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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 surveying, and its name is derived from Greek words meaning "Earth measurement." Eventually it was realized that geometry need not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms.
This article begins with a brief guidepost to the major branches of geometry and then proceeds to an extensive historical treatment. For information on specific branches of geometry, see Euclidean geometry, analytic geometry, projective geometry, differential geometry, non-Euclidean geometries, and topology.
Major branches of geometry
Euclidean geometry
In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. This geometry was codified in Euclid's Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. The Elements epitomized the axiomatic-deductive method for many centuries.
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.
Projective geometry
Projective geometry originated with the French mathematician Girard Desargues (1591–1661) to deal with those properties of geometric figures that are not altered by projecting their image, or "shadow," onto another surface.
Differential geometry
The German mathematician 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.
Non-Euclidean geometries
Beginning in the 19th century, various mathematicians substituted alternatives to Euclid's parallel postulate, which, in its modern form, reads, "given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line." They hoped to show that the alternatives were logically impossible. 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 1911, when the Dutch mathematician L.E.J. Brouwer (1881–1966) introduced methods generally applicable to the topic.
History of geometry
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. Beginning about the 6th century bce, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words geo ("Earth") and metron ("measure") for the measurement of the Earth.
In addition to describing some of the achievements of the ancient Greeks, notably Euclid's logical development of geometry in the Elements, this article examines some applications of geometry to astronomy, cartography, and painting from classical Greece through medieval Islam and Renaissance Europe. It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in the modern age.
Ancient geometry: practical and empirical
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. 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 2,300 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. In fact, the only thing known with a fair degree of confidence is that Euclid taught at the Library of Alexandria during the reign of Ptolemy I (323–285/283 bce). 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.
Euclid's Elements was so complete and clearly written that it literally obliterated the work of his predecessors. 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. 580–c. 500 bce) and his followers. 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.
Finding the right angle
Ancient builders and surveyors needed to be able to construct right angles in the field on demand. The method employed by the Egyptians earned them the name "rope pullers" in Greece, apparently because they employed a rope for laying out their construction guidelines. 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 square on the line opposite the right angle equals the sum of the squares on the other two sides. Similarly, the Vedic scriptures of ancient India contain sections called sulvasutras, or "rules of the rope," for the exact positioning of sacrificial altars. The required right angles were made by ropes marked to give the triads (3, 4, 5) and (5, 12, 13).
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. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of incommensurability.
Locating the inaccessible
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. The ancient Chinese arrived at measures of inaccessible heights and distances by another route, using "complementary" rectangles, as seen in the next figure, which can be shown to give results equivalent to those of the Greek method involving triangles.
Estimating the wealth
A Babylonian cuneiform tablet written some 3,500 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise on circular enclosures with an implied value of π = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23), used the same value. However, the Hebrews should have taken their π from the Egyptians before crossing the Red Sea, for the Rhind papyrus (c. 2000 bce; our principal source for ancient Egyptian mathematics) implies π = 3.1605.
Knowledge of the area of a circle was of practical value to the officials who kept track of the pharaoh's tribute as well as to the builders of altars and swimming pools. Ahmes, the scribe who copied and annotated the Rhind papyrus (c. 1650 bce), has much to say about cylindrical granaries and pyramids, whole and truncated. He could calculate their volumes, and, as appears from his taking the Egyptian seked, the horizontal distance associated with a vertical rise of one cubit, as the defining quantity for the pyramid's slope, he knew something about similar triangles.
Ancient geometry: abstract and applied
The three classical problems
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 2,000 years. In 1837 the French mathematician Pierre Laurent Wantzel proved that doubling the cube and trisecting the angle are impossible, and in 1880 the German mathematician Ferdinand von Lindemann showed that squaring the circle is impossible, as a consequence of his proof that π is a transcendental number.
Doubling the cube
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. If the sides of the original and derived altars are a and b, respectively, then b3 = 2a3. 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. He replied that the oracle did not mean that the gods wanted a larger altar but that they had intended "to shame the Greeks for their neglect of mathematics and their contempt for geometry." With this blend of Vedic practice, Greek myth, and academic manipulation, the problem of the duplication of the cube took a leading place in the formation of Greek geometry.
Hippocrates of Chios, who wrote an early Elements about 450 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:x = x:y = y:2. After the intervention of the Delian oracle, several geometers around Plato's Academy found complicated ways of generating mean proportionals.
A few generations later, Eratosthenes of Cyrene (c. 276–c. 194 bce) devised a simple instrument with moving parts that could produce approximate mean proportionals.
Trisecting the angle
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.
Several geometers of Plato's time tried their hands at trisection. 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. (See Sidebar: Trisecting the Angle: 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. 285–212/211 bce), exemplifies the method of angle trisection. (See Sidebar: Trisecting the Angle: Archimedes' Method.)
Squaring the circle
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: Hippocrates' dodge of substituting one problem for another; the application of a mechanical instrument, as in Hippias's device for trisecting the angle; and the technique that proved the most fruitful, the closer and closer approximation to an unknown magnitude difficult to study (e.g., the area of a circle) by a series of known magnitudes easier to study (e.g., areas of polygons)—a technique known in modern times as the "method of exhaustion" and attributed by its greatest practitioner, Archimedes, to Plato's student Eudoxus of Cnidus (c. 408–c. 355 bce).
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. (See Sidebar: Quadrature of the Lune.) A contemporary of Hippias's discovered that the quadratrix could be used to almost rectify circles. 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. As the number of sides of a regular polygon inscribed in a circle increases indefinitely, its perimeter and area "exhaust," or take up, the circumference and area of the circle to within any assignable error of length or area, however small. In Archimedes' usage, the method of exhaustion produced upper and lower bounds for the value of π, the ratio of any circle's circumference to its diameter. This he accomplished by inscribing a polygon within a circle, and circumscribing a polygon around it as well, thereby bounding the circle's circumference between the polygons' calculable perimeters. He used polygons with 96 sides and thus bound π between 310/71 and 31/7. (See animation.)
Idealization and proof
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. 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. (See Sidebar: 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. 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. 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 2,000 years to purge the Elements of what pure deductivists deemed imperfections.
The Euclidean synthesis
Euclid, in keeping with the self-conscious logic of Aristotle, began the first of his 13 books of the Elements with sets of definitions ("a line is breadthless length"), common notions ("the whole is greater than the part"), and axioms, or postulates ("all right angles are equal"). 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. Euclid saw farther, for coherent geometries (known as non-Euclidean geometries) can be produced by replacing the fifth postulate with other postulates that contradict Euclid's choice.
