Einstein, Albert (1879-1955)

German-American physicist who, in 1905, published three papers, each of which had a profound effect on the development of physics. In one paper, he proposed the theory of special relativity, Eric Weisstein's World of Physics which provides a correct description for particles traveling at high speeds. The two postulates of the special theory of relativity were that the speed of light Eric Weisstein's World of Physics in a vacuum is constant and that the laws of physics are the same for all inertial reference frames. Einstein did know about the Michelson-Morley experiment Eric Weisstein's World of Physics null result, but was not familiar with Lorentz's work after 1895, so he reinvented the Lorentz transformation Eric Weisstein's World of Math for himself (Pais 1982, p. 133).

While special relativity required a modification of the laws of mechanics, the Maxwell equations Eric Weisstein's World of Physics were found to already satisfy the requirements of special relativity. Eric Weisstein's World of Physics Using special relativity, Eric Weisstein's World of Physics Einstein derived the equivalence of rest mass Eric Weisstein's World of Physics m0 and energy Eric Weisstein's World of Physics E, expressible as E2-p2c2 = m02c4, where c is the speed of light Eric Weisstein's World of Physics and p is the (relativistic) momentum Eric Weisstein's World of Physics. When relativistic mass Eric Weisstein's World of Physics is used instead , the equation reduces to the famous E = mc2.

In another 1905 paper, Einstein also explained the photoelectric effect Eric Weisstein's World of Physics by hypothesizing that light consisted of particles (called photons Eric Weisstein's World of Physics) with energy Eric Weisstein's World of Physics equal to , where h is a constant known as Planck's constant Eric Weisstein's World of Physics (named after the physicist Max Planck) and is the frequency Eric Weisstein's World of Physics of the photon. This represented an extension of Planck's quantization to light. The equation Einstein derived was verified experimentally by Millikan in 1916.

Also in 1905, Einstein provided an explanation of Brownian motion Eric Weisstein's World of Physics using kinetic theory, Eric Weisstein's World of Physics stating that it was caused by random collisions of molecules. Einstein furthermore derived an equation stating that a suspension Eric Weisstein's World of Physics of small particles should arrange itself in an exponentially decreasing manner from bottom up. Using Einstein's equations for Brownian motion Eric Weisstein's World of Physics and the distribution of particles, Perrin was able to experimentally measure the value of Boltzmann's constant. Eric Weisstein's World of Physics

Einstein subsequently developed general relativity, Eric Weisstein's World of Physics which postulated that uniform acceleration Eric Weisstein's World of Physics and a gravitational field were equivalent, a statement known as the equivalence principle of gravitation. Eric Weisstein's World of Physics It interpreted gravity Eric Weisstein's World of Physics as a warping of space-time. The general theory of relativity made extensive use of Ricci-Curbastro's tensor calculus. Eric Weisstein's World of Math Einstein investigated cosmological Eric Weisstein's World of Physics modeling, but found that the general theory of relativity would not satisfy the conditions of homogeneity, isotropy, and staticity unless an additional "cosmological constant Eric Weisstein's World of Physics" was added.

Einstein spent the latter portion of his life in an unsuccessful attempt to create a unified theory which would explain all known forces in nature as manifestions of a single fundamental force. Eric Weisstein's World of Physics Einstein's theories were highly controversial for years after he proposed them. In a recommendation for Einstein's membership in the Prussian Academy of Science, the sponsors wrote "In sum, one can say that there is hardly one among the great problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the targeting his speculations, as, for example, in his hypothesis of light-quanta, cannot really be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk" (Pais 1982, p. 382).

A recent study of Einstein's preserved brain (for details, see Regis 1991) has discovered that the inferior parietal region--the part thought to be related to mathematical reasoning--was 15% wider than normal (Witelson et al. 1999). In additional, the groove normally running from the front to the back did not extend all the way in Einstein's brain. However, it is unclear what the true significance of these anatomical anomalies are in connection with Einstein's scientific creativity.

