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Topics - Mohammad Hassan Murad

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Science Discussion Forum / Maria Gaëtana Agnesi and her witch
« on: May 30, 2012, 12:38:12 PM »
Maria Gaëtana Agnesi (1718–1799)
Italian Mathematician and Philosopher

Agnesi (pronounced ahn-YAY-zee or anyesi) was born in Milan, Italy. By the time she was 5 years old, she could speak both Italian and French. Her father was a professor of mathematics in Bologna, and Agnesi enjoyed a childhood of wealth and privilege. Her father provided her with tutors, and she participated in evening seminars, speaking to the guests in languages as varied as Latin, Greek, Hebrew, and Spanish.
            In her teens, Agnesi mastered mathematics. She became a prolific writer and an advocate for the education of women. After her mother died, Agnesi managed the household of eight children, and educated her brothers. Her father remarried and after her stepmother died, Maria became housekeeper for twenty siblings.
            Agnesi continued studying mathematics, mostly at night, and often to the point of exhaustion. In 1748 her mathematical compendium, Instituzioni analitiche ad uso della gioventù italiana (Analytical Institutions), derived from teaching materials she wrote for her brothers, was published in two volumes. In this work, Agnesi had not only written clearly about algebra, precalculus mathematics, differential calculus, and integral calculus, but she had also added her conclusions and her own methods. Analytical Institutions remains the first surviving mathematical work written by a woman.
             So successful was Agnesi’s textbook that it became the standard text on the subject for the next 100 years. Her book was studied and admired not only in Agnesi’s native Italy but also in France and Germany, and was translated into a number of other languages.
             In his 1801 English translation of Agnesi’s work, John Colson, the Cambridge (England) Lucasian Professor of Mathematics, made the mistake of confusing the Italian word for a versed sine curve, aversiera, with another Italian word for witch or wife of the devil, avversiere. Although 200 years have passed, the curve Colson misnamed the “Witch of Agnesi” still bears that name in many calculus texts as an illustration of an “even” function with a “maximum” value. For a = 2, for example, the maximum value of y will be 2. The curve illustrates many basic concepts in calculus.
Witch of Agnesi

Agnesi was recognized during her lifetime with election to the Bologna Academy of Science. The Pope at the time was interested in mathematics and in 1750 made certain that Agnesi was invited to be an honorary lecturer in mathematics at the University of Bologna. However, Agnesi turned down the appointment and instead adopted a religious life, spending much of her time working among the elderly poor and sick women of Milan.
           Although a hospice Agnesi founded has become famous throughout Italy, and Italian streets, a school, and scholarships have been named in her honor, Agnesi is perhaps best known today for her contributions to mathematics.

Maor, E. Trigonometric Delights, Princeton University Press, 1998
Brandenberger, Barry M. Jr. (editor in chief) Mathematics The Macmillan Science Library Volume 1-4, Macmillan Reference USA, Thomson/Gale, 2002.
Bradley, M. J. Pioneers in Mathematics The Age of Genius 1300 to 1800 Vol. 2, Facts on File, 2006.

Science Discussion Forum / Do Mathematicians get Nobel Prize?
« on: May 27, 2012, 06:02:18 PM »
He [Abel] has left mathematicians something to keep them busy for five hundred years.

Do Mathematicians get Nobel Prize? No. They get Abel Prize. It is not a joke. The name comes from a Norwegian mathematician who died at the age of 26.

NIELS HENRIK ABEL, one of the foremost mathematicians of the 19th century, was born in Norway on August 5, 1802. At the age of 16, he began reading the classic mathematical works of Newton, Euler, Lagrange, and Gauss. When Abel was 18 years old, his father died, and the burden of supporting the family fell upon him. He took in private pupils and did odd jobs, while continuing to do mathematical research. At the age of 19, Abel solved a problem that had vexed leading mathematicians for hundreds of years. He proved that, unlike the situation for equations of degree 4 or less, there is no finite (closed) formula for the solution of the general fifth-degree equation. Although Abel died long before the advent of the subjects that now make up abstract algebra, his solution to the quintic (fifth degree equation) problem laid the groundwork for many of these subjects. Just when his work was beginning to receive the attention it deserved, Abel contracted tuberculosis. He died on April 6, 1829, at the age of 26. In recognition of the fact that there is no Nobel Prize for mathematics, in 2002 Norway established the Abel Prize as the Nobel Prize in mathematics in honor of its native son. The amount of money that comes with the prize is usually close to US$ 1 million, similar to the Nobel Prizes (10 million kr. or US$ 1.46 million), which are awarded in Sweden and Norway and do not have a category for mathematics. Norway gave the prize an initial funding of NOK (Norwegian Krone) 200,000,000 (about US$23,000,000) in 2001. The prize is an attempt at creating publicity for mathematics, making the discipline more prestigious, especially for young people. It is a prize of 6 million kronor or approximately (in 2012) 1.06 million US dollars. The Abel Prize is now seen as an award equivalent to the Nobel Prize.

