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Historians in the early 20th century such as J. Franklin Jameson examined the class composition of the Patriot cause, looking for evidence of a class war inside the revolution. More recent historians have largely abandoned that interpretation, emphasizing instead the high level of ideological unity. Both Loyalists and Patriots were a "mixed lot", but ideological demands always came first. The Patriots viewed independence as a means to gain freedom from British oppression and to reassert their basic rights. Most yeomen farmers, craftsmen, and small merchants joined the Patriot cause to demand more political equality. They were especially successful in Pennsylvania but less so in New England, where John Adams attacked Thomas Paine's Common Sense for the "absurd democratical notions" that it proposed.
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King George III The war became a personal issue for the king, fueled by his growing belief that British leniency would be taken as weakness by the Americans. He also sincerely believed that he was defending Britain's constitution against usurpers, rather than opposing patriots fighting for their natural rights.
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Although Prime Minister Lord North was not an ideal war leader, George III managed to give Parliament a sense of purpose to fight, and Lord North was able to keep his cabinet together. Lord North's cabinet ministers, the Earl of Sandwich, First Lord of the Admiralty, and Lord George Germain, Secretary of State for the Colonies, however, proved to lack leadership skills suited for their positions, which in turn, aided the American war effort.
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George III is often accused of obstinately trying to keep Great Britain at war with the revolutionaries in America, despite the opinions of his own ministers. In the words of the British historian George Otto Trevelyan, the King was determined "never to acknowledge the independence of the Americans, and to punish their contumacy by the indefinite prolongation of a war which promised to be eternal." The King wanted to "keep the rebels harassed, anxious, and poor, until the day when, by a natural and inevitable process, discontent and disappointment were converted into penitence and remorse". Later historians defend George by saying in the context of the times no king would willingly surrender such a large territory, and his conduct was far less ruthless than contemporary monarchs in Europe. After the surrender of a British army at Saratoga, both Parliament and the British people were in favor of the war; recruitment ran at high levels and although political opponents were vocal, they remained a small minority.
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With the setbacks in America, Lord North asked to transfer power to Lord Chatham, whom he thought more capable, but George refused to do so; he suggested instead that Chatham serve as a subordinate minister in North's administration, but Chatham refused. He died later in the same year. Lord North was allied to the "King's Friends" in Parliament and believed George III had the right to exercise powers. In early 1778, Britain's chief rival France signed a treaty of alliance with the United States, and the confrontation soon escalated from a "rebellion" to something that has been characterized as "world war". The French fleet was able to outrun the British naval blockade of the Mediterranean and sailed to North America. The conflict now affected North America, Europe and India. The United States and France were joined by Spain in 1779 and the Dutch Republic, while Britain had no major allies of its own, except for the Loyalist minority in America and German auxiliaries (i.e. Hessians). Lord Gower and Lord Weymouth both resigned from the government. Lord North again requested that he also be allowed to resign, but he stayed in office at George III's insistence. Opposition to the costly war was increasing, and in June 1780 contributed to disturbances in London known as the Gordon riots.
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As late as the Siege of Charleston in 1780, Loyalists could still believe in their eventual victory, as British troops inflicted defeats on the Continental forces at the Battle of Camden and the Battle of Guilford Court House. In late 1781, the news of Cornwallis's surrender at the siege of Yorktown reached London; Lord North's parliamentary support ebbed away and he resigned the following year. The King drafted an abdication notice, which was never delivered, finally accepted the defeat in North America, and authorized peace negotiations. The Treaties of Paris, by which Britain recognized the independence of the United States and returned Florida to Spain, were signed in 1782 and 1783 respectively. In early 1783, George III privately conceded "America is lost!" He reflected that the Northern colonies had developed into Britain's "successful rivals" in commercial trade and fishing.
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When John Adams was appointed American Minister to London in 1785, George had become resigned to the new relationship between his country and the former colonies. He told Adams, "I was the last to consent to the separation; but the separation having been made and having become inevitable, I have always said, as I say now, that I would be the first to meet the friendship of the United States as an independent power." Patriots
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Those who fought for independence were called "Revolutionaries" "Continentals", "Rebels", "Patriots", "Whigs", "Congress-men", or "Americans" during and after the war. They included a full range of social and economic classes but were unanimous regarding the need to defend the rights of Americans and uphold the principles of republicanism in rejecting monarchy and aristocracy, while emphasizing civic virtue by citizens. The signers of the Declaration of Independence were mostly - with definite exceptions - well-educated, of British stock, and of the Protestant faith. Newspapers were strongholds of patriotism (although there were a few Loyalist papers) and printed many pamphlets, announcements, patriotic letters, and pronouncements.
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According to historian Robert Calhoon, 40– to 45-percent of the white population in the Thirteen Colonies supported the Patriots' cause, 15– to 20-percent supported the Loyalists, and the remainder were neutral or kept a low profile. Mark Lender analyzes why ordinary people became insurgents against the British, even if they were unfamiliar with the ideological reasons behind the war. He concludes that such people held a sense of rights which the British were violating, rights that stressed local autonomy, fair dealing, and government by consent. They were highly sensitive to the issue of tyranny, which they saw manifested in the British response to the Boston Tea Party. The arrival in Boston of the British Army heightened their sense of violated rights, leading to rage and demands for revenge. They had faith that God was on their side.
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Thomas Paine published his pamphlet Common Sense in January 1776, after the Revolution had started. It was widely distributed and often read aloud in taverns, contributing significantly to concurrently spreading the ideas of republicanism and liberalism, bolstering enthusiasm for separation from Great Britain and encouraging recruitment for the Continental Army. Paine presented the Revolution as the solution for Americans alarmed by the threat of tyranny. Loyalists
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The consensus of scholars is that about 15– to 20-percent of the white population remained loyal to the British Crown. Those who actively supported the king were known at the time as "Loyalists", "Tories", or "King's men". The Loyalists never controlled territory unless the British Army occupied it. They were typically older, less willing to break with old loyalties, and often connected to the Church of England; they included many established merchants with strong business connections throughout the Empire, as well as royal officials such as Thomas Hutchinson of Boston. There were 500 to 1,000 Black Loyalists, enslaved African-Americans who escaped to British lines and supported Britain's cause via several means. Many of them succumbed to various diseases, but the survivors were evacuated by the British to their remaining colonies in North America.