Einstein was attacked by some with anti-Jewish leanings. When a pamphlet was published entitled 100 Authors Against Einstein, Einstein retorted "If I were wrong, one would be enough." Some famous Einstein quotes about God include

"Whoever undertakes to set himself up as judge in the field of Truth and Knowledge is shipwrecked by the laughter of the gods."
"I do not believe in immortality of the individual, and I consider ethics to be an exclusively human concern with no superhuman authority behind it."
"I want to know how God created this world. I am not interested in this or that phenomenon, Eric Weisstein's World of Physics in the spectrum of this or that element. I want to know His thoughts, the rest are details."
"God is subtle, but he is not malicious."
"God does not play dice with the world."
"Science without religion is lame, religion without science is blind" (Pais 1982, p. 319).

Einstein also had many insightful things to say about scientific discovery.

"Do not pride yourself on the few great men who, over the centuries, have been born on your earth through no merit of yours. Reflect, rather, on how you treated them at the time and how you have followed their teachings."
"Innovation is not the product of logical thought, even though the final product is tied to a logical structure" (Pais 1982, p. 131).
"Nature shows us only the tail of the lion. But I do not doubt that the lion belongs to it even though he cannot at once reveal himself because of his enormous size" (Pais 1982, p. 235).
"Above stands the marble smile of implacable Nature which has endowed us more with longing than with intellectual capacity" (Pais 1982, p. 343).
"One has been endowed with just enough intelligence to be able to see clearly how utterly inadequate that intelligence is when confronted with what exists. If such humility could be conveyed to everybody, the world of human activities would be more appealing."
"Politics is a pendulum whose swings between anarchy and tyranny are fueled by perennially rejuvenated illusions."
"It is a mistake often made in this country to measure things by the amount of money they cost."
"Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."
"The more success the quantum theory has, the sillier it looks. How nonphysicists would scoff if they were able to follow the odd course of developments!" (Pais 1982, p. 399).
"A practical profession is a salvation for a man of my type; an academic career compels a young man to scientific production, and only strong characters can resist the temptation of superficial analysis."

Riemann, Bernhard (1826-1866)

German mathematician who studied mathematics under Gauss and physics under Wilhelm Weber. Riemann did important work in geometry, Eric Weisstein's World of Math complex analysis, Eric Weisstein's World of Math and mathematical physics. He was also a friend of Dedekind, who was later Riemann's biographer. In his thesis, Riemann urged a global view of geometry Eric Weisstein's World of Math as a study of manifolds Eric Weisstein's World of Math of any number of dimensions Eric Weisstein's World of Math in any kind of space. He defined space by a metric Eric Weisstein's World of Math

Riemann's work laid the foundations on which general relativity Eric Weisstein's World of Physics was built. He investigated the Riemann zeta function, Eric Weisstein's World of Math about which he stated the famous (and still not completely proven) Riemann hypothesis. Eric Weisstein's World of Math He also refined the definition of the integral, Eric Weisstein's World of Math invented Riemann surfaces, Eric Weisstein's World of Math and invented a function which was discontinuous at an infinite number of points in an interval, but the integral Eric Weisstein's World of Math of which still existed (Segal 1978). In Über eine Frage der Wärmeleitung, Riemann developed the theory of quadratic forms. Eric Weisstein's World of Math

Riemann maintained that the transmission of electricity Eric Weisstein's World of Physics not was instantaneous, but occurred through the luminiferous ether Eric Weisstein's World of Physics at the speed of light. Eric Weisstein's World of Physics

Riemann died of tuberculosis at the tragically early age 39.

Pythagoras of Samos (ca. 560-ca. 480 BC)

Greek philosopher and mathematician who founded the mystic Pythagorean cult. The cult he founded was devoted to the study of numbers, which the Pythagoreans saw as concrete, material objects. They studied figurate numbers, Eric Weisstein's World of Math defining them as triangular numbers, Eric Weisstein's World of Math pentagonal numbers, Eric Weisstein's World of Math hexagonal numbers, Eric Weisstein's World of Math etc., based on the patterns that numbers of regularly spaced dots formed (Boyer 1968, p. 59-61). Pythagoras's biographer Proclus ascribed two specific mathematical discoveries to Pythagoras: construction of the regular solids (known today as the Platonic solids Eric Weisstein's World of Math), and the theory of proportionals.