The first six books contain most of what Euclid delivers about plane geometry. Book I presents many propositions doubtless discovered by his predecessors, from Thales' equality of the angles opposite the equal sides of an isosceles triangle to the Pythagorean theorem, with which the book effectively ends. (See Sidebar: Euclid's Windmill.)
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. (See Sidebar: Incommensurables.)
Books XI–XIII deal with solids: 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.
Gnomonics and the cone
During its daily course above the horizon the Sun appears to describe a circular arc. Supplying in his mind's eye the missing portion of the daily circle, the Greek astronomer could imagine that his real eye was at the apex of a cone, the surface of which was defined by the Sun's rays at different times of the day and the base of which was defined by the Sun's apparent diurnal course. Our astronomer, using the pointer of a sundial, known as a gnomon, as his eye, would generate a second, shadow cone spreading downward. The intersection of this second cone with a horizontal surface, such as the face of a sundial, would give the trace of the Sun's image (or shadow) during the day as a plane section of a cone. (The possible intersections of a plane with a cone, known as the conic sections, are the circle, ellipse, point, straight line, parabola, and hyperbola.)
However, the doxographers ascribe the discovery of conic sections to a student of Eudoxus's, Menaechmus (mid-4th century bce), who used them to solve the problem of duplicating the cube. His restricted approach to conics—he worked with only right circular cones and made his sections at right angles to one of the straight lines composing their surfaces—was standard down to Archimedes' era. Euclid adopted Menaechmus's approach in his lost book on conics, and Archimedes followed suit. Doubtless, however, both knew that all the conics can be obtained from the same right cone by allowing the section at any angle.
The reason that Euclid's treatise on conics perished is that Apollonius of Perga (c. 262–c. 190 bce) did to it what Euclid had done to the geometry of Plato's time. 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.
Astronomy and trigonometry
In an inspired use of their geometry, the Greeks did what no earlier people seems to have done: they geometrized the heavens by supposing that the Sun, Moon, and planets move around a stationary Earth on a rotating circle or set of circles, and they calculated the speed of rotation of these supposititious circles from observed motions. Thus they assigned to the Sun a circle eccentric to the Earth to account for the unequal lengths of the seasons.
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. 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. If a mathematical description fit the facts, as did Ptolemy's explanation of the unequal lengths of the seasons by the eccentricity of the Sun's orbit, should the description be taken as true of nature? The answer, with increasing emphasis, was "no." Astronomers remarked that the eccentric orbit representing the Sun's annual motion could be replaced by a pair of circles, a deferent centred on the Earth and an epicycle the centre of which moved along the circumference of the deferent. 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.
Ancient geometry: cosmological and metaphysical
Pythagorean numbers and Platonic solids
The Pythagoreans used geometrical figures to illustrate their slogan that all is number—thus their "triangular numbers" (n(n−1)/2), "square numbers" (n2), and "altar numbers" (n3), some of which are shown in the figure. This principle found a sophisticated application in Plato's creation story, the Timaeus, which presents the smallest particles, or "elements," of matter as regular geometrical figures. 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 tetrahedron, or pyramid (with 4 triangular faces); the cube (with 6 square faces); the octahedron (with 8 equilateral triangular faces); the dodecahedron (with 12 pentagonal faces); and the icosahedron (with 20 equilateral triangular faces). (See animation.)
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"), in which each of the known six planets revolved around the Sun on spheres separated by the five Platonic solids, as shown in the photograph. Although 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.
Measuring the Earth and heavens
Geometry offered Greek cosmologists not only a way to speculate about the structure of the universe but also the means to measure it. South of Alexandria and roughly on the same meridian of longitude is the village of Syene (modern Aswān), where the Sun stands directly overhead at noon on a midsummer day. At the same moment at Alexandria, the Sun's rays make an angle α with the tip of a vertical rod, as shown in the figure. Since the Sun's rays fall almost parallel on the Earth, the angle subtended by the arc l (representing the distance between Alexandria and Syene) at the centre of the Earth also equals α; thus the ratio of the Earth's circumference, C, to the distance, l, must equal the ratio of 360° to the angle α—in symbols, C:l = 360°:α. Eratosthenes made the measurements, obtaining a value of about 5,000 stadia for l, which gave a value for the Earth's circumference of about 250,000 stadia. Because the accepted length of the Greek stadium varied locally, we cannot accurately determine Eratosthenes' margin of error. However, if we credit the ancient historian Plutarch's guess at Eratosthenes' unit of length, we obtain a value for the Earth's circumference of about 46,250 km—remarkably close to the modern value (about 15 percent too large), considering the difficulty in accurately measuring l and α. (See Sidebar: Measuring the Earth, Classical and Arabic.)
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." The great difficulty of making the observations resulted in an underestimation of the solar distance about 20-fold—he obtained a solar distance, σ, roughly 1,200 times the Earth's radius, r. Possibly Aristarchus' inquiry into the relative sizes of the Sun, Moon, and Earth led him to propound the first heliocentric ("Sun-centred") model of the universe.
Aristarchus' value for the solar distance was confirmed by an astonishing coincidence. 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.
The post-classical period
Passage through Islam
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.
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. The Banū Mūsā directed a House of Wisdom in Baghdad sponsored by the caliph. There they presided over translations of the Greek classics. Thābit became an ornament of the House of Wisdom. 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. One of the desirable mathematical features of this method (the stereographic projection) is that it converts circles into circles or straight lines, a property proved in the first pages of Apollonius's Conics. 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.
Europe rediscovers the classics
Contacts among Christians, Jews, and Arabs in Catalonia brought knowledge of the astrolabe to the West before the year 1000. During the 12th century many manuals for its use and construction were translated into Latin along with geometrical works by the Banū Mūsā, Thābit, and others. Some of the achievements of the Arab geometers were rediscovered in the West after wide and close study of Euclid's Elements, which was translated repeatedly from the Arabic and once from the Greek in the 12th and 13th centuries. The Elements (Venice, 1482) 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. Ptolemy's Almagest appeared in Latin manuscript in 1175. 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.
