Math Notes, Symbols

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I
INTRODUCTION
Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems.
This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today.
II.
ANCIENT MATHEMATICS
The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs.
The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number. For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers to be added. Multiplication was based on successive doublings, and division was based on the inverse of this process.
The Egyptians used sums of unit fractions (0), supplemented by the fraction 0, to express all other fractions. For example, the fraction 0 was the sum of the fractions 0 and 0. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle, the Egyptians used the square on 0 of the diameter of the circle, a value of about 3.16-close to the value of the ratio known as pi pages,_{pi notes, 1}which is about 3.14. View pi pages, ().

The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system-which, when using clay tablets, consisted of various wedge-shaped marks-a single wedge indicated 1 and an arrow like wedge stood for 10 (see table).	Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used. That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 × 602 + 27 × 60 + 0. This principle was extended to the representation of fractions as well, so that
	1 the above sequence of numbers could equally well represent 2 × 60 + 27 + 10 × (0), or 2 + 27 × (0) + 10 × (0-2). With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system. The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (see Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest. They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about 0. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for ?. In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders. However, the Babylonians did not arrive at the correct formula for the volume of a pyramid.
A

Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century BC with Thales of Miletus and Pythagoras of Sámos, the latter a religious leader who taught the importance of studying numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras.
In the 5th century BC, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle-that is, constructing a square equal in area to a given circle. Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube-that is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than a straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone.
In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, ?, would today be called irrational.) As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations.
Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes.
The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes' writings on them are the earliest to survive. Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist René Descartes in the 17th century.
After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century AD show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers. Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century AD. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis).
B.
Applied Mathematics in Greece Paralleling the studies described in pure mathematics were studies made in optics, mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system for recording fractions and, at about the same time, composed tables of chords in a circle. For a circle of some fixed radius, such tables give the length of the chords subtending a sequence of arcs increasing by some fixed amount. They are equivalent to a modern sine table, and their composition marks the beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150BC-the arcs increased by steps of 7?°, from 0° to 180°. By the time of the astronomer Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his Almagest a table of chords in a circle for steps of ?°, which, although expressed sexagesimally, is accurate to about five decimal places.
In the meantime, methods were developed for solving problems involving plane triangles, and a theorem-named after the astronomer Menelaus of Alexandria-was established for finding the lengths of certain arcs on a sphere when other arcs are known. These advances gave Greek astronomers what they needed to solve the problems of spherical astronomy and to develop an astronomical system that held sway until the time of the German astronomer Johannes Kepler.
III.
MEDIEVAL AND RENAISSANCE MATHEMATICS
Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centers of Greek learning. The preservation of such works as have survived to modern times began with this tradition. It was continued in the Islamic world, where original developments based on these masterpieces first appeared.
A Islamic and Indian Mathematics After a century of expansion in which the religion of Islam spread from its beginnings in the Arabian Peninsula to dominate an area extending from Spain to the borders of China, Muslims began to acquire the results of the "foreign sciences." At centers such as the House of Wisdom in Baghdâd, supported by the ruling caliphs and wealthy individuals, translators produced Arabic versions of Greek and Indian mathematical works.
By the year 900AD the acquisition was complete, and Muslim scholars began to build on what they had acquired. Thus mathematicians extended the Hindu decimal positional system of arithmetic from whole numbers to include decimal fractions, and the 12th-century Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the algebra of polynomials of Muhammad al-Khwarizmi. Al-Karaji included polynomials with an infinite number of terms. (Al-Khwarizmi's name, incidentally, is the source of the word algorithm, and the title of one of his books is the source of the word algebra.) Geometers such as Ibrahim ibn Sinan continued Archimedes' investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve optical problems. Using the Hindu sine function and Menelaus's theorem, mathematicians from Habas al-Hasib to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical trigonometry. These did not become mathematical disciplines in the West, however, until the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.
Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. The Latin West acquired much of this learning during the 12th century, the great century of translation. Together with translations of the Greek classics, these Muslim works were responsible for the growth of mathematics in the West during the late Middle Ages. Italian mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for their knowledge.
B

Western Renaissance Mathematics Although the late medieval period saw some fruitful mathematical considerations of problems of infinity by writers such as Nicole Oresme, it was not until the early 16th century that a truly important mathematical discovery was made in the West. The discovery, an algebraic formula for the solution of both the cubic and quartic equations, was published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The discovery drew the attention of mathematicians to complex numbers and stimulated a search for solutions to equations of degree higher than 4. It was this search, in turn, that led to the first work on group theory (see Group) at the end of the 18th century, and to the theory of equations developed by the French mathematician Évariste Galois in the early 19th century. The 16th century also saw the beginnings of modern algebraic symbolism (see Mathematical Symbols), as well as the remarkable work on the solution of equations by the French mathematician François Viète. His writings influenced many mathematicians of the following century, including Pierre de Fermat in France and Isaac Newton in England.
IV.