To see the winners of Abel Prize please visit the following webpage

JEAN-PIERRE SERRE of the College de France has been awarded the first Abel Prize of the Norwegian Academy of Science and Letters. Serre is honored “for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.” The prize amount is 6 million Norwegian kroner (approximately US$825,000).

The Norwegian Academy of Science and Letters in Oslo awarded Endre Szemer the one million dollar Abel prize on 21st march for "fundamental contributions to discrete mathematics and theoretical computer science". His specialty was combinatorics, a field that deals with the different ways of counting and rearranging discrete objects, whether they be numbers or playing cards.

Holden, H., Piene, R. (ed.) The Abel Prize 2003-2007 The First Five Years, Springer, 2010.
Jackson, A. Norway Establishes Abel Prize in Mathematics, Notices of the AMS, 49, 39-40,
Jackson, A. Serre Receives Abel Prize, Notices of the AMS, 50, 693, June/July 2003
Atiyah & Singer Receive 2004 Abel Prize, Notices of the AMS, 51, 650-651, June/July 2004.

Further reading

Science Discussion Forum / Life of Pythagoras
« on: May 26, 2012, 06:28:56 PM »
Pythagoras (ca. 569??-475?? B.C.E.) Born in Samos, Ionia, Greek mathematician and mystic Pythagoras is remembered as founder of the Pythagorean School, which claimed to have found universal truth through the study of number. As such, the Pythagoreans are credited as the first to have taken mathematics seriously as a study in its own right without regard to possible application and practical need. What precisely Pythagoras contributed himself is no longer clear, but the school is acknowledged to have discovered the role of ratios of whole numbers in the musical scale, the properties of figurate numbers, amicable numbersperfect numbers, and the existence of irrational numbers. Although Pythagoras's Theorem was known to the Babylonians over 1,000 years earlier, the Pythagoreans are the first to have provided a general proof of the result.
            Very little is known of Pythagoras's life. The sect he founded was half scientific and half religious and followed a code of secrecy that certainly promoted great mystery about the man himself. Many historical writings attribute godlike qualities to Pythagoras and are generally not regarded as accurate portrayals. It is understood that Pythagoras visited with Thales of Miletus (ca. 625-547 B.C.E.) as a young man who, most likely, contributed to Pythagoras' interest in mathematics. Around 535 B.C.E., Pythagoras traveled to Egypt and was certainly influenced by the secret sects of the Egyptian priests. (Many of the practices the Pythagoreans followed were the same as those practiced in Egypt refusal to eat beans or to wear cloth made from animal skins, for instance.)
           In 525 B.C.E., Egypt was invaded by Persian forces, and Pythagoras was captured and taken to Babylon. Eventually, after being released, Pythagoras settled in southern Italy, where he founded his famous school around 518 B.C.E. Both men and women were welcomed as members.
           Pythagoras believed that all physical (and metaphysical) phenomena could be understood through numbers. This belief is said to have stemmed from his observation that two vibrating strings produce a harmonious combination of tones only when the ratios of their lengths can be expressed in terms of whole numbers. (Pythagoras was an accomplished musician and made significant contributions to the theory of music.) Pythagoras, and his followers, studied whole numbers and their ratios, and even went as far as to assign mystical properties to numbers. For instance, they believed that the first natural number, 1, acted as the divine source of all numbers and so was of different stature than an ordinary number. (The number 2 was deemed the first number.) All even numbers were assumed feminine and all odd numbers masculine, and all odd numbers (except 13) represented good luck. The number 4 stood for justice, being the first perfect square, and 5 marriage, as the union (sum) of the first even number (2) and the first odd number (3). The number 6 was perfect since it equals the sum of its factors different from itself.
             In Geometry, the Pythagoreans knew that the interior angles of a triangle always sum to 180° (and, more generally, that the interior angle of an n-sided figure sum to (n - 2)180°), and knew how to construct three of the five Platonic Solids. They could solve equations of the form x2 = a(a-x) using geometrical methods, and they developed a number of techniques for constructing figures of a given area. They attributed great mystical significance to the pentagram because of the geometric ratios it contains. In astronomy, the Pythagoreans were aware that the Earth is round, but believed it lay at the center of the universe. They were aware that the orbit of the Moon is inclined to the equator of the Earth and were the first to realize that the evening star and the morning star were the same heavenly object (namely, the planet Venus).
             The exact date, location, and circumstance of Pythagoras's death are not known. Despite his passing, the factions of the original Pythagorean order endured for more than 200 years. Members of the Pythagorean brotherhood so revered their founder that it was considered impious for any individual to claim a discovery for his own glory without referring back to Pythagoras himself. It is said, for instance, that member Hippasus of Metapontum (ca. 470 B.C.E.) was put to death by drowning for announcing his own discovery of the regular dodecahedroan, the fifth Platonic solid. (Other versions of this popular story submit that he was executed for claiming to have discovered that the square root of 2 is an irrational number.)