Linear perspective
The theory of linear perspective, the brainchild of the Florentine architect-engineers Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72) and their followers, was to help remake geometry during the 17th century. The scheme of Brunelleschi and Alberti, as given without proofs in Alberti's De pictura (1435; On Painting), exploits the pyramid of rays that, according to what they had learned from the Westernized versions of the optics of 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. 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. A distinctive feature of his system was the "point at infinity" at which parallel lines in the painting appear to converge, as shown in the photograph.
Alberti's procedure, as developed by Piero della Francesca (c. 1410–92) and 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. Coincidentally with these explorations, mapmakers recovered Ptolemy's Geography, in which he had recorded by latitude (sometimes near enough) and longitude (usually far off) the principal places known to him and indicated how they could be projected onto a map (see photograph).
The discoveries that enlarged the known Earth did not fit easily on Ptolemy's projections. Cartographers therefore adopted the stereographic projection that had served astronomers. Several projected the Northern Hemisphere onto the Equator just as in the standard astrolabe, but the most widely used aspect, popularized in the world maps made by Gerardus Mercator's son for later editions of his father's atlas (beginning in 1595), projected points on the Earth onto a cylinder tangent to the Earth at the Equator. 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 (1591–1661).
French circles
Desargues was a member of intersecting circles of 17th-century French mathematicians worthy of Plato's Academy of the 4th century bce or Baghdad's House of Wisdom of the 9th century ce. They included René Descartes (1596–1650) and Pierre de Fermat (1601–65), inventors of analytic geometry; Gilles Personne de Roberval (1602–75), a pioneer in the development of the calculus; and Blaise Pascal (1623–62), a contributor to the calculus and an exponent of the principles set forth by Desargues.
Projective geometry
Two main directions can be distinguished in Desargues's work. Like Renaissance artists, Desargues freely admitted the point at infinity into his demonstrations and showed that every set of parallel lines in a scene (apart from those parallel to the sides of the canvas) should project as converging bundles at some point on the "line at infinity" (the horizon). With the addition of points at infinity to the Euclidean plane, Desargues could frame all his propositions about straight lines without excepting parallel ones—which, like the others, now met one another, although not before "infinity." A farther-reaching matter arising from artistic perspective was the relation between projections of the same object from different points of view and different positions of the canvas. 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. The statement that accompanied this example became known as Desargues's theorem.
Desargues's second direction was to "simplify" Apollonius's work on conic sections. 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. (See figure.) 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.
Cartesian geometry
In 1619, as part of the great illumination that inspired Descartes to assume the modest chore of reforming philosophy as well as mathematics, he devised "compasses" made of sticks sliding in grooved frames to duplicate the cube and trisect angles. Descartes esteemed these implements and the constructions they effected as (to quote from a letter of 1619) "no less certain and geometrical than the ordinary ones with which circles are drawn." By the use of apt instruments, he would bring ancient mathematics to perfection: "scarcely anything will remain to be discovered in geometry."
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.
Geometrical 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 (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. 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.
This revitalized approach to exhaustion received a preliminary systematization in the Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota (1635; "A Method for the Determination of a New Geometry of Continuous Indivisibles") by the Italian 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. Cavalieri's method may be stated as follows: if two figures (solids) of equal height are cut by parallel lines (planes) such that each pair of lengths (areas) matches, then the two figures (solids) have the same area (volume). (See figure.) Although not up to the rigorous standards of today and criticized by "classicist" contemporaries (who were unaware that Archimedes himself had explored similar techniques), Cavalieri's method of indivisibles became a standard tool for solving volumes until the introduction of integral calculus near the end of the 17th century.
A second geometrical inspiration for the calculus derived from efforts to define tangents to curves more complicated than conics. Fermat's method, representative of many, had as its exemplar the problem of finding the rectangle that maximizes the area for a given perimeter. Let the sides sought for the rectangle be denoted by a and b. Increase one side and diminish the other by a small amount ε; the resultant area is then given by (a + ε)(b − ε). 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. On this principle, Fermat equated the areas ab and (a + ε)(b − ε) to obtain the stationary values: ab = ab − εa + εb − ε2. By canceling the common term ab, dividing by ε, and then setting ε at zero, Fermat had his well-known answer, a = b. 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).
The world system
Part of the motivation for the close study of Apollonius during the 17th century was the application of conic sections to astronomy. Kepler not only replaced the many circles of the old planetary system with a few ellipses, he also substituted a complicated rule of motion (his "second law") for the relatively simple Ptolemaic rule that all motions must be compounded of rotations performed at constant velocity. Kepler's second law states that a planet moves in its ellipse so that the line between it and the Sun placed at a focus sweeps out equal areas in equal times. 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. The inspired geometer was Isaac Newton (1642 [Old Style]–1727), who made planetary dynamics a matter entirely of geometry by replacing the planetary orbit by a succession of infinitesimal chords, planetary acceleration by a series of centripetal jerks, and, in keeping with Kepler's second law, time by an area.
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.
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.
A final example of early modern applications of geometry to the physical world is the old problem of the size of the Earth. (See Sidebar: 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.
Relaxation and rigour
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 (1788–1867) and his self-taught disciple Jakob Steiner (1796–1863), vehemently urged the claims of geometry over analysis. The early 19th-century revival of pure geometry produced the discovery that Euclid had devoted his efforts to only one of several comprehensive geometries, the others of which can be created by replacing Euclid's fifth postulate with another about parallels.
Projection again
Poncelet, who was an officer in the French corps of engineers, learned scraps of Desargues's work from his teacher Gaspard Monge (1746–1818), who developed his own method of projection for drawings of buildings and machines. Poncelet relied on this information to keep himself alive. Taken captive during Napoleon's invasion of Russia in 1812, he passed his time by rehearsing in his head the things he had learned from Monge. The result was projective geometry.
Poncelet employed three basic tools. One he took from Desargues: the demonstration of difficult theorems about a complicated figure by working out equivalent simpler theorems on an elementary figure interchangeable with the original figure by projection. 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 (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.
Poncelet's third tool was the "principle of duality," which interchanges various concepts such as points with lines, or lines with planes, so as to generate new theorems from old theorems. Desargues's theorem allows their interchange. So, as Steiner showed, does Pascal's theorem that the three points of intersection of the opposite sides of a hexagon inscribed in a conic lie on a line; thus, the lines joining the opposite vertices of a hexagon circumscribed about a conic meet in a point. (See figure.)
