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GEOMETRY.

While the origins of geometry are likely to remain a matter of pure speculation, the elaborate written cultures of ancient Egypt and Babylon provide a wealth of information about the uses of geometry. Area and volume measurements abound in work connected with taxation, the provision of cities, and large-scale building works. Sometimes the Babylonians' evidence (which survives because they wrote on durable clay tablets) spills over into purer matters, and reveals methods for finding the areas of circles, and an impressive calculation of the length of the diagonal of a unit square. The so-called Pythagorean Theorem for right-angled triangles was used to find sides and diagonals of rectangles, and approximate methods for finding square roots. Other tablets display a cut-and-paste method for dealing with questions that could be formulated as quadratic equationsthe origins of the method of completing the squarethat depends for its validity on a certain amount of elementary geometry.

Antiquity and the Middle Ages

Unfortunately there is little evidence of the transmission of geometrical knowledge from either Egypt or Babylonia to the emerging Greek culture. Significantly, the Greeks seem to have been interested in proof, and the nature of mathematical knowledge, in a way that these other cultures were not. Plato's dialogues display these features in a dramatic way. In the Meno, for example, Plato has Socrates ask a slave boy about the diagonal of a square. What Socrates elicits is a comparison between the square of the diagonal and the square on the side; not a numerical answer, and not an approximation to 2, but an argument accompanied by a proof.

The advent of proof permitted an important discovery: 2 is what we would call an irrational number: there are no integers p and q such that 2 p/q. Historians used to present this discovery as momentous. Allegedly the mathematics of earlier, Pythagorean, times was based on the idea that everything could be counted, any two lengths could be regarded as multiples of a common measure. The incommensurability of the side and diagonal of a square put an end to that belief and caused a crisis in Greek mathematics. In the late twentieth century, however, historians retreated from this position. The only evidence for it is very late, and no contemporaneous evidence suggests a crisis. Rather, as Plato's dialogues suggest, there might have been surprise and excitement. The slave boy, after all, gave an acceptable answer. The existence of incommensurable pairs of lengths greatly complicated the theory of proportion, which is attributed to Eudoxus of Cnidus (c. 400c. 350 b.c.e.) and presented in books 5 and 6 of Euclid's (fl. c. 300 b.c.e.) Elements, but again there is no suggestion of a crisis.

The paradoxes of Zeno.

Further evidence of the sophistication of Greek thought is found in Zeno of Elea's (c. 495c. 430 b.c.e.) paradoxes, which survive only in the form of a somewhat hostile account by Aristotle (384322 b.c.e.). Zeno aimed to show that the analysis of motion led inevitably to contradictions. Achilles may never catch a tortoise, because each time he runs to where the tortoise was it is still ahead, committing Zeno to an infinite chase. Indeed, by a somewhat similar argument he cannot get started. An arrow cannot move in an instant; therefore, it is at rest in every instant of its flight and therefore always at rest. Whatever Zeno's intention in proclaiming them, his paradoxes testify to a deep-seated interest in logical reasoning, and they continued to attract interest.

The notion of proof.

Much of Greek mathematics would be impossible without good notions of proof. The simplest form of proof was proof by showing, in which arrangements of pebbles were used to show such results as the sum of two odd numbers is even. Zeno's paradoxes display another form of reasoning, called reductio ad absurdum, in which a proposition is refuted by showing that it leads logically to a self-contradiction or other evident impossibility. This method was used extensively by Archimedes in his estimation of areas and volumes, and also earlier by Euclid in his Elements, for example when he showed that there are infinitely many prime numbers. For, if there are not, then there are only finitely many prime numbers, p 1, p 2, , p n say, in which case the number p 1 p 2 p n 1 is larger than any of these, so it cannot be prime, and yet it is divisible by none of them, so it must be prime.

Proofs in geometry turn approximate estimates based on a finite number of cases into certain knowledge. For example, the assumptions made at the start of Euclid's Elements, including the parallel postulate as described below, permitted Euclid to prove that the angle sum of a triangle is exactly two right angles by exhibiting a suitable pair of parallel lines, to prove Pythagoras's theorem by moving areas around, and, ultimately, to show that there are exactly five regular solids.

