Math Is Fun Forum

Oh! Thank you thickhead, that is correct. I made a logical error. Which means all my expectations were calculated to be 1/3 of the investment better than they in fact are. You would actually expect to lose 17/30, 8/15 and 11/30 of any amount invested in 30%, 40% or 90% profit ventures respectively. Much worse!

Hey (:

I'm going to set

and

So, for example, for the original problem:

For further example, if we were to order the terms the opposite way, we would get:

For the first paragraph (p,q,a/2 or a/2,q,p):

Or:

For the second paragraph (a/2,p,q or q,p,a/2):

Or:

which is a more difficult expression, but turns out to have a maximum of 54 when x is positive or negative a logarithmic expression near 0.974

#28. Ludwig Boltzmann

Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms (such as mass, charge, and structure) determine the physical properties of matter (such as viscosity, thermal conductivity, and diffusion).

Boltzmann's most important scientific contributions were in kinetic theory, including the Maxwell–Boltzmann distribution for molecular speeds in a gas. In addition, Maxwell–Boltzmann statistics and the Boltzmann distribution over energies remain the foundations of classical statistical mechanics. They are applicable to the many phenomena that do not require quantum statistics and provide a remarkable insight into the meaning of temperature.

Much of the physics establishment did not share his belief in the reality of atoms and molecules — a belief shared, however, by Maxwell in Scotland and Gibbs in the United States; and by most chemists since the discoveries of John Dalton in 1808. He had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908–1909), based on Einstein's theoretical studies of 1905, confirmed the values of Avogadro's number and Boltzmann's constant, and convinced the world that the tiny particles really exist.

To quote Planck, "The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases". This famous formula for entropy S is

where kB is Boltzmann's constant, and ln is the natural logarithm. W is Wahrscheinlichkeit, a German word meaning the probability of occurrence of a macrostate or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — number of (unobservable) "ways" in the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann's paradigm was an ideal gas of N identical particles, of which Ni are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

where i ranges over all possible molecular conditions. (! denotes factorial.) The "correction" in the denominator is because identical particles in the same condition are indistinguishable.

Boltzmann was also one of the founders of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.

The equation for S is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof — his second grave.

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Arithmetic series:

Geometric series:

#27. Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was an influential French scholar whose work was important to the development of mathematics, statistics, physics, and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.

Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.

Laplace became a count of the First French Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

Laplace's early published work in 1771 started with differential equations and finite differences but he was already starting to think about the mathematical and philosophical concepts of probability and statistics. However, before his election to the Académie in 1773, he had already drafted two papers that would establish his reputation. The first, Mémoire sur la probabilité des causes par les événements was ultimately published in 1774 while the second paper, published in 1776, further elaborated his statistical thinking and also began his systematic work on celestial mechanics and the stability of the Solar System. The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge." Laplace's work on probability and statistics is discussed below with his mature work on the analytic theory of probabilities.

Sir Isaac Newton had published his Philosophiae Naturalis Principia Mathematica in 1687 in which he gave a derivation of Kepler's laws, which describe the motion of the planets, from his laws of motion and his law of universal gravitation. However, though Newton had privately developed the methods of calculus, all his published work used cumbersome geometric reasoning, unsuitable to account for the more subtle higher-order effects of interactions between the planets. Newton himself had doubted the possibility of a mathematical solution to the whole, even concluding that periodic divine intervention was necessary to guarantee the stability of the Solar System. Dispensing with the hypothesis of divine intervention would be a major activity of Laplace's scientific life. It is now generally regarded that Laplace's methods on their own, though vital to the development of the theory, are not sufficiently precise to demonstrate the stability of the Solar System, and indeed, the Solar System is understood to be chaotic, although it happens to be fairly stable.