MATHEMATICS SINCE THE 16TH CENTURY Europeans dominated in the development of mathematics after the Renaissance.
A17th Century During the 17th century, the greatest advances were made in mathematics since the time of Archimedes and Apollonius. The century opened with the discovery of logarithms by the Scottish mathematician John Napier, whose continued utility prompted the French astronomer Pierre Simon Laplace to remark, almost two centuries later, that Napier, by halving the labors of astronomers, had doubled their lifetimes. (Although the logarithmic function is still important in mathematics and the sciences, logarithmic tables and their instrumental form-slide rules-are of much less practical use today because of electronic calculators.)
The science of number theory, which had lain dormant since the medieval period, illustrates the 17th-century advances built on ancient learning. It was Arithmetica by Diophantus that stimulated Fermat to advance the theory of numbers greatly. His most important conjecture in the field, written in the margin of his copy of the Arithmetica, was that no solutions exist to an + bn = cn for positive integers a, b, and c when n is greater than 2. This conjecture, known as Fermat's last theorem, stimulated much important work in algebra and number theory before it was finally proved in 1994.
Two important developments in pure geometry occurred during the century. The first was the publication, in Discourse on Method (1637) by Descartes, of his discovery of analytic geometry, which showed how to use the algebra that had developed since the Renaissance to investigate the geometry of curves. (Fermat made the same discovery but did not publish it.) This book, together with short treatises that had been published with it, stimulated and provided the basis for Isaac Newton's mathematical work in the 1660s. The second development in geometry was the publication by the French engineer Gérard Desargues in 1639 of his discovery of projective geometry. Although the work was much appreciated by Descartes and the French philosopher and scientist Blaise Pascal, its eccentric terminology and the excitement of the earlier publication of analytic geometry delayed the development of its ideas until the early 19th century and the works of the French mathematician Jean Victor Poncelet.
Another major step in mathematics in the 17th century was the beginning of probability theory in the correspondence of Pascal and Fermat on a problem in gambling, called the problem of points. This unpublished work stimulated the Dutch scientist Christiaan Huygens to publish a small tract on probabilities in dice games, which was reprinted by the Swiss mathematician Jakob Bernoulli in his Art of Conjecturing. Both Bernoulli and the French mathematician Abraham De Moivre, in his Doctrine of Chances in 1718, applied the newly discovered calculus to make rapid advances in the theory, which by then had important applications in the rapidly developing insurance industry.
Without question, however, the crowning mathematical event of the 17th century was the discovery by Sir Isaac Newton, between 1664 and 1666, of differential and integral calculus (see Calculus). In making this discovery, Newton built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as Descartes, Francesco Bonaventura Cavalieri, Johann van Waveren Hudde, and Gilles Personne de Roberval. About eight years later than Newton, who had not yet published his discovery, the German Gottfried Wilhelm Leibniz rediscovered calculus and published first, in 1684 and 1686. Leibniz's notation systems, such as dx, are used today in calculus.
B18th Century The remainder of the 17th century and a good part of the 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas to solving a variety of problems in physics, astronomy, and engineering. In the course of doing so they also created new areas of mathematics. For example, Johann and Jakob Bernoulli invented the calculus of variations, and French mathematician Gaspard Monge invented differential geometry. Also in France, Joseph Louis Lagrange gave a purely analytic treatment of mechanics in his great Analytical Mechanics (1788), in which he stated the famous Lagrange equations for a dynamical system. He contributed to differential equations and number theory as well, and he originated the theory of groups. His contemporary, Laplace, wrote the classic Celestial Mechanics (1799-1825), which earned him the title the French Newton, and The Analytic Theory of Probabilities (1812).
The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas. That is, Newton's own accounts were based on kinematics and velocities, Leibniz's explanation was based on infinitesimals, and Lagrange's treatment was purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century.
C19th Century In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of "real number." Although Cauchy's explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T. W. Weierstrass. A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician Peter G. L. Dirichlet who proposed the definition in terms of a correspondence between elements of the domain and the range. This is the definition that is found in texts today.
In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B. Riemann. Another important advance in analysis was Fourier's study of infinite sums in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantor's theory, which was considered quite abstract and even attacked as a "disease from which mathematics will soon recover," now forms part of the foundations of mathematics and has more recently found applications in the study of turbulent flow in fluids.
A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry. In non-Eculidean geometry, more than one parallel can be drawn to a given line through a given point not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication. The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian János Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics.
Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying.
Of more importance for algebra itself than Gauss's proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems. A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann Günther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange. Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula.
Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology.
Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantor's theory of sets, was the foundations of mathematics (see Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantor's theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set (see Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising. They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt Gödel, that in any axiom system complicated enough to be interesting to most mathematicians, it is possible to frame propositions whose truth cannot be decided within the system.
D-Current Mathematics At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at Göttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert's address at Göttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the "Hilbert problems" has been solved, mathematicians all over the world await the details of the story with impatience.
Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (see Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbage's imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois.
Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.

Contributed By:
J. Lennart Berggren

Further Reading
"Mathematics," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

Cornell Theory Center Math and Science Gateway http://www.tc.cornell.edu/Edu/MathSciGateway/index.html
The Cornell Theory Center and the Cornell University Department of Education provide a directory of mathematics and science Internet resources for high school students and teachers.

EDU: Education Division http://www.ncsa.uiuc.edu/Edu/
The National Center for Supercomputing Applications provides information about its projects, along with links to K-12 education resources and to information regarding the teaching profession and Internet usage.

Education Place http://www.eduplace.com/
This commercial site offers educational resources aimed at kindergarten through eighth grade, including lesson plans, classroom activities, and online games; there are sections for parents, children, and teachers.

Education Program http://ep.llnl.gov/
Lawrence Livermore National Laboratory's Education Programs aim to encourage and enhance science, math, engineering, and technology education; this site offers projects for students and teachers as well as other resources.