Pythagoras also investigated the ratios of lengths corresponding to musical harmonies, undertook studies in number theory, and developed methods of geometric proof. Among the results attributed to Pythagoras and his followers is the proof that the number (Pythagoras's constant Eric Weisstein's World of Math) is irrational, Eric Weisstein's World of Math usually attributed to Hippasus.

Pythagoras developed a modern theory of vision much simpler than that of Plato. This theory maintained that light is emitted from luminous bodies, can suffer reflections, and causes the sensation of sight when it enters the eyes. He was the first Greek to realize that the morning star and evening star were both the planet Venus. Eric Weisstein's World of Astronomy Pythagoras postulated that the Earth Eric Weisstein's World of Astronomy was spherical, and added more crystalline spheres to Anaximander's model, one for each planet, Eric Weisstein's World of Astronomy to account for the motions of the various planets.

In keeping with the assumed magical properties of the number ten, some of the Pythagoreans, led by Philolaus, added a tenth "wanderer" and proposed that there existed a counter-earth which, together with the earth and other "planets," orbited a central fire. Pythagoras believed that the planets produced sounds while tracing out their orbits, producing the "harmony of the spheres." While much of their studies were sheer mysticism, the Pythagoreans were the first to mathematicize the universe.

Fibonacci, Leonardo da Pisa (ca. 1170-ca. 1240)

Italian mathematician who was the first great Western mathematician after the decline of Greek science. The son of a merchant, Fibonacci drew the motivation to mathematical inquiry from his commercial trips to the the Orient. It was somewhere between Barbary (Maghreb) and Constantinople (now Istanbul) that he got acquainted with the Hindu-Arabic number system and discovered its enormous practical advantages compared to the Roman numerals, Eric Weisstein's World of Math which were still current in Western Europe.

Performing even the simplest arithmetical operations with a non-positional notation was a difficult endeavor: for this task the merchants were forced to resort to the abacus, a device where the numbers were represented by moving balls. Fibonacci exposed the new alternate computing method--based on written algorithms rather than on counting objects--in his Liber Abaci, first issued in 1202. The book began with a presentation of what he called the ten "Indian figures" (0, 1, 2, ..., 9). It was intended as an algebra manual for commercial use, and explained the arithmetical rules using numerical examples derived, for example, from measure and currency conversion, which were translated into proportions and solved by multiplication (rule of three). The so-called Fibonacci sequence Eric Weisstein's World of Math arose in this book from a concrete question concerning the growth of a rabbit population. Geometric progressions Eric Weisstein's World of Math also appeared in problems related to legacy and interest.

The treatise Practica Geometriae, published in 1225, is mainly inspired by Greek mathematics; it contains theorems from Euclid's Elements and also Heron's formula Eric Weisstein's World of Math for the area Eric Weisstein's World of Math of a triangle. Eric Weisstein's World of Math

Fibonacci distinguished himself in the mathematical competitions proposed at the court of Emperor Frederick II of Hohenstaufen, King of the Two Sicilies, who had his royal seat in Palermo. His striking ability in solving algebraic equations of higher degree clearly emerges from his works entitled Liber Quadratorum and Flos, both of which appeared in 1225. The first contains formulas and equations involving perfect squares, the second owes its fame to the irrational solution of a cubic equation, which Fibonacci determined with an accuracy of 10-9. Most of his solving techniques seem to be based on the algebraic works of al-Khwarizmi.

Fibonacci initiated the tradition of the maestri d'abaco, experts in practical algebra and arithmetic, who flourished in Italy during the 14th century, and can be considered as the forerunners of Cardano, Tartaglia, and Ferrari.

Newton, Isaac (1642-1727)

English physicist and mathematician who was born into a poor farming family. Luckily for humanity, Newton was not a good farmer, and was sent to Cambridge to study to become a preacher. At Cambridge, Newton studied mathematics, being especially strongly influenced by Euclid, although he was also influenced by Baconian and Cartesian philosophies. Newton was forced to leave Cambridge when it was closed because of the plague, and it was during this period that he made some of his most significant discoveries. With the reticence he was to show later in life, Newton did not, however, publish his results.