Euclid's Elements and the axiomatization of geometry.

The most impressive form of proof, however, in Greek mathematics is the axiomatic method, developed at length in Euclid's Elements. The aim, not perfectly honored but impressively so, was to state definitions of the fundamental terms, gives rules for what may be said about them, and then to derive truths successively from this base of assumptions (the axioms). The result is that later propositions in each book of the Elements depend in an elaborate, tree-like way, on the earlier ones, and confidence in these results depends on the transparency of the proofs and the quality of the original axioms.

Apollonius and Archimedes.

One of the intellectual high points of Greek mathematics is undoubtedly Apollonius of Perga's (c. 262c. 190 b.c.e.) theory of conic sections. It is forbiddingly austere, but it goes a long way to creating a unified theory of all (nondegenerate) plane sections of a cone: the ellipse, parabola, and hyperbola. The names derive from the way their construction is shown to produce an area that falls short, is equal to, or exceeds another area (compare the terms for figures of speech: elliptical, or of few words; a parable is exact, hyperbole an exaggeration). The comparisons of areas yield a proportion, which is modernized as the equation of the curves, and Apollonius shows in some detail how the equation may be simplified by suitable geometric choices and how properties of the conic sections may be obtained, such as the focal properties of conics and the construction of tangents.

Archimedes (c. 287212 b.c.e.), a near contemporary of Apollonius, has earned a reputation as the greatest of the Greek geometers not only for the brilliance of his achievements, but also perhaps because they are easier to admire. He found volumes of sections of cones and various solid figures, he was the first to show that the constant () that enters the formulas for the circumference and the area of a circle is in fact the same, and he also made a number of practical and mechanical discoveries. He also left a unique account, known as the Method, of how he came to some of his discoveries by heuristic means, regarding areas as made up of lines that could be moved around. A tenth-century copy of this account was discovered in 1906 in a monastery in Istanbul. It was then lost again, but reappeared in 1998, when it was put on auction and sold for the surprisingly small sum of \$2.2 million.

Arabic and Islamic work on geometry.

Islamic scholars did much more than simply transmit Greek ideas to the later West, as some accounts have suggested. They far surpassed all previous cultures in geometric design. They also produced the most penetrating analyses of the single most obvious weakness in all of Euclid's Elements : the parallel postulate. Euclid had assumed that if two lines m and n cross a third, k, and the angles and the lines m and n make with k are less than two right angles on one side of the line (in the figure 2 right angles) then the lines will meet on that side of the line if they are produced sufficiently far (see fig. 1).

Unlike all Euclid's other assumptions, the parallel postulate makes claims about what happens arbitrarily far away and so could be false. However, very few theorems can be proved unless the parallel postulate is known, so mathematicians were in a quandary. Greek and still more Islamic commentators took the view that it would be better to drop the parallel postulate from the list of axioms, and to derive it instead from the other axioms as a theorem.

Remarkably, from Thabit ibn Qurrah (c. 836901) to Nasir ad-Din at-Tusi (12011274), they all failed. To give just one example, Ibn al-Haytham (Alhazen; 9651039) assumed that if a segment of fixed length and perpendicular to a given line moves with one endpoint on the line then the other end point draws a straight line, parallel to the given line. Certainly, the parallel postulate follows as a theorem if one may make this assumption, and the parallel postulate implies it, but this only invites the question: how can the assumption itself be proved, or is it merely an alternative assumption to the parallel postulate? Some years later, Omar Khayyám (1048??1131) objected to the assumption on just these grounds, arguing that it was an illegitimate use of motion in geometry to attempt to define a curve this way, still more to assume that the curve so produced was a straight line.

Modern Era

Significant Western interest in mathematics ebbed for a long time during and after the Roman Empire, before flowing at times in the Middle Ages. Only in the sixteenth century did a continual process of growth begin, aided by the rediscovery of Greek and Arabic texts and the publication of editions of Euclid's Elements and the works of Apollonius and Archimedes. At the same time, the discovery of the method of single-focused perspective transformed first architecture and then the practice of painting, where it produced a dramatically heightened realism. The technique proved eminently teachable, although few painters apart from Piero della Francesca (c. 14201492) also understood the mathematics involved.

