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## GEOMETRY Defined for English Language Learners

❶You are aware that students of geometry , arithmetic, and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them either to themselves or others; but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion?
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Definition of geometry plural geometries 1 a: See geometry defined for English-language learners See geometry defined for kids. Examples of geometry in a Sentence the geometry of Sydney's famed opera house is suggestive of some modernistic sailing ship. Recent Examples of geometry from the Web Moreover, geometry helped Lewis not only think, but also imagine. David Calvis Opinion ," 6 May The latest Metroid side-scroller and first for the Nintendo 3DS is built on exactly the kind of 3D- geometry , climb-and-clamber engine that speedrunners love to exploit.

There was some futuristic geometry happening—a lot of shapes and a lot of insane lines. Remember when your math teacher said geometry would come in handy one day? While those types of information are mutable -- even Social Security numbers can be changed -- biometric data for retinas, fingerprints, hands, face geometry and blood samples are unique identifiers. The combination of technologies would enable companies to make metal or composite objects with the anfractuosities of art or the geometries of biology, parts with new functions and properties.

Near Antonyms composition , material , matter , raw material , stuff , substance ;. Related Words contour , outline , profile , silhouette ; frame , framework , shell , skeleton ; arrangement , design , format , layout , makeup , organization , pattern , plan , setup ;. Other Mathematics and Statistics Terms abscissa , denominator , divisor , equilateral , exponent , hypotenuse , logarithm , oblique , radii , rhomb.

Quadrature of the Lune. These were the substitution and mechanical approaches. The method of exhaustion as developed by Eudoxus approximates a curve or surface by using polygons with calculable perimeters and areas. The last great Platonist and Euclidean commentator of antiquity, Proclus c. 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.

The theorem may have earned its nickname from the Euclidean figure or from the commonsense notion that only an ass would require proof of so obvious a statement. The 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.

By the time of Plato, geometers customarily proved their propositions. Their compulsion and the multiplication of theorems it produced fit perfectly with the endless questioning of Socrates and the uncompromising logic of Aristotle. Perhaps the origin, and certainly the exercise, of the peculiarly Greek method of mathematical proof should be sought in the same social setting that gave rise to the practice of philosophy—that is, the Greek polis.

There citizens learned the skills of a governing class, and the wealthier among them enjoyed the leisure to engage their minds as they pleased, however useless the result, while slaves attended to the necessities of life.

Greek society could support the transformation of geometry from a practical art to a deductive science. Despite its rigour, however, Greek geometry does not satisfy the demands of the modern systematist. Euclid himself sometimes appeals to inferences drawn from an intuitive grasp of concepts such as point and line or inside and outside, uses superposition, and so on.

It took more than 2, years to purge the Elements of what pure deductivists deemed imperfections. Of this preliminary matter, the fifth and last postulate, which states a sufficient condition that two straight lines meet if sufficiently extended, has received by far the greatest attention. In effect it defines parallelism. Many later geometers tried to prove the fifth postulate using other parts of the Elements. The first six books contain most of what Euclid delivers about plane geometry.

Book VI applies the theory of proportion from Book V to similar figures and presents the geometrical solution to quadratic equations. As usual, some of it is older than Euclid. Books VII—X, which concern various sorts of numbers, especially primes, and various sorts of ratios, are seldom studied now, despite the importance of the masterful Book X, with its elaborate classification of incommensurable magnitudes, to the later development of Greek geometry.

XI contains theorems about the intersection of planes and of lines and planes and theorems about the volumes of parallelepipeds solids with parallel parallelograms as opposite faces ; XII applies the method of exhaustion introduced by Eudoxus to the volumes of solid figures, including the sphere; XIII, a three-dimensional analogue to Book IV, describes the Platonic solids.