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The revolution could divide families, such as William Franklin, son of Benjamin Franklin and royal governor of the Province of New Jersey who remained loyal to the Crown throughout the war. He and his father never spoke again. Recent immigrants who had not been fully Americanized were also inclined to support the King, such as Flora MacDonald, a Scottish settler in the backcountry.
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After the war, the most of the approximately 500,000 Loyalists remained in America and resumed normal lives. Some became prominent American leaders, such as Samuel Seabury. Approximately 46,000 Loyalists relocated to Canada; others moved to Britain (7,000), Florida, or the West Indies (9,000). The exiles represented approximately two percent of the total population of the colonies. Nearly all black loyalists left for Nova Scotia, Florida, or England, where they could remain free. Loyalists who left the South in 1783 took thousands of their slaves with them as they fled to the British West Indies. Neutrals
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A minority of uncertain size tried to stay neutral in the war. Most kept a low profile, but the Quakers were the most important group to speak out for neutrality, especially in Pennsylvania. The Quakers continued to do business with the British even after the war began, and they were accused of supporting British rule, "contrivers and authors of seditious publications" critical of the revolutionary cause. Most Quakers remained neutral, although a sizeable number nevertheless participated to some degree. Role of women
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Women contributed to the American Revolution in many ways and were involved on both sides. Formal politics did not include women, but ordinary domestic behaviors became charged with political significance as Patriot women confronted a war which permeated all aspects of political, civil, and domestic life. They participated by boycotting British goods, spying on the British, following armies as they marched, washing, cooking, and mending for soldiers, delivering secret messages, and even fighting disguised as men in a few cases, such as Deborah Samson. Mercy Otis Warren held meetings in her house and cleverly attacked Loyalists with her creative plays and histories. Many women also acted as nurses and helpers, tending to the soldiers' wounds and buying and selling goods for them. Some of these camp followers even participated in combat, such as Madam John Turchin who led her husband's regiment into battle. Above all, women continued the agricultural work at home to feed their families and the armies. They maintained their families during their husbands' absences and sometimes after their deaths.
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American women were integral to the success of the boycott of British goods, as the boycotted items were largely household articles such as tea and cloth. Women had to return to knitting goods and to spinning and weaving their own cloth—skills that had fallen into disuse. In 1769, the women of Boston produced 40,000 skeins of yarn, and 180 women in Middletown, Massachusetts wove of cloth. Many women gathered food, money, clothes, and other supplies during the war to help the soldiers. A woman's loyalty to her husband could become an open political act, especially for women in America committed to men who remained loyal to the King. Legal divorce, usually rare, was granted to Patriot women whose husbands supported the King.
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Other participants France and Spain In early 1776, France set up a major program of aid to the Americans, and the Spanish secretly added funds. Each country spent one million "livres tournaises" to buy munitions. A dummy corporation run by Pierre Beaumarchais concealed their activities. American Patriots obtained some munitions from the Dutch Republic as well, through the French and Spanish ports in the West Indies. Heavy expenditures and a weak taxation system pushed France toward bankruptcy.
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In 1777, Charles François Adrien le Paulmier, Chevalier d’Annemours, acting as a secret agent for France, made sure General George Washington was privy to his mission. He followed Congress around for the next two years, reporting what he observed back to France.
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Spain did not officially recognize the United States, but it was a French ally and it separately declared war on Britain on June 21, 1779. Bernardo de Gálvez y Madrid, general of the Spanish forces in New Spain, also served as governor of Louisiana. He led an expedition of colonial troops to capture Florida from the British and to keep open a vital conduit for supplies. Germans
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Ethnic Germans served on both sides of the American Revolutionary War. As George III was also the Elector of Hanover, many supported the Loyalist cause and served as allies of the Kingdom of Great Britain; most notably rented auxiliary troops from German states such as the Landgraviate of Hessen-Kassel.
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American Patriots tended to represent such troops as mercenaries in propaganda against the British Crown. Even American historians followed suit, in spite of Colonial-era jurists drawing a distinction between auxiliaries and mercenaries, with auxiliaries serving their prince when sent to the aid of another prince, and mercenaries serving a foreign prince as individuals. By this distinction the troops which served in the American Revolution were auxiliaries.
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Other German individuals came to assist the American revolutionaries, most notably Friedrich Wilhelm von Steuben, who served as a general in the Continental Army and is credited with professionalizing that force, but most who served were already colonists. Von Steuben's native Prussia joined the League of Armed Neutrality, and King Frederick II of Prussia was well appreciated in the United States for his support early in the war. He expressed interest in opening trade with the United States and bypassing English ports, and allowed an American agent to buy arms in Prussia. Frederick predicted American success, and promised to recognize the United States and American diplomats once France did the same. Prussia also interfered in the recruiting efforts of Russia and neighboring German states when they raised armies to send to the Americas, and Frederick II forbade enlistment for the American war within Prussia. All Prussian roads were denied to troops from Anhalt-Zerbst, which delayed reinforcements that Howe had hoped to receive during the winter of 1777–1778.
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However, when the War of the Bavarian Succession erupted, Frederick II became much more cautious with Prussian/British relations. US ships were denied access to Prussian ports, and Frederick refused to officially recognize the United States until they had signed the Treaty of Paris. Even after the war, Frederick II predicted that the United States was too large to operate as a republic, and that it would soon rejoin the British Empire with representatives in Parliament. Native Americans
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Most indigenous people rejected pleas that they remain neutral and instead supported the British Crown. The great majority of the 200,000 indigenous people east of the Mississippi distrusted the colonists and supported the British cause, hoping to forestall continued expansion of settlement into their territories. Those tribes closely involved in trade tended to side with the Patriots, although political factors were important as well. Some indigenous people tried to remain neutral, seeing little value in joining what they perceived to be a "white man's war", and fearing reprisals from whichever side they opposed.