Education Week on the Web http://www.edweek.org/
Education Week on the Web is an online publication providing the latest news about education-related topics.

Education World http://www.education-world.com/
This commercially hosted site provides articles on modern education and child development, information about state educational systems, and an index of related resources.

EdWeb http://edweb.cnidr.org/
This privately maintained site offers a guide to technology's impact on education reform, along with Internet resources for educators.

Electronic Games for Education in Math and Science http://www.cs.ubc.ca/nest/egems/home.html
This site provides information about the Electronic Games for Education in Math and Science (E-GEMS) project, a joint effort to create teaching materials that incorporate computers and computer games into elementary school classrooms.

ENC Online http://www.enc.org/
The Eisenhower National Clearinghouse for Mathematics and Science Education (ENC) provides K-12 teachers with a central source for mathematics and science curriculum materials on the Internet.

Explore Science http://www.explorescience.com/
This privately maintained site offers a variety of science-themed Shockwave modules for students and teachers.

Federal Resources for Educational Excellence http://www.ed.gov/free/
Federal Resources for Educational Excellence (FREE), maintained by the United States Department of Education, is a searchable and browsable interface to United States government sites that can be used as teaching resources.

For Kids Only-Earth Science Enterprise http://kids.mtpe.hq.nasa.gov/
The National Aeronautics and Space Science Administration presents a Web site just for children; the site includes online activities, answers to frequently asked questions about space, information about different science topics, and other resources.

GirlTECH: A Teacher Training and Student Council Program http://www.crpc.rice.edu/CRPC/Women/GirlTECH/
Sponsored by the Center for Research on Parallel Computation (CRPC) and the National Science Foundation Science and Technology Center, this site provides parent and teacher resources related to fostering girls' interest in computers.

Halls of Academia http://www.tenet.edu/academia/main.html
The Texas Education Network (TETNET) provides a directory of education resources, organized by subject area, for both parents and teachers.

Image Processing for Teaching Program http://ipt.lpl.arizona.edu/
The Image Processing for Teaching Program provides information about using image processing to teach math and science concepts, as well as sample lesson plans and other resources for teachers.

IPL Ready Reference Collection: Education http://www.ipl.org/ref/RR/static/edu0000.html
The Internet Public Library provides an annotated directory of education-related Web resources for all grade levels.

Kaplan Educational Centers http://www.kaplan.com/
Kaplan Educational Centers, a commercial service, presents this site about educational testing; they offer information on standardized tests, financial aid and career information, sample tests, and other resources.

Little Shop of Physics http://brianjones.ctss.colostate.edu/default.html
The Little Shop of Physics, Colorado State University's Hands-On Science Outreach Program, provides dozens of physics experiments the user can do online or at home, advice on science fair projects, and several physics brainteasers.

Mathematics Lessons http://math.rice.edu/~lanius/Lessons/
This privately maintained site provides a variety of mathematical activities for elementary, middle, and high school students and teachers.

MathWorld Interactive http://www.mathworld-interactive.com/
This is an online mathematics project for K-12 students; it offers challenges for different grade levels, interactive message boards, and other resources.

Microsoft in Education http://www.microsoft.com/education/
This commercial site offers education resources for parents, students, and teachers in both elementary and higher education.

Montessori Education http://www.amshq.org/
The American Montessori Society provides information on Italian educator Maria Montessori and on Montessori educational techniques.

National Center for Education Statistics http://nces.ed.gov/
The National Center for Education Statistics collects and reports statistics and information showing the condition and progress of education in the United States and other nations.

National School Boards Association http://www.nsba.org/
The National School Boards Association offers information about its organization and activities, as well as statistics on education in the United States and other resources.

National Service-Learning Cooperative Clearinghouse http://www.nicsl.coled.umn.edu/
This site provides resources to assist educators and community agencies in developing or expanding service-learning programs in K-12 classrooms nationwide.

Ocean of Know http://www.oceanofk.org/
This commercially-sponsored site provides a distance-learning marine biology curriculum, including an online biology lab, lesson plans, a 3-D chat environment, and other resources related to the study of ocean environments and creatures.

Online Educator http://www.ole.net/ole/
This site offers selections from the print publication of the same name, as well as featured weekly sites for teachers, an archive of past weeks' sites, discussion forums, and lesson ideas.

OnlineDelivery.com http://www.onlinedelivery.com/
As a public service, OnlineDelivery.com provides resources related to distance learning and education, including links to other Web resources, open discussions about online education, and information about streaming media as a teaching tool.

Pathways to School Improvement http://www.ncrel.org/sdrs/pathwayt.htm
A project of the North Central Regional Educational Laboratory, this site contains resources to improve educational methods and environment.

Reading Online http://www.readingonline.org/
The International Reading Association provides this online journal as part of an effort to increase communication between literacy professionals; the site includes articles, discussion forums, professional materials, and an archive of past issues.

Teacher Talk http://education.indiana.edu/cas/tt/tthmpg.html
Published by the Center for Adolescent Studies at the School of Education, Indiana University, this site for secondary school teachers offers tips and information on a variety of teaching and curriculum topics.

TeachersFirst - A Web Resource for K-12 Teachers http://www.teachersfirst.com/
This site offers resources for teachers, including links to lesson plans and Web resources arranged by subject and grade level, an Internet tutorial, and links to professional resources.