One particular problem from observational astronomy was the apparent instability whereby Jupiter's orbit appeared to be shrinking while that of Saturn was expanding. The problem had been tackled by Leonhard Euler in 1748 and Joseph Louis Lagrange in 1763 but without success. In 1776, Laplace published a memoir in which he first explored the possible influences of a purported luminiferous ether or of a law of gravitation that did not act instantaneously. He ultimately returned to an intellectual investment in Newtonian gravity. Euler and Lagrange had made a practical approximation by ignoring small terms in the equations of motion. Laplace noted that though the terms themselves were small, when integrated over time they could become important. Laplace carried his analysis into the higher-order terms, up to and including the cubic. Using this more exact analysis, Laplace concluded that any two planets and the sun must be in mutual equilibrium and thereby launched his work on the stability of the Solar System. Gerald James Whitrow described the achievement as "the most important advance in physical astronomy since Newton".

Laplace had a wide knowledge of all sciences and dominated all discussions in the Académie. Laplace seems to have regarded analysis merely as a means of attacking physical problems, though the ability with which he invented the necessary analysis is almost phenomenal. As long as his results were true he took but little trouble to explain the steps by which he arrived at them; he never studied elegance or symmetry in his processes, and it was sufficient for him if he could by any means solve the particular question he was discussing.

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#26. Gilbert N. Lewis

Gilbert Newton Lewis ForMemRS (October 23, 1875 – March 23, 1946) was an American physical chemist known for the discovery of the covalent bond and his concept of electron pairs; his Lewis dot structures and other contributions to valence bond theory have shaped modern theories of chemical bonding. Lewis successfully contributed to thermodynamics, photochemistry, and isotope separation, and is also known for his concept of acids and bases.

G. N. Lewis was born in 1875 in Weymouth, Massachusetts. After receiving his PhD in chemistry from Harvard University and studying abroad in Germany and the Philippines, Lewis moved to California to teach chemistry at the University of California, Berkeley. Several years later, he became the Dean of the college of Chemistry at Berkeley, where he spent the rest of his life. As a professor, he incorporated thermodynamic principles into the chemistry curriculum and reformed chemical thermodynamics in a mathematically rigorous manner accessible to ordinary chemists. He began measuring the free energy values related to several chemical processes, both organic and inorganic.

In 1916, he also proposed his theory of bonding and added information about electrons in the periodic table of the elements. In 1933, he started his research on isotope separation. Lewis worked with hydrogen and managed to purify a sample of heavy water. He then came up with his theory of acids and bases, and did work in photochemistry during the last years of his life. In 1926, Lewis coined the term "photon" for the smallest unit of radiant energy. He was a brother in Alpha Chi Sigma, the professional chemistry fraternity.

Though he was nominated 35 times, G. N. Lewis never won the Nobel Prize in Chemistry. On March 23, 1946, Lewis was found dead in his Berkeley laboratory where he had been working with hydrogen cyanide; many postulated that the cause of his death was suicide. After Lewis' death, his children followed their father's career in chemistry.

Most of Lewis’ lasting interests originated during his Harvard years. The most important was thermodynamics, a subject in which Richards was very active at that time. Although most of the important thermodynamic relations were known by 1895, they were seen as isolated equations, and had not yet been rationalized as a logical system, from which, given one relation, the rest could be derived. Moreover, these relations were inexact, applying only to ideal chemical systems. These were two outstanding problems of theoretical thermodynamics. In two long and ambitious theoretical papers in 1900 and 1901, Lewis tried to provide a solution. Lewis introduced the thermodynamic concept of activity and coined the term "fugacity". His new idea of fugacity, or "escaping tendency", was a function with the dimensions of pressure which expressed the tendency of a substance to pass from one chemical phase to another. Lewis believed that fugacity was the fundamental principle from which a system of real thermodynamic relations could be derived. This hope was not realized, though fugacity did find a lasting place in the description of real gases.