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The great majority of indigenous people did not participate directly in the war, with the notable exceptions of warriors and bands associated with four of the Iroquois tribes in New York and Pennsylvania which allied with the British, and the Oneida and Tuscarora tribes among the Iroquois of central and western New York who supported the American cause. The British did have other allies, particularly in the regions of southwest Quebec on the Patriot's frontier. The British provided arms to indigenous people who were led by Loyalists in war parties to raid frontier settlements from the Carolinas to New York. These war parties managed to kill many settlers on the frontier, especially in Pennsylvania and New York's Mohawk Valley.
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In 1776, Cherokee war parties attacked American Colonists all along the southern Quebec frontier of the uplands throughout the Washington District, North Carolina (now Tennessee) and the Kentucky wilderness area. The Chickamauga Cherokee under Dragging Canoe allied themselves closely with the British, and fought on for an additional decade after the Treaty of Paris was signed. They would launch raids with roughly 200 warriors, as seen in the Cherokee–American wars; they could not mobilize enough forces to invade settler areas without the help of allies, most often the Creek.
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Joseph Brant (also Thayendanegea) of the powerful Mohawk tribe in New York was the most prominent indigenous leader against the Patriot forces. In 1778 and 1780, he led 300 Iroquois warriors and 100 white Loyalists in multiple attacks on small frontier settlements in New York and Pennsylvania, killing many settlers and destroying villages, crops, and stores.
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In 1779, the Continental Army forced the hostile indigenous people out of upstate New York when Washington sent an army under John Sullivan which destroyed 40 evacuated Iroquois villages in central and western New York. Sullivan systematically burned the empty villages and destroyed about 160,000 bushels of corn that composed the winter food supply. The Battle of Newtown proved decisive, as the Patriots had an advantage of three-to-one, and it ended significant resistance; there was little combat otherwise. Facing starvation and homeless for the winter, the Iroquois fled to Canada. The British resettled them in Ontario, providing land grants as compensation for some of their losses.
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At the peace conference following the war, the British ceded lands which they did not really control, and which they did not consult about with their indigenous allies during the treaty negotiations. They transferred control to the United States of all the land south of the Great Lakes east of the Mississippi and north of Florida. Calloway concludes: The British did not give up their forts until 1796 in the eastern Midwest, stretching from Ohio to Wisconsin; they kept alive the dream of forming an allied indigenous nation there, which they referred to an "Indian barrier state". That goal was one of the causes of the War of 1812. Black Americans
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Free blacks in the New England Colonies and Middle Colonies in the North as well as Southern Colonies fought on both sides of the War, but the majority fought for the Patriots. Gary Nash reports that there were about 9,000 black veteran Patriots, counting the Continental Army and Navy, state militia units, privateers, wagoneers in the Army, servants to officers, and spies. Ray Raphael notes that thousands did join the Loyalist cause, but "a far larger number, free as well as slave, tried to further their interests by siding with the patriots." Crispus Attucks was one of the five people killed in the Boston Massacre in 1770 and is considered the first American casualty for the cause of independence.
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The effects of the War were more dramatic in the South. Tens of thousands of slaves escaped to British lines throughout the South, causing dramatic losses to slaveholders and disrupting cultivation and harvesting of crops. For instance, South Carolina was estimated to have lost about 25,000 slaves to flight, migration, or death which amounted to a third of its slave population. From 1770 to 1790, the black proportion of the population (mostly slaves) in South Carolina dropped from 60.5 percent to 43.8 percent, and from 45.2 percent to 36.1 percent in Georgia.
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During the War, the British commanders attempted to weaken the Patriots by issuing proclamations of freedom to their slaves. In the November 1775 document known as Dunmore's Proclamation, royal Virginia, governor Lord Dunmore recruited black men into the British forces with the promise of freedom, protection for their families, and land grants. Some men responded and briefly formed the British Ethiopian Regiment. Historian David Brion Davis explains the difficulties with a policy of wholesale arming of the slaves:
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Davis underscores the British dilemma: "Britain, when confronted by the rebellious American colonists, hoped to exploit their fear of slave revolts while also reassuring the large number of slave-holding Loyalists and wealthy Caribbean planters and merchants that their slave property would be secure". The Americans, however, accused the British of encouraging slave revolts, with the issue becoming one of the 27 colonial grievances.
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The existence of slavery in the American colonies had attracted criticism from both sides of the Atlantic as many could not reconcile the existence of the institution with the egalitarian ideals espoused by leaders of the Revolution. British writer Samuel Johnson wrote "how is it we hear the loudest yelps for liberty among the drivers of the Negroes?" in a text opposing the grievances of the colonists. Referring to this contradiction, English abolitionist Thomas Day wrote in a 1776 letter that "if there be an object truly ridiculous in nature, it is an American patriot, signing resolutions of independency with the one hand, and with the other brandishing a whip over his affrighted slaves". African-American writer Lemuel Haynes expressed similar viewpoints in his essay Liberty Further Extended where he wrote that "Liberty is Equally as pre[c]ious to a Black man, as it is to a white one". Thomas Jefferson unsuccessfully attempted to include a section in the Declaration of Independence which asserted that King George III had "forced" the slave trade onto the colonies. Despite the turmoil of the period, African-Americans contributed to the foundation of an American national identity during the Revolution. Phyllis Wheatley, an African-American poet, popularized the image of Columbia to represent America. She came to public attention when her Poems on Various Subjects, Religious and Moral appeared in 1773, and received praise from George Washington.