Analytic and projective geometry.

Girard Desargues (15911661) brought together projective ideas from both architecture and painting to create the first fully unified theory of conic sections (all nondegenerate conic sections are projections of a circle). This theory naturally highlights those aspects that are projective (such as tangency questions) and it led directly or indirectly to a number of novel discoveries over the next century before it petered out. It was then rediscovered by Gaspard Monge (17461818) and Jean-Victor Poncelet (17881867) at the time of the French Revolution. In the form of simple horizontal and vertical projections it became the core technique of descriptive geometry or engineering drawing, a mainstay of French mathematical education throughout the nineteenth century, and, of course, it is still in use in the early twenty-first century.

Poncelet's breakthrough at the start of the nineteenth century was to see that, for many geometric properties a curve is equivalent to any of its "shadows" (its images under central projection). His own way of doing this was not found to be acceptable by later mathematicians, but Michel Chasles (17931880) in France and August Ferdinand Möbius (17901868) and Julius Plücker (18011868) in Germany all independently found more rigorous ways of making his insight work, and the resulting subject of projective geometry became the fundamental geometry of the nineteenth century. Although the details remained obscure for some time, the key point was that projective geometry discussed geometric properties of figures that do not involve the concept of distance. Any theorem in projective geometry is true in Euclidean geometry, but not vice versa, and so projective geometry is more basic than Euclidean geometry.

Desargues's contemporary, René Descartes (15961650) was much more successful with a work that was ruthlessly modern in its approach, and entirely eclipsed earlier attempts. Descartes took contemporary algebra, rewrote it in simpler notation, and proceeded to solve geometric problems by recasting them in algebraic terms and solving them by algebraic means, then reinterpreting the solution in geometric terms. Typically, the algebraic solution will be a single equation between two unknowns. Descartes interpreted this as defining a curve, and gave an elaborate discussion of how, given an equation, the corresponding curve can be drawn. He published his ideas as an appendix (entitled La géométrie ) to a longer philosophical work in 1637.

Descartes did not explain the more elementary parts of his approach. This was done by a number of Dutch scholars who came after him, and the study of geometry by means of algebra (Cartesian, analytic, or coordinate geometry) was swiftly established. It took about a century for mathematicians to decide that the algebraic equation was an acceptable answer to a geometric problem, and to drop Descartes's search for geometric constructions, but the idea that geometric figures are naturally and fruitfully described by algebra has remained central to much of mathematics ever since.

Differential geometry.

Differential geometry, on the other hand, began as the study of curves and surfaces where the calculus is allowed. It is connected to such questions as: when a map of the earth's surface (assumed to be a sphere) is made on a plane, what geometric properties can be preserved? The decisive reformulation of the early nineteenth century came when Carl Friedrich Gauss (17771855) investigated the curvature of surfaces in space. The curvature of a surface at a point (and generally the curvature of a surface varies from point to point) is a measure of the best fitting sphere, plane, or saddle at that point (see figs. 2 and 3).

Gauss found that this quantity is intrinsic: it can be determined by measurements taken in the surface itself without reference to the ambient Euclidean space. This property was so unexpected Gauss called the result an exceptional theorem.

Gaussian curvature and the emergence of non-Euclidean geometries.

After Gauss's death it emerged that he, alone of the mathematicians of his time, had had some sympathy with efforts to show that Euclidean geometry was not the only possible geometry of space, and indeed his astronomer friends Friedrich Wilhelm Bessel (17841846) and Heinrich Wilhelm Matthäus Olbers (17581840) had also inclined that way. This leads back to the question of the parallel postulate.