Lewis’ early papers also reveal an unusually advanced awareness of J. W. Gibbs’s and P. Duhem’s ideas of free energy and thermodynamic potential. These ideas were well known to physicists and mathematicians, but not to most practical chemists, who regarded them as abstruse and inapplicable to chemical systems. Most chemists relied on the familiar thermodynamics of heat (enthalpy) of Berthelot, Ostwald, and Van’t Hoff, and the calorimetric school. Heat of reaction is not, of course, a measure of the tendency of chemical changes to occur, and Lewis realized that only free energy and entropy could provide an exact chemical thermodynamics. He derived free energy from fugacity; he tried, without success, to obtain an exact expression for the entropy function, which in 1901 had not been defined at low temperatures. Richards too tried and failed, and not until Nernst succeeded in 1907 was it possible to calculate entropies unambiguously. Although Lewis’ fugacity-based system did not last, his early interest in free energy and entropy proved most fruitful, and much of his career was devoted to making these useful concepts accessible to practical chemists.

At Harvard, Lewis also wrote a theoretical paper on the thermodynamics of blackbody radiation in which he postulated that light has a pressure. He later revealed that he had been discouraged from pursuing this idea by his older, more conservative colleagues, who were unaware that W. Wien and others were successfully pursuing the same line of thought. Lewis’ paper remained unpublished; but his interest in radiation and quantum theory, and (later) in relativity, sprang from this early, aborted effort. From the start of his career, Lewis regarded himself as both chemist and physicist.

About 1902 Lewis started to use unpublished drawings of cubical atoms in his lecture notes, in which the corners of the cube represented possible electron positions. Lewis later cited these notes in his classic 1916 paper on chemical bonding, as being the first expression of his ideas.

A third major interest that originated during Lewis’ Harvard years was his valence theory. In 1902, while trying to explain the laws of valence to his students, Lewis conceived the idea that atoms were built up of a concentric series of cubes with electrons at each corner. This “cubic atom” explained the cycle of eight elements in the periodic table and was in accord with the widely accepted belief that chemical bonds were formed by transfer of electrons to give each atom a complete set of eight. This electrochemical theory of valence found its most elaborate expression in the work of Richard Abegg in 1904, but Lewis’ version of this theory was the only one to be embodied in a concrete atomic model. Again Lewis’ theory did not interest his Harvard mentors, who, like most American chemists of that time, had no taste for such speculation. Lewis did not publish his theory of the cubic atom, but in 1916 it became an important part of his theory of the shared electron pair bond.

In 1916, he published his classic paper on chemical bonding "The Atom and the Molecule" in which he formulated the idea of what would become known as the covalent bond, consisting of a shared pair of electrons, and he defined the term odd molecule (the modern term is free radical) when an electron is not shared. He included what became known as Lewis dot structures as well as the cubical atom model. These ideas on chemical bonding were expanded upon by Irving Langmuir and became the inspiration for the studies on the nature of the chemical bond by Linus Pauling.

In 1923, he formulated the electron-pair theory of acid–base reactions. In this theory of acids and bases, a "Lewis acid" is an electron-pair acceptor and a "Lewis base" is an electron-pair donor. This year he also published a monograph on his theories of the chemical bond

Based on work by J. Willard Gibbs, it was known that chemical reactions proceeded to an equilibrium determined by the free energy of the substances taking part. Lewis spent 25 years determining free energies of various substances. In 1923 he and Merle Randall published the results of this study, which helped formalize modern chemical thermodynamics.

#25. Hermann von Helmholtz

Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist who made significant contributions to several widely varied areas of modern science. In physiology and psychology, he is known for his mathematics of the eye, theories of vision, ideas on the visual perception of space, color vision research, and on the sensation of tone, perception of sound, and empiricism. In physics, he is known for his theories on the conservation of energy, work in electrodynamics, chemical thermodynamics, and on a mechanical foundation of thermodynamics. As a philosopher, he is known for his philosophy of science, ideas on the relation between the laws of perception and the laws of nature, the science of aesthetics, and ideas on the civilizing power of science. The largest German association of research institutions, the Helmholtz Association, is named after him.

His first important scientific achievement, an 1847 treatise on the conservation of energy, was written in the context of his medical studies and philosophical background. He discovered the principle of conservation of energy while studying muscle metabolism. He tried to demonstrate that no energy is lost in muscle movement, motivated by the implication that there were no vital forces necessary to move a muscle. This was a rejection of the speculative tradition of Naturphilosophie which was at that time a dominant philosophical paradigm in German physiology.