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The 1779 Philipsburg Proclamation expanded the promise of freedom for black men who enlisted in the British military to all the colonies in rebellion. British forces gave transportation to 10,000 slaves when they evacuated Savannah and Charleston, carrying through on their promise. They evacuated and resettled more than 3,000 Black Loyalists from New York to Nova Scotia, Upper Canada, and Lower Canada. Others sailed with the British to England or were resettled as freedmen in the West Indies of the Caribbean. But slaves carried to the Caribbean under control of Loyalist masters generally remained slaves until British abolition of slavery in its colonies in 1833–38. More than 1,200 of the Black Loyalists of Nova Scotia later resettled in the British colony of Sierra Leone, where they became leaders of the Krio ethnic group of Freetown and the later national government. Many of their descendants still live in Sierra Leone, as well as other African countries.
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Effects of the Revolution
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After the Revolution, genuinely democratic politics became possible in the former American colonies. The rights of the people were incorporated into state constitutions. Concepts of liberty, individual rights, equality among men and hostility toward corruption became incorporated as core values of liberal republicanism. The greatest challenge to the old order in Europe was the challenge to inherited political power and the democratic idea that government rests on the consent of the governed. The example of the first successful revolution against a European empire, and the first successful establishment of a republican form of democratically elected government, provided a model for many other colonial peoples who realized that they too could break away and become self-governing nations with directly elected representative government.
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Interpretations
Interpretations vary concerning the effect of the Revolution. Historians such as Bernard Bailyn, Gordon Wood, and Edmund Morgan view it as a unique and radical event which produced deep changes and had a profound effect on world affairs, such as an increasing belief in the principles of the Enlightenment. These were demonstrated by a leadership and government that espoused protection of natural rights, and a system of laws chosen by the people. John Murrin, by contrast, argues that the definition of "the people" at that time was mostly restricted to free men who passed a property qualification. This view argues that any significant gain of the revolution was irrelevant in the short term to women, black Americans and slaves, poor white men, youth, and Native Americans.
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Gordon Wood states:
The American Revolution was integral to the changes occurring in American society, politics and culture .... These changes were radical, and they were extensive .... The Revolution not only radically changed the personal and social relationships of people, including the position of women, but also destroyed aristocracy as it'd been understood in the Western world for at least two millennia.
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Edmund Morgan has argued that, in terms of long-term impact on American society and values:
The Revolution did revolutionize social relations. It did displace the deference, the patronage, the social divisions that had determined the way people viewed one another for centuries and still view one another in much of the world. It did give to ordinary people a pride and power, not to say an arrogance, that have continued to shock visitors from less favored lands. It may have left standing a host of inequalities that have troubled us ever since. But it generated the egalitarian view of human society that makes them troubling and makes our world so different from the one in which the revolutionists had grown up.
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Inspiring other independence movements and revolutions The first shot of the American Revolution at the Battle of Lexington and Concord is referred to as the "shot heard ‘round the world" due to its historical and global significance. The ensuing Revolutionary War not only established the United States as the first modern constitutional republic, but marked the transition from an age of monarchy to a new age of freedom by inspiring similar movements worldwide. The American Revolution was the first of the "Atlantic Revolutions": the French Revolution, the Haitian Revolution, and the Latin American wars of independence. Aftershocks contributed to rebellions in Ireland, the Polish–Lithuanian Commonwealth, and the Netherlands.
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The U.S. Constitution, drafted shortly after independence, remains the world's oldest written constitution, and has been emulated by other countries, in some cases verbatim. Some historians and scholars aruge that the subsequent wave of independence and revolutionary movements has contributed to the continued expansion of democratic government; 144 countries, representing two-third of the world's population, are full or partially democracies of same form.
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The Dutch Republic, also at war with Britain, was the next country after France to sign a treaty with the United States, on October 8, 1782. On April 3, 1783, Ambassador Extraordinary Gustaf Philip Creutz, representing King Gustav III of Sweden, and Benjamin Franklin, signed a Treaty of Amity and Commerce with the U.S.
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The Revolution had a strong, immediate influence in Great Britain, Ireland, the Netherlands, and France. Many British and Irish Whigs in Parliament spoke glowingly in favor of the American cause. In Ireland, the Protestant minority who controlled Ireland demanded self-rule. Under the leadership of Henry Grattan, the Irish Patriot Party forced the reversal of mercantilist prohibitions against trade with other British colonies. The King and his cabinet in London could not risk another rebellion on the American model, and so made a series of concessions to the Patriot faction in Dublin. Armed volunteer units of the Protestant Ascendancy were set up ostensibly to protect against an invasion from France. As had been in colonial America, so too in Ireland now the King no longer had a monopoly of lethal force.
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The Revolution, along with the Dutch Revolt (end of the 16th century) and the 17th century English Civil War, was among the examples of overthrowing an old regime for many Europeans who later were active during the era of the French Revolution, such as the Marquis de Lafayette. The American Declaration of Independence influenced the French Declaration of the Rights of Man and of the Citizen of 1789. The spirit of the Declaration of Independence led to laws ending slavery in all the Northern states and the Northwest Territory, with New Jersey the last in 1804. States such as New Jersey and New York adopted gradual emancipation, which kept some people as slaves for more than two decades longer.
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Status of African Americans
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During the revolution, the contradiction between the Patriots' professed ideals of liberty and the institution of slavery generated increased scrutiny of the latter. As early as 1764, the Boston Patriot leader James Otis, Jr. declared that all men, "white or black", were "by the law of nature" born free. Anti-slavery calls became more common in the early 1770s. In 1773, Benjamin Rush, the future signer of the Declaration of Independence, called on "advocates for American liberty" to oppose slavery, writing, "The plant of liberty is of so tender a nature that it cannot thrive long in the neighborhood of slavery.". The contradiction between calls for liberty and the continued existence of slavery also opened up the Patriots to charges of hypocrisy. In 1775, the English Tory writer Samuel Johnson asked, "How is it that we hear the loudest yelps for liberty among the drivers of negroes?"