TEAMSnet http://teams.lacoe.edu/
The Los Angeles County Office of Education sponsors this site for elementary and middle school students, teachers, and parents; it provides suggestions for classroom projects and links to educational Internet resources.

The American Association of Physics Teachers http://www.aapt.org/
The American Association of Physics Teachers is dedicated to improving physics teaching at all levels; it offers information about its activities and publications.

The Homeschooling Zone http://www.homeschoolzone.com/
This privately maintained site offers information for parents on a variety of home-schooling issues; it includes teaching materials, lesson ideas, newsletters, online discussions, and links to educational resources.

The Incredible Art Department http://www.artswire.org/kenroar/
This site, aimed primarily at art educators, includes lesson plans for students of all ages, samples of student work, an overview of types of jobs for artists, and links to school and college art departments on the Internet.

The Magnificent Moose Project http://www3.northstar.k12.ak.us/schools/awe/moose/moosepage.html
Created by elementary school students in Fairbanks, Alaska, this site offers a variety of resources about moose, including a description of their physical characteristics, habitat and behavior information, _images, and guides to moose-human interactions.

The MBA Page http://www.cob.ohio-state.edu/dept/fin/oldmba.htm
Ohio State University's Max M. Fisher School of Business offers this guide for business school students, including school directories and rankings, lists of corporations, class and case materials, and many other resources.

The National Association for the Education of Young Children http://www.naeyc.org/
The National Association for the Education of Young Children offers practical information for parents and teachers.

Virtual Reference Desk http://www.vrd.org/
The National Library of Education sponsors this digital reference service for students in grades K-12; the site features "Ask An Expert" forums, a database of past questions and answers, and other related resources.

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single-precision

single-precision (seng'l pr?-sizh`?n) adjective
Of or pertaining to a floating-point number having the least precision among two or more options commonly offered by a programming language. See also floating-point notation, precision.

Microsoft Press® Computer and Internet Dictionary © & ? 1997, 1998 Microsoft Corporation. All rights reserved. Portions, The Microsoft Press® Computer Dictionary, 3rd Edition, Copyright © 1998 by Microsoft Press. All rights reserved.

differentiate (dif´?-ren?she-at´) verb
differentiated, differentiating, differentiates
verb, transitive

1.To constitute the distinction between: subspecies that are differentiated by the markings on their wings.
2.To perceive or show the difference in or between; discriminate.
3.To make different by alteration or modification.
4.Mathematics. To calculate the derivative or differential of (a function).

verb, intransitive

1.To become distinct or specialized; acquire a different character.
2.To make distinctions; discriminate.
3.Biology. To undergo a progressive, developmental change to a more specialized form or function. Used especially of embryonic cells or tissues.

dif´feren´tia?tion noun

Excerpted from The American Heritage Dictionary of the English Language, Third Edition Copyright © 1992 by Houghton Mifflin Company. Electronic version licensed from Lernout & Hauspie Speech Products N.V., further reproduction and distribution restricted in accordance with the Copyright Law of the United States. All rights reserved.

integrate (in?ti-grat´) verb
integrated, integrating, integrates

verb, transitive

1.To make into a whole by bringing all parts together; unify.
2.a. To join with something else; unite. b. To make part of a larger unit: integrated the new procedures into the work routine.
3.To open to people of all races or ethnic groups without restriction; desegregate.
4.Mathematics. a. To calculate the integral of. b. To perform integration on.
5.Psychology. To bring about the integration of (personality traits).

verb, intransitive
To become integrated or undergo integration.

[From Middle English, intact, from Latin integratus past participle of integrare, to make whole, from integer, complete.]

- in?tegra´tive adjective

The 6th-century-BC Greek mathematician and philosopher Pythagoras was not only an influential thinker, but also a complex personality whose doctrines addressed the spiritual as well as the scientific. The following is a collection of short excerpts from studies of Pythagorean teachings and from anecdotes about Pythagoras written by later Greek thinkers, such as the philosopher Aristotle, the historians Herodotus and Diodorus Siculus, and the biographer Diogenes Laërtius.

Early Greek Writings on Pythagoras
Pythagoras
Some of the legends about Pythagoras were collected by Aristotle in his lost work On the Pythagoreans. Here is a representative sample:

Pythagoras, the son of Mnesarchus, first studied mathematics and numbers but later also indulged in the miracle-mongering of Pherecydes. When at Metapontum a cargo ship was entering harbour and the onlookers were praying that it would dock safely because of its cargo, he stood up and said: 'You will see that this ship is carrying a corpse.' Again, in Caulonia, as Aristotle says, he foretold the appearance of the white shebear; and Aristotle in his writings about him tells many stories including the one about the poisonous snake in Tuscany which bit him and which he bit back and killed. And he foretold to the Pythagoreans the coming strife-which is why he left Metapontum without being observed by anybody. And while he was crossing the river Casas in company with others he heard a superhuman voice saying 'Hail, Pythagoras'-and those who were there were terrified. And once he appeared both in Croton and in Metapontum on the same day and at the same hour. Once, when he was sitting in the theatre, he stood up, so Aristotle says, and revealed to the audience his own thigh, which was made of gold. Several other paradoxical stories are told of him; but since I do not want to be a mere transcriber, enough of Pythagoras.
(Apollonius, Marvellous Stories 6)
A large body of teachings came to be ascribed to Pythagoras. They divide roughly into two categories, the mathematico-metaphysical and the moral-as the poet Callimachus put it, Pythagoras

was the first to draw triangles and polygons and *to bisect* the circle-and to teach men to abstain from living things.
(Iambi fragment 191.60-62 Pfeiffer)