Drawing on the earlier work of Sadi Carnot, Émile Clapeyron and James Prescott Joule, he postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single force (energy in modern terms). He published his theories in his book Über die Erhaltung der Kraft (On the Conservation of Force, 1847). Whether or not Helmholtz knew of Julius Robert von Mayer's discovery of the law of conservation of energy in the beginning of the 1840s is a point of controversy. Helmholtz did not quote Mayer in his work and was accused by contemporaries of plagiarism.

In the 1850s and 60s, building on the publications of William Thomson, Helmholtz and William Rankine popularized the idea of the heat death of the universe.

In fluid dynamics, Helmholtz made several contributions, including Helmholtz's theorems for vortex dynamics in inviscid fluids.

The sensory physiology of Helmholtz was the basis of the work of Wilhelm Wundt, a student of Helmholtz, who is considered one of the founders of experimental psychology. He, more explicitly than Helmholtz, described his research as a form of empirical philosophy and as a study of the mind as something separate. Helmholtz had, in his early repudiation of Naturphilosophie, stressed the importance of materialism, and was focusing more on the unity of "mind" and body.

In 1849, while at Königsberg, Helmholtz measured the speed at which the signal is carried along a nerve fibre. At that time most people believed that nerve signals passed along nerves immeasurably fast. He used a recently dissected sciatic nerve of a frog and the calf muscle to which it attached. He used a galvanometer as a sensitive timing device, attaching a mirror to the needle to reflect a light beam across the room to a scale which gave much greater sensitivity. Helmholtz reported transmission speeds in the range of 24.6 - 38.4 meters per second.

In 1851, Helmholtz revolutionized the field of ophthalmology with the invention of the ophthalmoscope; an instrument used to examine the inside of the human eye. This made him world famous overnight. Helmholtz's interests at that time were increasingly focused on the physiology of the senses. His main publication, entitled Handbuch der Physiologischen Optik (Handbook of Physiological Optics or Treatise on Physiological Optics), provided empirical theories on depth perception, color vision, and motion perception, and became the fundamental reference work in his field during the second half of the nineteenth century. In the third and final volume, published in 1867, Helmholtz described the importance of unconscious inferences for perception. The Handbuch was first translated into English under the editorship of James P. C. Southall on behalf of the Optical Society of America in 1924-5. His theory of accommodation went unchallenged until the final decade of the 20th century.

Helmholtz continued to work for several decades on several editions of the handbook, frequently updating his work because of his dispute with Ewald Hering who held opposite views on spatial and color vision. This dispute divided the discipline of physiology during the second half of the 1800s.

#23. Rudolf Clausius

Rudolf Julius Emanuel Clausius (born Rudolf Gottlieb; 2 January 1822 – 24 August 1888), was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis. His most important paper, On the Moving Force of Heat, published in 1850, first stated the basic ideas of the second law of thermodynamics. In 1865 he introduced the concept of entropy. In 1870 he introduced the virial theorem which applied to heat.

Clausius's PhD thesis concerning the refraction of light proposed that we see a blue sky during the day, and various shades of red at sunrise and sunset (among other phenomena) due to reflection and refraction of light. Later, Lord Rayleigh would show that it was in fact due to the scattering of light, but regardless, Clausius used a far more mathematical approach than some have used.

His most famous paper, "Über die bewegende Kraft der Wärme" ("On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom") was published in 1850, and dealt with the mechanical theory of heat. In this paper, he showed that there was a contradiction between Carnot's principle and the concept of conservation of energy. Clausius restated the two laws of thermodynamics to overcome this contradiction (the third law was developed by Walther Nernst, during the years 1906–1912). This paper made him famous among scientists.

Clausius' most famous statement of the second law of thermodynamics was published in German in 1854, and in English in 1856: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time."

During 1857, Clausius contributed to the field of kinetic theory after refining August Krönig's very simple gas-kinetic model to include translational, rotational and vibrational molecular motions. In this same work he introduced the concept of 'Mean free path' of a particle.