Around 1830 János Bolyai (18021860) in present-day Hungary and Nicolai Ivanovich Lobachevsky (17921856) in remote Kazan in Russia, wrote down and published detailed accounts of what a geometry would look like in which the parallel postulate was false and the angle sum of a triangle is always less than two right angles (reprinted in English translation in Bonola, 1912). Although independent, their work is remarkably similar and can be described together. They studied geometry in two and three dimensions, and found formulas for triangles in the plane analogous to the formulas of spherical trigonometry for triangles on the sphere. These new formulas showed that small regions of the new geometry differed only slightly from small regions of the Euclidean plane, thus explaining why the new geometry had not been noticed hitherto, but they also showed what many a previous defence of the parallel postulate had hinted atthat the new geometry was different from Euclidean geometry in many respects.

Such was the novelty of Bolyai's and Lobachevsky's work that few read it and the published responses to it were extreme in their hostility. Most people instinctively found it incoherent; they "knew" it was wrong but were not willing to find out where. Gauss, however, wrote to Bolyai to say that he agreed with János's presentation but implying that he knew it all already (a claim for which there is little surviving evidence). János was so enraged he never published again. In 1840 Gauss nominated Lobachevsky to the Göttingen Academy of Sciences, but did nothing else to promote the new geometry. The result was that both men died without getting the acclaim their discoveries undoubtedly merited.

Riemann's generalization for spaces of higher dimensions.

The hegemony of Euclidean geometry came to an end not with the discoveries of Bolyai and Lobachevsky, but in stages, starting with Riemann's wholly novel approach to geometry that severely undercut it. Bernhard Riemann (18261866) was a student of mathematics at the University of Göttingen in the mid-1850s. In his postdoctoral thesis he set out the view that geometry was the study of any "space" of points upon which one could talk about lengths, and he indicated a variety of ways in which the techniques of the calculus could do such service. This is a rather natural and elementary idea, the problems come in spelling out the details in any useful way.

Riemann concentrated on intrinsic properties of the space, such as Gauss's idea of the curvature of a surface, and he noted that there would be different geometries on spaces with different intrinsic properties. That includes spaces of different dimensions, and also spaces of dimension two, say, but different curvatures. What it does not do is say that Euclidean space of some dimension is the source of geometric concepts, thus Euclidean geometry is overthrown.

Riemann's thesis was published posthumously in 1867, just in time to resolve the doubts of an Italian mathematician, Eugenio Beltrami (18351899), who had come to some of the same ideas. He immediately published his reworking of the geometry of Bolyai and Lobachevsky as the geometry of a surface of constant negative curvature, of which he had a description in a disc of unit (Euclidean) radius. Beltrami's map and Riemann's philosophy of geometry convinced mathematicians, but not all philosophers, of the validity of non-Euclidean geometry, as the Bolyai-Lobachevsky geometry became known.

Twentieth Century

There, curiously, Riemann's ideas remained for more than a generation. There was some interest in novel three-dimensional geometries, almost none in geometries, in Riemann's sense, of higher dimensions, except to show that mechanics could be done in such a setting, and in simplifying the formidable algebraic complexity of the subject (today handled by means of the tensor calculus). The decisive change came with the work of Albert Einstein (18791955).

Einstein's special theory of relativity of 1905 was a thorough reworking of the mathematics of motion, and at first Einstein was unsympathetic to the geometrical reworking given to his ideas by Hermann Minkowski (18641909) in 1908. But when Einstein started to think of a general theory of relativity that would find an equivalence between forces and accelerations, he found the ideas of Riemannian differential geometry invaluable. The theory he came to in 1915 formulated gravitation as a change in the metric of space-time. On this theory, matter changes the shape of space, which is how it exerts its attractive force.

Felix Klein's Erlangen program.

By the 1870s, projective geometry had established itself as the fundamental geometry, with Euclidean geometry as a special case, along with some other geometries not described in this essay. The young German mathematician Felix Klein (18491925) then proposed to unify the subject, by treating projective geometry as the main geometry, and deriving the other geometries as special cases. Every geometry Klein was interested in, most strikingly non-Euclidean geometry, was defined on projective space or a subset of it, and the relevant geometric properties were those kept invariant by the action of a suitable subgroup of the group of all projective transformations. This view, called the Erlangen Program, after the university where Klein first published it, has remained the orthodoxy since the 1890s, when Klein republished it, but in its day the novelty was the explicit introduction of the then-novel concept of a group, and the shift of attention from properties of figures to the idea that these properties are geometric precisely because they are invariant under the appropriate group of transformations.