Most modern scholars are properly sceptical of these ascriptions, and their scepticism is nothing new. The best ancient commentary on Pythagoras' doctrines is to be found in a passage of Porphyry:

Pythagoras acquired a great reputation: he won many followers in the city of Croton itself (both men and women, one of whom, Theano, achieved some fame), and many from the nearby foreign territory, both kings and noblemen. What he said to his associates no-one can say with any certainty; for they preserved no ordinary silence. But it became very well known to everyone that he said, first, that the soul is immortal; then, that it changes into other kinds of animals; and further, that at certain periods whatever has happened happens again, there being nothing absolutely new; and that all living things should be considered as belonging to the same kind. Pythagoras seems to have been the first to introduce these doctrines into Greece.
(Porphyry, Life of Pythagoras 19)

The theory of metempsychosis, or the transmigration of the soul, is implicitly ascribed to Pythagoras by Xenophanes in the text quoted above. Herodotus also mentions it:

The Egyptians were the first to advance the idea that the soul is immortal and that when the body dies it enters into another animal which is then being born; when it has gone round all the creatures of the land, the sea and the air, it again enters into the body of a man which is then being born; and this cycle takes it three thousand years. Some of the Greeks-some earlier, some later-put forward this idea as though it were their own: I know their names but I do not transcribe them.
(Herodotus, Histories II 123)

The names Herodotus coyly refrains from transcribing will have included that of Pythagoras. Two later passages are worth quoting even though they belong to the legendary material.

Heraclides of Pontus reports that [Pythagoras] tells the following story of himself: he was once born as Aethalides and was considered to be the son of Hermes. Hermes invited him to choose whatever he wanted, except immortality; so he asked that, alive and dead, he should remember what happened to him. Thus in his life he remembered everything, and when he died he retained the same memories. Some time later he became Euphorbus and was wounded by Menelaus. Euphorbus used to say that he had once been Aethalides and had acquired the gift from Hermes and learned of the circtilation of his soul-how it had circulated, into what plants and animals it had passed, what his soul had suffered in Hades and what other souls experienced. When Euphorbus died, his soul passed into Hermotimus, who himself wanted to give a proof and so went to Branchidae, entered the temple of Apollo and pointed to the shield which Menelaus had dedicated (he said that he had dedicated the shield to Apollo when he sailed back from Troy); it had by then decayed and all that was left was the ivory boss. When Hermotimus died, he became Pyrrhus, the Delian fisherman; and again he remembered everything-how he had been first Aethalides, then Euphorbus, then Hermotimus, then Pyrrhus. When Pyrrhus died, he became Pythagoras and remembered everything I have related.
(Diogenes Laertius, Lives of the Philosophers VIII 4-5)
Pythagoras believed in metempsychosis and thought that eating meat was an abominable thing, saying that the souls of all animals enter different animals after death. He himself used to say that he remembered being, in Trojan times, Euphorbus, Panthus' son, who was killed by Menelaus. They say that once when he was staying at Argos he saw a shield from the spoils of Troy nailed up, and burst into tears. When the Argives asked him the reason for his emotion, he said that he himself had borne that shield at Troy when he was Euphorbus. They did not believe him and judged him to be mad, but he said he would provide a true sign that it was indeed the case: on the inside of the shield there had been inscribed in archaic lettering EUPHORBUS. Because of the extraordinary nature of his claim they all urged that the shield be taken down-and it turned out that on the inside the inscription was found.
(Diodorus, Universal History X vi 1-3)…

The idea of eternal recurrence had a wide currency in later Greek thought. It is ascribed to 'the Pythagoreans' in a passage from Simplicius:

The Pythagoreans too used to say that numerically the same things occur again and again. It is worth setting down a passage from the third book of Eudemus' Physics in which he paraphrases their views:

One might wonder whether or not the same time recurs as some say it does. Now we call things 'the same' in different ways: things the same in kind plainly recur-e.g. summer and winter and the other seasons and periods; again, motions recur the same in kind-for the sun completes the solstices and the equinoxes and the other movements. But if we are to believe the Pythagoreans and hold that things the same in number recur-that you will be sitting here and I shall talk to you, holding this stick, and so on for everything else-then it is plausible that the same time too recurs.
(Simplicius, Commentary on the Physics 732.23-33)
Source: Barnes, Jonathan. Early Greek Philosophy. Penguin Books.

"From Early Greek Philosophy," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