Poncelet's followers realized that they were hampering themselves, and disguising the true fundamentality of projective geometry, by retaining the concept of length and congruence in their formulations, since projections do not usually preserve them. Similarly, parallelism had to go. Efforts were well under way by the middle of the 19th century, by Karl George Christian von Staudt (1798–1867) among others, to purge projective geometry of the last superfluous relics from its Euclidean past.
Non-Euclidean geometries
The Enlightenment was not so preoccupied with analysis as to completely ignore the problem of Euclid's fifth postulate. In 1733 Girolamo Saccheri (1667–1733), a Jesuit professor of mathematics at the University of Pavia, Italy, substantially advanced the age-old discussion by setting forth the alternatives in great clarity and detail before declaring that he had "cleared Euclid of every defect" (Euclides ab Omni Naevo Vindicatus, 1733). Euclid's fifth postulate runs: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles." Saccheri took up the quadrilateral of Omar Khayyam (1048–1131), who started with two parallel lines AB and DC, formed the sides by drawing lines AD and BC perpendicular to AB, and then considered three hypotheses for the internal angles at C and D: to be right, obtuse, or acute (see figure). 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.
On the way to this spurious demonstration, Saccheri established several theorems of non-Euclidean geometry—for example, that according to whether the right, obtuse, or acute hypothesis is true, the sum of the angles of a triangle respectively equals, exceeds, or falls short of 180°. He then destroyed the obtuse hypothesis by an argument that depended upon allowing lines to increase in length indefinitely. If this is disallowed, the hypothesis of the obtuse angle produces a system equivalent to standard spherical geometry, the geometry of figures drawn on the surface of a sphere.
As for the acute angle, Saccheri could defeat it only by appealing to an arbitrary hypothesis about the behaviour of lines at infinity. One of his followers, the Swiss-German polymath Johann Heinrich Lambert (1728–77), observed that, based on the acute hypothesis, the area of a triangle is the negative of that of a spherical triangle. Since the latter is proportional to the square of the radius, r, the former appeared to Lambert to be the area of an imaginary sphere with radius ir, where i = Square root of√−1.
Although both Saccheri and Lambert aimed to establish the hypothesis of the right angle, their arguments seemed rather to indicate the unimpeachability of the alternatives. Several mathematicians at the University of Göttingen, notably the great Carl Friedrich Gauss (1777–1855), then took up the problem. Gauss was probably the first to perceive that a consistent geometry could be built up independent of Euclid's fifth postulate, and he derived many relevant propositions, which, however, he promulgated only in his teaching and correspondence. The earliest published non-Euclidean geometric systems were the independent work of two young men from the East who had nothing to lose by their boldness. Both can be considered Gauss's disciples once removed: the Russian Nikolay Ivanovich Lobachevsky (1792–1856), who learned his mathematics from a close friend of Gauss's at the University of Kazan, where Lobachevsky later became a professor; and János Bolyai (1802–60), an officer in the Austro-Hungarian army whose father also was a friend of Gauss's. Both Lobachevsky and Bolyai had worked out their novel geometries by 1826.
Lobachevsky and Bolyai reasoned about the hypothesis of the acute angle in the manner of Saccheri and Lambert and recovered their results about the areas of triangles. They advanced beyond Saccheri and Lambert by deriving an imaginary trigonometry to go with their imaginary geometry. Just as Desargues's projective geometry was neglected for many years, so the work of Bolyai and Lobachevsky made little impression on mathematicians for a generation and more. It was largely the posthumous publication in 1855 of Gauss's ideas about non-Euclidean geometry that gave the new approaches the cachet to attract the attention of later mathematicians.
A grand synthesis
Another of the profound impulses Gauss gave geometry concerned the general description of surfaces. Typically—with the notable exception of the geometry of the sphere—mathematicians had treated surfaces as structures in three-dimensional Euclidean space. However, as these surfaces occupy only two dimensions, only two variables are needed to describe them. This prompted the thought that two-dimensional surfaces could be considered as "spaces" with their own geometries, not just as Euclidean structures in ordinary space. For example, the shortest distance, or path, between two points on the surface of a sphere is the lesser arc of the great circle joining them, whereas, considered as points in three-dimensional space, the shortest distance between them is an ordinary straight line.
The shortest path between two points on a surface lying wholly within that surface is called a geodesic, which reflects the origin of the concept in geodesy, in which Gauss took an active interest. His initiative in the study of surfaces as spaces and geodesics as their "lines" was pursued by his student and, briefly, his successor at Göttingen, Bernhard Riemann (1826–66). Riemann began with an abstract space of n dimensions. That was in the 1850s, when mathematicians and mathematical physicists were beginning to use n-dimensional Euclidean space to describe the motions of systems of particles in the then-new kinetic theory of gases. Riemann worked in a quasi-Euclidean space—"quasi" because he used the calculus to generalize the Pythagorean theorem to supply sufficient flexibility to provide for geodesics on any surface.
When this very general differential geometry came down to two-dimensional surfaces of constant curvature, it revealed excellent models for non-Euclidean geometries. Riemann himself pointed out that, merely by calling the geodesics of a sphere "straight lines," the maligned hypothesis of the obtuse angle produces the geometry appropriate to the sphere's surface. Similarly, as shown by Eugenio Beltrami (1835–1900), who ended his teaching career in Saccheri's old post at Pavia, the geometry defined in the plane by the hypothesis of the acute angle fits perfectly a surface of revolution of constant negative curvature now called a pseudosphere (see figure)—again, provided that its geodesics are accepted as the straight lines of the geometry.
Since the hypothesis of the obtuse angle correctly characterizes Euclidean geometry applied to the surface of a sphere, the non-Euclidean geometry based on it must be exactly as consistent as Euclidean geometry. The case of the acute angle treated by Lobachevsky and Bolyai required a sharper tool. Beltrami found it in a projection into a disc in the Euclidean plane of the points of a non-Euclidean space, in which each geodesic from the non-Euclidean space corresponds to a chord of the disc. Geometry built on the hypothesis of the acute angle has the same consistency as Euclidean geometry.