To explore more see
Cornelius Lanczos Space Through the Ages Geometrical Ideas From Pythagoras to Hilbert & Einstein, Academic Press, 1970.
James Tanton Encyclopedia Of Mathematics, Facts on File, 2005.
Carl B. Boyer A History of Mathematics, 2nd ed. revised by Uta C. Merzbach and foreword by Isaac Asimov, John Wiley and Sons Inc., 1991.
George F. Simmons Calculus Gems Brief Lives and Memorable Mathematics, McGraw Hill 1992.

Charles Babbage is best remembered for his work on the design and manufacture of a mechanical calculator, the forerunner of a computer. After first constructing a “difference machine,” Babbage devoted the remainder of his life to the construction of a superior “analytic engine” capable of performing all mathematical operations. His work toward this goal laid the foundations of computer design used today. Partly due to lack of funding, however, the machine was never completed.....

To know more on him please see the attached file.

As mathematicians believe most of the mysteries of Nature are hidden in numbers. Now I shall talk about such a mysterious problem in number theory. At first I begin by quoting one of the most greatest mathematician of England,

The famous problem of which I propose to speak tonight is probably as difficult as any of the unsolved problems of mathematics.
G. H. Hardy on Goldbach's conjecture

What is Goldbach's Conjecture?
The Prussian mathematician Christian Goldbach noticed a pattern in the following sums:
4=2+2,                        6=3+3,                  8=3+5
10=3+7,12=5+7,         14=3+11=7+7,
16=3+13=5+11,          18=5+13=7+11,     20=3+17=7+13

(Do you see a pattern here?) Based on his observations, Goldbach, in a letter to Euler in 1742, conjectured that every even integer > 2 can be expressed as the sum of two primes. Euler could not prove it, and his conjecture still remains an unsolved problem.

Is Goldbach'ss Conjecture true for all even integer?
If not, up to when it is true?

To know more on this see the following attached file

If global warming is a reality, as most now agree it is, then its consequences must be considered. The possible consequences of global warming are controversial and difficult to predict. However, most of the scientists who study the atmosphere agree that the following are likely results of increased global warming, although the extent to which each of these phenomena may manifest itself is debatable.

  • Oceans may warm and release more carbon dioxide due to the reduced solubility of gases in warm water. This would increase warming even more and exacerbate the problem. However, warmer oceans might allow more phytoplankton to grow, which could consume more carbon dioxide.
  • Solid, frozen methane deposits at the bottom of the oceans may become gaseous as the ocean warms, and the gaseous methane would be released into air. This would increase warming even more.
  • Ice and snow cover may decrease significantly in parts of the world. This would cut down on reflected light so Earth would absorb even more radiation and warm up at an accelerated rate.