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In the late 1760s and early 1770s, a number of colonies, including Massachusetts and Virginia, attempted to restrict the slave trade, but were prevented from doing so by royally appointed governors. In 1774, as part of a broader non-importation movement aimed at Britain, the Continental Congress called on all the colonies to ban the importation of slaves, and the colonies passed acts doing so. In 1775, the Quakers founded first antislavery society in the Western world, the Pennsylvania Abolition Society.
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In the first two decades after the American Revolution, state legislatures and individuals took actions to free slaves, in part based on revolutionary ideals. Northern states passed new constitutions that contained language about equal rights or specifically abolished slavery; some states, such as New York and New Jersey, where slavery was more widespread, passed laws by the end of the 18th century to abolish slavery by a gradual method. By 1804, all the northern states had passed laws outlawing slavery, either immediately or over time. In New York, the last slaves were freed in 1827. Indentured servitude (temporary slavery), which had been widespread in the colonies (Half the population of Philadelphia had once been bonded servants) dropped dramatically, and disappeared by 1800.
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No southern state abolished slavery, but for a period individual owners could free their slaves by personal decision, often providing for manumission in wills but sometimes filing deeds or court papers to free individuals. Numerous slaveholders who freed their slaves cited revolutionary ideals in their documents; others freed slaves as a reward for service. Records also suggest that some slaveholders were freeing their own mixed-race children, born into slavery to slave mothers. The number of free blacks as a proportion of the black population in the upper South increased from less than 1 percent to nearly 10 percent between 1790 and 1810 as a result of these actions. Nevertheless, the slavery continued in the South, transforming slavery into a "peculiar institution", and setting the stage for future sectional conflict between North and South over the issue.
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Thousands of free Blacks in the northern states fought in the state militias and Continental Army. In the south, both sides offered freedom to slaves who would perform military service. Roughly 20,000 slaves fought in the American Revolution. Status of American women The democratic ideals of the Revolution inspired changes in the roles of women.
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The concept of republican motherhood was inspired by this period and reflects the importance of revolutionary republicanism as the dominant American ideology. It assumed that a successful republic rested upon the virtue of its citizens. Women were considered to have the essential role of instilling their children with values conducive to a healthy republic. During this period, the wife's relationship with her husband also became more liberal, as love and affection instead of obedience and subservience began to characterize the ideal marital relationship. In addition, many women contributed to the war effort through fundraising and running family businesses without their husbands.
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The traditional constraints gave way to more liberal conditions for women. Patriarchy faded as an ideal; young people had more freedom to choose their spouses and more often used birth control to regulate the size of their families. Society emphasized the role of mothers in child rearing, especially the patriotic goal of raising republican children rather than those locked into aristocratic value systems. There was more permissiveness in child-rearing. Patriot women married to Loyalists who left the state could get a divorce and obtain control of the ex-husband's property.
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Whatever gains they had made, however, women still found themselves subordinated, legally and socially, to their husbands, disfranchised and usually with only the role of mother open to them. But, some women earned livelihoods as midwives and in other roles in the community not originally recognized as significant by men. Abigail Adams expressed to her husband, the president, the desire of women to have a place in the new republic: "I desire you would remember the Ladies, and be more generous and favourable to them than your ancestors. Do not put such unlimited power into the hands of the Husbands."
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The Revolution sparked a discussion on the rights of woman and an environment favorable to women's participation in politics. Briefly the possibilities for women's rights were highly favorable, but a backlash led to a greater rigidity that excluded women from politics. For more than thirty years, however, the 1776 New Jersey State Constitution gave the vote to "all inhabitants" who had a certain level of wealth, including unmarried women and blacks (not married women because they could not own property separately from their husbands), until in 1807, when that state legislature passed a bill interpreting the constitution to mean universal white male suffrage, excluding paupers.
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Loyalist expatriation
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Tens of thousands of Loyalists left the United States following the war, and Maya Jasanoff estimates as many as 70,000. Some migrated to Britain, but the great majority received land and subsidies for resettlement in British colonies in North America, especially Quebec (concentrating in the Eastern Townships), Prince Edward Island, and Nova Scotia. Britain created the colonies of Upper Canada (Ontario) and New Brunswick expressly for their benefit, and the Crown awarded land to Loyalists as compensation for losses in the United States. Nevertheless, approximately eighty-five percent of the Loyalists stayed in the United States as American citizens, and some of the exiles later returned to the U.S. Patrick Henry spoke of the issue of allowing Loyalists to return as such: "Shall we, who have laid the proud British lion at our feet, be frightened of its whelps?" His actions helped secure return of the Loyalists to American soil.
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Commemorations The American Revolution has a central place in the American memory as the story of the nation's founding. It is covered in the schools, memorialized by a national holiday, and commemorated in innumerable monuments. George Washington's estate at Mount Vernon was one of the first national pilgrimages for tourists and attracted 10,000 visitors a year by the 1850s.
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The Revolution became a matter of contention in the 1850s in the debates leading to the American Civil War (1861–1865), as spokesmen of both the Northern United States and the Southern United States claimed that their region was the true custodian of the legacy of 1776. The United States Bicentennial in 1976 came a year after the American withdrawal from the Vietnam War, and speakers stressed the themes of renewal and rebirth based on a restoration of traditional values.
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Today, more than 100 battlefields and historic sites of the American Revolution are protected and maintained by the government. The National Park Service alone owns and maintains more than 50 battlefield parks and many other sites such as Independence Hall that are related to the Revolution, as well as the residences, workplaces and meeting places of many Founders and other important figures. The private American Battlefield Trust uses government grants and other funds to preserve almost 700 acres of battlefield land in six states, and the ambitious private recreation/restoration/preservation/interpretation of over 300 acres of pre-1790 Colonial Williamsburg was created in the first half of the 20th century for public visitation.