Among the jewels in Book XII is a proof of the recipe used by the Egyptians for the volume of a pyramid. During its daily course above the horizon the Sun appears to describe a circular arc. Our astronomer, using the pointer of a sundial, known as a gnomon, as his eye, would generate a second, shadow cone spreading downward. The possible intersections of a plane with a cone, known as the conic sections , are the circle, ellipse, point, straight line, parabola , and hyperbola. Doubtless, however, both knew that all the conics can be obtained from the same right cone by allowing the section at any angle.

Apollonius reproduced known results much more generally and discovered many new properties of the figures. He first proved that all conics are sections of any circular cone, right or oblique. Apollonius introduced the terms ellipse , hyperbola , and parabola for curves produced by intersecting a circular cone with a plane at an angle less than, greater than, and equal to, respectively, the opening angle of the cone.

In an inspired use of their geometry, the Greeks did what no earlier people seems to have done: Thus they assigned to the Sun a circle eccentric to the Earth to account for the unequal lengths of the seasons. Ptolemy flourished — ce in Alexandria, Egypt worked out complete sets of circles for all the planets. Contrary to the Elements , however, the Almagest deploys geometry for the purpose of calculation. Among the items Ptolemy calculated was a table of chords, which correspond to the trigonometric sine function later introduced by Indian and Islamic mathematicians.

The table of chords assisted the calculation of distances from angular measurements as a modern astronomer might do with the law of sines. The application of geometry to astronomy reframed the perennial Greek pursuit of the nature of truth.

That gave two observationally equivalent solar theories based on two quite different mechanisms. Geometry was too prolific of alternatives to disclose the true principles of nature. The Greeks, who had raised a sublime science from a pile of practical recipes, discovered that in reversing the process, in reapplying their mathematics to the world, they had no securer claims to truth than the Egyptian rope pullers.

Since the ancients recognized four or five elements at most, Plato sought a small set of uniquely defined geometrical objects to serve as elementary constituents. He found them in the only three-dimensional structures whose faces are equal regular polygons that meet one another at equal solid angles: The cosmology of the Timaeus had a consequence of the first importance for the development of mathematical astronomy.

It guided Johannes Kepler — 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. Geometry offered Greek cosmologists not only a way to speculate about the structure of the universe but also the means to measure it.

Measuring the Earth, Classical and Arabic. Aristarchus of Samos c. Ptolemy equated the maximum distance of the Moon in its eccentric orbit with the closest approach of Mercury riding on its epicycle; the farthest distance of Mercury with the closest of Venus; and the farthest of Venus with the closest of the Sun.

Thus he could compute the solar distance in terms of the lunar distance and thence the terrestrial radius. His answer agreed with that of Aristarchus. The Ptolemaic conception of the order and machinery of the planets, the most powerful application of Greek geometry to the physical world, thus corroborated the result of direct measurement and established the dimensions of the cosmos for over a thousand years.

As the ancient philosophers said, there is no truth in astronomy. Two centuries after they broke out of their desert around Mecca, the followers of Muhammad occupied the lands from Persia to Spain and settled down to master the arts and sciences of the peoples they had conquered. They admired especially the works of the Greek mathematicians and physicians and the philosophy of Aristotle.

By the late 9th century they were already able to add to the geometry of Euclid, Archimedes, and Apollonius. In the 10th century they went beyond Ptolemy. Stimulated by the problem of finding the effective orientation for prayer the qiblah , or direction from the place of worship to Mecca , Islamic geometers and astronomers developed the stereographic projection invented to project the celestial sphere onto a two-dimensional map or instrument as well as plane and spherical trigonometry.

Here they incorporated elements derived from India as well as from Greece. Their achievements in geometry and geometrical astronomy materialized in instruments for drawing conic sections and, above all, in the beautiful brass astrolabes with which they reduced to the turn of a dial the toil of calculating astronomical quantities.