The key role of Euclidean geometry in proofs of the consistency of non-Euclidean geometries exposed the Elements to ever-deeper scrutiny. The old blemishes—particularly appeals to intuition and diagrams for the meaning of concepts like "inside" and "between" and the use of questionable procedures like superposition to prove congruency—became intolerable to mathematicians who laboured to clarify the foundations of arithmetic and the calculus as well as the interrelations of the new geometries. The German mathematician Moritz Pasch (1843–1930), in his Vorlesungen über neuere Geometrie (1882; "Lectures on the New Geometry"), identified what was wanting: undefined concepts, axioms about those concepts, and more rigorous logic based on those axioms. The choice of undefined concepts and axioms is free, apart from the constraint of consistency. Mathematicians following Pasch's path introduced various elements and axioms and developed their geometries with greater or lesser elegance and trouble. The most successful of these systematizers was the Göttingen professor David Hilbert (1862–1943), whose The Foundations of Geometry (1899) greatly influenced efforts to axiomatize all of mathematics. (See Sidebar: Teaching the Elements.)
The real world
Euclid's Elements had claimed the excellence of being a true account of space. Within this interpretation, Euclid's fifth postulate was an empirical finding; non-Euclidean geometries did not apply to the real world. Bolyai apparently could not free himself from the persuasion that Euclidean geometry represented reality. Lobachevsky observed that, if there were a star so distant that its parallax was not observable from the Earth's orbit, his geometry would be indistinguishable from Euclid's at the point where the parallax vanished. By his calculation, based on stellar parallaxes then just detected, his geometry could be physically meaningful only in gargantuan triangles spanning interstellar space.
In fact, non-Euclidean geometries apply to the cosmos more locally than Lobachevsky imagined. In 1916 Albert Einstein (1879–1955) published "The Foundation of the General Theory of Relativity," which replaced Newton's description of gravitation as a force that attracts distant masses to each other through Euclidean space with a principle of least effort, or shortest (temporal) path, for motion along the geodesics of a curved space. Einstein not only explained how gravitating bodies give this surface its properties—that is, mass determines how the differential distances, or curvatures, in Riemann's geometry differ from those in Euclidean space—but also successfully predicted the deflection of light, which has no mass, in the vicinity of a star or other massive body. This was an extravagant piece of geometrizing—the replacement of gravitational force by the curvature of a surface. But it was not all. In relativity theory time is considered to be a dimension along with the three dimensions of space. On the closed four-dimensional world thus formed, the history of the universe stands revealed as describable by motion within a vast congeries of geodesics in a non-Euclidean universe.
Learn More in these related Britannica articles:
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Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid ( c.300 bce). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th…
mathematics: The foundations of geometryBy the late 19th century the hegemony of Euclidean geometry had been challenged by non-Euclidean geometry and projective geometry. The first notable attempt to reorganize the study of geometry was made by the German mathematician Felix Klein and published at Erlangen in 1872. In…
mathematics: Geometric and algebraic problemsIn a Babylonian tablet now in Berlin, the diagonal of a rectangle of sides 40 and 10 is solved as 40 + 102/(2 × 40). Here a very effective approximating rule is being used (that the square root of the sum…
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More About Geometry31 references found in Britannica articles
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The Panthers are using the weekend to get ahead on some roster moves, crossing off 12 names in advance of next Tuesday's deadline to get rosters from 90 to 75.
They waived-injured three players, who will revert to IR if not claimed: Tackle Bruce Campbell, wide receiver Kealoha Pilares and fullback Michael Zordich.
Pilares injured his knee in Thursday's game, while Campbell (shoulder) and Zordich (knee) were injured earlier in camp.
Campbell was their backup left tackle, but his rotator cuff surgery was going to keep him out this year, and exposed how thin they were up front.
Pilares was scrambling to find a roster spot even before he was injured, as the former fifth-rounder was in a crowded group of returner-receivers.
They also waived nine players: Quarterback Colby Cameron, punter Jordan Gay, cornerback Nick Hixson, kicker Morgan Lineberry, wide receiver Dale Moss, defensive end Louis Nzegwu , linebacker Ryan Rau, safety Ricardo Silva and guard Justin Wells.

subset: commoncrawl


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Indignant reader defends Idiot 2.0™
'Bugger off out the office'
FoTW Oh dear, oh dear - it appears CNET's blogging pundits are not actually a legitimate target for us proper hacks, those of us who've done English O-Level and read a couple of novels.
In case you missed it, this hack yesterday took exception to this crap by CNET's Steve Tobak.
Ok, I admit it - I was at the time bombed on whiskey, mince pies and methamphetamine. Nonetheless, through the fug I did see that unless someone draws a line in the sand, we're all condemned to a cyberdiet of complete and utter bollocks in a world where anyone with access to a keyboard can give forth at will.
On the other hand, who am I to pass judgement on Tobak's right to rant, eh?
You may have disabled comments on this one, but you'll read this out if you care about your 'journalistic abilities'. I suspect though that you knew you'd face a ton of shit for that 'crock of shite' you wrote so disabled them. Now I may have read that whole article back to front - as if it were some vastly out of season april fools joke. I don't know. I thought that blogger actually had a fantastic point. Just how the hell do journalists really sit in their chairs and slag off CEOs when they've most likely never sat in any form of management chair in their lives? And how is that not significant a factor? I dont know if you were trying to be funny or not in that article so I'll let most of it off, its near christmas, and yes I was down the pub till about 2am last night knowing all too well I had to be in work for 9am, I'm guessing you were / are in a similar drunken stupour. If anything, you are writing the crock of shite. Self-righteous - How exactly was he being self righteous, he was merely pointing out about something a lot of people already know. Take sports journalism and commentating - snooker for instance is well dealt with by the likes of Dennis Taylor and Steve Davis. These people were in the same situations in the 80s that the new boys are in right now. These people have a right to comment and slag off bad shots because they've been there and done it themselves. You simply do not have similar 'authority'. Sanctimonious - I'm not sure he was really displaying moral superiority - but merely pointing out how other journos do. So its either your drunkenness, or I'm guessing you're winding up your audience to gauge the best response and post it up on El-Reg counter-rants or something. Good luck on that. So take an aspirin and bugger off out the office / work room till your hangover clears. You're no use to us right now. Really REALLY unfunny rant.
Thanks, Stuart, you're right about the hangover. However, to paraphrase one Brit wit: "Sir, in the morning I shall be sober and Tobak will still be ugly."
That's figuratively ugly, btw, for the benefit of any US lawyers thinking of chasing this particular ambulance. Now, if you'll excuse me, I think I'll pop off to the bogs for a pre-Yule line of Bolivian marching powder which will better enable to me to set the world to rights. Happy Xmas. ®
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NASCAR Driver Tony Stewart Kills Fellow Driver Kevn Ward Jr. in Tragic Dirt Track Accident
On Saturday, NASCAR driver accidentally killed dirt track racer Kevin Ward, Jr. in a tragic incident in New York.