  • Ocean levels may rise as ice and snow at the polar caps melt. The consequences for islands and low-lying areas along coasts could be disastrous. There is already evidence of this occurring. Ice shelves that have been frozen for thousands of years are breaking away. The Arctic Climate Impact Assessment, a study by over 300 scientists released in late 2004, suggested that, among other consequences of such warming, polar bears could become extinct by 2099.

  • More water might evaporate from the oceans as they warm. This could increase global warming because water is a greenhouse gas. However, this could be offset by the reflective effect that additional cloud cover might have on incoming solar radiation.

  • Warming of the oceans could cause major shifts in ocean currents, leading to unpredictable influences on weather, fisheries, etc.

  • Rainfalls would become much heavier in some areas as a result of increased evaporation. This would lead to increased flooding.

  • The tree line would move farther north, causing trees to grow over a broader region. This could consume more carbon dioxide.

  • Deserts and arable regions of the world could shift location and change size as a function of shifts in areas of rainfall. The consequences are difficult to predict.

  • Tropical diseases might become more prevalent and widespread as Earth warms.

  • The intensity of storms and weather systems would probably increase.

  • Increased heat may mean worsened air pollution, damaged crops, and depleted resources in some areas.

  • Some scientists have speculated that when a certain atmospheric concentration of greenhouse gases is reached, it may trigger a relatively rapid and cataclysmic climate system reorganization.

  • Possibly, nothing of consequence, that is, nothing we can’t handle, will happen.
Excerpted from the book
Joesten, M., et al. The World of Chemistry Essentials, 4th edition, Thomson, Brooks/Cole, 2007.

« on: May 22, 2012, 01:35:04 PM »
        The world has become increasingly dependent on the use of fossil fuels (coal, oil, and natural gas) to support consumer-driven societies and their associated lifestyles. Enormous quantities of these fuels are used daily. The fuels may be burned in a variety of vehicles to propel them down the road, through the air, or across the oceans. The fuels may be utilized to drive industrial manufacturing processes directly, such as by powering an engine, or they may be used indirectly to generate electricity to power the economy by producing the consumer products upon which we’ve become so dependent. Our ability to heat and cool our homes; grow, harvest, transport, and cook our food; provide health care and medicines; and entertain ourselves during our free time all depend on fossil fuels in one way or another. Some activities depend on burning the fuels whereas others depend on utilizing the fossil fuels as raw materials in the chemical industry. Fossil fuels are nonrenewable resources, however, and the supply will eventually run out. It is becoming increasingly apparent that the products of fossil fuel combustion may be causing Earth to warm beyond what is natural and desirable. It is a known fact that both carbon dioxide and water vapor absorb and trap heat radiation. It is also known that the levels of carbon dioxide and some other heat-trapping substances have increased significantly in the atmosphere over the past hundred or so years. The debate among scientists, politicians, and concerned citizens centers on whether the warming of Earth is due to the increase in concentration of these substances. If so, what will be the consequences of this warming? And is there anything we can or should do about it?
        Some people believe this global warming will be good for Earth, some think it will be inconsequential, some think we will adjust to it, and some think it threatens life as we know it. Some people think there is nothing that can or should be done about this warming, whereas others feel it is imperative that we do everything within our power to halt this phenomenon. The average citizen must become knowledgeable enough about this issue to make informed decisions concerning factors affecting this global warming. Such knowledge could influence the type of automobile you purchase, the size of the house you live in, and your lifestyle in general. It could eventually influence where and how you live or, possibly, even if you live. Knowledge is power and only by obtaining, properly assessing, and utilizing this knowledge can we adjust to changing times.

Excerpted from the book
Joesten, M., et al. The World of Chemistry Essentials, 4th edition, Thomson, Brooks/Cole, 2007.

Science Discussion Forum / Amicable Numbers
« on: May 16, 2012, 12:54:08 PM »
Amicable Numbers

A pair of numbers is amicable if each is the sum of the proper divisors of the other. The smallest pair is 220 and 284. The proper divisors of 220 sum to,
                                           1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
and similarly,
                                                              1 + 2 + 4 + 71 + 142 = 220.
According to the philosopher Iamblichus (c. AD 250–330), the followers of Pythagoras “call certain numbers amicable numbers, adopting virtues and social qualities to numbers, such as 284 and 220; for the parts of each have the power to generate the other,” and Pythagoras described a friend as “one who is the other I, such as are 220 and 284.”