Mathematicians, persons skilled in mathematics, the study of relationships among quantities, magnitudes, and properties, and of logical operations by which unknown quantities, magnitudes, and properties may be deduced.
For information on:
ancient Greek mathematicians, see Anaximander; Apollonius of Perga; Archimedes; Eratosthenes; Euclid (mathematician); Eudoxus; Pythagoras; Zeno of Elea
ancient Arab mathematicians, see Alhazen; Al-Battani; Al-Khwarizmi
Chinese mathematicians, see Qin Jinshao; Zhu Shijie
medieval mathematician, see Leonardo Fibonacci
17th- and 18th-century American and European mathematicians, see Leonhard Euler; Pierre de Fermat; Joseph Louis Lagrange; Gottfried Wilhelm Leibniz; Gaspard Monge; John Napier; Blaise Pascal
19th- and 20th-century American and European mathematicians, see Niels Henrik Abel; Charles Babbage; Johann Jakob Balmer; Friedrich Wilhelm Bessel; János Bolyai; George Boole; Georg Cantor; Augustin Louis Cauchy; Arthur Cayley; Jean Baptiste Fourier; Gottlob Frege; Évariste Galois; Carl Friedrich Gauss; David Hilbert; Sir Fred Hoyle; Karl Gustav Jacobi; Sir James Hopwood Jeans; William Thomson Kelvin; Nikolay Lobachevsky; Benoit Mandelbrot; John von Neumann; Emmy Noether; Jules Henri Poincaré; , Siméon-Denis Poisson; Bernhard Riemann; Paolo Ruffini; Bertrand Russell; Alan Mathison Turing; Karl Theodor Wilhelm Weierstrass; Hermann Weyl; Alfred North Whitehead; Norbert Wiener

Anaximander (circa 611-c. 547BC), Greek philosopher, mathematician, and astronomer, born in Miletus in what is now Turkey. He was a disciple and friend of the Greek philosopher Thales. Anaximander is said to have discovered the obliquity of the ecliptic, that is, the angle at which the plane of the ecliptic is inclined to the celestial equator. He is credited with introducing the sundial into Greece and with inventing cartography. Anaximander's outstanding contribution was his authorship of the earliest prose work concerning the cosmos and the origins of life. He conceived of the universe as a number of concentric cylinders, of which the outermost is the sun, the middle is the moon, and the innermost is the stars. Within these cylinders is the earth, unsupported and drum-shaped. Anaximander postulated the origin of the universe as the result of the separation of opposites from the primordial material. Hot moved outward, separating from cold, and then dry from wet. Further, Anaximander held that all things eventually return to the element from which they originated.
"Anaximander," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

Pi, Greek letter () used in mathematics as the symbol for the ratio of the circumference of a circle to its diameter. Its value is approximately 22/7; the approximate value of ð to five decimal places is 3.14159. The formula for the area of a circle, A = r2 (r is the radius), uses the constant. Various approximations of the numerical value of the ratio were used in biblical times and later. In the Bible, the value was taken to be 3; the Greek mathematician Archimedes correctly asserted that the value was between 3 10/70 and 3?. With computers, the value has been figured to more than 100 million decimal places, although this has no practical value. The ratio is actually an irrational number, so the decimal places go on infinitely without repeating or ending in zeros. The symbol ð for the ratio was first used in 1706 by the English mathematician William Jones, but it became popular only after its adoption by the Swiss mathematician Leonhard Euler in 1737. In 1882 the German mathematician Ferdinand Lindemann proved that ð is a transcendental number-that is, it is not the root of any polynomial equation with rational coefficients (for example, ?x3 - 5/7x2 - 21x + 17 = 0). Consequently, Lindemann was able to demonstrate that it is impossible to square the circle algebraically or by use of the ruler and compass.

Contributed By:
James Singer, Reviewed by:
J. Lennart Berggren

Floating Point Math

float

float = noun
The data type name used in some programming languages, notably C, to declare variables that can store floating-point numbers. See also data type, floating-point number, variable.

floating-point arithmetic

floating-point arithmetic = noun
Arithmetic performed on floating-point numbers. See also floating-point notation, floating-point number.

floating-point number

floating-point number = noun
A number represented by a mantissa and an exponent according to a given base. The mantissa is usually a value between 0 and 1. To find the value of a floating-point number, the base is raised to the power of the exponent, and the mantissa is multiplied by the result. Ordinary scientific notation uses floating-point numbers with 10 as the base. In a computer, the base for floating-point numbers is usually 2.

floating-point notation = noun
A numeric format that can be used to represent very large real numbers and very small real numbers. Floating-point numbers are stored in two parts, a mantissa and an exponent. The mantissa specifies the digits in the number, and the exponent specifies the magnitude of the number (the position of the decimal point). For example, the numbers 314600000 and 0.0000451 are expressed respectively as 3146E5 and 451E-7 in floating-point notation. Most microprocessors do not directly support floating-point arithmetic; consequently, floating-point calculations are performed either by using software or with a special floating-point processor. See also fixed-point notation, floating-point processor, integer. Also called exponential notation.

floating-point processor

floating-point processor = noun
A coprocessor for performing arithmetic on floating-point numbers. Adding a floating-point processor to a system can speed up math and graphics dramatically if the software is designed to recognize and use it. The i486DX and 68040 and higher microprocessors have built-in floating-point processors. See also floating-point notation. Also called floating-point number, math coprocessor, numeric coprocessor.

floating-point operation

floating-point operation = noun
Acronym FLOP.

An arithmetic operation performed on data stored in floating-point notation. Floating-point operations are used wherever numbers may have either fractional or irrational parts, such as spreadsheets and computer-aided design (CAD). Therefore, one measure of a computer's power is how many millions of floating-point operations per second (MFLOPS or megaflops) it can perform. See also floating-point notation, MFLOPS.

floating-point constant

floating-point constant (flo`teng-point` kon'st?nt) noun
A constant representing a real, or floating-point, value. See also constant, floating-point notation.

floating-point register

floating-point register = noun
A register designed to store floating-point values. See also floating-point number, register.