Italians, Hilbert, and the axiomatization of geometry.

Klein's geometries do not include many of the geometries Riemann had pointed toward. It included only those that have large groups of transformations, which, however, is most of those of interest in physics and much of mathematics for a long time, including, it was to transpire, Einstein's special theory of relativity. The first step beyond what Klein had done, and for that matter Riemann, was proposed by David Hilbert (18621943), starting in 1899, although a number of Italian geometers had had similar ideas in the decade before.

Hilbert was insistent that theorems in geometry should only use the properties of objects that entered into their definitions, and to this end he formulated elementary geometry carefully in terms of five families of axioms. What distinguished his work from his Italian predecessors was his insistence that there was an interesting new branch of mathematics to be explored, which studied axiom systems. It would aim at showing the independence of certain axioms from others, and establishing what sorts of axioms were needed to deduce particular results. Whereas the Italian mathematicians had mostly aimed at axiomatizing projective and perhaps Euclidean geometry once and for all, Hilbert saw the axiomatic method as both creative and of wide applicability. He even indicated ways in which it could work in mathematical physics.

By 1915, the axiomatization of geometry had begun to spread to other branches of mathematics as well, with a consequent improvement in the standards of formal proof, and Einstein's general theory of relativity had similarly improved the ideas about the applications of geometry.

See also Cosmology: Cosmology and Astronomy ; Greek Science ; Islamic Science ; Mathematics ; Physics ; Relativity .

bibliography

Berggren, J. Len. Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag, 1986.

Bonola, R. History of Non-Euclidean Geometry. Translated by H. S. Carslaw, preface by F. Enriques. Chicago: Open Court, 1912.

Bos, Henk. J. M. Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction. New York: Springer-Verlag, 2001.

Field, Judith V., and Jeremy J. Gray, eds. The Geometrical Work of Girard Desargues. New York: Springer-Verlag, 1987.

Fowler, David H. The Mathematics of Plato's Academy: A New Reconstruction. 2nd ed. Oxford: Clarendon, 1999.

Gray, Jeremy J. The Hilbert Challenge. Oxford: Oxford University Press, 2000.

. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic. 2nd edition. Oxford: Oxford University Press, 1989.

Hilbert, David. Foundations of Geometry. 10th English edition, translation of the second German edition by L. Unger. Chicago: Open Court, 1971.

Høyrup, Jens. Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. New York: Springer-Verlag, 2002.

Jaouiche, K. La théorie des parallèles en pays d'Islam: Contributions à la préhistoire des géométries non-euclidiennes. Paris: Vrin, 1986.

Robson, Eleanor. "Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322." Historia Mathematica 28 (2001): 167206.

Salmon, Wesley C., ed. Zeno's Paradoxes. Indianapolis: Bobbs-Merrill, 1970.

Jeremy Gray

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Geometry

The term geometry is derived from the Greek word geometria, meaning "to measure the Earth." In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. Most people today know what those features are: lines, circles, angles, triangles, squares, trapezoids, spheres, cones, cylinders, and the like.

Humans have probably used concepts from geometry as long as civilization has existed. But the subject did not become a real science until about the sixth century b.c. At that point, Greek philosophers began to express the principles of geometry in formal terms. The one person whose name is most closely associated with the development of geometry is Euclid (c. 325270 b.c.), who wrote a book called Elements. This work was the standard textbook in the field for more than 2,000 years, and the basic ideas of geometry are still referred to as Euclidean geometry.

Elements of geometry

Statements. Statements in geometry take one of two forms: axioms and propositions. An axiom is a statement that mathematicians accept as being true without demanding proof. An axiom is also called a postulate. Actually, mathematicians prefer not to accept any statement without proof. But one has to start somewhere, and Euclid began by listing certain statements as axioms because they seemed so obvious to him that he couldn't see how anyone would disagree.

One axiom is that a single straight line, and only one, can be drawn through two points. Another axiom is that two parallel lines (lines running next to each other like train tracks) will never meet, no matter how far they are extended into space. Indeed, mathematicians accepted these statements as true without trying to prove them for 2,000 years. Statements such as these form the basis of Euclidean geometry.