The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368)

In the Bible (Genesis 32:14), Jacob gives 220 goats (200 female and 20 male) to Esau on their reunion. There are other biblical references at Ezra 8:20 and 1 Chronicles 15:6, while 284 occurs in Nehemiah 11:18. These references are all to the tribe of Levi, whose name derives from the wish of Levi’s mother to be amicably related to his father. They were also used in magic and astrology. Ibn Khaldun (1332–1406) wrote that “the art of talismans has also made us recognize the marvelous virtues of amicable (or sympathetic) numbers. These numbers are 220 and 284. . . . Persons who occupy themselves with talismans assure that these numbers have a particular influence in establishing union and friendship between individuals.” (Ore 1948, 97)

To see the full article download the following attached file.

Science Discussion Forum / Grand Unified Theory (GUT)
« on: May 15, 2012, 07:48:20 PM »
                                                                                   Toward a Grand Unified Theory

Ever since Sir Isaac Newton identified the gravitational force, all the forces of nature have come very close to unification in theory. Electricity and magnetism were found to be linked as the same force early in the twentieth century. Then, after gravity and electromagnetism, two more forces were discovered at the beginning of the twentieth century: the “strong” and “weak” nuclear forces of the atomic nucleus.
Whereas the electromagnetic and gravitational forces operate over relatively large distances, the strong and weak nuclear forces operate over distances confined to the diameter of an atom. The strong force holds the individual parts of the nucleus together, and the weak force, through “beta radiation,” is responsible for the decay of the nucleus itself. A subatomic particle called a “neutrino” reacts within the core of the atom and participates in some nuclear reactions. Although neutrinos do not react with mass, they interact with other subnuclear particles through the weak force. Most of the neutrinos that affect Earth are generated in the center of the Sun’s thermonuclear core.
The theory developed by Sheldon Glashow (Born 1932), Abdus Salam (1926-1996), and Steven Weinberg (Born 1933) is called the electroweak theory. For the first time it precisely detailed the interaction of the electromagnetic (large) force and the weak (small) force. It also stated that the neutrino and the electron are members of the same family of particles; in effect, the neutrino is the electron’s “little brother.” The theory predicted the existence of what are called “neutral currents.” When the electron changes its identity to a neutrino and vice versa, the theory predicts that a “charged current” will be manifested in advance of the change of charge. Concurrently, a “neutral current” is present as the neutrino “acts without changing identity.”

The unification of all the forces into a single, unified mathematical theory has been a dream of physicists. The development of the electroweak theory brought that “grand unified theory” one step closer to being discovered.

Excerpted from
Magill’s Choice Science and Scientists, Salem Press Inc., 2006.
Weinberg, S. A Unified Physics by 2050?, Scientific American, Special edition 2003.

Science Discussion Forum / Newton & the apple story
« on: April 26, 2012, 06:36:10 PM »
     Sir Isaac Newton (1642-1727) was born on Christmas Day of 1642 at the farm of his mother’s parents near Grantham in Lincolnshire, after his father had died. He was raised by his maternal grandparents and then enrolled in Trinity College, Cambridge, in 1661, to study mathematics under Isaac Barrow (1630-1677). After completing his degree in 1665, he returned home for nearly two years to escape an outbreak of the plague. During this isolation, he began to formulate his ideas about universal gravitation after making a connection between the fall of an apple and the motion of the Moon.

Legend has it that Newton’s interest turned to gravity when he was struck on the head by a falling apple. Is it an apocryphal story?

I'll be saying more on this topic in my forthcoming post.

Science Discussion Forum / Galileo & the leaning tower of Pisa
« on: April 26, 2012, 06:22:32 PM »
Aristotle’s Errors
     One of Aristotle’s predictions, which was passed down to the seventeenth century, concerned the behavior of falling bodies. Aristotle, adhering to his philosophy that all effects require a cause held that all motion (effect) required a force (cause), and hence that falling (a motion) required a force (the weight, or what we now know as mass, of the object falling). The Italian astronomer and mathematician Galileo Galilei was one of a number of scientists who questioned the Aristotelian view of the workings of nature and turned to experimentation to find answers. According to legend, Galileo dropped a bullet and a cannonball from the tower to show that all objects fall with the same acceleration.