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See also
Founding Fathers of the United States
List of George Washington articles
List of plays and films about the American Revolution
List of television series and miniseries about the American Revolution
Museum of the American Revolution
Timeline of the American Revolution Notes References General sources Bibliography
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Reference works
Barnes, Ian, and Charles Royster. The Historical Atlas of the American Revolution (2000), maps and commentary excerpt and text search
Cappon, Lester J. Atlas of Early American History: The Revolutionary Era, 1760–1790 (1976)
Fremont-Barnes, Gregory, and Richard A. Ryerson, eds. The Encyclopedia of the American Revolutionary War: A Political, Social, and Military History (5 vol. 2006) 1000 entries by 150 experts, covering all topics
Gray, Edward G., and Jane Kamensky, eds. The Oxford Handbook of the American Revolution (2013) 672 pp; 33 essays by scholars
Greene, Jack P. and J. R. Pole, eds. A Companion to the American Revolution (2004), 777 pp – an expanded edition of Greene and Pole, eds. The Blackwell Encyclopedia of the American Revolution (1994); comprehensive coverage of political and social themes and international dimension; thin on military
Herrera, Ricardo A. "American War of Independence" Oxford Bibliographies (2017) annotated guide to major scholarly books and articles
online
Kennedy, Frances H. The American Revolution: A Historical Guidebook (2014) A guide to 150 famous historical sites.
Purcell, L. Edward. Who Was Who in the American Revolution (1993); 1500 short biographies
Resch, John P., ed. Americans at War: Society, Culture and the Homefront vol 1 (2005), articles by scholars
Selesky, Harold E. ed., Encyclopedia of the American Revolution (3 vol. Gale, 2006)
Symonds, Craig L. and William J. Clipson. A Battlefield Atlas of the American Revolution (1986) new diagrams of each battle
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Surveys of the era
Alden, John R. A history of the American Revolution (1966) 644pp online free to borrow, A scholarly general survey
Allison, Robert. The American Revolution: A Concise History (2011) 128 pp excerpt and text search
Atkinson, Rick. The British Are Coming: The War for America, Lexington to Princeton, 1775-1777 (2019) (vol 1 of his 'The Revolution Trilogy'); called, "one of the best books written on the American War for Independence," [Journal of Military History Jan 2020 p 268]; the maps are online here
Axelrod, Alan. The Real History of the American Revolution: A New Look at the Past (2009), well-illustrated popular history
Bancroft, George. History of the United States of America, from the discovery of the American continent. (1854–78), vol 4–10 online edition, classic 19th century narrative; highly detailed
Black, Jeremy. War for America: The Fight for Independence 1775–1783 (2001) 266pp; by leading British scholar
Brown, Richard D., and Thomas Paterson, eds. Major Problems in the Era of the American Revolution, 1760–1791: Documents and Essays (2nd ed. 1999)
Christie, Ian R. and Benjamin W. Labaree. Empire or Independence: 1760-1776 (1976)
Cogliano, Francis D. Revolutionary America, 1763–1815; A Political History (2nd ed. 2008), British textbook
Ellis, Joseph J. American Creation: Triumphs and Tragedies in the Founding of the Republic (2008) excerpt and text search
Higginbotham, Don. The War of American Independence: Military Attitudes, Policies, and Practice, 1763–1789 (1983) Online in ACLS Humanities E-book Project; comprehensive coverage of military and domestic aspects of the war.
Jensen, Merrill. The Founding of a Nation: A History of the American Revolution 1763–1776. (2004)
Knollenberg, Bernhard. Growth of the American Revolution: 1766–1775 (2003)
Lecky, William Edward Hartpole. The American Revolution, 1763–1783 (1898), older British perspective online edition
Mackesy, Piers. The War for America: 1775–1783 (1992), British military study online edition
Middlekauff, Robert. The Glorious Cause: The American Revolution, 1763–1789 (Oxford History of the United States, 2005). online edition
Miller, John C. Triumph of Freedom, 1775–1783 (1948) online edition
Miller, John C. Origins of the American Revolution (1943) online edition, to 1775
Rakove, Jack N. Revolutionaries: A New History of the Invention of America (2010) interpretation by leading scholar excerpt and text search
Taylor, Alan. American Revolutions: A Continental History, 1750-1804 (2016) 704 pp; recent survey by leading scholar
Weintraub, Stanley. Iron Tears: Rebellion in America 1775–83 (2005) excerpt and text search, popular
Wood, Gordon S. Revolutionary Characters: What Made the Founders Different (2007)
Wrong, George M. Washington and His Comrades in Arms: A Chronicle of the War of Independence (1921) online short survey by Canadian scholar online
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Specialized studies
Bailyn, Bernard. The Ideological Origins of the American Revolution. (Harvard University Press, 1967).
Becker, Carl. The Declaration of Independence: A Study on the History of Political Ideas (1922)
Becker, Frank: The American Revolution as a European Media Event, European History Online, Mainz: Institute of European History, 2011, retrieved: October 25, 2011.
Breen, T. H. The Marketplace of Revolution: How Consumer Politics Shaped American Independence (2005)
Breen, T. H. American Insurgents, American Patriots: The Revolution of the People (2010) 337 pages; examines rebellions in 1774–76 including loosely organized militants took control before elected safety committees emerged.
Brunsman, Denver, and David J Silverman, eds. The American Revolution Reader (Routledge Readers in History, 2013) 472 pp; essays by leading scholars
Chernow, Ron. Washington: A Life (2010) detailed biography; Pulitzer Prize
Crow, Jeffrey J. and Larry E. Tise, eds. The Southern Experience in the American Revolution (1978)
Fischer, David Hackett. Paul Revere's Ride (1995), Minutemen in 1775
Fischer, David Hackett. Washington's Crossing (2004). 1776 campaigns; Pulitzer prize.
