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## GEOMETRY Defined for English Language Learners

❶Attention was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. Besides the problem of planetary motion, questions in optics pushed 17th-century natural philosophers and mathematicians to the study of conic sections.
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The first notable attempt to reorganize the study of geometry was made by the German mathematician Felix Klein and published at Erlangen in In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. 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. 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.

Ancient builders and surveyors needed to be able to construct right angles in the field on demand. One way that they could have employed a rope to construct right triangles was to mark a looped rope with knots so that, when held at the knots and pulled tight, the rope must form a right triangle.

The simplest way to perform the trick is to take a rope that is 12 units long, make a knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop, as shown in the animation. However, the Egyptian scribes have not left us instructions about these procedures, much less any hint that they knew how to generalize them to obtain the Pythagorean theorem: The required right angles were made by ropes marked to give the triads 3, 4, 5 and 5, 12, In Babylonian clay tablets c.

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. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of incommensurability.

By ancient tradition, Thales of Miletus , who lived before Pythagoras in the 6th century bce , invented a way to measure inaccessible heights, such as the Egyptian pyramids. Although none of his writings survives, Thales may well have known about a Babylonian observation that for similar triangles triangles having the same shape but not necessarily the same size the length of each corresponding side is increased or decreased by the same multiple.

A determination of the height of a tower using similar triangles is demonstrated in the figure. A Babylonian cuneiform tablet written some 3, years ago treats problems about dams, wells, water clocks, and excavations. Ahmes , the scribe who copied and annotated the Rhind papyrus c.

In addition to proving mathematical theorems, ancient mathematicians constructed various geometrical objects. Euclid arbitrarily restricted the tools of construction to a straightedge an unmarked ruler and a compass. The restriction made three problems of particular interest to double a cube, to trisect an arbitrary angle, and to square a circle very difficult—in fact, impossible.

Various methods of construction using other means were devised in the classical period, and efforts, always unsuccessful, using straightedge and compass persisted for the next 2, years. The Vedic scriptures made the cube the most advisable form of altar for anyone who wanted to supplicate in the same place twice.

The rules of ritual required that the altar for the second plea have the same shape but twice the volume of the first. The problem came to the Greeks together with its ceremonial content.

An oracle disclosed that the citizens of Delos could free themselves of a plague merely by replacing an existing altar by one twice its size. The Delians applied to Plato. Hippocrates of Chios , who wrote an early Elements about bce , took the first steps in cracking the altar problem. He reduced the duplication to finding two mean proportionals between 1 and 2, that is, to finding lines x and y in the ratio 1: A few generations later, Eratosthenes of Cyrene c.

The Egyptians told time at night by the rising of 12 asterisms constellations , each requiring on average two hours to rise. In order to obtain more convenient intervals, the Egyptians subdivided each of their asterisms into three parts, or decans. That presented the problem of trisection. It is not known whether the second celebrated problem of archaic Greek geometry, the trisection of any given angle, arose from the difficulty of the decan, but it is likely that it came from some problem in angular measure.

Although no one succeeded in finding a solution with straightedge and compass, they did succeed with a mechanical device and by a trick. The mechanical device, perhaps never built, creates what the ancient geometers called a quadratrix.

Invented by a geometer known as Hippias of Elis flourished 5th century bce , the quadratrix is a curve traced by the point of intersection between two moving lines, one rotating uniformly through a right angle, the other gliding uniformly parallel to itself.

The Quadratrix of Hippias. The trick for trisection is an application of what the Greeks called neusis , a maneuvering of a measured length into a special position to complete a geometrical figure.

A late version of its use, ascribed to Archimedes c. The pre-Euclidean Greek geometers transformed the practical problem of determining the area of a circle into a tool of discovery. Three approaches can be distinguished: While not able to square the circle, Hippocrates did demonstrate the quadratures of lunes; that is, he showed that the area between two intersecting circular arcs could be expressed exactly as a rectilinear area and so raised the expectation that the circle itself could be treated similarly.

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. He is studying geometry. ABCs affine affine geometry Alhacen Al-Hasan ibn al-Haytham al-Haytham Alhazen analytic geometry analytical geometry angle of inclination Archimedes arcograph Barycentric calculus Bernhard Riemann bothrodesy Circle at infinity congruent conic conic section.

References in classic literature? The geometry , for instance, they taught you at school is founded on a misconception. Thanks be, I'm done with geometry , learning or teaching it," said Anne Shirley, a trifle vindictively, as she thumped a somewhat battered volume of Euclid into a big chest of books, banged the lid in triumph, and sat down upon it, looking at Diana Wright across the Green Gables garret, with gray eyes that were like a morning sky.

You 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?

For in sciences which use demonstration there is that which is prior and that which is posterior in order; in geometry , the elements are prior to the propositions; in reading and writing, the letters of the alphabet are prior to the syllables.

He himself undertook his daughter's education, and to develop these two cardinal virtues in her gave her lessons in algebra and geometry till she was twenty, and arranged her life so that her whole time was occupied. Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry , I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts presentations which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects presentations that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams.

It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time. Of this nature are the maxims in geometry , that "the whole is greater than its part; things equal to the same are equal to one another; two straight lines cannot enclose a space; and all right angles are equal to each other.

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.

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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.