Did Galileo really drop two objects from the tower?

I'll discuss this topic in details in my forthcoming post.

Science Discussion Forum / Tale of an Imaginary Number
« on: April 26, 2012, 04:23:48 PM »
     In our college mathematics probably we all have learned about the imaginary number i. We were simply introduced i as a number.
Is it a number?
What kind of number is i ?
Where does the name imaginary come from?
Is it really an imaginary number? If yes then how did this number get involved into the real mathematics?
Can imaginary number do the same thing or different thing as real number do or more?
British mathematical physicist Roger Penrose calls i magic number. What is magical about i?

To explore this readers are requested to download and read the attached file. Author hopes to have some constructive comments from the scholars.

Science Discussion Forum / What is an Imaginary Number?
« on: April 26, 2012, 02:31:18 PM »
In our college mathematics probably we all have learned about the imaginary number. We were simply introduced i as a number. Our traditional introduction to imaginary number does not make us see the inner beauty. And it does not fascinate us. Have we ever thought about imaginary number as follows?
1. Is imaginary number a number?
2. What kind of number is i?
3. Where did the name imaginary come from?
4. Is it really an imaginary number? If yes then how did this number get involved into the real mathematics?
5. Can imaginary number do the same thing or different thing as real number do or more?
British mathematical physicist Sir Roger Penrose calls i magic number. What is magical about i?

I am preparing an article on this beautiful fascinating subject and hope to post it soon.

Science Discussion Forum / What is gravitation?
« on: April 25, 2012, 04:50:58 PM »
What is gravitation? Is it simply one of the four fundamental forces of nature, or something else?
What is gravitation according to Sir Isaac Newton?
What is gravitation according to Albert Einstein?
What major change Einstein had brought us about Newtonian picture of gravitation?
Lets have a look.

          Gravitation is a universal attractive force exerted by any two physical bodies on each other, even though they may be separated by a large distance. Gravitation is responsible for making objects fall to the surface of the Earth (gravitational attraction of the object by the Earth), for the nearly circular motions of the planets around the sun (gravitational attraction of the planets by the sun), for the structure of stars and planets (gravitational attraction balanced by pressure forces of constituent particles towards each other), and for the structure of star clusters and galaxies (hundreds of millions of stars would fly apart from each other if not held together by gravity). Gravitation also controls the rate at which the universe expands, and is responsible for the growth of small inhomogeneities in the expanding universe into galaxies and clusters of galaxies. Gravity is the weakest of the four fundamental forces known to physics, but it dominates on large scales because it is a long-distance force that is locally always attractive (in contrast to the far stronger electromagnetic force, which can both attract and repel, and cancels itself out on large scales). Thus gravitation is a dominant force in every day life, as well as in the motions of stars and planets and in the evolution of the cosmos. Indeed, it is one of the forces that makes our existence possible by enabling the formation and stability of plants like Earth that are hospitable to life. Without gravity (at approximately the strength it has on Earth) evolution of life would be difficult if not impossible. This fact can naturally lead to speculation that the existence and specific nature of gravitation could be part of a grand design allowing self-assembling structures to come into existence and lead to intelligent life. In this way, gravity can have theological significance.