Freeman, Douglas Southall. Washington (1968) Pulitzer Prize; abridged version of 7 vol biography
Horne, Gerald. The Counter-Revolution of 1776: Slave Resistance and the Origins of the United States of America. (New York University Press, 2014).
Kerber, Linda K. Women of the Republic: Intellect and Ideology in Revolutionary America (1979)
Kidd, Thomas S. God of Liberty: A Religious History of the American Revolution (2010)
Langley, Lester D. The Long American Revolution and Its Legacy(U of Georgia Press, 2019) online review emphasis on long-term global impact.
Lockwood, Matthew. To Begin the World Over Again: How the American Revolution Devastated the Globe. (Yale University Press; 2019)
McCullough, David. 1776 (2005). ; popular narrative of the year 1776
Maier, Pauline. American Scripture: Making the Declaration of Independence (1998) excerpt and text search
Nash, Gary B. The Unknown American Revolution: The Unruly Birth of Democracy and the Struggle to Create America. (2005).
Nevins, Allan; The American States during and after the Revolution, 1775–1789 1927. online edition
Norton, Mary Beth. Liberty's Daughters: The Revolutionary Experience of American Women, 1750–1800 (1980)
Norton, Mary Beth. 1774: The Long Year of Revolution (2020) online review by Gordon S. Wood
O'Shaughnessy Andrew Jackson. The Men Who Lost America: British Leadership, the American Revolution, and the Fate of the Empire (Yale University Press; 2013) 466 pages; on top British leaders
Palmer, Robert R. The Age of the Democratic Revolution: A Political History of Europe and America, 1760–1800. vol 1 (1959) online edition
Resch, John Phillips and Walter Sargent, eds. War and Society in the American Revolution: Mobilization and Home Fronts (2006)
Rothbard, Murray, Conceived in Liberty (2000), Volume III: Advance to Revolution, 1760–1775 and Volume IV: The Revolutionary War, 1775–1784. , libertarian perspective
Van Tyne, Claude Halstead. American Loyalists: The Loyalists in the American Revolution (1902) online edition
Volo, James M. and Dorothy Denneen Volo. Daily Life during the American Revolution (2003)
Wahlke, John C. ed. The Causes of the American Revolution (1967) primary and secondary readings online
Wood, Gordon S. American Revolution (2005) [excerpt and text search] 208 pp excerpt and text search
Wood, Gordon S. The Radicalism of the American Revolution: How a Revolution Transformed a Monarchical Society into a Democratic One Unlike Any That Had Ever Existed. (1992), by a leading scholar
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Historiography
Allison, David, and Larrie D. Ferreiro, eds. The American Revolution: A World War (Smithsonian, 2018) excerpt
Breen, Timothy H. "Ideology and nationalism on the eve of the American Revolution: Revisions once more in need of revising." Journal of American History (1997): 13–39. in JSTOR
Countrymen, Edward. "Historiography" in Harold E. Selesky, ed., Encyclopedia of the American Revolution (Gale, 2006) pp 501–508.
Gibson, Alan. Interpreting the Founding: Guide to the Enduring Debates over the Origins and Foundations of the American Republic (2006).
Hattem, Michael D. "The Historiography of the American Revolution" Journal of the American Revolution (2013) online outlines ten different scholarly approaches
Morgan, Gwenda. The Debate on the American Revolution (2007).
Schocket, Andrew M. Fighting over the Founders: How We Remember the American Revolution (2014), how politicians, screenwriters, activists, biographers, museum professionals, and reenactors portray the American Revolution. excerpt
Sehat, David. The Jefferson Rule: How the Founding Fathers Became Infallible and Our Politics Inflexible (2015) excerpt
Shalhope, Robert E. "Toward a republican synthesis: the emergence of an understanding of republicanism in American historiography." William and Mary Quarterly (1972): 49–80. in JSTOR
Waldstreicher, David. "The Revolutions of Revolution Historiography: Cold War Contradance, Neo-Imperial Waltz, or Jazz Standard?." Reviews in American History 42.1 (2014): 23–35. online
Wood, Gordon S. "Rhetoric and Reality in the American Revolution." William and Mary Quarterly (1966): 4–32. in JSTOR
Young, Alfred F. and Gregory H. Nobles. Whose American Revolution Was It? Historians Interpret the Founding (2011).
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Primary sources
The American Revolution: Writings from the War of Independence (2001), Library of America, 880 pp
Dann, John C., ed. The Revolution Remembered: Eyewitness Accounts of the War for Independence (1999) excerpt and text search, recollections by ordinary soldiers
Humphrey, Carol Sue ed. The Revolutionary Era: Primary Documents on Events from 1776 to 1800 (2003), 384 pp; newspaper accounts excerpt and text search
Jensen, Merill, ed. Tracts of the American Revolution, 1763–1776 (1967). American pamphlets
Jensen, Merill, ed. English Historical Documents: American Colonial Documents to 1776: Volume 9 (1955), 890pp; major collection of important documents
Morison, Samuel E. ed. Sources and Documents Illustrating the American Revolution, 1764–1788, and the Formation of the Federal Constitution (1923). 370 pp online version
Tansill, Charles C. ed.; Documents Illustrative of the Formation of the Union of the American States. Government Printing Office. (1927). 1124 pp online version
Martin Kallich and Andrew MacLeish, eds. The American Revolution through British eyes (1962) primary documents
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Contemporaneous sources: Annual Register
Murdoch, David H. ed. Rebellion in America: A Contemporary British Viewpoint, 1769–1783 (1979), 900+ pp of annotated excerpts from Annual Register Annual Register 1773, British compendium of speeches and reports
Annual Register 1774
Annual Register 1775
Annual Register 1776
Annual Register 1777
Annual Register 1778
Annual Register 1779
Annual Register 1780
Annual Register 1781
Annual Register 1782
Annual Register 1783
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External links
American Revolution, US National Park Service website portal
American Independence (Teaching with Historic Places) Teaching with Historic Places (TwHP) uses historic places in National Parks and in the National Park Service's National Register of Historic Places to enliven history, social studies, geography, civics, and other subjects. TwHP has created a variety of products and activities that help teachers bring historic places into the classroom.