There they presided over translations of the Greek classics. He translated Archimedes and Apollonius, some of whose books now are known only in his versions. In a notable addition to Euclid, he tried valiantly to prove the parallel postulate discussed later in Non-Euclidean geometries. Among the pieces of Greek geometrical astronomy that the Arabs made their own was the planispheric astrolabe , which incorporated one of the methods of projecting the celestial sphere onto a two-dimensional surface invented in ancient Greece.

As Ptolemy showed in his Planisphaerium , the fact that the stereographic projection maps circles into circles or straight lines makes the astrolabe a very convenient instrument for reckoning time and representing the motions of celestial bodies. The earliest known Arabic astrolabes and manuals for their construction date from the 9th century. The Islamic world improved the astrolabe as an aid for determining the time for prayers, for finding the direction to Mecca, and for astrological divination.

Contacts among Christians, Jews, and Arabs in Catalonia brought knowledge of the astrolabe to the West before the year The Elements Venice, was one of the first technical books ever printed.

Archimedes also came West in the 12th century, in Latin translations from Greek and Arabic sources. Apollonius arrived only by bits and pieces. Not until the humanists of the Renaissance turned their classical learning to mathematics, however, did the Greeks come out in standard printed editions in both Latin and Greek. These texts affected their Latin readers with the strength of revelation. Europeans discovered the notion of proof, the power of generalization, and the superhuman cleverness of the Greeks; they hurried to master techniques that would enable them to improve their calendars and horoscopes, fashion better instruments, and raise Christian mathematicians to the level of the infidels.

It took more than two centuries for the Europeans to make their unexpected heritage their own. By the 15th century, however, they were prepared to go beyond their sources. The most novel developments occurred where creativity was strongest, in the art of the Italian Renaissance. The theory of linear perspective, the brainchild of the Florentine architect-engineers Filippo Brunelleschi — and Leon Battista Alberti —72 and their followers, was to help remake geometry during the 17th century.

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. Supposing this decorated window to be the canvas, Alberti interpreted the painting-to-be as the projection of the scene in life onto a vertical plane cutting the visual pyramid. At the same time, cartographers tried various projections of the sphere to accommodate the record of geographical discoveries that began in the midth century with Portuguese exploration of the west coast of Africa.

Cartographers therefore adopted the stereographic projection that had served astronomers. After cutting the cylinder along a vertical line and flattening the resulting rectangle, the result was the now-familiar Mercator map shown in the photograph. The intense cultivation of methods of projection by artists, architects, and cartographers during the Renaissance eventually provoked mathematicians into considering the properties of linear perspective in general.

The most profound of these generalists was a sometime architect named Girard Desargues — Desargues observed that neither size nor shape is generally preserved in projections, but collinearity is, and he provided an example, possibly useful to artists, in images of triangles seen from different points of view.

Despite his generality of approach, Apollonius needed to prove all his theorems for each type of conic separately. Desargues saw that he could prove them all at once and, moreover, by treating a cylinder as a cone with vertex at infinity, demonstrate useful analogies between cylinders and cones. Following his lead, Pascal made his surprising discovery that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic lie on a straight line.

What Descartes had in mind was the use of compasses with sliding members to generate curves. To classify and study such curves, Descartes took his lead from the relations Apollonius had used to classify conic sections, which contain the squares, but no higher powers, of the variables. To describe the more complicated curves produced by his instruments or defined as the loci of points satisfying involved criteria , Descartes had to include cubes and higher powers of the variables.

He thus overcame what he called the deceptive character of the terms square , rectangle , and cube as used by the ancients and came to identify geometric curves as depictions of relationships defined algebraically. By reducing relations difficult to state and prove geometrically to algebraic relations between coordinates usually rectangular of points on curves, Descartes brought about the union of algebra and geometry that gave birth to the calculus.

The familiar use of infinity, which underlay much of perspective theory and projective geometry, also leavened the tedious Archimedean method of exhaustion.

Not surprisingly, a practical man, the Flemish engineer Simon Stevin — , 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.

