved in Herodotus’s History (5th century BCE), credits the Egyptians with inventing surveying in order to reestablish property values after the annual flood of the Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to evaluate tribute, store oil and grain, and build dams and pyramids. -
Period: 310 BCE to 230 BCE
Aristarchus of Samos
Aristarchus of Samos (c. 310–230 BCE) has garnered the credit for extending the grip of number as far as the Sun. Using the Moon as a ruler and noting that the apparent sizes of the Sun and the Moon are about equal, he calculated values for his treatise “On the Sizes and Distances of the Sun and Moon.” -
300 BCE
Euclid's Elements
The next great advancement in geometry came from Euclid in 300 BC when he wrote a text titled ‘Elements.’ In this text, Euclid presented an ideal axiomatic form (now known as Euclidean geometry) in which propositions could be proven through a small set of statements that are accepted as true. In fact, Euclid was able to derive a great portion of planar geometry from just the first five postulates in ‘Elements.’ -
Period: 127 to 145
Ptolemaic System
Ptolemy (flourished 127–145 CE in Alexandria, Egypt) worked out complete sets of circles for all the planets. In order to account for phenomena arising from the Earth’s motion around the Sun, the Ptolemaic system included a secondary circle known as an epicycle, whose centre moved along the path of the primary orbital circle, known as the deferent. Ptolemy’s Great Compilation, or Almagest after its Arabic translation, was to astronomy what Euclid’s Elements was to geometry. -
Period: 836 to 901
Thābit ibn Qurrah
Thābit ibn Qurrah (836–901) had precisely the attributes required to bring the geometry of the Arabs up to the mark set by the Greeks. As a member of a religious sect close but hostile to both Jews and Christians, he knew Syriac and Greek as well as Arabic; as a money changer, he knew how to calculate; as both, he recommended himself to the Banū Mūsā, a set of mathematician brothers descended from a robber who had diversified into astrology. -
Period: 965 to 1040
Ibn Al-Haytham
Ibn Al-Haytham (c. 965–1040), proceeds from the object to the painter’s eye. Imagine, as Alberti directed, that the painter studies a scene through a window, using only one eye and not moving his head; he cannot know whether he looks at an external scene or at a glass painted to present to his eye the same visual pyramid. -
Period: 1471 to 1528
Albrecht Dürer
Albrecht Dürer (1471–1528), was used by many artists who wished to render perspective persuasively. At the same time, cartographers tried various projections of the sphere to accommodate the record of geographical discoveries that began in the mid-15th century with Portuguese exploration of the west coast of Africa. -
Period: 1546 to
Tycho Brahe
Tycho Brahe (1546–1601), the world’s greatest observational astronomer before the invention of the telescope, rejected the Copernican model of the solar system, he invited Kepler to assist him at his new observatory outside of Prague. In trying to resolve discrepancies between his original theory and Brahe’s observations, Kepler made the capital discovery that the planets move in ellipses around the Sun as a focus. -
Period: 1548 to
Simon Stevin
Simon Stevin (1548–1620), who wrote on perspective and cartography among many other topics of applied mathematics, gave the first effective impulse toward redefining the object of Archimedean analysis. -
Period: 1571 to
Johannes Kepler
The cosmology of the Timaeus had a consequence of the first importance for the development of mathematical astronomy. It guided Johannes Kepler (1571–1630) to his discovery of the laws of planetary motion. Kepler deployed the five regular Platonic solids not as indicators of the nature and number of the elements but as a model of the structure of the heavens. In 1596 he published Prodromus Dissertationum Mathematicarum Continens Mysterium Cosmographicum (“Cosmographic Mystery”). -
Analytic Geometry
Analytic geometry was initiated by the French mathematician René Descartes (1596–1650), who introduced rectangular coordinates to locate points and to enable lines and curves to be represented with algebraic equations. Algebraic geometry is a modern extension of the subject to multidimensional and non-Euclidean spaces. -
Period: to
Bonaventura (Francesco) Cavalieri
mathematician Bonaventura (Francesco) Cavalieri (1598–1647). Cavalieri, perhaps influenced by Kepler’s method of determining volumes in Nova Steriometria Doliorum (1615; “New Stereometry of Wine Barrels”), regarded lines as made up of an infinite number of dimensionless points, areas as made up of lines of infinitesimal thickness, and volumes as made up of planes of infinitesimal depth in order to obtain algebraic ways of summing the elements into which he divided his figures. -
Rene Descartes Coordinate Geometry
The next tremendous advancement in the field of geometry occurred in the 17th century when René Descartes discovered coordinate geometry. Coordinates and equations could be used in this type of geometry in order to illustrate proofs. The creation of coordinate geometry opened the doors to the development of calculus and physics. -
Invention of the Telescope
The figuring of telescope lenses likewise strengthened interest in conics after Galileo Galilei’s revolutionary improvements to the astronomical telescope in 1609. Descartes emphasized the desirability of lenses with hyperbolic surfaces, which focus bundles of parallel rays to a point (spherical lenses of wide apertures give a blurry image), and he invented a machine to cut them—which, however, proved more ingenious than useful. -
Period: to
Philippe de la Hire
In 1685, in his Sectiones Conicæ, Philippe de la Hire (1640–1718), a Parisian painter turned mathematician, proved several hundred propositions in Apollonius’s Conics by Desargues’s efficient methods. -
Period: to
Gaspard Monge
Gaspard Monge (1746–1818), who developed his own method of projection for drawings of buildings and machines. -
Carl Friedrich Gauss
Carl Friedrich Gauss (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry. Using differential calculus, he characterized the intrinsic properties of curves and surfaces. For instance, he showed that the intrinsic curvature of a cylinder is the same as that of a plane, as can be seen by cutting a cylinder along its axis and flattening, but not the same as that of a sphere, which cannot be flattened without distortion. -
Period: to
Michel Chasles
Michel Chasles (1793–1880) extended the principle of continuity into the domain of the imagination by considering constructs such as the common chord in two circles that do not intersect. -
The Development of Non-Euclidean Geometry
In the 19th century, Carl Friedrich Gauss, Nikolai Lobachevsky, and János Bolyai formally discovered non-Euclidean geometry. In this kind of geometry, four of Euclid’s first five postulates remained consistent, but the idea that parallel lines do not meet did not stay true. This idea is a driving force behind elliptical geometry and hyperbolic geometry.