Library of Congress Guide to the American Revolution
Museum of the American Revolution
Revolution! The Atlantic World Reborn, Revolution! explores the enormous transformations in the world's politics that took place from 1763 to 1815, with particular attention to three globally influential revolutions in America, France, and Haiti. Linking the attack on monarchism and aristocracy to the struggle against slavery, Revolution!shows how freedom, equality, and the sovereignty of the people became universal goals.New-York Historical Society
132 historic photographs dealing with the personalities, monuments, weapons and locations of the American Revolution; these are pre-1923 and out of copyright.
Pictures of the Revolutionary War: Select Audiovisual Records, National Archives and Records Administration selection of images, including a number of non-military events and portraits
The Democratic Revolution of the Enlightenment. Legacy of the struggle for independence and democracy.
PBS Television Series Liberty
Chickasaws Conflicted by the American Revolution – Chickasaw.TV
Smithsonian study unit on Revolutionary Money
The American Revolution, the History Channel (US cable television) website
Black Loyalist Heritage Society
Spanish and Latin American contribution to the American Revolution
American Archives: Documents of the American Revolution at Northern Illinois University Libraries
Gayle Olson-Ramer, "Half a Revolution", 16-page teaching guide for high school students, Zinn Education Project/Rethinking Schools
"Counter-Revolution of 1776": Was U.S. Independence War a Conservative Revolt in Favor of Slavery? Democracy Now! June 27, 2014.
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1770s conflicts
1780s conflicts
1770s in the United States
1780s in the United States
18th-century rebellions
18th-century revolutions
Legal history of the United States
Age of Enlightenment
1760s conflicts
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Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
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Algebraic geometry
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The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
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Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
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In the 20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
Real algebraic geometry is the study of the real points of an algebraic variety.
Diophantine geometry and, more generally, arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the study of properties of explicitly given algebraic varieties.
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Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach.
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Basic notions Zeros of simultaneous polynomials In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points (x,y,z) with A "slanted" circle in R3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations Affine varieties
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First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k, denoted An(k) (or more simply An, when k is clear from the context). When one fixes a coordinate system, one may identify An(k) with kn. The purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries.
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A function f : An → A1 is said to be polynomial (or regular) if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(M) = p(t1,...,tn) for every point M with coordinates (t1,...,tn) in An. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in An.
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When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the regular functions on An is a ring, which is denoted k[An]. We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[An]. The vanishing set of S (or vanishing locus or zero set) is the set V(S) of all points in An where every polynomial in S vanishes. Symbolically,
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A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).
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Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an ideal of the polynomial ring k[An].
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Two natural questions to ask are:
Given a subset U of An, when is U = V(I(U))?
Given a set S of polynomials, when is S = I(V(S))?
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The answer to the first question is provided by introducing the Zariski topology, a topology on An whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k[An]. Then U = V(I(U)) if and only if U is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I(V(S)) is the radical of the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection.
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For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[An] are always finitely generated.
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An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed. Regular functions
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Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in An is the restriction to V of a regular function on An. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic.
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It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V]. This ring is called the coordinate ring of V.
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Since regular functions on V come from regular functions on An, there is a relationship between the coordinate rings. Specifically, if a regular function on V is the restriction of two functions f and g in k[An], then f − g is a polynomial function which is null on V and thus belongs to I(V). Thus k[V] may be identified with k[An]/I(V).
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Morphism of affine varieties
Using regular functions from an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space: Let V be a variety contained in An. Choose m regular functions on V, and call them f1, ..., fm. We define a regular map f from V to Am by letting . In other words, each fi determines one coordinate of the range of f.
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If V′ is a variety contained in Am, we say that f is a regular map from V to V′ if the range of f is contained in V′. The definition of the regular maps apply also to algebraic sets.
The regular maps are also called morphisms, as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets.
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Given a regular map g from V to V′ and a regular function f of k[V′], then . The map is a ring homomorphism from k[V′] to k[V]. Conversely, every ring homomorphism from k[V′] to k[V] defines a regular map from V to V′. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k-algebras. This equivalence is one of the starting points of scheme theory. Rational function and birational equivalence
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In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.
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If V is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted k(V) and called the field of the rational functions on V or, shortly, the function field of V. Its elements are the restrictions to V of the rational functions over the affine space containing V. The domain of a rational function f is not V but the complement of the subvariety (a hypersurface) where the denominator of f vanishes.
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As with regular maps, one may define a rational map from a variety V to a variety V'. As with the regular maps, the rational maps from V to V' may be identified to the field homomorphisms from k(V') to k(V). Two affine varieties are birationally equivalent if there are two rational functions between them which are inverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic.
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An affine variety is a rational variety if it is birationally equivalent to an affine space. This means that the variety admits a rational parameterization, that is a parametrization with rational functions. For example, the circle of equation is a rational curve, as it has the parametric equation which may also be viewed as a rational map from the line to the circle.
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The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also smooth completion). It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in 1964 and is yet unsolved in finite characteristic. Projective variety
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Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number i, a root of the polynomial , projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet.
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To see how this might come about, consider the variety . If we draw it, we get a parabola. As x goes to positive infinity, the slope of the line from the origin to the point (x, x2) also goes to positive infinity. As x goes to negative infinity, the slope of the same line goes to negative infinity.
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Compare this to the variety V(y − x3). This is a cubic curve. As x goes to positive infinity, the slope of the line from the origin to the point (x, x3) goes to positive infinity just as before. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of V(y − x3) is different from the behavior "at infinity" of V(y − x2).
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The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the line at infinity, while the point at infinity of the cubic curve is a cusp. Also, both curves are rational, as they are parameterized by x, and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.
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Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry.
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