Following a collision between Ward and Stewart during the race, Ward climbed out of his damaged car and tried to confront Stewart on the track while the race was still underway. Stewart's car struck and killed Ward.
But why would a driver get out of his car during a race? That is the subject of this Outside the Oval.


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The best dating site for equestrian singles & horse lovers!
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Learn Angamardhana - The Peak of Physical Fitness
The word "Angamardhana" means gaining complete mastery over the limbs, organs and other parts of the body. True to its name, this practice revitalizes all the systems of the body including the muscular system, skeletal structure, nervous system, circulatory system and respiratory system.
It offers to everyone the opportunity to invigorate the body and to reach peak physical fitness.
The advanced form of Angamardhana is still being practiced by Martial Arts Experts. Sadhguru has designed Angamardhana with a series for 31 dynamic processes to be of universal appeal and application, Isha Angamardana needs no fitness equipment. It involves only the body and floor exercises that can be practiced anywhere, even during travel.
True to its name, Regular practice would:
>Revitalizes the body on all levels including the muscles, blood circulation, skeletal structure,nervous systems.
>It strengthens the spine, builds physical strength fitness and tenacity.
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Watch Introduction video to Angamardhana
From 28th Oct to 2nd November , 2014.
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e) Bring Yoga mat (compulsary) and cushion (if necessary).


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Freedom to Read Foundation
The Freedom to Read Foundation is a sister organization to the American Library Association that promotes and defends the constitutional guarantees of freedom of expression and freedom to receive information.
Midwinter Events
Freedom to Read Foundation Board of Trustees meeting At this meeting, the Board conducts the business of the Foundation and receives an overview of the litigation in which FTRF is involved and discusses federal and state legislation and other updates on intellectual freedom issues. Visitors are welcome.
Date: Friday, January 19, 2007
Time: 9:00 a.m. - 5:00 p.m.
Location: Washington State Convention Center, 800 Convention Place, Room 2A
Fhe Trustees produce a report to the ALA Council following each Board meeting that is presented at each Annual and Midwinter Meeting. Here are two examples of recent Reports to Council from 2005 Annual Conference and 2006 Midwinter Meeting.
Freedom to Read Foundation fundraiser
FTRF will be holding a fundraiser during Midwinter in support our litigation efforts. The event will be a reading and book signing featuring author Chris Crutcher, who is consistently one of the most frequently challenged authors in schools and libraries today. There will also be a reception with hors d'oeuvres and a cash bar in conjunction with the event. Several books by Crutcher will be available for sale at the event.
Date: Sunday, January 21, 2007
Time: 5:45 - 8:00 p.m.
Location: Seattle Public Library, 1000 Fourth Ave.
Cost: $25. Space is limited. To make your donation now, visit FTRF's secure online donation form (indicate in comments that the donation is for the Chris Crutcher event) or call (800) 545-2433 x4226.
Visit http://www.ala.org/chriscrutcher for further details.
Carolyn Forsman Conversation Piece Jewelry
At booth #958 in the Exhibitors Area, jeweler Carolyn Forsman will be selling her jewelry again. Get your guilt gift here! Upbeat, upscale, memorable, and affordable. tween too. Credit cards accepted.
This jewelry is sold in museums! ALL profits go to the Freedom to Read Foundation. Carolyn has been exhibiting at conference for many years and has raised over $65,000 for FTRF. She's a librarian turned jewelry designer. Flipflop charm necklace, blinking earrings, rainbow, helix bracelet, and much, much more!
You can also purchase Carolyn's jewelry online at carolynforsman.com. If you type "Freedom to Read Foundation" in the "additional instructions" box, FTRF will receive a portion of the proceeds.
Join the Freedom to Read Foundation
The Freedom to Read Foundation has a separate membership structure from the American Library Association. Regular individual membership begins at $35.00 per year and organizational membership begins at $100.00 per year. Student membership is $10.00. To join FTRF or renew your membership visit http://www.ftrf.org/joinftrf.html.
Members receive the quarterly newsletter, Freedom to Read Foundation News. Organizations can order a subscription to the newsletter for $35.00 per year.
More information
For more information, visit http://www.ftrf.org or contact us at (800) 545-2433 x4226 or firstname.lastname@example.com.
International Read A Burned Book Campaign (Not an activity of FTRF)
FREADOM invites all members of the Freedom to Read Foundation to sign onto the Read A Burned Book project, a unique reading and human rights campaign aimed at getting high school and college students to read the books which tyrants burn. Join Lifetime ALA Member, Sandy Berman, the Founders of Cuba's Independent Libraries, Nat Hentoff, Andrei Codrescu, Carlos Eire and many famous authors as they seek to illuminate the history and practice of book burning. See the FREADOM statement at:[1](Note: FREADOM will soon apply for membership in the FTRF; See Topic Below)


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The practice of relieving oneself of semen into a condom (see posh-wank), while in a public (normally road-side) toilet, then leaving the offending article for someone else to drink in the style of the popular fromage-frais product as alluded to in the title of this activity. Popular in the undergraound gay community of Oxford.
"Robbie enjoyed frubin'in the loo's near Wheatley before leaving a frube for his mate Danny who was expected soon"
by The Convenience Store Kid April 23, 2009
buy the domain for your art site


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[Page 429]
The sign and the stamp
Standing: Liberman (from Smorgon), Itskhak Perski
Sitting: Noah Perski, Ishayahu Kaganovitsh, Benyamin Shishko.
The building contained four rooms and accordingly four learning classes only. There was also a fifth one, the Mekhina (preparation) class in which the new coming pupils were prepared to use the school's main language, the hitherto unknown Hebrew. A second after-noon shift was organized for this additional class.
Hayim Goloventshitz was invited to manage the School and also to teach Bible and Hebrew literature. M. Gurevitsh from Sol, Hayim Levin from Ivye, Benyamin Shishko and Noah Perski both of them from Volozhin served as teachers.
From its foundation until closure the School was short of resources. It did not receive any allowances from the town council. The authorities claimed that in the existing Polish Elementary School, which was supported by the Government, the learning was free and there is place enough for all. Parents who would like to teach their children Hebrew and educate them in Zionist spirit had to pay for it.