However, the vast majority of statements in geometry are not axioms but propositions. A proposition is a statement that can be proved or disproved. In fact, it is not too much of a stretch to say that geometry is a branch of mathematics committed to proving propositions.

Proofs. A proof in geometry requires a series of steps. That series may consist of only one step, or it may contain hundreds or thousands of steps. In every case, the proof begins with an axiom or with some proposition that has already been proved. The mathematician then proceeds from the known fact by a series of logical steps to show that the given proposition is true (or not true).

Constructions. A fundamental part of geometric proofs involves constructions. A construction in geometry is a drawing that can be made with the simplest of tools. Euclid permitted the use of a straight edge and a compass only. An example of a straight edge would be a meter stick that contained no markings on it. A compass is permitted in order to determine the size of angles used in a construction.

Many propositions in geometry can be proved by making certain kinds of constructions. For example, Euclid's first proposition was to show that, given a line segment AB, one can construct an equilateral triangle ABC. (An equilateral triangle is one with three equal angles.)

Plane

A plane is a geometric figure with only two dimensions: width and length. It has no thickness. The flatness of a plane can be expressed mathematically by thinking about a straight line drawn on the plane's surface. Such a line will lie entirely within the plane with none of its points outside of the plane.

A plane extends forever in both directions. Planes encountered in everyday life (such as a flat piece of paper with certain definite dimensions) and in mathematics often have a specific size. But such planes are only certain segments of the infinite plane itself.

Plane and solid geometry

Euclidean geometry dealt originally with two general kinds of figures: those that can be represented in two dimensions (plane geometry) and those that can be represented in three dimensions (solid geometry). The simplest geometric figure of all is the point. A point is a figure with no dimensions at all. The points we draw on a piece of paper while studying geometry do have a dimension, of course, but that condition is due to the fact that the point must be made with a pencil, whose tip has real dimensions. From a mathematical standpoint, however, the point has no measurable size.

Perhaps the next simplest geometric figure is a line. A line is a series of points. It has dimensions in one direction (length) but in no other. A line can also be defined as the shortest distance between two points. Lines are used to construct all other figures in plane geometry, including angles, triangles, squares, trapezoids, circles, and so on. Since a line has no beginning or end, most of the "lines" one deals with in geometry are actually line segmentsportions of a line that do have a limited length.

In general, lines can have one of three relationships to each other. They can be parallel, perpendicular, or at an angle to each other. According to Euclidean geometry, two lines are parallel to each other if they never meet, no matter how far they are extended. Perpendicular lines are lines that form an angle of 90 degrees (a right angle, as in a square or aT) to each other. And two lines that cross each other at any angle other than 90 degrees are simply said to form an angle with each other.

Closed figures. Lines also form closed figures, such as circles, triangles, and quadrilaterals. A circle is a closed figure in which every part of the figure is equidistant (at an equal distance) from some given point called the center of the circle. A triangle is a closed figure consisting of three lines. Triangles are classified according to the sizes of the angles formed by the three lines. A quadrilateral is a figure with four sides. Some common quadrilaterals are the square (in which all four sides are equal), the trapezoid (which has two parallel sides), the parallelogram (which has two pairs of parallel sides), the rhombus (a parallelogram with four equal sides), and the rectangle (a parallelogram with four right- or 90-degree angles).

Solid figures. The basic figures in solid geometry can be visualized as plane figures being rotated through space. Imagine that a circle is caused to rotate around its center. The figure produced is a sphere. Or imagine that a right triangle is rotated around its right angle. The figure produced is a cone.

Area and volume

The fundamental principles of geometry involve statements about the properties of points, lines, and other figures. But one can go beyond those fundamental principles to express certain measurements about such figures. The most common measurements are the length of a line, the area of a plane figure, or the volume of a solid figure. In the real world, length can be determined using a meter stick or yard stick. However, the field of analytic geometry provides a way to determine the length of a line by using principles adapted from geometry.

