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    The Snark Knight: s resources and Israeli-Palestinian monomers tend not 90 way of his size. The Elements epitomized the axiomatic-deductive method for many centuries. Algebraic geometry is a modern extension of the subject to multidimensional and non-Euclidean spaces. The German mathematician Carl Friedrich Gauss — , 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. Instead, they discovered that consistent non-Euclidean geometries exist. Topology, the youngest and most sophisticated branch of geometry, focuses on the properties of geometric objects that remain unchanged upon continuous deformation—shrinking, stretching, and folding, but not tearing.

    The continuous development of topology dates from , when the Dutch mathematician L. Brouwer — introduced methods generally applicable to the topic. The earliest known unambiguous examples of written records—dating from Egypt and Mesopotamia about 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.

    It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in the modern age. The origin of geometry lies in the concerns of everyday life. 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. Even the three abstruse geometrical problems of ancient times—to double a cube , trisect an angle, and square a circle , all of which will be discussed later—probably arose from practical matters, from religious ritual, timekeeping, and construction , respectively, in pre-Greek societies of the Mediterranean.

    And the main subject of later Greek geometry, the theory of conic sections , owed its general importance, and perhaps also its origin, to its application to optics and astronomy. While many ancient individuals, known and unknown, contributed to the subject, none equaled the impact of Euclid and his Elements of geometry, a book now 2, years old and the object of as much painful and painstaking study as the Bible.

    Much less is known about Euclid , however, than about Moses. Euclid wrote not only on geometry but also on astronomy and optics and perhaps also on mechanics and music. Only the Elements , which was extensively copied and translated, has survived intact. What is known about Greek geometry before him comes primarily from bits quoted by Plato and Aristotle and by later mathematicians and commentators.

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    Among other precious items they preserved are some results and the general approach of Pythagoras c. The Pythagoreans convinced themselves that all things are, or owe their relationships to, numbers. The doctrine gave mathematics supreme importance in the investigation and understanding of the world. Plato developed a similar view, and philosophers influenced by Pythagoras or Plato often wrote ecstatically about geometry as the key to the interpretation of the universe. Thus ancient geometry gained an association with the sublime to complement its earthy origins and its reputation as the exemplar of precise reasoning.

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