Classical physics

            Italian astronomer Galileo Galilei (1564–1642) first recognized in the early seventeenth century that when air resistance can be neglected, objects accelerate at the same rate towards the surface of the earth, irrespective of their physical composition. Thus a feather and a cannon ball will arrive at the same time at the earth’s surface if simultaneously released from rest at the same height in a vacuum chamber. This means there is a universal rate of acceleration downwards caused by the earth’s gravitational field—approximately 32 feet per second squared—irrespective of the nature of the object considered. Gravitational potential energy can be converted to kinetic energy, with total energy conserved, as for example in a roller coaster or a pole vaulter. This enables gravity to do useful work, as in a clock driven by weights or a water mill, but it also means people must work to go uphill. Gravity can also be a danger to people, who can fall or be hurt by falling objects. Despite this danger, gravity is an essential part of the stability of every day life—it is the reason that objects stay firmly rooted on the ground rather than floating into the air.
In the late seventeenth century, Isaac Newton (1642–1727) showed that the gravitational attraction of objects towards the earth and the motion of the planets around the sun could be described accurately by assuming a universal attractive force between any two bodies, proportional to each of their masses and to the inverse of the square of the distance between them. The attractive nature of gravity results because masses are always positive. On this basis he was able to explain both the universal acceleration towards the surface  of the earth observed by Galileo and the laws of motion of planets around the sun that had been observationally established earlier in the century by Johannes Kepler (1571–1630).
This was the first major unification of explanation attained in theoretical physics, showing that two phenomena that initially appeared completely unrelated had a unified origin. Newton’s account of gravitation also explained why the direction of gravity varies at different places on the surface of the earth (always being directed towards its center), allowing “up” to be different directions at different places on the earth’s surface (Australia and England, for example). In conformity with the rest of theoretical physics, Newton’s theory of gravity can be reformulated as a variational principle (Hamilton’s principle or Lagrange’s equation s) based on minimization of particular combinations of kinetic energy and gravitational potential energy along the trajectory followed by a particle. Gravity by itself is a conservative theory (energy is conserved), so there is no friction associated with the motion of stars and planets in the sky, and their motion is fully reversible; the past and future directions of time are indistinguishable, as far as gravity is concerned. Newton was puzzled as to how the force of gravity, as described by his equations, could succeed in acting at a distance when there was no apparent contact between the bodied concerned. Pierre Laplace (1749-1827), a French physicist and mathematician, essentially resolved this puzzle by introducing the idea of a gravitational force field that fills the empty space between massive bodies and mediates the gravitational force between them. The concept of such fields became one of the major features of classical physics, particularly in the case of electromagnetism. In quantum theory the idea gravitational fields is revised and understood as a force mediated by the interchange of force carrying particles.

Einstein and after

          In the early twentieth century, Albert Einstein (1879–1955) radically reshaped the understanding of gravity through his proposal of the general theory of relativity, based on the idea that space-time is curved, with the space-time curvature determined by the matter in it. This theory predicts the motion of planets round the sun more accurately than Newtonian theory can, and also predicts radically new phenomena, in particular, black holes and gravitational radiation. Insofar as science has been able to test these predictions, they are correct. A problem with the theory is that it predicts that under many conditions (for example, at the start of the universe and at the end of gravitational collapse to form a black hole), space-time singularities will occur. Scientists still do not properly understand this phenomenon, but presumably it means that they will have to take the effect of quantum theory on gravity into account. General Relativity does not do so; it is a purely classical theory. Quantum gravity theories try to develop a theory of gravity that generalizes Einstein’s theory and is also compatible with quantum theory. Even the way to start such a project is unclear. Approaches include twistor theory, lattice theories, noncommutative geometries, loop variable theories, and superstring theories. None has reached a satisfactorily developed state, however, much less been tested and shown to be correct. Indeed, in many ways such theories are likely to be untestable. The most ambitious are the superstring theories, now extended into a metatheory of uncertain nature known as M-theory, which promises to provide a unified theory of all fundamental forces and particles. M-theory still has far to go before making good on that promise. Despite the lack of a definite quantum theory of gravity, various attempts have been made to develop quantum theories of cosmology. These theories also face considerable conceptual and calculational problems. The satisfactory unification of quantum theory and general relativity theory, perhaps in some unified theory of all the fundamental forces, remains one of the most significant outstanding problems of theoretical physics. The desire to develop a practical antigravity machine remains one of humanity’s outstanding wishes. No present theory offers a way to such a machine, but the negative gravitational effect of the vacuum energy will continue to inspire some to hope that one day such a machine might exist. 

Based on the entry by George F. R. Ellis published in Encyclopedia of Science and Religion ed. J. Wentzel Vrede van Huyssteen, Thompson Gale, 2003.

To explore more
D’Inverno, R. Introducing Einstein’s Relativity. Oxford: Oxford University Press, 1996.
Ellis, G. F. R., and Williams, R. M. Flat and Curved Spacetimes. Oxford: Oxford University Press, 2000.
Hawking, S. W., Ellis, G. F. R. The Large-scale Structure of Spacetime. Cambridge, UK: Cambridge University Press, 1973.
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