Mathematical formulas are available for determining the area of any figures in geometry, such as rectangles, squares, various kinds of triangles, and circles. For example, the area of a rectangle is given by the formula A = l · h, where l is the length of the rectangle and h is its height. One can find the areas of portions of solid figures as well. For example, the base of a cone is a circle. The area of the base, then, is A = π · r2, where π is a constant whose value is approximately 3.1416 and r is the radius of the base. (Pi [π] is the ratio of the circumference of a circle to its diameter, and it is always the same, no matter the size of the circle. The circumference of a circle is its total length around; its diameter is the length of a line segment that passes through the center of the circle from one side to the other. A radius is a line from the center to any point on the circle.)

Words to Know

Axiom: A mathematical statement accepted as true without being proved.

Construction: A geometric drawing that can be made with simple tools, such as a straight edge and a compass.

Euclidean geometry: A type of geometry based on certain axioms originally stated by Greek mathematician Euclid.

Line: A collection of points with one dimension onlythat of length.

Line segment: A portion of a line.

Non-Euclidean geometry: A type of geometry based on axioms other than those first proposed by Euclid.

Plane geometry: The study of geometric figures that can be represented in two dimensions only.

Point: A figure with no dimensions.

Proposition: A mathematical statement that can be proved or disproved.

Proof: A mathematical statement that has been demonstrated logically to be correct.

Solid geometry: The study of geometric figures that can be represented in three dimensions.

Formulas for the volume of geometric figures also are available. For example, the volume of a cube (a three-dimensional square) is given by the formula V = s3, where s is equal to the length of one side of the cube.

Other geometries

With the growth of the modern science of mathematics, scholars began to ask whether Euclid's initial axioms were necessarily true. That is, would it be possible to imagine a world in which more than one straight line could be drawn through two points. Such ideas often sound bizarre at first. For example, can you imagine two parallel lines that do eventually meet at some point far in the distance? If so, what does the term parallel really mean?

Yet, such ideas have turned out to be very productive for the study of certain special kinds of spaces. They have been given the name non-Euclidean geometries and are used to study certain kinds of mathematical, scientific, and technical problems.

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geometry

geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.

Types of Geometry

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 BC). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

Their Relationship to Each Other

The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.

The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy ( "betweenness" ) but not that of measurement.

An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). Topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.

The Axiomatic Approach to Geometry

Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts. Euclid first defined his basic terms, such as point and line, then stated without proof certain axioms and postulates about them that seemed to be self-evident or obvious truths, and finally derived a number of statements (theorems) from the postulates by means of deductive logic. This axiomatic method has since been adopted not only throughout mathematics but in many other fields as well. The close examination of the axioms and postulates of Euclidean geometry during the 19th cent. resulted in the realization that the logical basis of geometry was not as firm as had previously been supposed. New axiom and postulate systems were developed by various mathematicians, notably David Hilbert (1899).

Bibliography

See H. G. Forder, The Foundations of Euclidean Geometry (1927); H. S. M. Coxeter, Introduction to Geometry (2d ed. 1969).

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geometry

geometry Branch of mathematics concerned with shapes. Euclidean geometry deals with simple plane and solid figures. Analytic geometry (co-ordinate geometry), introduced by Descartes (1637), applies algebra to geometry and allows the study of more complex curves. Projective geometry, introduced by Jean-Victor Poncelet (1822), concerns itself with projection of shapes and with properties that are independent of such changes. More abstraction occurred in the early 19th century with formulations of non-Euclidean geometry by Janos Bolyai and N. I. Lobachevsky, and differential geometry, based on the application of calculus. See also topology

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geometry

ge·om·e·try / jēˈämətrē/ • n. the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. ∎  (pl. -tries) a particular mathematical system describing such properties: non-Euclidean geometries. ∎  [in sing.] the shape and relative arrangement of the parts of something: the geometry of spiders' webs.

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geometry

geometry the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues; in the Middle Ages, one of the subjects of the quadrivium.

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geometry

geometry XIV. — (O)F. géométrie — L. geōmetria — Gr. geōmetríā; see GEO-, -METRY.
So geometric XVII, geometrical XVI.

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geometry

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