Perfect Numbers

From: Rune@mail1.stofanet.dk
Date: 30 Aug 1999
Time: 15:04:10
Remote Name: 212.10.22.175

Comments

I once posted an article "The Great Pyramid and its perfect numbers" - Nobody seems to bother, that's why I now tell you that I have now discovered that the Egyptians, who worshipped the sun and built the great solar symbols - the pyramids ,- also knew the 6th perfect number. But how do I know? Well - in my article I have shown that they knew the 5 first perfect numbers and I have now discovered that the royal cubit (only used for sacred buildings) was created this way: The circumference of the sun including its chromosphere divided by the 6th perfect number (8589869056) is almost equal to 1 royal cubit. You may think this is just a coincidense, but think of the almost empty space in our solar system. Consider that our metric-system is is related to the circumference of the earth. The earth is not our center and it certainly is not a perfect sphere like the sun - our center.

Perhaps this will make you wonder.

ARK Notes
http://www.mmlab2.rlc.dcccd.edu/nunley/HOMEPAGE/RANDOLPH/
Noah's Ark, Where is it?

"And God said unto Noah, the end of all flesh is come before me...Make thee an ark," Gen.6:13.

There are over 80,000 stories about the flood, in 72 different languages. Cultures from ancient Armenia
(Urartu) to the American Indians have stories about the flood. One of the oldest stories containing
mention of the flood is The Gilgamesh Epic. http://www.nd.edu/~theo/glossary/gilgamesh.epic.html search on http://search.nd.edu:8765/query.html?qt=Gilgamesh+Epic&submit2=Search&col=cwwwnded+endsport+zasknd

The Gilgamesh Epic was written on stone tablets as long ago as 2,000 BC. Here it was translated by
George Smith in 1872.

"I caused to embark within the vessel all my family made my relations. The beasts of the field, the
cattle of the field, the craftsmen, I made them all embark. I entered the vessel and closed the door...From
the foundations of heaven a black cloud arose...All that is bright turned into darkness...The gods feared
the flood, they fled, they climbed into heaven of Anu. The gods crouched like a dog on a wall, they lay
down...For six days and nights wind and flood marched on, the hurricane subdued the land. When the
seventh day dawned, the hurricane was abated, the flood which had waged war like an army; the sea was
still, the ill wind was calmed, the flood ceased. I beheld the sea, its voice was silent, and all mankind was
turned into mud! As high as the roofs reached the swamp! ... I beheld the world, the horizon of sea;
Twelve measures away an island emerged; Unto Mount Nisir came the vessel and let it not budge...
When the seventh day came, I sent forth a dove, I released it; it went, the dove, it came back, as there
was no place it came back. I sent forth a swallow, it came back, as there was no place, it came back. I
sent forth a raven, I released it, it went, the raven, and beheld the subsidence of the waters; it eats, it
splashes about, it caws, it comes not back."

The Cherokee Indians, along with the Tlinget Indians and Acagchemem Indians also have a version of the
flood. These sections are taken from a book entitled Noah's Ark by L. Patricia Kite. History of the Cherokee People: Cherokee Indians
         "In the tribal tales of the Cherokee Indians of the southeastern United States, the coming of a
      flood was told by a dog to his master. 'You must build a boat,' the dog said, 'and put in it all that
      you would save; for a great rain is coming that will flood the land.' "

           "The Tlinget tribe of northwestern Alaska told of a great flood which, driven by wind, covered all
      dwelling places. The Tlingets saved themselves by tying several boats together to make a great raft.
      They floated on this, huddling together for warmth under a tenet until Anodjium, a magician,
      ordered the sea to be calm and the flood to recede."
This section about the Acagchemem Indians was taken from a book entitled Chinigchinich by Friar
Geronimo Boscana.

"The Acagchemem Indians, near San Juan Capistrano in Southern California, were not entirely
destitute of knowledge of the universal Deluge, but how, or from whence, they received the same, I
could never understand. Some of their songs refer to it; and they have a tradition that, a time very remote,
the sea began to swell and roll in upon the plains, and fill the valleys, until it had covered the mountains;
and thus nearly all the human race and animals were destroyed, except a few, who had resorted to a very
high mountain which the waters did not reach."

In September of 1969, Life Magazine published an article about a strange boat-shaped formation in the
mountains of Ararat in eastern Turkey.

"While routinely examining aerial photos of his country, a Turkish army captain suddenly
gaped at the picture shown above. There, on a mountain 20 miles south of Ararat, the biblical landfall of
Noah's ark, was a boat - shaped form about 500 feet long. The captain passed on the word. Soon an
expedition including American scientists set out for the site."

"At 7,000 feet, in the midst of crevasses and landslide debris, the explorers found a clear,
grassy area shaped like a ship and rimmed with steep, packed- earth sides.Its dimensions are close to
those given on Genesis: 'The length of the ark shall be 300 (700 wds) cubits, the breadth of it 50 cubits,
and the height of it 30 cubits,' that is, 450x75x45 feet. A quick two day survey revealed no sign that the
object was man made. Yet a scientist in the group says nothing in nature could create such a symmetrical
shape. A thorough excavation may be made in another year to solve the mystery."

" And the ark rested in the seventh month on the seventeenth day of the month upon the mountains of
Ararat," Gen. 8:4.

Located in the northeastern corner of Turkey is Mount Ararat. It is close to the borders of Iran and
Armenia (formerly the Soviet Union). Due to it's location and the political conditions that surround that
region, there have been numerous problems in exploration of the ark.