Clausius deduced the Clausius–Clapeyron relation from thermodynamics. This relation, which is a way of characterizing the phase transition between two states of matter such as solid and liquid, had originally been developed in 1834 by Émile Clapeyron.

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#21. Josiah Willard Gibbs

Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period).

In 1863, Yale awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics from 1871 until his death. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics".

Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England and the great international impact of his ideas. Though his work was almost entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to Robert A. Millikan, in pure science Gibbs "did for statistical mechanics and for thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."

Together with James Clerk Maxwell and Ludwig Boltzmann, Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of ensembles of all possible physical states of a system composed of many particles. He introduced the concept of "phase of a mechanical system". He used the concept to define the microcanonical, canonical, and grand canonical ensembles, thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him.

According to Henri Poincaré, writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his Elementary Principles of Statistical Mechanics." Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century.

Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates, could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries. His resolution of the so-called "Gibbs paradox", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics.

#19. Thomas Jefferson

Thomas Jefferson (April 13 [O.S. April 2] 1743 – July 4, 1826) was an American Founding Father who was principal author of the Declaration of Independence (1776). He was elected the second Vice President of the United States (1797–1801), serving under John Adams and in 1800 was elected third President (1801–09). Jefferson was a proponent of democracy, republicanism, and individual rights, which motivated American colonists to break from Great Britain and form a new nation. He produced formative documents and decisions at both the state and national level.

Primarily of English ancestry, Jefferson was born and educated in Virginia. He graduated from the College of William & Mary and practiced law. During the American Revolution, he represented Virginia in the Continental Congress that adopted the Declaration, drafted the law for religious freedom as a Virginia legislator, and served as a wartime governor (1779–1781). He became the United States Minister to France in May 1785, and subsequently the nation's first Secretary of State in 1790–1793 under President George Washington. Jefferson and James Madison organized the Democratic-Republican Party to oppose the Federalist Party during the formation of the First Party System. With Madison, he anonymously wrote the Kentucky and Virginia Resolutions in 1798–1799, which sought to embolden states' rights in opposition to the national government by nullifying the Alien and Sedition Acts.

While President Jefferson pursued the nation's shipping and trade interests against Barbary pirates and aggressive British trade policies respectively he also organized the Louisiana Purchase almost doubling the country's territory. As a result of peace negotiations with France, his administration reduced military forces. He was reelected in 1804. Jefferson's second term was beset with difficulties at home, including the trial of former Vice President Aaron Burr. American foreign trade was diminished when Jefferson implemented the Embargo Act of 1807, responding to British threats to U.S. shipping. In 1803, Jefferson began a controversial process of Indian tribe removal to the newly organized Louisiana Territory, and, in 1807, signed the Act Prohibiting Importation of Slaves. Historians generally rank Jefferson as one of the greatest U.S. Presidents.

Jefferson mastered many disciplines which ranged from surveying and mathematics to horticulture and inventions. He was a proven architect in the classical tradition. Jefferson's keen interest in religion and philosophy earned him the presidency of the American Philosophical Society. He shunned organized religion, but was influenced by both Christianity and deism. Besides English, he was well versed in Latin, Greek, French, Italian, and Spanish. He founded the University of Virginia after retiring from public office. He was a skilled writer and correspondent. His only full-length book, Notes on the State of Virginia (1785), is considered the most important American book published before 1800.

Jefferson married Martha Wayles Skelton whose marriage produced six children, but only two daughters survived to adulthood. He owned several plantations and owned many slaves. Most historians believe that after the death of his wife in 1782, he had a relationship with his slave Sally Hemings and fathered at least some of her children. Jefferson died at his home in Charlottesville, Virginia, on July 4, the fiftieth anniversary of the adoption of the Declaration of Independence.