Newton suffered a mental breakdown in 1675 and was still recovering through 1679. In response to a letter from Hooke, he suggested that a particle, if released, would spiral in to the center of the Earth. Eric Weisstein's World of Astronomy Hooke wrote back, claiming that the path would not be a spiral, but an ellipse. Eric Weisstein's World of Math Newton, who hated being bested, then proceeded to work out the mathematics of orbits. Again, he did not publish his calculations. Newton then began devoting his efforts to theological speculation and put the calculations on elliptical motion aside, telling Halley he had lost them (Westfall 1993, p. 403). Halley, who had become interested in orbits, finally convinced Newton to expand and publish his calculations. Newton devoted the period from August 1684 to spring 1686 to this task, and the result became one of the most important and influential works on physics of all times, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) (1687), often shortened to Principia Mathematica or simply "the Principia."

In Book I of Principia, Newton opened with definitions and the three laws of motion now known as Newton's laws Eric Weisstein's World of Physics (laws of inertia, action and reaction, and acceleration proportional to force). Book II presented Newton's new scientific philosophy which came to replace Cartesianism. Finally, Book III consisted of applications of his dynamics, including an explanation for tides and a theory of lunar motion. To test his hypothesis of universal gravitation, Newton wrote Flamsteed to ask if Saturn Eric Weisstein's World of Astronomy had been observed to slow down upon passing Jupiter. Eric Weisstein's World of Astronomy The surprised Flamsteed replied that an effect had indeed been observed, and it was closely predicted by the calculations Newton had provided. Newton's equations were further confirmed by observing the shape of the Earth Eric Weisstein's World of Astronomy to be oblate spheroidal, Eric Weisstein's World of Math as Newton claimed it should be, rather than prolate spheroidal, Eric Weisstein's World of Math as claimed by the Cartesians. Newton's equations also described the motion of Moon Eric Weisstein's World of Astronomy by successive approximations, and correctly predicted the return of Halley's Comet. Newton also correctly formulated and solved the first ever problem in the calculus of variations Eric Weisstein's World of Math which involved finding the surface of revolution which would give minimum resistance to flow (assuming a specific drag law).

Newton invented a scientific method which was truly universal in its scope. Newton presented his methodology as a set of four rules for scientific reasoning. These rules were stated in the Principia and proposed that (1) we are to admit no more causes of natural things such as are both true and sufficient to explain their appearances, (2) the same natural effects must be assigned to the same causes, (3) qualities of bodies are to be esteemed as universal, and (4) propositions deduced from observation of phenomena should be viewed as accurate until other phenomena contradict them.

These four concise and universal rules for investigation were truly revolutionary. By their application, Newton formulated the universal laws of nature with which he was able to unravel virtually all the unsolved problems of his day. Newton went much further than outlining his rules for reasoning, however, actually describing how they might be applied to the solution of a given problem. The analytic method he invented far exceeded the more philosophical and less scientifically rigorous approaches of Aristotle and Aquinas. Newton refined Galileo's experimental method, creating the compositional method of experimentation still practiced today. In fact, the following description of the experimental method from Newton's Optics could easily be mistaken for a modern statement of current methods of investigation, if not for Newton's use of the words "natural philosophy" in place of the modern term "the physical sciences." Newton wrote, "As in mathematics, so in natural philosophy the investigation of difficult things by the method of analysis ought ever to precede the method of composition. This analysis consists of making experiments and observations, and in drawing general conclusions from them by induction...by this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general from effects to their causes, and from particular causes to more general ones till the argument end in the most general. This is the method of analysis: and the synthesis consists in assuming the causes discovered and established as principles, and by them explaining the phenomena preceding from them, and proving the explanations."

Newton formulated the classical theories of mechanics and optics and invented calculus Eric Weisstein's World of Math years before Leibniz. However, he did not publish his work on calculus Eric Weisstein's World of Math until afterward Leibniz had published his. This led to a bitter priority dispute between English and continental mathematicians which persisted for decades, to the detriment of all concerned. Newton discovered that the binomial theorem Eric Weisstein's World of Math was valid for fractional powers, but left it for Wallis to publish (which he did, with appropriate credit to Newton). Newton formulated a theory of sound, but derived a speed which did not agree with his experiments. The reason for the discrepancy was that the concept of adiabatic propagation did not yet exist, so Newton's answer was too low by a factor of , where is the ratio of heat capacities Eric Weisstein's World of Physics of air. Newton therefore fudged his theory until agreement was achieved (Engineering and Science, pp. 15-16).