The authorities compelled the school to teach the Polish language and history. Its teachers were Polish in the beginning. Later they were replaced by Jews.
The first remarkable happening we remember was the Jerusalem Hebrew University's opening celebration. Lord Balfour standing on the Mount Scopus announced that with the opening of the University a source of wisdom had been founded. Hayim Nakhman Bialik said at the end of his oration that redemption for all of mankind will come from this place.
Volozhin citizens congratulated one another with the Shehekheyonu benediction. They felt it was a great victory for the Hebrew Language. The windowsills were lighted with candles like at Hanukah. The Tarbut students gathered on the streets singing Zionist songs and speaking Hebrew.
During the first years of the school's function its pupils wore homogeneous uniforms, blue skirt, black apron - the girls, Blue shirt with blue pants the boys. Girls as boys wore blue hats decorated with the Tarbut symbol.
The management decided to introduce Latin language lessons to enable students' attending government high schools in which Latin was obligatory. Mr. Mirer from Vilna was invited to teach this language.
The school took also care of artistic education. Mr. Ratner founded a string orchestra and became its first teacher. The band participated in all festivity events of the school and of the community.
The Tarbut String Orchestra 1926
From Right to left, Standing: Berman, Lea Shvartsberg, Fruma Rogovin, Efrayim Rogovin, Shlomo Gurevitsh, Israel Berkovitsh, Yakov Rogovin, Shlomo Liberman.
From left to left Right, Seated: Bela Potashnik, Mina, Rashl Vaysbord, Shoshana Berkovtsh, Avram Gurevitsh, Hayim Ratner, Hayim Goloventshits, Noah Perski, Benyamin Shishko, Peshke Rogovin,Bela Kramnik, Berman, Tsirke Bunimovitsh.
JewishGen, Inc. makes no representations regarding the accuracy of
the translation. The reader may wish to refer to the original material
for verification.
JewishGen is not responsible for inaccuracies or omissions in the original work and cannot rewrite or edit the text to correct inaccuracies and/or omissions.
Our mission is to produce a translation of the original work and we cannot verify the accuracy of statements or alter facts cited.
Volozhin, Belarus Yizkor Book Project JewishGen Home Page
Copyright © 1999-2019 by JewishGen, Inc.
Updated 23 Sep 2006 by LA


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Bestselling author Tess Gerritsen talks about her latest Rizzoli & Isles medical thriller, The Killing Place.
The night before I am to meet Tess Gerritsen at the Balmoral in Edinburgh, her publicist texts me to say that they are on a "smog schedule" so could I meet them at Gerritsen's hotel instead? I agree, although I have no idea what a smog schedule is. Some kind of industry insider code? Or is Gerritsen, who will be travelling on to Newcastle after our interview, concerned about poor visibility affecting the next leg of her journey? This is Scotland, after all.
The answer is neither. The apologetic publicist explains the following day that they are in fact on a tight schedule and the word mix-up was the result of texting while tired.
Gerritsen herself is bright and lively when she meets me at the hotel reception. Although she only arrived in the UK from her native Maine a few days earlier, she has miraculously avoided jet-lag and was on top form the previous evening, entertaining an audience at the Edinburgh International Book Festival by recounting the real life incidents that have inspired medical thrillers. Dinner conversations, news stories and even the antics of her sons have given Gerritsen the glimpse of the macabre she needs to spin out a terrifying plot. "I go for the dark stuff. I'm always looking for things that are disturbing because I think that people are interested in those topics."
The inspiration for The Killing Place came from declassified U.S. federal government reports about an incident in the sixties where thousands of sheep were found dead in a valley. Gerritsen was shocked by the reports because she realised the same thing could happen again to an entire city. She knew she had hit on the idea for a great story because "the emotion around the inciting incident was like a punch in the gut."
The Killing Place is the eighth book to feature police detective Jane Rizzoli and medical examiner Dr Maura Isles. Gerritsen turns the traditional relationship between the crime solving duo on its head in this novel when Maura goes missing, leaving Jane to track her down.
After the GPS leads Maura and her travelling companions along a road that is closed in winter, they find themselves stranded in the mountains during a snow storm. They think they have been saved when they stumble upon an isolated village but it is soon clear that something sinister has happened in the tiny settlement of Kingdom Come: the houses have been abandoned, meals left untouched on kitchen tables and cars still parked in garages. With nowhere else to go, the group sets up shelter in one of the deserted houses but it isn't long before they realise that there is someone out there in the woods, watching them. Days later, a burned out car is found with four bodies inside, one of which is identified as Maura Isles. Jane Rizzoli is determined to prove the identification wrong and to find out what really happened to her friend.
"I think that the best mysteries are when the character has an emotional reason to want the case to be solved," Gerritsen explains. "What sort of creeped me out about this idea was the abandoned houses and the meals left on the table. I like to take something that looks like a horror story on the surface but in fact there's a logical explanation for it."
As with all of her medical thrillers, The Killing Place has its share of gory drama which Gerritsen's background in medicine left her more than equipped to deal with. (She trained and worked as a physician before becoming a full time writer.) "This particular book did not have a lot of research involved. I went online to find cases of GPS disasters. There have been a lot of people who have had accidents or who have died, in the US especially, because we have a lot of very solitary places, a lot of wilderness. There have been people who have been stuck in the snow for weeks because they've driven down a seasonal road."
The Killing Place sees the introduction of two new characters, troubled teenage runaway Rat and his dog Bear, who Gerritsen reveals will be making a comeback in a future novel. "Rat and Bear are going to be more a part of Maura's life in the future. The next book is not going to have them but I think in the one after that I may show them at their new school."
The Killing Place is out now in Bantam paperback (£7.99).


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To use all features of KLM.com safely, we recommend that you update your browser, or that you choose a different one. Continuing with this version may result in parts of the website not being displayed properly, if at all. Also, the security of your personal information is better safeguarded with an updated browser.
Hot stone massages, panoramic saunas and soothing whirlpools: there's nothing more relaxing than enjoying a day of wellness. Vancouver Island is the perfect wellness destination. Most of the spas here are located amidst the island's overwhelmingly beautiful nature. How about soaking in a bubble bath in the middle of a rainforest? Or enjoying a sauna with ocean views? These are only a few of the possibilities.
Not yet feeling Zen after a massage or steam bath? The soothing sounds of the ocean and sighing of the wind as you unwind in one of these spas will definitely make you feel just that. A single day of relaxation does wonders. However, to become completely stress-free, book a multi-night stay.