Jefferson was the primary author of the Declaration of Independence. At age 33 he was one of the youngest delegates to the Second Continental Congress beginning in 1775 at the outbreak of the American Revolutionary War where a formal declaration of independence from Britain was overwhelmingly favored. Jefferson chose his words for the Declaration in June 1775 shortly after the war had begun where the idea of Independence from Britain had long since become popular among the colonies. He was also inspired by the Enlightenment ideals of the sanctity of the individual as well as the writings of Locke and Montesquieu.

He sought out John Adams who, along with the latter's cousin Samuel, had emerged as a leader of the Congress. Jefferson and Adams established a permanent friendship which led to Jefferson's work on the Declaration of Independence. Adams supported Jefferson's appointment to the Committee of Five formed to write the Declaration in furtherance of the Lee Resolution passed by the Congress. After discussing the general outline of the document, the committee decided that Jefferson would write the first draft. The committee, including Jefferson particularly, initially thought Adams should write the document, but Adams persuaded the committee to choose Jefferson.

Consulting with other committee members over the next seventeen days, Jefferson drew on his own proposed draft of the Virginia Constitution, George Mason's draft of the Virginia Declaration of Rights, and other sources. The other committee members made some changes. A final draft was presented to the Congress on June 28, 1776.

The declaration was introduced on Friday, June 28, and congress began debate over its contents on Monday, July 1. resulting in the omission of a fourth of the text, including a passage critical of King George III and the slave trade. While Jefferson resented the changes, he did not speak publicly about the revisions. On July 4, 1776, the Congress ratified the Declaration, and delegates signed it on August 2 and in doing so were committing an act of treason against the Crown. Jefferson's preamble is regarded as an enduring statement of human rights, and the phrase "all men are created equal" has been called "one of the best-known sentences in the English language" containing "the most potent and consequential words in American history".

Jefferson was sworn in by Chief Justice John Marshall at the new Capitol in Washington, D.C. on March 4, 1801. In contrast to his predecessors, Jefferson exhibited a dislike of formal etiquette; he arrived alone on horseback without escort, dressed plainly and after dismounting, retired his own horse to the nearby stable. His inaugural address struck a note of reconciliation, declaring, "We have been called by different names brethren of the same principle. We are all Republicans, we are all Federalists." Ideologically Jefferson stressed "equal and exact justice to all men", minority rights, freedom of speech, religion and press. Jefferson said that a free and democratic government was "the strongest government on earth." Jefferson nominated moderate Republicans to his cabinet: James Madison as Secretary of State, Henry Dearborn as Secretary of War, Levi Lincoln as Attorney General, and Robert Smith as Secretary of Navy.

Upon assuming office, he first confronted an $83 million national debt. He began dismantling Hamilton's Federalist fiscal system with help from Secretary of Treasury Albert Gallatin. Jefferson's administration eliminated the whiskey excise and other taxes after closing "unnecessary offices" and cutting "useless establishments and expenses". They attempted to disassemble the national bank and its effect of increasing national debt, but were dissuaded by Gallatin. Jefferson shrank the Navy, deeming it unnecessary in peacetime. Instead he incorporated a fleet of inexpensive gunboats used only for defense with the idea that they would not provoke foreign hostilities. After two terms, he had lowered the national debt from $83 million to $57 million.

Jefferson pardoned several of those imprisoned under the Alien and Sedition Acts. Congressional Republicans repealed the Judiciary Act of 1801, which removed nearly all of Adams' 'midnight judges' from office. A subsequent appointment battle led to the Supreme Court's landmark decision in Marbury v. Madison, asserting judicial review over executive branch actions. Jefferson appointed three Supreme Court justices: William Johnson (1804), Henry Brockholst Livingston (1807), and Thomas Todd (1807).

Jefferson strongly felt the need for a national military university, producing an officer engineering corps for a national defense based on the advancement of the sciences, rather than having to rely on foreign sources for top grade engineers with questionable loyalty. He signed the Military Peace Establishment Act on March 16, 1802, thus founding the United States Military Academy at West Point. The Act documented in 29 sections a new set of laws and limits for the military. Jefferson was also hoping to bring reform to the Executive branch, replacing Federalists and active opponents throughout the officer corps to promote Republican values.