In Optics (1704), whose publication Newton delayed until Hooke's death, Newton observed that white light could be separated by a prism Eric Weisstein's World of Physics into a spectrum of different colors, each characterized by a unique refractivity, and proposed the corpuscular theory of light. Newton's views on optics were born out of the original prism Eric Weisstein's World of Physics experiments he performed at Cambridge. In his "experimentum crucis" (crucial experiment), he found that the image produced by a prism Eric Weisstein's World of Physics was oval-shaped and not circular, as current theories of light would require. He observed a half-red, half-blue string through a prism, Eric Weisstein's World of Physics and found the ends to be disjointed. He also observed Newton's rings, Eric Weisstein's World of Physics which are actually a manifestation of the wave nature of light which Newton did not believe in. Newton believed that light must move faster in a medium when it is refracted Eric Weisstein's World of Physics towards the normal, in opposition to the result predicted by Huygens's wave theory.

Newton also formulated a system of chemistry in Query 31 at the end of Optics. In this corpuscular theory, "elements" consisted of different arrangements of atoms, and atoms consisted of small, hard, billiard ball-like particles. He explained chemical reactions in terms of the chemical affinities of the participating substances. Newton devoted a majority of his free time later in life (after 1678) to fruitless alchemical experiments.

Newton was extremely sensitive to criticism, and even ceased publishing until the death of his arch-rival Hooke. It was only through the prodding of Halley that Newton was persuaded at all to publish the Principia Mathematica. In the latter portion of his life, he devoted much of his time to alchemical researches and trying to date events in the Bible. After Newton's death, his burial place was moved. During the exhumation, it was discovered that Newton had massive amounts of mercury in his body, probably resulting from his alchemical pursuits. This would certainly explain Newton's eccentricity in late life. Newton was appointed Warden of the British Mint in 1695. Newton was knighted by Queen Anne. However, the act was "an honor bestowed not for his contributions to science, nor for his service at the Mint, but for the greater glory of party politics in the election of 1705" (Westfall 1993, p. 625).

Newton singlehandedly contributed more to the development of science than any other individual in history. He surpassed all the gains brought about by the great scientific minds of antiquity, producing a scheme of the universe which was more consistent, elegant, and intuitive than any proposed before. Newton stated explicit principles of scientific methods which applied universally to all branches of science. This was in sharp contradistinction to the earlier methodologies of Aristotle and Aquinas, which had outlined separate methods for different disciplines.

Although his methodology was strictly logical, Newton still believed deeply in the necessity of a God. His theological views are characterized by his belief that the beauty and regularity of the natural world could only "proceed from the counsel and dominion of an intelligent and powerful Being." He felt that "the Supreme God exists necessarily, and by the same necessity he exists always and everywhere." Newton believed that God periodically intervened to keep the universe going on track. He therefore denied the importance of Leibniz's vis viva as nothing more than an interesting quantity which remained constant in elastic collisions and therefore had no physical importance or meaning.

Although earlier philosophers such as Galileo and John Philoponus had used experimental procedures, Newton was the first to explicitly define and systematize their use. His methodology produced a neat balance between theoretical and experimental inquiry and between the mathematical and mechanical approaches. Newton mathematized all of the physical sciences, reducing their study to a rigorous, universal, and rational procedure which marked the ushering in of the Age of Reason. Thus, the basic principles of investigation set down by Newton have persisted virtually without alteration until modern times. In the years since Newton's death, they have borne fruit far exceeding anything even Newton could have imagined. They form the foundation on which the technological civilization of today rests. The principles expounded by Newton were even applied to the social sciences, influencing the economic theories of Adam Smith and the decision to make the United States legislature bicameral. These latter applications, however, pale in contrast to Newton's scientific contributions.

It is therefore no exaggeration to identify Newton as the single most important contributor to the development of modern science. The Latin inscription on Newton's tomb, despite its bombastic language, is thus fully justified in proclaiming, "Mortals! rejoice at so great an ornament to the human race!" Alexander Pope's couplet is also apropos: "Nature and Nature's laws lay hid in night; God said, Let Newton be! and all was light."

Several interesting Newton quotes are given by Misner et al. (1973, pp. 40-41).