      In 1978 a man named John McIntosh attempted to climb Mt. Ararat. Terrorists overran the base
      camp, looted items they wanted and burned the rest. The exploration party was marched to a wall
      and faced with terrorists holding guns. McIntosh and his party felt they would be shot. Instead one
      of the terrorists took pictures of the group. The group was then forced to leave the site.

According to the Bible, Noah's Ark was a large barge constructed of wood and sealed with bitumen. It's
overall dimensions were 450 feet long, 75 feet wide, and 45 feet high with three interior decks. A long
"window" was constructed around the top. However, there is some speculation as to whether it was the
Royal Egyptian cubit or the Hebrew cubit used in the measurements of the ark.

"The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it
thirty cubits," Gen. 6:15

When Moses wrote the Genesis account of the flood story the Royal Egyptian cubit was the most
universal standard of measurement at that time. There was no "Hebrew cubit" during Moses' time. Three
hundred Egyptian cubits equals 515 feet. However, 300 Hebrew cubits equals 450 feet, which is more
widely accepted as the length of the ark.

      In 1979 an amateur archeologist by the name of Ron Wyatt climbed Mt. Ararat and saw what
      looked like a giant shipwreck. He described evenly spaced indentations that could be seen all the
      way around the object. While there, Ron took some samples from deep within the cracks, and more
      samples of material from outside the object for comparison. He then measured the object to 512
      feet in length, but he saw what looked like a section about a yard long that was broken off from the
      lower end. The total length was 515 feet, which again equals 300 Egyptian
      cubits.http://www.anchorstone.com/wyatt

Along with numerous eyewitness accounts of the ark being on Mt. Ararat, some dating as far back as the
sixth century B.C., are oral legends handed down and even aerial photographs.

      Most requests for satellite photos of Mt. Ararat are denied due to bureaucratic grounds though some
      have been released. These photos clearly show what scientists refer to as an "anomaly" on a glacier
      shelf near the Ahora Gorge, on Mt. Ararat. Computer enhanced photos and measurements by ark
      expeditions verify that the anomaly matches the size and shape of the ark described in the bible.
      http://www.beachin.net/~bjcorbin/noahsark/porcher.htm
This is a section taken from Noah's Book written around the third century B.C.
http://www.arminco.com/hayknet/tapan.htm

        "...Noah's Ark drifts on the surface of the water and while approaching the mountains of Armenia it,
as a human being, prays to receive. The mountains refuse and point at the biggest of the mountains,
Ararat. And Noah's Ark moors there."

According to the Bible the ark landed on the mountains of Ararat, not specifically Mount Ararat.
However, according to the Qu'ran the ark landed specifically on Mount Judi, a mountain located
approximately 200 miles south of Mt. Ararat in southern Turkey. At times Mt. Judi has been mistaken for
Mt. Ararat. The exact altitude of the mountain is unknown, however the Encyclopedia Of Islam lists is as
"over 13,000 feet and largely unexplored."
There have been several historical references to Mount Judi, or "Cudi Dagh." Mt. Judi has also been
referred to as "the Gordyaen mountains", " The Mountains of the Kurds", "Mt. Quardu",and " Mt.
Nipur." The Nestorians, a sect of Christianity, built several monasteries around the mountain including
one on the summit called " The Cloister of the Ark." It was destroyed by lightening in 766 A.D. The
Muslims later built a mosque on the site. http://user.aol.com/mkneisler/noah/nadanger.htm

In 1910 Gertrude Bell explored the area and found a stone structure still at the summit with the shape of a
ship called by the locals "Sefinet Nebi Nuh", "The ship of Noah." She also reported that every year on
September 14 Christians, Jews, Muslims, Sabians, and Yezidis gather on the mountain to commemorate
Noah's sacrifice. And as late as 1949 two Turkish journalists claimed to have seen the Ark on this
mountain, a ship 500 feet in length.
      Berossus', who was a Chaldean priest and historian in the third century B.C., account of the flood is
      similar to the Babylonian Flood account. He notes that the Ark "...grounded in   Armenia some part
      still remains in the mountains of the Gordyaens in Armenia, and some get pitch from the ship by
      scraping off, and use it for amulets. The location of Mt. Judi is both in the Gordyaen Mountains and
      with in the borders of ancient Armenia.
      The Samaritan Pentateuch is the manuscript that contains the first five books of the Old Testament,
      and the Bible used by the Samaritans. It puts the landing place of the Ark in the Kurdish mountains
      north of Assyria. The Samaritans were a Jewish people who separated from the Jews about the 5th
      Century B.C. The intermarriage of Assyrian colonists and these Jews became known as Samaritans.
      The Targums are paraphrases in Aramaic which were made for the Jews after they returned from
      their captivity in Babylon. (Nehemiah 8:8) Three of the Targums put the landing place of the ark in
      the Qardu mountains. This was not far from where some of these Jews spent their captivity.
      http://www.fni.com/cim/technicals/noah.txt

The search for Noah's Ark will continue until the actual Ark is brought down from atop which ever
mountain it sits. Through out history it has been one of the most sought after artifacts of the Bible. If
nothing else, finding the Ark will prove the story in Genesis about the flood to those who question and
search for proof. It will be a way of proving there is an after life and a heaven to those who are skeptical.
And though I can not offer any advice to which mountain the ark came to berth, either Ararat or Judi, I
can say that having faith is believing without proof.

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