Instead of confining the circle between an inscribed and a circumscribed polygon, the new view regarded the circle as identical to the polygons, and the polygons to one another, when the number of their sides becomes infinitely great.

A second geometrical inspiration for the calculus derived from efforts to define tangents to curves more complicated than conics. Let the sides sought for the rectangle be denoted by a and b. Fermat observed what Kepler had perceived earlier in investigating the most useful shapes for wine casks, that near its maximum or minimum a quantity scarcely changes as the variables on which it depends alter slightly.

The figure with maximum area is a square. To obtain the tangent to a curve by this method, Fermat began with a secant through two points a short distance apart and let the distance vanish see figure.

Part of the motivation for the close study of Apollonius during the 17th century was the application of conic sections to astronomy. His astronomy thus made pressing and practical the otherwise merely difficult problem of the quadrature of conics and the associated theory of indivisibles.

With the methods of Apollonius and a few infinitesimals, an inspired geometer showed that the laws regarding both area and ellipse can be derived from the suppositions that bodies free from all forces either rest or travel uniformly in straight lines and that each planet constantly falls toward the Sun with an acceleration that depends only on the distance between their centres. Besides the problem of planetary motion, questions in optics pushed 17th-century natural philosophers and mathematicians to the study of conic sections.

As Archimedes is supposed to have shown or shone in his destruction of a Roman fleet by reflected sunlight, a parabolic mirror brings all rays parallel to its axis to a common focus. The story of Archimedes provoked many later geometers, including Newton, to emulation. Eventually they created instruments powerful enough to melt iron. 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.

A final example of early modern applications of geometry to the physical world is the old problem of the size of the Earth. Measuring the Earth, Modernized. On the hypothesis that the Earth cooled from a spinning liquid blob, Newton calculated that it is an oblate spheroid obtained by rotating an ellipse around its minor axis , not a sphere, and he gave the excess of its equatorial over its polar diameter. During the 18th century many geodesists tried to find the eccentricity of the terrestrial ellipse.

At first it appeared that all the measurements might be compatible with a Newtonian Earth. By the end of the century, however, geodesists had uncovered by geometry that the Earth does not, in fact, have a regular geometrical shape. The dominance of analysis algebra and the calculus during the 18th century produced a reaction in favour of geometry early in the 19th century. Fundamental new branches of the subject resulted that deepened, generalized, and violated principles of ancient geometry.

The cultivators of these new fields, such as Jean-Victor Poncelet — and his self-taught disciple Jakob Steiner — , vehemently urged the claims of geometry over analysis. Poncelet relied on this information to keep himself alive. The result was projective geometry. Poncelet employed three basic tools. One he took from Desargues: The second tool, continuity , allows the geometer to claim certain things as true for one figure that are true of another equally general figure provided that the figures can be derived from one another by a certain process of continual change.

Poncelet and his defender Michel Chasles — 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. Similarly, parallelism had to go.

Efforts were well under way by the middle of the 19th century, by Karl George Christian von Staudt — among others, to purge projective geometry of the last superfluous relics from its Euclidean past.

The first possibility gives Euclidean geometry. Saccheri devoted himself to proving that the obtuse and the acute alternatives both end in contradictions, which would thereby eliminate the need for an explicit parallel postulate.

He then destroyed the obtuse hypothesis by an argument that depended upon allowing lines to increase in length indefinitely.

Geometry (from the Ancient Greek: γεωμετρία; geo-"earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.. Geometry arose independently in a number of early cultures as .

Geometry. Geometry is all about shapes and their properties.. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles shapes that can be drawn on a piece of paper.

Learn high school geometry for free—transformations, congruence, similarity, trigonometry, analytic geometry, and more. Full curriculum of exercises and videos. Set students up for success in Geometry and beyond! Explore the entire Geometry curriculum: angles, geometric constructions, and more. Try it free!

the geometry of Sydney's famed opera house is suggestive of some modernistic sailing ship. Geometry: Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in.