The Wickaninnish Inn in Tofino is one of the most idyllic hotels on Vancouver Island. In the Ancient Cedars Spa, the hotel offers treatments based on the cleaning rituals of the First Nations people, including a seaweed scrub and water therapy. And, instead of a mud treatment, you can get rubbed with kelp, a sea plant primarily found in the waters around the island. Enjoy views of the Pacific Ocean waves crashing against the rocks from your massage bed.
500 Osprey Lane, Tofino
The power of water is unprecedented. The people at Kingfisher Ocean Resort & Spa are well aware of this fact. The main attraction at this wellness centre is the Pacific Mist Hydropath. You follow a path along a simulated coastline through 8 different baths, caves and waterfalls. The route starts in the Swiss Shower, where water jets clean and warm your body. You then enjoy a mineral massage bath, a waterfall that relieves tension in tired shoulder muscles, and an aromatic steam cave that cleanses skin. After this, the Glacial Waterfall is a welcome rush of cold. The path continues to the River Walk, where alternating hot and cold baths stimulate blood circulation. In the Sea Mineral Soak, you float light as a feather in a tidal bath carved out of rock, where you enjoy a home-made sea salt scrub to remove dead skin cells. After so much wellness, you can relax and enjoy a fruit smoothie or herbal tea in the lounge, while enjoying the fabulous ocean views.
4330 Island Highway South, Courtenay
A spa that can only be reached by plane or boat has got to be unique. And that is exactly what the Clayoquot Wilderness Resort is. This resort is an enclave of luxuriously furnished canvas tents with remote controlled fireplaces, opulent rugs and beautiful bathrooms. You can heat up in one of the relaxing saunas or wood-fired hot tub or enjoy a massages or treatment in a tent at the edge of the rain forest, as you slowly drift into dreamland to the singing of birds and rustling of trees.
Tofino, Vancouver Island


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GSMOK Home page
Why us?
Delivery Price
How much shipping will cost to my country?
New Products
Completely NEW Products!
Last Deliveries
Last deliveries, what we have now on stock.
Register on GSMOK website.
Contact with GSMOK Company.
Please be advised that on Friday (19.04) the company will be open until 12 noon.
© Copyright 2019 by GSMok.pl . All rights reserved.Generation time: 1.761 sec


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Their sound is gone out, belts out a choir
visible and audible here in the transept
North East West South: acronym, N E W S
gone out andante into all lands
turf and twig. (An OED or google game
[tran—across; septum—
Caroline Knox
Caroline Knox earned her AB from Radcliffe College and her MA and PhD from University of Wisconsin–Milwaukee. She is the author of several books of poetry, most recently To Drink Boiled Snow (Wave Books, 2015), Flemish (Wave Books, 2013), and Nine Worthies (Wave Books, 2010).
C. D. Wright called Knox "a most inquisitive poet who relishes living inside her expansive vocabulary; one who has been faxing by the midnight oil while so many others were dipping their quills into dry sockets. Caroline Knox reminds us how whangy and interesting it all is."
Knox is the recipient of grants from The Fund for Poetry, the Ingram Merrill Foundation, the Massachusetts Cultural Council, and the National Endowment for the Arts, among others. She lives in Massachusetts.
Selected Bibliography
To Drink Boiled Snow (Wave Books, 2015)
Flemish (Wave Books, 2013)
Nine Worthies (Wave Books, 2010)
He Paves the Road with Iron Bars (Verse Press, 2004)
A Beaker: New and Selected Poems (Verse Press, 2002)
Sleepers Wake (Tinken Publishers, 1994)
To Newfoundland (University of Georgia Press, 1989)
The House Party (University of Georgia Press, 1984)


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Bringing home the classic arcade brawler, Bart, Lisa, Marge and Homer smack their way through Springfield in this 2D beat 'em up game based on the hilarity of The Simpsons TV series. Play through 8 levels of original arcade gameplay and join up to three of your friends in a journey to help the Simpsons on their quest to save Maggie! Chase down Smithers and Mr. Burns as you battle through classic Springfield locations in this memorable blast from the past!
This downloadable incarnation finally brings the coin-op excitement home, plus packs in an HD graphic remastering and four player online/local co-op.
Also known as: The Simpsons Arcade Game, The Simpsons HD
Release Region: United States
Release Date: February 8, 2012
Publisher: Konami
Release Region: United Kingdom
Release Date: February 3, 2012
Publisher: Konami
Release Region: Australia
Release Date: February 3, 2012
Publisher: Konami
Also available on: PlayStation 3, Commodore 64/128, Arcade, PC
The Simpsons: The Arca... at IGN
The Simpsons: The Arca... at GameSpy
The Simpsons: The Arca... at GameStats


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The usual term for this phenomenon is "UGC"—user-generated content. I prefer the term customer-contributed content because it rolls off the tongue better and fits our definition of a customer as an end-user of the goods and services you provide (whether they or someone else pays for those services).
On February 6, 2008, the Professional and Scholarly Publishing (PSP) division of the American Association of Publishers held its annual symposium. For the second year, I was the moderator for a day of fascinating talks and case studies. The topic at hand was Cyberscholarship: Where are our users taking us? The objectives for the meeting were to gain a clearer sense of:
1) How do today's readers, scholars, and professionals want to interact around professional and scholarly content?
2) How do they want to use and interact with published research?
3) How do they want to learn from each other?
The panelists included Kevin McKean, VP and Editorial Director of Consumers Union; Paris Patton, VP of consumer research firm Sachs Insights; Mark Ranalli, President and CEO of Helium; Melissa Burke, Director of Strategic Partnerships for Sermo; and Bryce Johnson, President and CEO of CafeScribe. What all of the panelists had in common was a passion to make it easy for customers to interact with and to contribute content.
Continue reading "Making Customer-Contributed Content Work for Your Business" »


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For recruitment or application questions or information, please contact us at the number below.
Commissioned Corps Officer Opportunities at the USU School of Medicine
The Uniformed Services University of the Health Sciences (USU) is a health science university run by the Federal government, providing training to military physicians, nurses, and educators. Currently, you can attend the USU F. Edward Hébert School of Medicine as a Public Health Service officer through one of two agencies:
For additional information about USPHS opportunities at USU please visit the USU Web site or contact:
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